THE ARCHÉ PAPERS ON THE MATHEMATICS OF ABSTRACTION
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Roy T. Cook

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THE ARCHÉ PAPERS ON THE MATHEMATICS OF ABSTRACTION

THE WESTERN ONTARIO SERIES IN PHILOSOPHY OF SCIENCE A SERIES OF BOOKS IN PHILOSOPHY OF SCIENCE, METHODOLOGY, EPISTEMOLOGY, LOGIC, HISTORY OF SCIENCE, AND RELATED FIELDS

Managing Editor WILLIAM DEMOPOULOS

Department of Philosophy, University of Western Ontario, Canada Department of Logic and Philosophy of Science, University of Californina/Irvine Managing Editor 1980–1997 ROBERT E. BUTTS

Late, Department of Philosophy, University of Western Ontario, Canada

Editorial Board JOHN L. BELL,

University of Western Ontario

JEFFREY BUB,

University of Maryland

PETER CLARK,

St Andrews University

DAVID DEVIDI,

University of Waterloo

ROBERT DiSALLE,

University of Western Ontario

MICHAEL FRIEDMAN, MICHAEL HALLETT, WILLIAM HARPER,

McGill University

University of Western Ontario

CLIFFORD A. HOOKER, AUSONIO MARRAS,

Indiana University

University of Newcastle

University of Western Ontario

JÜRGEN MITTELSTRASS, JOHN M. NICHOLAS,

Universität Konstanz

University of Western Ontario

ITAMAR PITOWSKY,

Hebrew University

VOLUME 71

THE ARCHÉ PAPERS ON THE MATHEMATICS OF ABSTRACTION Edited by

ROY T. COOK

123

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978–1–4020–4264–5 (HB) ISBN 978–1–4020–4265–2 (e-book) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

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All Rights Reserved c 2007 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

For Alice, who has kindly tolerated the company of many abstractionists, and one in particular

Contents

Foreword

ix

Notes on the Contributors

xi

Acknowledgements

xiii

Introduction

xv

Part I The Philosophy and Mathematics of Hume’s Principle Is Hume’s Principle Analytic? G. Boolos

3

Is Hume’s Principle Analytic? C. Wright

17

Frege, Neo-Logicism and Applied Mathematics P. Clark

45

Finitude and Hume’s Principle R. G. Heck, Jr.

61

On Finite Hume F. MacBride

85

Could Nothing Matter? F. MacBride

95

On the Philosophical Interest of Frege Arithmetic W. Demopoulos

105

Part II The Logic of Abstraction “Neo-logicist” Logic is not Epistemically Innocent S. Shapiro & A. Weir

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119

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Contents

Aristotelian Logic, Axioms, and Abstraction R. T. Cook

147

Frege’s Unofficial Arithmetic A. Rayo

155

Part III Abstraction and the Continuum Reals by Abstraction R. Hale

175

The State of the Economy: Neo-logicism and Inflation R. T. Cook

197

Frege Meets Dedekind: A Neo-logicist Treatment of Real Analysis S. Shapiro

219

Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege’s Constraint C. Wright

253

Part IV Basic Law V and Set Theory NewV, ZF and Abstraction S. Shapiro & A. Weir

275

Well- and Non-well-founded Extensions I. Jané & G. Uzquiano

303

Abstraction & Set Theory Bob Hale

331

Prolegomenon to Any Future Neo-logicist Set Theory: Abstraction and Indefinite Extensibility S. Shapiro

353

Neo-Fregeanism: An Embarassment of Riches A. Weir

383

Iteration One More Time R. T. Cook

421

Foreword

In September 2000 the Arché Centre launched a five-year research project entitled the Logical and Metaphysical Foundations of Classical Mathematics. Its goal was to study the prospects, philosophical and technical, for abstractionist foundations for the classical mathematical theories of the natural, real and complex numbers and standard set theory. Funding was provided by the then Arts and Humanities Research Board (now the Arts and Humanities Research Council) for the appointment of full-time postdoctoral research fellows and PhD students to collaborate with more senior colleagues in the project, and at the same time the British Academy awarded the Centre additional resources to establish an International Network of scholars to be associated with the work. This was the beginning of the serial ‘Abstraction workshops’ of which the Centre had staged no less than eleven by December 2006. We gratefully acknowledge the generous support of the Academy and Council, sine qua non. The project seminars and Network meetings generated—and continue to generate—a large number of leading-edge research papers on all aspects of the project agenda. The present volume is the first of what we hope will be a number of anthologies of these researches. With two exceptions,—the contribution by the late George Boolos and the co-authored paper by Gabriel Uzquiano and Ignacio Jané,—the papers that Roy Cook has collected in the present volume are all authored by sometime members of the project team or of the British Academy Network. Their broad focus, as he explains, is on some of the more technical issues thrown up by the Abstractionist project, and it is anticipated that subsequent volumes may have a more purely metaphysical or epistemological emphasis. I would like to thank Roy Cook for all his hard work putting the volume together, and Bill Demopoulos for sponsoring its publication in the Western Ontario Series in Philosophy of Science. Special thanks go to the members of the core team and the Network not just for their direct contributions to the researches of the project but for their continuing affirmation, by their active participation, of the wider interest and importance of

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Foreword

the neo-Fregean enterprise in the landscape of contemporary philosophy of mathematics. CJGW St. Andrews 6/07 The Logical and Metaphysical Foundations of Classical Mathematics Sometime project team members: Crispin Wright, Peter Clark, Roy Cook, Philip Ebert, Bob Hale, Fraser MacBride, Paul McCallion, Darren McDonald, Nikolaj Jang Pedersen, Agustin Rayo, Marcus Rossberg, Andrea Sereni, Stewart Shapiro, Chiara Tabet, Robert Williams Auditor: Kit Fine British Academy International Network members: Alexander Bird, Robert Black, Robin Cameron, William Demopoulos, Richard Heck, Keith Hossack, Daniel Isaacson, John Mayberry, Michael Potter, Adam Rieger, Ian Rumfitt, Peter Simons, William Stirton, Peter Sullivan, Alan Weir

Notes on the Contributors

George Boolos was Professor of Philosophy at Massachusetts Institute of Technology, and the co-author of Computability and Logic (with Richard Jeffrey, Cambridge 2007) and the author of The Logic of Provability (Cambridge 1995). Peter J. Clark is Professor of the Philosophy of Science and Head of the School of Philosophical and Anthropological Studies in the University of St Andrews. He works primarily in the philosophy of physical sciences and mathematics and was editor of the British Journal for the Philosophy of Science 1999–2005. Roy Cook is Visiting Assistant Professor of Philosophy at Villanova University, and an associate fellow of Arché. He has published papers in the philosophy of language, logic, and mathematics, focusing primarily on semantic, soritical, and set-theoretic paradoxes, and Fregean and neo-Fregean philosophies of mathematics. William Demopoulos is a member of the Department of Philosophy of the University of Western Ontario and the Department of Logic and Philosophy of Science of the University of California, Irvine. He has published articles in diverse fields in the philosophy of the exact sciences, and on the development of analytic philosophy in the twentieth century. Bob Hale is Professor of Philosophy at the University of Sheffield, and an Associate Director of Arché. He works mainly on topics in the epistemology and metaphysics of mathematics and modality. His publications include Abstract Objects (Blackwell 1987), and, together with Crispin Wright, The Blackwell Companion to the Philosophy of Language (Blackwell 1997) and The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics (Oxford 2001). Richard Heck is Professor of Philosophy at Brown University and an associate fellow of Arché. He has published extensively on historical, conceptual, and technical issues emerging from Frege’s philosophy of mathematics. Philosophy of language and philosophy of logic are his other main areas of interest. He is now working on a book on philosophy of language and another on the development of Frege’s mature philosophy (co-authored with Robert May). xi

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Notes on the Contributors

Ignacio Jané is Professor of Philosophy in the Department of Logic and the History and Philosophy of Science of the University of Barcelona. His main interests are in the foundations of mathematics, philosophy of mathematics, and philosophy of logic. His recent papers include “Reflections on Skolem’s Relativity of Set-Theoretical Concepts” (Philosophia Mathematica, 2001), “Higher-Order Logic Reconsidered” (The Oxford Handbook of Philosophy of Mathematics and Logic, 2005), and “What is Tarski’s Common Concept of Consequence” (The Bulletin of Symbolic Logic, 2006). Fraser MacBride is a Reader in the School of Philosophy at Birkbeck College, London. He previously taught in the Department of Logic & Metaphysics at the University of St. Andrews and was a research fellow at University College, London. He has written several articles on the philosophy of mathematics, metaphysics, and the history of analytic philosophy, and is the editor of Identity & Modality (Oxford, 2006) and The Foundations of Mathematics and Logic (special issue of The Philosophical Quarterly, vol. 54, no. 214, 2004). Agustin Rayo is Associate Professor of Philosophy at MIT and an associate fellow of Arché. He works mainly in the philosophy of language and the philosophy of logic. Stewart Shapiro is the O’Donnell Professor of Philosophy at The Ohio State University and a Professorial Fellow in the Research Centre Arché at the University of St. Andrews. His publications include Foundations without foundationalism: a case for second-order logic (Oxford, 1991), Philosophy of mathematics: structure and ontology (Oxford, 1997), and Vagueness in context (Oxford, 2006). Gabriel Uzquiano is a Tutorial Fellow in Philosophy at Pembroke College and a CUF lecturer in Philosophy at the University of Oxford. He has published articles in metaphysics, philosophical logic, and the philosophy of mathematics. Alan Weir is Professor of Philosophy, University of Glasgow. He has published articles on logic and philosophy of mathematics in a number of journals including Mind, Philosophia Mathematica, Notre Dame Journal of Formal Logic, and Grazer Philosophische Studien and contributed chapters to a number of volumes devoted to these areas. Crispin Wright is Bishop Wardlaw Professor at the University of St Andrews, Global Distinguished Professor at New York University, and Director of the Research Centre, Arché. His writings in the philosophy of mathematics and logic include Wittgenstein on the Foundations of Mathematics (Harvard 1980); Frege’s Conception of Numbers as Objects (Aberdeen 1983); and, with Bob Hale, The Reason’s Proper Study (Oxford 2001). His most recent books, Rails to Infinity (Harvard 2001) and Saving the Differences (Harvard 2003), respectively collect his writings on central themes of Wittgenstein’s Philosophical Investigations and those further developing themes of his Truth and Objectivity.

Acknowledgements

The Editor wishes to thank the following: Oxford University Press, Kluwer Academic Publishers, Analysis, The British Journal for the Philosophy of Science, The Journal of Philosophical Logic, The Journal of Symbolic Logic, The Notre Dame Journal of Formal Logic, Philosophia Mathematica, and Philosophical Books for permission to reprint the papers that follow. Detailed individual citations are included with the papers. Crispin Wright, Director of Arché: Philosophical Research Centre for Logic, Language, Metaphysics, and Epistemology, for providing the foreword. William Demopoulos for proposing, and securing, the publication of this work in the Western Ontario Series in the Philosophy of Science. Charles Erkelens and Lucy Fleet at Springer for their guidance and encouragement. The administrative staff at the Arché Centre in St Andrews (Gill Gardner, Sylvia Rescigno, and Sharon Coull) and at Villanova University (Elvia Beach and Terry DiMartino) for constant assistance in the practical aspects of preparing this volume. Marguerite Nesling for converting a number of the papers from hardcopy to electronic format. The Arts and Humanities Research Board (now the Arts and Humanities Research Council) for support in the form of a postdoctoral research fellowship, which was held by the editor during the initial stages of this volume.

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Introduction

As noted in the preface, the papers included in this volume concentrate (much of the time, at least) on philosophical questions that are intimately tied up with the interesting, and sometimes puzzling, mathematical properties of abstraction principles. As a result, the introduction you are about to read will follow suit – concentrating on philosophical issues that have their roots in the mathematical characteristics of abstraction principles as well as philosophical problems whose solution would seem to require a somewhat technical approach. This focus should not be read as any sort of value judgment regarding the worth of technical versus non-technical work on abstraction principles, or within the philosophy of mathematics more generally. Instead, this focus on philosophical problems that are linked to mathematical aspects of abstraction reflects the fact that there has, in the last decade or two, been an immense amount of valuable work on Fregean-inspired abstraction principles and their philosophical importance. To attempt to cover all of this work, or even all such work that has some connection to the Arché Centre, would require several volumes the size of the present one. Hence the narrower focus. The volume is divided into four sections. The first contains papers of a general sort (which can also serve as a helpful introduction to the subject for those less familiar with the literature), although the majority of these nevertheless address distinctly technical issues, at least indirectly. The remaining three sections are devoted to three topics which have come under increasing study and scrutiny after the apparent success of the account of arithmetic based on abstraction. The second section (“The Logic Of Abstraction”) contains three papers that examine the role of logic (in particular, higher-order logic) within the abstractionist framework. The third section (“Abstractionism and the Continuum”) contains papers that attempt to extend the abstractionist account to the theory of the real numbers, as well as papers critically evaluating such attempts. The fourth and final section (“Basic Law V and Set Theory”) is devoted to attempts to reconstruct set theory (or something like it) within the abstractionist framework – usually by adopting some consistent variant of Frege’s original Basic Law V. Even with our focus narrowed to the more technical aspects of abstraction principles, however, the range of topics and problems addressed in the papers

xv

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Introduction

to follow is vast. Therefore, in the interest of providing a reasonably concise and easily digestible introduction to the subject, there will be no attempt to discuss every issue that arises in the following chapters. Instead, the remainder of this introduction will proceed as follows: First, a brief sketch of the origin of interest in abstraction principles, i.e. Frege’s logicism and its failure, will be provided. Second, we will briefly examine the philosophical framework underlying the resurrection of interest in abstraction principles, a view often called Neo-Fregeanism, Neo-Logicism, or Abstractionism. Next we will look at brief sketches of the philosophical and technical work underlying the abstractionist reconstructions of arithmetic, analysis, and set theory. Then we shall survey three general types of problem that such reconstructions face, and conclude with a brief discussion of indefinite extensibility, a notion that has become of central importance in much of the work attempting to solve problems of the sort covered in the previous sections. Before moving on, a comment needs to be made regarding terminology. As already noted, the philosophical view (or views) under discussion in the remainder of this volume have been called, at various times and places, Neo-Fregeanism, Neo-Logicism, and Abstractionism. In the remainder of this essay the term “Abstractionism” will be used. The reasons for this are simple: “Neo-Logicism” is misleading, since it would seem to imply that the view is a new version of logicism, while, as we shall see, it is no such thing. “Neo-Fregeanism”, while perhaps not misleading in this way, is, in the editor’s opinion, better reserved for the more general view in the philosophy of language, clearly Fregean in nature, that (usually) underlies the philosophy of mathematics discussed in the chapters to follow (although even here there is further confusion, since this term is also used to refer to a collection of views associated with the work of certain Oxford philosophers such as Gareth Evans and John McDowell). One could presumably hold such Fregean views regarding language without believing in the fundamental importance of abstraction principles (and vice versa – see Agustin Rayo’s “Frege’s Unofficial Arithmetic” [2002], reprinted as chapter 10 below). When reading the essays collected in this volume, however, one should keep in mind that these terms are (unfortunately) used for the most part interchangeably.

1.

Abstraction and logicism An abstraction principle is any formula of the form: (∀α)(∀β)(@(α) = @(β) ↔ E(α, β)

where “@” denotes a unary function mapping entities of the type ranged over by α (usually concepts, objects, or sequences of such) to objects, and “E( , )” is an equivalence relation on those same entities. The general idea behind abstraction principles is that they allow us to introduce new

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terms (and thus presumably to gain privileged epistemological access to the corresponding objects) by defining the identity conditions for the referents of the novel terms using linguistic resources that are already understood (i.e. those resources occurring in the equivalence relation “E( , )” – in most cases “E( , )” is either a purely logical formula, or one composed of logic plus previously introduced abstraction operators). Thus, an abstraction principle is meant to act as an implicit definition of sorts, providing (so the story goes) an account of the meaning of novel terms of the form “@(α)”. Perhaps the first notable occurrence of an abstraction principle occurs in Frege’s attempt at a logicist reconstruction of arithmetic (and, in fact, all of mathematics). Frege notes, in the Grundlagen [1974] that the standard (higherorder) Peano axioms for arithmetic follow from the abstraction principle now known as Hume’s Principle (the explicit derivation of the Peano axioms from Hume’s Principle was “extrapolated” from Frege’s comments in Crispin Wright [1983], George Boolos [1990a], and Richard Heck [1993], and Boolos & Heck [1998], among others). Hume’s Principle (which Hume himself did not state, and whose name derives from a rather charitable reading of a comment in Hume’s Treatise) is the claim that, given two arbitrary concepts P and Q, the number of P’s is identical to the number of Q’s if and only if the P’s and the Q’s can be put in a one-to-one correspondence. More formally, we have: HP : (∀P)(∀Q)(NUM(P) = NUM(Q) ↔ (P ≈ Q)) where P ≈ Q abbreviates the second-order formula stating that P and Q are equinumerous. We can formulate rather natural definitions of arithmetical notions such ‘natural number’, ‘successor’, and ‘addition’ in terms of the numerical operator “NUM”. The fact that, given these definitions, the second-order Peano axioms for arithmetic follow from Hume’s Principle is quite notable as a mathematical theorem independent of any philosophical motivation, and the result has come to be called Frege’s Theorem (for a detailed examination of this result, and various streamlined versions of it, see Richard Heck’s “Finitude and Hume’s Principle” [1997a], reprinted as chapter 4 below). Frege, of course, wanted to reduce all of arithmetic to logic, thus defending (at least some of) mathematics from the Kantian charge of being synthetic a priori (logic, presumably, being analytic if anything is!). Thus, he rejected Hume’s Principle as the ultimate foundation for arithmetic, since it contains ineliminable occurrences of the cardinal number operator. (More famously, he also rejected Hume’s Principle, in its primitive form, since it was susceptible to the Caesar Problem. See section 7 of this introduction for further discussion of this issue.) As a result, Frege formulated a second abstraction principle, one that mapped each concept onto a unique object – its extension. Unlike Hume’s

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Introduction

Principle, Frege’s Basic Law V: BLV : (∀P)(∀Q)(EXT(P) = EXT(Q) ↔ (∀x)(Px) ↔ Qx)) contains only logical vocabulary (as long as talk of extensions is logical). Basic Law V was, in essence, an early attempt at formulating (in an a priori, logical manner) the foundations of what we would now call set theory. Using Basic Law V, Frege was able to reconstruct Peano Arithmetic on what appeared to be a purely logical basis. The first step was to define numbers to be certain sorts of extensions – the number of a concept P is the extension of the concept “(extension of a) concept equinumerous with P”, or, more formally: NUM(P) =df EXT((∃Y)(x = EXT(Y) ∧ Y ≈ P)) Given this definition, Frege was able to prove Hume’s Principle (now a theorem of Frege’s logic, and not a primitive non-logical definition of cardinal number) and thus prove the second-order Peano axioms for arithmetic. So if Basic Law V was, as Frege hoped, a logical truth, then logicism (at least regarding arithmetic) would be demonstrated. Since not all of us are convinced logicists, something must have gone wrong – something discovered by Bertrand Russell. In a letter dated June 16, 1902, Russell wrote to Frege, humbly pointing out that the crucial axiom (Basic Law V) that provided the power needed to reconstruct arithmetic within logic also seemed to allow for the derivation of a contradiction. Although Russell’s actual presentation of the paradox that now bears his name is a bit muddled in the original missive, the derivation of a contradiction from Basic Law V is well known, and need not be rehashed here. Frege attempted to fix the problem, but failed to find a convincing replacement for Basic Law V. Russell, meanwhile, along with Alfred North Whitehead, attempted his own reconstruction of mathematics from basic, a priori principles in the monumental Principia Mathematica [1910–13]. Although the Principia was (likely) consistent, in the long run it turned out to be no more convincing than Frege’s Grundgesetze.

2.

Abstractionism

After Frege’s failed attempt at utilizing abstraction principles in a logicist framework, this sort of principle lay unstudied for three-quarters of a century. Interest in abstraction of this sort was rekindled, however, by the publication of Crispin Wright’s Frege’s Conception of Numbers as Objects [1983]. Wright noted (perhaps among others, see Parsons [1965] and Hodes [1984]), in essence, that Frege’s project consisted of four basic steps: (1) Recognize Basic Law V as an axiom of logic. (2) Formulate suitable definitions of numerical notions in terms of the extensions provided by Basic Law V

Introduction

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(3) Derive Hume’s Principle (4) Derive arithmetic from Hume’s Principle (i.e. Frege’s Theorem)

Wright revived interest in Frege’s project, founding a new project that has come to be called Neo-Logicism, Neo-Fregeanism, and Abstractionism, variously, replacing the above blueprint with the following alternate plan: (1) Lay down Hume’s Principle as an implicit definition of cardinal number (2) Derive arithmetic from Hume’s Principle (i.e. Frege’s Theorem)

Of course, as was already noted, such a view (misleading nomenclature such as “Neo-Logicism” notwithstanding) does not deserve the title ‘logicism’, at least not in the traditional sense of the word as used by Frege, his fans, and his critics. Hume’s Principle, with its primitive and ineliminable occurrences of arithmetical terms (i.e. “NUM”), just does not have the character of a logical law or theorem (a point made strenuously and convincingly by George Boolos in “Is Hume’s Principle Analytic?”, the essay that opens this volume). Thus, abstractionists have had to look elsewhere for their defense of Hume’s Principle as something suitably basic as to provide the foundations of arithmetic. The answer, according to abstractionists, is to note that what is important about logicism is not so much the reduction of mathematics to logic, but rather the fact that this reduction (had it been successful) would have gone a long ways towards providing an account of certain aspects of mathematical knowledge that, ideally, we would like to be able to explain. In particular, the true advantages of logicism were that it purported to explain the a priori character of mathematical knowledge (assuming that the a priori character of purely logical knowledge is unproblematic) and it purported to explain the analyticity of mathematical truths (at least, this is important for those nonQuineans that retain a fondness for the analytic/synthetic distinction in the first place). The solution, then, is to retain these goals, while widening the scope of our means for achieving these goals to something more than pure logic. Along these lines, Wright and those that follow him deny that Hume’s Principle is a logical truth. Instead, Hume’s Principle, so it is argued, is (or is something like) an implicit definition of the “NUM” operator – one that explains the meaning of statements of identity of cardinal numbers. Since Hume’s Principle is a definition, we can come to know its consequences a priori in the same manner (or, at least, in a suitably similar manner) in which we obtain a priori knowledge of the consequences of more pedestrian definitions. Frege’s Theorem insures that all of second-order Peano arithmetic follows from Hume’s Principle plus standard second-order logic, so (since presumably second-order logic is a priori knowable and second-order consequence preserves a priori knowledge) it follows that we can, using the abstractionist recipe, obtain a priori knowledge of all of second-order arithmetic. (The question of analyticity is strictly speaking separate from that of aprioricity, and

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Introduction

has been less of a focus for the abstractionists than it was for Frege himself, although Bob Hale has rekindled interest in this issue.) There are, unsurprisingly, deep questions regarding how Hume’s Principle and Frege’s Theorem accomplish this epistemological feat. In particular, there are deep worries regarding the connection between our reconstruction of arithmetic within the abstractionist framework and actual arithmetic practice: How do we know that the knowledge gained from Frege’s Theorem is, in fact, knowledge about the ordinary natural numbers (and not some isomorphic surrogate)? And how do we determine whether our (supposed) a priori knowledge of the former allows for an explanation of the a priori status of everyday mathematics? William Demopoulos’ “On the Philosophical Interest of Frege Arithmetic” [2003] (reprinted below as chapter 7) develops a sustained examination and critique of this aspect of the project (although the reader is encouraged to consult Fraser MacBride’s two contributions to this volume as well). Of course, even if the view in question is not, really, a version of logicism, the above sketch makes it clear that logic plays a crucial role in the abstractionist account of mathematical truth and mathematical knowledge. Defending the claim that second-order logic preserves the relevant epistemological properties is one outstanding lacuna in the abstractionist literature, although it is not one they are unaware of. The most sustained discussion of the issues is to be found in Stewart Shapiro and Alan Weir’s “Neo-Logicist Logic Is Not Epistemically Innocent” [2000], reprinted below as chapter 8. Setting the role of logic aside, however, there is much of interest to be said regarding: (a) the notion of implicit definition required for such an abstractionist project, (b) the more general abstractionist accounts of meaning and reference which might allow for such implicit definitions to succeed, and (c) the metaphysical account of abstract objects that would allow for our epistemological access to them to proceed via such stipulations. Although all of these and more are touched on (and often discussed in some depth) in the essays that follow, they are not the primary focus of this volume or the papers included in it. Instead, we are here interested in those philosophical problems that stem from mathematical issues arising within the abstractionist project. Thus, we shall move on to examine those aspects of abstractionism that are of a more technical nature (the reader interested in more straightforwardly philosophical aspects of the abstractionist project, such as the topics mentioned at the top of this paragraph, can do no better than to consult Fraser MacBride’s “Speaking with Shadows: A Study of Neo-logicism” [2003], although the first two essays in this volume, both titled “Is Hume’s Principle Analytic?”, by George Boolos [1997] and Crispin Wright [1999], also provide much useful philosophical background material). Arithmetic, it would seem, is, in one sense, the big success story of abstractionism, since the technical results, at least, seem to be for the most part settled – all that remains is sorting out the philosophical problems and issues

Introduction

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that result from this abstractionist reconstruction. As we shift our attention to abstractionist accounts of other mathematical theories, however, we shall see that things are not always so successful even within the purely technical aspects of the project. Thus, our next task is to quickly survey the extension of this project to set theory and real analysis.

3.

Abstractionist real numbers

One of the two obvious test cases for extending any philosophy of mathematics past an initial account of the natural numbers is to attempt to reconstruct the continuum (the other test case is to provide an adequate account of sets or something like them, the subject of the next section). Abstractionism is no exception here, and it did not take long for both believers and critics to wonder what shape an abstractionist account of real analysis might take. Although various accounts differ in the details (and this difference tends to depend on varying attitudes towards Frege’s Constraint, see below), Bob Hale’s initial reconstruction (as found in his “Reals By Abstraction” [2000], reprinted as chapter 11 below) and those that follow are similar to the following, at least from a mathematical perspective. The first step in an abstractionist account of the real numbers is to note that we are already provided with the natural numbers via Hume’s Principle. We can obtain the integers from these by adding an additional abstraction principle to our theory – something like the following Difference Abstraction Principle (the universal quantifiers here are restricted to natural numbers and the arithmetical operators on the right-hand side of the biconditional are the standard operations on the natural numbers): DAP : (∀x)(∀y)(∀z)(∀w)(DIFF(x, y) = DIFF(z, w) ↔ x + w = y + z) This principle provides us with an object corresponding to the difference between two natural numbers – in other words, DAP provides us with (a priori access to) the integers. With the integers in hand, we can obtain the rational numbers from these by adding another abstraction principle to our theory – the following Quotient Abstraction Principle will do the trick (here, the initial universal quantifiers are restricted to integers, i.e. objects that are in the range of the DIFF operator, and the arithmetic operators on the right-hand side of the biconditional are the standard operations on the integers – these are definable in terms of DIFF and second-order logic): QAP : (∀x)(∀y)(∀z)(∀w)(QUO(x, y) = QUO(z, w) ↔ ((y = 0 ∧ w = 0) ∨ (y = 0 ∧ w = 0 ∧ x × w = y × z))) Note that the Quotient Abstraction Principle provides us, not only with the rational numbers, but also with an extra, ‘bad’ object: QUO(a, 0) for any integer a. This object results from the fact that we assume that our abstraction

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operators are total functions, and thus certain unintended instances (such as division by zero in the present instance) nevertheless result in abstracts. The presence and role of ‘bad’ objects will be discussed in section 6 of this introduction. Now that we have the integers, we can obtain the reals by applying an abstraction principle that simulates Dedekind-style cuts on the rationals, such as the following Cut Abstraction Principle (here the universal quantifiers are restricted to non-empty bounded concepts holding only of non-‘bad’ rationals, i.e. objects in the range of the QUO operator other than the ‘bad’ object): CAP : (∀P)(∀Q)(REAL(P) = REAL(Q) ↔ (∀x)((∀y)(P(y) → y < x) ↔ (∀y)(Q(y) → y < x))) It is possible (although non-trivial) to prove that the objects provided by CAP are a complete ordered field, i.e. that they are isomorphic to the standard classical continuum (see Shapiro’s “Frege Meets Dedekind: A Neologicist Treatment of Real Analysis” [2000], reprinted as chapter 14 below, for details). Thus, the abstractionist position can account for not only the natural numbers, but the classical theory of the real numbers as well (from a technical perspective, at least). As a technical note, there seems to be no reason why, at this last step, we could not have applied, instead of CAP which encodes Dedekind’s notion of cut within the abstractionist framework, an abstraction principle that, when applied to sequences of rationals, provides the real numbers along the lines of Cauchy’s methodology. There is no formal reason why we could not formulate such a principle (e.g. let the abstraction principle in question map functions from the naturals to the rationals onto objects). The possibility of such alternate constructions raises a host of philosophical issues, however. Not least among them are the following questions: If it turns out that both CAP and an appropriate Cauchy-sequence principle are legitimate abstraction principles, then how are we to determine whether they provide us with access to the same objects? If not, then which one delivers the genuine real numbers (as opposed to merely an isomorphic copy)? Such questions are intimately tied up both with Frege’s Constraint and with the Caesar Problem, both of which will be discussed below.

4.

Abstractionist sets

The second natural extension of a foundational account of mathematics is to produce some account of set theory (or, at the very least, to provide some other theory that can do the work for which we normally invoke the theory of sets). The most notable attempt to provide such an account within the abstractionist framework is due to George Boolos, one of the most outspoken critics of the abstractionist view itself.

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In “Iteration Again” [1989], Boolos compared and contrasted the iterative and limitation-of-size conceptions of sets. The former proposes to solve the problem posed by Russell’s paradox by claiming that sets must be formed in an infinitary step-by-step process, while the latter avoids paradoxes by claiming that only collections that are (in some sense) not too ‘big’ determine sets. One version of the limitation-of-size conception (the one Boolos used) can be formulated by defining ‘X is too big’ as ‘there is a bijection between X and the entire domain’. Boolos formulated an abstractionist version of the limitation-of-size conception of set along these lines. Letting “Big(P)” abbreviate the second-order formula asserting that there is an onto function from P to the entire domain, Boolos’ abstraction principle for extensions, called New V, is: New V : (∀P)(∀Q)(EXT(P) = EXT(Q) ↔ ((∀x)(Px ↔ Qx) ∨ (Big(P) ∧ Big(Q)))) New V provides a distinct object (an extension, or, more loosely, set) for each collection of objects provided that collection is smaller than the entire domain – concepts that hold of as many objects as there are in the domain, however, all receive the same abstract, the ‘Bad’ object (again, see below for discussion of ‘bad’ objects). Given New V, we can define a set to be the extension of a small concept: Set(x) =df (∃P)(x = EXT(P) ∧ ¬Big(P)) One object is the member of another object if and only if the second object is the extension of a concept which holds of the first object, or, in symbols: x ∈ y =df (∃P)(y = EXT(P) ∧ P(x)) (Note that ‘Bad’ objects can have, and be, members.) Given these definitions, New V entails many of the standard set theoretic axioms – extensionality, empty set, pairing, separation, replacement, and choice all follow (the union axiom does not follow on the above definitions, since the union of the singleton of the ‘bad’ object is not a set. Slight reformulations of this axiom do follow, however – for details see the chapters in section IV of this volume). In addition, the axiom of foundation holds if restricted to the pure sets (i.e. those sets that can be ‘built up’ from the empty set – see Gabriel Uzquiano and Ignacio Jané’s “Well- and Non-Well-Founded Extensions” [2004], reprinted as chapter 16 of this volume, for an in-depth examination of non-well-founded sets within the abstractionist framework). Thus, the only axioms that fail to follow, in some sense or another, are the powerset axiom and the axiom of infinity. It is easy to see why the axiom of infinity fails – if we take as our domain the hereditarily finite sets built from a single urelement (to serve as the ‘bad’ object), then the resulting model satisfies New V (since all ‘small’, i.e. finite, concepts receive extensions – the corresponding sets – while all ‘big’, i.e. infinite, concepts can be mapped onto our single urelement). In other words,

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although NewV entails that there must be infinitely many objects, it does not entail that there need be any non-‘Bad’ concept that holds of infinitely many objects. The proof that powerset fails is non-trivial, however – readers are encouraged to consult chapter 15 of this volume, “New V, ZF, and Abstraction” [1999] by Stewart Shapiro and Alan Weir, for the technical details. Given that New V does not allow us to reconstruct all of standard Zermelo Fraenkel set theory, work has been done exploring other abstractionist routes to set theory. Among these are Roy T. Cook’s “Iteration One More Time” [2004], which formulates an abstractionist version of the iterative conception of set based on an abstraction principle called Newer V. Newer V entails the extensionality, empty set, pairing, separation, powerset, and choice axioms, but fails to imply both the axiom of infinity and the replacement axiom. Other approaches include Bob Hale’s “Abstraction and Set Theory” [2000], which formulates an alternative version of the limitation of size conception, and Stewart Shapiro’s “Prolegomenon to Any Future Neo-Logicist Set Theory: Abstraction and Indefinite Extensibility” [2003], which examines the general conditions under which a restricted version of Basic Law V (such as New V) will entail various set-theoretic principles (these three papers are reprinted below as chapters 20, 17, and 18 respectively). Although one could debate how much we should worry about abstraction principles failing to imply powerset or replacement, there is no ignoring the fact that the failure of natural abstractionist accounts of set theory to provide a proof of the axiom of infinity is just that, a failure. Presumably, a successful defense of abstractionism will require a development of a set theory (or surrogate for it) that is stronger than any of the existing proposals, since any set theory which fails to guarantee the existence of any infinite sets is unlikely to be adequate to our needs. To be fair, there are abstraction principles that imply all the axioms of second-order Zermelo–Fraenkel set theory (Alan Weir considers such principles in his “Neo-Fregeanism: An Embarrassment of Riches” [2004], reprinted here as chapter 19). Unlike New V or even Newer V, however, these principles do not seem to codify plausible ‘definitions’ of the notion of set or collection – the sort of conception that could underlie successful a priori introduction of the notion of set or extension into our discourse. Instead, these principles seem tailor made to provide all of the set theoretic axioms, and it is unlikely that anyone could have conceived of them without extensive prior knowledge of advanced set theoretic methods (e.g. formulation of typical ‘distractions’ requires an understanding of notions such as strong inaccessible cardinal). Thus, unlike the case of arithmetic and real analysis, there seems to be much more work needed of a purely technical nature before the abstractionist can make any claim to have explicated the aprioricity and analyticity of set theory. The main technical problem is to find an appropriate abstraction principle for extensions that is satisfied only on uncountable domains of the right sort

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(presumably, something like an inaccessible rank). As of the time of writing this introduction, there does not seem to be any plausible abstraction principle that will do the job, although there is interesting work leading in this direction (e.g. see Shapiro’s “Prolegomenon to Any Future Neo-Logicist Set Theory. . . ” and the later sections of Cook’s “Iteration One More Time”).

5.

The first problem: too many abstraction principles

The first general problem plaguing the abstractionist project is that there seem to be too many abstraction principles. What is required, and what we, at present, fail to have, is some general criteria for distinguishing between acceptable and unacceptable abstraction principles. Clearly, Basic Law V, being inconsistent, is on the unacceptable side of the field, while Hume’s Principle, the pride and joy of abstractionism, is (it is hoped) on the acceptable side (if not, then presumably some suitably modification of it is, such as Finite Hume, discussed in the next section). The problem, however, is that mere consistency is not enough for acceptability, and as a result, we need some further guide to distinguishing the good from the bad. The initial formulation of this problem is (as is almost always the case in these debates) due to George Boolos (in Boolos [1990a]), who pointed out that there are abstraction principles that are consistent, but which are nevertheless incompatible with Hume’s Principle (or, in fact, with any abstraction principle guaranteeing the existence of infinitely many objects). Assuming that Hume’s Principle is acceptable if anything is, it follows that inconsistency, while sufficient for rejecting an abstraction principle as unacceptable, is not necessary. Crispin Wright [1997] provided perhaps the most well-known example of such an abstraction principle: his aptly-named Nuisance Principle (here FSD(P,Q) abbreviates the second-order formula asserting that the symmetric difference of P and Q, that is, the collection of objects that are either P-andnot-Q or are Q-and-not-P, is finite): NP : (∀P)(∀Q)[NUI(P) = NUI(Q) ↔ FSD(P, Q)] The Nuisance Principle can be satisfied on domains of any finite cardinality (in which case all objects receive the same nuisance), but can be satisfied on no infinite domain. Thus, the Nuisance Principle, although consistent, is as unacceptable an abstraction principle as is Basic Law V. The reason for the unacceptability is different, however. At first glance, the Nuisance Principle appears to derive its unacceptability, not solely in terms of its own formal properties, but rather in terms of its interaction with other principles (such as Hume’s Principle). The existence both of inconsistent abstraction principles, and of pairs of individually consistent but incompatible abstraction principles, has given rise to a collection of problems that have been labeled The Bad Company Objection. Chief amongst the concerns falling under this heading are:

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(1) The Existential Challenge: Given the existence of problematic principles of the same general form as Hume’s Principle (such as Basic Law V and the Nuisance Principle), what reason do we have for thinking that there are any good abstraction principles (including Hume) which have the privileged status the abstractionist claims for them? (2) The Epistemological Challenge: Even if one is convinced that there are good abstraction principles that can play a foundational role such as the one envisioned for Hume’s Principle, in general how do we tell the good principles from the bad?

The epistemological challenge, although clearly important, will be sidestepped here, since we are interested in those problems that are intimately connected to the mathematics of abstractionism. The existential challenge, however, is, or at least can be easily approached as, a logical/mathematical issue – i.e. what proof- or model-theoretic features will guarantee that an abstraction principle is acceptable? One common response to this version of the Bad Company Objection (one first put forward by Wright’s “Is Hume’s Principle Analytic?” [1999] and finessed by Shapiro and Weir in “New V, ZF, and Abstraction” [1999], both reprinted below) is to require that an abstraction principle be conservative in a certain sense. The intuitive philosophical idea is this: An acceptable abstraction principle is meant to be a definition of the abstracts that it introduces, but it is also meant to be no more than this. As a result, the principle in question should have no substantial consequences for those objects in the domain that are not abstracts. Put simply, Hume’s Principle might entail all sorts of interesting claims about numbers, and even interesting claims regarding the numbers corresponding to certain collections of cats, but Hume’s Principle should not imply any substantial non-numerical claim about cats (a numerical claim would be one containing at least one occurrence of the NUM operator). Hume’s Principle can be proven to be conservative, as we would expect. On the other hand, the Nuisance Principle turns out to be non-conservative, as we would hope (since it entails, for example, that there must be only finitely many cats). So at first glance the conservativeness constraint would seem to be doing the job that it was designed to do. There are problems, of course. For one, New V, the most promising abstractionist reconstruction of set theory so far (even if far from fully satisfactory), is non-conservative. The technical details can be found in Shapiro and Weir’s [1999] paper, but the informal idea is easy to grasp. Within the language of New V we can define the ordinals in the usual way – ordinals are just transitive pure sets, well-ordered by membership. By the familiar reasoning of the Burali-Forti paradox, we can conclude as usual that there is no set of all ordinals. Within the context of New V, however, this means that the collection of ordinals is ‘Big’ – i.e. there is an onto function from the ordinals to the entire universe. But, since the ordinals are well-ordered by membership, this

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imposes a well-ordering on the entire universe. So New V is not conservative, since it implies that the universe can be well-ordered (and, within second-order logic, we can express this claim using no set-theoretic terminology). It is worth noting that Cook’s [2004] iterative variant of abstractionist set theory fares no better on this score. The problems with the conservativeness requirement do not stop with the fact that it would seem to rule out principles (such as New V) that we might otherwise have wished to be acceptable, In addition, it turns out, as Alan Weir shows in his “Neo-Fregeanism: An Embarrassment of Riches” [2004], that there are consistent yet incompatible abstraction principles that pass the conservativeness constraint. Weir calls such principles distractions, and he shows, further, that if we try to strengthen the conservativeness constraint in various natural ways in order to avoid such pairs of distractions, analogous problems arise in the meta-theory. There are a number of other criteria that have been proposed for narrowing down the list of potentially good abstraction principles. One suggestion is that the equivalence relation on the right-hand side of the biconditional accurately reflect the mathematical content mastered when we actually first learn the mathematical theory in question. In other words, the criterion for identity of the mathematical objects in question, provided by the abstraction principle, should clearly reflect the criterion by which we actually learned to identify and distinguish the objects in question. (The various abstractionist set theories are of particular relevance here, since there does not seem to be one single notion of set underlying our mathematical practice, but a number of competing notions, which are reflected in the competing reconstructions such as New V and Newer V.) Critics of this approach, however, have suggested that such requirements confuse something like the order of discovery with the order of explanation (e.g. see MacBride’s “On Finite Hume” [2002] and “Could Nothing Matter” [2003], reprinted as chapters 5 and 6 below). According to this line of thought, abstraction principles are intended to provide a story about how we might come to know mathematical truths a priori, but there is no reason to think that the actual route that we took in first coming to know these truths is necessarily anything like the privileged route provided by the abstraction (since the initial knowledge could even have been a posteriori!).

6.

The second problem: Too many objects

One of the main (supposed) advantages of abstractionism is that abstraction principles imply the existence of more objects than we would expect from logic and definitions alone. Some (including, of course, Boolos) have objected to this, on the grounds that logic (or analytic statements, or a priori knowledge more generally) should not imply the existence of all (or most) of the objects studied by working mathematicians:

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Introduction . . . It was a central tenet of logical positivism that the truths of mathematics were analytic. Positivism was dead by 1960 and the more traditional view, that analytic truths cannot entail the existence either of particular objects or of too many objects, has held sway ever since. (Boolos [1997], pp. 249–250)

Nevertheless, abstractionism is hopeless without the assumption that at least some existential claims are analytic, or a priori knowable, or something similar – the position in question is (on one reading) nothing more than a detailed philosophical account of how such is possible. So for our purposes here we will ignore such general worries regarding ontological excess. This ontological success seems to come at a price, however, since the very abstraction principles (or, sometimes, natural generalizations of them) that provide us with the ontology of standard mathematics have a tendency to imply the existence of more objects than are strictly needed for the reconstruction of the mathematical theory in question. Unfortunately, these additional objects are often unwanted or inconvenient. The first such unwanted object is ‘anti-zero’. Hume’s Principle implies that, in addition to the countable infinity of finite numbers, at least one other number exists, namely the number of the universal concept denoted by “x = x” – this ‘number’ is anti-zero. The standard account of cardinal numbers as developed in ZFC implies that there is no largest cardinal number, however. Thus, as was first pointed out by George Boolos, the theory of cardinals derived from Hume’s Principle seems to contradict the spirit, if not the letter, of the standard theory of cardinality as derived in Zermelo-Fraenkel set theory, where there can be no cardinal number of all objects. It is important to note that there is no formal contradiction here. One can easily construct a model which satisfies both Hume’s Principle and the (second-order) axioms of Zermelo–Fraenkel set theory – just take any settheoretic model of second-order ZFC, and interpret the numerical operator in Hume’s Principle as mapping each concept onto the appropriate ZFC cardinal, if the concept’s extension is set-sized, and mapping all other concepts onto some other object. Boolos’ point, rather, must be that there is no model of Hume’s Principle plus second-order ZFC where the ZFC cardinal numbers are exactly the cardinal numbers as defined by Hume’s Principle (under the same ordering). There are a number of obvious moves one could make here, although each has its problems. Among them are: (a) We might deny that ZFC provides an account of all the cardinal numbers, arguing instead that through this means we only get a model of the cardinal numbers corresponding to setsized concepts (while Hume’s Principle provides us with a theory of all the cardinal numbers). While attractive, this option seems to challenge the idea that set theory (however it is formalized) can play the foundational role traditionally ascribed to it (a role that abstractionists presumably would prefer it to retain, hence the interest in set-theoretic abstraction principles such as New V).

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(b) We might adopt a (positive) free logic, so that some instances of the numerical operator fail to designate objects (such as the instance that purports to refer to anti-zero). This strategy, however, seems open to two problems. First, given the abstractionist’s rather lenient criteria for when a term refers (that it occur in a true statement of the appropriate sort), this response seems somewhat ad hoc. Second, if abstractionists make it part of their official view that some numerical terms can fail to refer, then this allows the critic of abstraction to ask why it is not possible that all numerical terms fail to refer (or to argue that since some numerical terms fail to refer, then it seems unlikely that we can know a priori that other numerical terms do refer). For a detailed discussion of free logic within the abstractionist context, see Shapiro and Weir’s “NeoLogicist Logic Is Not Epistemically Innocent” [2000], reprinted below. A final strategy, however, is to replace Hume’s Principle with some suitably modified version, such as Finite Hume (here “Inf(P)” abbreviates the secondorder claim that there are infinitely many P’s): FHP : HP : (∀P)(∀Q)(NUM(P) = NUM(Q) ↔ ((Px ≈ Q) ∨ (Inf(P) ∧ Inf(Q)))) Finite Hume’s Principle provides a cardinal number for each finite concept, but maps any concept with an infinite number of instances onto the same, ‘Bad’ object (assuming that the logic is not free). Frege’s Theorem still holds for Finite Hume’s Principle (since the finite cardinals, i.e. the natural numbers, behave just as they do in the case of Hume’s Principle). There is no largest cardinal number, however, since the ‘Bad’ object that is the value “NUM” assigns to any infinite concept cannot be interpreted coherently as a number at all (to see this, it is enough to note that Finite Hume maps concepts of differing cardinalities onto the ‘Bad’ object in any uncountable model). So Finite Hume does not entail the existence of any strange cardinal numbers, such as anti-zero. It does, however, provide us with a generic ‘Bad’ object, just as the Quotient Principle QAP and New V were seen to do earlier. While such an additional object does not, like anti-zero, seem to violate any intuitions regarding the order-type of the cardinal numbers, they do bring with them different, yet equally serious problems of their own. In examining such ‘Bad’ objects, however, let us return our attention to the ‘Bad’ object provided by New V and similar extensions-forming principles, as it is this object that has attracted the most attention in the literature. Now, all existent abstraction principles that purport to provide us with something like extensions or sets also provide at least one unwanted, ‘Bad’ object – in fact, if the extensions-forming operator EXT is a total function then they must, since the claim that each concept receives a unique extension is contradictory. Typically (as is the case with New V) all of the concepts which are too ‘badly behaved’ to determine sets get mapped on to a single object, the ‘Bad’ extension, and in the case of New V, the ‘Bad’ object is the extension of any concept that is equinumerous to the entire domain.

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Now, unlike anti-zero, where the jury is still out regarding whether or not it is a genuine number, the ‘Bad’ extension is clearly not a genuine extension or set at all. It is merely an artifact of the particular abstractionist means for obtaining the things that we do want, i.e. the other extensions. Since treating the ‘Bad’ extension as some novel, until now unrecognized, yet real set seems implausible, the other option would be to treat it exactly as just described – as an artifact of the fact that we are treating our abstraction operators as defining total functions. The first problem is, of course, the seeming unavoidability of such ‘Bad’ objects in the first place. Why should our account of set theory (or rational numbers, or perhaps natural or cardinal numbers) seem to require the existence of an additional, and unwanted, object in the first place? Shouldn’t it be possible to provide a foundational account of any mathematical theory that entails the existence of all, and crucially, only, the objects required by that theory? This sort of question, while important and intuitively quite troubling, is also rather loosely formulated. There are other problems associated with the existence of ‘Bad’ objects (in particular, the ‘Bad’ extension) that are a good bit more precise, however. If the ‘Bad’ extension is merely an artifact of our theory, and not a ‘real’ extension in some sense, then presumably any proof of a crucial set-theoretic result based on New V should not depend on the existence of the ‘Bad’ extension. In other words, any set theoretic axiom that turns out to be true (given New V) should be true solely in virtue of the settheoretic ‘behavior’ of the genuine extensions (and thus should not depend for its truth on the existence or ‘behavior’ of the ‘Bad’ extension). This requirement seems reasonable. Unfortunately, at least for the present attempts at reconstructing set theory within an abstractionist framework, it seems like a requirement that cannot be met. The problem is that, if we are not allowed to make use of the existence of the ‘Bad’ extension, we lose the proof that there are infinitely many sets. Boolos provides the following proof that New V entails the existence of at least two objects (this is the initial part of his proof that New V entails the existence of infinitely many objects): Let Ø be the concept [x : x = x] . . . since there is at least one object (e.g. EXT (x = x) or EXT (x = x)), Ø is small, Ø = V, and EXT(x = x) = EXT(x = x). ([1989], p. 90, notation modified to fit that used here)

Notice that the proof makes explicit reference to the ‘Bad’ extension (i.e. EXT (x = x)). This is not accidental – the result depends on the existence of the ‘Bad’ extension in order to guarantee that finite concepts are not ‘Big’. There does not exist any proof from New V to the pairing axiom that does not, in some way, make use of the ‘Bad’ extension. To see why, consider New V interpreted in a free logic that allows EXT to fail to take on a value when applied to ‘Big’ concepts. In such a logic, there will be

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a one-element model of New V – just let the domain contain (for example) the empty set. Then there are two concepts – the empty one, and the one holding of the empty set. Since the latter is ‘Big’ (it is, in fact, the entire universe), it need not receive an extension, so we can just map the empty concept onto the empty set, and we have our model. Thus, there seems to be a real problem in the way that abstraction principles for extensions such as New V behave. On the one hand, they seem to imply the existence of unwanted objects – in particular, the ‘Bad’ object. On the other hand, this object seems necessary in order to ‘bootstrap’ our way up to a proof that there are infinitely many sets. No satisfactory solution to this dilemma has been presented as of yet. Although the reader might be forgiven for thinking that this is, already, more than enough problems, a look at abstractionist reconstructions of real analysis is in order. As noted above, the abstraction principle generating quotients of integers also generated a ‘Bad’ object, but this seems less worrisome than anti-zero or the ‘Bad’ extension, since the traditional theory of the rationals was already plagued with a similar problem (i.e. the ill-definedness of 1/0 . More troubling, however, is the use of cut abstraction in the final step of the construction. Now, applying this particular abstraction principle to the domain does not, at first glance, seem to present any problems – we obtain, in fact, exactly the real numbers (or something isomorphic to them) and nothing else. The problem possibly arises, however, when we ask the following question: If we can use abstraction to take cuts on the rationals, as we did to obtain the reals, then is it permissible to take, as objects, the cuts on any linear order, by applying an appropriate abstraction principle? If the answer is “No”, the we seem faced with another particularly difficult instance of the Bad Company Objection – how are we to determine when we can, and when we cannot, apply an abstraction principle to a linear order to take cuts as objects? If the answer is “Yes”, however, then we are besieged by another worry: Generalizing such cut abstraction to any linear ordering whatsoever generates a large ontology (in the worse case, proper class sized). This is, at best, extremely surprising in a view that emphasizes its epistemic conservativeness. Additionally, some of the more powerful versions of generalized cut abstraction are incompatible with other, somewhat attractive abstraction principles, such as New V and variants of it (for a discussion of generalized versions of cut abstraction see Cook’s “The State of the Economy: Neo-Logicism and Inflation” [2002], reprinted as chapter 12 below, and criticism of it in Bob Hale’s “Reals by Abstraction”, [2000] chapter 11). Thus, in a number of ways the ontological power of abstractionism seems to backfire – the very strength of the view, the fact that it purports to provide us with an account of how we can have a priori knowledge of the existence and properties of those abstract objects studied by mathematicians, also is one of its weaknesses, since it also seems to provide us with a priori knowledge of the existence of until now unrecognized and, once recognized, unwanted

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objects such as anti-zero and the ‘Bad’ extension. Like the Bad Company objection before it, a satisfactory solution to this problem (or, better, this family of problems) would seem to be a matter of determining where to draw certain lines: How do abstractionists determine which principles (and which formulations of certain principles) will provide them with access to the objects required for mathematics without also entailing the existence of additional objects that are both unnecessary and, at times, inconvenient?

7.

The third problem: What objects?

The final major problem of interest here is the notorious Caesar Problem. Frege first points out the problem in the Grundlagen, where he considers an abstraction principle introducing directions (here the initial quantifiers range over lines, and “//” is the relation of parallelism): (∀a)(∀b)(DIR(a) = DIR(b) ↔ a//b) After pointing out that this definition provides us with the means for identifying directions, and distinguishing distinct directions from one another, he points out that: . . . this means does not provide for all cases. It will not, for instance, decide for us whether England is the same as the direction of the earth’s axis – if I may be forgiven an example which looks nonsensical. Naturally no one is going to confuse England with the direction of the Earth’s axis; but that is no thanks to our definition of direction. (Frege, [1974], §66, pp. 77–78)

Looking at this from a technical perspective, we can see the problem as follows: Abstraction principles, such as Hume’s Principle and New V, whose right-hand side can be expressed in purely logical vocabulary, place no constraints on which object in a particular domain plays the role, say, of seven, or the empty set (things are slightly more complicated in the case of abstraction principles, such as those used to construct the reals, where the equivelence relation on the right contains other abstraction operators). All that determines whether a particular set can serve as the domain of a model of either of these principles is the cardinality of the set – if the set is the right size, then any object in the set can be any number or set (the only requirement is that each object can play the role of at most one number, or one set). Much has been written on the Caesar Problem, but approaches to it generally take one of three routes: First, we can deny it is a problem, adopting a sort of structuralist approach to abstractionism where it does not matter whether Caesar turns out to be the number two, as long as we are guaranteed that some object plays this role. (Although the Caesar problem is not his main target, Crispin Wright’s “Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege’s Constraint” [2000], chapter 13 below, draws connections between ante rem structuralism and abstractionism, and is particularly relevant

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here.) Second, we can attempt to reformulate our abstraction principles in more complicated ways (e.g. by inserting modal operators in appropriate places or the like) so that the reference of numerical terms is more determinate. Third, we might argue that although abstraction principles alone do not determine which object, in particular, is picked out by a certain numeral, abstraction principles plus other background constraints do determine numerical reference uniquely. Which of these approaches is most promising has yet to be determined. In fact, as the literature grows, new variations on the Caesar Problem seem to sprout up at least as fast as attempts to solve them. Most important among these are: The Counter-Caesar Problem: How do we guarantee that particular Fregean numerals denote the same object as their natural language counterparts (e.g. does NUM(x = x) denote the same thing as the English locution “zero”)? The Julio Cesar Problem: How do we guarantee that the cardinal numbers provided by Hume’s Principle denote the same kind of objects as are denoted by mathematical terms occurring in natural language (e.g. does NUM(x = x) denote the same kind of thing as the English locution “zero”)? The C-R Problem: How do we determine whether abstracts provided by distinct abstraction principles are identical or distinct (e.g. is the complex number 0, provided by the appropriate abstraction principle, identical to the real number 0, provided by a different abstraction principle)? Although the Caesar Problem (and its cousins) results from certain formal characteristics of abstraction principles, responses to it tend to be less technical. Nevertheless, a number of the chapters included below contain extended discussions of it. (The reader is also encouraged to consult MacBride [2005] and Cook and Ebert [2005] for further discussion of variants of the Caesar Problem.)

8.

Indefinite extensibility

As the literature on these problems and other issues has grown, the notion of indefinite extensibility has become more and more central to purported solutions. One promising line of attack on both the ‘too-many-abstraction’ principles class of problems and the ‘too-many-objects’ class of problems has been to suggest that we restrict our attention to those abstraction principles that provide abstracts only for concepts which are not indefinitely extensible. Of course, this does little to help us until we know what indefinite extensibility is. Bertrand Russell seems to be the first person to discuss this notion when considering the cause of the various set-theoretic paradoxes:

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Introduction The contradictions result from the fact that . . . there are what we may call selfreproductive processes and classes. That is, there are some properties such that, given any class of terms all having such a property, we can always define a new terms also having the property in question. Hence we can never collect all of the terms having the said property into a whole; because, whenever we hope we have them all, the collection which we have immediately proceeds to generate a new term also having the said property. ([1906], p. 144)

The term “indefinite extensibility” is due to Michael Dummett, however, who extended Russell’s idea as follows: An indefinitely extensible concept is one such that, if we can form a definite conception of a totality all of whose members fall under the concept, we can, by reference to that totality, characterize a larger totality all of whose members fall under it. ([1993], p. 441)

It has become standard to use the term ‘definite’ for those concepts that are not indefinitely extensible. The ordinal numbers provide perhaps the clearest example of an indefinitely extensible collection. Consider any definite collection of ordinals (i.e. a set of ordinals). Given such a collection, we can immediately form a conception of an ordinal not in that collection (i.e. the ‘next’ ordinal, (i.e. either the successor of the greatest ordinal in the collection in question, or the supremum of the collection in question). As a result, there seems to be a sense in which we can never collect together all of the ordinals into a definite totality, since we could repeat this reasoning on such a collection to obtain an ordinal that is not in such a collection of all ordinals – contradiction (this is essentially just the reasoning behind the Burali-Forti paradox). An indefinitely extensible concept is thus one which allows for a certain sort of iteration – any time we have collected together some definite sub-collection of things falling under that concept, we can find a new object that is not in that collection. In fact, the ordinals are not only a clear example of the notion in question, but their structure seems to be fundamental to indefinite extensibility itself, since this iterability suggests that any indefinitely extensible collection will contain a structure isomorphic to the ordinals (it is worth noting, however, that Dummett would reject this Russellian characterization of indefinite extensibility). Thus, one way of characterizing indefinitely extensible concepts is “those concepts that are like the ordinals in relevant ways”. As Peter Clark points out in his contribution to this volume (“Frege, Neo-logicism and Applied Mathematics” [2004], chapter 3 below), another way of picking out the indefinitely extensible concepts is to just note that they are the ones whose extensions do not form sets. But neither of these suggestions, intuitively helpful as they are, do the abstractionist any real good. The abstractionist, remember, wishes to use the notion of indefinite extensibility in order to formulate a restricted version of Basic Law V (and other abstraction principles) which will provide an adequate

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set theory. As a result, no characterization of indefinite extensibility (such as those above) which uses set-theoretic notions can be of use, since using set theoretic notions to formulate one’s implicit definition of set would introduce a rather vicious circle into the picture. Thus, the abstractionist needs some neutral formulation of the notion in question. As of yet, no completely adequate account of indefinite extensibility has been found, at least none that is of the sort that could be mobilized by the abstractionist wishing to use it in formulating various abstraction principles. This is not to say, of course, that no work of interest has been carried out – on the contrary, at least half of the papers in the present volume make at least passing reference to the importance of this problem, and almost all of the papers in the last section, on set theory, contain detailed discussion of the issue. Of particular interest is Stewart Shapiro’s “Prolegomenon to Any Future Neo-Logicist Set Theory. . . ” (chapter 18 below), which contains both a detailed examination of indefinite extensibility as discussed by philosophers such as Dummett and Russell, as well as a sustained technical examination of what formal characteristics a successful abstractionist account of the notion requires.

9.

One last thing

Although the bulk of the literature on abstraction and its mathematics, and the majority of the papers to follow, focus either on the actual formalization of arithmetic, analysis, and set theory, or on the three major sorts of problem just outlined, there are of course many other crucial questions regarding abstraction to be answered and many other avenues to be explored. While space considerations preclude detailed discussion of them here, at least one of them deserves brief mention before moving on to the papers themselves. The question in question is this: In what ways can the abstractionist’s formal results be adopted or adapted by their philosophical opponents? For example, in “Frege’s Unofficial Arithmetic” [2002] (chapter 10 below) Agustin Rayo utilizes Frege’s Theorem (and corollaries of it) to provide a distinctly non-Fregean (in fact, somewhat Quinean) account of arithmetic (and, in particular, applied arithmetic). While Rayo suggests that the account he provides is at least inspired by Frege’s own views (views Frege held after he abandoned logicism), the project he sketches is worked out against a philosophical background quite different from the one assumed by most abstractionists. The point, to put it bluntly, is this: even if abstraction principles are not definitions in the sense the abstractionist suggests, they might nevertheless play some crucial role in our epistemological account of mathematics. It is thus hoped that this collection will serve as a repository of work on the technical aspects of abstraction principles which can be utilized by both the abstractionist himself and also by adherents of different, competing philosophical

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accounts of mathematics (even if the majority of the actual papers are working within the standard Fregean abstractionist picture).

References Black, M. [1965], Philosophy in America, Ithaca, Cornell University Press. Boolos, G. [1989], “Iteration Again”, Philosophical Topics 17: 5–21. Boolos, G. [1990a], “The Standard of Equality of Numbers”, in Boolos [1990b]: 3–20. Boolos, G. (ed.) [1990b], Meaning and Method: Essays in Honor of Hilary Putnam, Cambridge, Cambridge University Press. Boolos, G. [1997], “Is Hume’s Principle Analytic?”, in Heck [1997b]: 245–261, reprinted below as chapter 1. Boolos, G. [1998], Logic, Logic, and Logic, Cambridge, MA, Harvard University Press. Boolos, G. and Heck, R. [1998] “Die Grundlagen der Arithmetik §82–83”, in Boolos [1998]: 315–338. Burgess, J. [1984], Review of Wright [1983], Philosophical Review 93: 638–640. Clark, P. [2004], “Frege, Neo-logicism and Applied Mathematics”, in Stadler [2004]: 169–183, reprinted below as chapter 3. Cook, R. [2002], “The State of the Economy: Neologicism and Inflation”, Philosophia Mathematica 10: 43–66, reprinted below as chapter 12. Cook, R. [2003], “Aristotelian Logic, Axioms, and Abstraction”, Philosophia Mathematica 11: 195–202, reprinted below as chapter 9. Cook, R. [2004], “Iteration One More Time”, Notre Dame Journal of Formal Logic 44: 63–92, reprinted below as chapter 20. Cook, R. & P. Ebert [2005], “Abstraction and Identity”, Dialectica 59: 121–139. Demopoulos, W. [2003], “On the Philosophical Interest of Frege Arithmetic” Philosophical Books 44: 220–228, reprinted below as chapter 7. Fine, K. [2002], The Limits of Abstraction, Oxford, Clarendon Press. Frege, G. [1974], Die Grundlagen Der Arithmetic, J.L. Austin (trans.), Oxford, Basil Blackwell. Frege, G. [forthcoming], Grundgesetze der Arithmetik, C. Wright et al. (trans.), Oxford, Oxford University Press. Hale, R. [2000], “Reals by Abstraction”, Philosophia Mathematica 8: 100–123, reprinted below as chapter 11. Hale, R. [2000], “Abstraction and Set Theory”, Notre Dame Journal of Formal Logic 41: 379– 398, reprinted below as chapter 17. Hale, B. C. Wright [2001], The Reason’s Proper Study. Oxford, Oxford University Press. Heck, R. [1993], “The Development of Arithmetic in Frege’s Grundgesetze der Arithmetik”, Journal of Symbolic Logic 10: 153–174. Heck, R. [1997a], “Finitude and Hume’s Principle”, Journal of Philosophical Logic 26: 589– 617, reprinted below as chapter 4. Heck, R. (ed.) [1997b], Language, Thought, and Logic, Oxford, Oxford University Press. Hodes, H. [1984], “Logicism and the ontological commitments of arithmetic”, The Journal of Philosophy 81: 123–149. MacBride, F. [2000], “On Finite Hume”, Philosophia Mathematica 8: 150–159, reprinted below as chapter 5. MacBride, F. [2002], “Could Nothing Matter?”, Analysis 62: 125–135, reprinted below as chapter 6. MacBride, F. [2003], “Speaking with Shadows: A Study of Neo-logicism”, British Journal for the Philosophy of Science 54: 103–163. MacBride, F. [2005], “The Julio César Problem”, Dialectica 59: 223–236. Parsons, C. [1965], “Frege’s Theory of Number”, in Black [1965]: 180–203. Rayo, A. [2002], “Frege’s Unofficial Arithmetic”, Journal of Symbolic Logic 67: 1623–1638, reprinted below as chapter 10. Russell, B. [1902], “Letter to Frege” in van Heijenoort [1967]: 124–125.

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Russell, B. [1906], “On Some Difficulties in the Theory of Transfinite Numbers and Order Types”, Proceedings of the London Mathematical Society 4: 29–53. Shapiro, S. [2000], “Frege Meets Dedekind: A Neologicist Treatment of Real Analysis”, Notre Dame Journal of Formal Logic 41: 335–364, reprinted below as chapter 14. Shapiro, S. [2003], “Prolegomenon to Any Future Neo-Logicist Set Theory: Abstraction and Indefinite Extensibility”, British Journal for the Philosophy of Science 54: 59–91, reprinted below as chapter 18. Shapiro, S. & A. Weir [1999], “New V, ZF and Abstraction”, Philosophia Mathematica 7: 293– 321, reprinted below as chapter 15. Shapiro, S. & Weir [2000], “Neo-Logicist Logic Is Not Epistemically Innocent”, Philosophia Mathematica 8, 160–189, reprinted below as chapter 8. Stadler, F. (ed.) [2004], Induction and Deduction in the Sciences, Dordrecht, Kluwer Academic Publishers. Uzquiano, G. & I. Jané [2004], Well- and Non-Well-Founded Extensions”, Journal of Philosophical Logic 33: 437–465, reprinted below as chapter 16. van Heijenoort, J., (ed.) [1967], From Frege to Gödel: A Sourcebook in Mathematical Logic, Cambridge, MA, Harvard University Press. Weir, A. [2004], “Neo-Fregeanism: An Embarassment of Riches”, Notre Dame Journal of Formal Logic 44: 13–48, reprinted below as chapter 19. Whitehead, A. N. & B. Russell [1910–1913], Principia Mathematica, 3 vols., Cambridge, Cambridge University Press. Wright, C. [1983], Frege’s Conception of Numbers as Objects, Aberdeen, Aberdeen University Press. Wright, C. [1997], “On the Philosophical Significance of Frege’s Theorem”, in Heck [1997b]: 201–244. Wright, C. [1999], “Is Hume’s Principle Analytic?”, Notre Dame Journal of Formal Logic 40: 6–30, reprinted below as chapter 2. Wright, C. [2000], “Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege’s Constraint”, Notre Dame Journal of Formal Logic 41: 317–334, reprinted below as chapter 13.

IS HUME’S PRINCIPLE ANALYTIC?1 George Boolos

The reduction, however, cuts both ways. It is not easy to see how Frege can avoid the seemingly frivolous argument that if his reduction is really successful, one who believes firmly in the synthetic character of arithmetic can conclude that Frege’s logic is thus proved to be synthetic rather than that arithmetic is proved to be analytic. Hao Wang 2

There are a number of issues on which Crispin Wright and I disagree, some of them substantive and some merely terminological. For example, we disagree over whether the term “analytic” can be suitably applied to HP and whether a derivation of arithmetic from HP would establish a doctrine appropriately called “logicism.” I also have certain reservations, which I shall set out later, about his notions of explanation and reconceptualization. However, I think the areas of agreement about the interest of Frege’s derivation of arithmetic are both wide-ranging and far more significant than those of disagreement. In particular I want to endorse Wright’s closing suggestion that “the problems and possibilities of a Fregean foundation for mathematics remain [wide?] open” and the remark made earlier in his paper that “The more extensive epistemological programme which Frege hoped to accomplish in the Grundgesetze is still a going concern.” I also want to emphasize that I consider Wright to have made a great scientific contribution in showing contemporary readers 1 This article first appeared in Richard G. Heck Jr., ed., Logic, Language, and Thought, Oxford, Oxford University Press (1997). Reprinted by kind permission of Oxford University Press. A version of this paper was presented to a 1994 American Philosophical Association symposium on the topic of logicism. Crispin Wright was the co-symposiast and Charles Parsons the commentator. Michael Dummett much dislikes the designation “Hume’s Principle” because the remark in Hume’s Treatise (I, III, I, para. 5) which Frege cited with approval and from which the name derives, presupposes the doctrine that a number is an item composed of units, a doctrine which Frege is presumed to have refuted. Since this paper first appeared in a Festschrift for Michael, I used the designation “HP” instead. Cf. Chomsky and “LF.” 2 Wang, Hao, “The Axiomatization of Arithmetic,” Journal of Symbolic Logic 22 (1957), pp. 145–158, reprinted in Wang, Hao, A Survey of Mathematical Logic, Peking, Science Press (1963), pp. 68–81. The quotation, together with other extremely interesting observations, appears on p. 80.

3 Roy T. Cook (ed.), The Arché Papers on the Mathematics of Abstraction, 3–15. c 2007 Springer.

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how the deduction of the Peano postulates from HP could be carried out and in formulating the conjecture, subsequently verified, that HP is consistent. 3 The first issue I want to take up is whether a derivation of arithmetic from HP vindicates logicism. My view is: no logic, no logicism. It is clear what has to be established in order to show the truth of something we can call logicism with a clear conscience. Arithmetic has to be shown to be provable from an extension by definitions of a theory that is logically true. In technical parlance, arithmetic has to be interpreted in a logically true theory. It cannot be, trivially: Arithmetic implies that there are two distinct numbers; were the relativization of this statement to the definitions of the predicate “number” provable by logic alone, logic would imply the existence of two distinct objects, which it fails to do (on any understanding of logic now available to us). Wright states that if it has to be made out that HP is a truth of logic, then “the prospects are unimproved,” the prospects, I take it, being those for establishing a species of logicism. I infer that he does not consider HP to be a truth of logic. Nor do I: the principle implies the existence of too many objects. So I do not conclude, as Wright does, that the proof of Frege’s theorem by itself establishes logicism. It only shows the beautiful, deep, and surprising result that arithmetic is interpretable in Frege arithmetic, a theory whose sole nonlogical axiom is HP. Wright argues, though, that since HP is analytic, the proof yields “an upshot still worth describing as logicism, albeit rather different from the conventional understanding of the term.” I might be prepared to agree that something describable as logicism in a different understanding of that term would have been established if HP had been shown to be analytic or akin to something properly called a definition. But I doubt that it can be. Having to discuss whether HP is analytic is rather like having to consider whether hydrogen sulfide is deflogisticated. One can certainly see reasons why one might be tempted to call H2 S dephlogisticated: but if I am right in thinking that to deflogisticate is to combine with oxygen, there are conclusive reasons for not doing so. The main reason why the notion of analyticity is all but useless is discussing propositions of mathematics like HP is that, although an analytic statement is supposed to be one that is true in virtue of the meanings of the terms contained in sentences expressing it (and syntactic features of those sentences), the phrase “true in virtue of meanings” leaves it indeterminate how much mathematics may be used to get from facts about meanings to the truth of the statement, or, more exactly, how much mathematics it is allowable to use 3 Wright, Crispin, Frege’s Conception of Numbers as Objects, Scots Philosophical Monographs, vol. 2, Aberdeen, Aberdeen University Press (1983). The derivation is on pp. 154–168. The discussion of numbertheoretic logicism III is on pp. 153–154.

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in deriving the statement (or the statement that that statement is true) from reports of meanings. In brief, we are not told how strong the mathematics is that “in virtue of” permits. The stronger the mathematics permitted, the greater the number of analytic mathematical truths, of course. The point, in essence, is due to Gödel and is different from the objection raised by the question “Why mathematics rather than geology?” In the interest of trying to get at what’s really at issue between Wright and myself, however, I shall ignore the standard difficulties presented by “analytic,” including the uncertainty what the interest or point of classifying a statement as analytic is and the worry that complex logical argumentation might itself create semantic content, 4 and suppose that I understand the concept sufficiently well, well enough at least to know what’s meant by calling “all vixens are foxes,” etc. analytic and by saying that there is a semantic connection between “vixen” and “fox.” At the outset, let me acknowledge that I have no knock-down argument that will persuade a diehard defender of the claim that HP is analytic to abandon the view. All I shall offer are what strike me as some rather, and perhaps sufficiently, weighty considerations against that position. At first glance, HP might certainly seem analytic. In its statement “number” means “cardinal number” and, one would naturally wonder, isn’t it a matter of the semantic connection between “cardinal number” and “one–one correspondence” that two concepts have the same cardinal number just when things falling under one of them can be put in one–one correspondence with those falling under the other? Isn’t the cardinal of x the same as that of y just when there’s a one–one correspondence between x and y, and that because of what “cardinal number” means? So isn’t the left-hand side of HP close enough in meaning to the right-hand side for it to count as analytic? Doesn’t the left-hand side have the same sense as the right? 5 Let me begin to respond to this argument by recalling two features that analytic statements have been traditionally supposed to enjoy: first, they are true; secondly and roughly speaking, they lack content, i.e., they make no significant or substantive claims or commitments about the way the world is; in particular, they do not entail the existence either of particular objects or of more than one object. (It may be held that some analytic statement might entail the existence of at least one object, as will be the case if every logical truth counts as analytic.) Some have been tempted by the idea of analytic statements that happen not to be true, e.g., “the present king of France is a royal.” On the view in question, the semantic connection between “king” and “royal” suffices to ensure the analyticity of the entire statement, despite the failure of its subject to denote. But analytic statements are, and (since we are playing along) are analytically, 4 This possibility is suggested by a remark of Frege about condensation in §23 of his Begriffsschrift. 5 Thanks here to Arthur Skidmore.

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analytic truths, and the view may be put aside. The example is worth noting, however, for, as I am going to suggest later, HP suffers from a defect similar to that of “the present king of France is a royal,” which would not be analytic even if there were presently a (unique) king of France, since, of course, it would not be analytic that there is one. The main significant worry for the defender of the analyticity of HP concerns the quite strong content that it appears to possess. HP has consequences having to do with certain features of the domain of objects over which its firstorder variables range, in particular with the number of those objects there are. Much of the most interesting work in mathematical logic in the last 20 years or so has dealt with comparisons of strength of various logical and mathematical statements, examining which well-known theorems of mathematics can be derived from which logical principles (and vice versa!) in which background theories. We now know that Frege arithmetic is equi-interpretable with full second-order arithmetic, “analysis,” and hence equi-consistent with it. Learning that HP is analytic would not help us in the slightest with the problem of assessing the strength of various theorems, fragments, and subtheories of analysis, all of which would, I suppose, have to count as analytic. The first part of my worry about content is that HP, when embedded into axiomatic second-order logic, yields an incredibly powerful mathematical theory. Wright will say: Hooray! Math is analytic after all. But we don’t know what follows from its being so and we will have to study the subanalytic to see what (logically) entails what just as hard as before. It is known that HP does not follow (a word I will not surrender) from the conjunction of two of its strongest consequences: the (interesting) statements that nothing precedes zero and that precedes is a one–one relation. If HP is analytic, then it is strictly stronger (another non-negotiable term) than some of its strong consequences. It is also known that arithmetic follows from these two statements alone, and that arithmetic is strictly weaker than even their disjunction. 6 Faced with these results, how can we really want to call HP analytic? Frege, for a lengthy stretch of his career, held that the existence of infinitely many objects could be seen to follow from a set of principles and definitions that could, by his lights, be counted as analytic. He abandoned the view in 1906, according to Dummett, when he realized that his attempted patch to Basic Law V would not work. It is doubtful that Russell could be considered a logicist in the full sense of the term while writing Principia, whose stated aim is to analyze the notions employed in mathematics, not to show arithmetic to be a branch of logic. Despite the Gödel incompleteness theorems and Russell’s protestations that the axiom of infinity was no logical truth, it was a central tenet of logical positivism that the truths of mathematics were analytic. Positivism was dead by 1960 and the more traditional view, that analytic truths 6 For proofs of these results, see my “On the Proof of Frege’s Theorem,” in Benacerraf and His Critics, Adam Morton and Stephen Stich, eds., Cambridge, MA, Blackwell (1996), pp. 143–159.

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cannot entail the existence either of particular objects or of too many objects, has held sway since. Wright wishes to overthrow the tradition, but it should be asked how a statement that cannot hold if there are only finitely many objects can possibly be thought to be analytic, a matter of meanings or “conceptual containment.” On the symbolization that I prefer, HP reads: ∀F∀G(#F = #G ↔ F ≈ G) where “F ≈ G” is an abbreviation for a second-order formula expressing that there is a one–one correspondence between the objects falling under the concept F and those falling under the concept G. We need not here write out the formula, but must remember that it contains some first-order quantifiers. We must also remember the grammatical category of “#,” “octothorpe”: it is a function-sign, which when attached to a monadic second-order variable like “F,” produces a term of the same type as individual variables that occur in “F ≈ G.” It is essential to the proof of Frege’s theorem that octothorpe be so construed. Thus octothorpe denotes a total function from concepts to objects. Logic, plus the convention that function signs like octothorpe denote total functions, will guarantee that ∀F∃!x #F = x is true. It will not guarantee that HP is. HP entails, as Wright has put it with exemplary force and Cartesian clarity, that there is a partition of concepts into equivalence classes, in which two concepts belong to the same class if and only if they are equinumerous. If there are only k objects, k a finite number, then, since there are k + 1 natural numbers ≤ k, there will be k + 1 equivalence classes, viz. a class containing each concept under which zero objects fall, a class containing each concept under which exactly one objects falls, . . . , and a class containing each concept under which all k objects fall. (We need not here assume that concepts are individuated extensionally.) Thus, if there are only k objects, there is no function mapping concepts to objects that takes non-equinumerous concepts to different objects, for there won’t be enough objects around to serve as the values of the function, since k + 1 are needed. So if HP holds—even if only the left–right direction (the same direction as in the fatal Basic Law V) holds— there must be infinitely many objects. One person’s tollens is another’s ponens, and Wright happily regards the existence of infinitely many objects, and indeed, that of a Dedekind infinite concept, as analytic, since they are logical consequences of what he takes to be an analytic truth. He would also regard the existential quantification of HP (over the positions occupied by octothorpe) as analytic. But what guarantee have we that there is such a function from concepts to objects as HP and its existential quantification claim there to be? I want to suggest that HP is to be likened to “the present king of France is a royal” in that we have no analytic guarantee that for every value of “F,” there is an object that the open definite singular description “the number

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belonging to F” denotes. I shall also suggest that there may be some analytic truths in the vicinity of HP with which it is being confused. I hope that the suggestions will do justice both to the thought that there is a strong semantic connection between “the number of . . . ” and “one–one correspondence” and to the traditional idea that analytic truths do not entail the existence of a lot of objects. Our present difficulty is this: just how do we know, what kind of guarantee do we have, why should we believe, that there is a function that maps concepts onto objects in the way that the denotation of octothorpe does if HP is true? If there is such a function then it is quite reasonable to think that whichever function octothorpe denotes, it maps non-equinumerous concepts to different objects and equinumerous ones to the same object, and this moreover because of the meaning of octothorpe, the number-of-sign or the phrase “the number of.” But do we have any analytic guarantee that there is a function that works in the appropriate manner? Which function octothorpe denotes and what the resolution is of the mystery how octothorpe gets to denote some one definite particular function that works as described are questions we would never dream of trying to answer. (Harold Hodes’ article “Logicism and the ontological commitments of arithmetic” 7 contains much wisdom about these mysteries of mathematical reference.) Nevertheless, it would seem that if there is such a function, then whichever function octothorpe does denote, it also does the trick. 8 Thus, I am moved to suggest, very tentatively and playing along, that the conditional whose consequent is HP and whose antecedent is its existential quantification might be regarded as analytic. The conditional will hold, by falsity of antecedent, in all finite domains. By the axiom of choice, the antecedent will be true in all infinite domains, but then, we may suppose, nothing will prevent the consequent from being true. I also find plausible the suggestion that the right-to-left half of HP, which states that if F and G are equinumerous, then their numbers are identical, is analytic. It is the left-to-right half, which states that if F and G are not equinumerous, then their numbers are distinct, that blows up the universe. (E.g, consider the concept non-self-identical; call its number zero. Now consider the concept identical with zero; call its number one. By the left-to-right half of HP: since the concepts are not equinumerous, zero is not one.) The analogy with Basic Law V is obvious. Frege divided Basic Law V into Va, the left-to-right half, and its converse Vb. It was the left-to-right half that gave rise to Russell’s paradox. Vb has considerable claim to being regarded as a logical truth: (a) it is valid under standard semantics, thanks to the axiom of extensionality; (b) if the Fs are the Gs, as the antecedent asserts, then whatever 7 Hodes, Harold “Logicism and the Ontological Commitments of Arithmetic,” Journal of Philosophy 81 (1984), pp. 123–149. 8 Hartry Field has made a similar suggestion in his review of Wright’s Frege’s Conception of Numbers as Objects, which is reprinted in Field, Hartry, Realism, Mathematics, and Modality, Oxford, Blackwell (1989), pp. 147–170.

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“extension” may mean, the extension of the Fs is the extension of the Gs; and (c) if the antecedent holds, then the concepts Fand G bear a relation to each other that Frege called the analogue of identity. Thus under each of three familiar systems of formula-evaluation, Vb can never turn out false. In the case of both HP and Basic Law V, we have a principle whose left-to-right half requires that there be a function from concepts to objects respecting certain non-equivalences of those concepts. Unless enough objects exist, these nonequivalences cannot be respected. All that the right-to-left halves demand is that the equivalences be respected, as they can be trivially, by mapping all concepts to one and the same object. ∀F∀G(∀x(Fx ↔ Gx) → #F = #G), which has the same form as Basic Law Vb, can equally justifiably be claimed to be a logical truth, and the stronger ∀F∀G(F ≈ G → #F = #G) much more plausibly thought analytic, in virtue of the meaning of “#,” than its converse. There is a further difficulty, or at any rate a further aspect of the same difficulty: If numbers belonging to concepts F and G are supposed to be identical if and only if F and G are equinumerous, then how do we know that, for every concept, there is such a thing as a number belonging to that concept? We should not be led astray by the concision, symmetry, and apparent familiarity and obviousness of #F = #G ↔ F ≈ G into ignoring the fact that octothorpe is a function sign (for a function of higher type). Like constants and the usual sort of function sign, it may help in concealing significant existential commitments. (Perhaps because of that danger, Quine, concerned with ontology and logic’s role in its study, almost entirely avoids constants and function signs in his textbook Methods of Logic.) An analogy may help: if volumes are supposed to be translation- and rotation-invariant, finitely additive, and non-trivial, with singletons and balls of radius r having volumes 0 and 4πr 3 /3, respectively, then, as the “paradoxical” Banach–Tarski theorem shows, not every bounded set of points in three-space has a volume. It would thus be illegitimate to introduce a sign for a totally defined function from bounded sets of points in three-space to real numbers and assume that the function was translation-invariant, etc. And one had therefore better not say: it is analytic that volume is translation-invariant, etc., and it is analytic that there is always such a thing as the volume of any bounded set of points in three-space, for the conjunction of the two statements claimed to be analytic is false. Similarly, if numbers are supposed to be identical if and only if the concepts they are numbers of are equinumerous, what guarantee have we that every concept has a number? 9 Or, if we take ourselves to know that with every concept there is functionally associated some object, then how do we know that the associated object is a number belonging to F? 9 Profound thanks here to Peter Clark.

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It will be useful here to formulate HP in a way that expressly brings out its existential commitments. Let Numbers be the statement: for every concept F, there is a unique object xsuch that for every concept G, x is a number belonging to G if and only if F is equinumerous with G. Is Numbers analytically true? I see no reason at all to believe that it is analytic that for every F, there is such a (unique) object x. To reply that it is, since Numbers follows from HP, and HP is analytic, would seem to beg a question that ought not to be begged. Even more strongly, I don’t see any reason to think that it’s analytic that objects can be so assigned to concepts that any two concepts are assigned the same object if and only if they are equinumerous. It is not only the existence of a function of higher type making such an assignment of objects to concepts that seems synthetic to me: the weaker modal claim that objects can be so assigned strikes me as synthetic as well. I repeat that one person’s ponens is another’s tollens and admit again that I don’t have a knock-down argument against Wright’s view. I now want to raise some objections to Wright’s notion of a reconceptualization and his use of the term “explanation.” Discussing Frege’s (more-or-less) analogous case of directions and parallelism, Wright says, “we have the option . . . of re-conceptualizing, as it were, the state of affairs which is described on the right. That state of affairs is initially given to us as the obtaining of a certain equivalence relation . . . ; but we have the option, by stipulating that the abstraction is to hold, of so reconceiving such states of affairs that they come to constitute the identity of a new kind of thing . . . of which, by this very stipulation, we introduce the concept.” Part of the problem with this suggestion is this: in HP, numbers belonging to concepts are themselves among the objects over which the first-order variables on the right-hand side range. Talk of reconceptualizing a state of affairs would be in order only if the objects supposedly introduced by stipulation were new, objects that had not been previously quantified over. Whether old objects can be chosen to be identical or not under the right conditions would not seem to be a matter that it could be up to us to decide. It is here that the analogy between directions and lines and numbers and concepts breaks down: no one supposes that directions are any sort of constituent of lines, but on the Fregean treatment of number, numbers quite definitely are objects that both fall under concepts and are associated with concepts, as their numbers. However, when the objects allegedly introduced by this sort of stipulation are already objects quantified over in the equivalence relation, unexpected, and sometimes unwelcome, results can occur when we attempt to identify certain of them. We can’t, for example, stipulate that old objects be assigned to concepts in such a way that if some old object falls under one concept but not another then the two concepts are to be assigned different objects.

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Wright says, “The concept of direction is thus so introduced that that two lines are parallel constitutes the identity of their direction. It is in no sense a further substantial claim that directions exist and are identical under the described circumstances. But nor is it the case that, by stipulating that the principle is to hold, we thereby forfeit the right to a face-value construal of its left-hand side and thereby to the type of existential generalization which a face-value construal would license.” All well and good for directions, maybe, but what if the objects introduced on the left are already among those discussed on the right? Could there not then be a danger that a “substantial further claim” about those very objects, taken together, would be entailed? And of course there is such a danger: the generalized biconditional, or the biconditional with its free variables, taken as an axiom, might then entail that, e.g., there are many, many objects, too many for it to be capable of being regarded any longer as analytic. One might think: but does that not automatically show that HP isn’t analytic? How can an analytic truth be false in certain domains, indeed false in all the finite domains? There is of course a reply that is ready to hand, viz. that it’s analytically false that the objects that exist constitute any one of those finite domains. The response strikes me as incredible, but again, I don’t have a knock-down argument against the analyticity of HP, only a bunch of considerations. (Heidegger would hardly have welcomed the response, “Because, analytically, there is always the number of things that there are; so there couldn’t have been nothing rather than something.”) One final remark on reconceptualization. How can one call the left-hand side of HP a reconceptualization of the right if it can’t always be made to hold whenever the right-hand side does? Of course if the variables range over a set, one can always pick some new objects to play the role of the numbers belonging to subsets of that set, but why is one so sure one can do this if there is no set of objects over which the variables range? Wright’s idea that the role of HP is that of an explanation also worries me. In Frege’s Conception of Numbers as Objects, Wright writes: “the fundamental truths of number theory would be revealed as consequences of an explanation: [note the colon] a statement whose role is to fix the character of a certain concept.” 10 In the present paper, Wright calls HP “a principle whose role is to explain, if not exactly to define, the general notion of cardinal number.” Wright is impressed by the form of HP: a biconditional whose right limb is a formula defining an equivalence relation between concepts F and G and whose left limb is a formula stating when the cardinal numbers of F and G are the same. Since the sign for cardinal numbers does not occur in the right 10 Wright (1983), p. 153.

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limb, can one not appropriately say that HP explains the concept of a cardinal number by saying what it is for two cardinal numbers, both referred to by expressions of the form “the number of . . . ” to be identical? 11 Certainly. HP states a necessary and sufficient condition for an identity #F= #G to hold. Moreover the formula defining this condition doesn’t contain #. So if one wants merely to sum up this state of affairs by saying that HP explains the concept of cardinal number, I would not object. However, it is hard to avoid the impression that more is meant, that Wright holds that to call a statement an explanation of a concept is to assign it an epistemological status importantly similar to the one it was once thought analytic judgments, including definitions, enjoy. It is to this further suggestion that I wish to demur. I can’t help suspecting that Wright is using “explanation,” at least in the phrase “explanation of a concept,” as a term of art, as a member of the same family circle as “analytic,” “definition,” or “conceptual truth,” that the only reason he does not call HP an “analytic definition” is that it is not of the form: Definiendum(x) ≡ Definiens(x), and that he supposes it to be a super-hard truth like “all bachelors are unmarried” or “all equivalence relations are transitive.” The phrase “whose role” occurs in both quotations and may suggest that Wright thinks that HP has one and only one [pre-eminent] role, for “whose” seems in both places to mean “of which the” rather than “of which a.” This thought seems to me to be incorrect. HP might be taken as an axiom, the sole (non-logical) axiom in some axiomatization of arithmetic. It might be a sentence we want to show to be needlessly strong for some purpose, e.g., deriving arithmetic. It might serve as something to be obtained from Basic Law V. It might be used as an example of a beautiful proposition. Etc. etc. But there’s no such thing as the [unique] role of HP. It is certainly true that one of the ways in which HP can be used is to fix the character of a certain concept. Here’s how: lay Hume down. Then the concept the number of . . . will have been fixed to be such that numbers belonging to concepts will be the same if and only if the objects falling under one of the concepts are in one–one correspondence with those falling under the other. But Hume is no different in this regard from any other statement that we might choose to take as an axiom. The axiom of choice fixes the concept of set in a similar manner. Laid down, it determines that for any set of disjoint nonempty sets, there is a set with exactly one member in common with each of those sets. The principle of mathematical induction fixes the character of the natural numbers. The statement that bananas are yellow fixes the character of the concept of a banana. So nothing is said when it is said that one of the roles of HP is to fix the character of the concept of cardinal number. And HP doesn’t have a unique role. 11 I am grateful to Wright and Richard Heck for helpful comments on the whole of this paper but am particularly grateful to them here.

Is Hume’s Principle Analytic?

13

Let me now defend myself about the “bad company” argument. What I think I was doing was illustrating that what is called (unfortunately, as Wright has stated) “contextual definition” is not, in general, a permissible way of introducing a concept. I didn’t mean to be arguing that it never was and gave the example of the principle governing truth-values as another example of a legitimate contextual definition. Different examples had different purposes. I cited Hodes splendid observation that the relation-number principle (the relation-number belonging to R is identical with that belonging to S if and only if R and S are isomorphic relations) leads to the Burali-Forti paradox in order to point out that Basic Law V was not an isolated case and that HP might well be expected to be powerful if consistent (as it is). I gave the example of parities in order to show that one couldn’t say that a contextual definition is OK if only it is consistent. (I had thought of nuisances, but I seemed to recall actually having heard of the “parity” of a set, and the notion is in any case a natural one.) The example of a principle true iff there are no more than two members was designed to show that one didn’t need heavy involvement with set theory to find a contextual definition incompatible with HP. And did I ever say that it would be impossible to demarcate the good contextual definitions from the bad? I merely said that it would seem to be a problem we have no hope of solving at present. I have to reserve judgment on the question whether Wright has solved the problem, but I certainly hope he has. Wright says I was wrong to say that there is no notion that V**, my revision of Basic Law V, is analytic of; what is true, he says, is “that there is no prior, no intuitively entrenched notion, no notion given independently, which V** is analytic of.” I happily accept the correction. I now want to make a somewhat conciliatory remark. I have been aspersing, at great length, the idea that HP is an analytic truth, all the while taking “analytic” to bear something like the sense it has in current philosophical discourse, namely, “truth in or by virtue of meanings.” I think that is the sense in which Wright uses the term too. But there may be another notion of analyticity on which the analyticity of HP might well be more plausible. It is the idea of Gödel’s, as outlined in both his paper “Russell’s mathematical logic” and his 1951 Gibbs lecture to the American Mathematical Society, 12 according to which a proposition is analytic if it is true “owing to the nature of the concepts occurring therein.” 13 Concepts, he says in the Gibbs lecture, “form an objective reality of their own, which we cannot create or change, but only perceive and describe.” By reflection, which of course 12 See also George Boolos, “Introductory Note to Kurt Gödel’s ‘Some Basic Theorems on the Foundations of Mathematics and their Implications’ ,” in Kurt Gödel, Collected Works, vol. III, Unpublished Essays and Lectures, Solomon Feferman et al., eds., Oxford, Oxford University Press (1995), pp. 290–304. 13 Gödel also describes propositions as analytic if they are true in virtue of the meanings of the terms expressing them, but it should be understood that his notion of meaning is much broader than that of “linguistic” meaning. For example, Gödel held that it is a matter of the meanings of “set” and “∈” that the axioms of set theory hold. The difference between the sense he attached to “meaning” and “concept” would not seem to be particularly significant.

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includes philosophical or mathematical or other intellectual work, we can sometimes arrive at an understanding of the natures of certain concepts that is sufficient to enable us to see the truth of certain propositions in which they occur. With the passage of time, our understanding of those concepts may improve and the truth of ever more analytic propositions become evident to us. Perhaps, as Schoenfield has ironically suggested, the rejection of the “axiom” of constructibility is one example of improvement in our perception of the meaning (in Gödel’s sense) of “set” or of the nature of sets. The thought that understanding of abstract objects may be achieved through a sort of perception of them, which is crucial to Gödel’s conception of the analytic, will certainly strike many contemporary philosophers as unacceptably mystical and at any rate highly implausible. (Perhaps, paradoxically, there is even a tinge of materialism in the suggestion that our knowledge of abstract objects arises from “something like a perception” of them: could there not be ways in which we interact with abstracta that yield knowledge of them that are not at all like perception?) But if—IF—a Gödelian notion of analyticity could be made out, then HP might well be among the first candidates for this new sort of analytic truth. Perhaps by taking the thought in the right way, we can “see” that if nothing exists, then zero, at least, has to exist, for it is then the number of things there are, and therefore that something does exist after all, but then there have to exist two things, for . . . This Fregean argument may strike one, as it does me, as a good example of the kind of reflection Gödel might have thought showed that the proposition that there are infinitely many natural numbers is analytic, on his understanding of “analytic,” if not on that of most of us who use the word. Maybe in the end we would also thus “see” the truth of HP. But even on such a Gödelian view of the analytic, at least two difficulties would confront the view that HP is analytic. The first is that (it is not neurotic to think) we don’t know that second-order arithmetic, which is equi-consistent with Frege Arithmetic, is consistent. Do we really know that some hotshot Russell of the 23rd century won’t do for us what Russell did for Frege? The usual argument by which we think we can convince ourselves that analysis is consistent—“Consider the power set of the natural numbers . . . ”—is flagrantly circular. Moreover, although we may think Gentzen’s consistency proof for PA provides sufficient reason to think PA consistent, we have nothing like a similar proof for the whole of analysis, with full comprehension. We certainly don’t have a constructive consistency proof for ZF. And it would seem to be a genuine possibility that the discovery of an inconsistency in ZF might be refined into that of one for analysis. Saying exactly which theories are known to be consistent is a difficult problem made even more difficult when one hears of respected mathematicians telling of their failed attempts to prove Q inconsistent, but ZF and analysis, and therefore also Frege arithmetic, are theories that are surely in the black area, not the grey. While we may regret that these theories may well be consistent and that it

Is Hume’s Principle Analytic?

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would probably be wise to bet on their consistency, we must not despair: we do not know that they are and need not yet give up hope that someone will one day prove in one of them that 0 = 1. Uncertain as we are whether Frege arithmetic is consistent, how can we (dare to) call HP analytic? One final worry, perhaps the most serious of all, although one that may at first appear to be dismissible or silly or trivial: as there is a number, zero, of things that are non-self-identical, so, on the account of number we have been considering, there must be a number of things that are self-identical, i.e., the number of all the things that there are. Wright has usefully dubbed this number, #[x: x = x], anti-zero. On the definition of ≤, according to which m ≤ n iff ∃F∃G(m = #F ∧ n = #G∧ there is a one–one map of F into G), anti-zero would be a number greater than any other number. 14 Now the worry is this: is there such a number as anti-zero? According to Zermelo–Fraenkel set theory, there is no (cardinal) number that is the number of all the sets there are. The worry is that the theory of number we have been considering, Frege Arithmetic, is incompatible with Zermelo–Fraenkel set theory plus standard definitions, on the usual and natural readings of the non-logical expressions of both theories. To be sure, as Hodes once observed in conversation, if #α is taken to denote the cardinal number of α when α is a set and some favorite objects that is not a cardinal number when α is a proper class, then HP will be a theorem of von Neumann set theory. But on that definition of #, # will not be translatable as “the cardinal number of.” ZF and Frege arithmetic make incompatible assertions concerning what cardinal numbers there are. And of course, the response “Well, these are just formalisms; the question of their truth or falsity doesn’t arise or makes no sense” is hardly available to one claiming that HP is analytic, i.e., an analytic truth. So one who seriously believes that it is has to be bothered by the incompatibility of the consequence of Frege arithmetic that there is such a number as anti-zero with the claim made by ZF + standard definitions (on the natural reading of its primitives) that there is no such number. It is thus difficult to see how on any sense of the word “analytic,” the key axiom of a theory that we don’t know to be consistent and that contradicts our best-established theory of number (on the natural readings of its primitives) can be thought of as analytic.

14 By the Schröder–Bernstein theorem, which can be proved in second-order logic, ≤ is anti-symmetric: if m ≤ n ≤ m, then m = n.

IS HUME’S PRINCIPLE ANALYTIC? 1 Crispin Wright

1. It was George Boolos who, following Frege’s somewhat charitable lead at Grundlagen §63, first gave the name, “Hume’s Principle”, to the constitutive principle for identity of cardinal number: that the number of F’s is the same as the number of G’s just in case there exists a one-to-one correlation between the F’s and the G’s. The interest—if indeed any—of the question whether the principle is analytic is wholly consequential on what has come to be known as Frege’s Theorem: the proof, prefigured in Grundlagen §§82–3 and worked out in some detail in Wright [1983] 2 that second-order logic plus Hume’s Principle as sole additional axiom suffices for a derivation of second-order arithmetic— or, more cautiously, for the derivation of a theory which allows of interpretation as second-order arithmetic. (Actually I think the caution is unnecessary— more of that later.) Analyticity, whatever exactly it is, is presumably transmissible across logical consequence. So if second-order consequence is indeed a species of logical consequence, the analyticity of Hume’s Principle would ensure the analyticity of arithmetic—at least, provided it really is second-order arithmetic, and not just a theory which merely allows interpretation as such, which is a second-order consequence of Hume’s Principle. What significance that finding would have would then depend, of course, on the significance of the notion of analyticity itself. Later I shall suggest that the most important 1 This paper first appeared in the Notre Dame Journal of Formal Logic 40, [1999], pp. 6–30. Reprinted by kind permission of the editor and the University of Notre Dame. 2 At pp. 158–169. An outline of a proof of the Peano Axioms from Hume’s Principle is also given in the Appendix to Boolos [1990]. The derivability of Frege’s Theorem is first explicitly asserted in Charles Parsons [1964]; see remark at p. 194. My own ‘rediscovery’ of the theorem was independent. I do not know what form of proof Parsons had in mind, but the reconstruction of the theorem is trickier than Frege’s own somewhat telegraphic sketch suggests. For an excellent recent overview of the ins and outs of the matter— early on, they remark that

§§82–3 offer severe interpretative difficulties. Reluctantly and hesitantly, we have come to the conclusion that Frege was at least somewhat confused in these two sections and that he cannot be said to have outlined, or even to have intended, any correct proof there (p. 407) —see Boolos and Heck [1998].

17 Roy T. Cook (ed.), The Arché Papers on the Mathematics of Abstraction, 17–43. c 2007 Springer.

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issues here are ones which are formulable without recourse to the notion of analyticity at all—so that much of the debate between Boolos and me could have finessed the title question. Boolos wrote that “having to discuss whether Hume’s Principle is analytic is rather like having to consider whether hydrogen sulphide is dephlogisticated” 3 —a question formulated, I suppose he meant, in a discredited theoretical vocabulary. That would be consistent, of course, with there being a good question nearby of which that was merely a theoretically unfortunate expression; it would also be consistent with there being enough sense to the theoretically unfortunate question to allow of a negative answer in any case. I myself do not believe that when the dust settles on analytical philosophy’s first century, our successors will find that the notion of analyticity was discredited by any of the well-known assaults. In particular, the two core lines of attack in “Two Dogmas of Empiricism”, namely. that the notion resists all non-circular explanation and that no statement participating in general empirical theory can be immune to revision, set an impossible—Socratic—standard for conceptual integrity and confuse analyticity with indefeasible certainty, respectively. What is undeniable, though, is that the status and provenance of analytic truths, and the cognate class of a priori necessary truths, would have to be a lot clearer than philosophers have so far managed to make them before a positive answer to our title question could be justified and shown to have the sort of significance which early analytical philosophy would have accorded to it. Boolos thought the situation was of the second kind noted: that the question is theoretically flawed but allows of well-motivated—though less than “knockdown”—arguments for a negative answer. To the best of my knowledge— I’m drawing just on three of his papers 4 which are reprinted in the excellent Demopoulos collection, 5 plus his ipsonymous paper in Richard Heck’s volume for Michael Dummett 6 —he proffered exactly five such arguments. In what follows I shall briefly explore how a character I shall call the neo-Fregean might respond to each of these arguments. Each is interesting, some are very searching, but—if I’m right—none does irreparable damage.

2. 2.1

The ontological concern

The ontological concern is epitomized in the following passage: I want to suggest that Hume’s Principle is to be likened to ‘the present King of France is a royal’ in that we have no analytic guarantee that for every value of ‘F’, there is an object that the open definite singular description, ‘the 3 Boolos [1997], p. 247. 4 Boolos [1987, 1990, 1986].

5 Demopoulos [1995]. 6 Boolos [1997], Heck [1997].

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Is Hume’s Principle Analytic? number belonging to F’ denotes . . . Our present difficulty is this: just how do we know, what kind of guarantee do we have, why should we believe, that there is a function that maps concepts to objects in the way that the denotation of octothorpe [that is: ‘#’, Boolos’s symbol for the numerical operator] does if HP is true? . . . do we have any analytic guarantee that there is a function that works in the appropriate manner? 7

The basic thought is that Hume’s Principle says too much to be an analytic truth. As normally conceived, analytic truth must hold in any possible domain. On a (purportedly) more relaxed conception, some analytic truths are allowed to hold in any non-empty domain. But how can a principle which entails— indeed, is strictly stronger than is necessary to entail—that there are infinitely many objects—indeed infinitely many objects of a special sort—possibly count as analytic? Here is the neo-Fregean reply. There is, to be sure, a perfectly good sense in which whatever is entailed by certain principles together with truths of logic may be regarded as entailed by those principles alone. In this sense it is undeniable that Hume’s Principle does entail the existence of infinitely many objects—at least if second-order consequence is a species of entailment. But the manner of the entailment is important. Hume’s Principle is a second-order universally quantified biconditional. As such, we are not going to be able to elicit the existence of any objects at all out of it save by appropriate input into (instances of ) its right-hand side. Thus we get the number zero by taking the instance of Hume’s Principle: Nx : x = x = Nx : x = x ↔ x = x

1≈1

x = x

(1)

together with its right-hand side as a minor premise. Compare the fashion in which we derive the direction of the line a from an instantiation of Frege’s illustrative equivalence for directions: (DE)

Da = Da ↔ a//a

together with its right-hand side as a minor premise: the necessary truth, modulo the existence of line a, that that line is parallel to itself. Sure, in the case of zero the minor premise: x = x

1≈1

x = x

(2)

can be established in second-order logic. So the existence of zero follows from this truth of logic, together with Hume’s Principle. If, accordingly, the latter can be regarded as, in all relevant respects, having a status akin to that of a definition, then the existence of zero is a consequence of logic and definitions. But that was exactly the classical account of analyticity: the analytical truths were to be those which follow from logic and definitions. So the existence of zero would be an analytic truth. And now with that in the bag, as it were, 7 Boolos [1997], p. 251. See more generally pp. 248–254, ibid.; also Boolos [1987] at p. 231 and Boolos [1990] at pp. 246–8. (The latter references are to the pagination in Demopoulos [1995].)

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nothing stands in the way of regarding x = 0 1≈1

x =0

(3)

as also an analytic truth, since it follows in second-order logic given only that there is such a thing as zero. But that is the right-hand side for the application of Hume’s Principle which, following Frege, we use to obtain the number one. So its existence is also analytic. We may now proceed in similar fashion to obtain each of the finite cardinals from putatively analytic premises, in second-order logic. Our result is thus not quite—when done this way— that it is analytic that there is an infinity of finite cardinals, but rather that of each of the finite cardinals, it is analytic that it exists. Doubtless this will be equally offensive to the traditional understanding of analyticity—the (as nearly as possible) existentially neutral understanding of analyticity—called forth in the above quotation from Boolos. But my point now is simply that, for the reasons just sketched, that understanding of analyticity had to be in jeopardy all along provided there is a starting chance that Hume’s Principle has an epistemic status relevantly similar to that of a definition. In sum: on the classical account of analyticity, the analytical truths are those which follow from logic and definitions. So if the existence of zero, one, and so on. follows from logic plus Hume’s principle, then provided the latter has a status relevantly similar to that of a definition, it will be analytic, on the classical account, that n exists, for each finite cardinal n. The idea which standardly accompanies the classical conception, that—with perhaps a very few, modest exceptions—existential claims can never be analytically true, is thus potentially in tension with the classical conception. If Hume’s Principle has a status not relevantly different from that of a definition, then we learn that the classical conception will not marry with this standardly accompanying idea. The core of the neo-Fregean stance is that Hume’s Principle does have such a status: that it may be seen as an explanation of the concept of cardinal number in general, covering the finite cardinals as a special case. Boolos asks, “If numbers are supposed to be identical if and only if the concepts they are numbers of are equinumerous, what guarantee have we that every concept has a number?” 8 Earlier he suggested, in the passage quoted, that there is no such guarantee—or anyway no “analytic guarantee”—proposing a parallel between the principle and the statement, “The present King of France is a royal”— something which is analytically true, modulo its existential presupposition. This is also Hartry Field’s position in his critical notice of Wright [1983]. 9 But I think this seemingly sane and reserved position is unstable. Consider the case of direction again. How do we know there are any objects which behave in the way that the referents of direction terms ought to behave, 8 Boolos [1997], p. 253. 9 Field [1984].

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given their introduction by the direction equivalence (DE), that is, given that they are identical just in case the associated lines are parallel and distinct just in case they are not? Shouldn’t we just say that provided there are such things as directions in the first place, that will be the condition for their identity and distinctness? Well, if this were the right view of the matter, there could be no objection to making the presupposition explicit. The following principle would then count as absolutely analytic: that for any lines a and b, ((∃x)(∃y)(x = Da & y = Db)) → (Da = Db ↔ a//b)

(4)

But think: how are we to understand the antecedent of this? The condition for its truth must now incorporate some unreconstructed idea of what it is for contexts of the form, ‘ p = Da’ and ‘q = Db’ to be true—unreconstructed because Field and Boolos have just rejected the proposed sufficient conditions for the truth of such contexts, where ‘ p’ and ‘q’ are, respectively, direction terms, incorporated in DE. However, no other such sufficient condition has been proposed. So, if we side with Field and Boolos, we don’t have the slightest idea, actually, of what satisfaction of the antecedent of the supposedly more modest and reserved formulation could consist in. True, the reserved formulation could be made to raise an intelligible issue if relativised to an antecedently given domain of quantification—the issue would be whether any of the objects thereby already recognised, perhaps certain equivalence classes, are appropriately identified and distinguished in the light of relations of parallelism among lines. But Frege, remember, was trying to address the question how we come by and justify the conception of a domain of abstracta in the first place. If it is insisted that abstraction principles always stand in need of justification by reference to an antecedently given domain of entities, that’s just to presuppose—not argue—that they are useless in that project. And it is so far to offer no alternative conception of how the project might be accomplished. The neo-Fregean contention, by contrast, is that, under the right conditions, such principles are available to fix the truth-conditions of contexts of identity for a certain kind of thing and thereby—given appropriate input on their right-hand sides—to contribute towards determining that, and how it is possible for us to know that things of that kind exist. Boolos’s question “If numbers are supposed to be identical if and only if the concepts they are numbers of are equinumerous, what guarantee have we that every concept has a number?”, raises a doubt in the way he presumably wished to do only if it is granted that the existence of numbers is a further fact, something which the (mere) equinumerosity of concepts may leave unresolved. But the neo-Fregean’s intention in laying down Hume’s Principle as an explanation is so to fix the concept of cardinal number that the equinumerosity of concepts F and G is itself to be necessary and sufficient, without further ado, for the identity of the number of Fs with the number of Gs, so that nothing more is required for the existence of those numbers beyond the equinumerosity of the concepts. This idea is discussed more fully in the early sections of

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Wright [1997] and in Bob Hale’s [1997]. The key idea is that an instance of the left-hand side of an abstraction principle is meant to embody a reconceptualisation of the type of state of affairs depicted on the right. Here is not the place to pursue this crucial idea further. My point is merely that Boolos’s question either ignores this aspect of the neo-Fregean position or assumes it is ill-conceived.

2.2

The epistemological concern

A recurrent element in Boolos’s misgivings about Hume’s Principle concerns its proof-theoretic strength—more accurately, the strength of the system which results from its addition to axiomatic second-order logic. In part this concern relates to the ontological issues just reviewed. But there is a separate strand, nicely captured by a passage towards the end of “Is Hume’s Principle Analytic?” Boolos was the first to show that second-order logic plus Hume’s Principle is equi-interpretable with second-order arithmetic, and hence that each is consistent if the other is. 10 But he was not himself inclined to take that result as settling the question of the consistency of Hume’s Principle. He writes: . . . (it is not neurotic to think) we don’t know that second-order arithmetic . . . is consistent. Do we really know that some hotshot Russell of the 23rd Century won’t do for us what Russell did for Frege? The usual argument by which we think we can convince ourselves that analysis is consistent—“Consider the power set of the set of natural numbers . . . ”—is flagrantly circular . . . Uncertain as we are whether Frege arithmetic is consistent, how can we (dare to) call HP analytic? 11

Now, I do not myself know whether disclaiming knowledge of the consistency of Frege arithmetic is neurotic or not. But we must surely look askance at the presupposition of the concluding question, which arguably—as did Quine— confuses analyticity and certainty, or anyway insists that certainty is a precondition for warranted analyticity claims. That seems to me a great mistake. There is nothing incoherent in the idea that we can be defeasibly justified in believing or claiming to know that a proposition is true which, if true, is analytic. The neo-Fregean claim, remember, is that Hume’s Principle serves as an explanation of the concept of cardinal number. If it harbours some subtle inconsistency, then of course it fails as such an explanation—just as Basic Law V failed as an explanation of a coherent notion of set. But we can surely be fairly confident—though by all means with our eyes open—that Hume’s Principle is successful in that regard, and correspondingly confident that it enjoys the kind of truth possessed by any successful implicit definition—and hence is analytic in whatever may be the attendant sense. 10 Boolos [1987]. For a detailed proof, see the first appendix to Boolos and Heck [1998]. 11 Boolos [1997], pp. 259–60.

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23

The concern about the universal number

The construction of the finite cardinals on the basis of Hume’s Principle relies entirely on the legitimacy of applying the numerical operator to some necessarily empty concept at the first stage, the concept not self-identical being the standard choice. On the face of it there should accordingly be no obstacle to applying the operator to the complement of any such concept, so arriving at the universal number, anti-zero—the number of absolutely everything that there is. Certainly Hume’s Principle as standardly formulated poses no obstacle to such an application. As Boolos puts it, As there is a number, zero, of things that are non-self-identical, so, on the account of number we have been considering, there must be a number of things that are self-identical, i.e., the number of all the things that there are. 12

Now, Hume’s Principle can be no less dubious than any of its consequences, one of which is the claim then that there is such a number. But . . . the worry is this: is there such a number as anti-zero? According to [ZF] there is no cardinal number that is the number of all the sets there are. The worry is that the theory of number we have been considering, Frege arithmetic, is incompatible with Zermelo–Frankel set theory plus standard definitions . . . one who seriously believes that [HP is an analytic truth] has to be bothered by the incompatibility of the consequence of Frege arithmetic that there is such a number as anti-zero with the claim made by ZF plus standard definitions . . . that there is no such number. 13

This objection, Boolos wrote, although it “may at first appear to be dismissible as silly or trivial”, is “perhaps the most serious of all”. It’s certainly an arresting objection, about which there is a good deal to say. Clearly there would be great discomfort in regarding any principle as analytically true if the cost of doing so was regarding Zermelo–Frankel set theory as analytically false. A first rejoinder would be that any such upshot would depend on cross-identification of the referents of terms in Frege arithmetic and terms in Zermelo–Frankel set theory—the “standard definitions” to which Boolos alludes. Who said numbers like anti-zero had to be sets, after all? However the more general worry underlying Boolos’s point—the worry about the coherence of Hume’s Principle with standard set theory—need not depend on such cross-identification. Grant the plausible principle (to which I shall return below) that there is a determinate number of F’s just provided that the F’s compose a set. Zermelo–Frankel set theory implies that there is no set of all sets. So it would follow that there is no number of sets. Yet for all we have so far seen, the property, set, lies within the range of the secondorder quantifiers in Hume’s Principle and the usual proof, via the reflexivity of equinumerosity, should therefore serve to establish, to the contrary, that there 12 Boolos [ibid.], p. 260. 13 Ibid.

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is such a number. So there would seem to be a collision with Zermelo-Frankel set theory in any case, whether or not anti-zero is identified with a set. However I think there is good reason to expect a principled and satisfying response to this general trend of objection. Consider the direction equivalence, DE, again. The reflexivity of the relation, . . . is parallel to . . . , ensures in the presence of DE that a has a direction, no matter what straight line a may be. But the question arises: what of the implications of DE for the case where a and b fail to be parallel because they are not even lines, as for example my hat fails to be parallel to my shoe. We might have been tempted to allow that the D-operator is totally defined—to allow that every object, without restriction, has a direction: in the case of an object which fails to be parallel to anything else because it is merely not a line, this would then be a direction that nothing else has. But a moment’s reflection shows that is not an option: if the failure of parallelism between my hat and my shoe is down to the unsuitability of either object to be parallel to anything, then by the same token they are not self parallel, and DE provides no incentive to regard either as having a direction at all. Moral: just as not every object is suitable to determine a direction, so we should not assume without further ado that every concept—every entity an expression for which is an admissible substituen for the bound occurrences of the predicate letters in Hume’s Principle—is such as to determine a number. That’s only a first step, of course. What is wanted for the exorcism of antizero is nothing less than grounds for affirming that whereas the concept, not self-identical, or any other self-contradictory concept, is a suitable case for application of the numerical operator, its complement is not. Here are two, independent such lines of thought: The first line is directed specifically at anti-zero. To accept Frege’s insight that statements of number are higher-level—that they state things of concepts—is quite consistent with the familiar observation that a restriction is needed which he does not draw. The basic case in which the question, how many F’s are there? makes sense—or at least has a determinate answer— is that of a special class of substitutions for ‘F’: what are sometimes called ‘count nouns’, or expressions for ‘sortal concepts’. While it is by no means the work of a moment to make this notion sharp, the usual intuitive understanding is that a sortal concept is one associated both with a criterion of application— a distinction between the things to which it applies and those to which it does not —and a criterion of identity: some principle determining the truth values of contexts of the form, ‘X is the same F as Y ’. ‘Tree’, ‘person’, ‘city’, ‘river’, ‘number’, ‘set’, ‘time’, ‘place’, are all, in at least certain uses, sortal concepts in the intended sense. By contrast, ‘red’, ‘composed of gold’, ‘large’—in general, purely qualitative predicates, predicates of constitution, and attributive adjectives—although syntactically admissible substituents for occurrences of the predicate letters in higher-order logic, are not. Call the latter class of expressions: mere predicables. Where F is a mere predicable, then, the suggestion is that the question, how many F’s are there? is ceteris paribus

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deficient in sense and ‘the number of F’s’, accordingly, has no determinate reference. It is easy to see that ‘is self-identical’ is a mere predicable. For reflect that—prescinding from any cases of vagueness—mere predicables do nevertheless subserve determinate questions of cardinal number when their scope is restricted to that of some specific sortal concept: thus there can be a determinate number of red apples in the bowl, of gold rings in the jeweller’s window, and of large women at the reception. So if ‘self-identical’ were a sortal concept, it should follow that there can be determinate numbers of red self-identicals in the bowl, golden self-identicals in the jeweller’s window, and large self-identicals at the reception. However since ‘F and self-identical’ is equivalent to ‘F’, it follows that there can be no such determinate number wherever there is no determinate number of Fs. So self-identity is not a sortal concept. If we take it that, save where F is assured an empty extension on purely logical grounds, 14 only sortal concepts, and concepts formed by restricting a mere predicable to a sortal concept, have cardinal numbers, it follows that there is no universal number. To be sure, this first consideration will of course not engage the question whether we may properly conceive of a number of all ordinals, or all cardinals, or all sets—in general, cases where we are concerned with the results of applying the numerical operator to concepts which are (presumably) sortal but “dangerously” big. And as we saw, a variant of Boolos’s objection, that there is a potential clash of Hume’s Principle with Zermelo–Frankel set theory, does equally arise in those cases. However, a principled objection to the idea that there should be determinate numbers associated with these concepts may be expected to issue from the second line of thought, which concerns the tantalising notion of indefinite extensibility. As noted a little while ago, it seems natural and well motivated to suppose that the Fs should have a determinate cardinal number just when they compose a set. But a long tradition in foundational studies would argue that set-hood cannot be the right way to conceive of Frege’s intentionally all-inclusive domain of objects: that Cantor’s paradox shows, in effect, that there can be no universal set—no absolutely all-embracing totality which is subject, for example, to the operations and principles that provide for the proof of Cantor’s theorem. That is not the same as saying that unrestricted first-order quantification is illegitimate—a concession which would, of course, be fatal to Frege’s 14 A plausible general principle (suggested to me by Bob Hale) of which this exception would be a special case would be this: that a non-sortal concept, F, may nevertheless have a determinate cardinal number if every sortal restriction of it has the same cardinal number. This would not, of course, legitimate anti-zero, since the cardinality of sortal restrictions of the form, Gx & x = x, will vary with that of G. But it would save the standard Fregean definition of zero. (Would there be any instances of this principle other than those mere predicables which are necessarily uninstantiated?) More generally, we might—indeed, ought to—allow that a non-sortal F may determine a number if we know that all and only F-things are G, where G is sortal and non-indefinitely extensible. (But again, are there any such cases?)

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whole project. The point is rather that the objects that lie in the range of such unrestricted quantification compose not a determinate totality but one that is, in the phrase coined by Michael Dummett, “indefinitely extensible” 15 —a totality of such a sort that any attempt to view it as a determinate collection of objects will merely subserve the specification of new objects which ought, intuitively, to lie within the totality but cannot, on pain of contradiction, be supposed to do so. I do not know how best to sharpen this idea, still less how its best account might show that Dummett is right, both to suggest that the proof-theory of quantification over indefinitely extensible totalities should be uniformly intuitionistic and that the fundamental classical mathematical domains, like those of the natural numbers, or the reals, should also be regarded as indefinitely extensible. But Dummett could be wrong about both those points and still be emphasising an important insight concerning certain very large totalities— ordinal number, cardinal number, set, and indeed “absolutely everything”. If there is anything at all in the notion of an indefinitely extensible totality—and there are signs that the issue is now being taken up in productive ways 16 — one principled restriction on Hume’s Principle will surely be that F and G not be associated with such totalities. So that is a second definite programme for understanding how, in particular, not self-identical might determine a cardinal number even though self-identical does not. Indeed, when the range of both individual and higher-order variables is unrestricted, the complement of any determinate finite concept is presumably always an indefinitely extensible totality.

2.4

The concern about surplus content

This is the objection I find it hardest to be sure I properly understand. Here is one of Boolos’s expressions of it: It is known that Hume’s Principle does not follow . . . from the conjunction of two of its strong consequences: . . . that nothing precedes zero and that precedes is a one–one relation. If HP is analytic, then it is strictly stronger . . . than some of its strong consequences. It’s also known that arithmetic follows from these two statements alone . . . faced with these results, how can we really want to call HP analytic? 17

The objection is developed and endorsed by Richard Heck in recent work, 18 and I shall rely on his interpretation of it. Heck emphasises that there is a long conceptual leap involved in advancing to the concept of cardinal number enshrined in Hume’s Principle in full generality for one whose previous 15 Dummett first introduced this notion—which of course ultimately derives from one strand in Russell’s

Vicious Circle Principle—in his [1963] (reprinted in M. Dummett, Truth and Other Enigmas, London: Duckworth 1978, pp. 186–201). It is central to the argument of the concluding chapter of Dummett [1991]. See also his “What is Mathematics About?” in Dummett [1993] at pp. 429–45. 16 See for instance Clark [1998], Oliver [1998], and Shapiro [1998]. 17 Boolos [1997], p. 249. 18 Heck [1997a].

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acquaintance with cardinal number—a pre-Cantorian as it were—is restricted to finite arithmetic and its applications. The length of the leap is reflected in the results about the proof-theoretic strength of various systems, including Fregean arithmetic—i.e., Hume’s Principle plus second-order logic—secondorder Peano arithmetic and certain intermediaries which, building on work of Boolos’s, Heck demonstrates. 19 Here is his conclusion: . . . HP, conceptual truth or not, cannot be what underlies our knowledge of arithmetic. And no amount of reflection on the nature of arithmetical thought could ever convince one of HP, nor even of the coherence of the concept of cardinality of which it is purportedly analytic. Granted, any rationalist project of this sort will have to invoke a distinction between the ‘order of discovery’ and the ‘order of justification’. But the objection is not that Hume’s Principle is not known by ordinary speakers, nor that there was a time when the truths of arithmetic were known, but HP was not. It is that, even if HP is thought of as ‘defining’ or ‘introducing’ or ‘explaining’ our present concept of cardinality, the conceptual resources required if one is so much as to recognise the coherence of this concept (let alone HP’s truth) vastly outstrip the conceptual resources employed in arithmetical reasoning. Wright’s version of logicism is therefore untenable. 20

Heck goes on to consider whether some form of Hume’s Principle restricted to finite concepts might be resistant to the particular objection, that is, whether such a principle might be appreciable as a correct digest of its constitutive principles by one possessed just of the conceptual resources deployed in finite cardinal arithmetic and its applications. That is an interesting question, on which he offers interesting formal and informal reflection. But I have a prior difficulty in seeing that the original objection, concerning the conceptual excess of Hume’s Principle over second-order Peano arithmetic, does any serious damage to any contention that the neo-Fregean should want to make. Grant that a recognition of the truth of Hume’s Principle cannot be based purely on analytical reflection upon the concepts and principles employed in finite arithmetic. The question, however, surely concerned the reverse direction of things: it was whether access to those concepts and validation of those principles could be achieved via Hume’s Principle, and whether Hume’s Principle might in its own right enjoy a kind of conceptual status that would make that result interesting. The latter is, in effect, exactly the question raised by our title. But no particular view of it can be motivated merely by the reflection that the conceptual resources involved in Hume’s Principle, insofar as an extension of the notion of cardinal number to the infinite case is involved, considerably exceed those involved in ordinary arithmetical competence. More: it is unclear how anyone wishing to demonstrate the analyticity of arithmetic could clear-headedly acquiesce in the rules of debate implicit 19 See Heck [1997a], Section 4. 20 Heck [1997a], pp. 597–8.

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in Heck’s discussion. Those rules require that one canvasses some principle which is supposedly analytic of ordinary arithmetical concepts in the precise sense that it could be recognised by reflection as systematising those ordinary concepts and their proof theory. But, of course, an axiom could, in that sense, be analytic of a thoroughly synthetic theory, and itself as synthetic as that theory. (There might be a single such axiom which could be reflectively recognised as systematising exactly Euclidean geometry.) To be sure, it is a necessary condition of the success of the neo-Fregean project that the relevant principle does more than generate a theory within which arithmetic can be interpreted—there has to be a tighter conceptual relationship than that. But it is no necessary condition for the satisfaction of this necessary condition that there be no conceptual surplus of the axiom over the theory. And it is no sufficient condition of the analyticity of such an axiom that there be none; for again, a reflectively correct digest of a synthetic theory will be itself synthetic.

2.5

The concern about bad company

Boolos’s final objection is perhaps the most interesting and challenging of all. It begins with the excellent observation that there are close analogues of Hume’s Principle, specifically, principles taking the form of second-order abstractions, linking the obtaining of a (second-order logically definable) equivalence relation on concepts to the identity condition for certain associated objects, which are self-consistent (that is, the systems consisting of secondorder logic plus one of these principles are, arguably, consistent) yet which are inconsistent with Hume’s Principle. A nice example is what I have elsewhere called the Nuisance Principle (NP). The nuisance associated with the concept F is the same as the nuisance associated with the concept G just in case the symmetric difference between F and G—the range of things which are either F or G but not both—is finite. Straightforward set-theoretic reasoning leads to the conclusion that any universe in which NP is satisfied must be a finite one. 21 But it is, apparently, a self-consistent principle—it does have finite models. If Hume’s Principle is analytic, then NP is analytically false. But with what right could we make that claim—isn’t the analogy between the two principles near enough perfect? This challenge—there dubbed the ‘Bad Company’ objection—is treated in some detail in Wright [1997] on which Boolos’s “Is Hume’s Principle Analytic?” was commentary. My suggestion in that paper was that the first step to disarming it is a deployment of (something very close to) Hartry Field’s notion of conservativeness. A principle, or set of principles, is conservative with respect to a given theory when, roughly, its addition to that theory results

21 For details see Wright [1997] at pp. 221–5.

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in no new theorems about the old ontology. 22 One could hope that Hume’s Principle will be conservative with respect to any theory for which secondorder Peano arithmetic is conservative (that is, one would hope, any theory whatever). By contrast, the consistent augmentation of any theory, T , by NP will result in a theory of which it is a consequence that all categories of the original ontology of T are at most finitely instantiated. No pure definition could permissibly have that effect. So no merely conceptual-explanatory principle— no principle whose role, as that of abstractions is supposed to be, is merely to fix the truth-conditions of a range of contexts featuring a new kind of singular-term forming operator and is otherwise to be as close as possible to that of a pure definition—can permissibly have it either. Since it has consequences for the size of extensions of concepts which are quite unrelated to that which it purportedly serves to introduce, NP thus cannot be viewed as such a conceptual-explanatory principle. Moreover, any abstraction principle which clashes with Hume’s Principle by requiring the finitude of any domain in which it is to hold will be in like case. And indeed any abstraction principle which places an upper bound, finite or infinite, on the size of the universe will be non-conservative with respect to some consistent theory of things other than the abstracts it concerns. The particular analogy is therefore broken: Hume’s Principle, there is undefeated reason to hope, is conservative with respect to every consistent theory concerning things other than its own special ontology—the cardinal numbers. (That is, note, a kind of weak analyticity: if there were a possible world in which Hume’s Principle failed, it would have to be by dint of its misrepresentation of the nature of the cardinals in that world.) NP and its kin, by contrast, come short by this constraint. An abstraction is acceptable only if it is conservative with respect to every consistent theory whose ontology does not include its proper abstracts. It is a logical abstraction just in case its abstractive relation is definable in higherorder logic. The company kept by Hume’s Principle is thus, we may presume, that of conservative, logical abstractions. But are these all Good Companions? 22 A tidied version of the characterisation offered in Wright [1997] (at note 49, p. 232) would be as follows. Let:

()

(∀αi )(∀α j ) ((αi ) = (α j ) ↔ αi ≈ α j ),

be any abstraction. Introduce a predicate, Sx, to be true of exactly the referents of the -terms and no other objects. Define the -restriction of a sentence T to be the result of restricting the range of each objectual quantifier in T to non-S items—thus each sub-formula of T of the form (∀x)Ax is replaced by one of the form (∀x)(¬ Sx → Ax) and each sub-formula of the form (∃x)Ax is replaced by one of the form (∃x)(¬ Sx & Ax). The -restriction of a theory θ is correspondingly the theory containing just the -restrictions of the theses of θ . Let θ be any theory with which is consistent. Then is conservative with respect to θ just in case, for any T expressible in the language of θ , the theory consisting of the union of () with the -restriction of θ entails the -restriction of T only if θ entails T. The requirement on acceptable abstractions is, then, that they be conservative with respect to any theory with which they are consistent. (The tidying referred to, for which I am indebted to Alan Weir, consists in having the reference to the -restriction of θ , rather than as originally one simply to θ , in the clause for ‘conservative with respect to θ ’.)

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Recent critics of neo-Fregeanism have observed that they are not, so that the fifth concern extends beyond the point that Boolos himself took it to. I pursue the matter in the first Appendix.

3. It should now be apparent why I suggested earlier that my debate with Boolos could as well have proceeded, near enough, without recourse to the notion of analyticity. The point is simply that each of Boolos’s objections is, in effect, independent of the problematical aspects of that notion: what was really bothering him was not whether Hume’s Principle is analytic, but whether it is true, and whether and how we might be warranted in regarding it as being so. Thus, without any really significant loss, the five points of concern might be formulated as: 1. With what right do we regard ourselves as warranted in accepting a principle with such rich ontological implications—how do we know that there is any function which behaves as the referent of octothorpe must? 2. What warrant do we have for confidence that the strong theory—Fregean arithmetic—to which Hume’s Principle gives rise is a consistent theory? 3. Is not its inconsistency with Zermelo–Frankel set theory (plus standard definitions) a strong ground for doubting the truth of Hume’s Principle? 4. What warrant is there for accepting a principle which is supposed to provide a foundation for arithmetic yet has so much surplus content over arithmetic? 5. With what right do we accept a principle which seems to be on all fours with other consistent principles which are inconsistent with it?

These are all good concerns, and I hope I have indicated, point-by-point, something of the direction in which the neo-Fregean should try to launch respective responses to them. The crucial point remains that the notion of analyticity is not required to formulate the concerns. What is really at stake, rather, is the nature of our entitlement to Hume’s Principle. A worked-out account of the notion of analyticity, in all its varieties, might well provide an answer to the question. But the answer the neo-Fregean wants to give is not hostage to the provision of such an account. Let me rapidly recapitulate that answer. The neo-Fregean thesis about arithmetic is that a knowledge of its fundamental laws (essentially, the Dedekind-Peano axioms)—and hence of the existence of a range of objects which satisfy them—may be based a priori on Hume’s Principle as an explanation of the concept of cardinal number in general, and finite cardinal number in particular. More specifically, the thesis involves four ingredient claims: 23 (i) that the vocabulary of higher-order logic plus the cardinality-operator, octothorpe or ‘Nx: . . . x . . . ’, provides a sufficient definitional basis for a statement of the basic laws of arithmetic; 23 I here rely again on formulations given in Wright [1998a].

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(ii) that when they are so stated, Hume’s Principle provides for a derivation of those laws within higher-order logic; (iii) that someone who understood a higher-order language to which the cardinality operator was to be added would learn, on being told that Hume’s Principle governs the meaning of that operator, all that it is necessary to know in order to construe any of the new statements that would then be formulable; (iv) finally and crucially, that Hume’s Principle may be laid down without significant epistemological obligation: that it may simply be stipulated as an explanation of the meaning of statements of numerical identity, and that—beyond the issue of the satisfaction of the truth-conditions it thereby lays down for such statements—no competent demand arises for an independent assurance that there are objects whose conditions of identity are as it stipulates.

The first and third of these claims concern the epistemology of the meaning of arithmetical statements, while the second and fourth concern the recognition of their truth. With which of them would Boolos disagree? Even with a qualification I will come to in a minute, I think he had no quarrel with the first; nor, of course, with the second, which is just the point proved by Frege’s Theorem. And to accept just these two claims, of course, is already to acknowledge a substantial Fregean achievement: the analytical reduction of the primitive vocabulary of arithmetic to a base that contains just one nonlogical expression, the cardinality operator; and a demonstration that, on that basis, the fundamental laws of arithmetic can be reduced to just one: Hume’s Principle itself. The qualification concerning the first claim concerns the interpretation of the phrase “sufficient definitional basis”. No question of course but that Frege shows how to define expressions which comport themselves like those for successor, zero, and the predicate ‘natural number’, thus enabling the formulation of a theory which allows of interpretation as Peano arithmetic. But—as we remarked right at the start—it is one thing to define expressions which, at least in pure arithmetical contexts, behave as though they express those various notions, another to define those notions themselves. And it is the latter point, of course, that is wanted if Hume’s Principle is to be recognised as sufficient for a theory which not merely allows of pure arithmetical interpretation but to all intents and purposes is pure arithmetic. How is the stronger point to be made good? Well, I imagine it will be granted that to define the distinctively arithmetical concepts is so to define a range of expressions that the use thereby laid down for those expressions is indistinguishable from that of expressions which do indeed express those concepts. The interpretability of Peano arithmetic within Fregean arithmetic ensures that has already been accomplished as far as all pure arithmetical uses are concerned. So any doubt on the point has to concern whether the definition of the arithmetical primitives which Frege

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offers, based on Hume’s Principle and logical notions, are adequate to the ordinary applications of arithmetic. Did Frege succeed in showing how the concepts of arithmetic, as understood both in their pure and applied uses, can be understood simply on the basis of second-order logic and the numerical operator, as constrained by Hume’s Principle, or could someone fully understand the entirety of the construction without having the slightest inkling of the ordinary meaning of arithmetical claims? The matter needs more detail than I will offer here, but I think it’s clear that Frege did succeed in the more ambitious task, and a crucial first step in seeing that he did so is to realise that Hume’s Principle provides for the proof of a very important principle, dubbed N q by Bob Hale, to the effect that for each numeral, ‘n f ’, defined in Frege’s way, we can establish that n f = Nx : Fx ↔ there are exactly n Fs where the second occurrence of ‘n’ is schematic for the occurrence of an arabic numeral as ordinarily understood. 24 It follows that each Fregean numeral has exactly the meaning in application which it ought to have. That seems to me sufficient to ensure that Hume’s Principle itself enforces the interpretation of Fregean arithmetic as genuine arithmetic, and not merely a theory which can be interpreted as such. If this is right, then the key philosophical issues must concern the third and fourth claims. The importance of the third claim derives from the consideration that Hume’s Principle is not, properly speaking, an eliminative definition— it allows the construction of uses of the numerical operator which it does not in turn provide the resources eliminatively to define. Its claim to serve as an explanatory basis for arithmetic must therefore depend on its ability somehow to explain such uses in a non-strictly definitional fashion. Arguing the point requires stratifying occurrences of the numerical operator in sentences of Fregean arithmetic according to the degree of complexity of the embedding context, and making a quasi-inductive case: first, that a certain range of basic uses are unproblematic, and second, that at every subsequent stage, the type of occurrence distinctive of that stage may be understood on the basis of an understanding of the mode of occurrence exemplified at the immediately preceding stage. There are some complications with this; I’ve tried to work through the point in some detail elsewhere, 25 and will not repeat the detail here. For what it’s worth, it is Michael Dummett, rather than Boolos, who has been the most vociferous opponent of the third neo-Fregean claim. 24 I reproduce in the second Appendix the proof of this claim given at pp. 366–8 of Wright [1998]. 25 See Section V of Wright [1998] and—for a supplementary consideration in response to an objection

of Dummett’s—Section VI of Wright [1998a].

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It is the fourth claim—the claim that Hume’s Principle can be laid down as an explanatory stipulation, without further epistemological obligation— which seems to me to be the heart of the issue. Boolos was indeed uncomfortable with this claim, suspecting that more had been smuggled into the notion of explanation in this setting than was consistent with the seeming modesty of the explanatory thesis. But I do not feel that I have understood his reservations terribly well. If nominalism is a misconception—if it is possible to know of abstract entities and their properties at all—then it has to be because we have so fixed the use of statements involving reference to and quantification over such entities as to bring the obtaining of their truthconditions somehow within our powers of recognition. And whatever this fixing consisted in, it has to have been something we did by way of determination of meaning, and it should therefore have involved no epistemological obligations which are not involved in the construction of concepts and the determination of meanings generally. I really do not see why the fashion in which Hume’s Principle—if it indeed succeeds in doing so—determines the truthconditions of statements which configure the cardinality operator with secondorder logical concepts, should be epistemologically any more problematical than any definition or other form of stipulation whose effect is to fix the truthconditions of statements containing a targeted (type of ) term. It is of course— always—another question whether those truth-conditions are satisfied: something which a definition, without supplementary considerations, is powerless to determine. But a good abstraction principle always determines very explicitly what those supplementary considerations are to be—you have only to look at its right-hand side. If there are good reservations about this way of looking at Hume’s Principle, I do not think that they have yet been compellingly formulated. Whatever the ultimate assessment of that issue may prove to be, it is my hope that the foregoing overview of Boolos’s misgivings about the analyticity of Hume’s Principle may serve as a reminder of two things: first (we owe it to Frege to recognise) that there is still an unresolved debate to be had about the viability of something that is, in all essential respects, a Fregean philosophy of arithmetic and real and functional analysis; 26 second, that the progress made in the modern debate is owing in very considerable measure to George’s brilliant and unique articles on the issues. 26 This is a point that Boolos enthusiastically accepted:

. . . I want to endorse Wright’s . . . suggestion that the problems and possibilities of a Fregean foundation for mathematics remain [wide?] open and [his] remark . . . that ‘the more extensive epistemological programme which Frege hoped to accomplish in Grundgesetze is still a going concern. (Boolos [1997], p. 246). For interesting preliminary steps towards the extension of the neo-Fregean programme to the classical theory of the reals, see Bob Hale [2000].

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Appendix A. Conservativeness and modesty In their [2000], Shapiro and Weir observe that there are pairs of abstractions which result by various kinds of selection for φ in (D)

(∀F)(∀G)( F = G ↔ (φ F & φG) ∨ (∀x)Fx ↔ Gx)27

which are jointly unsatisfiable yet which are presumably conservative in the germane sense. For instance, take φ respectively as ‘is the size of the universe and some limit inaccessible’ and ‘is the size of the universe and some successor inaccessible’. (The Neo-Fregean should resist any tendency to impatience at the rarefied character of the example. These notions are definable in higherorder logic.) Any instance of schema (D) entails that some F is φ. So the two indicated abstractions respectively entail that the universe is limit-inaccessible sized and that it is successor-inaccessible sized. It cannot be both. Yet neither implication places any overall bound on the size of the universe—so these abstractions do not involve the kind of non-conservativeness which NP entrained. Still, they cannot both be in good standing. And if either is not, then it seems that neither should be. But by what (well-motivated) principle might they be excluded? What virtue does Hume’s principle have which they lack? What is intuitively salient about any D-schematic abstraction (henceforward “Distraction” 28 ) is that, the entailment notwithstanding, it provides no motive to believe that there is a concept which falls under its particular selection for ‘φ’—the result is obtained merely by exploitation of the embedded antinomy. For on the assumption of (∀F)¬(φ F) any Distraction entails Basic Law V: (∀F)(∀G)( F = G ↔ (∀x)(Fx ↔ Gx)) and thereby Russell’s Paradox. Such abstractions thus have no more bearing on the truth of the relevant ‘(∃F)φ F’ than instances of the following schema have: (∀F)F is φ-terological ↔ F does not apply to itself or φ F which likewise, on the assumption of (∀F)¬φ F entail the well-known Heterological paradox: (∀F)F is heterological ↔ Fdoes not apply to itself. 27 This is schema (D) discussed in some detail in Wright [1997]; see pages 216 and following. 28 Alan Weir’s puckish term.

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Again, we can select for ‘φ F’ that F is the size of the universe and some limit inaccessible, or the size of the universe and some successor inaccessible, or that F applies to God, or the Devil . . . and proceed to infer that the universe is limit-inaccessible in cardinality, or successor inaccessible, or that God, or the Devil, exists. It is long familiar how Liar-family paradoxes can occur not merely in contexts of self-contained aporia but may be exploited to yield unmotivated a priori resolutions of intuitively unrelated issues. The Cretan and the Curry Paradox are the best known examples of the latter. The schema for φ-terologicality, and Distractions as a class, merely provide two more. This perspective offers the option of a ‘holding’ response to the Shapiro/ Weir objection: “You persuade me”, the neo-Fregean may say, “that the general idea that a concept may be defined by stipulation of its satisfactionconditions is somehow confounded by the possibility of pairwise incompatible yet consistent instances of the rubric for φ-terologicality and I will concede that the neo-Fregean conception of an abstraction principle is put in similar difficulties by conservative yet pairwise incompatible instances of (D).” This response is dialectically strong. Who would suppose that roguish cases like “heterological” and instances of φ-terologicality somehow show that we may no longer in good intellectual conscience regard the general run of definitions of the form: X is F if and only if . . . X . . . as successful in fixing concepts? But then someone who had no other objection to the claim of Fregean abstractions to play the role of truth-condition fixers for the kinds of context that feature on their left-hand sides should not be fazed by roguish instances of (D). 29 It is only a holding response, however. It refurbishes one’s confidence that it has to be possible to draw the distinction which the neo-Fregean needs, but it does not draw it. The fact remains that just as a general explanation is owing of which are the pukka definitions of satisfaction-conditions, and which may be dismissed as rogues, so we still need a characterisation of which are the good abstractions and which are the (conservative but still) bad Distractions. In Wright [1997], motivated in part by the desire to legitimate Boolos’s axiom New V: (∀F)(∀G)( F = G ↔ (Big(F) & Big(G) ∨ (∀x)Fx ↔ Gx)) (where F is Big just if it has a bijection with self-identity) I ventured an additional conservativeness constraint which would be tolerant of at least some instances of schema (D) but would reject the majority. Roughly, it was that those consequences of such an abstraction which follow by exploitation of its “paradoxical component” have to be in ‘independent good standing’. I shall here attempt briefly to clarify and assess this proposal. 29 Cf. Wright [1997], pp. 220–1.

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Distractions entail conditionals of the form: ¬(∃F)(φ F) → (∀F)(∀G)( F = G ↔ (∀x)(Fx ↔ Gx)) The immediate intent of the proposed constraint is that anything derivable by the reductio of the antecedent of such a conditional afforded by its paradoxical consequent should be in independent good standing. New V fares well by this proposal: that there is a concept which is Big should presumably be a result in ‘independent good standing’ however that idea is filled out—for that selfidentity itself is Big follows from the definition of ‘Big’ in second-order logic. Of course, any abstraction will entail some such conditional. So the proposed constraint is quite general. How does Hume’s principle fare by it? Well enough, presumably, though in a different way. We may, for instance, obtain a relevant conditional by selecting ‘at least countably infinite’ for φ. But this time the resources required to make good the consequences of the denial of the antecedent are afforded not just by second-order logic but by Hume’s Principle itself, via its independent proof of the infinity of the number series. Indeed, it is just because it independently entails that denial that we are able to show that Hume’s Principle entails the selected conditional in the first place. By contrast, the kinds of roguish Distraction illustrated presumably fail the test. The only resources they have to show, e.g. that the universe is limit-inaccessible, or successor inaccessible, or whatever, are those furnished by the inconsistency of Basic Law V and the consequent modus tollens on the relevant conditional. So: an abstraction is good only if any entailed conditional whose consequent is Basic Law V (or, therefore, any other inconsistency) is such that all further consequences which can be obtained by discharging the antecedent are in independent good standing, as may be attested by their derivation in pure higher-order logic (like the case of New V) or their independent derivability from the abstraction in question (like the case of Hume’s Principle). But this is unclear in a crucial respect: what is the relevant sense of ‘independent derivability’? Clearly it would not be in keeping with the intended constraint if there were merely some collateral derivation of just the same suspect kind. The ‘independent derivation’ must be bona fide, must not proceed by “paradoxexploitative” means, as I expressed the matter. But what does that mean? In particular, how might it be characterised so as not to outlaw any proof by reductio ad absurdum? One possible response—the one I offered in Wright [1997]—was that a relevantly narrow sense of “paradox-exploitative” may be captured by reinvoking the previous (Fieldian) notion of conservativeness in the following way: a derivation from a conservative abstraction is paradox-exploitative just if there is a representation of its form of which any instance is valid and of which some instance amounts to a proof of the non-conservativeness of another abstraction. For instance, the derivation of the successor-inaccessibility of the universe from the Distraction canvassed above is paradox-exploitative because it may be schematised under a valid form of which another instance is a

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derivation, from the appropriately corresponding Distraction, that the universe contains exactly 144 objects. The only Distractions which are good are those which are both conservative and such that any of their consequences which may validly be derived by paradox-exploitative means, in the stipulated sense, may also validly be derived by non-paradox-exploitative means. Otherwise put: the second conservativeness constraint is that the paradox-exploitative derivations from an abstraction have to be conservative with respect to the results obtainable from it non-paradox-exploitatively. That was the essence of my previous proposal. In practice, its application would work like this. We would be defeasibly entitled to accept any (presumably) conservative abstraction, A, from which we had so far been able to construct no paradox-exploitative derivation—no proof of a valid form of which another instance demonstrated the non-conservativeness of another abstraction. But once we had such a derivation, it would then be inadmissible to accept A until we had found another non-paradox-exploitative derivation from it of the same conclusion: a formally valid derivation of which, so far as we could tell, no other instance was a proof of the non-conservativeness of another abstraction. That is apt to seem uneasily complex and less clearly motivated than one would wish. And one might worry about its reliance on our ability to judge non-paradox-exploitative derivations. However the play with ‘paradoxexploitation’, and its characterisation in terms of non-conservativeness may now seem inessential. The basic idea was that some abstractions—the Distractions and some others—are at the service of non-cogent proofs. We can tolerate this in particular cases so long as such proofs are matched by cogent ones of the same things. The natural—surely correct—objection to the derivation of, say, the successor inaccessibility of the universe from the appropriate Distraction is that it is unconvincing because “You could just as well prove the opposite—or anything—like that”, where “like that” means: by laying down a different (presumably consistent) Distraction and reasoning in just the same way. So a natural thought would be that we should ban those distractions—or abstractions generally—some of whose consequences are such as to deserve that complaint. That would suggest the following stipulation: that an abstraction A is unacceptable, at least pro tempore, if every proof it has yielded of some consequence C is such that, schematised so that any instance of it is valid, some other (conservative) abstraction yields a proof of the same form of something inconsistent with C. But there are still a number of salient concerns. First, it is not clear that any purpose is served by the continuing insistence on derivations of a given valid form. Why not just say that pairwise incompatible but individually conservative abstractions are ruled out—however the incompatibility is demonstrated— and have done with it? For think: if each such pair can be shown to be incompatible by proofs of a given single form, then the more complex formulation of the constraint is unnecessary; but if some pair cannot—if no derivation of C

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from A is of a valid form shared by some derivation of not-C from A*—then there will still be pairwise incompatible but conservative abstractions which survive the new test. So there will still be Bad Company, for which some further treatment will need to be devised. Besides, how is the proposed constraint meant to be applied to semantical—model-theoretic—demonstrations of consequence—for which of course, in the case of higher-order abstractions, there need be no effectively locating a corresponding derivation in higher-order logic? This whole direction was stimulated by the desire to save some ‘good’ Distractions, par excellence New V. It is therefore germane that, as Shapiro has since observed, New V itself is in any case non-conservative!—specifically that it entails that the universe can be well-ordered, and hence that the nonabstracts can. 30 This result, to be sure, does not show that there is nothing to be gained from attempting to refine the second conservativeness constraint of my [1997]—that it has no point. But it should occasion a re-think of the motivation for the general direction. I think there is something else amiss with the rogue Distractions— something which the second proposed constraint may well indirectly approximate but does not bring out with sufficient clarity. Start from the point that definitions proper should be innocent of substantive implications for the universe over which they range. Abstractions cannot in general match that, since in conjunction with logically (or other forms of metaphysically) necessary input, they may carry substantive implications for the abstracts whose concept they serve to introduce and hence—since those abstracts will be viewed, at least by neo-Fregeans, as full-fledged participants in the universe—at least some substantive implications for the universe as a whole. But to the extent that it is proposed to regard them as meaning-constituting stipulations, and hence as approximating definitions as nearly as possible, the character and scope of such implications needs to be curtailed. In brief: the requirement has to be that the only implications they may permissibly carry for the, as it were, enlarged universe in which their own abstracts participate must originate in what they imply—whether proof- or model-theoretically—about the abstracts they specifically concern. Hume’s Principle, for instance, implies of any object whatever that it participates in an at least countably infinite universe; but it carries that implication only via its entailment of the infinity of the cardinal numbers. This is a different requirement to Field conservativeness. A non-Field conservative abstraction—one that, as we put it intuitively above, entails new results about the prior ontology—may of course violate it. But it is possible for an abstraction to be immodest—for it to carry implications for other objects in 30 See Weir and Shapiro [1999]. In rough outline: we can derive the Burali-Forti paradox on the assumption that the concept, Ordinal, is not Big; but if Ordinal is Big, then there is a 1–1 correlation between Ordinal and x = x. So x = x may be well ordered by that correlation.

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the universe which cannot be shown to originate in implications it carries for its own proper abstracts—without thereby being demonstrably non-conservative. Consider the Limit-inaccessible Distraction again. As noted, this entails that the universe is limit-inaccessibly sized. But because, unlike NP, it places no upper bound on the size of the universe, it is not non-conservative in the way that NP is—it limits the extension of no other concept. And if it is nonconservative in any other way, we have yet to see how. However it is immodest. For its requirement that the universe have a certain kind of cardinality does not originate in any requirement that it imposes on its own abstracts. It is easy to overlook the force of “originate” here. The Limit-inaccessible Distraction, for instance, entails that any finite concept is non-φ. So it will allow singleton concepts to generate ‘well-behaved’ abstracts—abstracts whose identity and distinctness is governed by ordinary extensionality—of which there should therefore be no fewer than there are objects in the universe. 31 Thus this particular Distraction will indeed entail that its own abstracts are limit-inaccessible in number, from which the limit inaccessibility of the universe follows. 32 But—this is the crux—the result about the abstracts is not needed for the proof of the limit-inaccessibility of the universe. The Distraction provides no way of recognising the limit-inaccessibility of the universe which goes via a prior recognition of what it entails about its own proper abstracts. Rather the inference is the other way about: the proof that the Distraction entails that result about the universe as a whole is needed in order to obtain the result about its own abstracts. That is immodesty. Conservativeness constrains the kind of consequences which an acceptable abstraction is allowed to have: it is not allowable that there be any claim exclusively concerning the non-abstracts which was previously unprovable but which the abstraction, coupled with previous theory—now explicitly restricted to the previous ontology—enables us to prove. Modesty, by contrast, constrains the kind of ground which an acceptable abstraction can provide for consequences, not per se non-conservative, about the ontology of a theory in which that abstraction participates: such consequences must be grounded in what it requires of its own proper abstracts. But although the two constraints may seem different in character in this way, they are aspects of a single point au fond. Remember that the role of a legitimate abstraction, as I have repeatedly stressed, is merely to fix the truth-conditions of a class of contexts featuring a novel term-forming operator. It cannot have more than that role and yet retain the epistemically undemanding character of a meaning-stipulation. Logical abstractions, to be sure, are so designed that, consistently with their playing just this role, logical resources may enable us to show that there are abstracts of the kind they concern and to establish things about them. But no abstraction can be deemed to discharge the intended limited role successfully 31 Assuming that there no fewer singleton concepts than there are objects. 32 On standard cardinal-arithmetical assumptions.

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if, in conjunction with some consistent theory, it carries implications for the combined ontology which cannot be shown to derive from implications it has for its own abstracts. Non-conservativeness is (normally) one graphic way of failing that test. But if even a conservative abstraction entails some conclusion about the combined ontology which cannot be justified by reference to what it entails about its own abstracts, then knowledge of the truth of the abstraction cannot be founded in stipulation. Such an abstraction implicitly claims something about the world which might—for all we have shown to the contrary—be justified by reference to what it entails about its own abstracts; that is why we cannot accuse it of non-conservativeness. But equally, so long as we have no such justification, we have no defence against the suggestion that the abstraction is known only if we know that the world must be that way in any case, whether or not the abstracts themselves make it so. That would seem to demand knowledge about how the world would be even if the abstracts did not make it so. And that in turn is a substantial piece of collateral information which, by being prerequisite if we are to claim to be justified in laying down the abstraction in the first place, gives the lie to any claim that the abstraction is justified merely as a meaning-stipulation. In sum: an abstraction is modest if its addition to any theory with which it is consistent results in no consequences (whether proof- or model-theoretically established) for the ontology of the combined theory which cannot be justified by reference to its consequences for its own abstracts. And again, justification is the crucial point: an abstraction may fail this constraint even though every consequence it has for the ontology of a combined theory may be seen to follow from things it entails about its proper abstracts; in particular, it will not count if, as in the case of the Limit-inaccessible Distraction, a consequence for the combined ontology is needed as a lemma in the proof that the abstracts have a property from which that very consequence follows. Further clarification is needed of several matters: what kinds of proof should count in favour of the modesty of an abstraction—what it is to show that an abstraction independently carries certain implications for its own abstracts; whether the modesty constraint is effective against the general run of pairwise incompatible but (presumptively) conservative abstractions illustrated by Shapiro and Weir; what other constraints on Good Companions may be properly motivated. At the time of writing, these are largely open issues.

B. B.1

Proof of the principle, Nq Stage-setting

We assume the standard recursive definitions of the numerically definite quantifiers: (∃0 x)Fx ↔ (∀x)¬Fx (∃n+1 x)Fx ↔ (∃x)(Fx & (∃n y)(Fy & y = x)),

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and let ‘n f ’ abbreviate Frege’s definiens for n. Define ‘Pxy’ (immediate predecession) as (∃F)(∃w)(Fw & y = Nv : Fv & x = Nz : [Fz & z = w]) Define ‘Nat(x)’ (x is a natural number) as x = 0 ∨ P ∗ 0x where ‘P*xy’ expresses ancestral predecession. Let ‘(∃R)(F 1–1 R G)’ express that there is a one–one correspondence between F and G. We take three lemmas from the proof of the Peano axioms from HP outlined in the concluding section of Frege’s Conception (numbering as there assigned): Lemma 51: (∀x)(Nat(x) → x = Ny : [Nat(y) & P ∗ yx]—every natural number is the number of its ancestral predecessors. Lemma 52: (∀x)(Nat(x) → ¬P ∗ xx)—no natural number ancestrally precedes itself. Lemma 5121: (∀x)(∀y)(Nat(x) & Nat(y) → (Pxy → (∀z)(Nat(z) & (P ∗ zx ∨ z = x) ↔ (Nat(z) & P ∗ zy)))—if one natural number immediately precedes another, then the natural numbers which ancestrally precede the second are precisely the first and those which ancestrally precede the first. Finally, recall that Frege’s 0 is Nx : x = x and that each successive n + 1 f is N x : [x = 0 ∨ · · · ∨ x = n f ]. Each of these objects qualifies as a natural number in the light of the above definition of ‘Nat(x)’. Proof: 0 f qualifies by stipulation; n + 1 f qualifies if n f does—take ‘F’ in the definition of ‘Pxy’ as ‘[x = 0 ∨ · · · ∨ x = n f ]’ and ‘w’ as ‘n f ’ to show that P(n f , n + 1 f ); then reflect that Pxy → P ∗ xy and that P ∗ xy is transitive. (Frege’s Conception, Lemmas 3 and 4, respectively.)

B.2

Proof of Nq for Frege’s natural numbers

Induction Base: To show Nx : Fx = 0 f ↔ (∃0 x)Fx, it suffices to reflect that the left-hand side holds just if (∃R)(Fx 1–1 R x = x), which in turn holds just if ¬ (∃x)Fx. 33 Induction Hypothesis: Suppose Nx : Fx = n f ↔ (∃n x)Fx. We need to show that it follows that Nx : Fx = (n + 1) f ↔ (∃n+1 x)Fx. 33 As George Boolos remarked to me, Frege himself observes, at Grundlagen §75 and §78, that he is in a position to obtain proofs of N q for 0 f and 1 f , respectively.

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(Left-to right) Consider any F such that Nx : Fx = (n + 1) f . By Lemma 51 and the reflection that Nat(n f ), n f = Nx : [Nat(x) & P ∗ xn f ]. So by the Hypothesis (∃n x)(Nat(x) & P ∗ xn f ). But by Lemma 52, ¬P ∗ n f , n f . So (∃n x)(Nat(x) & (P ∗ xn f ∨ x = n f ) & x = n f ). So (∃y)(Nat(y) & (P ∗ yn f ∨ y = n f ) & (∃n x)(Nat(x) & (P ∗ xn f ∨ x = n f ) & x = y)). So by the recursion for the quantifiers (∃n+1 x)(Nat(x) & (P ∗ xn f ∨ x = n f ). But by Lemma 5121 and since P(n f , n + 1 f ), we have that (∀x)(Nat(x) & (P ∗ xn f ∨ x = n f ) ↔ Nat(x) & P ∗ (x, n + 1 f )). So (∃n+1 x)(Nat(x) & P ∗ (x, n + 1 f )). That establishes the desired result for one concept of which (n + 1) f is the number. But by HP, any G such that (n + 1) f = Nx : Gx will admit a one-one correspondence with that concept. So a lemma to the following effect will now suffice: (∀F)(∀G)((∃R)(F 1−1 R G) → ((∃n+1 x)Fx ↔ (∃n+1 x)Gx) A proof by induction—strictly, at third-order—suggests itself: Base: It suffices to show (∀F)(∀G)((∃R)(F 1−1 R G) → ((∀x)¬Fx ↔ (∀x)¬Gx)) Hypothesis: Suppose (∀F)(∀G)((∃R)(F 1−1 R G) → ((∃n x)Fx ↔ (∃n x) Gx). Consider any H such that (∃n+1 x) Hx. Then (∃x)(Hx&(∃n y)(Hy&y = x)). Let a be such that Ha & (∃n y)(Hy & y = a). Let J be one–one correlated with H by R. Let b be such that Jb & Rab. Then R one–one correlates Hx & x = a with Jx & x = b. So, by the Hypothesis, (∃n x)(Jx & x = b). So (∃x)(Jx & (∃n x)(Jx & x = b)). So (∃n+1 x)Jx. (Right-to left) Consider any F such that (∃n+1 x)(Fx). Then there is some a such that Fa & (∃n y)(Fy & y = a). So by the Hypothesis Ny (Fy & y = a) = n f . So, by HP, there is an R such that (Fy & y = a) (1–1 R ) (Nat(x) & P ∗ xn f ). Let R # correlate (Fy & y = a) with (Nat(x) & P ∗ xn f ) in just the fashion of R, and let it also correlate a with n f . Then (Fy) 1–1 R # (Nat(x) & (P ∗ xn f ∨ x = n f )). But, as established above (∀x)(Nat(x) & (P ∗ xn f ∨ x = n f ) ↔ Nat(x) & P ∗ (x, n + 1 f )). So Nx : Fx = (n + 1) f .

References George Boolos [1986] “Saving Frege from Contradiction” in Proceedings of the Aristotelian Society 87, pp. 137–51; reprinted in Demopoulos, ed. [1995], pp. 438–52. George Boolos [1987] “The Consistency of Frege’s Foundations of Arithmetic” in Judith Jarvis Thompson, ed. [1987], pp. 3–20; reprinted in Demopoulos, ed. [1995], pp. 211–33. George Boolos [1990] “The Standard of Equality of Numbers” in Boolos, ed. [1990a], pp. 261– 77; reprinted in Demopoulos, ed. [1995] pp. 234–54. George Boolos, ed. [1990a] Meaning and Method: Essays in Honor of Hilary Putnam, Cambridge: Cambridge University Press. George Boolos [1997] “Is Hume’s Principle Analytic?” in Heck, ed. [1997], pp. 245–61. George Boolos and Richard. G. Heck, Jr. [1998] “Die Grundlagen der Arithmetik §§82–3” in Schirn, ed. [1998], pp. 407–28.

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Peter Clark [1998] “Dummett’s Argument for the Indefinite Extensibility of Set and Real Number” in Grazer Philosophische Studien 55, New Essays on the Philosophy of Michael Dummett, eds. J. Brandl and P. Sullivan (Vienna: Rodopi), pp. 51–63. William Demopoulos, ed. [1995] Frege’s Philosophy of Mathematics, Cambridge, Mass.: Harvard University Press. Michael Dummett [1963] “The Philosophical Significance of Gödel’s theorem”, Ratio 5, pp. 140–55. Michael Dummett [1991] Frege: Philosophy of Mathematics, London: Duckworth. Michael Dummett [1993] The Seas of Language, Oxford: The Clarendon Press. Hartry Field [1984] Critical Notice of Crispin Wright Frege’s Conception of Numbers as Objects, Canadian Journal of Philosophy 14, pp. 637–62; reprinted as “Platonism for Cheap? Crispin Wright on Frege’s Context Principle” in Field [1989], pp. 147–70. Hartry Field [1989] Realism, Mathematics and Modality, Oxford: Basil Blackwell. Bob Hale [1997] “Grundlagen §64”, Proceedings of the Aristotelian Society XCV11, pp. 243– 61. Bob Hale [2000] “Reals by Abstraction”, Philosophia Mathematica 8, pp. 100–23. Richard G. Heck, Jr., ed. [1997] Language, Thought and Logic, Oxford: The Clarendon Press. Richard G. Heck, Jr., ed. [1997a] “Finitude and Hume’s Principle”, Journal of Philosophical Logic 26, pp. 589–617. Alex Oliver [1998] “Hazy Totalities and Indefinitely Extensible Concepts: An Exercise in the Interpretation of Dummett’s Philosophy of Mathematics” in Grazer Philosophische Studien 55, New Essays on the Philosophy of Michael Dummett, eds J. Brandl and P. Sullivan (Vienna: Rodopi), pp. 25–50. Charles Parsons [1964] “Frege’s Theory of Number” in Philosophy in America, ed. Max Black, London: Allen and Unwin, pp. 180–203; reprinted in Demopoulos, ed. [1995], pp. 182–210. Matthias Schirn, ed. [1998] Philosophy of Mathematics Today, Oxford: The Clarendon Press. Stewart Shapiro [1998] “Induction & Indefinite Extensibility: The Gödel Sentence is True but Did Someone Change the Subject”, Mind 107, pp. 597–624. Stewart Shapiro and Alan Weir [1999] “New V, ZF and Abstraction”, Philosophia Mathematica 7, pp. 293–321. Judith Jarvis Thompson, ed. [1987] On Being and Saying: Essays in Honor of Richard Cartwright, Cambridge, Mass.: MIT Press. Crispin Wright [1983] Frege’s Conception of Numbers as Objects, Aberdeen: Aberdeen University Press. Crispin Wright [1997] “On the Philosophical Significance of Frege’s Theorem” in Heck [1997], pp. 201–44. Crispin Wright [1998] “On the Harmless Impredicativity of N= (‘Hume’s Principle’)” in Schirn [1998], pp. 339–68. Crispin Wright [1998a] “Response to Dummett” in Schirn [1998], pp. 389–405.

FREGE, NEO-LOGICISM AND APPLIED MATHEMATICS 1 Peter Clark Philosophy Department, School of Philosophical and Anthropological Studies, University of St Andrews, St Andrews, Fife, Scotland, KY16 9AL, UK

1.

Introduction—logicism and neo-logicism

A little over one hundred years ago (the letter is dated July 28, 1902) Frege wrote to Russell in the following terms: 2 I myself was long reluctant to recognize ranges of values and hence classes; but I saw no other possibility of placing arithmetic on a logical foundation. But the question is how do we apprehend logical objects? And I have found no other answer to it than this, We apprehend them as extensions of concepts, or more generally, as ranges of values of functions. I have always been aware that there are difficulties connected with this, and your discovery of the contradiction has added to them; but what other way is there?

Frege here poses an extremely good question, a recent answer to which this paper is really devoted. Whatever view one may finally adopt about whether the new answer succeeds one has to recognise that it is surely a very remarkable fact that one hundred years after the discovery of the Zermelo–Russell contradiction which follows from Basic Law Five of Frege’s Grundgesetze (Frege (1893)) we should now be actively discussing not as a purely historical enterprise but as a viable possibility in the foundations of mathematics, the programme more or less explicitly laid out in the earlier work of Frege Die Grundlagen der Arithmetik (Frege (1884)). This is because recent research has highlighted three crucial facts. First and most importantly that in full conformity with the spirit of Frege’s programme the deduction of the axioms of (Second Order) Peano Arithmetic from principles of higher order logic and “definition” 3 does not require appeal to Basic Law Five. Second that 1 This paper first appeared in Induction and Deduction in the Sciences, F. Stadler (ed.) [2004], Dordrecht, Kluwer Academic Publishers. Reprinted by kind permission of Springer Academic Publishers. 2 Frege (1902), pp. 140–41. 3 Actually one has to be careful how this result is stated. Formally the central result is that if a formalisation of the key ‘definition’ is added as an axiom to standard axiomatic second order logic, second

45 Roy T. Cook (ed.), The Arché Papers on the Mathematics of Abstraction, 45–60. c 2007 Springer.

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the principles of logic and “definition” (this definition has become known as Hume’s Principle) which are employed are consistent, 4 and third that even the proofs of the axioms for Peano Arithmetic given by Frege himself in the Grundgesetze der Arithmetik do not depend essentially upon Basic Law Five. 5 These technical facts, of which more later, open the way for the revival of Frege’s programme, they make it possible but they do not determine its form. What was Frege’s programme and what is its revived form explicitly? Frege’s programme was the result of an answer to the famous question raised in Section 62 of the Grundlagen viz.: “how then are numbers given to us, if we cannot have any ideas or intuitions of them?” The Fregean answer was “by explaining the senses of identity statements in which number words occur”. That explanation was to be provided at least in part by what has come to called Hume’s Principle: the claim that the cardinal numbers corresponding to two concepts are identical if and only if the two concepts are equinumerous. I say in part by Hume’s Principle because as Frege had already argued in another context at Section 56 of the Grundlagen whatever the merits of Hume’s Principle it can’t explain the senses of identity statements in which number words occur of the form “the number of F’s is n”, where n is not given in the form of “the number of G’s”, for some G. Frege then adopted the explicit definition of number in terms of classes or extensions “the number of F’s is the class of all concepts G, equinumerous with F”. But this explicit definition together with Basic Law Five, the comprehension axiom for class existence entails Hume’s Principle. With Axiom Five in place it looked as if Frege’s programme could be carried out. It was now possible to show that second order logic (Frege (1879)) together with Basic Law Five entails the Peano–Dedekind axioms for arithmetic. As such the truths of arithmetic could be seen to be analytic, they could all be seen to be consequences of general logical laws together with suitable implicit definitions (like Basic Law Five which implicitly defines the notion of an extension). Further arithmetic could be seen as a body of truths about independently existing objects—the finite cardinals—which were logical objects, order arithmetic (arithmetic with the full second order induction axiom) can be interpreted in the resulting theory, often called Frege Arithmetic. Certainly the result seems to have been known to Geach in the forties, Dummett in the fifties and was first recently explicitly noted by Charles Parsons in his 1965 paper “Frege’s Theory of Number” (Parsons (1965) reprinted in Demopoulos (1995), pp. 182–210). A very closely related result was published by Timothy Smiley in 1981 (Smiley (1981)). The full significance of the result as well as a well developed proof was given by Wright (1983). The result has been systematically investigated by George Boolos (who discovered it independently in the early eighties and Richard Heck (see especially Boolos (1987a), (1998), papers 17, 18 and 19, Heck (1993). The most accessible proof can be found in the Appendix to Boolos (1990a), Boolos (1998) paper 13). 4 In his 1983 Wright conjectured that the system of axiomatic second order logic together with Hume’s Principle is consistent but did not establish it. Burgess (1984), Hazen and Hodes provided elementary consistency proofs with ω and ω + 1 as domains. George Boolos however established the central consistency result which is that the theory is equi-consistent with analysis (see his (1987b), (1990b) and Boolos and Heck, paper 20 of Boolos (1998)). 5 This was certainly known to Frege (see especially Heck (1995)), but in a letter to Russell he dismissed the possibility of basing his system on Hume’s Principle, saying only that it faced difficulties which were different from those facing the attempt to use Basic Law Five (Frege (1902), letter xxxvi/7, p. 141).

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logical in the sense that knowledge of which requires nothing beyond knowledge of logic and definitions. If this could be extended to the Real numbers and other parts of mathematics then a foundation for mathematics would have been established on which mathematics was presented as uncontaminated by empirical notions, presented as a body of truths in its full classical form and shown to be applicable to reality, since it can be shown to be in fact the more and more elaborate drawing of consequences from meaning postulates. But deduction applies to everything that can be thought. In fact it seems to me that the fundamental logicist thought can be put simply as follows: there can be no thought without representation, there can be no representation without concepts and there can be no concepts without number. Of course the serpent had already entered this Eden, with the introduction of Basic Law Five which says (∀F)(∀G)(Ext(F) = Ext(G) ↔ (∀x)(Fx ↔ Gx)) But by the Comprehension Principle for Second Order Logic there is a property corresponding to the formula of Second order logic (∃F)(Ext (F) = x &¬ Fx). Russell’s paradox immediately results from allowing this property to fall under the universal quantifier (∀F) in Basic Law Five. Another way of putting the same point is to note that the Russell reasoning shows that it is a theorem of Second order logic that there is no function from properties to objects such that distinct properties (i.e. non-coextensive properties) are associated with distinct objects. This is just what Basic Law Five read from left to right in contrapositive form asserts there is. 6 So much for logicism. What about neo-logicism, the revived form of Frege’s programme? We should let Wright and Hale, the main proponents of this view speak for themselves. They say: 7 Neo-Fregeanism holds that Frege need not have taken the step which lead to this unhappy conclusion [The appearance of the Russell contradiction]. At least as far as the theory of natural number goes, it is possible to accomplish Frege’s central mathematical and philosophical aims by basing the theory on Hume’s Principle, adjoined as a supplementary axiom to a suitable formulation of second order logic. Hume’s Principle cannot, to be sure, be taken as a definition in any strict sense—any sense requiring that it provide for the eliminative paraphrase of its definiendum (the numerical operator, “the number of . . . ”) in every admissible type of occurrence. But this does not preclude its being viewed as an implicit definition, effecting an introduction of a sortal concept of cardinal number and, accordingly, as being analytic of the concept—and this, the neo-Fregean contends, coupled with the fact that Hume’s Principle so conceived requires a prior understanding only of second order logical vocabulary, is enough to sustain an account of the foundations of Arithmetic that deserved to be viewed as a form of logicism which, whilst not quite logicism in the sense of a reduction of arithmetic to logic, preserves the essential core and content of Frege’s two fundamental theses. 6 This is Frege’s own generalisation of the lesson of Russell’s paradox. (See also Boolos (1993)). 7 Hale and Wright (2000), Introduction.

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So two conditions will have to hold for this position to viable: first what has come to be known as Frege’s theorem (which certainly does hold), that is the mathematical claim that the Dedekind–Peano axioms postulates for number theory in their second order form can be derived from a combination of Second order logic and Hume’s Principle and second that it will have to be shown that Hume’s Principle and other so called abstraction principles which share its form constitute a legitimate means of introducing the names of numbers (of abstract objects in general) by, in effect, stipulation by implicit definition. One needs to be careful to state the claim of neo-logicism properly, it is: (i) Hume’s Principle is a stipulation which gives the truth conditions of a restricted class of statements of numerical identity (ii) The resulting explanation of the concept of number is complete however, in that it suffices for the second order derivation of the basic laws of arithmetic (iii) The existence of numbers is something discovered and not stipulated (the Platonism of Frege’s original theory is preserved) (iv) Our (a priori) knowledge of number is derived from a principle whose truth is a matter of stipulation. 8

Abstraction principles of which Hume’s Principle is a paradigm example come in two types conceptual abstractions and objectual ones, but all have the following form. There is a domain of entities, denoted say, by α, β, etc., and a relation R defined over them. Then an abstraction principle has the form ((α) = (β)) ↔ R(α, β) Where R( , ) is an equivalence relation among the α and β’s. An abstraction principle may be called a logical abstraction when the relation R( , ) is definable in purely logical vocabulary, e.g. equinumerosity among concepts or ordinal similarity among binary relations. Under the classical canonical interpretation (α) is the equivalence class of α under the relation R and exists (where it does) in virtue of a set existence axiom. That is the existence and uniqueness of (α) has in effect to be guaranteed by a separate principle of set or class existence. Wright and Hale however argue that in certain cases logical abstraction principles can play the role of stipulations and if the relation on the right hand side of the iff is ever satisfied then no further question concerning the existence of the (α) need arise. Conceptual abstraction principles are those in which α’s are concepts (as in the case of Hume’s Principle) and objectual abstraction principles are those in which the field of the equivalence relation comprises objects. In both cases and this is crucially so the abstracta the (α) are objects, so in the case of conceptual abstractions acts as a type down operation, from concepts to objects. 8 See also Demopoulos (1998), (2000).

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Of course Wright and Hale do not argue that it is always legitimate to introduce abstracta in this way. Two examples of conceptual logical abstraction principles which fail to introduce abstracta are Basic Law Five and what might be called Ordinal Hume which is the claim that (∀ R)(∀ S)(Ord R = Ord S ↔ R is similar to S) This has the form of an abstraction principle since similarity is an equivalence relation among binary relations. But Ordinal Hume leads directly to the Burali–Forti paradox. 9 Hume’s Principle in logical abstraction form says: (∀ F)(∀ G)(NxFx = NxGx ↔ (∃ R)(F1 − 1R G)) where (∃R)(F1−1R G) is an abbreviation for the standard formulation in the vocabulary of second order logic of the formula expressing that there is a relation R which establishes a one to one correspondence between the things falling under F and those falling under G, that is ((∃R)(∀x)(Fx→ ∃!y)(Gy & R(x,y)) & ∀z(Gz→ ∃!w)(Fw & R(w,z))) and the operator Nx . . . x is a term forming operator. Wright has argued that there are general principles which can distinguish between good and bad abstraction principles and in any case as is well known there is no similar problem about Hume’s Principle, since it is known to be consistent. Like Basic Law Five Hume’s Principle asserts the existence of a function from concepts to objects but unlike Basic Law Five it asserts that merely non-equinumerous concepts (not non-co-extensive concepts) can be sent to distinct objects and this is possible provided that the domain is (Dedekind) infinite. For a domain of k objects there are k + 1 non-equinumerous concepts definable over it, so no finite domain can satisfy Hume’s Principle. The values of the second order variables for Frege are concepts and objects are denoted by terms that may appear on either side of the identity sign, so the terms like NxFx denote objects. As such Nx . . . x will have to be thought of as a term forming operator and our theory of second order logic plus Hume’s Principle must be sufficiently strong to have as a theorem (∀F)(∃!x)(x = NxFx). In general one would expect that we would have to have an axiom asserting that for each F, NxFx was a term. This fact seems to me to have the profoundest significance for the claim that Hume’s Principle and other abstraction principles can be thought of analytic stipulations introducing the names of numbers (or other abstract objects). Nor do I see any of the various equivalent methods of adding Hume’s Principle to second order logic in order to derive Frege’s theorem as avoiding this difficulty, for they will all have to guarantee that (∀ F)(∃!x)(x = NxFx) holds in some way or other. If it is guaranteed then we may proceed as follows: consider the concept non-self identical. It is a truth of logic that the concept non-self identical is equinumerous with the concept non-self identical, so 9 See Hodes (1984) and Fine (1998).

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the right hand side of Hume’s principle for F and G both non-self identical reduces to a logical truth. So we may detach and assert as a theorem Nx ∼ (x = x) = Nx ∼ (x = x) and so infer (∃!y)(Nx ∼ (x = x) = y) and so introduce the name “zero” to designate that unique object. But that hinges on us being able to read the stipulation, to understand the stipulation, as providing a context into which it is appropriate to quantify in. But what guarantees that is the correct reading of the stipulation? This is my first sceptical argument then: it essentially concerns how the underlying logic is to function. On the one hand to presume NxFx is a term because it appears on one side of the identity sign seems to me to beg the question, while if the question is not begged and say a free logic is employed then I fail to see how the required existential postulates will form any intended contrast with “mere axiomatic postulation”. 10 But I will not dwell further on this matter here.

2.

Anti-zero Following Boolos we can write Hume’s Principle in the form ∗

F = ∗ G ↔ (F1 − 1G)

and understand it as asserting that there exists a total function from concepts to objects, call it *, such that non-equinumerous concepts are assigned distinct objects (that is the contrapositive of Hume’s principle read from left to right). Adding this principle to Second Order logic allows us to prove in the Fregean way that, with the successor relation defined in the usual manner viz., S(n, m) ↔ (∃ F)(∃y)(Fy & ∗ F = n & ∗ (x : Fx & x = y) = m) (i) the successor relation is functional and one–one (ii) but that with zero defined as *(x:x = x), (∀x) ∼ S(0,x) and n is Finite iff n is zero or Sˆ(n,0) where Sˆ is the ancestral relation of S. But that is just to say that the natural numbers form a Dedekind infinite sequence (that is (∃f)[(∀x)(∀y)(fx = fy → x = y) & (∃x)(∀y)(fy = x)]). The key step is clearly to be able to prove that every number has a successor and Frege’s proof works precisely because n is an object that can be proved not to fall under the concept “being less than n”. This was, one of, Frege’s triumphs in the Grundlagen. It might therefore seem 10 Wright and Hale consistently draw attention to the difference in methodology they see between mere axiomatic stipulation and their proposed methodology, whereby abstracta are taken as stipulations. I have not dwelt on what I regard in this respect as a separate issue namely that of the very strong existential import of second order logic. This is most clearly seen in a point that George Boolos constantly emphasised that is the very strong existential commitments embodied in the Second Order Comprehension Principle (∃X)(∀x)(Xx↔A(x)) where A(x) is a formula of second order logic not containing X free. The issue of the existential presuppositions of the Hume’s principle has been addressed by Shapiro and Weir (2000) and by Demopoulos in his review of M. Schirn (1998) in the Journal of Symbolic Logic. Shapiro and Weir conclude that “the neologicist has no non-question begging account of how there could be an epistemologically innocent route to the demonstration of platonistically construed mathematical existence claims.” (p. 188).

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little more than trite to claim that H.P. is an axiom of infinity, it clearly is, since it forces the domain to be infinite. No doubt it is, but the principle has many other odd entailments too. Some of which are surprisingly and worryingly strong. Since the function * is total we can take it that it is a theorem of Frege Arithmetic that (∀F)(∃!x)(x = *F). Call this theorem (**). Now as Frege noted if we let F be “. . . is a member of the natural sequence of numbers”, it follows that there is a number of finite numbers, Frege’s ∞1 and that since it succeeds itself it is not a finite number (since it is a theorem of the formal system implicit in the Grundlagen that if Finite(n) then ∼S(n,n)). Again it follows immediately from (**) that if we take F to be the concept “is self identical” then (∃!x)(x = *(x = x)), that is, that there is a number of all things that there are. This latter looks like a very strong claim indeed. But we should note immediately, as Boolos pointed out, that we cannot show within the theory (Frege Arithmetic) that the two numbers are distinct. 11 In what we might call the standard model of Hume’s Principle, with domain N, the natural numbers in which all infinite concepts are assigned the object zero, while all finite concepts are assigned the cardinality of the corresponding subset of N plus one, the object associated with *Finite (n) is zero, as is the object associated with *(x = x), since neither are finite but the set of all ordinals less than or equal to Aleph one is also a model for the theory (that is for Frege Arithmetic) and in this case *Finite (n) is assigned Aleph zero while *(x = x) is assigned Aleph one. So we certainly cannot prove in Frege Arithmetic that *Finite (n) = *(x = x). But a worry still remains for as Boolos asked “is there such a number as anti-zero?” 12 That seems to me a very good point indeed. For that, if anything is, seems to be a substantial matter, which cannot be decided by stipulation. Indeed this issue seems to generalise as Boolos pointed out into a general issue about the compatibility of two conceptions of cardinal number. One derived from what one might call the pure theory of cardinal number based on Hume’s Principle and one derived from set theory as we shall see below. However it is possible to dissolve this worry and Wright has done so. He points out that it was a prime tenet of Frege’s view that numbers were the numbers of sortal concepts. Clearly the concept “self identical” is not a genuinely sortal concept so it is not an appropriate instance of the theorem above (**). Thus as Wright remarks in his reply to Boolos: Moral: just as not every object is suitable to determine a direction, so we should not assume without further ado that every concept—every entity an expression for which is an admissible substituted for the bound occurrences of the predicate letters in Hume’s Principle—is such as to determine a number. 11 Boolos (1987b), p. 197. Page reference to the reprint in Boolos (1998). 12 Boolos (1997), p. 314. Page reference to the reprint in Boolos (1998).

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He goes on to say: So self-identity is not a sortal concept. If we take it that, save where F is assured an empty extension on purely logical grounds, only sortal concepts, and concepts formed by restricting a mere predicable to a sortal concept, have cardinal numbers, it follows that there is no universal number. 13 So the function * is not to be regarded as having as its domain all concepts, rather there is to be natural and principled restriction on its domain of interpretation. It is to be confined to sortal concepts. However it is now unclear whether we can any longer regard Hume’s Principle as a purely logical abstraction principle since it is not clear that the notion of being a sortal concept can be expressed in purely logical vocabulary. But as Wright was quick to note if we do accept the need to restrict the domain of the function * to genuinely sortal concepts then a second difficulty looms and to some of us it looms very large.

3.

The good company objection

Let us follow Wright’s lead and consider the sortal concepts “set”, “ordinal” and “cardinal number”. According to Hume’s Principle these concepts too ought to have a number associated with them by (**). But then we immediately invite the observation that there is bound to be conflict with the notion of number as embodied in set theory, say ZFC. According to ZFC there are no numbers associated with the concepts “set”, “ordinal” and “cardinal number” precisely because the extensions of such concepts do not form sets and ZFC embodies the principle: no set no cardinal number. The collections we have been considering are proper classes, not sets, so there is no number associated by ZFC (or set theory) with them. Considerations such as these suggested to Boolos that there were two conflicting conceptions of number in play. He remarks: 14 Two thoughts about the concept of number are incompatible: that any zero or more things have a (cardinal) number, and that any zero or more things have a number (if and) only if they are the members of some one set. It is Russell’s paradox that shows the thoughts incompatible: the sets that are not members of themselves cannot be the members of any one set. The thought that any (zero or more) things have a number is Frege’s; the thought that things have a number only if they are the members of a set may be Cantor’s and is in any case a commonplace of the usual contemporary presentations of the set theory that originated with Cantor and has become ZFC.

In a similar vein he elaborated on the issue: 15 The worry is that the theory of number we have been considering, Frege Arithmetic, is incompatible with Zermelo–Fraenkel set theory plus standard 13 Wright (2000) “Is Hume’s principle Analytic?” in the Notre Dame Journal of Formal Logic.

14 Boolos (1995) “Frege’s Theorem and the Peano Postulates”, in Boolos (1998), p. 291. 15 Boolos (1997), p. 314. Page reference is to Boolos (1998).

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definitions, on the usual and natural readings of the non-logical expressions of both theories. To be sure, as Hodes once observed in conversation, if *a is taken to denote the cardinal number of a when a is a set and some favourite object that is not a cardinal number when a is a proper class, then HP will be a theorem of Von Neumann set theory. But on that definition of *, *will not be translatable as ‘the cardinal number of.’ ZF and Frege arithmetic make incompatible assertions concerning what cardinal numbers there are. And of course, the response ‘Well, these are just formalisms; the question of their truth or falsity doesn’t arise or makes no sense’ is hardly available to one claiming that HP is analytic, i.e., an analytic truth. So one who seriously believes that has to be bothered by the incompatibility of the consequence of Frege arithmetic that there is such a number as anti-zero with the claim made by ZF + standard definitions (on the natural reading of its primitives) that there is no such number.

Since to be in the company of set theory is to be in very good company indeed, let us call this objection to the stipulatory nature of Hume’s Principle, the Good Company objection. Now there is a clear response which can be made by someone who wishes to defend the idea that the truth of Hume’s principle can be simply stipulated and that is that the conflict alluded to above is illusory. 16 After all set theory assigns numbers to sets, but Frege arithmetic assigns numbers to concepts. Frege arithmetic assigns a cardinal number to the Russell concept “non-self membered set” but in virtue of the way cardinal numbers are introduced in ZFC, no cardinal number can be assigned to a nonset. (It is worth recalling that in ZFC if X is any set, then there is an ordinal number α and a bijection f: α →X. For any set X the cardinality of X is the least ordinal α such that there is a bijection f: α →X. A cardinal number is an ordinal number α such that for no β < α is there a bijection f: β → α.) The response to the Good Company objection in short then is Frege arithmetic assigns a number to proper classes, ZFC is silent. In any case certainly there is no conflict. But this dismissal of the good company objection is too swift. Let us go back to the first quotation from Boolos. He rightly points out that two conceptions of number are incompatible. The first conception is embodied in Hume’s Principle and says that every sortal concept has a number and the other says that every sortal concept has a number if and only if the extension of that concept is a set. Call these the Fregean and non-Fregean conceptions respectively. Clearly Russell’s paradox does indeed show that these two conceptions are incompatible. “Non-self membered set” is certainly a sortal concept and so has a number, by the Fregean conception. But it is provably the case that the concept “non-self membered set” has no set as its extension so on the second conception by the contrapositive of the only if clause there can be no number corresponding to the concept. Now imagine someone, we had better call him “Anti-Hero” or simply “Villain” (in this context) who believes this: the conception of number we have is revealed in mathematical practice and he holds to the non-Fregean concept of number. Is it conceivable that he holds an analytically false belief, 16 This objection was put to me by Stewart Shapiro and Fraser MacBride.

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or a belief that a stipulation may show to be false? Our grasp of number is what we do with it in mathematics. Now it is certainly true that the nonFregean concept of number outlined above is entirely adequate for the whole of mathematics since it realised by ZFC (though ZFC does not entail it). The extra content which Anti-Hero in fact believes over ZFC is the claim that the mathematical universe is exhausted by ZFC. That claim certainly seems believable. What it certainly does not seem to be is analytically false and it seems to have enormous evidence in favour of it. It may seem that Anti-Hero is merely legislating on the content of all possible mathematics and that can’t be right. But Anti-Hero doesn’t believe that his view is analytically true, what he certainly believes is that it is not analytically false however. Perhaps it was something like this that Boolos had in mind when he wrote of Anti-Hero’s view that: “It is in any case a commonplace of the usual contemporary presentations of the set theory that originated with Cantor and has become ZFC”. Clearly this matter must be connected with Boolos’s rejection of the notion of proper class. He took the view that they are in fact just “a manner of speaking” in the sense that they really play the role of abbreviations whose use can always be eliminated in any formal theory by replacing them by their defining formulas and this latter view I think is directly connected with his support for the iterative conception of set and the claim that ZFC exhausts that iterative hierarchy. Whatever may be the case about this it does seem to me that Boolos’s Good Company objection is a compelling one, for the set theoretic conception of number is a perfectly viable one and it surely cannot be rejected as analytically false. A possible response would be to say that there are two conceptions of cardinal number one for concepts and one for sets, but then what would have happened to the Fregean foundational programme that this was the correct ontological and epistemological account of the nature of and of our knowledge of the cardinal numbers? We would then seem to be left with an impenetrable problem about reference, we have an account of Frege numbers, we have an account of “set” numbers but neither is apparently related to each other or the numbers of ordinary arithmetic. That response would amount to abandoning the foundational programme and would hardly be acceptable to the neo-Fregean.

4.

Indefinitely extensible concepts again—proper classes

Perhaps it was some considerations such as these which lead Wright to endorse the principle at the heart of the non-Fregean concept of number as characterised above. He writes: 17 Grant the plausible principle that there is a determinate number of F’s just provided the F’s compose a set 17 Wright (2000), reprinted in Hale and Wright (2000), pp. 307–32. Reference is to p. 314.

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and later . . . it seems natural and well-motivated to suppose that the F’s should have a determinate cardinal number just when they compose a set

If this is the line to follow then what we are looking for is some further principled restriction on range of the quantifiers in Hume’s Principle over and above that of the restriction to sortal concepts. Wright proposes that the concepts falling under the range of the universal quantifiers in Hume’s principle should be restricted to those which are “definite”. Recall that a concept is definite just when it is not “indefinitely extensible”. One sense of indefinite extensibility goes back to Russell and Poincaré. Russell, in 1903 had thought that the contradiction derivable from Basic Law V of Frege’s Grundgesetze showed that not every property determines a class simpliciter. 18 The fundamental question as he saw it was then “to determine, which propositional functions define classes which are single terms as well as many, and which do not?”. By 1906, after reading Poincaré he had changed his view. He wrote “the contradictions result from the fact that, according to current logical assumptions, there are what we may call selfreproductive processes and classes. That is, there are some properties such that, given any class of terms all having such a property, we can always define a new term also having the property in question.” 19 In formulating the Vicious Circle Principle he made a very similar claim: “Thus all our contradictions have in common the assumption of a totality such that, if it were legitimate, it would at once be enlarged by new members defined in terms of itself”. 20 Both the Russell of this period and Poincaré would have agreed that the objects falling under such self-reproductive properties like ordinal or set form no totality. As is very well known Dummett has recently revived this idea. 21 As Wright remarks of the idea: 22 I do not, myself, know how best to sharpen this idea, still less how its best account might show that Dummett is right both to suggest that the proof-theory of quantification over indefinitely extensible totalities should be uniformly intuitionistic and that the fundamental classical mathematical domains, like those of the natural numbers, or the reals, should also be regarded as indefinitely extensible. But Dummett could be wrong about both those points and still be emphasizing an important insight concerning certain very large totalities—ordinal number, cardinal number, set, and indeed ‘absolutely everything’.

So the idea that Wright is proposing then is this I take it. We should restrict the range of the quantifiers in Hume’s principle to definite concepts, that is 18 Russell (1903). 19 Russell (1906), p. 144.

20 Russell (1908), p. 63. 21 Dummett (1991), (1963) and Clark (1998).

22 Wright (2000), p. 316 of Hale and Wright (2000).

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those that are not indefinitely extensible. But one has to be careful here for there are at least two conceptions of indefinite extensibility, if there are any, one which is very closely related to the Russell–Wright line of thought which sees indefinite extensibility associated with concepts generating or holding of very large totalities, with in effect, proper classes (in the presence of strong versions of the axiom of choice all the proper classes we have been considering are equinumerous with the Universe) and one which Dummett has developed in which both N and R are indefinitely extensible. Deploying Dummett’s conception will of course have essentially the same effect as deploying a suggestion due to Heck 23 to the effect that analytic core of Hume’s Principle is Finite Hume. This is the Principle that (∀F)(∀G)(Fin(F) & Fin(G) → (∗ F = ∗ G ↔ (F1 − 1R G)) where the notion of finite can be spelled out in purely logical vocabulary (as the negation of the second order sentence expressing that F is (Dedekind) infinite). This is of course not in the form of an abstraction principle and so is unacceptable as an implicit definition for the neo-Fregean. The closely related principle (∀F)(∀G)((∗ F = ∗ G ↔ Not Fin(F)V Not Fin(G) V (F1 − 1G)) although it is an abstraction principle (the right hand side being an equivalence relation) would not suffice either since it conflicts with the Cantorian conception of cardinal number by assigning the same number to all infinite concepts. However what would seem to capture Wright’s restriction is precisely the notion of set as opposed to proper class. I take it what we want to say is that N and R are definite while “set”, “ordinal” and “cardinal number” are not. Can such a notion of definite be made out? Well of course it can: the natural candidate is, a concept F is definite iff it has a set as its extension. This will do exactly what Wright wants. But it would make the understanding of Hume’s principle parasitic upon the notion of set and our grasp of the set theoretic Universe. It could hardly then be argued that the truth of Hume’s Principle was guaranteed by stipulation. We have already noted that in the presence of a very strong choice principle the concepts we wish to exclude from the domain of Hume’s Principle are equinumerous with the Universe. We could use this fact to independently give a justification of Wright’s New Hume which would have the form (∀F)(∀G)(∗ F = ∗ G ↔ (InDef (F) & InDef(G)) v (F1 − 1G)) where InDef (F) would be a condition stating that F is equinumerous with the Universe. 23 Heck (1997c).

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This would still be a logical abstraction principle, provided that the condition InDef (F) were expressed by the formula: (∃R)((∀x)(Fx → (∃!y)Rxy) & (∀y)(∃!x) (Fx & Rxy)) 24 However one proceeds here there looks to be a dilemma: either restrictions on the range of the quantifiers in Hume’s principle are actually motivated by set theoretic considerations or some notion of definiteness can really be made out, which itself does not rely upon set existence principles. These would serve as a principled restriction but in either case clearly something other than the notion of (sortal) concept and of second order logic is motivating our understanding of Hume’s Principle and thus of our knowledge of number. Whatever route is the correct one, it seems hardly possible to regard the laying down of such principles as guaranteeing their truth by stipulation.

5.

Frege and the application of arithmetic

It is certainly true that Frege put the application problem at the heart of his philosophy of mathematics. To the question: “why does arithmetic apply to reality?”, the logicist provides the clear answer because it applies to everything that can be thought. It is the most general science possible. The partial contextual definition, provided by Hume’s Principle and the fundamental thought that numerical concepts are second level concepts yield Frege’s account of the applicability of mathematics. In the simplest case for which the question arises—the application of the cardinal numbers—the solution is that arithmetic is applicable to reality because the concepts, under which things fall, themselves fall under numerical concepts. Thus it is possible to prove in second order logic that ∃n xFx − F falls under the numerical property expressed by the numerically definite quantifier ∃n x if and only if the Frege numeral introduced by the partial contextual definition (Hume’s Principle) is indeed n. In other words the theorem that ∃n xFx ≡ n = NxFx can be obtained, from Hume’s Principle in second order logic. But there is a real difficulty with Frege’s solution to the application problem and that is that we are provided by Hume’s principle with at best a partial contextual definition. The principle cannot settle the truth conditions of sentences of the form q = NxFx where q is not given in the form NxHx for some H, this of course is the famous Julius Caesar problem. In the case of pure arithmetic the Julius Caesar problem can 24 Boolos suggested that we might regard the principle.

(∀F)(∀G)(EXT(F) = EXT(G) ↔ (InDef(F) & InDef(G)) v (∀x) (Fx ↔ Gx)) as a repair of Basic Law Five. This might well then be used as a justification of set theory based upon the Principle of the Limitation of size. But like that justification however the power set axiom might well prove a difficulty when taken together with the separation schema. Knowing that for example N is not “too big” does not help us with the claim that P(N) is not too big. Similarly knowing that X is definite might tell us very little about or nothing at all about the definiteness of P(X). The effect of introducing New Five has been very carefully studied by Shapiro and Weir (see Shapiro and Weir (1999)).

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properly be regarded as irrelevant, since there would be no other singular terms in the language but once the language is extended to include empirical singular terms, as it would in the language of applied arithmetic, Hume’s Principle will no longer settle the sense of numerical identities and the solution to the application problem will fail. Of course this issue does not arise in the formal language of the Grundgesetze, which is a language of pure arithmetic, since in that language all the objects it is possible to refer to are already given as extensions (value ranges) the identity conditions for which are given by Basic Law Five. But as Michael potter has put it: 25 “a formal language in which Julius Caesar cannot be spoken of is one in which he cannot be counted, and in such a language the applicability of arithmetic remains unexplained. At some stage in the development we shall have to extend the formal language by adding some empirical vocabulary, and we shall then have to address the Julius Caesar problem just as before.” The question naturally arises as to whether the neo-logicist fares any better than Frege with respect to the application problem. This seems very unlikely since the neo-logicist 26 relies exclusively upon Hume’s Principle and therein lays the real difficulty, as Frege long ago knew. In the same letter which I quoted at the beginning of this paper he wrote to Russell about the idea of letting his programme rest on Hume’s Principle alone the following: 27 We can also try the following expedient, and I hinted at this in my Foundations of Arithmetic. If we have a relation (ξ, ζ ) for which the following propositions hold (i) from (a, b) we can infer (b, a) and (2) from (a, b) and (b, c) we can infer ; then this relation can be transformed into an equality (identity), and can be replaced by writing, e.g., “§a = §b”. If the relation is, e.g., that of geometrical similarity, then “a is similar to b” can be replaced by saying “the shape of a is the same as the shape of b”. This is perhaps what you call “definition by abstraction”. But the difficulties here are not the same as in transforming the generality of an identity into an identity of range of values.

References Boolos, G. (1987a), “Saving Frege from Contradiction” Proceedings of the Aristotelian Society 87 (1987), pp. 137–51. Boolos, G. (1987b), “The Consistency of Frege’s Foundations of Arithmetic” in Thomson (1987), pp. 3–20. Boolos, G. (1990a), “The Standard of Equality of Numbers” in Boolos (1990b), pp. 261–77. Boolos, G. (Ed.) (1990b), Meaning and Method: Essays in Honor of Hilary Putnam (Cambridge: Cambridge University Press, 1990). 25 Potter (2000), p. 108. 26 Hale and Wright treat the Julius Caeser problem very seriously indeed, they say of it that it is “one

of the hardest the neo-Fregean must solve” (Hale and Wright 2000, pp. 14–16). They devote the whole of essay 14 of their (2000), (pp. 335–396) to the topic. 27 Frege (1902), p. 141.

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Boolos, G. (1993), “Whence the Contradiction?” Proceedings of the Aristotelian Society, Supp. Vol 67 (1993), pp. 213–33. Boolos, G. (1997), “Is Hume’s Principle Analytic?”, in Heck (1997a), pp. 245–62. Boolos, G. (1998), Logic, Logic and Logic (Cambridge MA: Harvard University Press, 1998). Boolos, G. & Heck, R. (1998), “Die Grundlagen der Arithmetik §§82–3”, in Schirn (1998), pp. 407–28. Brandl, J. & Sullivan, P. (Eds.) (1998), New Essays on the Philosophy of Michael Dummett (Vienna: Rodopi, 1998). Burgess, J. P. (1984), Review of Wright [1983], Philosophical Review 93, 1984, pp. 638–40. Clark, P. (1998), “Dummett’s Argument for the Indefinite Extensibility of Set and Real Number” in Brandl & Sullivan (1998), pp. 51–63. Demopoulos, W. (Ed.) (1995), Frege’s Philosophy of Mathematics (Cambridge: Harvard University Press, 1995). Demopoulos, W. (1998), “The Philosophical Basis of our Knowledge of Number”, Noûs 32 (1998), pp. 481–503. Demopulos, W. (2000), “On the origin and Status of Our Conception of Number” Notre Dame Journal of Formal Logic, 41 (2000), pp. 210–26. Dummett, M. (1963), “The Philosophical Significance of Gödel’s theorem”, Ratio 5 (1963), pp. 140–55. Dummett, M. (1991), Frege: Philosophy of Mathematics (London: Duckworth, 1991). Fine, K. (1998), “The Limits of Abstraction” in Schirn (1998), pp. 503–629. Frege, G. (1879), Begriffsschrift (Halle: L. Nebert, 1879). Frege, G. (1884), Die Grundlagen der Arithmetik (Breslau: W. Koebner, 1884); reprinted with English translation by J. L. Austin as The Foundations of Arithmetic (Oxford: Blackwell, 1950). Frege, G. (1893), Die Grundgesetze der Arithmetik vol. 1 (Jena: H. Pohle, 1893), part translated into English by Montgomery Furth in The Basic Laws of Arithmetic (Berkeley: University of California Press, 1964). Frege, G. (1902), Letter XV7 [xxxvi/7] in Frege (1980), pp. 139–42. Frege, G. (1980), Philosophical and Mathematical Correspondence, ed. G. Gabriel, 1980. Hale, R. and Wright, C. (2000), The Reason’s Proper Study (Oxford: Oxford University Press, 2001). Heck, R. Jr. (1993),“The Development of Arithmetic in Frege’s Grundgesetze der Arithmetik”, Journal of Symbolic Logic 58 (1993), pp. 579–601. Heck, R. Jr. (1995), “Frege’s principle” in J. Hintikka (ed.), From Dedekind to Godel, 1995, pp. 119–42. Heck, R. Jr. (1997a), Language, Thought and Logic (Oxford: Oxford University Press, 1997). Heck, R. Jr. (1997b), “The Julius Caesar Objection”, in Heck (1997a), pp. 273–308 (a). Heck, R. Jr. (1997c), “Finitude and Hume’s Principle”, Journal of Philosophical Logic 26, (1997), pp. 589–617. Hodes, H. (1984), “Logicism and the Ontological Commitments of Arithmetic”, Journal of Philosophy 81 (1984), pp. 123–49. Parsons, C. (1965), “Frege’s Theory of Number” in Mathematics to Philosophy, pp. 150–75. Potter, M. (2000), Reason’s Nearest Kin (Oxford: Oxford University Press, 2000). Russell, B. (1903), The Principles of Mathematics (London, George Allen and Unwin, 1903). Russell, B. (1906), “On some difficulties in the theory of transfinite numbers and order types”, reprinted in D. Lackey ed. Bertrand Russell Essays in Analysis (London, George Allen and Unwin, 1973), pp. 135–64. Russell, B. (1908), “Mathematical logic as based on the theory of types”, reprinted in R. C. Marsh ed. Logic and Knowledge (London, George Allen and Unwin, 1956), pp. 59–102. Schirn, M. (1998), Philosophy of Mathematics Today (Oxford: Clarendon Press, 1998). Shapiro, S. and Weir, A. (1999), “New V, ZF and Abstraction”, Philosophia Mathematica 1999, pp. 293–321. Shapiro, S. and Weir, A. (2000), “Neo-logicist Logic is not epistemically innocent” Philosophia Mathematica, pp. 160–89. Smiley, T. (1981), “Frege and Russell”, Epistemologica, 1981, pp. 51–6.

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Wright, C. (1983), Frege’s Conception of Numbers as Objects (Aberdeen: Aberdeen University Press, 1983). Wright, C. (1997), “The Philosophical Significance of Frege’s Theorem” in Heck (1997), pp. 201–45. Wright, C. (1998a), “On the Harmless Impredicativity of N=”, in Schirn (1998), pp. 339–68. Wright, C. (1998b), “Response to Dummett” in Schirn (1998), pp. 389–406. Wright, C. (2000), “Is Hume’s Principle Analytic”, Notre Dame Journal of Formal Logic, 40 (2000), pp. 6.

FINITUDE AND HUME’S PRINCIPLE 1 Richard G. Heck, Jr Brown University, Providence RI, U.S.A. E-mail: [email protected]

Abstract The paper formulates and proves a strengthening of ‘Frege’s The-

orem’, which states that axioms for second-order arithmetic are derivable in second-order logic from Hume’s Principle, which itself says that the number of Fs is the same as the number of Gs just in case the Fs and Gs are equinumerous. The improvement consists in restricting this claim to finite concepts, so that nothing is claimed about the circumstances under which infinite concepts have the same number. ‘Finite Hume’s Principle’ also suffices for the derivation of axioms for arithmetic and, indeed, is equivalent to a version of them, in the presence of Frege’s definitions of the primitive expressions of the language of arithmetic. The philosophical significance of this result is also discussed.

1.

Opening

In recent work, 2 George Boolos has, with an eye towards philosophical issues I shall discuss in Section 3, investigated the relative strengths of two sorts of systems of second-order arithmetic. The more familiar of these originates with the work of Dedekind and Peano; the less familiar, with that of Frege. Dedekind–Peano systems characterize the natural numbers in terms of properties of the sequence of natural numbers; these systems may be thought of as axiomatizations of finite ordinal arithmetic. The Fregean systems, on the other hand, characterize the natural numbers as finite cardinals. 3 Fundamental to such systems is an axiom specifying the condition under 1 This paper first appeared in the Journal of Philosophical Logic 26, [1997], pp. 589–61. Reprinted by kind permission of the editor and Springer Academic Publishers. 2 G. Boolos, “On the Proof of Frege’s Theorem”, in A. Morton and S. Stich, eds. Benacerraf and His Critics (Oxford: Blackwells, 1996), pp. 143–59. 3 For further discussion of this difference, see my “The Finite and the Infinite in Frege’s Grundgesetze der Arithmetik”, in M. Schirn, ed., Philosophy of Mathematics Today (Oxford: Oxford University Press, 1998), §5, pp. 429–66.

61 Roy T. Cook (ed.), The Arché Papers on the Mathematics of Abstraction, 61–84. c 2007 Springer.

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which two concepts 4 have the same cardinal number, together with another specifying under what conditions a cardinal number is finite. What is perhaps the most familiar (second-order) Dedekind–Peano system is axiomatized as follows: 1. 2. 3. 4. 5. 6. 7.

N0 Nx & Pxy → Ny ∀x∀y∀z(Nx & Pxy & Pxz → y = z) ∀x∀y∀z(Nx & Ny & Pxz & Pyz → x = y) ¬∃x(Nx & Px0) ∀x(Nx → ∃y Pxy) ∀F[F0 & ∀x∀y(Fx & Pxy → Fy) → ∀x(Nx → Fx)]

Let us call this system PA2 (for second-order Peano arithmetic). I have here formulated its axioms using a relational expression ‘Pξ η’, rather than the more usual functional expression ‘Sξ ’, to facilitate comparison with Fregean systems. The most familiar Fregean system has but one ‘non-logical’ axiom, Hume’s Principle, which states that the number of Fs is the same as the number of Gs just in case the Fs and Gs are in one–one correspondence. Taking ‘Eqx (Fx;Gx)’ to abbreviate one of the (many equivalent) second-order formulae which define ‘the Fs correspond one–one with the Gs’ (or, in Frege’s terminology, ‘the Fs are equinumerous with the Gs’), Hume’s Principle (HP) is then: Nx : Fx = Nx : Gx ≡ Eqx (F x, Gx) The second-order theory whose sole non-logical axiom is HP is FA (for ‘Frege arithmetic’). Note that ‘Nx : x’ is a unary, second-level, term-forming operator: The result of substituting any formula (possibly containing further occurrences of ‘Nx : x’) for ‘x’ in ‘Nx : x’ is a term. The definition of finite or natural number can be given in different ways. In Frege’s work, 5 zero and the relation of predecession are defined and, famously, the finite numbers are defined as those to which zero stands in the weak ancestral of this relation. The necessary definitions are thus: 0 = Nx : x = x Pmn ≡ ∃F∃y[Fy & n = Nx : Fx & m = Nx : (Fx & x = y)] Frege defines the strong ancestral of a relation Rξ η as follows: R ∗ ab ≡ ∀F[∀z(Raz → Fz) & ∀x∀y(Fx & Rxy → Fy) → Fb] 4 I shall use this term to denote whatever are in the range of the second-order variables. Though my choice of terminology certainly suggests a view about what these are, my remarks here do not depend upon it. It is, of course, essential to the logicist project that second-order logic is logic, but this is not at issue among those whose positions we shall be discussing. 5 See, of course, G. Frege, The Foundations of Arithmetic, 2nd. ed., trans. by J. L. Austin (Evanston, IL: Northwestern University Press, 1953), §§74, 76, 83.

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And he defines the weak ancestral of Rξ η thus: R ∗= ab ≡ R∗ ab ∨ a = b Frege’s definition of natural number is then: n is a natural number just in case P ∗= 0n. There are other ways to proceed, however. In sections K and of Grundgesetze der Arithmetik, Frege formulates a purely second-order definition of finitude, to state which we need an additional definition: 6 Btwx y (Rxy; a; b)(n) ≡ ∀x∀y∀z(Rxy & Rxz → y = z) & ¬R∗ bb & R ∗= an & R ∗= nb Thus, n is between a and b in the R-series if, and only if, Rξ η is a functional relation, in whose strong ancestral b does not stand to itself (i.e., there is no ‘loop’ from b to b), such that a stands in the weak ancestral of Rξ η to n, which in turn stands in the weak ancestral of Rξ η to b. Frege’s definition of finitude is then: 7 Finitex (Fx) ≡ ∃R∃x∃y∀z[Fz ≡ Btw(R; x; y)(z)] That is: A concept is finite just in case the objects falling under it may be ordered in a certain way, namely, as the objects between x and y in the Rseries, for some R, x and y. That this definition is correct follows from the central theorems of sections K and of Grundgesetze der Arithmetik, which are Theorems 327 and 348 of Grundgesetze: (327) Finite(F) → P ∗= (0, Nx : Fx) (348)P ∗= (0, Nx : Fx) → Finite(F) Thus, a concept is finite, in Frege’s sense, just in case its number is a natural number. Frege’s definition of natural number could, therefore, be replaced by: N(n) ≡ ∃F[Finite(F) & n = Nx : Fx] Of course, this definition will be adequate only in a theory strong enough to prove Theorems 327 and 348. 8 Analogues of these theorems are the crucial lemmas in the proofs of the main result of this paper (see Lemmas 3.1, 3.11, and 3.21). As we shall see, given Frege’s definitions of ‘0’ and ‘Pξ η’, Theorem 327 becomes a theorem of second-order logic. The proof of Theorem 348, however, must rely upon additional assumptions, for without additional assumptions, it is consistent that 6 G. Frege, Grundgesetze der Arithmetik (Hildesheim: Georg Olms Verlandsbuchhandlung, 1966). The definition is given in §158 of volume I. I shall insert the bound variables, such as ‘x’ and ‘y’ on the left-hand side here, into the definitions, but will drop them when doing so causes no confusion. 7 Frege does not explicitly formulate any such definition, but it is clear from the theorems proven in sections K and that this is what he intends. For further discussion, see my “The Finite and the Infinite”, op. cit. 8 Frege proves it in the system FA + FD. As we shall see, it is also provable in FAF + FD (and so in PAF + FD).

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all concepts have the number 0 (and, of course, it is consistent that some of these are not finite). In investigating the relative strengths of Dedekind–Peano and Fregean systems, there are two sorts of questions one might raise. First, one might inquire about the relative consistency of such theories. To ask whether FA is consistent relative to PA2 is to ask whether the consistency of FA would follow from that of PA2. One familiar sort of proof that it would consists in a demonstration that FA can be relatively interpreted in PA2. Roughly speaking, to interpret FA in PA2 is to give definitions of the primitives of FA in terms of the primitives of PA2, which definitions, when added to PA2, allow one to prove relativizations of the axioms of FA in PA2: By a relativization of a formula is meant, as usual, the result of restricting quantifiers occurring in the formula by means of some formula of PA2. 9 If FA can be interpreted in PA2, it follows immediately that, if there is a proof of a contradiction in FA, that proof can be mimicked in PA2, so that, if PA2 is (syntactically) consistent, so is FA. As it turns out, FA and PA2 are equi-interpretable—each can be interpreted in the other—and so equi-consistent—an inconsistency in either would imply an inconsistency in the other. Still, one might wonder whether FA is not, in some other sense, a stronger theory than PA2. This question is more easily understood when we have two theories formulated in the same language. Consider, for example, the following Dedekind–Peano system, which we shall call PAS (for ‘Strong’ Peano arithmetic): 1. 2. 3. 4. 5. 6. 7.

N0 Nx & Pxy →Ny ∀x∀y∀z(Pxy & Pxz → y = z) ∀x∀y∀z(Pxz & Pyz → x = y) ¬∃xPx0 ∀x(Nx → ∃yPxy) ∀F[F0 & ∀x∀y(Fx & Pxy → Fy) → ∀x(Nx→ Fx)]

Clearly, every axiom of PA2 is a theorem of PAS, but the converse does not hold. As far as the axioms of PA2 are concerned, zero could have as its predecessor Julius Caesar, so long as Caesar is not a natural number. Thus, PAS is strictly stronger than PA2. This is perfectly compatible with the fact that PA2 and PAS are equi-interpretable. (To interpret PAS in PA2, no ‘definitions’ are needed: Just restrict all the quantifiers in the axioms of PAS to the natural numbers, i.e., by the formula ‘Nx’.) 9 Of course, in the context of second-order logic, one must restrict not only the first-order, but also the second-order, quantifiers. As we replace ‘∀x A(x)’ by ‘∀x[R(x) → A(x)]’, so we replace ‘∀F A(F)’ by ‘∀F{∀x[Fx → R(x)] → A(F)}’.

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The question under discussion here concerns the proof-theoretic strength of the two systems. This question is harder to raise when the theories under discussion are not formulated in the same language: Obviously, the axioms of FA are not going to be theorems of PA2, since the axioms of FA are not even sentences in the language of PA2. Nor would expanding the language of PA2 to include such formulae help. To consider the relative strength of theories formulated in different languages, what we require is a bridge theory which relates (the referents of) the primitives of PA2 to those of FA. We can then ask whether, with the aid of one or another bridge theory, the theorems of FA can be proven in PA2. 10 One might wonder what difference there is between the question whether FA can be relatively interpreted in PA2, and the question whether the theorems of FA can be proven in PA2, with the aid of some bridge theory. For, one might ask, if FA can be relatively interpreted in PA2, will that not itself guarantee that there is some bridge theory with the aid of which the theorems of FA can be proven in PA2? namely, that theory whose axioms are exactly the definitions used in interpreting FA in PA2? The answer to this question is “No”. One must not overlook the fact that, in relatively interpreting one theory in another, it may be essential to relativize the axioms of the former theory: The usual relative interpretation of FA in PA2, for example, requires that the quantifiers occurring in Hume’s Principle be restricted to the natural numbers. There is no necessity that there should be a way of mimicking this restriction in any bridge theory, and there is certainly no need that any particular bridge theory should impose such a restriction. Our chief interest here is in the relative strength of various Fregean systems and various Dedekind–Peano systems. We thus must make use of a bridge theory which relates (the referents of) their primitives. The bridge theory in which we shall be interested is that which has the following three axioms: 0 = Nx : x = x Pmn ≡ ∃F∃y[Fy & n = Nx : Fx & m = Nx : (Fx & x = y)] Nn ≡ P ∗= 0n This theory we shall call FD—for ‘Frege’s definitions’, since these are the definitions Frege uses in deriving axioms for arithmetic (in particular, those of PAS) in FA. 10 One might well wonder what such a bridge theory must be like, if the provability of the theorems

of one system from those of another is to have the kind of interest it is here taken to have. I do not know how this question should be answered. Surely, however, it is sufficient if the axioms of the bridge theory are definitions of the primitives of one of the two theories in terms of the primitives of the other. The bridge theories we shall employ below are of this sort. The question we are considering is thus one of interpretability, rather than relative interpretability.

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The systems

We will here investigate the relative strengths of five different systems of arithmetic. The Dedekind–Peano systems at which we shall look are PA2 and PAS, mentioned above, and a third system, PAF, whose axioms are those of PA2 plus: ∀x∀y(Nx & Pyx → Ny) This axiom, which we also call PAF (for ‘predecessors are finite’), states that any predecessor of a natural number is a natural number. 11 As we shall see, PAF is stronger than PA2 and weaker than PAS. Before discussing the variations on FA at which we shall look, let me make a remark about the background logic in which we shall be working. In the case of the Dedekind–Peano systems, the logic is usually taken to be standard (axiomatic) second-order logic, with full, impredicative comprehension. In discussing FA and its relations to the Dedekind–Peano systems, however, it is convenient to take the logic also to contain the axiom Boolos calls FE, for ‘functional equivalence’: ∀x(Fx ≡ Gx) → Nx : Fx = Nx : Gx This axiom is clearly valid on any extensional semantics for second-order logic and so should itself be regarded as a truth of (extensional, higher-order) logic. 12 The system whose axioms are those of second-order logic, plus FE, Boolos calls Log. We shall suppose our background logic, throughout, to be Log. The Fregean systems at which we shall look are FA and a variation on it, in which Hume’s Principle has been weakened by restricting its range of application. The axiom is HPF (for Finite Hume’s Principle): Finite(F) ∨ Finite(G) → [Nx : Fx = Nx : Gx ≡ Eqx (Fx; Gx)] Here, the formula ‘Finite(F)’ may be defined via any of the equivalent secondorder definitions of finitude: We shall take it to be defined as Frege defines it. HPF states that finite concepts have the same number if, and only if, they are equinumerous and that no infinite concept has the same number as any finite one—making no further claim about the conditions under which infinite concepts have the same number. (For all that HPF says, all infinite concepts could have the same number, so long as no finite concept also has that number.) Call the theory whose sole non-logical axiom is HPF, FAF (for finite Frege arithmetic). 11 G. Boolos remarked to me that, when presenting “On the Proof of Frege’s Theorem”, he has heard it objected that PAF—or, more precisely, its consequence NPZ, to be mentioned below—cannot be true, since −1 surely precedes 0. But the theories in which we are interested here are theories of cardinal or ordinal numbers, and ‘Pξ η’ is defined as a relation between such numbers. Negative numbers are neither ordinals nor cardinals. 12 As should the axiom schema: ∀x(Fx ≡ Gx) → A(F) ≡ A(G).

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The results of this paper may now be summarized in the following diagram: FA ⇒ PAS ⇒ {PAF (⇒ FAF)} ⇒ PA2 Here, ‘⇒’ means: Is strictly stronger than, relative to the bridge theory FD; that is, ‘A ⇒ B’ means that every theorem of B is a theorem of A + FD, but that not every theorem of A is a theorem of B + FD. That FAF and PAF occur together in the braces indicates that they are equivalent, relative to FD: Every theorem of PAF is a theorem of HPF + FD, and every theorem of FAF is a theorem of PAF + FD. What we need to prove are thus the following: 1. 2. 3. 4.

FA ⇒ PAS PAS ⇒ PAF PAF is equivalent to FAF (modulo FD) PAF ⇒ PA2

Some of the required proofs have been discussed in detail by Boolos: I shall merely indicate how those proofs go. The main work of the present paper consists in establishing Theorem 3.

3.

On the philosophical significance of these results

Before turning to the proofs, let me make a couple of remarks about the inspiration for the present investigation and about its philosophical implications. In Frege’s Conception of Numbers as Objects, Crispin Wright rediscovered Frege’s Theorem, which states that the axioms of PAS are provable in FA + FD, proved it in some detail, and conjectured that FA is consistent, which it turned out to be. 13 On the basis of this result, Wright not only revived Frege’s logicist project, but claimed that it was substantially vindicated by the proof of Frege’s Theorem. If logicism were to be vindicated completely, of course, HP would have to be shown to be a logical truth, which it certainly cannot be, given our contemporary understanding of ‘logical truth’. Nevertheless, Wright argued, HP is ‘analytic’, whence the truths of arithmetic are logical consequences of an analytic truth and so, presumably, are themselves analytic. The sense in which HP is analytic is that, “even if inadequate as a definition, it nevertheless succeeds as an explanation; . . . it contrives to fix the meaning of the sorts of occurrence of [‘Nx : x’] which it fails to eliminate”. 14 In the paper mentioned at the outset, Boolos shows that FA is strictly stronger than PAS (and so PA2), relative to the bridge theory FD. His purpose is not primarily technical: He intends this to be one consideration in favor 13 C. Wright, Frege’s Conception of Numbers as Objects (Aberdeen: Aberdeen University Press, 1983). See especially Ch. 4. The consistency of the system was noted by Burgess, Hazen, and Hodes. Boolos later showed that FA and PA2 are equiinterpretable. For a proof, see Boolos and Heck, Jr., “Die Grundgesetze der Arithmetik §§82–3”, in Schirn, ed., pp. 407–28. 14 Wright, Frege’s Conception, p. 140. See also the statement of Number-theoretic Logicism (III) on p. 153.

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of the view that, contra Wright, HP is neither ‘analytic’, nor a ‘conceptual truth’, nor any such thing. 15 Boolos does not explain in detail why his result should trouble Wright, but his point seems to be that, since FA is significantly stronger than PAS (which is itself a stronger theory even than PA2, which is itself a very strong theory), it is implausible to claim that HP is a conceptual truth. It is difficult, however, to evaluate the force of this consideration: As Boolos recognizes, Wright would likely reply that since his view is the view that arithmetic is analytic—and, indeed, that the general theory of cardinality which FA embodies is analytic—he is simply being accused of holding that very view. Still, there is a stronger consideration in the vicinity. For the additional proof-theoretic strength of FA, as compared to PA2, reflects a very real, and very large, conceptual gap between second-order arithmetic and the general theory of cardinality. W. W. Tait has pointed out that ‘Hume’s Principle’ is something of a misnomer: In the passage Frege cites when introducing it, Hume is speaking not of cardinality in general but only of the cardinality of finite concepts (or sets, or whatever). 16 As of course he was. Prior to Cantor’s work on transfinite numbers, the view that all equinumerous concepts have the same number, whether they are finite or infinite, was almost universally rejected, because it gives rise to antinomies: For example, it implies that the number of natural numbers is the same as the number of even numbers, and that can seem absurd, because there are lots of natural numbers which are not even— indeed, according to Cantor, as many numbers as there are natural numbers. Cantor’s realization that one can coherently suppose, even in the infinite case, that all and only equinumerous sets have the same cardinality constituted as enormous a conceptual advance as his introduction of transfinite numbers was a mathematical advance. It is easy to forget this, so at home are we initiates with Cantor’s ideas. But it is just as easy to be reminded of it: One has an opportunity every time a student wanders into one’s office puzzled about these very antinomies. Indeed, my own work on this very paper was fundamentally altered by just such an experience. A friend of mine—a professional philosopher, and so no fool— was telling me about an objection one of his students had raised in lecture. The student had insisted that there is only one ‘kind’ of infinity, and my friend had been tempted to reply (but wanted to check with me first) that of course there was more than one kind of infinity, since both the natural numbers and the even numbers are infinite, and the infinities in question certainly cannot be of the same kind. He was troubled by my response. Not just philosophically troubled, mind you, but really bothered: As I conveyed Cantor’s ideas to him, 15 See G. Boolos, “Is Hume’s Principle Analytic?”, in R. Heck, ed., Language, Thought and Logic: Essays in Honor of Michael Dummett (Oxford: Oxford University Press, 1997), pp. 245–62. 16 Private communication. The term ‘Hume’s Principle’ should not confuse, however: It came into common use not because anyone thought Hume scooped Cantor, but because Frege introduced HP in §63 of Grundlagen by quoting from Treatise I iii 1.

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he kept saying, “That’s very worrying”, over and over again. I had to do a lot of explaining before he was again at ease. He made the leap, but my experience served to remind me how great a conceptual leap he made at that point—and so how great a conceptual leap Cantor himself had made. I am not going to argue that HP isn’t a conceptual truth: On that question I regard myself, to steal a phrase of David Wiggins’s, as a militant agnostic. My point is that, once one has recognized just how great a conceptual advance is required if one is to acknowledge the truth of Hume’s Principle, one can no longer accept that Frege’s Theorem has the sort of epistemological interest Wright and others have wanted it to have. What is required if logicism is to be vindicated is not just that there is some conceptual truth or other from which what look like axioms for arithmetic follow, given certain definitions: That would not show that the truths of arithmetic, as we ordinarily understand them, are analytic, but only that arithmetic can be interpreted in some analytically true theory. 17 To put the point differently, if we are so much as to evaluate logicism, we must first uncover the ‘basic laws of arithmetic’, laws which are not just sufficient to allow us to prove translations of arithmetical truths, but laws from which arithmetical truths themselves can be proven. (The distinction is not a mathematical one, but a philosophical one.) But, if these ‘basic laws’ are to be the basic laws of arithmetic, they had better be ones upon which ordinary arithmetical reasoning relies. If Frege’s Theorem is to have the kind of interest Wright suggests, it must be possible to recognize the truth of HP by reflecting on fundamental features of arithmetical reasoning—by which I mean reasoning about, and with, finite numbers, since the epistemological status of arithmetic is what is at issue. For what the logicist must establish is something like this: That there is, implicit in the most basic features of arithmetical thought, a commitment to certain principles, the (tacit) recognition of whose truth is a necessary precondition of arithmetical reasoning, and from which all axioms of arithmetic follow. Having identified these basic laws, we will then be in a position to discuss the question whether they are analytic, or conceptual truths, or what have you. What used to be my favorite argument for the analyticity of HP went roughly like this: HP is a conceptual truth, because it is part of the very concept of cardinality that equinumerous concepts have the same cardinal number. 18 Perhaps, but the argument overlooks the fact that, though this may be true of our present concept of cardinality, ‘we’ did not even have this concept of cardinality until about a 120 years ago. A recognition of the very coherence 17 If analysis were analytic, as Frege thought it was, then Euclidean geometry would be interpretable in an analytically true theory, via Cartesian co-ordinates. Are we to conclude that Frege’s position was inconsistent, since he held that geometry is not analytic, but synthetic a priori? Surely not. 18 Compare C. Wright, “On the Philosophical Significance of Frege’s Theorem”, in Heck, ed, pp. 201– 44. A somewhat different version of this claim is that, even if HP is not analytic of any preexisting concept of cardinality, it is perfectly in order to introduce such a concept by means of HP. This version of the claim becomes important in certain contexts.

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of our present concept of cardinality requires the conceptual leap I discussed above, whence, even if HP is analytic of our present concept of cardinality, it is extremely odd to attempt to ground our knowledge of arithmetic, of all things, upon it. Moreover, there is demonstrably no way in which a recognition of the truth of HP can arise simply from reflection on the nature of ordinary arithmetical thought—not, that is, if the principles governing ‘ordinary arithmetical thought’ are captured by the axioms of PA2 (or even of PAS) and the outcome of ‘reflection’ is something that could be written down as a proof. That is what follows from the fact that HP is proof theoretically stronger than PAS (and so PAF and PA2). The disparity of strength parallels the conceptual disparity remarkably well—so well as to remind one why well-conceived technical investigations can be so philosophically fruitful. To summarize and emphasize: HP, conceptual truth or not, cannot be what underlies our knowledge of arithmetic. For no amount of reflection on the nature of arithmetical thought could ever convince one of HP, nor even of the coherence of the concept of cardinality of which it is purportedly analytic. Granted, any rationalist project of this sort will have to invoke a distinction between the ‘order of discovery’ and the ‘order of justification’. But the objection is not that Hume’s Principle is not known by ordinary speakers, nor that there was a time when the truths of arithmetic were known, but HP was not. It is that, even if HP is thought of as ‘defining’ or ‘introducing’ or ‘explaining’ our present concept of cardinality, the conceptual resources required if one is so much as to recognize the coherence of this concept (let alone HP’s truth) vastly outstrip the conceptual resources employed in arithmetical reasoning. Wright’s version of logicism is therefore untenable. Of course, this does not imply that no form of logicism is defensible. And careful examination of Boolos’s proofs itself reveals a way forward. The important observation is that the distinction between finitude and infinitude plays a major role in these proofs. Consider, for example, the sort of model Boolos uses to show that FA is stronger than PAS. Take the domain of the model to be the natural numbers, together with Caesar and Brutus. Given any term of the form ‘Nx : Fx’, assign it a value according to the following scheme: Caesar, if there are infinitely many Fs and infinitely many non-Fs Brutus, if there are infinitely many Fs, but only finitely many non-Fs n, if there are exactly n Fs, for some natural number n Interpret the primitives of PAS according to the ‘definitions’ of the bridge theory FD (thus guaranteeing that all axioms of the bridge theory are true in the model): Thus, ‘0’ denotes the number 0; ‘Nξ ’ is true of xiff x is a natural number; and ‘Pξ η’ is true of the pair < x, y > i just in case either x = y = Caesar, or x = y = Brutus, or y = x + 1. It should be clear that the axioms of PAS are all true in this model. But HP is not: For example, ‘Even(ξ )’ having

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been appropriately defined, ‘Nx : [Nx & Even(x)] = Nx : Nx’ will be false in the model—the former term denoting Caesar, the latter, Brutus—even though the evens are equinumerous with the natural numbers. The important point is that, in this model, Hume’s Principle fails only for infinite concepts. Indeed, as Boolos essentially observes, HPF holds in every model of PAS + FD: So, in any model for PAS + FD, Hume’s Principle will fail to hold, if it does, only because there are some equinumerous infinite concepts which are assigned different numbers. The natural technical question is then: Is there a reasonable Dedekind–Peano system which, in the presence of FD, is equivalent to FAF? The answer is that there is: Relative to FD, FAF is equivalent to PAF. Now, in Grundgesetze, Frege actually derives the axioms of PAS in FA + FD, and these proofs do exploit the full power of HP (since the axioms of PAS are not provable in FAF + FD). But Frege’s proofs can easily be adapted to yield proofs, in FAF + FD, of the axioms of PAF: One need only relativize certain of the formulae appearing in those proofs to the natural numbers. Frege’s development of arithmetic thus does not depend essentially upon (though it may have been psychologically impossible without) the conceptual advance of which I have been speaking. This is striking enough, but it is all the more so since my objections to Wright’s attempt to ground arithmetic on Hume’s Principle simply cannot be raised against an attempt to ground it on HPF. For HPF’s weakness, as compared to HP, reflects the conceptual distance between them, too. There are two points to be made here: First, that recognizing the truth of HPF does not require making the conceptual advance made by Cantor; and, secondly, that one can be convinced of the truth of HPF merely by reflection on ordinary arithmetical thought. To take the first point: Just as HP may be thought of as the sole axiom of a general theory of cardinal numbers, HPF may be thought of as the sole axiom of a theory of finite cardinals. And since HPF makes no claims whatsoever about the conditions under which infinite concepts have the same cardinality, 19 it will not give rise to any of the antinomies generated by HP, whence one does not need to make Cantor’s leap before one can accept the truth of HPF. Indeed, not only could HPF have been recognized as true prior to Cantor’s work, it almost universally was. Bolzano, who was famously skeptical about HP in the infinite case, accepted HPF, 20 as did just about everyone else who considered the matter. For all that HPF says is that, in the finite case, all and only equinumerous concepts have the 19 This would be all the more clear were HPF formulated in a logic which allowed partial functions, so that it was defined only for finite concepts. But working in such a logic would complicate matters quite unnecessarily. Such a formulation would also answer the objection that, in its present form, HPF does not have the form of a Fregean abstraction. 20 See B. Bolzano, Paradoxes of the Infinite, trans., by F. Prihonsky (London: Routledge and Kegan Paul, 1950), §§21–2. In §22, Bolzano gives an argument for HPF similar to the one to be given in the next two paragraphs.

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same number—and who knows what we should say about the infinite ones, other than that none of them have got the same number as any of the finite ones. The second point is that this claim really is implicit in arithmetical reasoning and that one can convince oneself of its truth, come to understand why it is true, by (and perhaps only by) reflecting on basic aspects of arithmetical thought. Now, it is not initially obvious to what notion of finitude we might appeal in reflecting on our arithmetical thought and investigating whether a commitment to HPF is implicit in it. Nor is it clear whether that notion is itself a logical one. But I submit that the intuitive notion of a finite concept is that of one the objects falling under which can be counted, i.e., enumerated by means of some process which eventually terminates. Frege’s definition of finitude directly reflects this intuitive notion: For what the definition says is precisely that a concept is finite if, and only if, the objects falling under it can be ordered as a discrete sequence which has a beginning and an end. 21 Our intuitive notion of finitude can thus be straightforwardly transcribed into second-order logic—thereby showing, modulo the status of second-order logic itself, that this intuitive notion is a logical one. How then can one convince oneself of the truth of HPF? It suffices to realize that the process of counting, which lies at the root of our assignment of numbers to finite concepts, already involves the notion of a one–one correspondence: As Frege frequently points out, 22 to count is to establish a one–one correspondence between certain objects and an initial segment of a sequence of numerals, starting with ‘1’; the process ends with a numeral which names the number of objects counted. By the transitivity of ‘is equinumerous with’, concepts the objects falling under which are themselves equinumerous must be equinumerous with the same initial segments; 23 conversely, any concepts the objects falling under which can be put in one–one correspondence with the same initial segment must be equinumerous. So any two concepts the objects falling under which can be counted—i.e., any two finite concepts— will be assigned the same numeral by the process of counting—i.e., will have the same number—if, and only if, they are equinumerous. And, of course, no infinite concept will get assigned any number by the process of counting. That is enough to establish HPF. 21 That Frege intended his definition to correspond to this intuitive notion is, furthermore, clear from the way he proves Theorems 327 and 348 of Grundgesetze. See my discussion of this point in “The Finite and the Infinite”, op. cit. 22 See, e.g., G. Frege, “Review of E. G. Husserl, Philosophie der Arithmetik I”, in his Collected Papers, ed. B. McGuinness, trans., by H. Kaal (Oxford: Blackwell, 1984), p. 199, original page 319. 23 That there is only one such initial segment will follow from the finitude of the segments themselves, given Frege’s definition. For we shall be able to show that no distinct initial segments are in one– one correspondence. The proof will depend upon certain claims about the numerals themselves, claims corresponding to the axioms of PAF. See here Frege’s discussion of counting in Grundgesetze, Vol. I, §108. I should emphasize that the argument being given here can be formalized: Indeed, the proof of Theorem 3.1, below, can be read as a very rough formalization of it.

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Of course, one might yet have all kinds of worries about the claim that HPF is a conceptual truth. There are two broad classes of such worries: Those which arise from its impredicativity, and those which rest upon the fact that it implies the existence of a lot of objects (infinitely many). I am not going to say anything here about questions of the former sort. 24 But, with regard to the latter, let me say that one needs to be very careful with such objections. Any principle sufficient to ‘ground’ arithmetic in the relevant sense obviously has to imply the existence of infinitely many objects: So one cannot object to someone who is trying to establish that the truths of arithmetic are conceptual truths, or logical consequences of such, by saying that the principle on which he proposes to base arithmetic cannot be a conceptual truth, because no conceptual truth can imply that there are infinitely many objects. One might as well object that the principle yields arithmetic, that his premises imply his conclusion, i.e., accuse him of holding his view. Or, better, one should just say, flatfootedly, that arithmetic can’t be ‘analytic’, in any reasonable sense, since it implies the existence of lots of objects. But that is not so much an objection as a refusal even to discuss the matter. For no one interested in the question whether arithmetic is ‘analytic’ is likely to be moved by that thought. But, having said all of that, let me emphasize that the importance of the question whether HPF is ‘analytic’, in the context of discussions of logicism, should not be allowed to obscure the fact that how we answer it does not affect the philosophical interest of the modification of Frege’s Theorem to be presented below. If HPF really is the ‘basic law of arithmetic’, in the relevant sense, that is philosophically important, whatever its epistemological status might turn out to be.

4.

The relative strengths of the systems

We turn now to the proofs of the four results mentioned at the end of Section 2. In this section, we prove Theorems 1, 2, and 4. We will prove Theorem 3 in the following section. Theorem 1: FA ⇒ PAS. Proof: In the last section, we saw a countermodel which establishes that HP is not a theorem of PAS + FD. That all the axioms of PAS are theorems of FA + FD is the content of Frege’s Theorem, first proven by Frege in Grundgesetze der Arithmetik (though very nearly proven in Die Grundlagen der Arithmetik). 24 I mean to include so-called ‘bad company’ objections. For discussion of these, see the papers of Wright and Boolos in Heck, ed., op. cit. For discussion of general concerns about the impredicativity of HP, see M. Dummett, Frege: Philosophy of Mathematics (Cambridge, MA: Harvard University Press, 1991), pp. 187–89 and 217–22; C. Wright, “The Harmless Impredicativity of Hume’s Principle”, in Schirn, ed., op. cit.; Dummett’s reply, in the same volume.

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As I have discussed Frege’s proof of Frege’s Theorem in detail elsewhere, and as adaptations of Frege’s proofs will be employed below, we need not dwell on it here. 25 Theorem 2: PAS ⇒ PAF. Proof: Clearly, every axiom of PAF other than PAF itself is a theorem of PAS + FD (indeed, of PAS by itself). That PAF is can be proven by induction. If a = 0, then all of its predecessors are finite, since it has none, by Axiom 5 of PAS. Suppose, then, that Na, that, if Na and Pya, then Ny, and that Pab. We must show that, if Pxb, then Nx. So suppose Pxb. By Axiom 4 of PAS, x = a, so Nx. Done. That not every theorem of PAS is a theorem of PAF + FD should be obvious: The axioms of PAF make no claims whatsoever about what the predecessors of objects which are not natural numbers might be, whereas the axioms of PAS state that predecession is one–one, not just on the natural numbers, but universally. Construction of a model is left to the reader. Theorem 4: PAF ⇒ PA2. Proof: Clearly, every theorem of PA2 is a theorem of PAF + FD. Again, that the converse (roughly speaking) is not true should be obvious: PA2 is completely silent on the question whether zero, or any other natural number, has predecessors which are not natural numbers. To construct a model, let the domain consist of the natural numbers and Julius Caesar. Assign denotations to terms of the form ‘Nx : Fx’ according to the following scheme: 0, if there are no Fs or if everything is F n, if there are nFs, for some finite n > 0 JC, if there are infinitely many Fs, but not everything is F Interpret the primitives of PA2 and PAF according to the axioms of FD. The axioms of PA2 may be verified, but PAF fails: The sentence ‘P (Nx : x = 0,0)’ is true in the model. Now, since, by Theorem 3, to be proven in the next section, PAF is equivalent to FAF (in the presence of FD), it follows that FAF is strictly stronger than PA2. The model just given also shows this directly. For the sentence ‘Finitex (x = x) & Nx : x = x = Nx : x = x & ¬Eqx (x = x; x = x)’ is true in the model, whence HPF is false in the model. 25 See my “The Development of Arithmetic in Frege’s Grundgesetze der Arithmetik”, Journal of Symbolic Logic 58 (1993), pp. 579–601, reprinted, with a Postscript, in W. Demopoulos, ed., Frege’s Philosophy of Mathematics (Cambridge, MA: Harvard University Press, 1995), pp. 257–94. See also, of course, Wright, Frege’s Conception, Ch. 4.

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Theorem 3: PAF is equivalent to FAF

In the proofs to be given below, we shall appeal frequently to the following easy consequence of the second axiom of FD: Fa → P[Nx : (Fx & x = a); Nx : Fx] This is Theorem 102 of Grundgesetze, and I shall cite it as such below. The equivalence of PAF and FAF is, in essence, a consequence of the fact that Theorems 327 and 348 of Grundgesetze, mentioned above, can be proven both in PAF and in FAF, with the aid of the bridge theory FD. That is to say, Finite(F) ≡ P ∗= (0, Nx : Fx) can be proven in both theories. This fact will allow us to work back and forth between the condition of finitude, as it appears in HPF, and the claims about the natural numbers made in the axioms of PAF. The proof to be given here of the left-to-right direction—which is Theorem 327 of Grundgesetze—shows it to be a consequence simply of Frege’s definitions and the axiom FE of Log. ‘The number of a finite concept is natural number’ may thus be added to the list of arithmetical facts which are, modulo the status of second-order logic itself, undeniably logical truths. (Others are ‘0P1’ and ‘1P2’, which Boolos shows to be provable in FD.) We begin by noting that we can, without loss of generality, assume the relation which orders a finite set to be one–one (not just functional), and such that no object follows itself in the relevant series. We define: Betw(Q; a; b)(n) ≡ ∀x∀y∀z∀w(Qxy & Qzw → x = z ≡ y = w)& ¬∃x Q ∗ xx & Q ∗= an & Q ∗= nb Proposition : Log ⊢ Finite(F) ≡ ∃Q∃a∃b∀x[Fx ≡ Betw(Q; a, b)(x)]. Proof: Right-to-left: Trivial, since if Betw(Q; a; b)(x), then Btw(Q; a,b)(x). Left-to-right: Assume that Fξ is finite, i.e., that for some Rξ η, a, and b: ∀x[Fx ≡ Btw(R; a, b)(x)]. If ¬∃xFx, let Qξ η be the universal relation; let a and b be whatever you like. Then for no x Betw(Q; a, b)(x). If ∃xFx, say x, then we have that Btw(R; a; b)(x), and so: ∀x∀y∀z(Rxy & Rxz → y = z) & ¬R∗ bb& ∀x[Fx ≡ R ∗= ax & R ∗= xb] Define: Qxy ≡ Rxy & Fx & Fy. Then ∀x[Fx ≡ Betw(Q; a, b)(x)]. The proof is straightforward; I shall not present the details here. 26 26 The plan of the proof is as follows. First, Qξ η is functional, since Rξ η is. Second, if Q ∗ xy, then R ∗ xy; so, if Q ∗= ax & Q ∗= xb, then R ∗= ax & R ∗= xb, so Fx. Third, if Q ∗ xx, then R ∗ xx and Fx, so R ∗= xb; so it is enough to prove that, if R ∗= xb, then ¬Q ∗ xx; do this by induction on the converse of Rξ η. Fourth, prove that the converse of Qξ η is functional, using Theorem 133 of Begriffsschrift, the roll-forward theorem, and the fact that ¬∃x Q ∗ xx. Then prove that, if Fx, then Q ∗= ax, by noting that, if Fx, then R ∗= ax, so it is enough to prove that, if R ∗= ax, then, if Fx, then Q ∗= ax, which can be done by induction. Finally, prove that, if Fx, then Q ∗= xb, similarly.

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Lemma 3.1 (Theorem 327): FD ⊢ Finite(F) → P ∗= (0, Nx : Fx). Proof: If ¬∃xFx, then Nx : Fx = 0 (by FE and the first axiom of FD), whence certainly P ∗= (0, Nx : Fx). So we suppose throughout that ∃xFx. Suppose F is finite; by the proposition, there are objects aand band a relation, Rξ η, which is one–one, in whose (strong) ancestral no object stands to itself, and which is such that x is F iff R ∗= ax and R ∗= xb. It will thus suffice to prove that P ∗= [0, Nx : (R ∗= ax & R ∗= xb)]. Since ∃xFx, for some x, R ∗= ax and R ∗= xb, so R ∗= ab. So it will be enough to prove that R ∗= ay → P ∗= [0, Nx : (R ∗= ax & R ∗= xy)] which we can prove by (logical) induction. For whenever R ∗= mn, we can prove that Fn by showing that Fm and: ∀x∀y(R ∗= mx & Fx & Rxy → Fy) (This fact is an easy consequence of the definition of the weak ancestral.) By comprehension, we may take F ξ to be: P ∗= [0, Nx : (R ∗= ax & R ∗= xξ )]. It will thus suffice to prove: (i) P ∗= [0, Nx : (R ∗= ax & R ∗= xa)] (ii) ∀y∀z{R ∗= ay & P ∗= [0, Nx : (R ∗= ax & R ∗= xy)] & Ryz → P ∗= [0, Nx : (R ∗= ax & R ∗= xz)]}

We are assuming, of course, that Rξ η satisfies the conditions mentioned above. For (i): By (102), P[Nx : (x = a & x = a), Nx : x = a]. Since x = a & x = a iff x = x, Nx : (x = a & x = a) = 0, by FE. Hence, P(0, Nx : x = a) and so P ∗= (0, Nx : x = a). So it will suffice to show that Nx : (R ∗= ax & R ∗= xa) = Nx : x = a, for which, by FE, it suffices to show that R ∗= ax & R ∗= xa ≡ x = a. From right-to-left, this is obvious. For the other direction, suppose that R ∗= ax and R ∗= xa and x = a. Then R∗ ax and R∗ xa, so, by the transitivity of the ancestral, R∗ aa. Contradiction. For (ii): Suppose the antecedent. By comprehension, we may take Fξ and a in (102) to be, respectively, R ∗= a ξ & R ∗= ξ z and z, whence: P{Nx : [(R ∗= ax & R ∗= xz) & x = z], Nx : (R ∗= ax & R ∗= xz)} Since P ∗= [0; Nx : (R ∗= ax & R ∗= xy)], it will be enough to show that Nx : (R ∗= ax & R ∗= xy) = Nx : [(R ∗= ax & R ∗= xz) & x = z] for which, by FE, it is enough to show that: ∀x[(R ∗= ax & R ∗= xy) ≡ (R ∗= ax & R ∗= xz) & x = z] Left-to-right: If R ∗= ax and R ∗= xy, R ∗= ay; since Ryz, then R ∗= az. And if x = z, then R ∗= zy and Ryz, so R∗ zz, contradiction. Right-to-left: Since R ∗= xz and

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x = z, R∗ xz. We then have the following theorem of second-order logic, the roll-back theorem: 27 Q ∗ x y → ∃z(Qzy & Q ∗= xz) By the roll-back theorem, there is some w such that Rwz and R ∗= xw. Since Rξ η is one–one and Ryz, w = y, so R ∗= xy. Remark: The following are theorems of PAF: 28 N P Z : ¬∃xPx0 P1M F : ∀x yz(P ∗= 0z & Pxz & Pyz → x = y) Z E : Nx : Fx = 0 ≡ ¬ ∃x Fx Proof: Zero has no predecessor which is a natural number and, by PAF, only natural numbers precede natural numbers. So since zero is a natural number, it can have no predecessor at all. P1MF will follow immediately from Axiom 4 of PAF if we can show that x and y are themselves natural numbers. But this follows from PAF, since z is a natural number and x and y both precede it. For ZE: If ¬∃xFx, then ∀x(Fx ≡ x = x). So, by FE, Nx : Fx = Nx : x = x = 0. Suppose, then, that Nx : Fx = 0 and ∃x Fx, say, a. By (102), P[Nx : (Fx & x = a), Nx : Fx], so, by NPZ, Nx : Fx = 0. Contradiction. Lemma 3.11 (Theorem 348): PAF + FD ⊢ P ∗= (0, Nx : Fx) → Finite(F). Proof: We prove the equivalent: P ∗= 0n → ∀F[n = Nx : Fx → Finite(F)] The proof is by (logical) induction. We must thus establish that: (i) ∀F[0 = Nx : Fx → Finite(F)] (ii) P ∗= 0n & ∀F[n = Nx : Fx → Finite(F)] & Pnm → ∀F[m = Nx : Fx → Finite(F)]

For (i): Suppose 0 = Nx : Fx. By ZE, ¬∃xFx, so F is finite. For (ii): Suppose the antecedent, and suppose further that m = Nx : Fx. We must show that F is finite. Suppose ¬∃xFx. Then, by ZE, m = Nx : Fx = 0, so 27 The roll-back theorem is proved by induction. We must show

(i) Qxw → ∃z(Qzw & Q ∗= xz) (ii) ∃z(Qzw & Q ∗= xz) & Qwv → ∃z(Qzv & Q ∗= xz) The proof of (i) is trivial: Take z to be x. For (ii), assume the antecedent. Take z in the consequent to be w. By hypothesis, Qwv. And since, by the antecedent, Q ∗= xz & Qzw, certainly Q ∗= xw. 28 In fact, PA2 + NPZ + P1MF is deductively equivalent to PAF. The proof of PAF given above, in the proof of Theorem 2, depends only upon NPZ and P1MF, and not on the full force of Axioms 4 and 5 of PAS.

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Pn0, contradicting NPZ. So ∃xFx, say a, and P[Nx : (Fx & x = a); Nx : Fx]. But also, by hypothesis, P(n, Nx : Fx), and, since P ∗= 0n; P ∗= (0, Nx : Fx). So, by P1MF, n = Nx : (Fx & x = a). Hence, by the induction hypothesis, Finitex (Fx & x = a). It is then a simple matter to show that F too must be finite. Corollary 3.12: PAF + FD ⊢ Finite(F) ≡ P ∗= (0, Nx : Fx). Theorem 3.1: PAF+ FD ⊢ HPF. Proof: By Corollary 3.12, it suffices to show that P ∗= (0; Nx : Fx) → Nx : Fx = Nx : Gx ≡ Eqx (Fx, Gx) We prove the equivalent: P ∗= 0n → ∀F{n = Nx : Fx → ∀G[Nx : Fx = Nx : Gx ≡ Eqx (Fx, Gx)]} The proof is by induction. We must show that: (i) 0 = Nx : Fx→ ∀G[Nx : Fx = Nx : Gx ≡ Eqx (Fx, Gx)] (ii) P ∗= 0n & ∀F{n = Nx : Fx → ∀G[Nx : Fx = Nx : Gx ≡ Eqx (Fx, Gx)]} & Pnm→ ∀F{m = Nx : Fx→∀G[Nx : Fx = Nx : Gx ≡ Eqx (Fx, Gx)]}

For (i): Suppose that 0 = Nx : Fx. By ZE, ¬∃xFx. Now, if 0 = Nx : Gx, by ZE, ¬∃xGx, so Eq(F,G). Conversely, if Eq(F; G), then ¬∃xGx, so, by FE, Nx : Fx = Nx : Gx. For (ii): Suppose the antecedent, and suppose further that m = Nx : Fx. We must show that, for every G, Nx : Fx = Nx : Gx iff Eqx (Fx, Gx). Since P ∗= 0n and Pnm, P ∗= 0m and so P ∗= (0, Nx : Fx). Left-to-right: Suppose Nx : Fx = Nx : Gx. Since P(n, Nx : Fx), Nx : Fx = 0, by NPZ, and so, by ZE, ∃xFx, say a; similarly, ∃xGx, say b. By (102): P[Nx : (Fx & x = a), Nx : Fx] P[Nx : (Gx & x = b), Nx : Gx] Since Nx : Fx = Nx : G, Nx : (Fx & x = a) = Nx : (Gx & x = b), by P1MF. Moreover, since P(n, Nx : Fx), by P1MF, again, n = Nx : (Fx & x = a). So, by the induction hypothesis: Eqx (Fx & x = a, Gx & x = b) But then Eqx (Fx, Gx), since Fa and Gb. Right-to-left: Suppose Eqx (Fx;Gx). Once again, ∃xFx, say a, and ∃xGx, say b, and: P[Nx : (Fx & x = a), Nx : Fx] P[Nx : (Gx & x = b), Nx : Gx]

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Since P(n; Nx : Fx), n = Nx : (Fx & x = a), by P1MF. But, if Eqx (Fx, Gx), Fa, and Gb, certainly: Eqx (Fx & x = a; Gx & x = b) So, by the induction hypothesis, Nx : (Fx & x = a) = Nx : (Gx & x = b). But then, since P[Nx : (Fx & x = a), Nx : Fx] and P[Nx : (Gx & x = b), Nx : Gx], Nx : Fx = Nx : Gx, by Axiom 3 of PAF. That, then, establishes that HPF is a theorem of PAF + FD. We now turn to the proof that all axioms of PAF are theorems of FAF + FD. Our plan is simply to mimic Frege’s proofs of the axioms of arithmetic, relativized in the appropriate way to the natural numbers. To make these proofs work, we need to establish an analogue of Corollary 3.12. From this it will follow that, when talking about natural numbers, we are dealing only with finite concepts, so HPF will do the work HP does in Frege’s proofs. We divide the proof of Theorem 3.2 into two parts: The proof that all axioms other than Axiom 6 hold is relatively easy, and we prove this first; the proof that Axiom 6 holds is of special interest and so will be considered separately. First, we establish the corollary, by establishing an analogue of Lemma 3.11. Lemma 3.21 (Theorem 348, again): FAF + FD ⊢ P ∗= (0, Nx : Fx) → Finite(F). Proof: It will suffice to show that (∗ )P ∗= 0n → ∃F[Finite(F) & n = Nx : Fx] For then, suppose that P ∗= (0, Nx : Fx). Then, for some finite G, Nx : Fx = Nx : Gx. By HPF, Eqx (Fx, Gx), so F is finite. The proof of (∗ ) itself is by induction. We must show that: (i) ∃F[Finite(F) & 0 = Nx : Fx] (ii) P ∗= 0n & ∃F[Finite(F) & n = Nx : Fx] & Pnm→ ∃F[Finite(F) & m = Nx : Fx]

For (i): 0 = Nx : x = x and Finitex (x = x). For (ii): Suppose the antecedent, so that Finite(F) and n = Nx : Fx. Since Pnm, P(Nx : Fx, m), so by Axiom 2 of FD, for some G and b: Gb & m = Nx : Gx & Nx : Fx = Nx : (Gx & x = b) Since Finite(F), by HPF, Eqx [Fx, Gx & x = b], so Finitex (Gx & x = b). But then G too is finite. Corollary 3.22: FAF + FD ⊢ Finite(F) ≡ P ∗= (0, Nx : Fx). Lemma 3.23: FAF + FD ⊢ All axioms of PAF other than Axiom 6.

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Proof: Axioms 1, 2, and 7 do not require any special attention: Each of them is an immediate consequence of FD itself—indeed, of just the third axiom of FD. We thus need to prove Axioms 3, 4, and 5 and the Axiom PAF itself. Axiom 5 is: ¬∃x(Nx & Px0). Suppose that Pn0. By the second axiom of FD, there are Fand ysuch that: Fy & 0 = Nx : Fx & n = Nx : (Fx & x = y) But since 0 = Nx : x = x and Finitex (x = x), HPF yields that Eqx (x = x, Fx). But ∃yFy. Contradiction. (Note that this actually establishes NPZ.) Axiom 3 is: ∀x∀y∀z(Nx & Pxy & Pxz → y = z). So suppose that Na, i.e., that P ∗= 0a, and that Pab and Pac. By the second axiom of FD, there are Fand G, and yand z, such that: Fy & b = Nx : Fx & a = Nx : (Fx & x = y) Gz & c = Nx : Gx & a = Nx : (Gx & x = z) Since P 0a and a = Nx : (Fx & x = y), by Corollary 3.22, Finitex (Fx & x = y). Since Nx : (Fx & x = y) = a = Nx : (Gx & x = z), by HPF, Eqx (Fx & x = y, Gx & x = z). But then Eqx (Fx, Gx), since Fy and Gz, and certainly Finite(F). So, by HPF again, Nx : Fx = Nx : Gx and so b = c. Axiom 4 is: ∀x∀y∀z(Nx & Ny & Pxz & Pyz → x = y). So suppose that Naand Nb, and that Pac and Pbc. Note that P ∗= 0c. We shall make no further appeal to the assumptions that Na and Nb. Once again, there are F and G, and y and z, such that: ∗=

Fy & c = Nx : Fx & a = Nx : (Fx & x = y) Gz & c = Nx : Gx & b = Nx : (Gx & x = z) Since c = Nx : Fx and P ∗= 0c, by Corollary 3.22, Finite(F). And Nx : Fx = c = Nx : Gx, so by HPF, Eqx (Fx, Gx). But then, since Fy and Gz, Eqx (Fx & x = y, Gx & x = z); these are finite, since F and G are, so by HPF again, Nx : (Fx & x = y) = Nx : (Gx & x = z) and so a = b. (Note that this actually establishes P1MF.) PAF is: Nx & Pyx → Ny. As noted parenthetically above, the proofs of Axioms 4 and 5 in fact suffice to prove NPZ and P1MF, from which PAF follows. But it can also be proven directly. Suppose that P ∗= 0n and that Pmn. Since Pmn, there are Fand a such that: Fa & n = Nx : Fx & m = Nx : (Fx & x = a) Since P ∗= (0, Nx : Fx), F is finite; so Finitex (Fx & x = a), and so P ∗= [0, Nx : (Fx & x = a)]. But then P ∗= 0m. In the paper mentioned earlier, Boolos proves the surprising result that, in the bridge theory FD, Axiom 6, that is, ∀x(Nx → ∃yPxy), of PA2 is redundant, since it follows from Axioms 3, 4, and 5. More recently, he has observed that, in FD, Axiom 6 in fact follows from Axiom 3 alone. Boolos’s original proof

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of this extraordinary result is somewhat indirect: 29 I shall take the opportunity to give a direct proof here. Of course, since FAF + FD ⊢ Axiom 3, it follows that FAF + FD ⊢ Axiom 6. To prove Axiom 6, Frege proves P ∗= 0n → P[n, Nx : (P ∗= 0x&P ∗= xn)] to establish which he needs the crucial lemma: P ∗= 0n → ¬P ∗ nn It is for the proof of this lemma that Axiom 5 is needed. Our proof will differ from his, in the first instance, in that we do not make use of this lemma, but instead pack the necessary condition into the antecedent and prove the weaker: P ∗= 0n & ¬P ∗ nn → P[n, Nx : (P ∗= 0x & P∗= xn)] We complete our argument by also proving P ∗= 0n & P∗ nn → ∃yPny whence Axiom 6 follows by dilemma. Our proof will also differ in another way. Frege appeals to Axiom 4 at a crucial point, but we shall see that the necessary inference does not require it, even in his proof. What we shall use instead is the logicized version of the Law of Trichotomy: P ∗= 0b & P∗= 0c → P ∗ bc ∨ P∗ cb ∨ b = c The Law follows from Axiom 3 and the following strengthening of the famous Proposition 133 of the Begriffsschrift: 30 ∀x ∀ y∀z[R ∗= ax & Rxy & Rxz → y = z] & R ∗= ab & R∗= ac → R ∗ bc ∨ R∗ cb ∨ b = c Instantiating ‘R’ with ‘P’, ‘a’ with ‘0’, and noting that the first conjunct then follows from Axiom 3, the Law of Trichotomy follows immediately. 29 We have that: FD, 3, 4, 5 ⊢ 6. Boolos then observed that also: FD, 3, ¬6 ⊢ 4 & 5. But then, by truthfunctional logic: FD, 3, ¬6 ⊢ 6. And so: FD, 3 ⊢ 6. For a proof of a related result, see G. Boolos, “Frege’s Theorem and the Peano Postulates”, Bulletin of Symbolic Logic 1 (1995), pp. 317–26. 30 The strengthening lies in our assuming not that Rξ η is functional, but just that it is functional on the members of the R-series beginning with a.

Proof: We assume that ∀x∀y∀z[R ∗= ax & Rxy & Rxz → y = z] and R ∗= ab and prove that R ∗= ac → R ∗ bc ∨ R ∗ cb ∨ b = c by induction on c. We must thus prove: (i)

R ∗= aa → R ∗ ba ∨ R ∗ ab ∨ b = a

(ii)

R ∗= ax & (R ∗= ax → R ∗ bx ∨R ∗ xb ∨b = x) & Rxy → (R ∗= ay → R ∗ by ∨R ∗ yb ∨ b = y)

Since R ∗= ab, (i) follows from the definition of the weak ancestral. So suppose the antecedent in (ii). Note that R ∗= ay. Moreover, R ∗ bx or R ∗ xb or b = x. If R ∗ bx, then since Rxy, R ∗ by. Moreover, if b = x, then Rby, so R ∗ by. So suppose that R ∗ xb. By the roll-forward theorem, to be mentioned shortly, for some z, Rxz and R ∗= zb. But then Rxz and Rxy: And since Rξ η is functional on the R-series beginning with a and R ∗= ax, we have z = y. So R ∗= yb, hence R ∗ yb or b = y.

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Lemma 3.24: FD + Axiom 3 ⊢ Axiom 6 of PAF. Proof: We must establish ∀x[Nx → ∃yPxy] for which, in light of FD, it will suffice to establish: P ∗= 0n → ∃yPny We proceed by dilemma, proving each of: P ∗= 0n & P ∗ nn → ∃yPny P ∗= 0n & ¬P ∗ nn → ∃yPny The former follows immediately from the following theorem of second-order logic, the roll-forward theorem: 31 R ∗ ab → ∃y(Ray & R ∗= yb) For, then, if P∗ nn, for some y, Pny & P ∗= yn, so certainly ∃yPny. For the latter, we prove: P ∗= 0n & ¬P ∗ nn → P[n, Nx : (P ∗= 0x & P ∗= xn)] The proof is by induction. We thus need to establish: (i) FD ⊢ ¬P ∗ 00 → P[0, Nx : (P ∗= 0x & P ∗= x0)] (ii) FD + Axiom 3 ⊢ P ∗= 0a & {¬P∗ aa → P[a, Nx : (P ∗= 0x & P ∗= xa)] & Pab → {¬P∗ bb → P[b, Nx : (P ∗= 0x & P ∗= xb)]}

For (i): Suppose ¬P ∗ 00. Since P ∗= 00, by (102): P[Nx : (P ∗= 0x & P ∗= x0 & x = 0), Nx : (P ∗= 0x & P ∗= x0)] Now suppose that P ∗= 0x & P ∗= x0 & x = 0. Then, since P ∗= x 0 & x = 0, P ∗ x0. But then P ∗ x0 and P ∗= 0x, so P ∗ 00, contradicting our supposition. Thus ¬∃x(P ∗= 0x & P ∗= x0 & x = 0). By FE, Nx : (P ∗= 0x & P ∗= x0 & x = 0) = 0 and so P[0, Nx : (P ∗= 0x & P ∗= x0)]. For (ii): Suppose the antecedent and suppose further that ¬P∗ bb. Suppose, for reductio, that P∗ aa. By the roll-forward theorem, for some y, Pay and P ∗= ya. Since Pab, Axiom 3 implies that y = b. But then P ∗= ba and Pab, so P∗ bb. Contradiction. Hence ¬P∗ aa. By the induction hypothesis, then, P[a, Nx : (P ∗= 0x & P ∗= xa)]. Now, we need to show that P[b, Nx : (P ∗= 0x & P ∗= xb)]. Since P ∗= 0b and P ∗= bb, by (102): P[Nx : (P ∗= 0x & P ∗= xb & x = b), Nx : (P ∗= 0x & P ∗= xb)] So it is enough to show that b = Nx : (P ∗= 0x & P ∗= xb & x = b). And since Pab and P[a, Nx : (P ∗= 0x & P ∗= xa)], we have, by Axiom 3, that b = 31 The proof of this result is similar to that of the roll-back theorem, mentioned earlier.

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Nx : (P ∗= 0x & P ∗= xa). So we need only show that Nx : (P ∗= 0x & P ∗= xa) = Nx : (P ∗= 0x & P ∗= xb & x = b). By FE, this will follow from: ∀x[(P ∗= 0x & P ∗= xa) ≡ (P ∗= 0x & P ∗= xb & x = b)] Left-to-right: Suppose P ∗= 0x and P ∗= xa. Then since Pab, certainly P ∗= xb and, further, P ∗= 0b. Suppose x = b. Then P ∗= ba and Pab, so P∗ bb. Contradiction. Right-to-left: Suppose P ∗= 0x & P ∗= xb & x = b. Then P∗ xb. By the roll-back theorem, for some y, P ∗= xy and Pyb. Up to this point, we have been following Frege’s proof closely. Here, he uses Axiom 4 to conclude that, since Pab, a = y, whence P ∗= xa, and he is done. But we can in fact establish that a = y without appeal to Axiom 4. We have that P ∗= xy and Pyb. Since P ∗= 0x, certainly P ∗= 0y. Since, by the initial hypotheses of the inductive step, P ∗= 0a, the Law of Trichotomy yields that either P∗ ay or P∗ ya or a = y. Suppose that P∗ ay. By the roll-forward theorem, for some z, Paz & P ∗= zy. But since Pab, Axiom 3 implies that z = b. So P ∗= by & Pyb, so P∗ bb, contradiction. Similarly, if P∗ ya, then for some z, Pyz & P ∗= za. But since Pyb, Axiom 3 implies that z = b, so P ∗= ba Pab, so P∗ bb, once again. Hence a = y, and we are done. Theorem 3.2: FAF ⊢ FD ⊢ All axioms of PAF. Proof: By Lemmas 3.23 and 3.24.

6.

Closing

We have thus seen that FAF is equivalent, in the presence of the bridge theory FD, to PAF. By Theorem 4, then, FAF is strictly stronger than PA2. The following two questions now raise themselves: Whether there is some further weakening of HP which is provable in PA2 + FD and, if so, whether some such principle is equivalent, in FD, to the conjunction of the axioms of PA2. A natural axiom at which to look would be WHP (for Weak Hume’s Principle): Finitex (Fx) & Finitex (Gx) → [Nx : Fx = Nx : Gx ≡ Eqx (Fx, Gx)] WHP states only that finite concepts have the same number if, and only if, they are equinumerous and makes no claim whatsoever about the conditions under which infinite concepts have the same number as any concept, finite or otherwise. (As far as WHP is concerned, some infinite concepts could have the number zero, others one, and so forth.) Call the theory whose sole nonlogical axiom is WHP, WHP. It can be shown that, though WHP is provable in PA2 + FD, none of the axioms of PA2 which are not already theorems of FD itself are theorems of WHP + FD: Not even the disjunction of these axioms— that is, of Axioms 3, 4, 5, and 6—is a theorem of WHP + FD.

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Still, it is easy to see that there are no finite models of WHP + FD. For, in any model, there must be a number Nx : x = x; there must be a number Ny : (y = Nx : x = x), which, by WHP, will differ from Nx : x = x; there will be a number Nz : [z = Nx : x = x ∨ z = Ny : (y = Nx : x = x)], which again must differ from the first two, and so forth. Indeed, it is not terribly difficult to prove that PA2 can be relatively interpreted in WHP, and so that PA2 and WHP are equi-interpretable and therefore equi-consistent. As we saw earlier, however, the fact that a theory A is interpretable in another B is no guarantee that there is any reasonable bridge theory by using which one can prove the axioms of A in B. One might therefore wonder whether, in this case, there is some bridge theory other than FD, by appeal to which one could prove the axioms of PA2 in WHP. In fact there is, the necessary modification to the axioms of FD not being drastic. The bridge theory FDF has the same first and third axioms as FD, but we change the second axiom to: ∃F[Finite(F) & m = Nx : Fx] ∨ ∃F[Finite(F) & n = Nx : Fx] → Pmn ≡ ∃F∃y[Finite(F) & Fy & n = Nx : Fx & m = Nx : (Fx & x = y)] This axiom now states nothing at all about when numbers which are not the numbers of finite concepts precede one another. It requires, however, that, if a number is the number of a finite concept, then it will precede or be preceded by another number only if there is some finite concept which does the trick. The axiom, though complicated as stated, seems intuitive enough and is certainly true, since it is a theorem of FAF + FD, as can easily be seen. It may thus be considered a partial definition of one of the primitives of PA2 in terms of those of WHP. And it can be shown that, relative to FDF, WHP, PA2, and PAF are all equivalent. What philosophical interest this result might have for a logicist is a question I shall not pursue. 32

32 Thanks here to Charles Parsons, Alison Simmons, Jason Stanley, Jamie Tappenden, and Crispin Wright for discussion. The paper also benefitted from the comments of the Journal of Philosophical Logic’s referees. I owe a special debt to George Boolos, discussions with whom led to my work on this topic, as many others. George was a teacher, a colleague, a mentor, and a source of inspiration and courage—but most of all, he was a friend. This paper is dedicated to his memory.

ON FINITE HUME 1 Fraser MacBride†

Neo-Fregeanism declares there to be an a priori route that we may follow (guided by proofs and definitions) from an understanding of analytic truths to a grasp of the fundamental laws of arithmetic (see Wright [1997], pp. 202– 11). The purportedly analytic principle from which the neo-Fregean claims we may set out is Hume’s Principle, a principle that specifies the conditions under which concepts have the same cardinal number: (HP)∀F∀G[(N x : F x = N x : Gx) ↔ F1 − 1G] When this principle is adjoined to second-order logic, the system that results is Frege arithmetic. What makes it plausible to suppose that Hume’s principle provides a departure point from which we may successfully go onto grasp arithmetic a priori is the result called Frege’s theorem. For according to Frege’s theorem, the Peano postulates can be interpreted in Frege arithmetic and their interpretations proved in that system (see Boolos [1996]). There are, however, alternative departure points, other purportedly analytic principles, from which we may just as plausibly set out. As Richard Heck has shown (Heck [1997]), finite Hume’s principle is one such principle, a principle that states the conditions under which finite concepts have the same cardinal number: (HPF) ∀F∀G((Finite(F)vFinite(G)) → [(Nx : Fx = Nx : Gx) ↔ F1 − 1G]) The system that results from adjoining this principle to second-order logic may be called finite Frege arithmetic. Heck demonstrates that the Peano postulates have provable interpretations in finite Frege arithmetic just as they do in Frege arithmetic. 1 This paper first appeared in Philosophia Mathematica 8, [2000], pp. 150–9. Reprinted by kind permission of the editor and Oxford University Press. † I wish to thank Patrick Greenough, Katherine Hawley, Richard Heck, Alex Oliver, Stewart Shapiro, Crispin Wright and, especially, Peter Clark for discussion of this paper. I am also grateful to the audience at an Arché conference on abstraction held at the University of St. Andrews for their helpful comments.

85 Roy T. Cook (ed.), The Arché Papers on the Mathematics of Abstraction, 85–93. c 2007 Springer.

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To these technical results Heck adds the philosophical contention that it is from an understanding of finite Hume’s principle—rather than Hume’s principle—that the neo-Fregean should guide us to an a priori grasp of arithmetic. I will argue that Heck’s philosophical arguments are flawed. They do not give the neo-Fregean reason to lose nerve. An abstraction is a principle of the form: ((α j ) = (αk )) ↔ α j ≈ αk Abstraction principles tie the conditions under which entities of one kind (s) are identical to the obtaining of an equivalence relation (≈) amongst another kind of entity (αs). Such a principle may be read as offering a substantial, synthetic claim to the effect that facts about one kind of entity are necessarily connected to facts about another entirely distinct kind. But the neo-Fregean counsels that we need not always read abstractions in this way (Wright [1997], pp. 205–08). He suggests instead that an abstraction principle may be read (in certain circumstances) as offering an analytic claim. For where an abstraction contains an occurrence of a novel term-forming operator (“”) on expressions (“α1 ” · · · “αk ” . . . ) that are already understood, the abstraction may be read as embodying a stipulation that introduces the novel term into language. It may be read as stipulating that the truth conditions of identity statements featuring the novel operator (“(α j ) = (αk )”) coincide with the truth conditions of another familiar form of statement (“α j ≈ αk ”). Since one way to be analytic is to be stipulated, the neoFregean concludes, that under these circumstances, abstraction principles are analytic. According to neo-Fregean doctrine, Hume’s principle is an analytic abstraction. It introduces a novel cardinality operator by stipulating that the truth conditions of identity statements concerning cardinal numbers coincide with the conditions under which an equivalence relation amongst concepts may be familiarly said to obtain. Imagine a faultlessly rational character—call him ‘Hero’—who has mastered second-order logic but has yet to be introduced to any characteristically mathematical notions (see Wright [1998], p. 359). Despite his mathematical ignorance, Hero is able to grasp Hume’s principle because the familiar vocabulary (bound concept variables, the notion of one–one correspondence) which the principle uses to introduce the cardinality operator is second-order expressible. Hero is therefore able to appreciate a priori the conditions under which—according to the stipulation Hume’s principle effects—cardinal numbers are identical. Having come to appreciate Hume’s principle, Hero is then able to appreciate the truth of Peano’s postulates. For as Frege’s theorem shows, these postulates may be interpreted and their interpretations proved in a system generated from Hume’s Principle and the second-order logic that Hero has already mastered. The possible case of Hero—the neo-Fregean

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claims—shows how it is possible for mathematical knowledge to be acquired a priori. There are many challenges the neo-Fregean must overcome in order to substantiate the claim that Frege’s theorem may be invested with such epistemological significance. Amongst the most basic of these challenges are two. First, the neo-Fregean must establish that Hume’s Principle is nothing more than a stipulation even though in conjunction with second-order logic it generates a theory—Frege arithmetic—committed to the existence of infinitely many objects. Second, the neo-Fregean must establish—contrary to Quinean suspicions—that an understanding of second-order logic does not presuppose prior knowledge of mathematics. Nevertheless, provided such challenges can be met, it appears the neo-Fregean may legitimately claim that it is possible to travel a priori from an understanding of Hume’s Principle to a grasp of arithmetic. But the legitimacy of this final claim remains open to question. For even if it is granted that Hume’s principle may be employed to access some array or other of a priori truths, it remains to be established that these really are arithmetical truths and that the a priori knowledge acquired by employing Hume’s principle is genuinely arithmetical in character. Frege’s theorem, on its own, does not secure this result. All that Frege’s theorem strictly shows—given the assumption that abstraction principles can be used to generate a priori knowledge—is that there is a system of a priori truths (Frege arithmetic) that is capable of modelling arithmetic. It does not show that any of these a priori truths are arithmetical in character. According to Heck, the a priori truths that flow from Hume’s principle can only be arithmetical if Hume’s principle itself is a genuinely arithmetical principle. To establish that Hume’s principle is arithmetical, Heck claims, it must be shown to be a basic law “upon which ordinary arithmetical reasoning relies” (Heck [1997], p. 596). But, Heck goes onto argue, Hume’s principle does not underlie ordinary arithmetical reasoning and so the a priori knowledge which a grasp of Hume’s principle grounds cannot be arithmetical. Heck concludes that Frege’s theorem fails to map an a priori route to arithmetical knowledge. Heck bases his argument that Hume’s principle does not inform ordinary arithmetical reasoning on the contention that “no amount of reflection on the nature of arithmetical thought could ever convince one of HP” (Heck [1997], p. 597). Hume’s Principle says that all and only equinumerous concepts have the same cardinality. It follows that the number of natural numbers and the number of even numbers are the same (since the concepts natural number and even number can be put in one–one correspondence). But no amount of reflection on ordinary arithmetical reasoning—that is, “reasoning about, and with, finite numbers”—could ever convince one that these infinite cardinals are the same. Indeed it comes as something of a conceptual shock to discover that the concepts natural number and even number have the same cardinality.

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Since the evens form only a portion of the naturals, our naive inclination is to say that there are fewer evens than naturals. That is why it took not just an ordinary arithmetical thinker, but an intellect of great genius—Georg Cantor— to make the conceptual leap required to recognise the truth of what Hume’s principle tells us, that the concepts in question share the same cardinal. That is why it cannot be Hume’s principle that is implicit in ordinary arithmetical reasoning. By contrast, Heck continues, it is plausible to hold that Finite Hume’s principle informs habitual arithmetical practice. Finite Hume’s principle does not make any claim concerning the conditions under which infinite cardinals are the same or different. It is possible to accept Finite Hume without taking the conceptual leap that Cantor made. In fact, Heck claims, prior to the receipt of Cantor’s work, reflection on numerical practice led almost all thinkers to endorse finite Hume’s principle. Heck cites Bolzano as an example of such a thinker, a thinker who was sceptical about Hume’s principle (since he thought it possible for there to be infinite totalities that, even though equinumerous, nevertheless differed in multiplicity) but who endorsed Finite Hume’s principle. Heck goes so far as to claim that Finite Hume “really is implicit in arithmetical reasoning and that one can convince oneself of its truth, come to understand why it is true, by (and perhaps only by) reflecting on basic aspects of arithmetical thought”. We count by establishing a one–one correspondence between an initial segment of the sequence of numerals and the objects counted: we begin with “1” and end with a numeral n that stands for the number of objects. Consequently, if two finite concepts are equinumerous, then, by the transitivity of equinumerosity, the objects falling under one of those concepts will be one– one correspondent with the same initial segment of numerals as the objects falling under the other concept. In other words, those two finite concepts will have the same number. Conversely, if the objects falling under one concept are one–one correspondent with the same initial segment of the numerals as the objects falling under another concept—that is, if those two finite concepts share the same number—then, by the transitivity of equinumerosity, those concepts will be equinumerous. Reflection on the nature of counting thereby establishes that finite concepts have the same number if, and only if, they are equinumerous. Indeed—Heck claims—it is just such an argument that convinced Bolzano of the truth of finite Hume. Heck concludes that if the neo-Fregean claim to provide an epistemology of arithmetic is to have any legitimacy, the neo-Fregean should adopt finite Hume’s principle rather than Hume’s principle. The neo-Fregean account of how a priori knowledge may be generated applies just as well to the former principle as the latter. Just like its less modest cousin, finite Hume’s principle may be understood as an analytic abstraction, only a restricted one, stipulating the conditions under which finite cardinal numbers are identical. The additional vocabulary finite Hume’s principle uses to fix these conditions

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(the extra notion of finite) is also expressible in second-order logic. And, furthermore, interpretations of the Peano postulates are—as Heck shows— provable in the system that results from uniting finite Hume’s principle and second-order logic. But, unlike Hume’s principle, finite Hume is implicit in ordinary arithmetical practice. So, Heck claims, there is every reason to suppose that the a priori knowledge that finite Hume delivers—by contrast to the knowledge that flows from Hume’s principle—is arithmetical. Heck’s arguments are, however, flawed. To begin with, Heck’s claim that prior to the receipt of Cantor’s work it was finite Hume, rather than Hume’s principle, that informed arithmetical thinking is highly contentious. Prior to the receipt of Cantor’s work it seemed paradoxical to think that the number of natural numbers was the same as the number of even numbers even though the latter constitute only a portion of the former. But in order for this thought to appear a paradox, it is necessary not only to have an intellectual inclination to deny that those numbers are distinct, it is also necessary to have an intellectual inclination to affirm that they are the same. Yet if—as Heck supposes— pre-Cantorian thinkers only endorsed finite Hume, then they would have had no reason to affirm the identity of the number of natural numbers and the number of even numbers. They would have had no reason because finite Hume’s principle is entirely silent about identities amongst infinite numbers. So, it would not have seemed—as it did seem to them—a paradox that those numbers were distinct. Indeed, such thinkers as Heck describes would have been quite unable to frame a thought about the identity and distinctness of infinite numbers. If, prior to the reception of Cantor’s work, ordinary arithmetical reasoning had only been informed by finite Hume then Bolzano’s Paradoxes of the Infinite would have been a shorter book. Heck sketches an historical Cantor whose conceptual contribution was— through his work on the infinite—to clear the way for the introduction of a novel criterion of numerical identity that applied not only to finite but also to infinite numbers. According to Heck history, we fail to register the significance of Cantor’s contribution if we suppose Hume’s principle already informed ordinary arithmetical reasoning. But Heck’s sketch makes no sense of the fact that prior to Cantor it appeared paradoxical to affirm that equinumerosity amongst concepts sufficed for the identity of infinite numbers. In fact it makes better sense of history to describe the significance of Cantor’s contribution in a quite different way. According to this historical reconstruction, ordinary arithmetical thinking, prior to Cantor, was guided not only by Hume’s principle but also by the intuition that the cardinality of a collection of entities is always greater than the cardinality of any of its proper parts. The clash of principle and intuition made the identities of infinite numbers appear paradoxical to earlier thinkers. Hume’s principle, according to this reconstruction, led these thinkers to affirm that the natural numbers and the even numbers had the same cardinality, whereas intuition led them to suppose their cardinalities were different since the even numbers form only

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a proper part of the collection of natural numbers. Cantor’s contribution was to recognise this intuition was founded only on a parochial acquaintance with finite wholes and parts, and to enable us to resolve the paradox by persuading us to abandon the intuition. Cantor’s contribution was not to clear the way for the introduction of a novel principle of numerical identity, Hume’s principle. Rather—like a great moral leader—he persuaded us to see that a familiar principle should be applied in an unfamiliar case. Heck’s argument that finite Hume’s principle does inform ordinary arithmetical practice (because it may be arrived at by reflection on the process of counting) must also be questioned. First, if any abstraction principle can be arrived at by reflection on the process of counting, then that principle is not finite Hume. Finite Hume presupposes the intelligibility of the notion of an infinite cardinal. Suppose that concepts F and G fail to be equinumerous because F is infinite whereas G is finite. Then, by finite Hume, there is an infinite number belonging to the concept F that is distinct from the number belonging to the concept G. But, if the identity conditions for cardinals flow from reflection on the process of counting, then the notion of an infinite cardinal cannot make sense. For an infinite cardinal is a number that belongs to a concept the objects falling under which, by definition, cannot be counted; there is no initial segment of the numerals with which the objects falling under such a concept can be put in one–one correspondence. So finite Hume cannot be arrived at by the reflective route recommended by Heck. This suggests that if any abstraction principle arises from simple reflection on the counting process, it is another principle—weak Hume’s principle—that, unlike finite Hume, concerns only the identity conditions of finite cardinals: (WHP)∀F∀G((Finite(F) & Finite(G)) → [(Nx : Fx = Nx : Gx) ↔ F1 − 1G]) The suggestion receives historical support. Heck mentions Bolzano as an example of a thinker who endorsed finite Hume on the basis of reflection on the counting process (Bolzano [1950], §22). In fact, in the passage cited by Heck, it is weak Hume, rather than finite Hume, which Bolzano sanctions on that basis. 2 Second, it is far from evident that there is any abstraction even resembling finite Hume that is implicit in ordinary arithmetical practice. A character can readily be imagined who is capable of counting yet lacks the conceptual wherewithal to grasp finite Hume. He might, for instance, lack the notion of a relation that would be required to grasp the second-order definition of one– one correspondence embodied in finite Hume. Or he might be unable to comprehend the notion of an arbitrary property required to grasp the significance 2 Bolzano [1950], pp. 98–9 endorses the following principle: “Whenever, in fact, two finite sets are constituted so that every object a in the one corresponds to another object b in the other which can be paired off with it, no object in either set being without a partner in the other, and no object occurring in more than one pair: then indeed are the two finite sets always equal in respect of multiplicity”.

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of its second-order quantifiers. Alternatively, instead of having any thought of numerical identity, he might have a concern only for the employment of numerals. More generally, it is contentious to suppose that any theoretical principle is implicit in ordinary arithmetical practice. For not only is the notion of a theoretical principle implicit in practice a very murky and difficult notion to apply—as Kripke’s rule following paradox makes clear—the more radical possibility remains that our arithmetical practice should not be described in theoretical terms at all. Perhaps ordinary arithmetical practice, rather than being informed by a ghostly inner theory, is better understood as the exercise of a repertory of arithmetical techniques. Of course, if ordinary arithmetical reasoning is not informed by a secondorder abstraction, then a fortiori it is not informed by Hume’s principle. Heck assumes that the neo-Fregean can only succeed in providing an epistemology of arithmetic if the basic laws employed for that purpose actually inform ordinary arithmetical reasoning. So if Heck is correct to make this assumption it appears the epistemological project the neo-Fregean undertakes cannot succeed. Unfortunately, Heck makes no attempt to justify the assumption that an arithmetical epistemology must be derived from principles that we can retrieve by reflection on ordinary arithmetical reasoning. Heck’s remarks suggest the following argument. It is constitutive of arithmetical truths that they are derived from the basic laws that actually inform ordinary arithmetic. We might call these laws the ‘canonical sources’ of arithmetical truths. Since Hume’s principle is not a canonical source, the a priori truths that may be derived from it cannot be arithmetical in character. But this argument is far from convincing. The premise that an arithmetical truth can only be derived from canonical sources is unmotivated, and generalised it leads to the absurd conclusion that a stronger principle cannot be employed—perhaps for reasons of elegance—to prove the consequences of a weaker principle. Heck’s objections to neo-Fregeanism reflect a presupposition common amongst its critics. According to this presupposition, the neo-Fregean project is a hermeneutic one: it aims to show that what we ‘had in mind’ all along, when we reasoned arithmetically, is a priori. It is now generally recognised that Frege had no concern to determine that ordinary arithmetical reasoning is a priori (see Benacerraf [1981], Weiner [1984], and Dummett [1991], pp. 176– 79). Regrettably, it is not generally recognised that neo-Fregeans need have no hermeneutic concern either. Benacerraf exhibits such a lack of recognition in his discussion of Hempel: According to Hempel the Frege-Russell definitions of number, successor, and related concepts have shown the propositions of arithmetic to be analytic because they follow by stipulative definitions from logical principles. What Hempel has in mind here is clearly that in a constructed formal system of logic one may introduce by stipulative definition the expressions ‘Number’, ‘Zero’. . . in such

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Having outlined Hempel’s logicist project, Benacerraf proceeds to remark (next sentence) that the project fails because Hempel does not establish that the arithmetical truths we express in ordinary language are analytic: He [Hempel] is not entitled to that conclusion. Nor would he be even if the theorems of logic in their primitive notations were themselves analytic. For the only things that have been shown to follow from theorems of logic by stipulation are the abbreviated theorems of the logistic system. To parlay that into an argument about the propositions of arithmetic, one needs an argument that the sentences of arithmetic, in their preanalytic senses, mean the same (or approximately the same) as their homonyms in the logicistic system. That requires a separate and longer argument. I bring this up here not to berate Hempel but to use his views as an illustration of the epistemological motivation that drives twentieth century logicists.

But Benacerraf’s observation is misplaced. For neither Hempel nor any other neo-Fregean ever claimed to be putting forward a thesis about the ordinary senses of arithmetical expressions. Hempel’s expresses the very different nature of his epistemological project with exemplary clarity: The assertion that the definitions given above state the “customary” meaning of arithmetical terms involved is to be understood in the logical, not the psychological sense of the term “meaning”. It would obviously be absurd to claim that the above definitions express “what everybody has in mind” when talking about numbers and the various operations that can be performed with them. What is achieved by those definitions is rather a “logical reconstruction” of the concepts of arithmetic in the sense that if the definitions are accepted, then those statements in science and everyday discourse which involve arithmetical terms can be interpreted coherently and systematically in such a manner that they are capable of objective validation. Hempel [1945], p. 387

The objections that Heck (and Benacerraf) voice are ineffective because they misconstrue the nature of the neo-Fregean project. That project never was to uncover a priori truth in what we ordinarily think, but to demonstrate how a priori truth could flow from a logical reconstruction of arithmetical practice. By failing to recognise the nature of the beast, Heck fails to articulate a convincing objection to the neo-Fregean doctrine that there is an a priori route from Hume’s principle to knowledge of arithmetic. 3 3 Of course, there are a host of other difficulties the neo-Fregean must confront in order to make good their claims, difficulties concerning the ability of Hume’s principle to introduce objects and the tenability of the minimalist metaphysic that principle assumes. I explore these issues further in MacBride [2003].

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References Benacerraf, P. [1981]: “Frege: The Last Logicist” in P. French et al. (eds) Midwest Studies in Philosophy VI, pp. 17–35. Minneapolis: University of Minnesota Press. Reprinted in Demopoulos [1995], pp. 41–67. Benacerraf, P. and Putnam, H. (eds) [1983]: The Philosophy of Mathematics: Selected Readings. Cambridge: Cambridge University Press. Bolzano, B. [1950]: Paradoxes of the Infinite trans. F. Prinhonsky. London: Routledge and Kegan Paul. Boolos, G. [1996]: “On the Proof of Frege’s Theorem” in A. Morton and P. Stich (eds), Benacerraf and His Critics, pp. 143–59.Oxford: Blackwells. Demopoulos, W. (ed.) [1995]: Frege’s Philosophy of Mathematics. Harvard: Harvard University Press. Dummett, M. [1991]: Frege: Philosophy of Mathematics. London: Duckworth. Frege, G. [1950]: Foundations of Arithmetic, trans. J.L. Austin. Oxford: Basil Blackwell. Hempel, C. [1945]: “On the Nature of Mathematical Truth”, American Mathematical Monthly, 52, pp. 543–56. Reprinted in Benacerraf and Putnam [1983], pp. 377–93. Heck, R. [1997]: “Finitude and Hume’s Principle”, Journal of Philosophical Logic, 26, pp. 598–617. MacBride, F. [2003]: “Speaking with Shadows: A Study of Neo-Fregeanism”, British Journal for the Philosophy of Science, 54, pp. 103–63. Weiner, J. [1984]: “The Philosopher behind the Last Logicist” in C. Wright (ed.) Frege: Tradition & Influence. Oxford: Basil Blackwell. Wright, C. [1997]: “On the Philosophical Significance of Frege’s Theorem” in R. Heck (ed.), Language, Thought and Logic: Essays in Honour of Michael Dummett, pp. 201–44. Oxford: Oxford University Press. Wright, C. [1998]: “On the Harmless Impredicativity of N= (‘Hume’s Principle’)” in M. Schirn (ed.), The Philosophy of Mathematics Today, pp. 339–68. Oxford: Clarendon Press.

COULD NOTHING MATTER? 1 Fraser MacBride

According to the neo-Fregean we may acquire a priori knowledge of arithmetic’s fundamental laws by reflecting upon the (recognisable) second-order logical consequences of an a priori principle (Hume’s Principle) that specifies identity conditions for cardinal numbers: (HP)(∀ F)(∀ G)(Nx : Fx = Nx : Gx ↔ F 1 − 1 G) This epistemological contention receives mathematical support from Frege’s Theorem, the result that Peano’s axioms can be interpreted and their interpretations proved in the system (Frege arithmetic) that results from adjoining (HP) to second-order logic (see Wright 1997: 202–11). 2 Black (2000) brands this contention “implausible” and argues that (HP) provides the wrong sort of reason for believing in the infinity of the number series. I will argue that a central argument in Black’s paper is ineffective, relying upon a popular misconception of the epistemological character of the neo-Fregean project (see Black 2000: 233–36 and also Heck 1997: 597–98 and Lowe 1998: 49–50). It is important that this misconception is corrected and the concern Black voices assigned its proper place. Otherwise an accurate assessment will continue to evade us of the relative merits and demerits of the neo-Fregean philosophy of mathematics compared to any other.

1.

Black’s thought experiment

Black asks us to imagine a tribe of arithmeticians whose basic notion of number is that of finite ordinal. They arrive at this notion by reflecting upon their practice of counting. The tribe counts a totally ordered collection of objects by linking its members one by one with some other objects (the ‘numbers’) taken in a privileged order. The last number so assigned is the 1 This paper first appeared in Analysis 62, [2002], pp. 125–135. Reprinted by kind permission of the editor and Blackwell Publishing. 2 All references to Wright, and Hale & Wright are to the reprints of their papers in Hale & Wright 2001.

95 Roy T. Cook (ed.), The Arché Papers on the Mathematics of Abstraction, 95–104. c 2007 Springer.

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ordinal number of the collection. Black also asks us to imagine that the tribe does not recognise the number zero. Let “W(R)” stand for the second-order statement that the relation R well-orders its field and let “R ∼ = S” stand for the statement that the orderings R and S are isomorphic. Then, according to Black (2000: 234), the notion of ordinal number that informs the tribe’s practice may be encapsulated by a principle (let’s call it the Tribe’s Principle) that specifies identity conditions for ordinal numbers: (TP)(∀ R)(∀ S)[(W(R) & W(S) & ∃ xRxx & ∃ xSxx) → (oR = oS ↔ R ∼ = S)] Continuing the fantasy, the tribe progress to the notion of cardinal number by establishing that all orderings of a given non-empty set have the same ordinal number. Their notion of cardinal number may then be captured by the restriction of (HP) to non-zero cardinals: (RHP)(∀ X)(∀ Y) [(∃yXy & ∃zYz) → (Nx : Xx = Nx : Yx ↔ X1 − 1 Y)] On this basis the tribe develops finitary arithmetic (or at least that portion of finite arithmetic required for counting the objects that the tribe actually encounters). But despite the considerable advances made by the tribe they fail to take Cantor’s leap and only envisage the application of (RHP) to finite totalities. Black’s thought experiment is designed to reveal that it is possible to have a “coherent understanding” of finite arithmetic and its applications which is not epistemologically founded on (HP) (2000: 235–36). For even though the tribe possesses such an understanding the principle for cardinal identity (RHP) they employ eschews—by contrast to (HP)—a commitment to zero. Moreover, (HP) is inconsistent with the principle of ordinal identity (TP) upon which the tribe’s understanding is ultimately founded. For whereas (HP) may be satisfied only in infinite domains, (TP) may be satisfied only in domains of finite size (inducing the Burali-Forti paradox otherwise) (Hodes 1984: fn 16). Since a coherent understanding of finite arithmetic can be achieved without recourse to (HP) Black concludes that (HP) cannot perform the foundational epistemological role the neo-Fregean proposes for it. There are a number of distinct issues here that require to be disentangled. First, Black’s assumption that (TP) is a mathematical principle—a principle of ordinal identity—may be questioned. For (TP) fails to exhibit an arguably constitutive feature of any genuinely mathematical principle, namely the feature of being conservative with respect to non-mathematical theories (cf. Field 1980: 9–16). Roughly, a mathematical principle should not allow the derivation of any non-mathematical conclusions from non-mathematical premises that could not have been drawn from those premises already. (TP) fails to be conservative (in this sense) because it cannot be satisfied in any infinite domain, even if the domain in question is composed solely of nonmathematical objects. Consequently (TP) allows us to draw a conclusion that we might not otherwise have drawn, the conclusion that there are only

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finitely many non-mathematical objects. 3 If it is indeed a characteristic mark of mathematical principles that they are conservative, it is correspondingly doubtful whether the understanding of the tribe that is based solely upon (TP) is genuinely mathematical in character. 4 Of course, Black may reject the general assumption that mathematical principles are (by constitution) conservative—although Black will then have to undertake the burden of explaining away the implausibility of supposing that a purely general mathematical proposition rules out the possibility of so many non-mathematical objects. Alternatively, he may reject the more specific assumption that (TP) need be labelled ‘mathematical’ in the first place— although he will then have to explain why this manoeuvre is anything more than ad hoc. But in either case it is open to the neo-Fregean to question whether (TP) is capable of serving as any sort of epistemological foundation, never mind an alternative one to (HP). For they may argue (to take one plausible example) that it is an open epistemic possibility that there are infinitely many spacetime points. Since the truth of (TP) is incompatible with any such infinitary hypothesis (therein lies (TP)’s failure to be conservative) it also an open epistemic possibility that (TP) is false. It then becomes a mystery how grasp of a principle that—without Cartesian excess—may legitimately be doubted could serve as a foundation for acquiring knowledge of arithmetical truths. These difficulties may be avoided by weakening (TP) so as to render it relevantly conservative. This result may be achieved by strengthening the antecedent of (TP) to characterise only concepts of finite extension: (WTP) (∀ R)(∀ S)[(W(R) & W(S) & ∃xRxx & ∃xSxx & Finite(S) & Finite(R)) → (oR = oS ↔ R ∼ = S)] It may (in any case) be argued that it is (WTP) rather than (TP) that underwrites the counting practice of the tribe in Black’s thought experiment. If, as Black suggests, the tribe is chary of the infinite then it is just as plausible to suppose that their practice is described by a version of (TP) restricted to the finite. But (WTP) may be satisfied even in infinite domains and so—by contrast to (TP)— appears not only conservative (in the relevant sense) but also consistent with (HP). 3 Less roughly, let T be any non-mathematical theory and, for any sentence A, let A* be the result of restricting the quantifiers in A to non-mathematical objects. Similarly, let T* be the result of restricting all the quantifiers in the theory T to non-mathematicals. Then a mathematical principle N is conservative iff for any sentence A, if T* + N implies A* then T implies A (cf. Wright 2000: 319). Now let S be the sentence stating that the non-mathematical universe is finite. Since (TP) implies S, (TP) + T* implies S even if T does not. Therefore, (TP) fails to be conservative. 4 By failing to be conservative (TP) also broaches a (plausible) constraint on abstraction principles that—like (HP)—are intended to introduce novel concepts (Wright 1997: 297). However, it is unclear from Black’s discussion whether this constraint should apply to (TP). For Black does not distinguish between the case where (TP) is laid down as a piece of conceptual innovation (in which case it may be obliged to be conservative) and the case where (TP) is intended as a codification of existing numerical practice (that may already fail to be conservative).

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The modal-re-constructive character of neo-Fregean epistemology

The fact remains, however, that (HP) incorporates a commitment to zero not even (WTP) incurs. And, as Black remarks, it is not only fantastical tribes whose arithmetical understanding may be zero free. The Greeks did not recognise zero nor did Dedekind or Peano in their original formulations of the axioms of arithmetic. But, according to Black, if (HP) is to perform the foundational role the neo-Fregean intends then “we must say that the reason there are infinitely many numbers is that 0 counts as one of them”. Since none of the aforementioned thinkers would have endorsed such a claim Black reasons that (HP) “can no longer be regarded as making explicit the ideas which already underlay our use of the natural number system” (Black 2000: 236). Black concludes that—contrary to neo-Fregean intention—(HP) cannot perform any foundational role in the epistemology of arithmetic. Black’s criticism is an instance of a general form of objection to the neoFregean programme that bemoans (HP) for its strength. According to objections of this form, (HP) cannot provide an analysis of the concept number because it incorporates existential commitments that no ordinary arithmetical reasoner needs to countenance. Usually such objections focus upon the commitment of (HP) to infinite numbers, numbers that prior to Cantor went almost entirely unnoticed (Heck 1997: 597–98). Black’s criticism reveals that (HP) might also be faulted as an analysis because it is committed to zero and— although Black does not mention the possibility—perhaps other finite numbers too (after all (HP) is also committed to 1, another number that has not always been recognised as such). However, objections of this form are in general misguided. 5 This is because the epistemological success of the neo-Fregean programme need not rely upon the effectiveness of (HP) as an analysis of ordinary arithmetical notions. Of course, neo-Fregeans do sometimes speak of (HP) as an “analysis” of the ordinary notion of number or “analytic of” that concept (see, for example, Wright 1983: 106–07 and Hale 1997: 99). Nevertheless, an alternative epistemology may be gleaned (and extrapolated) from what the neo-Fregeans have to say that makes no relevant play with the notion of analysis and obviates Black’s criticism. Black’s criticisms fail to take proper account of the modal character of this epistemology. Neo-Fregean epistemology (so envisaged) offers an account of how it is possible to acquire knowledge of the fundamental laws of arithmetic (by deriving them from (HP)). It thereby undertakes to describe “an a priori route” (that goes via the recognition of zero) to knowledge of the laws of arithmetic (Wright 1997: 279–80). But is not thereby committed, as Black 5 There are, in addition, particular reasons why objections to (HP) that complain about the commitment of (HP) to infinite numbers are also misguided. I discuss some of these reasons elsewhere (MacBride 2000).

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assumes, to saying this route is “the” only one available. It is consistent with our coming to recognise arithmetical truths in one way that we could have, and perhaps do, come to recognise their truth by different means. So the mere fact that certain figures—historical or imaginary—could have achieved a coherent understanding of arithmetic from which a recognition of zero was absent does nothing to compromise the neo-Fregean contention that it is possible for (HP) to serve as a basis for acquiring arithmetical knowledge. Black’s criticisms also fail to take proper account of the re-constructive character of the neo-Fregean epistemology. According to Black, (HP) can only discharge a foundational role if it makes explicit the principles that actually underlie established arithmetical usage. But, according to the neo-Fregean epistemology, (HP) can only discharge its intended role because, in the first instance, it does not answer to existing usage. Rather, the neo-Fregean claims, (HP)—properly understood—is nothing more than a stipulation that serves to introduce a novel operator (“Nx”) into our language (Wright 1997: 278, Wright 2000: 317–18, Hale & Wright 2000: 142, Hale & Wright 2001: 14). The introduction is achieved by implicit definition: the meaning of the novel operator is fixed by stipulating that the truth conditions of identity statements (“Nx : Fx = Nx : Gx”) in which it occurs coincide with the conditions under which an equivalence relation amongst concepts may be said to obtain (“F 1 − 1 G”). And it is because (HP) is intended merely as a stipulation that the neoFregean feels able to legitimately claim that (HP) is a priori. Nevertheless, the neo-Fregean continues, (HP) provides a basis for grasping arithmetical truths a priori because (as Frege’s theorem demonstrates) the system that results from (HP) and second-order logic allows for a reconstruction of ordinary arithmetical practice in the following sense. It—Frege arithmetic—suffices for the interpretation of the laws of ordinary arithmetic and the proof of their interpretations. It is in virtue of the interpretative powers of the system (HP) engenders that the neo-Fregean takes himself to be retrospectively entitled to characterise (HP) an arithmetical principle, a principle of cardinal (in the usual sense) identity. To simply complain that (HP) fails to “make explicit the ideas which already underlay our use of the natural number system” (Black 2000: 236) is to fail to take into account the re-constructive character of the epistemology proposed and the crucial role Frege’s theorem performs in the envisaged reconstruction.

3.

The counter-Caesar problem

Whilst this characterisation of the neo-Fregean programme obviates the criticisms of Black (and others) it also brings into focus a potentially critical epistemological difficulty quite peculiar to that programme. For—strictly speaking—Frege’s theorem does not establish that the truths of ordinary arithmetic are themselves a priori. It only establishes that there is a system of truths (Frege arithmetic) capable of modelling (interpreting) the laws of arithmetic.

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Assume the best case scenario for the neo-Fregean and suppose the truths that comprise Frege arithmetic are a priori in character. An additional argument is still required to show that the truths of arithmetic inherit the epistemological status of their Fregean models (see Benacerraf 1981: 20, Heck 1997: 596). There appear to be at least two styles of strategy—re-constructive and hermeneutic—whereby the neo-Fregean might endeavour to address this issue. According to the re-constructive strategy, the neo-Fregean may accept that the truths expressed in Frege arithmetic concern an entirely novel subject matter and merely model the truths of ordinary arithmetic. Nevertheless, the neo-Fregean may still maintain that Frege’s theorem bears epistemological significance for ordinary arithmetic. For, the neo-Fregean may argue, the mappings that the theorem establishes a priori between the a priori truths of Frege arithmetic and ordinary arithmetic suffice to demonstrate the operational effectiveness (reliability), if not the truth, of ordinary arithmetical claims. 6 Alternatively, the neo-Fregean may adopt the hermeneutic strategy according to which all the truths of the ordinary arithmetic are expressed by truths of Frege arithmetic. Ordinary arithmetic will then automatically inherit the a priori status of the latter system. The difficulty attendant upon this second strategy is, in a sense, the reverse of the more familiar Caesar problem (or at least one member of that family of problems). The latter difficulty concerns our capacity (or lack of it) to establish that the terms and sentences occurring in two different theories (concerning, for example, persons and numbers respectively) are, as we might pre-theoretically suppose, terms for quite different entities (Julius Caesar, 2) and sentences expressive of very different truths. The former difficulty may appropriately be dubbed the ‘counter-Caesar’ problem. It concerns our capacity to establish that the terms and sentences figuring in two different theories (ordinary arithmetic, Frege arithmetic) are, as the neo-Fregean would have it, terms for the same entities (2, Nz:[z = Nx: x =x v z = Ny:(y = Nx: x = x)]) and sentences expressive of the same truths. In fact, the neo-Fregean appears to adopt the second strategy and is therefore obliged—if he is to realise the epistemological pretensions of his programme—to take on the counter-Caesar problem with all the seriousness usually reserved for the Caesar problem itself. The neo-Fregean proceeds upon the assumption “that to define the distinctively arithmetical concepts is to so define a range of expressions that the use thereby laid down for those expressions is indistinguishable from that of expressions which do indeed express those concepts” (Wright 2000: 322). He then claims that the stipulation of (HP) gives rise to a pattern of linguistic use that is ensured by the interpretability of Peano’s axioms in Frege arithmetic to be equivalent to ordinary arithmetic. 6 The neo-Fregean may also add that the existence of these mappings is in no way compromised by the commitment of Frege arithmetic to zero (and other numbers ordinary arithmetic fails to recognise). So, the neo-Fregean may conclude, a commitment to zero can hardly provide a reason for denying (HP) the status of an epistemological foundation.

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So, relying on the aforementioned assumption, the neo-Fregean concludes that regardless of the different underlying principles ((HP), Peano’s axioms or some other source) that gave rise to that same pattern of use the very same arithmetical truths are thereby expressed. 7 An analogy may help bring into relief the epistemological character now envisaged for the neo-Fregean project. According to Davidson, we may achieve insight into the nature of language by reflecting upon the possibility of a radical interpreter who (entirely ignorant of a given language L) constructs a theory knowledge of which suffices for interpreting a speaker of L (Davidson 1973). The radical interpreter does not, however, make any attempt to describe the inner psychological mechanisms that in fact account for the speaker’s mastery of L. Nonetheless, Davidson claims, the theory supplied by a radical interpreter provides insight into the character of the complex linguistic abilities displayed by ordinary speakers of L. This is because (very roughly) the theory is empirically adequate to their linguistic performances. In an analogous way, the neo-Fregean claims, we may achieve insight into the nature of arithmetical knowledge. This time we are asked to reflect upon the possibility of a character (call him “Hero”) who (entirely ignorant of arithmetic) seeks to construct a theory knowledge of which suffices for the competent use of ordinary arithmetical language. 8 Hero makes no efforts to uncover the psycho-genetic origins of our arithmetical abilities. Nonetheless, the neo-Fregean claims, the a priori theory provided by Hero provides insight into the character of arithmetical knowledge. This is because grasp of his theory provides Hero with the ability to engage in a practice of use equivalent to the arithmetical performance displayed by ordinary speakers. The a priori theory Hero supplies is Frege arithmetic. The assurance that knowledge of this theory engenders competence in arithmetic flows (very roughly) from an appreciation of Frege’s theorem.

4.

Meaning-theoretic foundations of neo-Fregeanism

How then should the neo-Fregean programme be assessed once it is liberated from the popular misconception that it was ever intended to characterise whatever principles in fact underlie ordinary arithmetical reasoning? If it is to carry conviction the neo-Fregean assumption—that ‘arithmetical’ systems which exhibit the same pattern of use refer to the same objects and express 7 More generally, the neo-Fregean must establish that the stipulation of (HP) provides for two distinct patterns of use that are respectively equivalent to the distinct uses of pure and applied arithmetical language. The neo-Fregean takes the former result to be established by demonstrating the intepretability of Peano’s axioms within Frege arithmetic. The latter result is secured by deriving from (HP) the principle (Nq) that relates (in the intuitive manner) pure occurrences of the numerals “nf ” of Frege arithmetic with appropriate applied occurrences of the numerals “n” of ordinary arithmetic (Wright 2000: 322, 330–32): (Nq) nf = Nx : Fx ↔ there are exactly n Fs. 8 The character Hero was introduced by Wright (1997: 247) for the heuristic purpose of showing how Frege’s definitions of zero and its successors might be grasped upon the basis of (HP) and second-order logic. Here I extend Hero’s role to show how ordinary numerals might similarly be grasped.

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the same truths about them-must be underwritten by a general and principled conception of how meaning and use relate. That conception must include a meaning-theoretic doctrine of (at least) the following strength: (MSU) Meaning (truth and reference) supervenes on use.

But by relying upon (MSU) the neo-Fregean undertakes a distinctive theoretical commitment that it is far from trivial. First, if the neo-Fregean employs (MSU) then he will confront a substantial explanatory challenge. On the one hand, the notion of ‘use’ figuring in (MSU) cannot be so behaviouristically conceived that it becomes implausible to suppose intentional facts could ever supervene on facts about use (a difficulty familiar to us from Kripke’s rule-following paradox). On the other hand, the relevant notion of ‘use’ cannot be so intensionally conceived that it becomes impossible for ordinary arithmetic and Frege arithmetic to exhibit the same pattern of use. To vindicate his stance it is incumbent upon the neoFregean to supply an account of use that navigates between these undesirable consequences. Second, a version of Black’s original concern resurfaces—this time in its proper place. For even if such a notion of ‘use’ can be supplied, the neoFregean faces the further difficulty that ordinary arithmetic and Frege arithmetic exhibit different patterns of use (in whatever sense of use that might turn out to be). This is because Frege arithmetic is a far richer language than ordinary arithmetic. It includes expressions putatively referring to numerical objects for which there may be no corresponding ordinary arithmetical terms: for example, the number of identical things, the number of natural numbers, the number of non-self-identical things, and so on. To accommodate this point, the neo-Fregean must claim that the meaning of a given expression does not supervene upon the global pattern of use associated with it, but rather upon some relevant local holism. More specifically, the neo-Fregean must claim that the meanings of expressions in Frege arithmetic that do correspond to terms in ordinary arithmetic are determined by a local pattern of use from which the more colourful, unfamiliar expressions of Frege arithmetic are excluded. The neo-Fregean will then seek to harness Frege’s theorem to show that ordinary arithmetical terms exhibit the very same local pattern of use, and so mean the same, as their Fregean counterparts. But to make good such claims the neo-Fregean is obliged to undertake a further explanatory task: to explain the significance of the notion ‘relevant local holism’ and to provide some principled account of why meaning should be taken to supervene on that sort of use. Part of this task will be to systematically specify—in the face of widespread Quinean scepticism that it cannot be done (see Quine 1976 and, more recenly, Fodor & LePore 1992: 163–83)—those aspects of the use of an expression that confer meaning (and belong to the relevant local holism) from those aspects that do not confer meaning (and

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belong only to a wider pattern of use). There is, however, reason to be sanguine about the neo-Fregean’s prospects of executing this task. After all, if some such contrast could not (at some level) be made out then the possibility of translating from a richer language into a more impoverished one, or acquiring greater knowledge of a language that we have already partially learnt, would appear to be foreclosed. Moreover, if it turns out that the explanatory task cannot be discharged it is always open to the neo-Fregean to avoid the counterCaesar problem by adopting the first strategy proposed and seeking only to model ordinary arithmetical usage. Black concludes with the suggestion that his real objection to neoFregeanism is not so much to do with the commitment of (HP) to zero (or the existence of some other object). He writes: “rather the problem is the way in which (HP) generates an infinity of numbers, generating new numbers to count the numbers already there with a tail biting circularity” (2000: 237). Black here raises an objection to the impredicative character of (HP), a form of objection familiar from Dummett’s critique of the neo-Fregean programme (Dummett 1991: 226–29). But this sort of objection must surely be distinguished from Black’s earlier concern that (HP) fails to make explicit the ideas that underlie ordinary arithmetic. For even supposing a 21st century Boolos were to uncover a predicative version of (HP) with sufficient strength to generate an ‘arithmetical’ system, the principle in question might still fail to underlie ordinary practice. Moreover, even if (HP) did underlie ordinary practice this would in no way address the objections to (HP) based upon its impredicative character. Finally, it is worth reflecting that impredicativity (specified in such general terms) appears to be a ubiquitous phenomenon. In the linguistic environment into which we are thrown the meaning of a given word can never—or so it seems—be determined independently of some significant portion of the sentential contexts in which that word occurs. It may well be that the terms introduced by (HP) are impredicative in a stronger and more objectionable sense than this. But until we have equipped ourselves with a more discerning means of saying just why this is so we—Black and Dummett included—should be more reticent to dismiss (HP) on such generic grounds. 9

References Benacerraf, P. 1981. Frege: The Last Logicist. In Midwest studies in philosophy VI, eds. P. French et al., 17–35. Minneapolis: University of Minnesota Press. Black, R. 2000. Nothing matters too much, or Wright is wrong. Analysis, 60: 229–37. Boolos, G. 1997. Is Hume’s Principle Analytic? In Heck: 1997a: 245–62. Davidson, D. 1973. Radical Interpretation. Dialectica 27: 313–28. 9 The neo-Fregean programme remains, however, beset by many other challenges, not least that of justifying the metaphysical assumption it presupposes of an intimate communion between language and reality. I explore these issues further in MacBride [2003]. For comments and discussion of the present paper thanks to Robert Black, Peter Clark, Roy Cook, Bill Demopoulos, Philip Ebert, Patrick Greenough, Jonathan Hesk, Stephanie Schlitt, Stewart Shapiro, Crispin Wright, and an anonymous referee for this journal.

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Dummett, M. 1991. Frege Philosophy of Mathematics. London: Gerald Duckworth & Co. Ltd. Field, H. 1980. Science without Numbers. Oxford: Basil Blackwell. Fodor, J. & LePore, E. 1992. Holism: A Shopper’s Guide. Oxford: Basil Blackwell. Hale, B. 1997. Grundlagen §64. Proceedings of the Aristotelian Society 97: 243–61. Reprinted in Hale & Wright 2001: 90–116. Hale, B. & Wright C., 2000. Implicit Definition and the a priori. In New Essays on the A Priori, ed. P. Boghossian and C. Peacocke, 286–319. Oxford: Clarendon Press. Reprinted in Hale & Wright 2001: 117–50. Hale, B. & Wright, C. 2001. The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics. Oxford: Clarendon Press. Heck, R. 1997. Finitude and Hume’s Principle. Journal of Philosophical Logic, 26: 598–617. Heck, R. 1997a. Language, Thought and Logic: Essays in Honour of Michael Dummett. Oxford: Oxford University Press. Hodes, H. 1984. Logicism and the Ontological Commitments of Arithmetic. Journal of Philosophy 81: 123–49. Lowe, E.J. 1998. The Possibility of Metaphysics: Substance, Identity and Time. Oxford: Clarendon Press. MacBride, F. 2000. On Finite Hume. Philosophia Mathematica, 8: 150–59. MacBride, F. 2003. Speaking with Shadows: A Study of Neo-Logicism. British Journal for the Philosophy of Science, 54: 103–63. Quine, W.V.O. 1976. Carnap and Logical Truth. In his Ways of Paradox and other essays, 107–32. Cambridge, Mass.: Harvard University Press. Wright, C. 1983. Frege’s Conception of Numbers as Objects. Aberdeen: Aberdeen University Press. Wright, C. 1997. On the Philosophical Significance of Frege’s Theorem. In Heck 1997a: 201–44. Reprinted in Hale & Wright 2001: 272–306. Wright, C. 1998. On the Harmless Impredicativity of N= (‘Hume’s Principle’). In M. Schirn (ed.), The Philosophy of Mathematics Today, ed. M. Schirn, 339–68. Oxford: Clarendon Press. Reprinted in Hale & Wright 2001: 229–55. Wright, C. 2000. Is Hume’s Principle Analytic? Notre Dame Journal of Formal Logic 40. Reprinted in Hale & Wright 2001: 307–22.

ON THE PHILOSOPHICAL INTEREST OF FREGE ARITHMETIC 1 William Demopoulos

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Fregean logicism: the laws of logic have an arithmetical content

Traditional “Fregean” logicism held that arithmetic could be shown free of any dependence on Kantian intuition if its basic laws were shown to follow from logic together with explicit definitions. It would then follow that our knowledge of arithmetic is knowledge of the same character as our knowledge of logic, since an extension of a theory (in this case the “theory” of secondorder logic) by mere definitions cannot have a different epistemic status from the theory of which it is an extension. If the original theory consists of analytic truths, so also must the extension; if our knowledge of the truths of the original theory is for this reason a priori, so also must be our knowledge of the truths of its definitional extension. The uncontroversial point for traditional formulations of the doctrine is that a reduction of this kind secures the sameness of the epistemic character of arithmetic and logic, while allowing for some flexibility as to the nature of that epistemic character. Thus, it is worth remembering that in Principles (p. 457), Russell concluded that a reduction of mathematics to logic would show, contrary to Kant, that logic is just as synthetic as mathematics. Nevertheless, the methodology underlying this approach to securing the aprioricity of arithmetic by a traditional logicist reduction has been challenged. For example, Paul Benacerraf, 2 who focuses on Hempel’s 3 classic exposition, tells us that 1 This paper first appeared in Philosophical Books 44, [2003], pp. 220–228. It is reprinted by kind permission of the editor and Blackwell Publishing. With the exception of section headings and a small number of minor stylistic changes, it is unaltered. A Postscript addresses Hale and Wright’s response to the original paper. 2 “Frege: The last logicist,” in William Demopoulos (ed.) Frege’s philosophy of mathematics (Harvard University Press: 1995), pp. 42 and 46. 3 C. G. Hempel, “On the nature of mathematical truth,” Hilary Putnam and Paul Benacerraf (eds.) The philosophy of mathematics: selected readings, second ed. (Cambridge University Press: 1983), 377–393.

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The Arché Papers on the Mathematics of Abstraction . . . logicism was . . . heralded by Carnap, Hempel . . . and others as the answer to Kant’s doctrine that the propositions of arithmetic were synthetic a priori . . . in reply to Kant, logicists claimed that these propositions are a priori because they are analytic—because they are true (or false) merely “in virtue of” the meanings of the terms in which they are cast. . . . According to Hempel, the Frege-Russell definitions . . . have shown the propositions of arithmetic to be analytic because they follow by stipulative definitions from logical principles. What Hempel has in mind here is clearly that in a constructed formal system of logic (set theory or second-order logic plus an axiom of infinity), one may introduce by stipulative definition the expressions ‘Number,’ ‘Zero,’ ‘Successor’ in such a way that sentences of such a formal system using these introduced abbreviations and which are formally the same as (i.e., spelled the same way as) certain sentences of arithmetic—e.g., ‘Zero is a Number’—appear as theorems of the system. He concludes . . . that these definitions show the theorems of arithmetic to be mere notational extensions of theorems of logic, and thus analytic. He is not entitled to that conclusion. Nor would he be even if the theorems of logic in their primitive notations were themselves analytic. For the only things that have been shown to follow from the theorems of logic by [stipulative definitions] are the abbreviated theorems of the logistic system. To parlay that into an argument about the propositions of arithmetic, one needs an argument that the sentences of arithmetic, in their preanalytic senses, mean the same (or approximately the same) as their homonyms in the logistic system. That requires a separate and longer argument.

Benacerraf is questioning whether the logicist can claim to have established any truth of arithmetic on the basis of a successful reduction. What is required according to Benacerraf, is a supplementary argument showing that the logicist theorems have the preanalytic meanings of their ordinary arithmetical analogues. But Benacerraf’s demand for a further argument is not justified. The philosophical interest elicited by traditional logicism derived from the fact that it was thought implausible that the concepts and laws of logic could have an “arithmetical content.” To have successfully dispelled this belief it would have been sufficient to have shown that the concepts of logic allow for the explicit definition of notions which, on the basis of logical laws alone, demonstrably satisfy the basic laws of arithmetic. The philosophical impact of the discovery that the concepts and laws of logic have an arithmetical content in this sense would not have been in any way diminished by the observation that the preanalytic meanings of the primitives of arithmetic were not the same as their logicist reconstructions. The sense in which the logicist thesis must be understood in order to be judged successful cannot therefore be the one for which Benacerraf claims Hempel must argue. Notice also that independently of one’s view of meaning and truth in virtue of meaning, it must be conceded that traditional logicism would have provided a viable answer to Kant had it succeeded in showing that arithmetical knowledge requires only an extension of logic by explicit definitions. Hempel’s appeal to these notions addresses a different issue: Frege left the

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problem of securing the epistemic basis of the laws of logic largely untouched. Benacerraf’s Hempel should be understood as proposing to fill this gap by suggesting that the laws of logic are true in virtue of the meanings of the logical constants they contain. Like Frege, Hempel seeks to secure the aprioricity of arithmetic by an argument that proceeds from its analyticity. But Hempel’s version of logicism differs from Frege’s, for whom “analytic” merely meant belonging to logic or a definitional extension of logic, by providing a justification for the analyticity of logical laws: logical laws are analytic, not by fiat as on Frege’s account, but because they are true in virtue of the meanings of the logical terms they contain. From this it would follow that if logical laws are true in virtue of meaning, so also is any proposition established solely on their basis, where “established solely on their basis” is intended to encompass the use of explicit definitions. The clarity of the thesis that the laws of logic are true in virtue of meaning is therefore central to Hempel’s presentation of the view. Also central is the substantive and further claim that the basic laws of arithmetic can be recovered within a definitional extension of logic. The implied criticism of Hempel’s appeal to truth in virtue of meaning gains its force from the difficulties that stand in the way of establishing the traditional logicist thesis that arithmetic is reducible to logic in the original sense of the doctrine. Certainly, the failure to sustain this thesis led to more ambitious applications of the notion of truth in virtue of meaning. But if the basic laws of arithmetic had been recovered as a part of logic—not merely shown to have analogues that are part of some formal system or other, but to be part of logic—what more would be needed to infer that they share the epistemic status of logical laws? Once Hempel is not represented as seeking to secure the truth of the basic laws of arithmetic by an appeal to the derivability of mere formal analogues or a blanket appeal to the notion of truth in virtue of meaning, it is clear that he simply doesn’t owe us the argument Benacerraf claims he does. The difficulties that attend traditional logicism are therefore not the methodological difficulties Benacerraf advances, but the simple failure to achieve the stated aim of showing arithmetic to be a definitional extension of logic. This point is obscured by Benacerraf’s suggestion that the reduction might proceed from second-order logic with an axiom of infinity or from some version of set theory. Neither theory supports the truth in virtue of meaning account that underlies Hempel’s formulation of logicism. A reduction to second-order logic with infinity would mean a reduction to a system augmented with an axiom like Whitehead and Russell’s; but no one ever thought such a system was true in virtue of meaning. As for a reduction to set theory, set theory is properly regarded as the arithmetic of the transfinite. Why should a reduction of the natural numbers to such a generalized arithmetic be regarded as a means of establishing its aprioricity on a less synthetic footing? The only coherent logicist methodology would therefore seem to be the one just outlined: to reduce arithmetic to a theory like Begriffsshrift’s. Unfortunately, such a theory is either too weak, or in the presence of Frege’s theory of classes, inconsistent.

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Wright and Hale’s neo-Fregean alternative

The renewed interest in logicism is based on the fact that the secondorder theory having Hume’s principle 4 as its only non-logical axiom—“Frege arithmetic”—has a definitional extension which contains the Dedekind–Peano axioms. The neo-Fregean program of Crispin Wright and Bob Hale seeks to imbed this logical discovery into a philosophically interesting account of our knowledge of arithmetic by subsuming Hume’s principle under a general method for introducing a concept by an “abstraction principle.” 5 This program explains the epistemological interest of the discovery that arithmetic is a part, not of second-order logic, but of Frege arithmetic, by the program’s account of concept introduction. The key to achieving this goal is the idea that abstraction principles have a distinguished status: they are a special kind of stipulation. Their stipulative character shows them to be importantly like explicit definitions even if their creativeness suggests an affinity with axioms; and it is a central tenet of neo-Fregean logicism that abstraction principles are sufficiently like definitions to yield an elegant explanation of why arithmetical knowledge is knowledge a priori. The neo-Fregean program has a methodological dimension that parallels the role of the theory of definition in traditional logicism. Frege accords a statement the status of a proper definition if it meets conditions of eliminability and conservativeness. The classical theory of definition is supplemented by the neo-Fregean methodology of good abstractions. Thus, the theory of definition mandates that a definitional extension must be conservative in the familiar sense of not allowing the proof of sentences formulated in the unextended vocabulary which are not already provable without the addition of the definitions which comprise the extension. But “extensions by abstraction” need not be conservative in this sense; indeed interesting extensions are interesting precisely because they are not conservative in the sense of the theory of definition. The neo-Fregean theory of good abstractions allows for classically non-conservative extensions—extensions which properly extend the class of provable sentences—while imposing a constraint on the consequences an extension by good abstraction principles can have for the ontology of the theory to which they are added. This methodology is constrained and principled, it is just not constrained in the same way as the classical theory of definition. We can, perhaps, put the difference by saying that the constraints on definition have a more purely epistemic motivation than do the constraints the neo-Fregean imposes on good abstractions. 4 Hume’s principle tells us that for any concepts F and G, the number of Fs is identical with the number of Gs if, and only if, the Fs and the Gs are in one–one correspondence. 5 By an abstraction principle, Hale and Wright mean the universal closure of an expression of the form (X) = (Y) ↔ X ᑬ Y, where ᑬ is an equivalence relation, the variables X and Y may be of any type, and the function may be of mixed type. In the case of Hume’s principle, the equivalence relation is the (second-order definable) relation on concepts of one–one correspondence, and the “cardinality function,” is a type-lowering map from Fregean concepts to objects.

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In my view, the reticence of the classical theory of definition to allow a mere definition to properly extend the theory to which it is added is wellfounded, and should also inform the epistemic basis of a principle as rich as Hume’s. My goal here is to consider whether the neo-Fregean account of Hume’s principle as a kind of stipulation can support the epistemological claim of neo-Fregeanism to have secured the aprioricity, if not the analyticity, in one or another traditional sense of the notion, of our arithmetical knowledge. The matter is taken up by Hale and Wright in their paper “Implicit definition and a priori knowledge”—and by Wright in his “Is Hume’s principle analytic?” which, notwithstanding its title, is not concerned to secure the analyticity of Hume’s principle but to address the question of its epistemic status within the neo-Fregean program and the light it sheds on our arithmetical knowledge. 6 Wright and Hale use the stipulative character of Hume’s principle as a premise in an argument for the aprioricity of our arithmetical knowledge. This becomes clear when we reflect on the fact that they are concerned to show that our knowledge of arithmetic can be represented as resting on a principle that introduces the concept of number. In acquiring the concept of number, we acquire a criterion of identity for number—a criterion for saying when the same number has been given to us in two different ways as the number of one or another concept. This criterion of identity—Hume’s principle—affords the only non-logical premise needed to derive the basic laws of arithmetic. Our arithmetical knowledge is secured, therefore, with our grasp of the concept of number and is based on nothing more than what we acquire when we are introduced to the concept. But since this knowledge rests on a stipulation, it is unproblematically knowledge a priori.

3.

Recovering the epistemic status of ordinary arithmetic by modeling

This is essentially the same account of the philosophical interest of Frege arithmetic that is elaborated by Fraser MacBride in two thoughtful papers 7 that address this issue. For MacBride the neo-Fregean explanation of the aprioricity of our knowledge of arithmetic runs as follows: We first stipulate a criterion of identity for a special kind of objects; call them cardinal numbers. That certain fundamental truths about these objects are established on the basis of a stipulation guarantees that our knowledge of those truths is knowledge a priori. This is to be contrasted with an account which would seek to infer the aprioricity of our knowledge of arithmetic from theses about meaning or truth in virtue of meaning. The neo-Fregean account does not depend on a traditional notion of analyticity: since neo-Fregeanism demands only 6 Both reprinted in their collection of their papers, The reason’s proper study (Oxford University Press: 2002), as chapters 5 and 13, respectively. 7 “Finite Hume,” Philosophia mathematica 8 (2000) 150–159, and “Can nothing matter?,” Analysis 62 (2002) 125–134.

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the relatively uncontentious concession of the aprioricity of a stipulation, it can claim that its explanation of the aprioricity of arithmetic need not address the difficulties associated with defending traditional conceptions of analyticity. The fact that the reduction to Frege arithmetic requires more than a merely definitional extension of second-order logic suggests an objection to neoFregean logicism that is closely related to the one we saw Benacerraf urge against to Hempel: How, one might ask, does our knowledge of the truths that hold of the objects the neo-Fregean has singled out—the Frege-numbers— bear on our knowledge of the numbers, on the subject matter of ordinary arithmetic? In so far as the epistemological issues are issues concerning ordinary arithmetic, have they even been addressed by the neo-Fregean? In this form, the objection presupposes only preservation of subject matter—a minimal requirement that it would be difficult to justify not meeting—and says nothing about preservation of meaning. The first of two neo-Fregean responses to this objection that I wish to review holds that it is because ordinary arithmetic can be “modeled” in Frege arithmetic that the epistemological status of the truths of Frege arithmetic is shared by the truths of ordinary arithmetic. Wright remarks (p. 322) that this answer is too weak. And although MacBride does not endorse this response, neither does he reject it as altogether unlikely. Nevertheless, I think it is worth recording exactly why such a straightforward answer, couched in terms of the relatively unproblematic relation of modeling, can’t be right. It is clearly possible to stipulate the conditions that must obtain for the properties of a purely hypothetical and imaginary “abstract” physical system to hold without in any way committing ourselves to the existence—or even the dynamical possibility—of such a system. Our knowledge that such abstract systems are configured in accordance with our stipulations is no less a priori than our knowledge that, for example, the four element Boolean algebra has a free set of generators of cardinality one. But it sometimes happens that abstract configurations “model” actual configurations, in the sense that there is a correspondence between the elements of the two systems that preserves fundamental properties. It is clear that in such circumstances we take ourselves to know more than that an imaginary example has the properties we stipulate it to have: if the example is properly constructed, we know the dynamical behavior of a part of the physical world. But of course the fact that an actual system is “modeled” by our imaginary system, together with the fact that our knowledge of the properties of our imaginary system is a priori knowledge because it depends only on our free stipulations, are completely compatible with the claim, obvious to preanalytic intuition, that our knowledge of the dynamical behavior of the actual system is a posteriori. Whatever role stipulation may have in fixing the properties of the abstract system by which the behavior of some real process is modeled, it lends no support to the idea—and

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would never be regarded as lending support to the idea—that our knowledge of the real process is knowledge a priori. There is a disanalogy between the number-theoretic case and our example that might seem to undermine its effectiveness as a criticism. In the numbertheoretic case the existence of the correspondence between ordinary numbers and the “Frege-numbers” that model them is known a priori. But the correspondence between the abstract system of our example and the actual system is not known a priori; it depends on the a posteriori knowledge that there are in reality configurations of particles having the postulated characteristics. This is of course entirely correct. However it is of no use to the neo-Fregean, since to know a priori that there is a mapping between the ordinary numbers and the Frege-numbers it is necessary to have a priori knowledge of the existence of the domain and co-domain of the mapping. To be of any use to the neoFregean, the fact that the Frege-numbers model the ordinary numbers therefore requires that our knowledge of the ordinary numbers be a priori. But if the modeling of the ordinary numbers by the Frege-numbers presupposes that our knowledge of the ordinary numbers is a priori, it cannot be part of a noncircular account of why ordinary arithmetic is known a priori. The general point may be put as follows: The fact that M models N , so that for any sentence s, s is true in M if and only if s is true in N , does not entitle us to infer that because the sentences true in M are known a priori, the sentences true in N are known a priori. Indeed it is perfectly possible that (with the obvious exception of the logical truths) our knowledge of sentences true in N is wholly a posteriori. So even if we grant that an assumption rich enough to secure an infinity of objects is correctly represented as a stipulation, it remains unclear how the neo-Fregean can use this fact to answer the question which motivates his account of arithmetical knowledge—it remains unclear how it yields an account of our knowledge of the numbers, knowledge that we have independently of the neo-Fregean analysis. Notice that this objection depends only on an observation about the modeling of one domain by another, and that, in particular, it does not require the resolution of various difficult issues in the theory of meaning.

4.

Recovering the epistemic status of ordinary arithmetic by preserving a pattern of use

There is an alternative to the response based on modeling. The idea is that since Frege arithmetic captures the “patterns of use” exhibited by our ordinary number-theoretic vocabulary, both in pure cases and in applications, we are justified in inferring not merely that the Frege-numbers model the ordinary numbers but that the Frege-numbers are the ordinary numbers. Showing that Frege arithmetic captures the patterns of use of our ordinary number-theoretic vocabulary constitutes a considerable strengthening of the claim that ordinary arithmetic is merely modeled by Frege arithmetic.

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Suppose we grant, both that Frege arithmetic captures the patterns of use of our ordinary number-theoretic vocabulary and that because of this, ordinary arithmetic and Frege arithmetic share the same subject matter. Vindicating the claim that ordinary arithmetic and Frege arithmetic share a subject matter is only one of the difficulties that the weaker understanding of the view in terms of modeling fails to address. If we are only modeling ordinary arithmetic, it is unproblematic to hold that as a statement of the modeling theory Hume’s principle is known a priori because it is a mere stipulation. The difficulty, as we saw, is that this fails to transfer to the aprioricity of the truths we are modeling—to the truths of ordinary arithmetic. Is this difficulty removed when the neo-Fregean account is extended to one which claims to capture the patterns of use of our ordinary number-theoretic vocabulary? And does it illuminate the epistemic status of the basic laws of arithmetic to observe that, in the neo-Fregean reconstruction of our patterns of use, Hume’s principle has the status of a stipulation? When neo-Fregeanism is understood to preserve our patterns of use, it becomes virtually indistinguishable from the traditional idea that the account of the numbers given by Frege arithmetic is analytic of the ordinary notion of number, so that a major burden of the account now falls on establishing the adequacy of an analysis in something very much like the traditional sense. This is a task the neo-Fregean had sought to avoid, since once the neoFregean has to defend the idea that the patterns of use of ordinary numerical expressions have been captured, the simplicity of urging the stipulational character of Hume’s principle, and then basing the aprioricity of arithmetic on this footing, has been lost: the principle no longer governs the introduction of a new concept but is constrained to capture an existing one. But let us grant both that sameness of pattern of use implies sameness of reference and that Frege arithmetic does in fact capture the pattern of use of our ordinary arithmetical vocabulary and, therefore, articulates a successful reconstruction of our arithmetical knowledge. Since we have given up the idea that Hume’s principle is being used simply to introduce a new concept, but forms part of an attempt to articulate principles that capture our numerical concepts as they are given by the patterns of use of our ordinary number-theoretic vocabulary, its justification does not consist merely in its being laid down as a stipulation. Rather, Frege arithmetic is justified because it captures the fundamental features of the judgements—pure and applied—that we make about the numbers. For the neo-Fregean, the reconstruction must not only capture an existing concept by recovering the patterns of use to which our arithmetical vocabulary conforms, it must also illuminate the epistemic status of our pure arithmetic knowledge. Having the status of a stipulation is not, of course, a characteristic of Hume’s principle that is recoverable from our use of our arithmetical vocabulary, but is something the reconstruction imposes on the principle in order to illuminate the basis for our knowledge of the propositions derivable

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from it. But it is unclear what is achieved if one has captured the pattern of use of an expression by a principle that—in the reconstruction of the knowledge claims in which that expression figures—is regarded as a stipulation. Does this confer the epistemological characteristics that the notion of a stipulation is supposed to enjoy on the knowledge claims that have been reconstructed? To establish that the epistemic basis for the knowledge these judgements express resides in the stipulative character the neo-Fregean analysis assigns to Hume’s principle, it is not enough to show that Frege arithmetic captures patterns of use. The essential point is not all that different from what we have already noted when discussing the response based on modeling, and it can be seen by an example that is not all that different from the one cited in that connection. Suppose the world were Newtonian. We could then give a reconstruction of our knowledge of the mechanical behavior of bodies by laying down Newton’s laws as stipulations governing our use of the concepts of force, mass and motion. 8 But the fact that in our reconstruction the Newtonian laws have the status of stipulations would never be taken to show that they are in any interesting sense examples of a priori knowledge. Why then should the fact that the neo-Fregean represents Hume’s principle as a stipulation be taken to show that arithmetic is known a priori? The neo-Fregean reconstruction of the patterns of use of expressions of arithmetic leaves the epistemic status of the basic laws of arithmetic as unsettled as it was on the suggestion that Frege arithmetic merely models ordinary arithmetic. Neither reconstruction supports the epistemological claim of the neo-Fregean to have accounted for the aprioricity of our knowledge of arithmetic. Whether that account is put forward as a theory within which ordinary arithmetic can be modeled, or whether it is said to capture the patterns of use of our number-theoretic vocabulary, it fails to have the direct bearing on the epistemic basis of our arithmetical knowledge that the neo-Fregean supposes it to have. Showing that Hume’s principle is correctly represented as a stipulation may be one route to securing it as a truth known a priori, but it is questionable whether, proceeding in this way, the task of revealing the proper basis for the aprioricity of arithmetic is made any easier than it would be by general reflection on why its basic laws are plausibly represented as known truths.

5.

Frege arithmetic and the analysis of number

Putting to one side the problem of establishing the aprioricity of arithmetic on a correct basis, a compelling argument that Frege arithmetic captures preanalytic intuitions about the numbers can be extracted from the neo-Fregean 8 There are of course well-known historical examples along these lines. Cp. Ernst Mach’s The science of mechanics (Open Court: 1960, sixth American edition, translated by Thomas J. McCormack), whose famous definition of mass (p. 266) even has the form of an abstraction principle. Thanks to Peter Clark for calling my attention to Mach’s rational reconstruction of Newtonian mechanics.

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corpus: Since the Dedekind–Peano axioms codify our pure arithmetical knowledge, their derivability constitutes a condition of adequacy which any account of our knowledge of number should fulfill. By Frege’s theorem, Frege arithmetic satisfies this condition of adequacy. But what makes Frege arithmetic an interesting analysis of the concept of number is that it not only yields the Dedekind–Peano axioms, but derives them from an account of the role of the numbers in our judgements of cardinality—from our foremost application of the numbers. As such, it is arguably a compelling philosophical analysis of the concept of number since, as Wright has observed, one can show that the Frege-number of Fs = n if, and only if, there are, in the intuitive sense of the numerically definite quantifier, exactly n Fs. 9 But once the project of securing a correct analysis is divorced from the project of securing a body of truths as analytic or a priori, neither the fact that Frege arithmetic satisfies our condition of adequacy nor the fact that it connects the pure theory of arithmetic with its applications—essential as each is to securing it as a correct analysis of number—addresses the question of the epistemic status of our knowledge of arithmetic. This conclusion is not particularly surprising. Both neo-Fregean strategies we have been considering are variants on the methodology of reconstruction associated with Carnap. For Carnap the thesis that arithmetical knowledge is non-factual, and therefore, a priori, was not in serious doubt. And since the aim of a reconstruction is simply to delimit more precisely the extension of a predicate, we should never have expected that a Carnapian reconstruction of arithmetical knowledge would in any way justify the claim that our knowledge of arithmetic is a species of a priori knowledge. It is precisely in respect of their epistemological significance that Fregean logicism and neo-Fregean logicism—reduction to logic by explicit definition vs. reconstruction by Frege arithmetic—come apart.

Postscript (added 2004) Hale and Wright have replied to the criticism raised in Section 4: Let ‘Newton’ denote the conjunction of Newton’s laws as ordinarily understood, and ‘NewStip’ denote the (perhaps typographically indistinguishable) conjunction of the corresponding stipulations taken as introducing certain concepts of force, mass and motion. Then Demopoulos’s claim is—or ought to be, if the parallel is to be damaging—that while we may, by laying down NewStip, acquire some a priori knowledge (in some sense, knowledge about (some things we are calling) force, mass and motion), we obviously do not thereby acquire a priori knowledge of Newton—as we ought to do, if we can, in just or essentially the same fashion, acquire a priori knowledge of truths of ordinary arithmetic by stipulating Hume’s Principle, etc. . . . [But clearly,] the mere possibility of regarding (the sentences which formulate) Newton’s laws as stipulations introducing concepts of force, mass and motion (as distinct from generalisations to which 9 See The reason’s proper study, p. 251 and pp. 330ff.

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bodies conform) does not, and cannot, by itself justify the claim that NewStip ‘captures a pattern of use’ exhibited by ‘ordinary’ statements of Newtonian dynamics. 10

But it was never claimed that regarding Newton’s laws as stipulations is what justifies the contention that NewStip captures a pattern of use. Rather, what justifies the contention that Frege arithmetic preserves a pattern of use is that it recovers the deductive structure of a body of pure and applied unreconstructed knowledge claims. The point at issue is whether, by representing certain principles as recoverable from a stipulation, a reconstruction sheds any light on their epistemic status. The comparison with the Newtonian case makes it transparent that from the fact that we can recover a pattern of use from a stipulation, nothing follows regarding the aprioricity or otherwise of our knowledge of the principles being reconstructed. The situation would, of course, be entirely different if, in accordance with the methodology of Fregean logicism, it had proved possible to recover arithmetic from logic plus explicit definitions.

10 “Responses to commentators,” Philosophical books 44 (2003) 245–263, pp. 248–249.

“NEO-LOGICIST” LOGIC IS NOT EPISTEMICALLY INNOCENT 1 Stewart Shapiro The Ohio State University E-mail: [email protected]

Alan Weir University of Glasgow E-mail: [email protected]

1. A number of philosophers in recent years, most notably Crispin Wright and Bob Hale, 2 have tried to revive something recognisably akin to the logicist programme of showing that mathematical truths are in some sense analytic, a priori or at any rate “epistemically innocent”. They do so in the face of widespread scepticism with respect to the “a priori” and especially with regard to the idea of a priori proofs of existence. Their neo-logicism seems to involve two main tenets: firstly that mathematical truths are not known a posteriori, in the way empirical truths are known, but neither are they known via some Kantian form of intuition; rather our knowledge of mathematics arises from our ability to derive mathematical truths from rules or principles which are “analytic” or “meaning-constitutive” or in some sense explanatory of key mathematical notions such as that of natural number. Secondly, the realist thesis that this mathematical knowledge is knowledge of a world which is in some sense mind-independent or objective. 1 This paper first appeared in Philosophia Mathematica 8, [2000], pp. 160–189. Reprinted by kind permission of the editor and Oxford University Press. 2 See for example Crispin Wright, Frege’s Conception of Numbers as Objects (Aberdeen University Press, 1983); ‘The Philosophical Significance of Frege’s Theorem’ in Richard Heck Jr. (ed.) Language, Thought and Logic (Oxford: Oxford University Press, 1997), pp. 201–244; ‘On the Harmless Impredicativity of N= ’ in Matthias Schirn (ed.) The Philosophy of Mathematics Today (Oxford: Clarendon, 1998), pp. 339–368. Bob Hale, ‘Dummett’s critique of Wright’s Attempt to Resuscitate Frege’, Philosophia Mathematica 2 (1994), pp. 122–147.

119 Roy T. Cook (ed.), The Arché Papers on the Mathematics of Abstraction, 119–146. c 2007 Springer.

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Clearly neo-logicism is a very attractive position for anyone sympathetic to a fairly traditional view of mathematics as a body of objective truths knowable a priori but who is worried by the standard epistemological problems faced by platonistic mathematics: how could we gain knowledge of a world of causally inert abstract objects, and so forth? The neo-Fregean answers, roughly speaking: by virtue of our knowledge of what we mean when we use mathematical expressions. More fully, we understand mathematical concepts 3 by following, in some sense, rules and following these rules is constitutive of understanding the expressions which express those concepts. These rules are entities which, like mathematical objects, are not part of the concrete, physical world; by tracing out the consequences of these rules we can find out truths concerning abstract objects. Even those such as Quine hostile to the notion of analyticity as glossed by the positivists recognise that our knowledge of the truth of sentences such as ‘if John is tall then either John is tall or Mary is short’ and ‘ “John is tall and Mary is short” entails that “John is tall” ’ may well proceed not through ordinary empirical means, nor through any special faculty of intuition but arise rather out of our understanding of operators such as the conditional, disjunction and conjunction, understanding which Quine claims is encapsulated, in the latter two cases, in his verdict matrices for the sentential connectives. 4 We will call such knowledge “epistemically innocent”. The phrase is deliberately somewhat vague, pending fuller amplification by neo-logicists. Certainly if there are truths which are analytic in the positivists’ sense, these will count as epistemically innocent; conversely any truths which require empirical verification (including holistic verification in Quinean fashion) or verification via some sort of Kantian intuition do not count as epistemically innocent. But we allow that the term may apply even if the positivist account of analyticity as truth by virtue of meaning fails. The neo-logicist claim we take then to be that there are principles which are epistemically innocent in something like the same way in which “if John is tall then either John is tall or Mary is short” is but which are strong enough to generate at least a sizeable body of standard mathematics, enough for the needs of the physical sciences, perhaps. The simple principles involved in elementary fragments of propositional logic are of course insufficient for the derivation of, for example, arithmetic. Neo-logicists have appealed instead to second-order abstraction principles of the form: αx(ϕx) = αx(ψ x) ↔ (ϕ, ψ) where α is some term-forming variable-binding operator which forms singular terms from open sentences ({x: ϕx} and nxϕx − the class of ϕ’s and the number of ϕ’s—are classic examples) and is an equivalence relation over 3 We use “concept” in the everyday, non-Fregean sense, unless otherwise indicated. 4 W. V. O. Quine, Word and Object (Cambridge, Mass. MIT Press, 1960) §13 and The Roots of Reference

(La Salle, Illinois: Open Court, 1974) §§20–21.

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properties. For the first of those two cases, the corresponding abstraction principle is “Hume’s Principle”: 5 ∀F∀G(nxFx = nxGx ↔ F1 − 1G) with F1 − 1G the second-order sentence which expresses the existence of a one–one correspondence between the F’s and the G’s. From this principle (plus suitable “bridging” definitions) one can derive, in standard second-order logic, all the theorems—including of course theorems expressing the infinitude of the natural numbers—of the usual Peano-Dedekind formulation of the theory of second-order arithmetic. This result—the derivability of second-order Peano arithmetic from Hume’s Principle—has become known as “Frege’s Theorem”. 6 For the second case, abstraction using the class operators, we get, much more problematically of course, Frege’s notorious Axiom V, which for the extensions of (Fregean) concepts takes the form ∀F∀G({x : Fx} = {x : Gx} ↔ ∀z(Fz ↔ Gz)). In the present paper, we accept for the sake of argument that abstraction principles such as Hume’s Principle are, indeed, epistemically innocent, at least on some natural readings (but see ahead Section 5). 7 We accept furthermore that not only are simple logical principles, such as ∨I and & E, or the related conditional theorems, epistemically innocent; so, too, are certain at least of the quantifier rules, for instance the natural deduction rules of universal generalisation and existential elimination. Moreover we include here these rules applied to second-order variables; that is, whilst some such as Quine would block derivation of Frege’s Theorem at the outset by refusing to accept the legitimacy of second-order logic, we do not object to second-order logic per se. Our claim will be, nonetheless, that Frege’s Theorem requires 5 The term is George Boolos’, following Frege, Grundlagen §63. See George Boolos, ‘The Standard of Equality of Numbers’ in George Boolos (ed.) Meaning and Method: essays in honour of Hilary Putnam (Cambridge Eng.: Cambridge University Press, 1990), pp. 261–277, see p. 267; the article is reprinted in the collection of Boolos’ papers Logic, Logic and Logic (Cambridge, Mass.: Harvard University Press, 1998), pp. 202–219. For the Grundlagen see The Foundations of Arithmetic translated by J. L. Austin, second edition (Oxford: Blackwell, 1980), p. 73. Frege’s rather honorific reference to the Treatise Book I, III.i is garnered from Baumann’s Die Lehren von Raum, Zeit und Mathematik (Berlin, 1868). 6 The phrase is Wright’s from a suggestion by Boolos—see ‘On The Philosophical Significance of Frege’s Theorem’, p. 203. 7 Boolos, Field, and Dummett have argued that the fact that Axiom V is classically inconsistent but is formally very similar to, e.g. Hume’s Principle, irredeemably vitiates the claim of abstraction principles to be analytic truths or at any rate to be epistemically unproblematic. See George Boolos, ‘The Standard Equality of Numbers’, p. 273, ‘Whence the Contradiction?’ in Proceedings of the Aristotelian Society, Supplementary Volume LXVII (1993), pp. 213–233 (reprinted in Logic, Logic and Logic, pp. 220–236), Hartry Field, Realism, Mathematics and Modality (Oxford: Blackwell, 1989), p. 158, Michael Dummett, Frege Philosophy of Mathematics (London: Duckworth, 1991), pp. 188–189, p. 208. For a more nuanced development (we claim!) of this objection see Stewart Shapiro and Alan Weir, ‘New V, ZF and Abstraction’, Philosophia Mathematica, 7 (1999), pp. 293–321. Wright’s ‘The Philosophical Significance of Frege’s Theorem’ is in large part a response to such objections. One radical response is to reject the logic (significantly weaker than classical logic) used in the derivation of antinomies and hold that Axiom V is not in fact inconsistent or at any rate not trivial. Cf. Alan Weir ‘Naïve Set Theory is Innocent!’, Mind 107 (1998), pp. 763–798.

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use of first- and second-order logical principles which are not epistemically innocent. More exactly, certain of the logical principles which are essential to the derivation of a theorem of infinity, when this is construed as expressing the existence of infinitely many mind-independent entities, are at least as problematic epistemologically as axioms of infinity laid down simply as postulates. Our supposed knowledge of these principles is, we will argue, every bit as mysterious as Kantian intuition of an infinity of numbers. We will look at two main cases in turn: the second-order axiom of comprehension applied to non-instantiated properties (Sections 2 and 3) and the first-order existential instantiation and universal elimination principles as applied in standard nonfree classical logic (Sections 4 and 5). We finish in the sixth section with a summary of our overall conclusions.

2. The standard second-order logic needed in the derivation of Frege’s Theorem includes straightforward generalisation of the first-order quantifier rules plus an Axiom Scheme of Comprehension. 8 Thus, for example, universal elimination and existential introduction become ∀FF F/ P F/ P ∃FF where P is any simple predicate constant or parameter and is any open sentence with F free. The (impredicative) Comprehension Scheme consists in all instances of: ∃R∀x 1 , . . . , ∀xn (Rx1 , . . . , xn ↔ ϕx1 , . . . , xn ) where R is an n-place relation variable and ϕ is any formula of the language in which R does not occur free. Finally we add a Substitution rule permitting substitution of co-extensional predicates: 9 ∀x(ϕx ↔ ψ x), θ . θ[ϕ/ψ] Now on the face of it, our neo-logicist, in taking second-order logic as characterised above as a body of epistemically innocent truths, is committed by the Axiom Scheme of Comprehension to a strong realism about properties, committed moreover to the a priori demonstrability of this strong realism. 8 It would be neater to dispense with Comprehension in favour of use of λ terms and λ conversion. But since most treatments of second-order logic use the comprehension scheme rather than λ abstracts we will follow suit. The points we will make in connection with Comprehension can be rephrased to apply to the λ term version of second-order logic. 9 This rule is a derivable rule in pure second-order logic but we will be concerned with expansions of the language to include complex singular terms formed by binding open formulae in one free variable x with operators such as the numerical operator nx(. . . x . . . ) or the class operator {x: . . . x . . . } and to handle these we need either a device such as λ abstraction or else a rule of the above sort. Since it could be dispensed with in a treatment with λ abstracts we will not question the epistemic status of this rule.

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At any rate, the neo-logicist seems committed to it being demonstrable that whatever it is that second-order variables range over exists, indeed exists in a mind-independent fashion. For the neo-logicist views mathematical theories such as number theory as arising (at least on a “rational reconstruction”) from the extension of an “empirical” or non-mathematical second-order language by the addition of new operators such as nxPx, thereby generating existential assertions with the same (or similar) degree of objectivity as pertains to the original “empirical” language. This has the advantage of treating the mathematical and empirical subfragments of language as semantically homogenous thereby easing, it is hoped, the problem of explaining the applicability of the allegedly “epistemically innocent” realm of pure mathematics to the epistemologically guilt-stained empirical realm. The downside to this for the neologicist is that the impredicative comprehension axiom must be assumed not only to be true in the same objective sense as other sentences of the original empirical sector, we must also be able to know or verify its truth in an epistemically innocent fashion. If comprehension is known via intuition or known a posteriori then no demonstration of a mathematical truth can be innocent if it relies essentially on comprehension just as no demonstration is innocent if it relies on an axiom of set theory conjecturally justified holistically in terms of the fruitfulness of its empirical consequences. We have agreed that our knowledge of the validity of simple rules such as &E and ∨I is epistemically innocent; we may well expect that there can be other less simple cases of epistemic innocence which do not share all the features of &E and ∨I. But one cannot help but notice the enormous leap from simple rules of the above type to complex principles such as: ∃R∀x 1 , . . . , ∀xn (Rx1 , . . . , xn ↔ ϕx1 , . . . , xn ) (especially when one reflects on those impredicative cases where ϕ can contain bound second-order variables). We can give a fairly plausible account of how we know &E is sound, grounded perhaps in nothing much more than the Quinean verdict matrices for &; the neo-logicist needs to come up with something along similar lines (more complex no doubt) which will explain how we know, of each instance of the above, that it is true. A Quinean might dismiss neo-logicism on these grounds alone: if one assumes that it is a priori (or innocently) demonstrable that a realm of properties exists independently of the mind, such a person may say, then maybe you can derive the existence of abstract objects. But anyone sympathetic to the “anti-Anselmian” intuition that one cannot derive existence from concepts alone will refuse to accept the above axiom scheme of comprehension as epistemically innocent. For each instance entails the existence of a property and, (in)famously, Quine argued that second-order logic is not logic at all, but is set theory in disguise. As early as 1941, he claimed that properties are too

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obscure to serve logic, and should be replaced with items like classes. 10 Once we invoke classes, however, we have crossed the border out of logic and into mathematics proper. Later, Quine wrote: 11 Set theory’s staggering existential assumptions are. . . . hidden. . . . in the tacit shift from schematic predicate letter to quantifiable set variable.

For Quine, then, second-order logic is a wolf in sheep’s clothing. It is set theory made to look like logic, by having variables ranging over properties/sets. But it is important to note that the Quinean is only the most extreme of the opponents the neo-logicist faces and even if the neo-logicist can confute the Quinean position, he or she is far from home and dry. To see this, let us assume for the sake of argument that properties exist in as mind-independent a fashion as scientific entities do but assume as little else as we can about properties so that we can be neutral as to what exactly the nature of properties is (perhaps they are just classes of objects, for example). Consider now a philosopher—Macari, let us say—who accepts second-order logic as correct subject to one, rather minor looking amendment. She accepts only the following form of the Comprehension Schema as logically valid: ∃x1 , . . . , ∃xn ϕx1 , . . . , xn → ∃R∀x1 , . . . , ∀xn (Rx1 , . . . , xn ↔ ϕx1 , . . . , xn ). That is, focusing on one-place open sentences for simplicity, Macari agrees that it is a logical truth that to every such sentence which is instantiated by something or other, there corresponds a co-extensional property. But she refuses to accept that logic alone tells us that there are uninstantiated properties, so refuses to conclude that to predicates such as x = x there corresponds a property. Macari’s attitude is not all that odd or lacking in motivation. Macari may have what one might loosely call “Aristotelian” reasons for being sceptical about uninstantiated properties. 12 Macari accepts, let us suppose, that it is just as sound to infer wisdom exists, because Socrates is wise, as it is to infer that Socrates exists, because that same proposition is true. But it is a far more contentious step to assume that if a sentence involving a predicate P is true, then a mind-independent property corresponds to P, even when P is the predicate x = x. For Macari, this is as substantive an assumption as the 10 W. V. O. Quine, ‘Whitehead and the rise of modern logic’, in P. A. Schilpp, The Philosophy of Alfred North Whitehead (New York: Tudor Publishing Company, 1941), pp. 127–163. 11 W. V. O. Quine, Philosophy of Logic (Englewood Cliffs, New Jersey: Prentice-Hall, 1970), p. 68. 12 There seems to be enough textual support for the idea that Aristotle was opposed to the idea of ‘ante rem’ universals to justify use of his name; but since Macari’s view of properties is doubtless not all that close to Aristotle’s we will refer to the position as “aristotelian”, with lower case “a” in analogy with “platonism” in philosophy of mathematics. The key tenet of Macari’s view is only that there is no epistemically innocent proof that uninstantiated properties exist; she need not believe, for example, in ‘concrete universals’ which are somehow both properties and extended physical objects but which, unlike mereological fusions, are ‘wholly present’—whatever that means—wherever their instances are present.

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assumption of the existence of an abstract object. It is certainly hard to see how this assumption can be known in as innocent a fashion as the soundness of &E and ∨I can be known. Macari’s views, we submit, err, if anything, on the side of generosity to neo-logicism, in allowing that to any arbitrary instantiated predicate there corresponds a property. Neo-logicists, if they are to convince Macari that abstract objects can be shown to exist in something like an a priori fashion must therefore do so by adding abstraction principles to a logic which does not already embody the assumption that uninstantiated properties exist, i.e. a logic which is no stronger than Macari’s “aristotelian” second-order logic (let’s call it “A2L”), that is standard second-order logic but with Comprehension restricted as above. But here they face a huge problem: arithmetic, in particular a theorem of infinity, is not derivable from Hume’s Principle in A2L. The stumbling block is the number zero, defined by the neo-logicist as nxPx for some predicate P x with ⊢ ∀x ∼ P x, e.g. with x = x for P x. On the face of it, Macari ought to hold that nx(x = x) is, or could in some situations be, an empty term standing for nothing rather than standing for something, namely the number zero. For the intended interpretation of the numerical operator nxϕx is as a function which maps properties to objects in a certain way; and where ϕx is x = x there is no property available as argument for the function. But in order to keep separate a number of distinct problems with neo-logicism we will leave discussion of free logics till later (Sections 4 and 5) and assume that all singular terms, simple or complex, have a reference. We will therefore ensure that there is at least a “dummy” referent assigned to all numerical terms of the form nxϕx, even where nothing satisfies ϕx. The aristotelian interpretation of the numerical operator is as follows. For simplicity let properties, the range of the monadic second-order quantifiers, just be non-empty subsets of the domain of individuals. In each model, the semantics, via the recursion theorem, will ensure that each open sentence ends up getting assigned a subset (empty or not) of the domain as its extension. To do this for the language containing the numerical operator, we first partition all the non-empty subsets into equivalence classes under the relation of equinumerosity and map these equivalence classes via a one:one function f into the domain of individuals. We then select some arbitrary member d of the individual domain as the image of the equivalence class whose sole member is the empty extension Ø and thus extend f to a function N by mapping {Ø} to d. Since d is arbitrary, there is no requirement that d not occur in the range of f; it might, for example, turn out to be the image of f applied to the equivalence class A of all unit sets. The numerical operator is then interpreted as a function which maps the extension of E each open sentence ϕxgo the N-image of the equivalence class in our partition to which E belongs. In the model envisaged, then, N(A), that is d, is the object which is, in the model, the number one so that d may number both the empty set and every unit set.

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On this extension of A2L to the language including the numerical operator, we can, of course, prove that zero—nx(x = x) on the Fregean definition— exists. For ∃x(x = nx(x = x)) is a simple consequence of the appropriate instance of the axiom schema of identity which holds in A2L. 13 But the rub is that we cannot prove the standard facts about zero which we need in order to prove a theorem of infinity. In particular, we cannot prove ∃x Suc(x, 0), where Suc(x, y) abbreviates the standard Fregean definition of x succeeds y: Suc(x, y) ≡ df. ∃F∃G(x = nw Fw & y = nwGw & ∃z(F z & nw(Fw & w = z) = nwGw)); and we cannot rule out the possibility that for some positive number j, 0 = S j 0, where S j 0 is the standard Fregean numeral for the number j. 14 In particular, as in the case set out above, we can have as true in an aristotelian model 0 = S0, i.e. 0 = 1. The assumption 0 = S j 0 does not lead to conflict with Hume’s Principle since in A2L, with its restriction on the Comprehension scheme, we cannot derive: nx(x = x) = nx J x < − > 1 − 1((x = x), J x) from Hume’s Principle, where J x is (x = 0 ∨ · · · ∨x= S j−1 0) (x= 0 in the example given). Since Jx is provably instantiated, we will be able to prove from the relevant instance of the Comprehension scheme: ∃G∀x(Gx ↔ J x)

(i)

If we had available full Comprehension we would also have for the instance (x = x): ∃F∀x(F x ↔ x = x)

(ii)

from which, after excising the initial second-order existential quantifiers for application of ∃E (or else after existential instantiation), we get: ∀x(Gx ↔ J x)

(iii)

∀x(F x ↔ x = x).

(iv)

∀E on Hume’s Principle yields nx F x = nx Gx ↔ F1 − 1G

(v)

S j 0 abbreviates nx J x so that two applications of the Substitution rule takes us (iii), (iv), and (v) to: 0 = S j 0 ↔ 1 − 1((x = x), J x). 13 If t = u is defined as ∀F(Ft → Fu) (equivalently ∀F(Ft ↔ Fu)) the derivation of each instance is fairly trivial. If it is primitive, we add each instance of t = t as an axiom. 14 i.e. takes the form nx(x = 0 ∨ . . . x = S j−1 0).

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Since we can disprove the right-hand side, (ii) along with the rest of our logical apparatus would yield a reductio ad absurdum of 0 = S j 0. But since (ii) is not forthcoming in aristotelian second-order logic, neither is this standard proof. As a consequence, the Fregean proof of infinity fails in A2L because we cannot prove there are n + 1 numbers less than or equal to n; for all the aristotelian knows, zero happens to be identical to some number Sk 0 between zero and n. As we have seen, it is possible for 0 = 1 to hold in a model in the aristotelian framework in which case the number of numbers less than or equal to one is just one. Nor can the neo-logicist hope that it is merely the standard proofs of the theorem of infinity that fail, that the aristotelian may be able to find some more complex ways of arriving at infinity. For there are aristotelian models of any finite size of Hume’s Principle. To see this, take any finite domain of individuals D = {d1 , . . . , dn }. As above, we interpret the numerical operator by partitioning the subsets of D into classes of equinumerous classes:– there will be n + 1 of these including the class containing the single zero-sized subset of D, namely Ø and with the range of the second-order quantifiers the non-empty subsets of D. 15 Interpret nxϕx by the map N which takes X ⊆ D to dk , where X is the extension of ϕ and is of size k = 0 and which takes the empty extension to d j for some j which is doubling up as “zero”. Then Hume’s Principle: ∀F∀G(nx F x = nx Gx ↔ F1 − 1G) is satisfied because for any non-empty subsets S1 and S2 of D assigned to F and G (remember assignment of Ø is not possible in this model) the left-hand side of the biconditional is true iff M assigns to F and G the same element di iff F and G belong to the same partition class under the equinumerosity equivalence relation (since F and G are non-empty) iff the right-hand side is true. 16 In the model given, all the other principles of second-order logic, including all the instances of aristotelian Comprehension and the Substitution rule are sound. Hence Hume’s Principle (HP) does not semantically entail, in models sound for A2L, a theorem of infinity and so such theorems are not derivable in A2L from HP. The neo-logicist cannot claim, then, that the existence of infinitely many numbers is demonstrable in an epistemically innocent fashion unless he or she can show that it is demonstrable, in such an epistemically innocent fashion, 15 Though the argument goes through just as well for a property realist who distinguishes the extension of a predicate from the property it stands for, so long as they allow some interpretations in which no property has an empty extension. 16 The aristotelian interpretation also yields a (not very exciting!) one-element model of Axiom V. If d is the sole individual, there are two subsets of domain D: Ø and {d} and so only one non-empty subset falling in the range of the predicate variables. We assign {d} to d and Axiom V comes out true for every assignment of predicate extensions to the two variables F and G of the embedded biconditional, since every assignment assigns {d} to both variables. This, however, is the only aristotelian model (up to isomorphism) of Axiom V .

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that there is an uninstantiated property, unless, in other words, they can show that aristotelianism on properties can be shown to be wrong by an epistemically innocent argument. The “aristotelian neo-logicist” will, it is true, be able to prove (∃F∀y ∼ Fy) → Inf where Inf is a theorem of infinity; that is, if an empty property exists then there are infinitely many numbers. Even if that conditional is demonstrable in an epistemically innocent fashion, this is of little use to the neo-logicist if our knowledge of the existence of an empty property is no better placed or explicable than our knowledge of the existence of infinitely many numbers. Certainly the neo-logicist cannot argue that the hypothesis that there is an empty property is holistically confirmed by the fact that it enables us to develop the mathematics needed by science. If we take this Quinean course, we may as well help ourselves to the standard axiom systems of number theory, analysis etc. Perhaps it will be felt that to credit us with an intuitive knowledge of the existence of one empty property is more plausible than crediting us with an intuitive grasp of infinitely many numbers, or even an intuitive grasp of the fact that infinitely many numbers exist. But the neo-Kantian need not appeal to direct intuition of every number, nor does there seem any relevant difference between admitting direct intuition of one non-concrete entity and admitting direct intuition of many; 17 any such appeal to intuition represents abandonment of neo-logicism. One patch the neo-logicist might try out is to add to Hume’s Principle each instance of the following “Zero” axiom scheme: ∼ ∃xϕx ↔ (nxϕx = 0) This ensures that if ϕ and ψ are both unsatisfied then nxϕx = nxψ x = 0, whereas if θ is satisfied then nxθ x = 0. Call the theory consisting of HP plus all instances of the above schema HPP · HPP entails that the universe is infinite. Model-theoretically we can show this as follows: if there are only finitely many individuals k in a domain then there are k equivalence classes under equinumerosity on the domain of non-empty sets so we need all k distinct individuals to number the non-empty properties so as to satisfy HP. But we also need an individual which is the referent of nxθ x, where θ has an empty extension and this individual, the referent of 0, is constrained by the Zero axiom scheme to be different from the k individuals which number the instantiated properties; hence there can be no finite model of HPP. A salient point to note here is that we run into problems if we replace the schematic variable ϕ with a second-order variable because in our aristotelian theory such 17 Compare Russell’s view that if one accepts that one universal—of resemblance—exists, one may as well accept a plurality: (Bertrand Russell, The Problems of Philosophy, London: Williams and Norgate, 1912, pp. 150–151).

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variables range over non-empty properties and the case we wish to capture here is one in which ϕ is uninstantiated. This brings to the fore how ungainly the theory HPP is; how different in form it is from an abstraction principle, formulable as a single sentence, such as HP, not to mention simple inference rules such as &E. What reason is there to suppose that the infinitary theory HPP is epistemically innocent? Since HPP contains infinitely many sentences, there is no way we can think of it as grounded in inferential practice in the way that &E is grounded in the verdict matrices for &. However, there is a single formula related to the theory HPP which will yield, even in aristotelian second-order logic, Frege’s theorem and that is HPP∗ : ∀F∀G((nx F x = nx Gx ↔ F1 − 1G) & (∼ ∃x F x ↔ (nx F x = 0))) In other words, we conjoin to HP an instance of the Zero axiom scheme but with an ordinary variable in place of a schematic one, a variable bound by one of our initial quantifiers. It follows from the right conjunct of HPP* that no property has number zero; since all properties are instantiated, in our aristotelian models, all falsify the left-hand side ∼ ∃ xFx of the right conjunct. Hence, as with HPP, there must be infinitely many numbers. Starting with our definition of zero as nx(x = x) we can then prove the Peano Postulates using the other “bridging” definitions of successor or predecessor and so on. What we cannot prove, though, and here is a key difference with HPP, is that there is exactly one “zero”, in the sense of one number which all uninstantiated properties have. This can be seen by adding to our interpretations “virtual” properties in addition to the “real” properties which exhaust the range of the second-order quantifiers and letting formulae in one individual variable whose extension is empty be assigned an arbitrary such virtual property. The numerical operator is then to be interpreted as a function on properties, real or virtual in which equinumerous real properties are assigned the same value but distinct virtual properties can be assigned distinct values. Suppose, for example, that we can prove ∼ ∃xϕx. The formula ϕx might, for example, be x < 0. We cannot go on to prove that nx(x < 0) = 0 because we cannot apply Hume’s Principle to empty properties. If we had full Comprehension we could prove ∃F∀x(Fx ↔ x < 0) and so assume for existential elimination (i) ∀x(Fx ↔ x < 0). We instantiate HPP* and derive the right conjunct by & E. Our generalised substitution rule yields, from (i) (∼ ∃xϕx ↔ (nxϕx = 0)).

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Having already proved that ∼ ∃x x < 0, we would then have been able to conclude nx(x < 0) = 0. But we do not have full Comprehension! We only have, for the instance in question: ∃x x < 0 → ∃F∀x(Fx ↔ x < 0) and we cannot discharge the antecedent. This plurality of “zeros” is a problem for the neo-logicist, even though Frege’s Theorem follows from HPP∗ . For PA is only one possible formulation of arithmetic (albeit a very important one) where we understand by “arithmetic” a theory investigated (in natural language plus some notation) by mathematicians; moreover mathematicians do not solely (or even mainly) deal with formalised theories of arithmetic, but also use arithmetic principles in order to prove theorems in various different areas, pure and applied. It is by their ability to account for and explain real mathematical theories that philosophies of mathematics, and the formal systems they utilise, must be judged. In “real arithmetic” we say things like “the number of numbers less than zero is itself zero”; such applications of “the number of Ps is zero” are also very commonplace in applications of arithmetic in real situations. If a philosophy of mathematics cannot account for the truth of such assertions, that is a serious defect. All this on the assumption that HPP∗ (and the aristotelian version of the Comprehension scheme) are epistemically innocent. But once again we can ask the question what grounds are there for thinking that HPP* is innocent? It is not even an abstraction principle. We will raise this question again with respect to a related principle which arises when neo-logicism is set in the framework of plural quantification.

3. George Boolos, the most influential advocate of plural quantification, proposed that one of its values is that we can use it to interpret second-order quantification without invoking either properties or classes. 18 Perhaps the neologicist can use this to bypass the issues concerning which properties exist. If Hume’s Principle can also be interpreted “pluralistically” in such a way as to retain its innocence (we have granted it innocence on the usual way of reading it) then the neo-logicist would seem to be home and dry. Boolos suggested that a monadic, existential second-order quantifier be considered a counterpart of a plural quantifier, “there are (objects)”, in natural language. The following illustration is called the Geach-Kaplan sentence: Some critics admire only one another. 18 George Boolos, ‘To Be is to Be a Value of a Variable (or to Be Some Values of Some Variables)’, Journal of Philosophy 81 (1984), pp. 430–449; ‘Nominalist Platonism’, Philosophical Review 94 (1985), pp. 327–344; ‘Reading the Begriffschrift’, Mind 94 (1985), pp. 331–344. These essays are reprinted in Logic, Logic and Logic at pages 54–72, 73–87, and 155–170, respectively.

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It has a (more or less) straightforward second-order rendering, taking the class of critics to be the domain of discourse: ∃F(∃xFx & ∀x∀y((Fx & Axy) → (x = y & Fy))). According to the usual reading, the formula would correspond to “there is a non-empty class (or a non-empty property) F of critics such that for any x in F and any y, if x admires y, then x = y and y is in F”. But this implies the existence of a class or property, while the original “some critics admire only one another” does not, at least prima facie. Boolos developed a rigorous, model-theoretic semantics for monadic, second-order quantification in these terms. Some philosophers with nominalist tendencies have invoked the Boolos semantics in order to obtain the benefits of second-order quantification without encumbering oneself with a second-order ontology. A good deal. Our neo-logicist might attempt a similar maneuver, in order to make the logic more tractable. In this case, however, there are troubles at every turn. Second-order logic enters into Hume’s Principle (and Frege’s theorem) in two places. Like any second-order abstraction, Hume’s Principle has prenex universal quantifiers binding monadic property variables. The Boolos plural construction originally was limited to existential second-order quantifiers but Boolos extended it to universal quantifiers. He glosses a second-order universal quantification of the form ∀F(F) along the lines: no matter which things the Fs are, holds of the Fs.

We will give an example later illustrating how he glosses the schematic phrase:– holds of the Fs. However, on the right-hand side of Hume’s Principle we find the notion of equinumerosity and the definition of equinumerosity invokes a variable over (binary) relations on the domain: two (monadic) properties are equinumerous if there is a one-to-one relation from the extension of one of them onto the other. So our first problem is that the Boolos plural construction is limited to monadic property variables, while the second-order definition of equinumerosity has a binary relation variable. However, if there is a (definable) pair function in the language, then relations can be introduced in the usual manner. Variables over binary relations are replaced with monadic variables ranging over pairs: ∃R is rendered ∃F′ , where ′ is obtained from by replacing each occurrence of Rtu with F< t, u>, where < t, u > is the ordered pair of t and u. This move is not available here, at least not without begging the crucial question. There can be no pair function on a finite domain with more than one member. If the domain has size n, then there are n 2 ordered pairs. So our neo-logicist cannot introduce a pair function until she has shown that the universe has either at most one object in it or is infinite. This throws a monkey wrench into the works. The neo-logicist wants to use Hume’s Principle and

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therewith Frege’s Theorem to establish that the universe is infinite, by showing that the natural numbers exist (and are distinct). But on the present plan, she cannot even formulate Hume’s Principle (via plural quantification with pairing) without first showing that the universe is either non-plural or infinite. Since we non-Hegelians know the first disjunct is false, this means showing that the universe is infinite. The plan is frustrated before it can even get started. The existence of a pair function on each infinite domain is equivalent to the full axiom of choice, since it amounts to κ 2 = κ, for each cardinality κ. How can our “pluralist” neo-logicist claim that the existence of pairs is epistemically innocent? This amounts to the epistemic innocence of full choice. Perhaps we can be helpful here. The neo-logicist might introduce pairs via an abstraction principle: ∀x∀y∀z∀w( (x, y) = (z, w) ≡ (x = z & y = w)). Call this the Pair Principle. Strictly speaking, it is not in the same form as other abstraction principles, since the right-hand side (symbolised as (x = z & y = w)) represents a four-place relation and so not an equivalence relation. But it has at least the flavor of an equivalence relation (symmetry, transitivity, reflexivity), and so is in the spirit of other abstraction principles. 19 The Pair Principle lies between first-order abstractions and second-order abstractions. If our neo-logicist can maintain that it is epistemically innocent, then an acceptable “plural” formulation of equinumerosity can be produced. The antecedent represents quite a big “if” since the principle will in effect give the neo-logicist an infinite universe. But for the rest of this section, we will make that concession and assume that the right-hand side of Hume’s Principle is kosher. What, next, of the left-hand side of the principle in which we find, crucially, the abstraction operator itself—“the number of”. On the standard reading, this is a function from properties (or classes) to objects, but we are considering here a philosopher who is trying to get by without acknowledging properties or classes at all. We thus cannot think of the abstraction term as denoting a function, for there is (or may be) nothing for it to operate on. This is not an insuperable stumbling block: English and other natural languages with a definite article construction allow that construction with respect to plurals. We speak, for example, of “the dogs in the room” or “the numbers less than 12”. So it seems to make sense to use the plural definite article with a second-order variable, to produce the locution, the F’s or, here, the number of F’s 19 Reflexivity: (x = x & y = y); symmetry: (x = z & y = w) entails (z = x & w = y); transitivity: (x = z & y = w); and (z = t & w = u) entail (x = t & y = u). Bob Hale invokes a pair abstraction in “Reals by abstraction” (Philosophia Mathematica (3) 8, 100–123; reprinted in Hale and Wright [2001], 399–420). Frege’s theorem only needs to invoke instances of Hume’s principle applied to finite properties with finite extensions and so it may be possible to formulate a “predicative” version of the Pair Principle in which no occurrences of occur (at least untyped) inside terms of the form (x, y). The (set-theoretic) statement that the Pair Principle is satisfiable on every infinite domain is equivalent to choice. However, as far as we know, the Pair Principle does not entail the axiom of choice, since it only produces a pair function on the universe.

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(we will attempt to be as neutral as possible on the right way or ways to approach the syntax and semantics of such locutions). Certainly Boolos found application of the number operator to plural phrases perfectly sensible. 20 Here is one of his glosses of Hume’s Principle, illustrating also his way of reading second-order universal quantifiers plurally: Hume’s Principle is the statement that no matter which things the Fs and Gs may be, the number of Fs is the same as the number of Gs just in case the Fs and Gs are in one–one correspondence. 21

So we finally have a plural-quantifier version of Hume’s Principle. Call this Plural Hume. Let us assume that, as written, Plural Hume is epistemically innocent. What does it say? In particular, what of the initial plural quantifiers? Macari, it will be recalled, agrees that instantiated properties exist, and so instantiated properties can be objects in the range of second-order variables; but Macari denies that empty properties exist (or at least that we can prove innocently that they do). We saw in the previous section that under Macari’s assumption, Frege’s Theorem does not go through using Hume’s Principle alone. In Macari’s system, Hume’s Principle is satisfied on finite domains. But if the neo-logicist goes the route of plurals, and formulates Plural Hume, then Frege’s Theorem does not go through for the same reason:—if our neo-logicist relies on plurals, she has played into Macari’s hands. The reason for this is fairly simple. The ordinary locution, there are F’s such that . . . entails that there is at least one F. Indeed, this is what the locution says. 22 In traditional terms, the plural quantifier declares that a certain predicate is instantiated. For example, I cannot announce that there are vicious elephants in our backyard, unless I have reason to believe that there are some. The empty property (if there is one) cannot instantiate a plural existential quantifier. Similarly if I say that no matter what things the Fs are, if everything which is an F is a G, then Hamish is an F, i.e. no matter what the F’s are: (∀x(Fx → Gx) → Fh) then I do not seem to be committed, absurdly, to Hamish being non-selfidentical and what I say seems perfectly coherent if, for example, only Hamish is a G. 23 Thus the plural quantifiers are analogues only of Macari’s aristotelian quantifiers, not the standard second-order quantifiers. If, therefore, the Boolos program is successful, it eliminates a commitment to instantiated properties 20 See, for example, Boolos [1993]. 21 Op. cit., p. 223. Perhaps “no matter which things the Fs and Gs are” is better than “no matter which

things the Fs and Gs may be” as the latter may seem to import a modal element foreign to straightforward universal quantification. 22 Boolos notes that the locution there are F’s might entail that there are at least two F’s, but one F is enough for present purposes. See, for example, Boolos [1984], p. 443. 23 If there is no way of reading universal second-order quantification without commitment to properties or classes, including empty ones, then someone sticking to Boolos’ original motivation for plural quantification would have to drop ∀F as a primitive and use the ∼ ∃F ∼ translation.

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or to non-empty sets. For that reason we cannot validate, in the way required by neo-logicists, the full axiom scheme of Comprehension; only the restricted form available to Macari is sound for plural quantifiers. And so Frege’s Theorem is blocked from the start. By reasoning parallel to that in the Macari case we can show that there are models in which the numbers form a finite subset of the infinite domain, with 0 = S n 0, for some number n. Indeed there will be a model of size one, one in which the pairing axiom holds. Plural Hume does not entail the Peano postulates. However, Boolos’ pluralist agenda included the program of rendering standard “non-aristotelian” second-order languages into the language of plural quantification. (This language is akin to lawyer’s English, with indexed pronouns and related constructions playing a role similar to “the party of the third part” and so on.) He does this by including in his translations a clause to handle explicitly what would be empty properties. Let (F) be a formula with the monadic, second-order variable F free. Let ∗ be the result of replacing each occurrence of Ft in (F) with t = t. In standard terms, ∗ states that holds of the empty property. Then Boolos renders the second-order ∃F(F) as something like: Either there are some F’s such that (F), or ∗

For example, he “translates” the second-order set-theoretic truth ∃F∀x(Fx ≡ x∈ / x) as (after some simplification) “Either there are some sets that are such that every set is one of them iff it is not a member of itself or every set is a member of itself” (ibid.). 24 In this case, the second disjunct does no work (being provably false), but it is part of the “translation”. Applying the same idea to ∀F(F) we get: no matter what the Fs are, (F) and ∗

with conjunction in place of disjunction. Since plural English can get rather complex we will carry out some translations into a formalised version of it but in order that the plural reading be borne in mind we will use (F) for no matter what the F’s are and (EF) for there are some F’s such that. So we translate, e.g. the sentence ∀F Ft, which is false in standard second-order logic, into: ((F)Ft & t = t) which is also false even in a one-element universe in which no uninstantiated property exists. Can we amend Hume’s Principle along those lines? The matter is complicated a bit in the present context. Since we have introduced a higherorder operator the number of F’s , we cannot just replace the “F” with something like t = t to handle the uninstantiated case. The phrase The 24 ∗ (F) here is ∀x(x = x ↔ x ∈ / x) which is provably equivalent to ∀x(x ∈ x).

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number of t = t is not grammatical. One way round this is to introduce λ terms and thereby a translation *λ in which [∀F(F)]∗λ = ((F)(F)) & (λx(x = x)) with a dual clause for ∃F; thus ∀F Ft becomes (F)Ft & λx(x = x)t. But since we can always eliminate λ terms by λ conversion, starting from λ terms with narrowest scope and working outwards substituting equivalents for equivalents, we can in this way arrive at a translation ∗∗ which is the result of applying λ conversion to the ∗ λ translation. Hence applying ∗ λ to ∃F(nxFx = nx(x = x)) (i.e. there is some non-zero number) gives us: (E F)(nxFx = nx(x = x)) ∨ n(λx(x = x) = n(λx(x = x)) 25 and then λ conversion yields the ∗∗ translation (E F)(nxFx = nx(x = x)) ∨ nx(x = x) = nx(x = x) in which the second disjunct is a logical falsehood. Monadic Comprehension: ∃F∀y(Fy ↔ ϕy)(whereFdoes not occur in ϕ) becomes, after λ conversion: (E F)(∀y)(Fy ↔ ϕy) ∨ (∀y)(y = y ↔ ϕy) which is equivalent to (∃y)ϕy → (E F)(∀y)(Fy ↔ ϕy), i.e. If there is a ϕ then there are some Fs such that anything is an F if it is a ϕ and is a ϕ if it is an F.

We submit that if this is formalised so as to be put to use in a derivational system, it must be formalised as Macari’s aristotelian (monadic) axiom of Comprehension, with the antecedent requiring the existence of a ϕ; 26 it ought not to be formalised as standard comprehension. In one sense, the Boolos program validates full second-order comprehension: each instance is translated into a truth of the plural quantification framework. But never a truth with which Macari would quibble. For example, where ϕ in the Comprehension scheme is instantiated by y = y, the translation yields (∃y)y = y → (E F)(∀y)(Fy ↔ y = y). What happens to Plural Hume under the Boolos translation? Here things are complicated by the fact that we have two initial universal quantifiers to 25 If we have λ terms, we can do all variable-binding using them, formalise the numerical operator as n(λxϕx) and abbreviate the latter as nxϕx. 26 As remarked above, we leave aside the complication that the plural reading is indeed plural, that is, seems to require more than one ϕ; even ‘at least two’ is arguably not quite right as a reading either.

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work on. Plural Hume, using our conventions about “pluralese” (and leaving unpacked the definition of one:one correspondence) is: (F)(G)(nx F x = nx Gx ↔ F1 − 1 G). Applying ∗∗ to (F) we get: ((F)[(G)(nxFx = nxGx ↔ F1 − 1G)]) & [(G)(nx(x = x) = nxGx ↔ (x = x)1 − 1G)]. We must now apply ∗∗ to the two (G) formulae in square brackets. For the first, it yields (using the standard definition of 0 as nx(x = x)): ((G)(nxFx = nx Gx ↔ F1 − 1G)) & (nx F x = 0 ↔ F1 − 1x = x). For the second we get ((G)(0 = nx Gx ↔ (x = x)1 − 1G)) & (0 = 0 ↔ (x = x)1 − 1(x = x)). Putting all this together the result is ((F)((G)(nx F x = nx Gx ↔ F1 − 1G) & (nx F x = 0 ↔ F1 − 1x = x)))& (G)((0 = nx Gx ↔ (x = x)1 − 1G) & (0 = 0 ↔ (x = x)1 − 1(x = x))). Call this Amended Plural Hume. Now the last conjunct is a logical truth and the second and third are symmetrical so that we can delete one of these which gives, with some further simplification: (F)(G)((nx F x = nx Gx ↔ F1 − 1G) & (∼ ∃x F x ↔ nx F x = 0)). That is: No matter what the Fs are and no matter what the Gs are, the number of Fs = the number of Gs just in case there is a one–one correspondence between them and there are no Fs iff the number of Fs = zero.

But this is none other than the non-schematic HPP∗ . We conjoin to Plural Hume a clause specifying that the number of Fs is zero iff there are no Fs. HPP∗ yields, we have seen, Frege’s theorem but fails to capture many applications of the numerical operator by allowing a plethora of zeros. This a form of plurality which supporters of plural quantification will be less keen on. In sum, the plural version of HP does not give us Frege’s theorem. Plural HPP∗ does but does not give us the applications we want. Moreover we can ask again why think plural HPP∗ , which is not an abstraction principle, is innocently true. After all, none of the formal versions of HP can be presumed to be explicitly part of the concept of number possessed by everyday speakers (or even most mathematicians); indeed it is doubtful whether any are even implicitly held by hoi polloi. The best case for the epistemic innocence of HP is surely that something like:

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the number of apples equals the number of oranges iff the apples and oranges can be paired off one:one

is part of competent users’ tacit understanding of number (and so on for other sortal concepts). If Boolos’ plural reading of quantifiers is on the right lines, the most faithful reading of these “platitudes” which jointly are constitutive of our notion of number (compare the conceptual functionalists’ idea that our folk psychological notions arise from psychological platitudes) is the original Plural Hume not Amended Plural Hume, and the original is certainly too weak to do the work the logicist requires of it. The neo-logicist might abandon the project of reconstructing arithmetic using Hume’s Principle and consider other abstraction principles, perhaps principles which are not constitutive of any ready-to-hand notion. 27 But given the equivalence in power of the plural quantification and aristotelian second-order logic, and the existence of oneelement models of Axiom V, this strategy will not work either. To conclude these two sections and the examination of logicism’s use of second-order logic: we have considered the effect of weakening second-order logic by allowing as epistemically innocent only the assumption of arbitrary instantiated properties, or equivalently, allowing only pluralist interpretations of second-order quantification. But the result, combined with Hume’s Principle, does not yield Frege’s Theorem. Moreover thus far we have been very generous to the neo-logicist. Many who are realist about properties, for example, assume only a “sparse” rather than an “abundant” theory of properties: that is, they do not assume that to each arbitrary predicate—(x is an electron or x is a baseball or x is a pleasant dream)—there corresponds a property. (And even those who favour an abundant theory do not always claim they know a priori that it is true.) This suggests that no form of the axiom scheme of Comprehension is epistemically innocent. But if we drop it completely, then it is easy to produce finite models of HP (and indeed of Axiom V), models 28 in which any number can be a “rogue” number in the same way that zero is a rogue number in the aristotelian version of Fregean arithmetic. Take any finite set of D individuals and any subset S of the power set of D such that there are at most n equivalence classes over S under the equivalence relation of equinumerosity. There is thus a one:one function f from this set of equivalence classes into D. We let S be the range of the monadic predicate quantifiers and select some member d of D as our rogue number; we then interpret the numerical operator as follows: where the extension s of ϕx belongs to S, the referent of nxϕx is f (E(s)), where E(s) is the equivalence class s belongs to; otherwise the referent of nxϕx is d. As before, this gives us a model of 27 Cf. Wright’s response to Boolos on the question of what notion “New V”, an abstraction principle which is a weakened form of Axiom V, is constitutive of:—“The Philosophical Significance of Frege’s Theorem”, pp. 239–240. 28 Non in standard ‘unfaithful Henkin’ models, of course: cf. Cf. Stewart Shapiro, Foundations without Foundationalism (Oxford: Clarendon, 1991), p. 89.

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second-order logic (minus Comprehension) plus Hume’s Principle, this time one in which (∃F∀y ∼ Fy) → Inf can fail, if the empty set belongs to S. Our overall conclusion, then, is that the neo-logicist program, as standardly developed, requires the use of the strongest form of the impredicative axiom scheme of Comprehension if it is to achieve its goals (e.g. the derivation of Frege’s theorem) but that this principle is on a par with outright stipulation of the Peano-Dedekind or ZF axioms in terms of epistemic innocence.

4. We turn now from the presuppositions the neo-logicist makes with respect to second-order logic to those made with respect to first-order. If one accepts that standard classical first-order logic is epistemically innocent then one already accepts that one can prove existence claims innocently, for one can prove ∃x(Fx ∨ ∼ Fx) or, in first-order logic plus identity, 29 ∃x(x = x). Even though this is not to prove very much about whatever it is which exists, the neo-logicist might take these standard theorems as revealing at least an ad hominem problem for those opponents who reject the idea of a priori or epistemically innocent existence proofs but nonetheless accept first-order logic. However, we conjecture that most contemporary logicians would respond by denying that standard non-free logic is epistemically innocent. The theorem ∃x(x = x) is harmless, they might say, since we know some things exist. And the quantifier rules ∀E and ∃I: ∀xϕx ϕx/t

ϕx/t ∃xϕx

are harmless if applied in a language in which one can reasonably assume or suppose that no singular term is empty, that is non-denoting. Making these assumptions certainly simplifies the proof theory and model theory. However, if one was interested in what could be derived by pure logic alone, or if one was working in a language containing complex singular terms which may not denote—for example, description terms or terms constructed using function terms standing for partial functions—then the above simple rules must be rejected in favour of more complex rules such as ∀xϕx, E(t) ϕx/t

ϕx/t, E(t) ∃xϕx

where E(t) represents some way of expressing the claim that t exists. Whether or not this would be the usual response of an “anti-Anselmian” logician, we submit it is the right response to make to the neo-logicist. Moreover Wright himself seems to presuppose some sort of free logic background at 29 From now on, include the standard theory of identity in “first-order logic”.

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least in the sense of allowing for empty domains. 30 Furthermore, to hold both that numerical terms such as nx(x = x) are genuine singular terms and that ∃I is unrestrictedly valid for all complex terms gives the neo-logicist a very easy victory but a Pyrrhic one since the key question is so obviously begged. Let us look, then, at Hume’s Principle and theorems of infinity against a free logic background. The restrictions on second-order ∀E and ∃I which blocked the proof of infinity in an unfree first-order background will, of course, a fortiori, do so in a free logic background. In order to isolate the problem with free logic, we grant the neo-logicist for these purposes standard second-order rules and axiom schemes, our sole amendment being altering first-order ∀E and ∃I as above, that is to ∀xϕx, E(t) ϕx/t

ϕx/t, E(t) ∃xϕx

Different policies on, and semantics for, “E” (it may, for example, be some complex formula containing t) will yield different frameworks for free logic. The problem for the neo-logicist is that in some of these the crucial proof of the infinity of the natural numbers is not forthcoming from Hume’s Principle. We cannot look at all possible free logics of course; rather we will look at one framework under which it is plausible that Hume’s Principle is epistemically innocent but under which Frege’s Theorem fails and at a rather different one on which the latter holds but it is very implausible that Hume’s Principle is epistemically innocent. It is then is up to the neo-logicist to refute our contention that these two approaches are representative in the sense that any free logic will fall on one or other horn of the dilemma. For the first example, we take the inner domain/outer domain framework— cf. Read Thinking about Logic (Oxford: Oxford University Press, 1994), pp. 134–7, 146–7. One divides the domain of individuals, of referents of individual constants, into two classes, the inner domain of “real” individuals, this domain being the range of the individual quantifiers, and an outer domain of “dummy” or “virtual” individuals, which do not belong in that range, so intuitively do not “exist”, though they can be assigned to nonvariable terms as referents. One can then adopt a number of policies for the value of an atomic sentence Pt in an interpretation in which t is assigned a merely virtual referent. One can always set such sentences as gappy, neither determinately true nor determinately false; or else always set them false, or (this option is rather unmotivated) always true, or let them vary freely as for sentences in which there is no failure of “real” reference. Predicate extensions then are assigned positive and negative extensions, each being divided into real and virtual components which together exhaust the particular extension. 30 Op. cit., pp. 235–6 where he writes, concerning the question whether the property λx(x = x) is too big, “But it will, presumably, be too big, however exactly that notion is defined, if the universe is empty”.

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Since we are working in a second-order framework, additional complications can arise if we want to mark failure of predicates to stand for properties. In a general treatment we would have domains of individuals, of properties and of n-ary relations. We then ask whether, e.g. the assignment to P in Pt of a merely virtual property induces truth-gaps or not; the semantics will proceed by assigning extensions: either of individual (positive and negative) satisfiers to properties or, dually, to each individual a set of properties which the individual has, and a negative set of properties which it lacks. To keep things as close as possible to the standard case, let us assume bivalence (so we can drop negative extensions) and confine the free logic component to the first-order case alone. So the domain of individuals D divides into exclusive and exhaustive subdomains I and O, the inner and outer domains; let us also take identity to be a primitive, its extension being all pairs α, α, for α in D, i.e. α virtual or real. So if terms t and u stand for the same individual, real or virtual, t = u is true, if they stand for distinct individuals, real or virtual, the sentence is false. This yields pretty much the standard theory of identity: t = t is valid for all t, Leibniz’s law—from ϕx/t and t = u conclude ϕx/u—is sound and so these two can be added as axiom scheme and inference rule respectively. This yields the standard theorems constraining identity to be an equivalence relation: ∀x x = x ∀x∀y(x = y → y = x) ∀x∀y∀z((x = y & y = z) → x = z). The main difference is that, since we are working in a free logic background, we cannot move from universal to existential generalisations as freely as before, i.e. cannot conclude ∃x x = x from ∀x x = x. As remarked, we are assuming the second-order quantifiers are interpreted as closely as possible to the standard way. Hence we can take the range of these quantifiers to be the power set of D thus allowing the “platonic” case of “properties” uninstantiated by a real individual. In fact, if the virtual domain is not empty we get a number of distinct empty properties even where we simply identify (which we do here, purely for convenience) properties with subsets of the domain of individuals; if v is in O, then Ø and {v} are two distinct properties both satisfying ∼ ∃yFy. Now in this framework we cannot prove, in any system sound for that semantics, ∃x(x = x), or indeed ∃x(ϕx) for any ϕ, since there are interpretations in which I is the empty set and all individuals are virtual (though we will be able to prove ∃F∀y ∼ Fy). This framework, then, does not build the neo-logicist assumption that there are a priori existence proofs, at least of the existence of individuals, into the underlying logic. Can we then show that by augmenting it with abstraction principles we can nonetheless prove, from

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epistemically innocent principles alone, that some things exist, indeed that an infinity of different numbers exist? The answer is no, for reasons analogous to the aristotelian case. There are finite, indeed empty models of Hume’s Principle in this framework, that is models in which I is finite or empty. For the latter case, take a “universally free” model in which D = O = {v} and I = Ø and so not even ∃x(x = x) true, far less a theorem of infinity is true. The Fregean proof of the infinity of the natural numbers breaks down right at the outset: we do not have ∃x(x= nx(x = x)). To see what happens to Hume’s Principle in this model, note that there are only two properties: {v} and Ø, both of them empty, that is both have empty real subextensions. Both properties, therefore, are equinumerous: with {v} assigned to F and Ø to G the formula which expresses the condition that a one:one correspondence holds between F and G, namely: ∃R∀x((Fx → ∃!y(Rxy & Gy))&(Gx → ∃ ! y(Ryx & Fy))) is true in the model. For remember that the quantifiers ∀x and the uniqueness quantifier ∃!y 31 on the right-hand side both range over the inner domain I, a domain which is empty in this case. Hence assigning to R any relation over D one likes satisfies the right-hand side of the biconditional. Every assignment to x trivially renders Fx → ∃!y(Rxy & Gy) true; for none renders it false, since there are no assignments from I to x; likewise for the second conditional. By assigning v as the number of each property with an empty real extension, thus to nxϕx, for every ϕ, we ensure the left-hand side is true so that the instance: nx F x = nx Gx ↔ ∃R∀x((F x → ∃ !y(Rx y & Gy)) &(Gx → ∃ !y(Ryx & F y))) of Hume’s Principle is true in the model; and the same holds for the other three possible assignments of pairs of properties to F and G. More generally, ensure there is at least one virtual individual and partition the properties of the universe, the subsets of D, into equivalence classes under equinumerosity (of the real extensions of the properties). If the inner domain is of size k, there will be k + 1 of these; 32 since D is of size at least k + 1, there is a function N which maps these equivalence classes into the total domain, inner plus outer. If we then interpret nxϕx via N, i.e. assign to it the value of N applied to the element of the partition to which the extension of ϕx belongs, then it is easy to see Hume’s Principle comes out as true in the model. Note that we could add as an additional requirement on admissible models that all simple singular terms have real referents. If there is some finite number n of simple singular terms, then will be able to construct finite models of Hume’s Principle in which standard ∀E and ∃I are sound where instantiation is for simple terms, finite models which can be of any size ≥ n. 31 i.e. ∃!yϕy is (∃yϕy & ∀x(ϕx → x = y)). 32 Since k can be infinite, we assume the Axiom of Choice in the metatheory.

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One might also note that Axiom V is satisfiable in this framework too; in fact, there are models of every cardinality of Axiom V and indeed of any abstraction principle. 33 This tolerance of arbitrary sets of abstraction principles affords no comfort to the neo-logicist since we still cannot prove any set, any non-proper class, exists. The outer domain of “virtual” objects, in other words, is best seen as a technical device. It gives us a semantics in which the free logic versions of first-order ∀E and ∃I are sound (with E(t) true just when t has a referent in the inner domain). And even if we were prepared to tolerate a baroque ontological slum in which virtual objects have some type of being, this will still not help the neo-logicist since there are models where the total domain D, including virtual individuals as well, is finite.

5. However, there are other frameworks for free logic than the inner/outer framework above. For instance, there are free logics in which the existence premiss E(t) is taken to be self-identity: t = t is true only if the term t refers. 34 This framework seems better suited to the neo-logicist position. The neologicist can hold that t = t entails ∃x x = t but deny that t = t is valid for every t. In some special cases, however, such an identity may be provable. For example, nx(x = x) = nx(x = x) is provable from the instance of Hume’s Principle in which we instantiate both predicate variables with x = x, for the right-hand side of this instance is a logical truth. Let us take E(t), then, to be t = t. But in order to prevent confusion with the identity relation in which t is identical with t is always true, we will retain “=” with the interpretation given above and expand our language to include in addition this second “existential” identity relation which we will express as “t ∼ = t”, evaluating such sentences as true in a model iff the referent of t belongs to the inner domain of the model. 35 The amendments to standard rules needed are then: ∀xϕx, t ∼ =t = t ϕx/t, t ∼ . ϕx/t ∃xϕx 33 Where I is of cardinality κ, let O be a disjoint set of cardinality 2κ . There is thus a one:one function B from the power set of I into D. Every property contains a real subextension (possibly empty). Interpret {x: ϕx} by a map which takes property P to B(X), where X ⊆ I is the real subextension of P. Given our interpretation of =, this is easily seen to satisfy Axiom V (for courses of values):

∀F∀G(({x : Fx} = {x : Gx}) ↔ ∀z(Fz ↔ Gz)). For, given any assignment of properties P1 and P2 to F and G, the left-hand side of the embedded biconditional is true iff B(rP1 ) = B(rP2 )—rPi being the real subextension of P—iff the real subextensions of P1 and P2 are identical iff (since ∀z ranges only over I), ∀z(Fx ↔ Gz) is true. The argument generalises to arbitrary abstraction principles. 34 See Dana Scott et al., Notes on the Formalization of Logic, Part II, Sub-Faculty of Philosophy Study Notes, University of Oxford. 35 Though when we leave wide identity = out of consideration, it does not matter whether we divide the individual domain into real and virtual components or simply let empty terms denote nothing at all.

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We can also lay down in the semantics that individual parameters can only be assigned real referents in I. 36 This ensures the soundness of standard ∀I and ∃E and also the logical truth of a ∼ = a,where a is any parameter (though t ∼ = ∼ t is not true in general). Since the = version of Leibniz’ law is also sound in this semantics—the premiss t ∼ = u can only be true when both t and u are nonempty and stand for the same referent—reflexivity, symmetry and transitivity of identity still hold in the generalised form above (i.e. ∀x x ∼ = x holds, even though t ∼ t may fail). = So we still have a fairly standard-looking theory of identity. The neo-logicist can then argue that the mere introduction of numerical and class terms alone does not prove that numbers and classes exist so that there is no questionbegging assumption from the outset that “cheap” proofs of existence theorems are available. However, it is still true, in this framework, that the identity relation is a one:one map from non-self-identity onto itself, reading identity as “wide” non-existential identity, i.e. (using λ abstracts for clarity here) (λx(x = x) 1 − 1 λx(x = x)) holds by dint of the relation λx(λy(x = y)). More fully this is: ∀x((x = x → ∃ !y(x = y & y = y)) & (x = x → ∃!y(y = x & y = y))). ∼ x) 1–1∗ λx ∼ (x = ∼ x)), where the ∗ indicates Indeed we also have λx ∼ (x = one:one functions expressed in terms of ∼ = rather than =. For although we do not have ⊢ t ∼ = t, for arbitrary t, we do have ⊢ a ∼ = a, where a is a parameter; hence we have both ⊢ a = a → ∃!y(a = y & y = y) and ⊢∼ (a ∼ =y&y∼ = y) 37 = a) → ∃ ! ∗ y(a ∼ and in general can prove anything from a = a (e.g. ∃!*y(a ∼ = y & y∼ = y)) ∼ and anything from ∼ (a = a) (e.g. ∃!y(a = y & y = y)) by ex contradictione quodlibet. Thus (λx ∼ (x ∼ = x)) also holds by dint of = x) 1 − 1∗ λx ∼ (x ∼ the relation λx(λy(x ∼ = y)). Now that we have two different notions of identity we have a number of different versions of Hume’s Principle, for instance HP∗ : ∀F∀G (nx F x ∼ = nx Gx ↔ ∃R∀x((F x → ∃!∗ y(Rx y & Gy)) &(Gx → ∃!∗ y(Ryx & F y)))) where ∃!∗ y(Rx y & Gy) is ∃y((Rx y & Gy) & ∀z((Rx z & Gz) → z ∼ = y)). 36 In models with empty I, atomic formulae containing parameters are all to be interpreted as true since there are no admissible assignments under which they are false. 37 With ∃!∗ ϕx abbreviating (∃yϕy & ∀x(ϕx → x ∼ = y)).

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Our original HP but with = interpreted in the wide non-existential fashion in our free logic setting is, for comparison. 38 ∀F∀G(nxFx = nxGx ↔ ∃R∀x((Fx → ∃!y(Rxy & Gy)) & (Gx → ∃!y(Ryx&Fy)))) ∃ !y(Rx y & Gy) being ∃y((Rxy & Gy) & ∀z((Rxz & Gz) → z = y)). Focusing on HP*, we can prove by instantiating F and G by λx(∼ (x ∼ = x)), (and using the Substitutivity Rule) the following: N x ∼ (x ∼ = x) ∼ = nx ∼ (x ∼ = x) ↔ ∃R∀x((∼ (x ∼ = x) → ∃ !∗ y(Rx y & ∼ (y ∼ = y))) & (∼ (x ∼ = x) → ∃ !∗ y(Ryx & ∼ (y ∼ = y)))). Since the right-hand side of the biconditional is, as we have seen, provable, we can deduce (nx ∼ (x ∼ = x) ∼ = nx ∼ (x ∼ = x)) (i.e. 0 ∼ = 0 with zero defined ∼ ∼ ∼ using =) and so prove ∃x(x = nx ∼ (x = x)) using our latest version of free ∃I where E(t) = df. T ∼ = t. And from here the Fregean proof of the infinity of the natural numbers can proceed as before. So HP∗ amounts to an axiom of infinity even in the context of free logic with existential identity. However, this is another place at which we must depart from our policy of not contesting the neo-logicist claim that abstraction principles such as Hume’s Principle are epistemically innocent. For the plea of innocence is more plausible with respect to HP than HP∗ . One key notion which neo-logicists have used to argue for the innocence of Hume’s Principle and against the idea that they beg any questions in teasing out ontological commitments from such abstraction principles is the idea of “reconceptualisation”. Taking as their text Grundlagen §64, the neo-logicists do not claim that in general one can generate objects from concepts; rather one shows how to “reconceptualise” some thoughts so as to generate new concepts which carve up the “state of affairs” represented in the thoughts in a different way (as involving an identity between directions rather than a parallelism between lines, for instance). More generally, the neo-logicist argues that one can “recarve” the concepts involved in thinking of a state of affairs in which an equivalence relation obtains in such a way as to generate new concepts, pertaining to abstracts. Moreover there is to be, using Frege’s directions example: 39 absolutely no gap between the existence of directions and the instantiation of properties and relations among lines. 38 And of course there are a number of other of versions of Hume’s Principle using permutations of = and ∼ =; the further variants produce no philosophically relevant new cases. 39 C. Wright, “On the philosophical significance of Frege’s theorem”, Language, thought, and logic, edited by Richard Heck, Jr., Oxford, Oxford University Press, 201–244.

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But what Wright must mean here is that there is absolutely no epistemic gap between the existence of directions (or the truth of identities between abstracts) and the instantiation of properties and relations among the original domain of objects. Reconceptualisation cannot do the job it is intended to do unless in addition to referring to a process of conceptual innovation or creation it also carries an epistemic punch. For suppose numbers do exist of necessity. Then “0 exists” and “0 ∼ = 0” will be semantically equivalent to (various readings of) “there is a one:one map from non-self-identity onto itself”. Indeed “0 exists” will be semantically equivalent to P → P , for any P. But the neo-logicist needs more than this, if neo-logicism is to answer the epistemological worries usually directed against platonism. The neo-logicist needs there to be an epistemic equivalence so that we can know that 0 ∼ = 0, know the truth of the left-hand side of the relevant instance of HP*, on the basis of our knowledge of the truth of the right-hand side (though we do not know it on the basis of our knowledge that P → P). So to say that we “reconceptualise” some “state of affairs” described in terms of relations among properties into one involving objects such as numbers is just to assert that we can know the objects exist on the basis of our knowledge of the relations among properties. It is not to show how we can know this. In the case of wide identities such as 0 = 0, the epistemic gap may indeed be small. But x = x is not, of course, an existence predicate and in knowing the truth of 0 = 0 we are not coming to know anything about the existence of objects. The identity t ∼ = t, on the other hand, has the sense or informational content of [t exists] or a sense very close to it: competent speakers, after all, are to use such identities as the existential premisses of the ∃I and ∀E rules. So in this case we are being asked to accept that the state of affairs consisting of there being a one:one mapping of the existential identity relation λx(x ∼ = x) onto ∼ nx(∼ x∼ x) itself can be reconceptualised into the thought [nx(∼ x ∼ = x)] = = in such a way that there is not even the slightest epistemic gap between the latter proposition, which is tantamount to “the number zero exists” and the former which is held to be a (second-order) logical truth. But this is to ask us to accept right at the outset that some epistemically innocent truths 40 are equivalent to existence claims. This is precisely the point on which we remain to be convinced:– it is no clearer how we can know zero exists on the basis of our knowledge of the one:one map on non-self-identity than we can know that it exists on the basis of our knowledge that if the moon is made of green cheese then it is made of green cheese. It may seem that there is a closer link between “0 ∼ = 0” and “there is a one:one map from non-self-identity onto itself” than there is between “0 ∼ = 0” 40 Let us for the sake of argument grant epistemic innocence to (λx ∼ (x ∼ x) 1 − 1∗ λx ∼ (x ∼ x)) = = though it is far from innocent on the aristotelian understanding of second-order logic.

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and P → P. The existence of zero is an “ingredient” of the state of affairs of the one:one map on the property, and not on the conditional state of affairs, if there is such a thing. Stripping away the metaphor, the claim of an especially close epistemic link between the two sides of the biconditional double instantiation of Hume’s Principle surely rests on the idea that the principle is analytic or in some sense constitutive of our notion of number. While this may be plausible for HP, applied to HP∗ it entails the claim that there is a purely conceptual link between a truth of (standard) second-order logic and an existence claim, i.e. it entails that one can get existence out of meanings alone, the very thing which has to be demonstrated not taken as a premiss. Far from there being no epistemic gap between the left and right sides of instances of HP∗ , there is a large chasm bridged, according to the neo-logicist, by “reconceptualisation”. But to anyone not already convinced that meaningconstitutive principles can generate mind-independent existence claims, this is a bridge too far.

6. Overall we have seen that the neo-logicist needs to show not only that some second-order principles are epistemically innocent, if the full neo-logicist programme is to be successfully accomplished; neo-logicism requires the full axiom scheme of Comprehension which, we argued, embodies substantive non-innocent ontological commitments (at least if the semantics for the pure mathematical sector is taken to be homogeneous with a broadly realist semantics for the non-mathematical sector). Moreover if the neo-logicist assumes the innocence of standard non-free first-order logic then he or she begs the question against opponents of neo-logicism. If not, then if identity does not have existential import, Frege’s Theorem fails whereas if it does have existential import, then Frege’s Theorem holds but the interpretation of the required abstraction principles, such as HP∗ , will beg the question in much the same way. Our conclusion, therefore, is that the neo-logicist has no non-questionbegging account of how there could be an epistemically innocent route to the demonstration of platonistically construed mathematical existence claims. 41

41 Thanks to the participants at the Abstraction Day conference, St. Andrews, Scotland, 14th November 1998 for discussion and to Michelle Friend for comments on an earlier draft.

ARISTOTELIAN LOGIC, AXIOMS, AND ABSTRACTION 1 Roy T. Cook

1.

Introduction

Neo-logicism is the view that various branches of mathematics can be reformulated in terms of abstraction principles that we can stipulate, and thus come to know the truth of, a priori. The main success story of neo-logicism so far is the derivation of arithmetic from Hume’s Principle: HP: (∀P)(∀Q)[Num(P) = Num(Q) ↔ P ≈ Q] where P ≈ Q is the second-order formula asserting that there is a one-toone correspondence between the P’s and the Q’s. Recently Stewart Shapiro and Alan Weir have criticized this view, arguing in ‘Neo-logicist logic is not epistemically innocent’ [2000] that abstraction principles do not provide us with a priori access to the objects necessary for mathematics: Frege’s theorem requires use of first- and second-order logical principles which are not epistemically innocent. More exactly, certain of the logical principles which are essential to the derivation of a theorem of infinity, when this is construed as expressing the existence of infinitely many mind-independent entities, are at least as problematic epistemologically as axioms of infinity laid down simply as postulates. Our supposed knowledge of these principles is, we will argue, every bit as mysterious as Kantian intuition of an infinity of numbers. (p. 162)

Shapiro and Weir suggest that someone skeptical of the strength of full secondorder logic could accept that: . . . it is a logical truth that to every . . . sentence which is instantiated by something or other, there corresponds a co-extensional property. But she refuses to accept that logic alone tells us that there are uninstantiated properties, so refuses to conclude that to predicates such as x = x there corresponds a property. (p. 165) 1 This paper first appeared in Philosophia Mathematica 11, [2003], pp. 195–202. Reprinted by kind permission of the editor and Oxford University Press.

147 Roy T. Cook (ed.), The Arché Papers on the Mathematics of Abstraction, 147–153. c 2007 Springer.

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To avoid such problematic assumptions, Shapiro and Weir suggest that the neo-logicist abandon standard second-order logic with the full comprehension schema and replace it with the following more restricted version of comprehension, guaranteeing the existence of a property for every instantiated predicate ( any (possibly complex) predicate): AristComp : (∃x1 , x2 , . . . , xn )((x1 , x2 , . . . , xn )) → (∃R)(∀x1 , x2 , . . . , xn ) (R(x1 , x2 , . . . , xn ) ↔ (x1 , x2 , . . . , xn )). Shapiro and Weir call the resulting logic Aristotelian. For the neo-logicism, the important difference between Aristotelian logic and standard second-order logic is that one cannot derive the existence of infinitely many numbers from HP in Aristotelian logic: . . . the Fregean proof of infinity fails . . . because we cannot prove that there are n + 1 numbers less than or equal to n; for all the Aristotelian knows, zero happens to be identical to some number Sk 0 between zero and n. As we have seen, it is possible for 0 = 1 to hold in a model in the Aristotelian framework in which case the number of numbers less than or equal to one is just one. (p. 168)

Shapiro and Weir’s result generalizes to any abstraction principle as long as the equivalence relation on the right-hand side of the biconditional is formulated in purely logical terminology—any such neo-logicist abstraction principle (including Frege’s notorious Basic Law V) will have a one-element model. Given an abstraction principle AP: AP: (∀P)(∀Q)[@(P) = @(Q) ↔ E(P, Q)] We can construct an Aristotelian model of AP by letting the domain consist of a single object, call it a. @ is then the function that maps any property or predicate onto a. Since there is only one property, {a}, AP is trivially satisfied. Thus, on an Aristotelian conception of logic, every second-order abstraction principle is consistent, but no second-order abstraction principle will imply that there are infinitely many objects. Shapiro and Weir conclude from all this that: . . . the neo-logicist has no non-question-begging account of how there could be an epistemically innocent route to the demonstration of platonistically construed mathematical existence claims. (p. 188)

Just because the neo-logicist project does not meet one of its goals (or even perhaps its most important goal) does not mean that it meets none of its goals, however. Even on the Aristotelian conception of logic, neo-logicism does provide us with a great deal, as the following two case studies demonstrate.

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Case study 1: Hume’s Principle

The first step in deriving second-order arithmetic from HP is the definition of the finite (or ‘natural’) numbers, the constant ‘0’, the binary successor relation ‘S’, and the ternary relations addition ‘+’ and multiplication ‘×’. In standard second-order logic HP implies that the definitions of S, + , and × define total functions, i.e. given these definitions, all of the Peano axioms for arithmetic follow from HP (for details see Wright [1983]). On the Aristotelian conception of logic we do not get all of this, but even so HP implies a significant chunk of Peano arithmetic. The successor and addition relations need not be total, but we can prove that they are partial functions. Of the seven Peano axioms, four are not implied by HP trivially since they contain ‘0’, the name of the number of the empty property, which does not (or might not) exist on the Aristotelian conception of logic. 2 The other three do follow. Of course, just because 0 does not exist does not mean that no numbers exist. We can define the number 1 as: 3 1 =df Num(x is a finite number ∧ (∀y)(S(x, y) ↔ + (x, x, y))) Assuming that we do not countenance empty models, 4 the existence and uniqueness of 1 follow from HP in Aristotelian logic. As a result we can formulate alternatives to each of the problematic Peano axioms in terms of ‘1’. For example, the problematic successor axiom can be expressed as: 5 (∃x)(x = 1) ∧ (∃x)(∀y)(¬(Sy, x)) Along similar lines, we can replace the axioms for addition, multiplication, and induction containing ‘0’ with: (∀x)(∀y)(S(x, y) → + (y, 1, x)) (∀x)(×(x, 1, x)) (∀P)[((∀w)((∀z)(¬S(z, w)) → P(w))) ∧ (∀x)(∀y)(S(x, y) → (P(x) → P(y)))) → (∀x)P(x)] Each of these follows from HP on the Aristotelian conception of logic. Interestingly, on the Aristotelian picture of logic, although we cannot prove that there are infinitely many numbers, we do get a substantial description of the behavior of the numbers that do exist. There are two main differences 2 Actually, as Shapiro and Weir point out, 0 is in a sense guaranteed to exist, since ‘Num(x = x)’ is guaranteed to have a referent. If there is no property P such that (Px ↔ x = x), however, then HP does not apply to 0. As a result, 0 could be identical to any object, including any other number. Thus, the real problem here is that, in Aristotelian logic, 0 may be very badly behaved. 3 In this formula and those below the quantifiers should be understood to be restricted to finite numbers. 4 In later sections of their paper Shapiro and Weir challenge this very assumption, examining the prospects for neo-logicist arithmetic in a free logic. Here, however, our concern is with Aristotelian logic and its motivations, such as Boolos’ plural reading of second-order quantification. As a result, assuming that the domain is non-empty seems both unproblematic and, since there might be no property to pick out the supposed empty domain, entirely natural. 5 Note that the number without a predecessor need not be identical to 1.

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between the consequences of HP in standard second-order logic and its consequences on the Aristotelian conception. First, the relations of successor and addition can turn out to be partial functions on Aristotelian logic. Second, we were forced to reformulate some of the axioms so that they did not rely explicitly on the existence (or, more accurately, on the good behavior) of 0. The axioms we obtain in Aristotelian second-order logic, however, are relatively natural 6 axioms for the non-zero natural numbers, and are defective only in the sense that we are not assured that successor, addition, and multiplication are total functions.

3.

Case study 2: NewV

A second case study is useful to convince us that the phenomenon in question is quite general. In this section we will investigate the neo-logicist treatment of set theory from the perspective of Aristotelian logic, based on Boolos’ [1989] NewV: 7 NewV: (∀P)(∀Q)[Ext(P) = Ext(Q) ↔ ((P is ‘Big’ ∧ Q is ‘Big’) ∨ (∀x)(Px ↔ Qx))] where ‘P is ‘Big’ is an abbreviation for the second-order formula asserting that the P’s are equinumerous with the entire domain. In standard secondorder logic, with ‘is a set’ and ‘∈’ defined in the standard way (see Boolos [1989] for details), NewV entails the extensionality, empty set, pairing, union, separation, and replacement axioms, but not the powerset axiom or the axiom of infinity. On the Aristotelian account of second-order logical consequence, however, NewV implies extensionality, pairing, union, and replacement, but fails to imply empty set or separation. The fact that the empty-set axiom does not follow on the Aristotelian conception of logic is unsurprising, since the empty-set axiom is equivalent to the claim that there is a property that is not ‘Big’ and has no instances. The failure of the axiom of replacement seems more surprising until one realizes that it (plus the claim that some set exists) implies the empty-set axiom. A revised version of separation that does not imply the existence of an empty set does follow from NewV in Aristotelian logic: Arist.Separation: (∀P)(∀x)((x is a set ∧ (∃w)(w ∈ x ∧ Pw)) → (∃y)(y is a set ∧ (∀z)(z ∈ y ↔ (z ∈ x ∧ Pz)))) 6 Historically, of course, 0 was not accepted as a legitimate number by many groups that were otherwise mathematically quite sophisticated, including the Greeks. In addition, both Peano and Dedekind formulated their original arithmetical axioms with 1 as the initial number. Thus, there is some reason for thinking that the axioms that do follow from HP on Aristotelian logic capture the (or an) intuitive conception of the natural numbers. Thanks are owed to Fraser MacBride for pointing this out. 7 I am ignoring the well-documented problems with NewV (see Shapiro and Weir [1999] and Boolos [1989]). Even if NewV is an inadequate foundation for a neo-logicist theory of sets, it nevertheless provides us with another nice example of how abstraction principles behave in Aristotelian contexts.

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Thus, with suitable reformulations, the only axiom that does not follow from NewV on the Aristotelian conception of logic that does follow on the standard conception is the empty-set axiom. Again, we see a division between existential principles such as the empty-set axiom and principles that just govern the behavior of and interactions between sets without implying the existence of any particular sets. On the Aristotelian conception of logic NewV does not guarantee that any sets exist (since it has a one-element model) but it does guarantee that any sets that do exist behave exactly as we would expect them to behave, i.e. they satisfy (many of) the standard axioms of set theory.

4.

What does neo-Fregean abstraction actually give us?

There are two initial reactions that one might have to these results. The first, optimistic, reading was suggested at the beginning of this essay. Even if the Aristotelian challenge is correct, the neo-logicist can retrench, arguing that he has still given us something useful. First, even in the Aristotelian context, HP and NewV provide us with enough for many of the basic applications of arithmetic and set theory. In Aristotelian logic HP plus the claim that there are n distinct objects implies that there are (at least) n distinct numbers, and HP plus the claim that there is a non-numerical object implies that there are infinitely many finite numbers. 8 Similarly, NewV plus the claim that there are two distinct objects implies that there are infinitely many sets. Thus, adding HP or NewV to a suitably robust physical theory (i.e. one that contained one or more of these additional claims) would surely allow us to carry out much of the mathematics necessary for science. Second, and perhaps more importantly, abstraction principles such as HP and NewV imply strong constraints on the behavior of the concepts that they purport to define, even if they do not (on their own) imply the existence of any (or many) of the objects supposedly falling under the scope of these concepts. If the neo-logicists can still defend the claim that Hume’s Principle in some sense defines the concept of cardinal number (or NewV defines the concept of set), independent of any ontological implications, then they will have provided the philosophy of mathematics with something valuable. Replacing a collection of axioms haphazardly compiled over decades or centuries with a single principle that tells us exactly what a mathematical concept means and how the objects falling under it must behave is certainly a step in the right direction. Shapiro and Weir’s objections do not in any way affect this part of the neo-logicist project, since it is perfectly conceivable that we could provide a suitable account of the meaning of a concept without thereby judging one way or another what objects fall under this definition, if in fact any do. 8 None of this is of help to the neo-logicist, however, unless ‘there are n objects’ or ‘there is a nonnumerical object’ is knowable a priori.

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It is this last point, however, that brings us to the pessimistic reading. On this interpretation, neo-logicism was doomed from the start, since we should not expect any definition of a concept, even an implicit one, to imply the existence of infinitely many objects. George Boolos has expressed something like this worry: Despite the Godel incompleteness theorems and Russell’s protestations that the axiom of infinity was no logical truth, it was a central tenet of logical positivism that the truths of arithmetic were analytic. Positivism was dead by 1960 and the more traditional view, that analytic truths cannot entail the existence of either particular objects or of too many objects, has held sway since . . . it should be asked how a statement that cannot hold if there are only finitely many objects can possibly be thought to be analytic, a matter of meanings or ‘conceptual containment’. ([1997], pp. 249–250)

According to this line of thinking, what we should expect from our definitions, at best, is constraints on when the concept being defined is applicable. Thus, there must have been something wrong with the original formulation of neologicism, and the results of Shapiro and Weir’s paper (and of the present essay) finally show us exactly what that flaw was. The pessimistic reading, however, seems a bit harsh. If we accept from the beginning the idea that logic and definitions cannot have existential consequences, then neo-logicism is a non-starter. What the pessimist has got right, however, is emphasizing that, if neo-logicism is to be successful in explaining how we come to know of the existence of infinitely many mathematical objects from definitions and logic alone, then the neo-logicists owe us an explanation of the role of logic and, more specifically, a justification of their particular choice of logic. The results of Shapiro and Weir demonstrate that the choice of logic is more crucial than one might initially think. Neo-logicism follows not merely from the conjunction of views about the nature of definition and stipulation alone, but follows from these claims plus a view about what the correct account of logic is. Wright and Hale have explicitly pointed out the importance logic has to play in the neo-logicist project and the relative lack of attention it has received: The logicist theory about a particular mathematical theory is that its fundamental laws are obtainable on the basis just of definitions and logic. It would at the time of writing be a justifiable complaint that while much attention has been paid by neo-Fregeans, and their critics, to the first component in the recipe— issues to do with abstractions in general and Hume’s Principle in particular— comparatively little has been given to the second component: the demands, technical and philosophical, to be made on the logical system which is to provide the medium for the proofs the neo-Fregeans need. ([2001], p. 429, emphasis added)

Shapiro and Weir’s paper represents an important first step in fleshing out the second of these issues, the requirements on the logic of neo-logicism, and

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the present discussion has, it is hoped, helped to sharpen their insights even further. While they are right to point out that: . . . the neo-logicist has no non-question-begging account of how there could be an epistemically innocent route to the demonstration of platonistically construed mathematical existence claims ([2000], p. 188, emphasis added) 9

it would be premature to conclude that no such non-question-begging account is possible. Rather, their arguments serve to point out that work remains to be done, and the direction that this work needs to take. 10

References Boolos, G. [1989], “Iteration Again”, Philosophical Topics 17: 5–21, reprinted in Boolos [1998], pp. 88–104. Boolos, G. [1997], “Is Hume’s Principle Analytic?”, in Heck [1997]: 245–261, reprinted in Boolos [1998], pp. 301–314. Boolos, G. [1998], Logic, Logic, and Logic, Cambridge Mass, Harvard University Press. Heck, R. [1997], Language, Thought, and Logic, Oxford, Clarendon Press. Shapiro, S. and A. Weir [1999], “NewV, ZF, and Abstraction”, Philosophia Mathematica 7: 293–321. Shapiro, S. and A. Weir [2000], “Neo-logicist Logic Is Not Epistemically Innocent”, Philosophia Mathematica 8: 160–189. Wright, C. [1983], Frege’s Conception of Numbers as Objects, Scots Philosophical Monographs, vol. 2, Aberdeen, Aberdeen University Press. Wright, C. and R. Hale [2001], The Reason’s Proper Study, Oxford, Oxford University Press.

9 Actually, they are not quite correct here, since, even in the Aristotelian context, HP implies ‘(∃x)(x = 1)’, which is, one would think, a ‘platonistically construed mathematical existence claim’. 10 A version of this note was presented to members of Arché: The Centre for the Philosophy of Logic, Language, Mathematics, and Mind at the University of St Andrews and benefited considerably from the resulting discussion. Thanks are also owed to Peter Clark, Philip Ebert, Fraser MacBride, Graham Priest, Agustín Rayo, Stewart Shapiro, Crispin Wright, and an anonymous referee for helpful comments and criticism.

FREGE’S UNOFFICIAL ARITHMETIC 1 A. Rayo

In The Foundations of Arithmetic and The Basic Laws of Arithmetic, Frege held the view that number-terms refer to objects. 2 Later in his life, however, he seems to have been open to other possibilities: Since a statement of number based on counting contains an assertion about a concept, in a logically perfect language a sentence used to make such a statement must contain two parts, first a sign for the concept about which the statement is made, and secondly a sign for a second-order concept. These second-order concepts form a series and there is a rule in accordance with which, if one of these concepts is given, we can specify the next. But still we do not have in them the numbers of arithmetic; we do not have objects, but concepts. How can we get from these concepts to the numbers of arithmetic in a way that cannot be faulted? Or are there simply no numbers in arithmetic? Could the numbers help to form signs for these second-order concepts, and yet not be signs in their own right? 3

To illustrate Frege’s point, let us consider the number–statement ‘there are three cats’. It might be paraphrased in a first-order language as: 4 (∃3 x)[C AT(x)].

(1)

If its logical form is to be taken at face value, (1) can be divided into two main logical components: first, the predicate ‘C AT(. . . )’, which for Frege refers to the (first-order) concept cat; and, second, the quantifier-expression ‘(∃3 x)[ . . . (x)]’, which for Frege refers to a second-order concept (specifically, the second-order concept which is true of the first-order concepts under which

1 This paper first appeared in Journal of Symbolic Logic 67, [2002], pp. 1623–1638. Reprinted by kind

permission of the editor and the Association for Symbolic Logic. 2 This is reflected in his definition of number. See, for instance Frege (1884) §67. 3 Notes for Ludwig Darmstaedter, pp. 366–7. I have substituted ‘second-order’ for ‘second-level’. 4 As usual, ‘(∃ x)[φ (x)]’ is defined as ‘∃x(φ (x) ∧ ∀y(φ (y) → x = y))’, and (for n > 1) ‘(∃ x)[φ (x)]’ n 1 is defined as ‘∃x(φ (x)∧ (∃n−1 y)[φ (y)∧ y = x])’.

155 Roy T. Cook (ed.), The Arché Papers on the Mathematics of Abstraction, 155–171. c 2007 Springer.

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precisely three objects fall). 5 Significantly, Frege would regard neither of these components as referring to an object. Let us now consider a close cousin of ‘there are three cats’, namely, ‘the number of the cats is three’. This sentence might be paraphrased as: the number of the cats = 3.

(2)

If its logical form is to be taken at face value, (2) cannot be divided into a predicate and a quantifier-expression, like (1). Instead, Frege would take ‘the number of the cats’ and ‘3’ to be names, referring to numbers (which he regarded as objects). Frege saw a deep connection between sentences like (1)—in which something is predicated of a concept—and sentences like (2)—in which something is predicated of the number associated with that concept. An effort to account for this connection was a main theme in his philosophy of arithmetic. But, after the discovery that Basic Law V leads to inconsistency, he found much reason for dissatisfaction with his original proposal. As evidenced by the quoted passage, he no longer felt confident about the possibility of getting from concepts to their numbers ‘in a way that cannot be faulted’. Towards the end of the passage, Frege considers an alternative: the view that there really are no numbers in arithmetic, and that—appearances to the contrary—numerals are not names of objects. They do not even instantiate a legitimate logical category, they are merely orthographic components of expressions standing for second-order concepts. The grammatical form of a sentence like (2) is therefore not indicative of its logical form. Presumably, ‘the number of the cats = 3’ is to be divided into two main logical components. First, the expression ‘. . . cats’, which refers to the (first-order) concept cat; and, second, the expression ‘the number of the . . . = 3’, which refers to a secondorder concept (specifically, the second-order concept which is true of the firstorder concepts under which precisely three objects fall). The numeral ‘3’ is merely an orthographic component of ‘the number of the . . . = 3’, in much the same way that ‘cat’ is an orthographic component of ‘caterpillar’. The outermost logical form of (2) is therefore identical to that of (1). If, in addition, it turns out that the logical form of ‘the number of the . . . = 3’ corresponds to that of ‘(∃3 x)[ . . . (x)]’, then the logical form of (1) is identical to that of (2). It is unfortunate that Frege never spelled out his unofficial proposal (as we shall call it) in any detail. In particular, he said nothing about how first-order arithmetic might be understood. Luckily, Harold Hodes has developed and defended a version of the Unofficial Proposal. 6 On Hodes’s reconstruction, a 5 For Frege, a first-order concept is a concept that takes objects as arguments, and an (n + 1)th-order concept is a concept that takes nth-order concepts as arguments. See Frege (1831903), §21. Unless otherwise noted, we shall use ‘concept’ to mean ‘first-order concept’. 6 See Hodes (1984). See also Wright (1983) pp. 36–40 and Bostock (1979), volume II chapter 1.

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sentence ‘F(n)’ of the language of first-order arithmetic is to be regarded as abbreviating a higher-order sentence ‘(FX )((∃n x)[Xx])’, where ‘(∃n x)[ . . . x]’ refers to a second-order concept, and ‘(FX )( . . . X . . . )’ refers to a third-order concept. For instance, the first-order sentence ‘PRIME(19)’ abbreviates a certain higher-order sentence ‘(PrimeX )((∃19 x)[Xx])’. On Hodes’s version of the Unofficial Proposal, quantified sentences involve quantification over second-order concepts. More specifically, they involve quantification over finite cardinality object-quantifiers: the referents of quantifier-expressions of the form ‘(∃n x)[ . . . x]’. 7 Thus, the first-order ‘∃zP RIME (z)’ would abbreviate the result of replacing the position occupied by ‘(∃19 x)[ . . . x]’ in ‘(PrimeX )((∃19 x)[Xx])’ by a variable ranging over finite cardinality object-quantifiers, and binding the new variable with an initial existential quantifier. Hodes’s account of first-order arithmetic therefore requires third-order quantification. And the obvious extension to nth-order arithmetic (for n ≥ 2) would call for (n + 2)th-order quantification. Such logical resources are increasingly problematic. 8 Here we shall see that more modest resources will do. We will develop a version of the Unofficial Proposal within a second-order language, and show that it can be used to account for nth order arithmetic (for any finite n). This, in itself, is a surprising result. But it is especially important in light of the fact that, although the use of higher-order languages is often considered problematic, recent work has done much to assuage concerns about certain second-order resources. 9 We will also see that the Unofficial Proposal has important applications in the philosophy of mathematics.

1.

Transformation

We will see that there is a general method for ‘nominalizing’ arithmetical formulas as second-order formulas containing no mathematical vocabulary. As an example, consider ‘The number of the cats is the number of the dogs’. This sentence might be nominalized as ‘The cats are just as many as the dogs’, or: ˆ OG(x)],10 x[C ˆ AT(x)] ≈ x[D where ‘≈’ expresses one–one correspondence. 11 7 See Hodes (1990) §3. 8 Hodes (1990), observation 5, offers a nominalization of second-order arithmetic which does not

exceed the resources of second-order logic. But it proceeds by encoding Ramsey sentences, and is therefore not a version of Frege’s Unofficial Proposal. 9 See Boolos (1984, 1985a, 1985b), McGee (2000), and Rayo and Yablo (2001). 10 Syntactically, an expression of the form ‘ x[φ ˆ (x)]’ takes the place of a monadic second-order variable. But the result of substituting ‘x[φ(x)]’ ˆ for ‘Y ’ in a formula ‘(Y )’ is to be understood as shorthand for: ∀W (∀x(W x ↔ φ (x)) → (W)). 11 That is, ‘X ≈ Y ’ abbreviates ∃R[∀w(Xw → ∃!v(Yv ∧ Rwv)) ∧ ∀w(Yw → ∃!v(Xv ∧ Rvw))]

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Consider now the sentence ‘the number of the cats is 3’. It can be nominalized as: ˆ AT(x]); 3 f (x[C where numeral-predicates are defined in the obvious way:

r 0 f (X ) ≡d f ∀ v¬X (v); r 1 f (X ) ≡d f ∃W ∃v(0 f (W ) ∧ ¬W (v) ∧ ∀ w(X (w) ↔ (W (w) ∨ w = v))); r 2 f (X ) ≡d f ∃W ∃v(1 f (W ) ∧ ¬W (v) ∧ ∀ w(X (w) ↔ (W (w) ∨ w = v))); r etc. This sort of nominalization can easily be generalized. In order to do so, we work within a two-sorted second-order language L containing the following variables: first-order arithmetical variables, ‘m 1 ’, ‘m 2 ’, . . . , monadic secondorder arithmetical variables ‘M1 ’, ‘M2 ’, . . . , first-order general variables, ‘x1 ’, ‘x2 ’, . . . , and, for n a positive integer, n-place second-order general variables X 1n , X 2 , . . . . 12 We assume that L has been enriched with a single higher-level predicate ‘N’ taking a monadic second-order general variable in its first argument-place and a first-order arithmetical variable in its second argument-place. 13 The well-formed formulas of L are defined in the usual way, with the proviso that an atomic formula can contain arithmetical variables only if it is of the form m i = m j , Mi m j or N(X i1 , m j ). 14 On the intended interpretation, arithmetical variables are taken to range over the natural numbers, and general variables are taken to have an unrestricted 12 As a precaution against variable clashes, we divide monadic second-order general variables in two: the 1 —which we abbreviate Z —will be paired with first-order arithmetical variables; the X 1 X 2i i 2i+1 — which we abbreviate X i —will be used for more general purposes. Also to avoid variable clashes, we 2 —which we abbreviate R —will be divide dyadic second-order general variables in two: the X 2i i 2 —which we abbreviate Ri2 —will be used paired with second-order arithmetical variables; the X 2i+1 3 — for more general purposes. Finally, we divide triadic second-order general variables in two: the X 2i 3 —which which we abbreviate Si —will be paired with third-order arithmetical variables; the X 2i+1 we abbreviate Ri3 —will be used for more general purposes. For n > 3, we use Rin as a terminological variant of X in . We will sometimes appeal to the introduction of unused variables. We employ ‘m’ as an unused first-order arithmetical variable, ‘w’, ‘v’, and ‘u’ as unused first-order general variables, ‘M’ as an unused second-order arithmetical variable, ‘W ’, ‘V ,’ and ‘U ’ as unused monadic second-order general variables, and, for each n > 1 (to be determined by context), we employ ‘R’ as an unused n-place secondorder general variable. (It is worth noting that appeal to unused variables could be avoided by renumbering subscripts.) It will often be convenient regard ‘x’, ‘y’, and ‘z’ as arbitrary first-order general variables and ‘X ’, ‘Y ’, and ‘Z ’ as arbitrary (monadic) second-order general variables. 13 For a discussion of higher-order predicates see Rayo, A. “Word and Objects.” Noûs 36, 436–464 (2002). 14 Formally, the well-formed formulas of L can be characterized as follows: (a) N(X 1 ,m ) and m = j i i m j are formulas; (b) for any n-place atomic predicate P other than ‘N’, P(xi 1 , . . . , xi n ) is a formula; (c) Mi m j and X in (x ji , . . . , x jn ) are formulas; (d) if φ and ψ are formulas, then ¬φ, (φ ∧ψ), ∃m i φ, ∃Mi φ, ∃xi φ, and ∃X in φ are formulas; and (e) nothing else is a formula.

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range. 15 In addition, ‘N(X i1 , m j )’ is true just in case the number of the X i1 is m j . Consider ‘The number of the cats is three’ as an example. It can be formalized in L as: ∃ m 1 (N(xˆ1 [C AT(x1 )], m 1 ) ∧ 3(m 1 ));

(3)

where, again, the number predicates are defined in the obvious way:

r 0(m) ≡d f ∃W (0 f (W ) ∧ N(W , m)); r 1(m) ≡d f ∃W (1 f (W ) ∧ N(W , m)); r 2(m) ≡d f ∃W (2 f (W ) ∧ N(W , m)); r etc. 16 Arithmetical predicates such as ‘S UCCESSOR’, ‘S UM’ and ‘P RODUCT’ can easily be defined in terms of ‘N’ and purely logical vocabulary. 17 So, without appealing to arithmetical primitives beyond ‘N’, the whole of pure and applied second-order arithmetic can be expressed within L. It will be convenient to introduce the following definitions, which are couched in purely logical vocabulary: Definition 1: F(X ) ≡d f ¬∃W (∃ w(¬Ww ∧ ∀ v(Xv ↔ (Wv ∨ v = w))) ∧ W ≈ X ) (there are at most finitely many Xs). Definition 2: ∃ f X φ (X ) ≡d f ∃ X (F(X ) ∧ φ(X )). 15 More precisely, first-order arithmetical variables are taken to range over the natural numbers, and first-order general variables are taken to have an unrestricted range. The range of the second-order variables is to be characterized accordingly. For instance, on a Fregean interpretation of second-order quantification, second-order arithmetical variables are taken to range over first-order concepts under which natural numbers fall, and second-order general variables are taken to range over first-order concepts under which arbitrary objects fall. 16 We use number-predicates rather than numerals for the sake of simplicity, but it is worth noting that our nominalization could be carried out even if L was extended to contain numerals. To see this, note that—using standard techniques—any formula φ of the extended language can be transformed into an equivalent formula φ ∗ of the original language in which numerals have been eliminated in favor of corresponding number-predicates (defined as above). One can then identify the nominalization of φ with that of φ ∗ . 17 The definitions run as follows:

S UCCESSOR(m i , m j ) ≡d f ∀V ∀U [(N(V , m i )∧ N(U , m j )) → ∃u(Uu ∧ wˆ [Uw ∧w = u] ≈ V )]; S UM(m i , m j , m k ) ≡d f ∀V ∀U ∀W [(N(V , m i )∧ N(U , m j ) ∧ N(W , m k ) ∧ ∀w(V w→ ¬Uw)) → w[V ˆ w ∨ Uw] ≈ W ]; P RODUCT(m i , m j , m k ) ≡d f ∀V ∀U ∀w[(N(V , m i )∧N(U , m j )∧N(W , m k )) → ∃R[∀v∀u((V v∧ Uu) → ∃!w(Ww ∧ Rvuw)) ∧ ∀w(Ww → ∃!v∃!u(V v∧ Uu ∧ Rvuw))]]:

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Our nominalization method can now be generalized to encompass the whole of first-order arithmetic by way of the following transformation: 18

r Tr(∃ m i (φ)) = ∃ f Z i ∧ Tr(φ); r Tr( m i = m j ) = Zi ≈Z j ; r Tr( N(Xi , m j )) = X i ≈ Z j . Intuitively, the transformation works by replacing talk of the number of the Fs by talk of the Fs themselves. As an example, let us return to ‘the number of the cats is three’. It can be formalized in L as: ∃ m 1 (N(xˆ1 [C AT(x1 )], m 1 ) ∧ 3(m 1 )); which Tr converts to: ∃ f Z 1 (xˆ1 [C AT(x1 )] ≈ Z 1 ∧ 3 f (Z 1 )); or, equivalently: 3 f (xˆ1 [C AT(x1 )]). For further illustration, note that ‘the number of the cats is the number of the dogs’ can be formalized in L as: ∃ m 1 [N(xˆ1 [C AT(x1 )], m 1 ) ∧ (N(xˆ1 [DOG(x1 )], m 1 )]. which Tr converts to: ∃ f Z 1 [xˆ1 [C AT(x1 )] ≈ Z 1 ∧ xˆ1 [DOG(x1 )] ≈ Z 1 ], or, equivalently: xˆ1 [C AT(x1 )] ≈ xˆ1 [DOG(x1 )]. It is worth emphasizing that mixed identity statements such as ‘m i = x j ’ are not well-formed formulas of L, so our transformation has not been defined for them. Intuitively, this means that the transformation is undefined for sentences along the lines of ‘The number 2 is Julius Caesar’, which do not express internal properties of a mathematical structure. We call such sentences Caesar sentences. This is as it should be. The view that numbers are objects led Frege to the uncomfortable question of whether the number belonging to the concept cat is, for instance, Julius Caesar. But in the context of our nominalizations, 18 The remaining clauses are trivial:

r r r r r r r r r

Tr(¬φ) = ‘¬’ ∧ Tr(φ); Tr(φ∧ψ) = ‘(’ ⌢ Tr(φ) ⌢ ‘⌢’ ⌢ Tr(ψ) ⌢ ‘)’; Tr(∃xi (φ)) = ∃xi ⌢ (Tr(φ)); Tr(∃X i (φ)) = ∃X i ∧ (Tr(φ)); Tr(X i x j ) = X i x j ; Tr(∃Rin (φ)) = ∃Rin ⌢(Tr(φ)); Tr(Rin (x j1 , . . . , x jn )) = Rin (x j1 , . . . , x jn ); Tr(xi = x j ) = xi = x j ; Tr(Pnj (xi 1 , . . . . , xi n )) = Pnj (xi 1 . . . , xi n ).

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such questions never arise, because number-terms do not refer to objects. ‘The number belonging to the concept cat is the number belonging to the concept dog’ is nominalized as ‘the objects falling under the concept cat are in one– one correspondence with the objects falling under the concept dog’, and ‘the number belonging to the concept cat is 3’ is nominalized as ‘there are three objects falling under the concept cat’. The question whether Julius Caesar is the number belonging to the concept cat isn’t only uncomfortable because it appears to be non-sensical. It also underscores a problem Paul Benacerraf made famous, that if mathematical terms refer to objects, then nothing in our mathematical practice determines which objects they refer to. 19 A remarkable feature of the Unofficial Proposal is that it avoids Benacerraf’s Problem altogether. It would, however, be a mistake to conclude from this that the Unofficial Proposal is the last word on Benacerraf’s Problem, since the inscrutability of reference pervades far beyond arithmetic.

2.

Second-order arithmetic

On the assumption that there are infinitely many objects in the range of the general variables of L, a certain kind of coding can be used extend Tr so that it encompasses second-order arithmetic (thanks here to . . . ). Intuitively, the coding works by representing each arithmetical concept Mi by a dyadic relation Ri . Specifically, we represent the fact that a number m j falls under Mi by having it be the case that some concept W under which precisely m j objects fall be such that some individual v bears Ri to all and only the individuals falling under W . 20 We implement the coding by enriching our transformation with the following two clauses: 21

r Tr(∃ Mi (φ)) = ∃ Ri ⌢ Tr(φ); r Tr( Mi m j ) = ∃ v(F(u[R ˆ i (v, u)]) ∧ Z j ≈ u[R ˆ i (v, u)]).

3.

Higher-order arithmetic

It is possible to express any (non-Caesar) formula in the language of n-th order arithmetic as a formula of L for which Tr is defined, provided that the range of the general variables contains at least גn−2 many objects. 19 See Benacerraf (1965). 20 We represent the fact that the number zero falls under M by having it be the case that some object i

bears Ri to nothing. Thus, in order to represent the fact that zero does not fall under Mi we must have it be the case that every object bears Ri either to n objects for some n > 0 falling under Mi , or to infinitely many objects. 21 Polyadic second-order quantification can be defined as monadic second-order quantification over sequences, which can be simulated within first-order arithmetic.

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Consider the case of third-order arithmetic. Intuitively, we proceed by pairing each second-order concept αi with a triadic relation Si in such a way that a set of numbers M j falls under αi just in case there is some object x with the following property: (*) For any number n, M j n holds just in case there is some object y such that there are exactly n vs. satisfying Si (x, y, v). 22 So that the ‘empty’ second-order concept (i.e. the second-order concept under which no first-order concept falls) may be represented, we let Si represent the fact that M j falls under αi only if there is an object x such that it is both the case that (*) is satisfied, and that there is no y such that Si (x, y, x). The ‘empty’ second-order concept can then be represented by any relation Si such that for every x there is some y such that Si (x, y, x). Formally, if ‘αi ’ is a monadic third-order variable restricted to the natural numbers, 23 we define a transformation C as follows: 24

r C(∃ αi φ ) = ∃ Si ⌢ C(φ); r C(αi (M j )) = ∃ x[∀ y(¬Si (x, y, x)) ∧ ∀ m(M j m ↔ ∃ y(N(v[S ˆ i (x, y, v)], m)))]

On the assumption that the range of the general variables contains least continuum many objects, it is easy to verify that, for any formula of third-order arithmetic, φ, on which C is defined, φ ↔ C(φ). By using n-adic relations instead of triadic ones, this procedure can be extended to n-th order arithmetic. And, on the assumption that the range of the general variables contains at least גn−2 objects, it will be the case that, for any formula of n-th order arithmetic, φ on which C is defined, φ ↔ C(φ).

4.

Numbering numbers

One would like to be able to number cats. But one would also like to be able to number numbers. One would like to say, for example, that the number of primes smaller than ten is four. And, unfortunately, an expression such as ‘N(mˆ i [P RIME-LESS - THAN-10(m i )], m j )’ is not well-formed formula of L because ‘N’ can only admit of a general variable in its first argument-place. 25 22 We represent the fact that the number zero falls under M by having it be the case that some object y j is such that there are no vs satisfying Si (x, y, v). Thus, in order to represent the fact that zero does not fall under M j we must have it be the case that every object y is either such that that there are n vs satisfying Si (x, y, v) for some n > 0 falling under M j , or such that there are infinitely many vs satisfying Si (x, y, v). 23 For instance, on a Fregean interpretation of third-order quantification, ‘α ’ ranges over second-order i concepts under which fall first-order concepts under which fall natural numbers. 24 The remaining clauses are trivial. 25 In analogy with the above, we let the result of substituting ‘m ˆ i [φ (m i )]’ for ‘M j ’ in a formula ‘ψ(M j )’ be shorthand for

∀M(∀m i (Mm i ↔ φ(m i )) → (M)).

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To remedy the situation, we may define a predicate ‘NN(Mi , m j )’, by appealing to the same sort of coding as before. Informally, ‘NN(Mi , m j )’ is to abbreviate a formula of L to the effect that there is a binary relation R with the following properties:

r for any number n, Mi n holds just in case some member of the domain of R is paired with exactly n+ 1 objects; 26

r every member of the domain of R is paired with finitely many objects; r for any x and y in the domain of R, if the objects paired with x are as many as the objects paired with y, then x = y;

r the domain of R contains exactly m j objects. 27 The new predicate allows us to say that the number of primes smaller than ten is four. It also allows us to say that the number of primes smaller than three is the number of objects falling under the concept cat: ∃ m 2 (NN(mˆ 1 [P RIME - LESS - THAN-6(m 1 )], m 2 ) ∧ N(xˆ1 [C AT(x1 )], m 2 )). 28 And, as desired, our any expression of the form NN(Mi , m j ) is definitionally equivalent to a well-formed formula of L.

5.

Formulas of L and their transformations

Our nominalization method is now complete. 29 Caesar sentences aside, any formula in the language of n-th order applied arithmetic can be expressed as a formula of L for which Tr is defined. And the result of applying Tr is always a formula with no mathematical vocabulary. We may now give a general characterization of the relationship between a formula and its transformation. In order to do so, consider the following five principles, all of which hold on the intended interpretation of L: 1. ∀ X (∃ m(N(X , m)) → ∃ !m(N(X , m))) (If m is a number of the Xs, then m is the number of the Xs.) 26 We require that a member of the domain of R be paired with n + 1 objects rather than n objects in order to accommodate the fact that the number zero might fall under Mi , since every member of the domain of R must be paired with at least one object. 27 More precisely, ‘NN(M , m )’ is to abbreviate: i j

∃R[∀m k (Mi m k ↔ ∃w∃W ∃u(Rwu ∧ ∀v(Wv ↔ (Rwv ∧ v = u)) ∧ N(W , m k ))) ∧ ∀w∀v(Rwv → ∃ f W ∀u(Wu ↔ Rwu)) ∧ ∀w∀v∀W ∀V ((∃u(Rwu) ∧ ∀u(Wu ↔Rwu) ∧ ∀u(V u↔Rvu) ∧ W ≈ V ) → w = v)∧ ∃W (∀v(Wv ↔ ∃u(Rvu)) ∧ N(W , m j )) ]; for m k an unused variable. 28 Whereas ‘C AT (. . . )’ may be regarded as an atomic predicate, ‘P RIME - LESS - THAN-6(. . . )’ abbreviates a complex formula constructed using the arithmetical predicates defined in Footnote 17. 29 So far we have only been concerned with the arithmetic of finite cardinals. But it is worth noting that a similar transformation could be applied to the language of infinite cardinal arithmetic.

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2. ∀ m∃ X N(X , m) (Given any number m, there are some objects such that m belongs to those objects.) 3. ∀ X (∃ m(N(X , m)) ↔ F(X )) (A number belongs to the Xs just in case they are at most finite in number.) 4. ∀ X ∀ Y [∀ m(N(X , m) → (Y , m)) ↔ X ≈ Y )]. (A number belonging to the Xs is also a number belonging to the Ys just in case the Xs are in one–one correspondence with the Ys.) 5. ∃ X ¬F(X ) (There are infinitely many things in the range of the general variables.)

Let Ꮽ be the conjunction of these five principles, and let φ T r be a notational variant for Tr(φ) . It is possible to show that, for any sentence φ of L, 30 Ꮽ ⊢ φ ↔ φT r

where ‘⊢’ expresses derivability in a standard second-order deductive system. In order to prove this result, a few preliminaries are necessary. ˆ Ri (v, u)], m))): Definition 3: N(Ri , m j ) ≡d f ∀ m(M j m ↔ ∃ v(N (u[ Definition 4: If mi1 , . . . , m ik , M j1 , . . . , M jl are arithmetical variables, we let ______________________ m i1 , . . . , m ik , M j1 , . . . , M jl abbreviate the following: (N(Z i1 , m i1 ) ∧ · · · ∧ N(Z ik , m ik )∧ N(R j , M j1 ) ∧ · · · ∧ N(R jl , M jl )). Definition 5: If φ is a formula of L, with free arithmetical variables mi1 , . . . , m ik , M j1 , . . . , M jl , we let φ ↔* φ T r abbreviate the universal closure of the following: ______________________ m i1 , . . . ,m ik ,M j1 , . . . ,M jl → (φ ↔ φ T r ). If φ contains no free arithmetical variables, we let φ ↔* φ T r be φ ↔ φ T r . Finally, we proceed to our main result: Theorem 1: If φ is a well-formed formula of L, then Ꮽ ⊢ φ ↔* φ T r. See appendix for proof. [An interesting feature of the proof is that the fifth conjunct of Ꮽ is required only to ensure the adequacy of the coding for second-order variables set forth in Section 2. In particular, the fifth conjunct is not required to prove a version of the theorem restricted to first-order 30 Here and in what follows I assume that, as a precaution against variable clashes, φ contains no variables for the form Z i , Ri or Si .

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arithmetic. On the other hand, without its fifth conjunct—or, alternatively, without a principle guaranteeing the existence of infinitely objects in the range of the arithmetical variables—the standard arithmetical axioms do not follow from Ꮽ.] Corollary 1: (Completeness of Ꮽ with respect to applied arithmetic) If φ is a sentence of L and T is the set of true sentences of L which do not contain ‘N’, then either A ∪T ⊢ φ or Ꮽ ∪T ⊢ ¬φ. Proof: Let φ be a sentence of L. It is easy to verify that φ T r does not contain ‘N’. Therefore, either T ⊢ φ T r or T ⊢ ¬φ T r , since either φ T r ∈ T or ¬φ T r ∈ T . But, since φ contains no free variables, it follows from our theorem that Ꮽ ⊢φ ↔ φ T r . So, either Ꮽ ∪ T ⊢ φ or Ꮽ ∪ T ⊢ ¬φ. Corollary 2: Suppose Ꮽ holds when ‘N(X , m)’ is interpreted as ‘the number of the Xs is m’. Let φ (m i ) be a well-formed formula of L, and let ψ(Zi ) be Tr(φ (m i )). If there are at most finitely many Fs, then φ(m i ) is true of the number of the Fs just in case ψ(Z i ) is true of the Fs. 31 Proof: Immediate from theorem.

6.

Interpreting second-order languages

We have taken care to ensure that the outputs of our transformation are always second-order formulas. So an interpretation for second-order quantifiers is all we need to make sense of our nominalizations. Frege took secondorder quantifiers to range over concepts, but Fregean concepts might be considered problematic on the grounds that they constitute ‘items’ which are not objects. Not any alternative will do. On Quine’s interpretation, second-order logic is ‘set-theory in sheep’s clothing’. So we would have succeeded in eliminating number-terms from arithmetic only by making use of set-terms. And, from the perspective of the Unofficial Proposal, set-terms are presumably no less problematic than number-terms. Nor is any progress made by interpreting second-order logic as Boolos has suggested. 32 Some of our definitions make essential use of polyadic second-order quantifiers, which Boolos treats as ranging (plurally) over ordered n-tuples. And, again, from the perspective of the Unofficial Proposal, ordered-pair-terms are presumably no less problematic than numbers-terms. 31 In fact, the result is slightly more general. Suppose φ (m , . . . , m ) is a formula of L and let ψ(Z , i1 in i1 . . . , Z i n ) be Tr(φ (m i 1 , . . . , m i n )); suppose, moreover, that there are at most finitely many F1 s, at most finitely many F2 s, . . . , and at most finitely many Fn s. Then φ(m i 1 , . . . , m i n ) is true when m i 1 is the number of the F1 s, m i 2 is the number of the F2 s, . . . , and m i n is the number of the Fn s just in case ψ(Z i 1 , . . . , Z i n ) is true when the Z i 1 s are the F1 s, the Z i 2 s are the F2 s, . . . , and the Z i n s are the Fn s. 32 See Boolos (1984) and Boolos (1985a).

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Some deviousness is needed to avoid Fregean concepts without betraying the spirit of the Unofficial Proposal. One way of doing so is by defining second-order quantifiers implicitly, in terms of an open-ended schema, as in McGee’s ‘Everything’. Another is by interpreting second-order logic as in Rayo and Yablo’s ‘Nominalism through De-Nominalization’. Alternatively, one might argue that genuine second-order quantification is to be accepted as a primitive.

7.

Applications

Frege’s Unofficial Proposal—the view that number–statements are to be eliminated in favor of their transformations—can take several different forms, depending on the sort of elimination one has in mind. On an approach like Hodes’s, number–statements are taken to abbreviate their transformations. As a result, number-terms do not refer to objects, and there is room for rejecting the existence of numbers altogether. The Unofficial Proposal might therefore provide a basis for a nominalist philosophy of arithmetic. It should be noted, however, that unless the universe is infinite, φ T r will not always have the truth-value that φ receives on its standard interpretation. In order to avoid infinity assumptions, a nominalist might claim that a number– statement φ abbreviates ‘necessarily, (ξ → φ T r )’, where ‘ξ ’ is a sentence stating that there are infinitely many objects, such as ‘∃ X ¬F(X )’. On the plausible condition that it is possible for the universe to infinite, ‘necessarily, (ξ →φ T r )’ is true if and only if φ is true on its standard interpretation. 33 A different approach towards the Unofficial Proposal might serve the purposes of the Neo-Fregean Program, championed by Bob Hale and Crispin Wright. Neo-Fregeans believe that Hume’s Principle allows us to reconceptualize the state of affairs which is described by saying that the Fs are as many as the Gs, and that, on the reconceptualization, that same state of affairs is rightly described by saying that the number of the Fs is the number of the Gs. 34 A version of the Unofficial Proposal might allow Neo-Fregeans to make the more general claim that every number–statement φ describes—on the appropriate reconceptualization—the state of affairs which is otherwise described by φ T r . Even if the Unofficial Proposal is to be abandoned altogether, it would be a mistake to neglect the connection between number–statements and their transformations described in Section 5. For non-nominalist accounts of mathematics must yield the result that there is no special mystery about how one might come to know what the truth-values of mathematical sentences are. But, on the assumption that A can be known to be true, our theorem ensures 33 For more on modal strategies, see part II of Burgess and Rosen (1997). Hodes discusses a modal strategy in Section III of Hodes (1984). 34 See Wright (1997), Section I, and Hale (1997).

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that this goal can be achieved for the case of pure and applied arithmetic. Let φ be an arithmetical sentence of L. When Ꮽ is known, it follows from our theorem that one is in a position to derive φ ↔ φ T r . So, insofar as one is in a position to know the truth of φ T r , which contains no arithmetical vocabulary, one is also in a position to know the truth of φ. 35 (Of course, one may not be in a position to know the truth of φ T r . In that case one is not, for all that has been said, in a position to know φ. But that cannot be used as an objection against a non-nominalist account of mathematical knowledge. Such an account is required to show that mathematical knowledge is no more mysterious than non-mathematical knowledge, not that all knowledge is unproblematic.)

8.

Logicism

Our theorem provides us with a partial vindication of Logicism. For whenever φ is a sentence of pure arithmetic (appropriately expressed in L), φ T r is a sentence of pure second-order logic. Moreover, Tr allows us to express formulas of pure arithmetic as formulas of pure second-order logic in a way which preserves compositionality. 36 This would constitute a complete vindication of Logicism if it were true as a matter of pure logic that, for every appropriate φ, Tr(φ) has the truth-value that φ would receive on its standard interpretation. Unfortunately, the general equivalence in truth-value holds only if the universe is big enough, and the size of the universe is not a matter of pure logic. Tr doesn’t reduce arithmetic to logic—but it comes close.

Appendix The theorem is proved by induction on the complexity of φ. Trivial cases are omitted.

r Assume φ = N(X i , m j ). Then φ ↔* φ T r is the universal closure of N(Z j , m j ) → (N(X i , m j ) ↔ X i ≈ Z j ), which is an immediate consequence of Ꮽ (first and fourth conjuncts). 35 For a more detailed treatment of this issue see Rayo, A. (2004) “ Frege’s Correlation.” Analysis 64, 119–122. It is worth noting that the completeness of the second-order Dedekind-Peano axioms yields a similar result for the case of pure second-order arithmetic, and that the quasi-categoricity result in McGee (1997) yields a similar result for the case of pure set-theory. 36 Unlike nominalization in terms of Ramsey sentences, Tr respects the logical connectives and quantifiers:

r r r r

Tr(¬φ) = ‘¬’ ⌢ Tr(φ), Tr(φ ∧ ψ) = Tr(φ) ⌢‘∧’ ⌢ Tr (ψ), Tr(∃m i φ) = ∃ f Zi ⌢ Tr(φ), Tr(∃Mi φ) = ∃Ri ⌢ Tr(φ).

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r Assume φ = m i = m j . Then φ ↔* φ T r is the universal closure of (N(Z i , m i ) ∧ N(Z j , m j )) → (m i = m j ↔ Z i ≈ Z j ), which is an immediate consequence of Ꮽ (first and fourth conjuncts).

r Assume φ = M j m i . Then φ ↔* φ T r is the universal closure of m i , M j → (M j m i ↔ ∃ v(F(u[R ˆ j (v, u)]) ∧ Z i ≈ u[R ˆ j (v, u])).

We make the following two assumptions: N(Z i , m i ),

(1)

N(R j , M j ).

(2)

Recall that (2) is shorthand for ˆ j (v, u)], m))), ∀ m(M j m ↔ ∃ v(N(u[R

(3)

from which it follows immediately that (M j m i ↔ ∃ v(N(u[R ˆ j (v, u)], m i ))).

(4)

From (1) and (4), together with Ꮽ (first and fourth conjuncts), it follows that M j m i ↔ ∃ v(Z i .u[R ˆ j (v, u)]),

(5)

And from (1) and (5), together with Ꮽ (first, third and fourth conjuncts), it follows that M j m i ↔ ∃ v(F(u[R ˆ j (v, u)]) ∧ Z i ≈ u[R ˆ j (v, u)]) :

(6)

Discharging assumptions (1) and (2) we get: m i , M j → (M j m i ↔ ∃ v(F(u[R ˆ j (v, u)]) ∧ Z i ≈ u[R ˆ j (v, u)])).

(7)

And the desired result follows from (7) by universal generalization.

r Assume φ = ∃ m i ψ(m i ). Let ψ have free arithmetical variables m i , . . . , 1 m ik ,M j1 , . . . , M jl distinct from m i . 37 Then φ ↔* φ T r is the universal closure of: m i1 , . . ., m ik , M j1 , . . ., M jl → (∃ m i ψ(m i ) ↔ ∃ f Z i ψ T r (Z i )).

By inductive hypothesis, the following is provable from HP: m i , m i1 , . . . , m ik , M j1 , . . . , M jl → (ψ(m i ) ↔ ψ T r (Z i )).

(1)

We make the following two assumptions: m i1 , . . . , m ik , M j1 , . . . , M jl ,

(2)

∃ m i ψ(m i ).

(3)

37 The case where ψ has no free arithmetical variables distinct from m , and the case where ψ does not i contain m i free require trivial differences in terminology. We ignore them for the sake of brevity.

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By Ꮽ (second and third conjuncts), it follows from (3) that ∃ m i ∃ W (F(W ) ∧ N(W, m i ) ∧ ψ(m i )).

(4)

So, by existential instantiation, F(C) ∧ N(C, c) ∧ ψ(c).

(5)

c, m i1 , . . ., m ik , M j1 , . . ., M jl → (ψ(c) ↔ ψ T r (C)).

(6)

But by (1) we have: And from (2), (5), and (6) we may conclude ψ T r (C).

(7)

∃ f Z i ψ T r (Z i ),

(8)

Thus, making again use of (5), and, discharging assumption (3), ∃ m i ψ(m i ) → ∃ f Z i ψ T r (Z i ).

(9)

∃ f Z i ψ T r (Z i ).

(10)

F(C) ∧ ψ T r (C).

(11)

Conversely, assume By existential instantiation: It is a consequence of (10) and Ꮽ (third conjunct) that ∃ mN(C, m).

(12)

From (12) we obtain the following, by existential instantiation: N(C, c).

(13)

c, m i1 , . . ., m ik , M j1 , . . ., M jl → (ψ(c) ↔ ψ T r (C)).

(14)

But by (1) we have: And from (2), the second conjunct of (11), (13), and (14) we may conclude ψ(c).

(15)

∃ m i ψ(m i ),

(16)

Thus, and, discharging assumption (10), ∃ f Z i ψ T r (Z i ) → ∃ m i ψ(m i ).

(17)

Finally, we combine (9) and (17), and discharge assumption (2): m i1 , . . . , m ik , M j1 , . . ., M jl → (∃ n m i ψ(m i ) ↔ ∃ f Z i ψ T r (Z i )). (18) The desired result is then obtained by universal generalization.

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r Assume φ = ∃ M j ψ (M j ). Let ψ have free arithmetical variables m i , . . . , 1 m ik , M j1 , . . . , M jl distinct from M j . 38 Then φ ↔* φ T r is the universal closure of: m i1 , . . ., m ik , M j1 , . . ., M jl → (∃ M j ψ(M j ) ↔ ∃ R j ψ T r (R j )).

By inductive hypothesis, the following is provable from HP: M j , m i1 , . . ., m ik , M j1 , . . ., M jl → (ψ(M j ) ↔ ψ T r (R j )).

(1)

We make the following two assumptions: m i1 , . . ., m ik , M j1 , . . ., M jl ,

(2)

∃ M j ψ(M j ).

(3)

By Ꮽ (second, third and fifth conjuncts), it follows from (3) that ∃ M j (∃ R(N(R, M j )) ∧ ψ T r (M j )).

(4)

So, by existential instantiation, N(P, C) ∧ ψ(C).

(5)

C, m i1 , . . ., m ik , M j1 , . . ., M jl → (ψ(C) ↔ ψ T r (P)).

(6)

But by (1) we have: And from (2), (5) and (6) we may conclude ψ T r (P).

(7)

∃ R j ψ T r (R j ),

(8)

Thus, and, discharging assumption (3), ∃ M j ψ T r (M j ) → ∃ R j ψ T r (R j ).

(9)

∃ R j ψ T r (R j ).

(10)

ψ T r (P).

(11)

∃ M∀ m(Mm ↔ ∃ v(N(u[P(v, ˆ u)], m))).

(12)

Conversely, assume By existential instantiation, The following is a logical truth: But (12) is definitionally equivalent to ∃ MN(P, M).

(13)

N(P, C).

(14)

So, by existential instantiation,

38 The case where ψ has no free arithmetical variables distinct from M , and the case where ψ does not j contain M j free require trivial differences in terminology. We ignore them for the sake of brevity.

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But by (1) we have: C, m i1 , . . ., m ik , M j1 , . . ., M jl → (ψ(C) ↔ ψ T r (P))

(15)

And from (2), (11), (14), and (15) we may conclude ψ(C).

(16)

∃ M j ψ(M j ),

(17)

Thus, and, discharging assumption (10), ∃ R j ψ T r (R j ) → ∃ M j ψ(M j ).

(18)

Finally, we combine (10) and (18), and discharge assumption (2): m i1 , . . ., m ik , M j1 , . . ., M jl → (∃ M j ψ(M j ) ↔ ∃ R j ψ T r (R j )).

(19)

The desired result is then obtained by universal generalization.

References Beaney, M., ed. (1997) The Frege Reader, Blackwell, Oxford. Benacerraf, P. (1965) “What Numbers Could not Be,” The Philosophical Review 74, 47–73. Reprinted in Paul Benacerraf and Hilary Putnam, Philosophy of Mathematics. Benacerraf, P., and H. Putnam, eds. (1983) Philosophy of Mathematics, Cambridge University Press, Cambridge, second edition. Boolos, G. (1984) “To Be is to Be a Value of a Variable (or to be Some Values of Some Variables),” The Journal of Philosophy 81, 430–49. Reprinted in George Boolos, Logic, Logic and Logic. Boolos, G. (1985a) “Nominalist Platonism,” Philosophical Review 94, 327–44. Reprinted in George Boolos, Logic, Logic and Logic. Boolos, G. (1985b) “Reading the Begriffsschrift,” Mind 94, 331–34. Reprinted in George Boolos, Logic, Logic and Logic. Boolos, G. (1998) Logic, Logic and Logic, Harvard, Cambridge, Massachusetts. Bostock, D. (1979) Logic and Arithmetic, Clarendon Press, Oxford. Burgess, J., and G. Rosen (1997) A Subject With No Object, Oxford University Press, New York. Frege, G. (1884) Die Grundlagen der Arithmetik. English Translation by J.L. Austin, The Foundations of Arithmetic, Northwestern University Press, Evanston, IL, 1980. Frege, G. (1893/1903) Grundgesetze der Arithmetik. Vol. 1 (1893), Vol. 2 (1903). English Translation by Montgomery Furth, The Basic Laws of Arithmetic, University of California Press, Berkeley and Los Angeles, 1964. Frege, G. (1919) “Notes for Ludwig Darmstaedter.” Reprinted in Michael Beaney, The Frege Reader. Hale, B. (1997) “Grundlagen x64,” Proceedings of the Aristotelian Society 97, 243–61. Reprinted in Bob Hale and Crispin Wright, The Reason’s Proper Study. Hale, B., and C. Wright (2001) The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics, Clarendon Press. Heck, R., ed. (1997) Language, Thought and Logic, Clarendon Press, Oxford. Hodes, H. T. (1984) “Logicism and the Ontological Commitments of Arithmetic,” Journal of Philosophy 81:3, 123–49. Hodes, H. T. (1990) “Where do Natural Numbers Come From?” Synthese 84, 347–407. McGee, V. (1997) “How We Learn Mathematical Language,” Philosophical Review 106:1, 35– 68. McGee, V. (2000) “Everything.” In Gila Sher and Richard Tieszen, Between Logic and Intuition.

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Rayo, A. (2002) “Word and Objects.” Noûs 36, 436–64. Rayo, A. (2004) “Frege’s Correlation.” Analysis 64, 119–22. Rayo, A., and S. Yablo (2001) “Nominalism Through De-Nominalization,” Noûs 35:1. Sher, G., and R. Tieszen, eds. (2000) Between Logic and Intuition, Cambridge University Press, New York and Cambridge. Wright, C. (1983) Frege’s Conception of Numbers as Objects, Aberdeen University Press, Aberdeen. Wright, C. (1997) “The Significance of Frege’s Theorem.” In Richard Heck, Language Thought and Logic.

REALS BY ABSTRACTION 1 Bob Hale

1.

General aim and basic ideas

1.1

Abstraction

A Fregean abstraction principle is now usually taken to be a principle of the general form: ∀α∀β(§α = §β ↔ α ≈ β) where ≈ is an equivalence relation on entities denoted by expressions of the type of α and β and § is an operator which forms singular terms when applied to constant expressions of the same type. The most prominent examples in Frege’s own writings are the Direction equivalence: the direction of line a = the direction of line b iff lines a and b are parallel together with what is now often called Hume’s principle: the number of Fs = the number of Gs iff the Fs and the Gs are 1–1 correlated and his ill-fated Basic Law V: the extension of F = the extension of G iff F and G are co-extensive In general, an abstraction principle seeks to give necessary and sufficient conditions for the identity of objects mentioned on its left-hand side in terms of the holding of a suitable equivalence relation between entities of some other sort. The Direction equivalence is a first-order abstraction, because its equivalence relation is a first-level relation on objects, whereas Hume’s principle and Basic Law V are second-order, their equivalence relations being second-level relations on concepts.

1.2

Frege’s logicism

Frege discusses at Grundlagen §§60–67 the suggestion that number might be contextually defined by means of Hume’s principle, but rejects it because 1 This paper first appeared in Philosophia Mathematica 8, [2000], pp. 100–123. Reprinted by kind permission of the editor and Oxford University Press.

175 Roy T. Cook (ed.), The Arché Papers on the Mathematics of Abstraction, 175–196. c 2007 Springer.

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he can see no way to solve what is now called the Caesar problem. The problem is that while Hume’s principle provides the means to settle, at least in principle, the truth-values of identity-statements linking terms for numbers when those terms are of the form ‘the number of Fs’(or definitional abbreviations of such terms), it appears not to enable us to answer questions of numerical identity, when one of the terms is not of that form, such as whether the number of Jupiter’s moons = Julius Caesar. Frege then immediately switches to his well-known explicit definition of number in terms of extensions (or classes): the number of Fs = the class of concepts 1–1 correlated with F. This requires him to provide a theory of extensions or classes, which he does by means of Basic Law V. As is well known, Basic Law V is inconsistent. Frege’s own attempt to arrive at a restricted axiom on classes which is both consistent and able to serve in its place as the basis for his hoped-for derivation of arithmetic from logic was unsuccessful and he eventually abandoned his belief that arithmetic could be provided with a purely logical foundation. Further, whilst we now know—or at least think we know—how to formulate a consistent theory of sets, this affords no comfort to anyone in sympathy with Frege’s logicist project, for two reasons. One is that this theory—Zermelo–Fraenkel set theory, say—is not plausibly viewed as a purely logical theory, owing to the very substantial existence assumptions it involves. The other is that Frege’s definition of number cannot be consistently embedded in the theory, because the objects with which it identifies cardinal numbers are too big to be treated as sets.

1.3

Neo-Fregean logicism

As far as elementary arithmetic goes, Frege’s only indispensable appeal, in Grundlagen and in Grundgesetze 2 to his explicit definition of number (and thence to Basic Law V) is in proving Hume’s principle from it. That is, once Hume’s principle has been established as a theorem, no further appeal need be made, either to the explicit definition or to Basic Law V, in deriving as theorems what are, near enough, the Dedekind–Peano axioms for arithmetic. These include, crucially, the axiom asserting that every natural number has another natural number as its successor, which amounts (in the presence of the others) to the assertion that there are infinitely many natural numbers. This fact is now, following a suggestion of the late George Boolos, 3 referred to as Frege’s Theorem. What Frege’s Theorem asserts, in effect, is that if Hume’s principle is added to a standard formulation of second-order logic as a further axiom, the resulting system suffices for the derivation of elementary 2 As far as Grundlagen goes, this is quite clear from a reading of §§68–83 and is emphasised by Crispin Wright in his Frege’s Conception of Numbers as Objects (Aberdeen University Press 1983). That the same is true of Grundgesetze is shown by Richard G. Heck, Jr., in “The Development of Arithmetic in Frege’s Grundgesetze der Arithmetik” The Journal of Symbolic Logic 58 (1993), pp. 579–601. 3 cf. “The Standard of Equality of Numbers” in George Boolos (ed.) Meaning and Method: Essays in Honor of Hilary Putnam (Cambridge University Press 1990), pp. 261–77.

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arithmetic. It is known that this system is consistent—or at least, that it is so, if second-order arithmetic is. Whether this fact supports any kind of logicism about arithmetic depends, of course, on the status of Hume’s principle. Boolos, along with many others, denies—plausibly, in my view—that it can be regarded as a truth of logic. Further, Hume’s principle cannot be taken as a definition, in any strict sense, because it does not permit the elimination of numerical terms in all contexts. This does not settle the issue, however, since it may be claimed that the principle is analytic, or a conceptual truth, in some sense broader than: either a truth of logic or reducible to one by means of definitions. That it can be so regarded is the view—now often called neo-Fregean logicism—of Crispin Wright and myself. 4 I do not intend, here, to defend this view of arithmetic against the many objections to our claim that Hume’s principle is a conceptual truth about numbers. Nor shall I offer a solution to the Julius Caesar problem 5 —though this must be (and we believe can be) done, if our view is to be viable. Nor, finally, shall I offer a general philosophical defence of the idea—which is again central to our view—that abstraction principles (provided they are consistent and perhaps meet certain other constraints) provide a legitimate means of introducing concepts of various kinds of abstract object in such a way that the existence of those objects depends only upon there being true instances of their right-hand sides. 6 Instead, what I want to do is explain one way in which I think it may be possible to extend our view beyond elementary arithmetic, to encompass the theory of real numbers. I say ‘one way’ because there are, on the face of it, several different ways in which one might try to do this.

1.4

Reals via Fregean set theory

In some ways, the most obvious approach—the one which has probably received most attention in recent work 7 —is a set-theoretic one. This would involve formulating a consistent Fregean axiom for sets to replace Basic Law V—an axiom which could form the basis of a theory of sets powerful enough 4 cf. Wright Frege’s Conception . . . , “The Philosophical Significance of Frege’s Theorem” in Richard G. Heck, Jr. (ed.) Language, Thought, and Logic: Essays in Honour of Michael Dummett (Oxford 1997) and “Is Hume’s principle analytic?” in Bob Hale & Crispin Wright The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics, Oxford: Clarendon Press 2001; Hale Abstract Objects (Blackwell 1987), “Dummett’s critique of Wright’s attempt to resuscitate Frege” Philosophia Mathematica (3) Vol. 2 (1994), pp. 122–47 and “Grundlagen §64” Aristotelian Society Proceedings 1997, pp. 243–62. For Boolos’s opposed view, see “Is Hume’s principle analytic?” in Heck (ed) op cit. 5 qv works cited in fn 4. 6 qv works cited in fn 4. 7 cf. George Boolos “Iteration Again” Philosophical Topics XVII, 2 (1989) pp. 5–21, also “Saving Frege from Contradiction” Aristotelian Society Proceedings 1987, pp. 137–51 and “Basic Law V” Aristotelian Society Supplementary Volume 67 (1993), pp. 213–34; Crispin Wright “The Philosophical Signficance of Frege’s Theorem”, Stewart Shapiro and Alan Weir “New V, ZF and Abstraction”, Philosophia Mathematica (3), vol. 7, 293–321.

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to support one or other of the usual set-theoretic constructions (Dedekind’s or Cantor’s) of the reals. The most obvious way to do this is by means of a suitably restricted version of Basic Law V, and a good deal of work has been done on one particular axiom of this sort, which builds in a restriction on the ‘size’ of concepts which are permitted to have sets corresponding to them which obey the principle of extensionality. 8 I shall not discuss this work here, save to remark that some of it seems to me to show that the prospects for obtaining a satisfactory treatment of the reals along this line are uncertain at best. In particular, as Boolos observed, 9 a theory based on secondorder logic plus this axiom alone, without further comprehension or existence assumptions, will not enable us to prove either an axiom of infinity or a power set axiom. So it will not yield sets large enough for the construction of the reals. This is not conclusive evidence against a broadly set-theoretic approach, of course, since it may be possible to formulate some other more powerful but still consistent Fregean axiom for sets which will give us large enough sets. Or again, it may be possible to justify supplementing this particular restricted version of Basic Law V with other principles to obtain a strong enough theory. I take no stand on that question here. 10 Instead, I want to pursue a quite different approach, which is in some respects much more like that taken by Frege in his incomplete treatment of the reals in Grundgesetze, although it differs from Frege’s in at least one quite fundamental way. This approach can roughly be described by saying that it tries (i) to minimise reliance on set theory and (ii) to obtain the reals very directly by means of abstraction principles, without any form of set-abstraction. In these respects, I think my approach may be seen as the most direct and natural way of extending the neo-Fregean position to the reals. Just as basing elementary arithmetic on Hume’s principle minimises (and, indeed, eliminates) reliance on set theory by avoiding a definition of cardinal numbers as certain equivalence classes, introducing them instead via a specifically numerical abstraction—so my approach to the arithmetic of real numbers will minimise (and indeed eliminate) reliance on set theory by avoiding a definition of reals as sets of one kind or another, introducing them instead via abstraction principles which— even if not happily described as purely numerical—are not distinctively set-theoretical. 8 The axiom (New V) is: ∀F∀G[*F= *G ↔ ((Small(F)∨ Small(G)) → ∀x(Fx↔ Gx))], where a concept is Small if fewer objects fall under it than fall under the universal concept ξ = ξ , and *F is what Boolos calls the ‘subtension’ of F (the subtensions of Small concepts being sets)—see Boolos “Saving Frege from contradiction”, Proceedings of the Aristotelian Society 87 (1986/87), pp. 137–51; also below p. 190ff. 9 cf. “Iteration again”. 10 For a brief discussion of this possibility, see Crispin Wright “On the Philosophical Significance of Frege’s Theorem”, section XI.

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179

Reals as ratios of quantities

Frege’s actual (incomplete) treatment of the reals in Grundgesetze Pt III 11 is, of course, unsatisfactory—if only because it relies, as does his theory of cardinal numbers, on an inconsistent theory of extensions, and cannot be simply relocated within any standard (and plausibly consistent) theory of sets such as ZF or NBG because the objects with which he proposes to identify the reals are too big to be treated as sets. In any case, such a relocation would obviously betray Frege’s philosophical aims, since it would leave our entitlement to the substantial existential commitments of the theory quite unaccounted for. From a philosophical standpoint, the most striking and most important features of Frege’s treatment of the reals are two: (i) the real numbers are to be defined as ratios of quantities [§§73,157] and (ii) in regard to the analysis of the notion of quantity, the fundamental question requiring to be answered is not: What properties must an object have, if it is to be a quantity? but: What properties must a concept have, if the objects falling under it are to constitute quantities of a single kind? [§§160–61]. Briefly and roughly, his insistence that reals be defined as ratios of quantities derives from his belief that the application of reals as measures of quantities is essential to their very nature, and so should be built into an adequate definition of them. It is this, more perhaps than any other single consideration, which underlies his dissatisfaction with the theories of Cantor and Dedekind, on which the applicability of the reals appears, in Frege’s view, merely as an incidental extra. As regards the second point, it is obvious to anyone that there are many different kinds of quantity (lengths, masses, volumes, angles, etc.) and that addition and comparison (as greater or less) make sense only as applied to quantities of the same kind. Since we may not simply take the notion of a kind of quantity for granted, as already understood and itself in no need of analysis, we cannot explain what a quantity is by saying that it is something which can be added to, or be greater or less than, (other) quantities of the same kind. If an explanation of quantity is not to be vitiated by circularity in this way, Frege thinks, it must take as its target the notion of a kind of quantity, and say what characteristics a collection of entities must, as a whole, possess if it is to form what he calls a quantitative domain [ein Grössengebiet]. When that has been done, what it is to be a quantity can be easily stated—an object is a quantity if it belongs, together with other objects, to a quantitative domain. I believe Frege was substantially right on both points. Here I shall simply assume as much, without argument. Where I disagree with him is over the analysis of what he calls quantitative domains. For reasons which I shall not go into, Frege decides that the elements of a quantitative domain should themselves be relations and—heavily influenced by a passage from Gauss [quoted 11 For expositions see Michael Dummett Frege Philosophy of Mathematics (Duckworth, 1991), ch. 22 and Peter Simons “Frege’s Theory of Real Numbers” History and philosophy of logic 8 (1987), pp. 25–44.

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in Grundgesetze §162]—analyses such a domain as an ordered group of permutations on an underlying set, with composition as its additive operation. Since quantities themselves are, on his approach, relations of a certain sort, real numbers, when defined as ratios of quantities, turn out to be relations of relations. One advantage of Frege’s approach is that it provides very easily for negative as well as positive real numbers. I do not have space to discuss Frege’s view properly here. Whilst there is justice in his criticism of earlier writers who simply help themselves to the notion of quantities being of the same kind, I think that the notions of addition and quantitative comparability are central and fundamental to the general notion of quantity in a way Frege fails to acknowledge. Accordingly, I shall propose a different account of quantitative domains—one which gives a central role to the idea that the elements of such a domain may always be added to yield further elements.

2. 2.1

Quantities and reals Types of quantitative domain

I distinguish between the entities (usually concrete objects) which may stand in various quantitative relations to one another—such as being longer than, or being as long as—and quantities themselves, which I take to be abstract objects introduced by abstraction on quantitative equivalence relations—for example: the length of a = the length of b ↔ a is as long as b This way of introducing (terms for) quantities makes no explicit mention of addition. However, a full analysis of the notion of a quantitative relation would, I claim, show that the notion of addition is nevertheless central to that of quantity. I do not have space to go into details here, but the essential idea is this. Among quantitative relations, we may distinguish—as conceptually basic— what may be called relations of simple quantitative comparison (e.g. longer than/as long as, heavier than/as heavy as, etc.) from relations of numerically definite or determinate comparison (e.g. twice as long as, 2.4 kg heavier than, etc.). A necessary condition for φ to denote a kind of quantity is that it be associated with a pair of relations of simple quantitative comparison: more φ than and as φ as. In virtue of this, things which are φ may be partially ordered with respect to φ-ness. However, the existence of an associated pair of such relations—a strict partial ordering relation and a cognate equivalence relation—is insufficient for φ-ness to be a kind of quantity. There are enormously many adjectives in ordinary use which may be substituted without violence to sense or syntax in the schemas: more φ than and as φ as—‘sweet’, ‘elegant’, ‘graceful’, ‘pretty’, ‘clumsy’, ‘ambitious’, ‘impatient’, ‘irrascible’, ‘probable’, . . . is clearly no more than the start of a potentially very long list. But in the case of only relatively few of them is it remotely plausible that they denote something properly describable as a quantity. It is therefore

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necessary to enquire what further condition needs to be satisfied, if such a pair of relations are properly to be viewed as quantitative. I contend that what makes the difference between quantitative ordering relations and others is that in the case of a quantitative ordering relation, but not otherwise, the entities which can significantly be asserted to stand in the relation can (at least in principle) be combined in such a way that compounds must come later in the relevant ordering than their components. In other words, for more φ than to be a quantitative ordering relation, there must be an operation of combination c on items lying in the field of more φ than, analogous to addition, such that c is more φ than a and ab c is more φ for any a,b in more φ than’s field, ab than b. 12 Quantitative domains are composed of (abstract) quantities. My aim in this section is to provide an informal axiomatic characterisation of such domains, on the basis of which it will be possible to introduce real numbers by means of an appropriate abstraction principle. Instead of simply laying down a single set of axioms for something to be a quantitative domain, I shall distinguish several—successively richer—types of quantitative domain. This will be helpful later, when I come to consider questions about the existence of quantitative domains. 1.1 A minimal q-domain is a non-empty collection Q of entities closed under an additive operation ⊕, which commutes, associates and satisfies the strong trichotomy law that for any a,b∈Q we have exactly one of: ∃c(a = b ⊕ c), ∃c(b = a ⊕ c) or a = b. Any minimal q-domain is strictly totally ordered by < , defined by: a < b ↔ ∃c(a ⊕ c = b). Multiplication of elements of Q by positive integers is easily defined—inductively—in terms of ⊕. 1.2 A normal q-domain is any minimal q-domain meeting the [Archimedean] comparability condition: ∀a,b∈Q ∃ (ma > b). Here and subsequently (unless explicitly indicated), m (and later n as well) ranges over positive integers. This requires quantities to be finite, in the sense that no quantity is infinitely greater (or smaller) than any other—it rules out infinitesimal 12 The basic idea is of course not new. It is, in particular, central to the theory of measurement advanced by N.R. Campbell in a number of works first published in the 1920s, the most important of them being Physics: the Elements (originally published by Cambridge University Press, 1919, and subsequently republished as Foundations of Science (Dover 1957)—see Part 2) and Measurement and Calculation (Longmans, Green & Co, London 1928). A briefer popular statement of his theory is given in What is Science? (Methuen, London 1921—see ch. VI). Whilst there is much in Campbell’s overall theory which I think we neither can nor need accept, I believe that Campbell was right, pace critics such as Brian Ellis (see Basic Concepts of Measurement, Cambridge University Press 1966, ch. IV), to insist upon the importance of a physical analogue of addition, and right too (at least in essentials) in taking there to be an important distinction between fundamental and derived measurement. More recent treatments of measurement—see, for example, the comprehensive text of Krantz, D.H., Luce, R.D., Suppes, P. and Tversky, A. (Foundations of Measurement New York and London: Academic Press 1971 (vol 1), 1989 (vols 2,3))—have not looked kindly on these distinctive features of Campbell’s approach. I need hardly emphasise that the very rough and dogmatic statement of my view, both here and in the text, requires both considerable qualification and further explanation, as well as defence.

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quantities. With his eye on Euclid’s Def.4 of Elements Bk.V, Howard Stein 13 describes it as the condition necessary and sufficient for a and b to have a ratio. It might be compared, in status, to the requirement on concepts presupposed by Hume’s principle, that the concepts through which it quantifies be sortal— which might be described as the condition for a concept to have a (cardinal) number. Where Q, Q* are any normal q-domains, not necessarily distinct, we introduce ratios of quantities by the abstraction principle: EM

∀a, bεQ∀c, dεQ∗ [a : b = c : d ↔ ∀m, n(ma < => nb ↔ mc< => nd)]

That is, ratios a:b and c:d are the same just if equimultiples of their numerators stand in the same order relations to equimultiples of their denominators. 14 The condition for identity of ratios is framed so as to allow that one and the same ratio may be at the same time a ratio of pairs of quantities of different kinds— belonging to different domains—such as masses and lengths. The operation in terms of which comparability is ultimately defined (i.e. addition of quantities) is, of course, domain specific—no sense is given to adding a length and a mass, for instance. But this does not preclude the introduction of ratios so that the same ratio may be found among, say, both masses and lengths. 1.3 A normal q-domain Q is full if ∀a,b,c∈Q∃q∈Q(a:b = q:c). This condition, which is a restricted form of the ancient postulate of ‘fourth proportionals’, ensures that, given a pair of ratios a:b and c:d, there is a quantity c′ such that c′ :b = c:d, so that we may always, without loss of generality, restrict attention to ratios with common denominators. I shall refer to it as CD. It is easy to see that CD ensures that there is no smallest quantity. 15 1.4 A full q-domain may be incomplete, in the sense that it may include only quantities which are rationally measurable; in consequence, the set of all ratios on a full domain is not guaranteed to include ratios corresponding

13 “Eudoxos and Dedekind: On the Ancient Greek Theory of Ratios and its Relation to Modern Mathematics” Synthese 84 (1990), pp. 163–82. Whilst the approach I pursue here differs quite radically from anything suggested by Stein, I have derived much benefit from this excellent paper. 14 This is, of course, the central principle in the ancient theory of proportion presented in Euclid’s Elements Book V (cf. Def.5) and standardly attributed to Eudoxos. I should perhaps emphasise that EM is not an abstraction principle of the form characterised at the outset. On the other hand, it should be clear that it is intended to work in essentially the same way as paradigm abstractions like the Direction equivalence and Hume’s principle and that it is reasonable to regard it as one. We might bring EM into line with the characterisation of abstraction principles with which I began by first defining an equivalence relation on ordered pairs of quantities: E[(a,b), (c,d)] ↔ ∀m,n (ma < => nb ↔ mc ⇔ nd), and then setting: Ratio(a,b) = Ratio(c,d) ↔ E[(a,b), (c,d)]. Alternatively, if it were felt desirable to avoid reliance on the notion of an ordered pair, we could introduce an extension of the notion of an equivalence relation so as to allow relations of arity greater than 2 to qualify as equivalence relations. Later we shall meet another abstraction principle which does not, as it stands, conform to the usual characterisation, but which may readily be brought into line in one or other of these ways. 15 Although I am not identifying quantities, as such, with numbers of any kind, it should be fairly clear that a full domain, and likewise the domain of ratios on it, is dense, and that we can develop an ‘arithmetic’ of ratios structurally analogous to that of the positive rationals.

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to any, much less all, (positive) irrational numbers. 16 If ratio-abstraction is to yield all the positive reals, we require a complete domain. Indulging—for convenience, but avoidably—in set-theoretic language, we say that a subset S of quantities belonging to a q-domain Q is bounded above by b iff for every quantity a in S, a ≤ b. A quantity b ∈ Q is a least upper bound of S ⊆ Q iff b bounds S above & ∀c(c bounds S above → b ≤ c), and finally that a q-domain Q is complete iff Q is full and every bounded above non-empty S ⊆ Q has a least upper bound.

2.2

Real numbers

We may straightforwardly define ‘bounded above’ ‘lub’ and ‘ordercomplete’ for ratios in a way that parallels our definitions of these notions for quantities and then prove, as an easy consequence of the completeness of the underlying domain, that where Q is any complete q-domain, the set RQ of ratios on Q is order-complete. 17 It can be shown that if Q and Q∗ are any ∗ complete q-domains, they are isomorphic, so that R Q = R Q , i.e. the set of ratios on Q is identical with the set of ratios on Q∗ . Thus provided there exists at least one complete q-domain, we can introduce the positive real numbers, by abstraction, as the ratios on that domain. In standard constructions of the various number systems, negative numbers make their entry at an early stage. The method by which this is accomplished—introducing a new, enlarged domain including negative numbers as certain ordered pairs (difference pairs) of numbers belonging to an underlying domain—is, however, perfectly general, in the sense that it is quite inessential to it that the numbers in the underlying domain should be natural numbers. Of course, we must start with the natural numbers if we want to get just the integers—but in general, all that is required for the application of the method itself is that the objects belonging to the underlying domain have the requisite arithmetic properties. There is, so far as I can see, no reason, either technical or philosophical, why this step may not just as well be taken at a (much) later stage. In particular, essentially the same construction can be used to get negative reals, starting from positive ones, as difference pairs of positive reals. Letting x, y, z, . . . range over, and ⊕ stand for addition of, positive reals, we obtain difference pairs of positive reals by the abstraction: D (x, y) = (z, w) ↔ x ⊕ w = y ⊕ z 16 Of course, since quantitative domains, as I have characterised them, do not include either a zero quantity or negative quantities, the ratios on such domains will not, in any case, have elements corresponding to all the reals. 17 Proof : Let S be any bounded above subset of R Q . By CD, each ratio in S can be expressed with a single common denominator, so that the members of S are: a1 :b, a2 :b, . . . , ai:b , . . . The set of numerators of these ratios is a non-empty subset of Q, and so—by the completeness of Q—have a least upper bound a◦ . Since every ai = a◦ , ai :b = a◦ :b for every ratio ai :b in S. And if some ratio p:q is less than a◦ :b, it follows [by CD] that p:q = p′ :b for some p′ , with p′ < a◦ . But then by the completeness of Q, there is some ak among the numerators of the ratios ai :b so that p′ < ak , and hence a ratio ak :b in S such that p′ :b < ak :b. So a◦ :b is a least upper bound of S.

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Defining < , > , addition, subtraction and multiplication and zero for dpairs in the obvious way, it can be shown that the collection R of d-pairs forms a field with the operations + and ×. Further, there is a subset P of R, namely the set of all pairs (x, y) such that (z, z) < (x, y), meeting the conditions: (i) if (x, y), (z, w) ∈ P then (x, y) + (z, w) ∈ P ∧(x, y) × (z, w) ∈ P and (ii) if (x, y) ∈ R, then exactly one of (x, y) ∈ P, (y, x) ∈ P or (x, y) = (z, z) holds

Thus R is an ordered field. There is an obvious isomorphism between the strictly positive subset P of R and the positive reals as previously defined. Using this, it can be shown without too much difficulty that R is complete.

3.

The existence of quantitative domains

Our result thus far is conditional: real numbers may be obtained by abstraction on quantities, if there exists at least one complete q-domain. Even if this were the best result that could be obtained, it is not completely obvious that this would signal the collapse of the neo-Fregean abstractionist approach to foundations. It might be possible to provide principled reasons for adopting different attitudes towards the question of the existence of reals and that of the natural numbers, holding that while the latter admits of resolution, a priori, in the affirmative, the existence of the reals is a matter on which no similar a priori assurance is to be expected. According to such a view, the existence of (at least) finite cardinal numbers would be a matter of necessity— whatever the universe might be like, its ingredient objects would be assignable to distinguishable sorts or kinds; there would be some sortal concepts or other, under which the objects fell, so that for various concepts F and cardinal numbers n, there would be facts of the form: the number of Fs = n. More importantly, for any such sortal concept F, there will be a sortal—F-andnot-F—logically guaranteed to have no objects falling under it, in terms of which 0 may be defined, thus giving the necessary toe-hold for a Fregean proof of the existence of an infinite collection of finite cardinals. But there can be no similar a priori guarantee that the physical universe comprises quantities which are real-valued—it is perfectly conceivable, even if in fact false, that the physical world should be discontinuous. So a result which says, in effect, that if it does exhibit continuity, the real numbers are available to measure it, might not appear utterly outrageous. Defending this position would, naturally, require speaking to the contrary intuition, that while it may be in some way an empirical question whether the physical universe is continuous, and so an empirical question whether the reals have ‘objective’ application [in the sense that there actually are real-valued quantities—contrast the idea that using the reals simply affords a useful simplification of applied mathematics], the existence of the reals should not itself be an empirical, a posteriori matter.

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Clearly, however, it is important to enquire whether a neo-Fregean can secure a stronger result. Evidently, the question of greatest interest is whether there can be proved to exist a complete q-domain. But it is worth emphasising that the question arises, not only for the case of complete q-domains, but equally for q-domains of the more modest kinds described—thus far, nothing has been done to establish the existence of a full q-domain, or even that of a normal, or even minimal, one. Even the question of the existence of a minimal domain is anything but trivial. A minimal domain is, by definition, non-empty. Since such a domain is closed under its addition operation and satisfies the additive trichotomy condition, it must comprise arbitrarily large quantities, and thus be at least countably infinite. To anyone who thinks of quantities as physical entities of some sort, the existence of such a domain must, for this reason, appear open to serious question. On my own view, quantities such as lengths, masses, angles, etc., should not be thought of a physical entities; they are, rather, abstract objects, ‘introduced’ via abstraction principles employing appropriate equivalence relations on the concrete objects whose lengths, masses, etc., they are. But this makes no essential difference, so far as the present question is concerned. At least, it will make no difference if the existence of a given length, say, is taken to be contingent upon the existence of a suitable concrete entity of which it is the length; for in that case, the ground for doubt about the existence of arbitrarily large quantities of any given kind remains. Clearly there must be an analogous doubt about the existence of arbitrarily small quantities, and hence about the existence of a full q-domain. However, it seems to me that these doubts may be assuaged and that we can actually prove the existence of at least one domain of each of the kinds I have distinguished, including complete domains. The crucial point here is to notice that whilst quantities as such are not identified, in my approach, with numbers, nothing in the characterisation of qdomains precludes such domains being composed of numbers. As previously remarked, Hume’s principle suffices for a derivation of the Dedekind–Peano axioms for elementary arithmetic, and hence for a proof of the existence of an infinite sequence of natural numbers—0, 1, 2, . . . . Omitting 0 to obtain the strictly positive naturals, N+ , and adjusting the usual recursive definitions of + and × to suit, we can easily show that N+ constitutes a minimal—and indeed a normal—q-domain. It is clear that N+ is not itself a full domain, i.e. it does not satisfy CD. + However, the collection R N of ratios on N+ does constitute a full domain. To see this, note first that since N+ is normal, there exists a ratio a:b for every a and b in N+ . Let a, b, c, d, e, f be any elements of N+ . Then what we must show is that there is a ratio g:h such that [a:b]:[c:d] = [g:h]:[e:f]. It is quite straightforward to verify that [a:b]:[c:d] = ad:bc = ade:bce = [ade:bcf]:[bce:bcf] = [ade:bcf]:[e:f]

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so that [ade:bcf] is our required ratio. 18 In the presence of CD, satisfaction + by R N of the minimality and normality conditions follows easily from their + satisfaction by the underlying domain N+ . Thus R N is a full domain. What we have, in effect, is a quite natural way of obtaining the positive rationals by abstraction on the positive natural numbers—each and every positive rational is simply a ratio positive natural numbers. Thus 3/4 just is the ratio 3:4. Of course, it is also the ratio 6:8 and the ratio 9:12, etc., but that is no problem, since these are all simply one and the same ratio in our sense (i.e. by the lights of EM). + It is clear that iteration of the abstractive procedure which yields R N from N+ will not yield any new kind of q-domain. The crucial point emerges above, in the observation that [a:b]:[c:d] = ad:bc. This holds quite generally—any ratio of ratios of positive natural numbers are simply ratios of positive natural numbers. In the same way, ratios of ratios of ratios of positive natural numbers collapse to ratios of positive natural numbers. Iteration of the abstraction to ratios of higher order thus merely gives us the positive rationals all over again. Thus the operation by which we obtained a full domain from an underlying normal one cannot, when re-applied to a full domain, yield a complete one. This is a special case of a quite general fact about first-order abstraction: no first-order abstraction on an infinite domain can generate a ‘new’ domain of greater cardinal size than that abstracted on. It follows that if a complete domain is to be obtained by abstraction, we must invoke a second-order abstraction. In this way—and only in this way—we may advance from a domain of objects of given cardinality to a strictly larger domain of abstracts. Given an initial domain comprising κ objects, there will be 2κ properties of those objects. By taking these properties, rather than the objects which have them, as our underlying domain for an abstraction, we may obtain a strictly larger collection of abstracts—up to (but not more than) 2κ of them. 19 We take as our initial domain the (at least countably infinite) full domain N+ R of ratios on N+ . Our goal is to obtain a complete domain Q# by cutabstraction, so-called because of its obvious correspondence to Dedekind’s + construction. 20 As anticipated, cut-abstraction operates, not directly upon R N itself, but upon properties of a certain kind defined over its elements, which I shall call cut-properties. These are defined by reference to the ordering on + R N . Informally, a cut-property is a non-empty property whose extension is 18 Recall that a, b, c, d, e, f are all positive integers. A ratio is unchanged by multiplying its numerator and denominator by the same positive integer. Hence a:b = ad:bd. Similarly, c:d = bc:bd. But the ratio to one another of ratios with a common denominator is simply the ratio of their numerators, so [ad:bd]:[bc:bd] = ad:bc, whence [a:b]:[c:d] = ad:bc. Further e:f = bce:bcf and ad:bc = ade:bce. Hence, since [ade:bcf]:[bce:bcf] = ade:bce, we have: [a:b]:[c:d] = ad:bc = ade:bce = [ade:bcf]:[bce:bcf] = [ade:bcf]:[e:f] 19 If κ is infinite and CH holds, then we shall, of course, get more than κ abstracts only if we get exactly 2κ of them; but I am not assuming CH, much less GCH. 20 cf. Richard Dedekind Stetigkeit und Irrationale Zahlen (1872), translated by Wooster Woodruff Beman as “Continuity and Irrational Numbers” in Richard Dedekind Essays on the theory of numbers, New York: reprint, Dover Publications (1963), pp. 1–27.

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a proper subset of R N and which is downwards closed [i.e. ∀a∀b(Fa→(b < a → Fb)) 21 ] and has no greatest instance [i.e. ∀a(Fa→ ∃b(b > a ∧ Fb))]. We now introduce objects—cuts—corresponding to cut-properties by the abstraction principle: Cut: #F = #G ↔ ∀a(Fa ↔ Ga) where F,G are any cut-properties on R N + and a ranges over R N .

+

+

Q# is the collection of all cuts, #F, for cut-properties F on R N . It may be shown that Q# constitutes a complete domain, in the sense previously explained. Obviously the main thing here is to verify that Q# has the least upper bound property, i.e. where φ varies over proper+ ties of cuts on R N , and bounds above and lub are defined in an obvious way, that if ∃Fφ(#F) and φ is bounded above then φ has a least upper bound. This can be done, mimicking the usual proof, by defining the property H by: H a ↔ ∃F(φ(#F) ∧ Fa)—we can then show that H is a cut-property and that #H is a lub of φ. We may define #F+ #G to be #H , where H a ↔ ∃b∃c(Fb ∧ Gc ∧ a= b⊕c), and #F × #G to be #P, where Pa ↔ ∃b∃c(Fb ∧ Gc ∧ a = b⊗c). With the aid of these and some supplementary definitions, it can then be proved that Q# is full, i.e. that it is a minimal q-domain which also meets the normality and common denominator conditions.

4.

Safe abstractions and safe sets

Are the abstraction principles which I have employed all in good standing? The question is urgent, since we know that not all abstraction principles are acceptable, if only because some—Basic Law V being the obvious example— are inconsistent. And there may be other constraints, besides consistency, with which good abstractions must comply. A thorough examination of the question lies well beyond the scope of this paper, but I should like to conclude by saying a little about it. Of the abstraction principles I have used, two—ratio-abstraction (EM) and difference abstraction—are first-order, while the other two – Hume’s principle and Cut—are second-order. In the case of first-order abstraction, we abstract upon a domain of objects of some kind, and thereby come to recognise objects of another kind; with a second-order abstraction, by contrast, we abstract upon a domain of concepts, themselves defined on some underlying domain of objects, and come to recognise ‘new’ objects, i.e. objects of a kind other than those belonging to this underlying domain. I shall call the field of an abstraction’s equivalence relation the domain for the abstraction, and in the case where this is a domain of (first-level) concepts, I shall call the domain of objects on which these concepts are defined the underlying domain. 21 Here and subsequently a,b, . . . range over elements of R N + .

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In the case of second-order abstractions, the underlying domain—if it has a determinate size at all—is much smaller than the domain for the abstraction; if the underlying domain has cardinality κ, then the domain for the abstraction (assuming it to comprise all the concepts defined on the underlying domain, and assuming concepts to be individuated extensionally) has cardinality 2κ . In consequence, the abstraction may ‘generate’ up to 2κ abstracts—and so many more abstracts than there are objects in the underlying domain. It is this feature of second-order abstractions which has led some writers to think that it is these abstractions—in contrast with first-order abstractions—which pose the greatest worry, as far as the risk of inconsistency is concerned. I think that is correct, and I shall therefore focus on the second-order abstractions. In fact, since Hume’s principle is known to be consistent, I shall concentrate upon the other second-order abstraction I have used—cut-abstraction. Cut—in contrast with Hume’s principle and Basic Law V—is a restricted abstraction principle, in the sense that the domain for the abstraction comprises only cut-properties on a certain specified underlying domain of objects. It is obvious that if the side constraints on it are ignored, Cut is just a notational variant on Basic Law V. Clearly, then, from unrestricted Cut, we could derive Russell’s contradiction. If we define a Russell property R by: Rx ↔ ∃F(x = #F ∧ ¬Fx), then by unrestricted Cut we have: #R = #R ↔ ∀x(Rx ↔ Rx), whence: #R = #R—so #R exists, and we may proceed: 1 1 3 3 3 3 3 3 1 12 12 12 12

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

R(#R) ∃F(#R = #F ∧ ¬F(#R)) #R = #F ∧ ¬F(#R) #R = #F #R = #F ↔ ∀x(Rx ↔ Fx) ∀x(Rx ↔ Fx) R(#R) ↔ F(#R) ¬F(#R) ¬R(#R) ¬R(#R) R(#R) → ¬R(#R)

assn 1, Def R assn 3∧E (unrestricted) Cut 4,5 ↔E 6 ∀E 3∧E 7,8 ↔ E 2, 3, 9∃E 1,10 →I

(12) (13) (14) (15) (16) (17) (18)

¬R(#R) #R = #R #R = #R ∧ ¬R(#R) ∃F(#R = #F ∧ ¬F(#R)) R(#R) ¬R(#R) → R(#R) R(#R) ↔ ¬R(#R)

assn =I 12, 13 ∧I 14 ∃I 15 Def R 12,16 →I 11,17 ↔I

With the constraints on Cut in place, however, this derivation will not go through without two further assumptions: to establish the existence of #R, and

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to justify the (second-order) ∀E step involved at line (5), we must assume that + R is a cut-property on R N ; and for the application of ∀E at line (7), we must + further assume that #R is in R N . Since the contradiction at line (18) depends upon these further assumptions, we may apply reductio to infer that either R + + isn’t a cut-property on R N , or #R is not an element of R N . Does that settle the matter? Well, no. The particular cut-abstraction principle I’ve used may be viewed as a special case of a general schema which runs: (#)#F = #G ↔ ∀a(Fa ↔ Ga) where F, G are any cut-properties on a suitable domain Q and a ranges over Q. A suitable domain Q here will be any domain with an at least dense linear ordering, with respect to which cut-properties are definable. Two obvious questions which may be raised about this general schema are: Are all its instances safe? If not, what distinguishes those which are from those which are not? I’ll venture a few somewhat tentative thoughts about these questions. Perhaps the first thing I should say is that I am not, so far as I can see, committed to endorsing all instances of (#)—i.e. to defending its universal closure with respect to Q—though I would think that, should it prove that some of its instances are either prone to Russell trouble or otherwise unsafe, it should be possible to provide some principled characterisation/explanation of the limitations here. It is clear that so long as the underlying domain Q for an instance of (#) is not inclusive of all objects whatever, any derivation of Russell’s contradiction can be seen, not as showing the inconsistency of that instance (#), but as a demonstration that either the Russell property R cannot be a cut-property on Q or the Russell cut #R cannot be an element of Q. If the universe of all objects whatever constitutes an admissible underlying domain for cut-abstraction, then the Russell cut, if there is such an object at all, must belong to that domain—so the second option lapses. But the first remains open. There will be such an object as the Russell cut only if the Russell property is a cut-property on the universe. But, at least in the absence of any compelling independent reason to think (#) defective, a derivation of the Russell contradiction would seem to give us ample reason to think that the Russell property cannot be a cut-property on the universe. If what I have said is right, it is possible to block Russell trouble without challenging the assumption that the universe constitutes an admissible underlying domain for cut-abstraction. The point is, however, somewhat academic since there are other worries—having more to do with Cantor’s paradox than with Russell’s—which are, I think, best answered by rejecting that assumption. Briefly, cut-abstraction, for all I have said thus far, may be applied to any domain on which cut-properties are definable – that is, any domain with an at least dense linear ordering. If the chosen domain is strictly dense (i.e. dense— like the rationals—but not complete—like the reals), then an instance of cutabstraction will inflate, in the sense that there are more abstracts ‘generated’

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than there are objects in the underlying domain (i.e. the domain on which the cut-properties are defined). 22 If it is dense but complete, then there will be no inflation—the collection of abstracts will be isomorphic to the underlying object domain. If the universe of all objects whatever admits of a strictly dense linear ordering and can be taken as a domain for cut-abstraction, we shall wind up with more abstracts (and so more objects) than there are objects altogether! How should we avoid this disastrous conclusion? The answer I shall tentatively commend makes crucial play with the contrast I drew previously between unrestricted abstractions, such as Hume’s principle, and restricted ones, such as cut-abstraction. In the case of Hume’s principle, it is essential that the first-order quantifiers on its right-hand side be allowed to range unrestrictedly over all objects whatever, including—crucially—the numbers themselves. In this sense, the first-order quantifiers in Hume’s principle must be understood impredicatively. If instead those quantifiers were restricted so as to range only over objects other than numbers, we could not prove the infinity of the sequence of finite numbers—at least, not without the additional assumption that there exist infinitely many objects of some other kind. With cut-abstraction, by contrast, it is unnecessary—in order to ensure that the abstraction delivers all the abstracts we require—to construe its first-order quantifier impredicatively in this way. Moreover, if we do allow that—in particular, if we allow an instance of the cut-schema whose first-order quantifier ranges over all objects whatever—then we will (provided the universe admits of a strictly dense ordering) run into Cantor-type trouble. But we do not have to allow this. As I have explained, cut-abstraction is—in contrast with Hume’s principle, and Basic Law V—a restricted abstraction, in the sense that each instance of the cut-schema (#) involves a restriction to a specified underlying domain, over which its first-order quantifier ranges. All I have said thus far about what constitutes a suitable underlying domain is that it shall be some densely ordered collection of objects. But as far as I can see, nothing stands in the way of imposing a further restriction which will preclude application of cut-abstraction to the universe as a whole. It may seem that the most obvious way to do this would be to incorporate a ‘limitation of size’ requirement in the conditions for a suitable domain for cut-abstraction—the idea would be to require that any suitable domain Q for cut-abstraction be smaller than the universe. This would bring cut-abstraction much closer to the modified version of Basic Law V which George Boolos dubbed New V. Following Boolos, say that a concept F is a subconcept of a concept G iff ∀x(Fx→ Gx), and that F goes into G iff F ≈ H for some subconcept H of G. Let V be the concept [x: x= x], and say that F is small iff V does not go into F. Define F to be similar to G iff (F is small ∨G is 22 As an anonymous referee, Stewart Shapiro and his student Roy Cook (independently) pointed out to me, cut-abstraction inflates at every cardinality, in the sense that, for every cardinal κ, there is a domain of size κ with a strictly dense linear order on it, so that cut-abstraction applies to yield a ‘new’ domain of size 2κ .

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small → ∀x(Fx↔ Gx)). Similarity is an equivalence relation. New V is then the abstraction: New V *F = *G ↔ F is similar to G If we agree—as I think we should—that the numbers may only properly be assigned to genuine sortal concepts—that is, roughly, concepts F with which are associated not only criteria of application but also criteria of identity—then we should be happy with this modification (of either cut-abstraction or Basic Law V) only if we are persuaded that self-identity is a genuine sortal. For if a concept F can have a number only if F is sortal, then, assuming Hume’s principle, F can be equinumerous with itself only if it is sortal. And if it can’t be equinumerous with itself, it can scarcely be equinumerous with any other concept. Since small is defined so that F is small iff self-identity doesn’t go into F, New V is a real restriction of Basic Law V only if self-identity is a genuine sortal. I do not think it is. A simple argument due to Crispin Wright shows, in effect, that if self-identity were a genuine sortal, many concepts which are plainly not sortal would qualify as such. The argument turns on the point that whenever a concept G is genuinely sortal, its restriction by any other (even merely adjectival) concept F—i.e. the conjunctive concept: F-and-G—will likewise be sortal. For example, since horse is, presumably, genuinely sortal, so is white horse, for all that the restricting concept white is no sortal. Thus if self-identical were a genuine sortal, so would be any restriction of it, such as white-and-self-identical. However, since white-and-self-identical is equivalent to white, it would follow that white is after all a sortal concept. Since white (or white thing) is not a genuine sortal, neither can self-identical be one. For the same reason, clearly, no concept which applies universally can be a genuine sortal concept. 23 If this is right, some other means of formulating the needed restriction is required. There is an obvious next thought. Why should we not simply stipulate that a predicate Q determines a suitable domain for cut-abstraction only if Q is genuinely sortal? Since neither self-identity, nor any other predicate (such as ‘F ∨ ¬F’) which is guaranteed application to all objects whatever, is a genuine sortal, this will ensure that the universe of objects as a whole— even if it admits of a strictly dense ordering—is not an admissible domain for cut-abstraction. 23 cf. Wright “Is Hume’s principle analytic?”. Wright formulates the argument slightly differently, as follows: “Call a concept that is not sortal a mere predicable. Where F is a mere predicable, the question: “How many F’s are there?”, is deficient in sense and “the number of F’s” has no determinate reference. However, attaching a mere predicable to a genuine sortal, G, produces a complex, restricted sortal, F-and-G, such that there can be, and normally will be, a determinate number of objects falling under it. Thus if Fis any mere predicable, and self-identity is a genuine sortal, there will be a determinate number of objects which are F and self-identical. But since F and self-identical is equivalent to F, it follows that there can be no such determinate number wherever there is no determinate number of F’s—i.e. wherever F is a mere predicable. So self-identity is not a sortal concept”.

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A thorough defence of this proposal requires more space than I have here. To conclude, I should like to comment briefly on three points. (i) It might be observed that a restriction of admissible domains to those specifiable by sortal concepts will not, on the face of it, exclude certain very large domains such as those comprising all ordinals, or all cardinals, or all sets (since the relevant concepts appear to qualify as genuinely sortal)—giving rise to concern that paradox may still be derivable from cut-abstraction by taking one or other of these collections as underlying domain. I think this might be met in either of two ways. First, any attempt to generate paradox from (#) by taking the ordinals, say, as domain will—so far as I can see—rely on the idea that the collection of all ordinals is universe-sized. That requires the assumption that the concept ordinal number is equinumerous with some concept under which every object—whether an ordinal number or not—falls. But if what I have already said is right, concepts can be equinumerous only if both are sortal, and there can be no universal sortal concept, so that this assumption can be rejected, and there will be no need to strengthen the restriction on cut-abstraction to preclude taking the ordinals, etc., as domains. But second, even if it should prove necessary to exclude the ordinals, etc., as admissible domains for cutabstraction, there is a quite natural way to do this. Instead of requiring simply that an admissible domain be given by a sortal concept, we might require that such a domain should have a determinate cardinal size. Since being the extension of a sortal concept is at least a necessary condition for a collection to have a determinate size, this restriction would encompass the one already proposed. If this necessary condition is not sufficient—i.e. if certain sortal concepts fail to have determinately-sized extensions – then those concepts will be excluded by the revised restriction. In particular, what Michael Dummett has called indefinitely extensible concepts, such as ordinal, cardinal and set itself, will be excluded. (ii) It may be objected that restricting admissible domains for cutabstraction in either of the ways suggested is arbitrary or ad hoc. And the objection might be thought to draw strength from the neo-Fregean’s willingness (and, indeed, need) to employ unrestricted abstractions such as Hume’s principle. I shall make just two quick points in reply, leaving—no doubt— much more to be said. First, as should by now be clear, it is in fact false that Hume’s principle is a completely unrestricted abstraction—although its first-order quantifiers are unrestricted, its initial second-order quantifiers are— crucially—restricted to sortal concepts. Second, my proposed restriction(s) on cut-abstraction appear to be no more arbitrary or ad hoc than the restriction which New V seeks to build into Basic Law V. It is true that the manner in which the restriction is imposed on (#) differs, formally, from what happens with New V—where what is done is not to restrict the range of any quantifier, but to complicate the equivalence relation—with the effect that when F and G are not small, *F and *G exist, but are identified irrespective of whether their

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concepts are co-extensive. But I think this difference is superficial. Provided that the conditions for a first-level concept to be sortal can be expressed (using only logical vocabulary) in a second- (or perhaps third-) order language, I can see no reason why (#) should not be recast in essentially the same mould as New V. And if they cannot be so expressed, that is bad news (if it really is bad) not only for (#) but for New V too, for reasons already mentioned. But I am not persuaded that it would be bad news—since I see no ground for assuming that every philosophically important concept must be capable of definitive expression in the purely logical vocabulary of a second- or thirdorder language. (iii) Finally, a quick word about the state of the economy. Some recent writers 24 have claimed—plausibly, in view of the obvious risk of some form of Cantor’s paradox—that acceptable abstractions should be, in some sense, non-inflationary. Is cut-abstraction inflationary, in any objectionable sense? Some care needs to be exercised in characterising the relevant notion of inflationariness, since a great part of the point and interest of abstractions lies in the fact that they ‘generate’ objects which are ‘new’, and so, in a certain sense, ‘expand’ the underlying domain. So that in one way, inflation—or at least domain-expansion—is just what the neo-Fregean wants. Of course, this way of putting the matter is potentially very misleading, since it gives the entirely false impression of ontological prestidigitation—in which abstraction creates objects out of nothing, as it were, much as a practised conjurer appears to pull pigeons out of thin air. The neo-Fregean can, and should, insist upon a more sober description of what is going on. What an abstraction does, if all goes well, is to set up a concept—of direction, or cardinal number, or whatever—by supplying necessary and sufficient conditions for the truth of identity-statements linking terms which purport reference to objects falling under it. It draws our attention to the possibility of redescribing—or reconceptualising—the state of affairs which consists in line a being parallel to line b, for example, in terms of the holding of the relation of identity between certain objects, the direction of a and the direction of b. 25 Accepting the proposed reconceptualisation does not—in and of itself—involve acknowledging the existence of these objects. What it involves, rather, is accepting that the question whether there are such objects reduces to the question whether suitable instances of the right-hand side of the abstraction principle are indeed true. So what an abstraction does is not to ‘create’ objects, but to equip us to recognise, identify and distinguish objects which we could not recognise, identify and distinguish before—i.e. in advance of grasping the concept which the abstraction introduces. 24 See Kit Fine “The limits of abstraction”, in M.Schirn ed. The Philosophy of Mathematics Today, Oxford University Press (1998), pp. 503–629. 25 For fuller discussion of this idea, see Wright “On the Philosophical Significance of Frege’s Theorem”, §I; Hale “Dummett’s critique of Wright’s attempt to resuscitate Frege”, §2; and Hale “Grundlagen §64” passim.

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If inflation of this kind is acceptable, what kind might not be? Kit Fine writes: Two necessary conditions for the truth of an abstraction principle hold as matter of logic. . . . In the first place, it follows from the truth of an abstraction principle that its underlying criterion of identity on concepts should be an equivalence relation . . . Secondly, it follows from the truth of an abstraction principle that the identity criterion should not be inflationary, the number of equivalence classes must not outstrip the number of objects. There must, that is to say, be a one–one correspondence between all of the equivalence classes, or their representatives, on the one hand, and some or all of the objects, on the other. It is, of course, on this score that Law V proves unacceptable; for where there are n objects, it demands that there be 2n abstracts. 26

There is, I think, some ambiguity or vagueness in these remarks which we need to resolve if avoidable confusion is to be avoided. Let us say that an abstraction A inflates on an underlying domain D if A’s equivalence relation partitions D into more equivalence classes than D has elements. Then one might say that an abstraction is weakly inflationary if there is some domain on which it inflates, and strongly inflationary if it inflates on every domain (or perhaps—a little less exiguously—on some domain of cardinality κ, for every cardinal κ). 27 To require of an acceptable abstraction that it should not be (even) weakly inflationary would stop the neo-Fregean project dead in its tracks, before it even got moving (as it were). It will be clear that I think there is no good ground to impose such a requirement, and I shall not discuss it further. It is much more plausible to require that acceptable abstractions should not be strongly inflationary. 28 Some of the neo-Fregean’s key abstractions, including the other crucial second-order abstraction, Hume’s principle, satisfy this requirement. 29 But whilst the requirement that abstractions not be strongly inflationary is 26 “The limits of abstraction”, p. 506. 27 This characterisation of weak and strong inflation applies directly only to abstractions—like Hume’s

principle and Basic Law V—which are not restricted abstractions in the sense previously explained, i.e. are not such that their formulation already involves a specification of a particular domain as the underlying domain for the abstraction. Since any particular cut-abstraction, such as Cut, is restricted in this sense, there can be no question of its being strongly inflationary. We can, however, properly ask of the corresponding general schema—(#) in the case of Cut—whether it is strongly inflationary. 28 More plausible, because it might seem that strong inflation is bound to give rise to a version of Cantor’s paradox. It might also be thought that if an abstraction is strongly inflationary, then there could be no hope of showing that it is satisfiable, i.e. has a model—for let D be any domain, of cardinality κ, say. Then any strong abstraction inflates on D, i.e. its equivalence relation partitions D into more than κ equivalence classes, and so ‘generates’ more than κ abstracts. Thus D cannot be a model for the abstraction. But D was any domain whatever, so our abstraction can have no models. On reflection, it should be apparent that this short argument involves an unstated assumption—that the domain of any putative model for an abstraction must be the underlying domain for the abstraction. As against this, I cannot see why, in setting up model for a restricted abstraction—such as cut-abstraction—we should not choose as the domain of the model some larger collection which properly includes the collection which is to play the rôle of the underlying domain for the abstraction. 29 Hume’s principle inflates, of course, on any finite domain, but can be shown—assuming Choice, but without assuming CH or GCH—that it does not inflate on any infinite domain.

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more plausible, I can see no compelling reason to accept it in full generality— that is, as applying both to unrestricted abstractions and restricted ones. It may be necessary to insist that no unrestricted abstraction can be strongly inflationary. But, as I have tried to make plausible, it is unnecessary to require this of restricted abstractions. The cut-schema, in particular, is strongly inflationary in the sense that for every cardinality κ, there is an admissible domain of cardinality κ on which an instance of (#) inflates. But that, so far as I can see, does no harm, provided admissible domains are restricted to those given by genuine sortal concepts (or perhaps, those of determinate cardinal size).

5.

Summary and concluding remarks

My aim in this paper has been to set forth one plausible way in which a neo-Fregean account of arithmetic may be extended to encompass the real numbers. I have followed Frege himself in suggesting that the reals should be introduced as ratios of quantities. This approach, as Frege perceived, demands a prior analysis of the notion of quantity. I have agreed with Frege, too, in thinking that this should be done by providing a general characterisation of what he called quantitative domains, but have offered a somewhat different account of them from that given in Grundgesetze. Ratios of quantities are introduced by an abstraction principle based on the ancient theory of proportion which comes down to us from Eudoxos. The positive reals are then obtainable as ratios of quantities in a complete quantitative domain, and zero and the negative reals by essentially the move by which the integers are standardly constructed as difference-pairs of natural numbers. My construction, taken by itself, establishes only a conditional result: if there exists a complete quantitative domain, then the reals may be introduced as ratios of quantities on it. However, as I argue in the second half of the paper, there is a route by which a neo-Fregean may establish the existence of at least one complete domain, starting with the natural numbers (as given by Hume’s principle), by successively applying ratio-abstraction to obtain a full domain and a suitably adapted version Dedekind’s method of cuts to obtain from this a complete domain. Two points deserve emphasis: first, quantities, though (on my account) abstract objects which are sharply to be distinguished from the concrete entities which stand in various quantitative relations to one another, are not themselves to be identified with numbers; and second, although I use a version of Dedekind’s method in proving the existence of a complete domain, there is no question, on the present approach, of defining the reals as or in terms of Dedekind cuts. Here is not the place to elaborate upon the significance of these points. The first is, I believe, integral to the defence of my approach against several more or less familiar objections to older attempts to treat real numbers as directly abstracted from quantitative relations among concrete entities— but that defence is best conducted in the context of a more searching analysis

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of the notion of quantity than I have had space for here. Such an analysis would also do much to motivate the axiomatic characterisation of quantitative domains which I have been obliged to state somewhat dogmatically, without the philosophical defence it surely requires. The second is essential to the claim of the present approach to respect Frege’s belief—I would say, insight— that a satisfying foundational account of the real numbers should introduce them in a way which expressly provides for their applications. 30, 31

30 If one disregards this constraint—as I think one should not—then it would, of course, be possible to obtain the reals by Fregean abstraction in a much simpler and more direct way than I have described. One might, for example, start with the natural numbers as given by Hume’s principle, obtain rationals by some form of ratio-abstraction (such as that employed here, but there are obviously other ways in which this might be done) and then directly introduce the reals as cuts by cut-abstraction (either as explained here, or in some similar way). 31 I am indebted to an anonymous referee for this journal, and to Roy Cook, Jim Edwards, Gary Kemp, Pierluigi Miraglia, Philip Percival, Stewart Shapiro, Neil Tennant and Crispin Wright for helpful discussion of earlier versions of this material, as well as to my audiences at presentations of parts of it in Cambridge, Columbus OH, Glasgow, L’Institut d’Histoire et Philosophie des Sciences et des Techniques in Paris and St. Andrews. Very special thanks are due to my colleague Adam Rieger. Work on this paper was carried out during my tenure of a British Academy Research Readership—I am most grateful to the Academy for its generous support.

THE STATE OF THE ECONOMY: NEO-LOGICISM AND INFLATION 1,2 Roy T. Cook

1.

Introduction

In recent years there has been a resurgence of interest in logicism as a viable philosophy of mathematics, stemming in great part from Crispin Wright’s Frege’s Conception of Numbers as Objects [1984] and the formal and philosophical work of George Boolos. Before this work it was generally accepted that Frege’s project of reducing mathematics to pure logic was devastated by Russell’s detection of a paradox produced by Frege’s notorious Basic Law V. Frege’s project has recently been reborn, with some modifications. In this paper I explore some of the landscape surrounding this project, concentrating on the prospects for a successful neo-logicist reconstruction of the real numbers. I focus on Bob Hale’s “Reals by Abstraction” [2000] and his use of a cut abstraction principle, as this approach seems to be the one most likely to be generalizable to complex analysis, functional analysis, etc. There is a serious problem that plagues Hale’s project. Natural generalizations of the sort of principle needed to construct the reals imply that there are far more objects than one would expect from a position that stresses its epistemological conservativeness. In other words, the sort of abstraction needed to obtain a theory of the reals is rampantly inflationary. After arguing for this claim with respect to Hale’s treatment, I will indicate briefly why this problem is likely to reappear in any neo-logicist reconstruction of real analysis. 1 This paper first appeared in Philosophia Mathematica 10[2002], pp. 43–66. Reprinted by kind permission of the editor and Oxford University Press. 2 This title is taken from a phrase used by Bob Hale [2000] in his own discussion of neo-logicist abstraction principles and domain inflation.

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Abstraction principles An abstraction principle 3 is any second-order formula 4 of the form: (∀P)(∀Q)[@(P) = @(Q) ↔ E(P, Q)]

“@” here is a function from properties (or relations) to objects, and E is an equivalence relation on the properties (or relations). Abstraction principles allow us to take, as objects, characteristics that the properties or relations have in common. Frege’s Basic Law V is: BLV : (∀P)(∀Q)[EXT(P) = EXT(Q) ↔ (∀x)(Px ↔ Qx)] Frege derives all of arithmetic from BLV plus second-order logic, but Russell’s discovery that BLV is inconsistent with the second-order comprehension axiom renders this result less noteworthy. The resurrection of logicism stems for the observation that Frege’s only ineliminable use of BLV occurs in his derivation of Hume’s Principle: HP : (∀P)(∀Q)[NUM(P) = NUM(Q) ↔ P ≈ Q] [P ≈ Q is the second-order formula asserting that there is a one-to-one correspondence between the P’s and the Q’s] 5 The “NUM” operator is, in effect, a number generating function, mapping properties onto the number corresponding to the cardinality of the extension of the property. Unlike BLV above, HP is consistent. It can be added to any theory that has an infinite model, and the new theory will have (infinite) models of the same cardinality 6 as the original theory. Frege’s derivation of arithmetic in the Grundgesetze can be reconstructed from second-order logic plus HP, thereby avoiding the troublesome BLV. 7 This result, quite remarkable as a mathematical fact independent of any philosophical implications, has come to be called Frege’s Theorem. 8 Of course, HP, with its explicit reference to numbers via the “NUM” function, is not a logical truth. Thus, the neo-logicist must abandon the hope that 3 I am ignoring “objectual” abstraction principles where the abstraction operator “@” maps objects onto objects, as the phenomena that interest us here involve only “conceptual” abstraction, where properties or relations are mapped onto objects. 4 I assume standard set theoretic semantics for second-order logic, where the second-order predicate variables range over the full powerset of the domain, and which therefore satisfies the comprehension scheme:

(∃R)(∀x1 , x2 , . . . , xn )(R(x1 , x2 , . . . , xn ) ↔ ) for each formula not containing R free. For details see Shapiro [1991]. 5 Although I often phrase the equivalence relation for an abstraction principle in everyday English, every abstraction principle considered in this paper can be expressed using only the resources of second-order logic (plus, in some cases, previously defined abstraction operators). 6 This result depends on the axiom of choice. 7 See Boolos [1987] and Heck [1993]. 8 Another abstraction principle which will be used in examples later in the paper is a size-restricted version of Basic Law V: NewV : (∀P)(∀Q)[EXT(P) = EXT(Q) ↔ ((Pis “Big” ∧ Qis “Big”) ∨ (∀x)(Px ↔ Qx))]

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one can reduce all of mathematics to truths of pure logic, but this is not surprising. The discovery of Russell’s paradox, coupled with the failure of Russell and Whitehead’s subsequent logicist attempt in the Principia Mathematica [1913], suffice to render the original logicist project implausible. In addition, Boolos argues that this sort of reduction of mathematics to pure logic is in principle impossible: mathematics has ontological commitments, while on the contemporary conception logic does not. 9 One can argue, however, that the crucial aspect of Frege’s logicism is not the reduction of all of mathematics to truths of logic. Instead, Frege’s main goal was to demonstrate the analyticity of mathematics, saving it from Kant’s charge of a priori yet synthetic (see Coffa [1991]): The problem becomes, in fact, that of finding the proof of the proposition, and of following it up right back to the primitive truths. If, in carrying out this process, we come only on logical laws and on definitions, then the truth is an analytic one . . . If however, it is impossible to give the proof without making use of truths which are not of a general logical nature, but belong to the sphere of some special science, then the proposition is a synthetic one. (Frege [1884], p. 4, emphasis added)

According to Frege, the aprioricity of mathematics is a direct consequence of its analyticity: For a truth to be a posteriori, it must be impossible to construct a proof of it without including an appeal to facts, i.e. to truths which cannot be proved and are not general. But if, on the contrary, its proof can be derived exclusively from general laws, which themselves neither need nor admit of proof, then the truth is a priori. ([1884], p. 4)

Thus, the reduction of mathematics to logic was just the particular strategy Frege adopted to secure the analyticity 10 and apriority of mathematics. Although Frege abandoned his project, Wright has revived it, stressing that the part of Frege’s project that is of interest is not the reduction of mathematics to logic but rather a demonstration of the analyticity, or at least a prioricity, of (much of) mathematics: Frege’s Theorem will still ensure . . . that the fundamental laws of arithmetic can be derived within a system of second-order logic augmented by a principle whose role is to explain, if not exactly to define, the general notion of identity of cardinal number, and that this explanation proceeds in terms of a notion which can be defined in terms of the concepts of second-order logic. If such an explanatory [where “P is Big” is an abbreviation for the second-order formula asserting that the P’s are equinumerous with the entire domain]. NewV is consistent, and satisfied by the hereditarily finite sets Vω . Many of the standard axioms of ZFC (but not infinity or powerset) can be reconstructed using NewV. 9 Except for (∃x)(x = x), the claim that there is at least one object. Even this, however, is only accepted for convenience. We could easily formulate a logic that countenanced the empty model. 10 The assumption that logic is analytic if anything is seems unproblematic. Even Quine admits something like this in “Two Dogmas of Empiricism” [1951]!

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The Arché Papers on the Mathematics of Abstraction principle . . . can be regarded as analytic, then that should suffice . . . to demonstrate the analyticity of arithmetic. Even if that term is found troubling, as for instance by George Boolos, it will remain that Hume’s principle – like any principle serving implicitly to define a certain concept – will be available without significant epistemological presupposition . . . So one clear a priori route to the recognition of the truth of . . . the fundamental laws of arithmetic will have been made out. And if in addition [Hume’s principle] may be viewed as a complete explanation – as showing how the concept of cardinal number may be fully understood on a purely logical basis – then arithmetic will have been shown up by Hume’s principle . . . as transcending logic only to the extent that it makes use of a logical abstraction principle – one [that] deploys only logical notions. So, always provided that concept formation by abstraction is accepted, there will be an a priori route from mastery of second-order logic to a full understanding and grasp of the truth of the fundamental laws of arithmetic. Such an epistemological route . . . would be an outcome still worth describing as logicism. ([1997], pp. 210–211, emphasis added)

Although the neo-logicists are a bit vague regarding exactly what the special status of abstraction principles is, the general idea seems to be something along the following lines: Acceptable abstraction principles provide something akin to an implicit definition of the abstracts generated by the principle, providing an explanation, although not necessarily a complete 11 explanation, of what it is to be an abstract of the relevant sort. This explanation provides us with a method by which we can come to know truths about these abstracts a priori. 12 Thus, the abstraction principles are meant, among other things, to provide some sort of epistemological advantage – the idea being that we can get all of arithmetic, for example, from the epistemologically unproblematic HP. Finally, although abstraction principles are not logical truths, the fact that they invoke only logical terminology on the right-hand side of the biconditional in giving the truth conditions of the identity on the left supports the claim that neo-logicism provides “an outcome still worth describing as logicism”. Of course, as is well known, most 13 of modern mathematics can be reconstructed quite nicely in Zermelo–Fraenkel set theory. In addition, some philosophers, such as Gödel [1947], argue that the axioms of set theory are a priori knowable. 14 The interest of the neo-logicist project, then, depends on the extent to which it can be argued that the necessary abstraction principles 11 Abstraction principles, according to many critics (and some defenders) of neo-logicism, notoriously fail to solve the “Caesar Problem”. 12 This emphasis on how we come to know the truths of mathematics seems to be what is crucial in the passages from Frege’s Grundlagen quoted above. 13 One standard textbook on set theory, Kunen [1980], contains the following as an exercise:

Verify that within ZC [ZFC minus replacement] one may develop at least 99% of modern mathematics. (p. 147) 14 Unlike the neo-logicists, Gödel would not have claimed that the axioms of set theory are analytic, as their truth depends not only on the meaning of the terms involved but also (in some manner) on our direct intuition of the set theoretic universe (see Gödel [1947]).

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are epistemically “cheap” in a way in which the axioms of ZFC are not. A first step towards this goal is an account of which abstraction principles are neo-logicistically acceptable. Mere consistency is not enough for an abstraction principle to be acceptable. Presumably if two abstraction principles are both acceptable, then their conjunction should be as well, yet we can formulate consistent abstraction principles that are only satisfiable on finite domains. 15 Such a principle and HP are not jointly satisfiable, so we need more stringent requirements on which abstraction principles are acceptable. The point of this paper is to examine one such constraint. Before moving on, a technical fact relevant to the satisfiability of abstraction principles needs to be noted: If the right-hand side of the biconditional in an abstraction principles contains no non-logical vocabulary, then the abstraction principle will be satisfiable on a domain of size κ if and only if it is satisfiable on any domain of size κ. A proof 16 of this result can be found in Fine [1998], although the reasoning behind it should be clear once one realizes that an abstraction principle only requires that there be a distinct object for each equivalence class of properties (or relations). It implies nothing regarding which object is associated with which class.

3.

Inflation

A number of requirements on abstraction principles have been proposed. The constraint that concerns us here is the idea that suitable abstraction principles should be non-inflationary. Informally this is just a requirement that the abstraction principles should not imply the existence of too many objects, reflecting the intuition, due to von Neumann, that the way to avoid the set theoretic paradoxes is by avoiding collections that are too large, i.e. what are now known as proper classes. 17 Consider BLV. The source of its inconsistency can be traced, at least in part, to the fact that it assigns a distinct object (an extension) to each collection of objects in the domain, violating Cantor’s theorem. To avoid this sort of contradiction, it is a good start to require that acceptable abstraction principles do not involve equivalence relations that partition the domain into more collections than there are objects. Kit Fine argues that: 15 The following, much discussed abstraction principle has come to be called the Nuisance Principle:

NP : (∀P)(∀Q)[NUI(P) = NUI(Q) ↔ (P, Q)] [where (P, Q) abbreviates the second-order formula asserting that the collection of objects that are either P-and-not-Q or are Q-and-not-P is finite]. NP is satisfiable on any finite domain, but on no infinite one. 16 Fine’s argument can be generalized to universal generalizations of logical abstraction principles. 17 I use the term “proper class” in the technical sense, referring to collections of sets that are too big to themselves be sets. Intuitively, at least, there are in fact collections that are (or at least might be) bigger than proper classes, such as various large collections of proper classes.

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The Arché Papers on the Mathematics of Abstraction . . . the identity criterion should not be inflationary, the number of equivalence classes must not outstrip the number of objects. There must, that is to say, be a one–one correspondence between all of the equivalence classes, or their representatives, on the one hand, and some or all of the objects, on the other. ([1998], p. 506)

Along similar lines, Wright writes that: The cells into which the relevant equivalence relation partitions the universe of Concepts must not outrun the population of objects which constitute the range of the first-order variables in the abstraction principle. ([1997], p. 222)

This way of phrasing the prohibition on domain inflation begs the question against the neo-logicist reconstruction of the reals, since the strategy is to add a suitable abstraction principle to a theory satisfied by a countable domain to get a theory that is only satisfied by an uncountable domain. The neo-logicist approach depends on some abstraction principles being at least somewhat inflationary. We can get at the spirit of the ban on inflationary abstraction principles with something akin to Boolos’ take on the matter: . . . it was a central tenet of logical positivism that the truths of mathematics were analytic. Positivism was dead by 1960 and the more traditional view, that analytic truths cannot entail the existence either of particular objects or of too many objects, has held sway ever since. ([1997], pp. 249–250, emphasis added)

Ignoring the issue of whether analytic principles should entail the existence of particular objects, we can assume for the sake of argument that some acceptable abstraction principles might be inflationary, i.e. their addition to a theory with models of size κ might result in a theory whose models all have domains larger than κ. This domain inflation should not be too rampant, however. Acceptable abstraction principles should not imply the existence of too many objects, at least not if they are to be “epistemologically cheap”. There may be no way to delineate exactly what “too many” means in the previous sentence. On the contrary, like the vague predicate “red”, there might be no sharp line marking off where the extension of “too many” begins. Even so, we can lay down a number of precise ways in which an abstraction principle might be inflationary, even if we cannot determine with certainty which of them are neo-logicistically acceptable and which are not. Given an abstraction principle AP and a set of object S, the restriction of AP to S is the result of replacing every first-order quantifier “∀x” (“∃x”) with “∀x ∈ S” (“∃x ∈ S”) and replacing every second-order quantifier “∀X ” (“∃X ”) with “∀X ⊆ S” (“∃X ⊆ S”). An abstraction principle AP generates κ objects when applied to the domain S iff, for every domain D such that S ⊆ D and D satisfies 18 the restriction of AP to S, the cardinality of D– S ≥ κ. (Here, 18 Since all that is relevant to the satisfaction of a logical abstraction principle is the cardinality of the domain, I say that a set D satisfies an abstraction principle AP if there is some model with D as domain that satisfies AP.

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and below, κ, γ , and λ are infinite cardinals.) 19 We now define the notion of κ-inflationary: 20 An abstraction principle AP is κ-inflationary if, for any domain S of cardinality κ, the application of AP to S generates γ objects where γ > κ. 21

Using the notion of κ-inflation, we can now define some more general senses in which an abstraction principle can be inflationary: Strictly Non-inflationary:

Locally Inflationary:

Boundedly Inflationary:

Unboundedly Inflationary: Universally Inflationary:21

An abstraction principle AP is strictly non-inflationary if there is no κ for which AP is κ-inflationary. An abstraction principle AP is locally inflationary if there are (only) finitely many k’s such that AP is κ – inflationary. An abstraction principle AP is boundedly inflationary if there are infinitely many κ‘s such that AP is κ – inflationary but there is some γ such that, for all λ > γ , AP is not λ-inflationary. An abstraction principle AP is unboundedly inflationary if, for every κ, there is a γ > κ, such that AP is γ – inflationary. An abstraction principle AP is universally inflationary if, for every κ, AP is κ-inflationary.22

HP is strictly non-inflationary, since it can be added to any theory with an infinite model and the result will have a model of the same cardinality. Shapiro and Weir [1999] show that, if the Generalized Continuum Hypothesis holds, then NewV (see note 7) is unboundedly inflationary, since on this assumption it is satisfied at every successor cardinal but at no singular cardinal. 24 Finally, it is clear that any abstraction principle used to obtain the real numbers must 19 Notice that, for any abstraction principle AP containing only logical vocabulary on the right-hand side of the biconditional and any set S of cardinality κ, if AP applied to S generates γ objects, then γ > κ. 20 I am ignoring cases where abstraction principles inflate on finite domains since they are irrelevant to the case at hand. Hume’s Principle inflates on finite domains, yet this inflation has rarely been the target of serious criticism. In addition this sort of inflation is critical to the success of the neo-logicist project given the possibility that there are only finitely many non-abstract objects in the world. 21 Since the satisfaction of an abstraction principle depends solely on the cardinality of the domain, we could have phrased this as:

An abstraction principle AP is k-inflationary if there is some domain S of cardinality κ such that the application of AP to S generates γ objects where γ > κ. 22 Locally inflationary and universally inflationary are equivalent (roughly) to Hale’s weakly and strongly inflationary, respectively (see Hale [2000], p. 121). 23 Universal inflation is one way of formalizing the intuition that some mathematical concepts given by abstraction principles are indefinitely extensible (see Dummett [1963] and note 47 below). 24 Shapiro and Weir also prove that it is consistent with ZFC that NewV has no uncountable models at all.

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be at least locally inflationary, since the point of the abstraction is to take us from a countable domain to an uncountable one where we can formulate real analysis. The question thus becomes: Where should we draw the line with respect to domain inflation? As has already been pointed out, abstraction principles that are locally inflationary must be acceptable. In addition, we can treat locally inflationary principles and boundedly inflationary principles as roughly on a par, since in both cases we are confronted with cases where the abstraction may blow up our ontology, but only so much. At some point we reach an upper limit beyond which the abstraction principle does not inflate. Along similar lines, we can think of unboundedly inflationary and universally inflationary principles as equally problematic, since in both cases the problem, if any, has to do with the fact that its (possibly repeated) application might multiply the underlying ontology without limit. Thus, we need to determine whether unboundedly or universally inflationary abstraction principles are neo-logicistically acceptable. A number of considerations can be brought to bear against unboundedly and universally inflationary abstraction principles, in addition to the points already canvassed against inflation more generally. I will give a different argument against each sort of inflation, although unbounded and universal inflation are sufficiently similar that problems with one are likely to indicate problems with the other. The neo-logicist is claiming that the abstraction principles implicitly define, or at least ground our use of mathematical concepts and theories. Definitions of the abstract objects of mathematics, even implicit ones, ought to determine a unique group of objects which necessarily fall under the definition. If this “defining” abstraction principle is unboundedly inflationary, however, then the neo-logicist has failed in his task. Assume that we have some unboundedly inflationary abstraction principle AP and there are κ objects in the universe 25 (including the abstracts guaranteed to exist by AP). Let γ be the least cardinal > κ such that AP is γ -inflationary. Then, had there been γ objects in the universe, there would have, by AP been more than γ (and thus more than κ) abstracts. But then the original abstracts are not all of the objects whose identity conditions are given by AP. This process can be repeated indefinitely (and transfinitely), so we never have all the objects that fall under the purview of AP. In other words, if AP is unboundedly inflationary then it fails to secure a definite collection of objects as the domain of its abstraction operator, but instead gives us different abstracts relative to how many objects exist. 26 25 This way of setting things up implies that AP is not κ-inflationary.

26 There is a tempting response at this point. One might point out that, in addition to the abstracts falling

under the principle AP, other sorts of abstract objects such as real numbers, sets, and groups will also exist and will exist necessarily. Therefore, the cardinality of the actual world is the same as the cardinality of every possible world, namely, however many objects could possibly exist. In other words, even if AP alone does not secure a definite extension for the abstraction operator, this unique extension might be secured by AP plus the fact that all of the sets of ZFC exist. While this is true, it does not save unboundedly inflationary

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This argument against unbounded inflation is quite compelling, especially if we think that abstraction principles should generate abstract mathematical objects that exist necessarily instead of providing an undetermined multitude of objects whose existence depends on the number of non-abstract objects present in the universe. The case against universal inflation is a bit different, but equally worrying. In moving from an abstraction principle that is unboundedly inflationary to one that is universally inflationary we have replaced one problem with another. With unbounded inflation there was no unique collection of abstracts generated by the abstraction. In the case of universal inflation, there might be a unique collection of objects generated by the abstraction principle, but if so, then it is an extremely badly behaved collection. In other words, if an abstraction principle is universally inflationary, then it will be satisfied (if satisfied at all) only by a structure that is at least the size of the smallest proper class. Assume that AP has a set-sized model M. Then there is some κ such that the domain of M has cardinality κ. If AP is universally inflationary, then application of AP to a domain of size κ produces γ objects for some γ > κ. M satisfies AP, so the domain of M contains at least γ objects, but then the cardinality of the domain is greater than κ. Contradiction. This sort of issue seems to be what Hale has in mind where he writes that, although boundedly inflationary abstraction principles are neo-logicistically acceptable: It is much more plausible to require that acceptable abstraction principles not be strongly inflationary [equivalent to my universally inflationary]. Some of the neo-Fregean’s key abstractions, including the other crucial second-order abstraction, Hume’s Principle, satisfy this requirement. ([2000], p. 120)

He adds in a footnote that: . . . it might seem that strong inflation is bound to give rise to a version of Cantor’s paradox. It might also be thought that if an abstraction is strongly inflationary, then there could be no hope of showing that it is satisfiable, i.e. has a model. (p. 120)

Hale touches on two main worries here, each of which deserve closer scrutiny. First, there is the claim that universally inflationary abstraction principles are likely to be susceptible to set-theoretic paradoxes such as Cantor’s paradox (or Russell’s or Burali-Forti’s). The idea is simple: If, for any κ-sized collection, the universally inflationary principle AP generates, say, 2κ new objects, then it seems plausible that when applied to a proper class, or any other sort of structure, it would also inflate. This is one way of explaining what goes wrong with Frege’s Basic Law V. The reasoning is not general, however. There could be abstraction principles that inflated on all sets but did not inflate on proper classes. For example, principle AP might inflate on any collection that can be abstraction principles from the force of the objection, because an adequate definition should determine a unique extension independently of the existence of any other objects.

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well-ordered, but on no structure that cannot. If this is the case, then, as long as there are proper classes too large to be well ordered, AP might be satisfiable even though it is universally inflationary. We will see a potential candidate for such a satisfiable yet universally inflationary abstraction principle below. Hale’s second worry is that we might be faced with insuperable difficulties when attempting to prove the consistency/satisfiability of universally inflationary abstraction principles. The standard definition states that a sentence is satisfiable if and only if there is a set theoretic model (read: set as domain plus appropriate assignments to various bits of language) such that the sentence is true in that model. Any universally inflationary abstraction principle fails to be satisfiable in this sense, yet it is still possible that some structure (such as a proper class) might make the sentence true. This is a serious problem for the neo-logicist. As we have seen, some abstraction principles are consistent while others that resemble the former a great deal are not. Thus, one of the most important parts of defending a neo-logicist abstraction principle as acceptable is to demonstrate its satisfiability. This will prove difficult, if not impossible, for universally inflationary abstraction principles since our methods for studying and manipulating proper classes are less powerful and less secure than our set theoretic machinery. There is a disturbing historical irony here. The notion of proper class was introduced as a result of, among other things, reflection on what exactly went wrong in Frege’s Grundgesetze. The idea was to draw a distinction between the logically safe sets and the problematic proper classes, which were in some sense too large to be safely manipulated like sets. If the neo-logicist reconstruction of Frege’s project pushes us once again into the realm of proper classes, historical sensitivity (or perhaps merely superstition) should cause some worry. 27 Thus, the neo-logicist should be extremely wary of unboundedly and universally inflationary abstraction principles. While these sorts of abstractions are not necessarily susceptible to the sorts of paradoxes usually associated with “bad” abstraction principles, and are even (possibly) satisfiable, they nevertheless take us far from the epistemically innocent implicit definitions that the neo-logicists argue acceptable abstraction ought to provide.

4.

Hale’s reconstruction of the reals

I take it as a desideratum of a successful philosophy of mathematics that it must account for enough mathematics to handle scientific applications. It follows that the neo-logicists need, at a minimum, to be able to reconstruct the theory of the real numbers. In fact, the success or failure of the neo-logicist project seems to hinge on their successful treatment of the real numbers. If this difficult case can be dealt with, then it is plausible that most other areas of 27 This point was brought to my attention by Jon Cogburn.

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contemporary mathematics can be handled by relatively unproblematic neologicist constructions building on the continuum. On the other hand, if the neo-logicist is unable to account for the reals, then the project has failed to provide a foundation 28 for mathematics. Reconstructing arithmetic from unproblematic abstraction principles might be interesting, and even mathematically important, but arithmetic is too simple a theory to allow us to conclude anything interesting about mathematics as a whole. It is at this point that the work of Bob Hale [2000] becomes relevant. Although others, including Simons [1987] and Dummett [1991] have written on Frege’s treatment of the reals, Hale is the first to attempt a full-scale neologicist account. Thus Hale’s work is of independent interest in an investigation of the prospects for a neo-logicist reconstruction of analysis. More importantly, given our purposes here, Hale’s account provides us with a useful case study of inflationary abstraction principles. Hale puts much stock in the fact that the reals are not just any sort of mathematical object but are, like the natural numbers and rational numbers, quantities: The most striking and most important features of Frege’s treatment of the reals are two: (i) the real numbers are to be defined as ratios of quantities . . . and (ii) in regard to the analysis of the notion of quantity, the fundamental question requiring to be answered is not: What properties must an object have if it is to be a quantity? but: What properties must a concept have, if the objects falling under it are to constitute quantities of a single kind? ([2000], p. 104)

Hale’s strategy is to, first, set up a general theory of quantity; second, to argue that if a certain sort of quantity exists, then ratios on those quantities can serve as the reals; and third, to prove, with the use of a novel abstraction principle, that the requisite sort of quantities exist. Hale begins by giving definitions of various sorts of “Quantitative Domain”. The series of definitions he proposes are intended to flesh out the second feature of Frege’s treatment of the reals – determining what properties a concept must have if the objects falling under that concept are quantities. A minimal q(uantitative)-domain is: . . . a non-empty collection Q of entities closed under an additive operation ⊕ which commutes, associates, and satisfies the strong trichotomy law that for any a, b ∈ Q, we have exactly one of: ∃c (a = b ⊕ c), ∃c (b = a ⊕ c), or 28 This is not to say that the work stemming from neo-logicism does not have other interesting philosophical applications. For example, Harold Hodes [1984] considers the following Order Type Abstraction Principle:

OTA : (∀P)(∀Q)[ORD(P, nd))] The new structure resulting from the application of EM to a normal q-domain Q is called R Q . 29 Finally, Hale defines a full q-domain to be a normal qdomain where we have: ∀a, b, c ∈ Q∃ q ∈ Q(RAT(a, b) =

RAT (q, c))(p. 107)

This can be reworded to avoid the reliance on abstraction. A normal q-domain is full iff: ∀a, b, c ∈ Q∃q ∈ Q(∀m, n(ma = nb ↔ mq = nc) ∧ ∀m, n(ma < nb ↔ mq < nc) ∧ ∀m, n(ma > nb ↔ mq > nc)) Finally, a q-domain Q is said to be complete iff: . . . every bounded above non-empty S ⊆ Q has a least upper bound. (p. 108)

This completes Hale’s first task – specifying what criteria a concept Q must meet for the objects falling under Q to be quantities of various varieties. Hale next points out that any two complete q-domains are isomorphic. 30 Applying the abstraction principle EM above, we get the result that, for any two complete domains Q and Q*, R Q = R Q∗ . Thus, according to Hale, the reals can be obtained as the ratios of any complete domain, as long as some such domain exists. All that remains is the third step in Hale’s argument – the proof that there is a complete q-domain. Hale’s argument is relatively straightforward: The 29 The principle EM, which is equivalent to a sort of pairing axiom, is strictly non-inflationary. 30 This fact depends on the explicit use of second-order quantifiers in the definition of complete q-

domains and their implicit use (in securing the fact that we are talking about the standard natural numbers and not some non-standard model of them) in the definition of normal q-domains.

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neo-logicist already has access to the positive natural numbers N + via HP. The natural numbers constitute a normal q-domain, but not a full one. An application of the abstraction principle EM to N + , however, gives us the ratios on the positive naturals R N + , which is a full domain (although not complete) and an obvious candidate to serve as the positive rational numbers. The next move is to apply the following Cut Abstraction Principle to R N + (reworded to fit the notation used here): C A : (∀P)(∀Q)[CUT(P) = ((∀x)((x ∈ R

N+

CUT (Q)

↔

∧ P and Q are cut properties30 on R N + ) → (Px ↔ Qx)))]31

This gives us the required complete q-domain, and we need only apply EM once more to obtain the reals. So far the neo-logicist project looks good, as long as HP, EM, and CA are acceptable. We will assume that HP and EM are acceptable, and concentrate on CA, since CA is responsible both for guaranteeing that there are uncountably many quantities and for producing the complete q-domain.

5.

Abstraction principles and abstraction processes

We have seen how CA generates a complete, uncountable q-domain from (something like) the rationals. There seems to be no principled reason for restricting this procedure, however. We should, prima facie, be able to apply cut abstraction to any linear ordering guaranteed to exist by previously accepted principles. To paraphrase Georg Kreisel, 33 one can argue that the evidence for the applicability of CA derives from the more general idea that cuts can be taken on any linear order whatsoever. In other words, we need to distinguish between abstraction principles (such as CA) and the more general 31 Hale defines a cut-property on R N + as follows:

. . . a cut property is a non-empty property whose extension is a proper subset of R N + and which is downwards closed [i.e., (∀a) (∀b) (Fa → (b < a → Fb)] and has no greatest instance [i.e., (∀a) (Fa → (∃b) (b > a ∧ Fb)]. (p. 112) We can generalize this to any linear ordering by replacing R N + with a name for the order in question. Hale’s definition, however, is a bit unwieldy, since it implies that the only linear orders with non-trivial cuts are dense. Thus, we can use a slightly modified version of Hale’s definition [for a linear order (Q, b, or a = b holds.

Notice that, in addition to its obvious similarity to Hale’s original definition of minimal q-domain, this definition respects Hale’s intuition that:

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. . . what makes the difference between quantitative ordering relations and others is that in the case of a quantitative ordering relation, but not otherwise, the entities which can significantly be asserted to stand in the relation can (at least in principle) be combined in such a way that compounds must come later in the relevant ordering than their components. In other words, for more than to be a quantitative ordering relation, there must be an operation of combination ⊕ on items lying in the field of more than, analogous to addition, such that for any a, b in more than’s field, a ⊕ b is more than a and a ⊕ b is more than b (p. 106)

In other words, the sum of two members of a quantitative domain must be larger than either member. 44 With this definition of minimal q*-domain in place, we get the following result: Theorem 3: (AC): Given an infinite cardinal κ, there is a minimal q*-domain (A, ⊕) such that: |A| ≤ κ and | Comp (A, ⊕)| > κ. Proof: Given an infinite cardinal κ, let λ be the least cardinal ≤ κ such that 2λ > κ. Let A be the subset of functions from λ (as an ordinal) into the set of positive rationals Q + such that f ∈ A iff there is an ordinal γ < λ such that for all ordinals αβ ≥ γ , f (α) = f (β). For f , g ∈ A, let h = f ⊕ g iff, for every α ∈ λ h(α) = f (α) + g(α). f < g iff at the least α such that (α) = g(α), f (α) < g(α). It is easy to verify that A is a minimal q-domain. The proof that |A| ≤ κ and |Comp(A, ⊕)| > κ is similar to the proof of Theorem 1. If we define normal q*-domain by substituting “minimal q*-domain” for “minimal q-domain” in Hale’s definition of normal q-domain, and similarly define full q*-domain by replacing “normal q-domain” with “normal q*domain”, then analogues of Theorem 3 hold for normal and full q*-domains. Given an f , g, h in the q*-domain A constructed in the proof of Theorem 3, let q be defined as: for all α < λ, q(α) = [ f (α) × h(α)] ÷ g(α). It follows that q ∈ A and f : g = q : h, so A is a full q*-domain. Universal inflation has returned. If the neo-logicist wishes to apply of cut abstraction to full q-domains but prohibit taking cuts on full q*-domains, then some relevant difference between the two sorts of structure needs to be explained. Neither definition appears, prima facie, to be more natural or intuitive than the other as an explication of our pre-formal notion of quantity, and the technical differences between the two definitions are subtle. 44 The main difference between a minimal q-domain and a minimal q*-domain is that a minimal q*domain does not have to be closed under subtraction. This does not seem absurd, at least if ordinary usage of language is our guide. It seems natural to say that “Cleopatra is as beautiful as Athena and Helen combined”, yet it is much less natural to say “Athena is as beautiful as Cleopatra is more beautiful than Helen” or “Athena’s beauty is equal to Cleopatra’s beauty minus Helen’s beauty”.

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There is one final move the neo-logicist might try in order to avoid the problems hovering around cut abstraction. Since the original Fregean idea of taking extensions as objects was salvaged via NewV by making use of the notion of “Big”-ness (see note 7), perhaps the neo-logicist can restrict cut abstraction to linear orders that are not “Big”, formulating something like the following Size-restricted Cut Abstraction Principle: SCA : (∀P)(∀Q)(∀H )(∀ λ, A has a model of size κ. So if A is unboundedly inflationary then for every cardinal λ there is a κ > λ such that A has no model of size κ. Say that A is unboundedly satisfiable if, for every cardinal λ, there is a κ > λ such that A has a model of size κ. Notice that if A is unboundedly satisfiable, then (assuming choice) we can turn any set into a model of A by adding more elements: for every set d, there is a model of A whose domain contains d. In the best cases, the “new” elements will be the new abstracts. Shapiro and Weir [1999] show that if the generalized continuum hypothesis is true, then New V is satisfiable at all regular cardinals, and so it is unboundedly satisfiable. However, it is independent of Zermelo–Fraenkel set theory (with choice) whether New V is actually unboundedly satisfiable. It might not have any uncountable models at all. So, again, how much inflation is too much? Cook [2001] argues that only strictly non-inflationary and boundedly inflationary abstraction principles should be acceptable to a neo-logicist. Let us examine the arguments, since they go to the heart of the goals of neo-logicism. 10 The usefulness of this notion turns on the axiom of choice, since in that case, for any distinct cardinals κ, κ ′ , either κ < κ ′ or κ ′ < κ.

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Concerning unbounded inflation, Cook writes: The neo-logicist is claiming that the abstraction principles implicitly define, or at least ground our use of mathematical concepts and theories. Definitions of the abstract objects of mathematics, even implicit ones, ought to determine a unique group of objects which necessarily fall under the definition. If this ‘defining’ abstraction principle is unboundedly inflationary, however, then the neo-logicist has failed in his task.

Assume, for example, that an abstraction principle A is unboundedly inflationary, and suppose that M is a model of both A and the background theory T . Let κ be the cardinality of the domain of M, and let γ be the smallest cardinal greater than κ such that A is γ -inflationary. Cook continues: . . . had there been γ objects in the universe, there would have, by [ A], been more than γ (and thus more than κ) abstracts. But then the original abstracts are not all of the objects whose identity conditions are given by [A]. This process can be repeated indefinitely (and transfinitely), so that we never have all the objects that fall under the purview of [ A]. In other words, if [A] is unboundedly inflationary then it fails to secure a definite collection of objects as the domain of its abstraction operator, but instead gives us different abstracts relative to how many objects exist.

In a note, Cook adds that “an adequate definition should determine a unique extension independently of the existence of any other objects”. Some abstraction principles do characterize a unique domain of objects, at least up to isomorphism. The present cut abstraction principle (CP), for example, yields all and only the real numbers, plus two extra abstracts. Another example would be the restriction of Hume’s principle to finite concepts (see Heck [1997]). This yields all and only (an isomorphic copy of) the natural numbers. Cook is correct that if an abstraction principle A is intended to characterize a unique structure (such as the natural numbers or the real numbers), then it should not be unboundedly inflationary. In this case, A should yield the required objects and no others. It should not inflate on any domain that is as large as or larger than the requisite structure. However, it is not true that every legitimate abstraction principle determines “a definite collection of objects” as the range of the defined operator. Some principles do yield “different abstracts relative to how many objects exist”. Consider Hume’s principle. Thanks to Frege’s theorem, it implies the existence of the natural numbers and the cardinality of the natural numbers (i.e., ℵ0 ). But what of other cardinalities? Since Hume’s principle has countable models, it does not, by itself, entail that the cardinality of the continuum exists. But Hume’s principle does entail that if there is a property that holds of continuummany objects, then the cardinality of the continuum exists. So, for example, Hume’s principle and (CP) together entail that the cardinality of the continuum exists. In general, which cardinal numbers exist depends on how many objects

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there are. I, at least, do not see this as a problem with Hume’s principle as an abstraction. Some acceptable abstractions are open-ended, in the sense that the abstracts they yield depend on the ontology of the background theory. Increasing the ontology might increase the abstracts. A different problem with unboundedly inflationary abstraction principles is that they may conflict with each other. Weir [2000] formulates a pair of “distraction” principles B, B ′ , such that B and B ′ are each unboundedly satisfiable, but are mutually inconsistent. Suppose that the background theory has a model of size κ0 . To extend this to satisfy B, we add κ1 > κ0 abstracts. But this new model does not satisfy B′ . To satisfy B′ , we add κ2 > κ1 abstracts. But once we add these abstracts to satisfy B ′ , we no longer satisfy B. To (re-)satisfy B, we have to add κ3 > κ2 more abstracts. But then we no longer satisfy B′ . In short, the conjunction B&B ′ is universally inflationary. Faced with such a pair of abstractions, the neo-logicist must find a principled way to choose among them. Or else she can play it safe and reject any unboundedly inflationary abstraction principle, and require that all acceptable abstractions be boundedly inflationary. Then, once we are satisfied that the universe is sufficiently large, the abstraction will be satisfied no matter how much larger we go on to recognize the universe to be. Let us turn to Cook’s treatment of universally inflationary principles—those that inflate on every cardinality. Of course, if an abstraction A is inconsistent, then it is unacceptable. Suppose that A is consistent, but universally inflationary. Let b be a set and κ = |b|. Since A is κ-inflationary, A cannot be satisfied on b. Since b is arbitrary, A has no models whose domain is a set (with a cardinality). As Cook puts it, A “will be satisfied (if satisfied at all) only by a structure that is at least the size of proper class”. This, he says, is problematic, since proper classes are “extremely badly behaved”. The idea is that if A can be satisfied only on a proper class, then it yields a proper class of abstracts (so to speak). Thus, the abstraction takes “us far from the epistemically innocent implicit definitions that the neo-logicists argue acceptable abstraction ought to provide”. In sum, Cook’s claim is that the “generation” of a proper class of abstracts is incompatible with the epistemic goals of neo-logicism. Notice that this judgement comes from the external perspective. Internally, the neo-logicist claims that we can come to know about the existence of some objects through deduction from principles in the neighborhood of implicit definitions or analytic truths. Externally, we use the set-theoretic meta-theory, accepted already by the established mathematician, to show that a certain abstraction principle yields more objects than there are members of any element of the set-theoretic hierarchy. Cook seems to hold that a proper class of abstracts is indeed “too many” objects to obtain this way. As noted above, for neo-logicism to have a chance, we have to temper the widely-held view that definitions, or principles much like definitions, have no ontological consequences. Cook’s claim, in effect, is that enough is enough. He presupposes that from the external

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perspective of an advocate of Zermelo–Fraenkel set theory, the abstracts must constitute (or be equinumerous with) a set. But the fact is that the objects of mathematics do not constitute a set, for well-known reasons. So Cook’s thesis entails that neo-logicism must fall short of its grand goal of providing an epistemic foundation for all of mathematics. A neo-logicist set theory and a neo-logicist theory of ordinals and cardinals is out of the question. Thus, the neo-logicist must rest content with an account of arithmetic, real and complex analysis, and perhaps a little more. The main (external) question that remains is just how big the neo-logicist’s ontology can be. What is the cardinality of the objects yielded by all acceptable abstractions together? Presumably, it will be ℵα , for some ordinal α. If the neo-logicist wants to avoid demanding revisions to established mathematics, she must provide some other epistemic foundation for those branches of mathematics— such as set theory, ordinal theory, and cardinal theory—whose ontology is not a set. For what it is worth, I believe that Cook’s view begs the question against the neo-logicist quest. So far as I know, no argument has been given that the objects yielded by an abstraction principle must constitute a definite, set-sized totality. The neo-logicist thesis is that an acceptable abstraction is akin to an implicit definition, providing an epistemic foundation of the theory of the objects it yields. There is no requirement that the objects be limited in any way, or that they constitute a definite totality. Perhaps further discussion of this should await either specific arguments concerning the limits of abstraction principles or the presentation of particular candidate principles that do yield a proper class of abstracts. We briefly revisit the issue at the end of the next section.

6.

The acceptability of cut abstraction

I now turn to the inflation and satisfiability of the abstraction principles presented here: (DIF), (QUOT), and (CP). Unlike Basic Law V and Hume’s principle, the quantifiers in all three of these principles are restricted. Since the right hand side of the difference principle (DIF) explicitly invokes addition on the natural numbers, (DIF) entails the existence of a difference-abstract for each pair of natural numbers, but that is all. It says nothing about “differences” of (pairs of) other objects, and in particular, it does not yield differenceabstracts for (pairs of) difference-abstracts. Similarly, the quotient principle (QUOT) yields a ratio for each pair of integers, but nothing else. And (CP) yields a cut for each property of rational numbers, but nothing else. Since there are only countably many integers, the difference principle is satisfiable on any domain that contains the natural numbers (using standard coding techniques if necessary). In a sense, (DIF) is universally satisfiable in that it is satisfiable on any domain over which it is defined. And so it does not inflate on any such domain. Since there are only countably many

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rational numbers, the quotient principle (QUOT) is satisfiable on any domain that contains the integers, i.e., on any domain on which it is defined. Since there are continuum-many distinct cuts, (CP) inflates on the rational numbers, but that is the end of its inflation. So (CP) is boundedly inflationary, in that it is satisfiable on any domain that is at least the size of the continuum and contains the rational numbers. Concerning inflation and satisfiability, the neo-logicist cannot do any better than this. If she hopes to recapture real analysis, she will need principles that yield continuum-many abstracts. The cut principle does that, and no more. Perhaps we should not be sanguine. The main reason why (CP) does not inflate beyond the real numbers is that it only defines cuts for properties of rational numbers. But we can mimic the development of (CP) for any linear order 11 “< ” defined on a set or class h. Let P be a property of items in h, and suppose that r ∈ h. Say that r is an upper bound of P, written P ≤ r , if for any s ∈ h, if Ps then either s < r or s = r . In other words, P ≤ r if r is greater than or equal to any object that P applies to (under the given linear order). Consider the following abstraction principle: (h,

THE WESTERN ONTARIO SERIES IN PHILOSOPHY OF SCIENCE A SERIES OF BOOKS IN PHILOSOPHY OF SCIENCE, METHODOLOGY, EPISTEMOLOGY, LOGIC, HISTORY OF SCIENCE, AND RELATED FIELDS

Managing Editor WILLIAM DEMOPOULOS

Department of Philosophy, University of Western Ontario, Canada Department of Logic and Philosophy of Science, University of Californina/Irvine Managing Editor 1980–1997 ROBERT E. BUTTS

Late, Department of Philosophy, University of Western Ontario, Canada

Editorial Board JOHN L. BELL,

University of Western Ontario

JEFFREY BUB,

University of Maryland

PETER CLARK,

St Andrews University

DAVID DEVIDI,

University of Waterloo

ROBERT DiSALLE,

University of Western Ontario

MICHAEL FRIEDMAN, MICHAEL HALLETT, WILLIAM HARPER,

McGill University

University of Western Ontario

CLIFFORD A. HOOKER, AUSONIO MARRAS,

Indiana University

University of Newcastle

University of Western Ontario

JÜRGEN MITTELSTRASS, JOHN M. NICHOLAS,

Universität Konstanz

University of Western Ontario

ITAMAR PITOWSKY,

Hebrew University

VOLUME 71

THE ARCHÉ PAPERS ON THE MATHEMATICS OF ABSTRACTION Edited by

ROY T. COOK

123

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978–1–4020–4264–5 (HB) ISBN 978–1–4020–4265–2 (e-book) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

Printed on acid-free paper

All Rights Reserved c 2007 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

For Alice, who has kindly tolerated the company of many abstractionists, and one in particular

Contents

Foreword

ix

Notes on the Contributors

xi

Acknowledgements

xiii

Introduction

xv

Part I The Philosophy and Mathematics of Hume’s Principle Is Hume’s Principle Analytic? G. Boolos

3

Is Hume’s Principle Analytic? C. Wright

17

Frege, Neo-Logicism and Applied Mathematics P. Clark

45

Finitude and Hume’s Principle R. G. Heck, Jr.

61

On Finite Hume F. MacBride

85

Could Nothing Matter? F. MacBride

95

On the Philosophical Interest of Frege Arithmetic W. Demopoulos

105

Part II The Logic of Abstraction “Neo-logicist” Logic is not Epistemically Innocent S. Shapiro & A. Weir

vii

119

viii

Contents

Aristotelian Logic, Axioms, and Abstraction R. T. Cook

147

Frege’s Unofficial Arithmetic A. Rayo

155

Part III Abstraction and the Continuum Reals by Abstraction R. Hale

175

The State of the Economy: Neo-logicism and Inflation R. T. Cook

197

Frege Meets Dedekind: A Neo-logicist Treatment of Real Analysis S. Shapiro

219

Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege’s Constraint C. Wright

253

Part IV Basic Law V and Set Theory NewV, ZF and Abstraction S. Shapiro & A. Weir

275

Well- and Non-well-founded Extensions I. Jané & G. Uzquiano

303

Abstraction & Set Theory Bob Hale

331

Prolegomenon to Any Future Neo-logicist Set Theory: Abstraction and Indefinite Extensibility S. Shapiro

353

Neo-Fregeanism: An Embarassment of Riches A. Weir

383

Iteration One More Time R. T. Cook

421

Foreword

In September 2000 the Arché Centre launched a five-year research project entitled the Logical and Metaphysical Foundations of Classical Mathematics. Its goal was to study the prospects, philosophical and technical, for abstractionist foundations for the classical mathematical theories of the natural, real and complex numbers and standard set theory. Funding was provided by the then Arts and Humanities Research Board (now the Arts and Humanities Research Council) for the appointment of full-time postdoctoral research fellows and PhD students to collaborate with more senior colleagues in the project, and at the same time the British Academy awarded the Centre additional resources to establish an International Network of scholars to be associated with the work. This was the beginning of the serial ‘Abstraction workshops’ of which the Centre had staged no less than eleven by December 2006. We gratefully acknowledge the generous support of the Academy and Council, sine qua non. The project seminars and Network meetings generated—and continue to generate—a large number of leading-edge research papers on all aspects of the project agenda. The present volume is the first of what we hope will be a number of anthologies of these researches. With two exceptions,—the contribution by the late George Boolos and the co-authored paper by Gabriel Uzquiano and Ignacio Jané,—the papers that Roy Cook has collected in the present volume are all authored by sometime members of the project team or of the British Academy Network. Their broad focus, as he explains, is on some of the more technical issues thrown up by the Abstractionist project, and it is anticipated that subsequent volumes may have a more purely metaphysical or epistemological emphasis. I would like to thank Roy Cook for all his hard work putting the volume together, and Bill Demopoulos for sponsoring its publication in the Western Ontario Series in Philosophy of Science. Special thanks go to the members of the core team and the Network not just for their direct contributions to the researches of the project but for their continuing affirmation, by their active participation, of the wider interest and importance of

ix

x

Foreword

the neo-Fregean enterprise in the landscape of contemporary philosophy of mathematics. CJGW St. Andrews 6/07 The Logical and Metaphysical Foundations of Classical Mathematics Sometime project team members: Crispin Wright, Peter Clark, Roy Cook, Philip Ebert, Bob Hale, Fraser MacBride, Paul McCallion, Darren McDonald, Nikolaj Jang Pedersen, Agustin Rayo, Marcus Rossberg, Andrea Sereni, Stewart Shapiro, Chiara Tabet, Robert Williams Auditor: Kit Fine British Academy International Network members: Alexander Bird, Robert Black, Robin Cameron, William Demopoulos, Richard Heck, Keith Hossack, Daniel Isaacson, John Mayberry, Michael Potter, Adam Rieger, Ian Rumfitt, Peter Simons, William Stirton, Peter Sullivan, Alan Weir

Notes on the Contributors

George Boolos was Professor of Philosophy at Massachusetts Institute of Technology, and the co-author of Computability and Logic (with Richard Jeffrey, Cambridge 2007) and the author of The Logic of Provability (Cambridge 1995). Peter J. Clark is Professor of the Philosophy of Science and Head of the School of Philosophical and Anthropological Studies in the University of St Andrews. He works primarily in the philosophy of physical sciences and mathematics and was editor of the British Journal for the Philosophy of Science 1999–2005. Roy Cook is Visiting Assistant Professor of Philosophy at Villanova University, and an associate fellow of Arché. He has published papers in the philosophy of language, logic, and mathematics, focusing primarily on semantic, soritical, and set-theoretic paradoxes, and Fregean and neo-Fregean philosophies of mathematics. William Demopoulos is a member of the Department of Philosophy of the University of Western Ontario and the Department of Logic and Philosophy of Science of the University of California, Irvine. He has published articles in diverse fields in the philosophy of the exact sciences, and on the development of analytic philosophy in the twentieth century. Bob Hale is Professor of Philosophy at the University of Sheffield, and an Associate Director of Arché. He works mainly on topics in the epistemology and metaphysics of mathematics and modality. His publications include Abstract Objects (Blackwell 1987), and, together with Crispin Wright, The Blackwell Companion to the Philosophy of Language (Blackwell 1997) and The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics (Oxford 2001). Richard Heck is Professor of Philosophy at Brown University and an associate fellow of Arché. He has published extensively on historical, conceptual, and technical issues emerging from Frege’s philosophy of mathematics. Philosophy of language and philosophy of logic are his other main areas of interest. He is now working on a book on philosophy of language and another on the development of Frege’s mature philosophy (co-authored with Robert May). xi

xii

Notes on the Contributors

Ignacio Jané is Professor of Philosophy in the Department of Logic and the History and Philosophy of Science of the University of Barcelona. His main interests are in the foundations of mathematics, philosophy of mathematics, and philosophy of logic. His recent papers include “Reflections on Skolem’s Relativity of Set-Theoretical Concepts” (Philosophia Mathematica, 2001), “Higher-Order Logic Reconsidered” (The Oxford Handbook of Philosophy of Mathematics and Logic, 2005), and “What is Tarski’s Common Concept of Consequence” (The Bulletin of Symbolic Logic, 2006). Fraser MacBride is a Reader in the School of Philosophy at Birkbeck College, London. He previously taught in the Department of Logic & Metaphysics at the University of St. Andrews and was a research fellow at University College, London. He has written several articles on the philosophy of mathematics, metaphysics, and the history of analytic philosophy, and is the editor of Identity & Modality (Oxford, 2006) and The Foundations of Mathematics and Logic (special issue of The Philosophical Quarterly, vol. 54, no. 214, 2004). Agustin Rayo is Associate Professor of Philosophy at MIT and an associate fellow of Arché. He works mainly in the philosophy of language and the philosophy of logic. Stewart Shapiro is the O’Donnell Professor of Philosophy at The Ohio State University and a Professorial Fellow in the Research Centre Arché at the University of St. Andrews. His publications include Foundations without foundationalism: a case for second-order logic (Oxford, 1991), Philosophy of mathematics: structure and ontology (Oxford, 1997), and Vagueness in context (Oxford, 2006). Gabriel Uzquiano is a Tutorial Fellow in Philosophy at Pembroke College and a CUF lecturer in Philosophy at the University of Oxford. He has published articles in metaphysics, philosophical logic, and the philosophy of mathematics. Alan Weir is Professor of Philosophy, University of Glasgow. He has published articles on logic and philosophy of mathematics in a number of journals including Mind, Philosophia Mathematica, Notre Dame Journal of Formal Logic, and Grazer Philosophische Studien and contributed chapters to a number of volumes devoted to these areas. Crispin Wright is Bishop Wardlaw Professor at the University of St Andrews, Global Distinguished Professor at New York University, and Director of the Research Centre, Arché. His writings in the philosophy of mathematics and logic include Wittgenstein on the Foundations of Mathematics (Harvard 1980); Frege’s Conception of Numbers as Objects (Aberdeen 1983); and, with Bob Hale, The Reason’s Proper Study (Oxford 2001). His most recent books, Rails to Infinity (Harvard 2001) and Saving the Differences (Harvard 2003), respectively collect his writings on central themes of Wittgenstein’s Philosophical Investigations and those further developing themes of his Truth and Objectivity.

Acknowledgements

The Editor wishes to thank the following: Oxford University Press, Kluwer Academic Publishers, Analysis, The British Journal for the Philosophy of Science, The Journal of Philosophical Logic, The Journal of Symbolic Logic, The Notre Dame Journal of Formal Logic, Philosophia Mathematica, and Philosophical Books for permission to reprint the papers that follow. Detailed individual citations are included with the papers. Crispin Wright, Director of Arché: Philosophical Research Centre for Logic, Language, Metaphysics, and Epistemology, for providing the foreword. William Demopoulos for proposing, and securing, the publication of this work in the Western Ontario Series in the Philosophy of Science. Charles Erkelens and Lucy Fleet at Springer for their guidance and encouragement. The administrative staff at the Arché Centre in St Andrews (Gill Gardner, Sylvia Rescigno, and Sharon Coull) and at Villanova University (Elvia Beach and Terry DiMartino) for constant assistance in the practical aspects of preparing this volume. Marguerite Nesling for converting a number of the papers from hardcopy to electronic format. The Arts and Humanities Research Board (now the Arts and Humanities Research Council) for support in the form of a postdoctoral research fellowship, which was held by the editor during the initial stages of this volume.

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Introduction

As noted in the preface, the papers included in this volume concentrate (much of the time, at least) on philosophical questions that are intimately tied up with the interesting, and sometimes puzzling, mathematical properties of abstraction principles. As a result, the introduction you are about to read will follow suit – concentrating on philosophical issues that have their roots in the mathematical characteristics of abstraction principles as well as philosophical problems whose solution would seem to require a somewhat technical approach. This focus should not be read as any sort of value judgment regarding the worth of technical versus non-technical work on abstraction principles, or within the philosophy of mathematics more generally. Instead, this focus on philosophical problems that are linked to mathematical aspects of abstraction reflects the fact that there has, in the last decade or two, been an immense amount of valuable work on Fregean-inspired abstraction principles and their philosophical importance. To attempt to cover all of this work, or even all such work that has some connection to the Arché Centre, would require several volumes the size of the present one. Hence the narrower focus. The volume is divided into four sections. The first contains papers of a general sort (which can also serve as a helpful introduction to the subject for those less familiar with the literature), although the majority of these nevertheless address distinctly technical issues, at least indirectly. The remaining three sections are devoted to three topics which have come under increasing study and scrutiny after the apparent success of the account of arithmetic based on abstraction. The second section (“The Logic Of Abstraction”) contains three papers that examine the role of logic (in particular, higher-order logic) within the abstractionist framework. The third section (“Abstractionism and the Continuum”) contains papers that attempt to extend the abstractionist account to the theory of the real numbers, as well as papers critically evaluating such attempts. The fourth and final section (“Basic Law V and Set Theory”) is devoted to attempts to reconstruct set theory (or something like it) within the abstractionist framework – usually by adopting some consistent variant of Frege’s original Basic Law V. Even with our focus narrowed to the more technical aspects of abstraction principles, however, the range of topics and problems addressed in the papers

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Introduction

to follow is vast. Therefore, in the interest of providing a reasonably concise and easily digestible introduction to the subject, there will be no attempt to discuss every issue that arises in the following chapters. Instead, the remainder of this introduction will proceed as follows: First, a brief sketch of the origin of interest in abstraction principles, i.e. Frege’s logicism and its failure, will be provided. Second, we will briefly examine the philosophical framework underlying the resurrection of interest in abstraction principles, a view often called Neo-Fregeanism, Neo-Logicism, or Abstractionism. Next we will look at brief sketches of the philosophical and technical work underlying the abstractionist reconstructions of arithmetic, analysis, and set theory. Then we shall survey three general types of problem that such reconstructions face, and conclude with a brief discussion of indefinite extensibility, a notion that has become of central importance in much of the work attempting to solve problems of the sort covered in the previous sections. Before moving on, a comment needs to be made regarding terminology. As already noted, the philosophical view (or views) under discussion in the remainder of this volume have been called, at various times and places, Neo-Fregeanism, Neo-Logicism, and Abstractionism. In the remainder of this essay the term “Abstractionism” will be used. The reasons for this are simple: “Neo-Logicism” is misleading, since it would seem to imply that the view is a new version of logicism, while, as we shall see, it is no such thing. “Neo-Fregeanism”, while perhaps not misleading in this way, is, in the editor’s opinion, better reserved for the more general view in the philosophy of language, clearly Fregean in nature, that (usually) underlies the philosophy of mathematics discussed in the chapters to follow (although even here there is further confusion, since this term is also used to refer to a collection of views associated with the work of certain Oxford philosophers such as Gareth Evans and John McDowell). One could presumably hold such Fregean views regarding language without believing in the fundamental importance of abstraction principles (and vice versa – see Agustin Rayo’s “Frege’s Unofficial Arithmetic” [2002], reprinted as chapter 10 below). When reading the essays collected in this volume, however, one should keep in mind that these terms are (unfortunately) used for the most part interchangeably.

1.

Abstraction and logicism An abstraction principle is any formula of the form: (∀α)(∀β)(@(α) = @(β) ↔ E(α, β)

where “@” denotes a unary function mapping entities of the type ranged over by α (usually concepts, objects, or sequences of such) to objects, and “E( , )” is an equivalence relation on those same entities. The general idea behind abstraction principles is that they allow us to introduce new

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terms (and thus presumably to gain privileged epistemological access to the corresponding objects) by defining the identity conditions for the referents of the novel terms using linguistic resources that are already understood (i.e. those resources occurring in the equivalence relation “E( , )” – in most cases “E( , )” is either a purely logical formula, or one composed of logic plus previously introduced abstraction operators). Thus, an abstraction principle is meant to act as an implicit definition of sorts, providing (so the story goes) an account of the meaning of novel terms of the form “@(α)”. Perhaps the first notable occurrence of an abstraction principle occurs in Frege’s attempt at a logicist reconstruction of arithmetic (and, in fact, all of mathematics). Frege notes, in the Grundlagen [1974] that the standard (higherorder) Peano axioms for arithmetic follow from the abstraction principle now known as Hume’s Principle (the explicit derivation of the Peano axioms from Hume’s Principle was “extrapolated” from Frege’s comments in Crispin Wright [1983], George Boolos [1990a], and Richard Heck [1993], and Boolos & Heck [1998], among others). Hume’s Principle (which Hume himself did not state, and whose name derives from a rather charitable reading of a comment in Hume’s Treatise) is the claim that, given two arbitrary concepts P and Q, the number of P’s is identical to the number of Q’s if and only if the P’s and the Q’s can be put in a one-to-one correspondence. More formally, we have: HP : (∀P)(∀Q)(NUM(P) = NUM(Q) ↔ (P ≈ Q)) where P ≈ Q abbreviates the second-order formula stating that P and Q are equinumerous. We can formulate rather natural definitions of arithmetical notions such ‘natural number’, ‘successor’, and ‘addition’ in terms of the numerical operator “NUM”. The fact that, given these definitions, the second-order Peano axioms for arithmetic follow from Hume’s Principle is quite notable as a mathematical theorem independent of any philosophical motivation, and the result has come to be called Frege’s Theorem (for a detailed examination of this result, and various streamlined versions of it, see Richard Heck’s “Finitude and Hume’s Principle” [1997a], reprinted as chapter 4 below). Frege, of course, wanted to reduce all of arithmetic to logic, thus defending (at least some of) mathematics from the Kantian charge of being synthetic a priori (logic, presumably, being analytic if anything is!). Thus, he rejected Hume’s Principle as the ultimate foundation for arithmetic, since it contains ineliminable occurrences of the cardinal number operator. (More famously, he also rejected Hume’s Principle, in its primitive form, since it was susceptible to the Caesar Problem. See section 7 of this introduction for further discussion of this issue.) As a result, Frege formulated a second abstraction principle, one that mapped each concept onto a unique object – its extension. Unlike Hume’s

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Principle, Frege’s Basic Law V: BLV : (∀P)(∀Q)(EXT(P) = EXT(Q) ↔ (∀x)(Px) ↔ Qx)) contains only logical vocabulary (as long as talk of extensions is logical). Basic Law V was, in essence, an early attempt at formulating (in an a priori, logical manner) the foundations of what we would now call set theory. Using Basic Law V, Frege was able to reconstruct Peano Arithmetic on what appeared to be a purely logical basis. The first step was to define numbers to be certain sorts of extensions – the number of a concept P is the extension of the concept “(extension of a) concept equinumerous with P”, or, more formally: NUM(P) =df EXT((∃Y)(x = EXT(Y) ∧ Y ≈ P)) Given this definition, Frege was able to prove Hume’s Principle (now a theorem of Frege’s logic, and not a primitive non-logical definition of cardinal number) and thus prove the second-order Peano axioms for arithmetic. So if Basic Law V was, as Frege hoped, a logical truth, then logicism (at least regarding arithmetic) would be demonstrated. Since not all of us are convinced logicists, something must have gone wrong – something discovered by Bertrand Russell. In a letter dated June 16, 1902, Russell wrote to Frege, humbly pointing out that the crucial axiom (Basic Law V) that provided the power needed to reconstruct arithmetic within logic also seemed to allow for the derivation of a contradiction. Although Russell’s actual presentation of the paradox that now bears his name is a bit muddled in the original missive, the derivation of a contradiction from Basic Law V is well known, and need not be rehashed here. Frege attempted to fix the problem, but failed to find a convincing replacement for Basic Law V. Russell, meanwhile, along with Alfred North Whitehead, attempted his own reconstruction of mathematics from basic, a priori principles in the monumental Principia Mathematica [1910–13]. Although the Principia was (likely) consistent, in the long run it turned out to be no more convincing than Frege’s Grundgesetze.

2.

Abstractionism

After Frege’s failed attempt at utilizing abstraction principles in a logicist framework, this sort of principle lay unstudied for three-quarters of a century. Interest in abstraction of this sort was rekindled, however, by the publication of Crispin Wright’s Frege’s Conception of Numbers as Objects [1983]. Wright noted (perhaps among others, see Parsons [1965] and Hodes [1984]), in essence, that Frege’s project consisted of four basic steps: (1) Recognize Basic Law V as an axiom of logic. (2) Formulate suitable definitions of numerical notions in terms of the extensions provided by Basic Law V

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(3) Derive Hume’s Principle (4) Derive arithmetic from Hume’s Principle (i.e. Frege’s Theorem)

Wright revived interest in Frege’s project, founding a new project that has come to be called Neo-Logicism, Neo-Fregeanism, and Abstractionism, variously, replacing the above blueprint with the following alternate plan: (1) Lay down Hume’s Principle as an implicit definition of cardinal number (2) Derive arithmetic from Hume’s Principle (i.e. Frege’s Theorem)

Of course, as was already noted, such a view (misleading nomenclature such as “Neo-Logicism” notwithstanding) does not deserve the title ‘logicism’, at least not in the traditional sense of the word as used by Frege, his fans, and his critics. Hume’s Principle, with its primitive and ineliminable occurrences of arithmetical terms (i.e. “NUM”), just does not have the character of a logical law or theorem (a point made strenuously and convincingly by George Boolos in “Is Hume’s Principle Analytic?”, the essay that opens this volume). Thus, abstractionists have had to look elsewhere for their defense of Hume’s Principle as something suitably basic as to provide the foundations of arithmetic. The answer, according to abstractionists, is to note that what is important about logicism is not so much the reduction of mathematics to logic, but rather the fact that this reduction (had it been successful) would have gone a long ways towards providing an account of certain aspects of mathematical knowledge that, ideally, we would like to be able to explain. In particular, the true advantages of logicism were that it purported to explain the a priori character of mathematical knowledge (assuming that the a priori character of purely logical knowledge is unproblematic) and it purported to explain the analyticity of mathematical truths (at least, this is important for those nonQuineans that retain a fondness for the analytic/synthetic distinction in the first place). The solution, then, is to retain these goals, while widening the scope of our means for achieving these goals to something more than pure logic. Along these lines, Wright and those that follow him deny that Hume’s Principle is a logical truth. Instead, Hume’s Principle, so it is argued, is (or is something like) an implicit definition of the “NUM” operator – one that explains the meaning of statements of identity of cardinal numbers. Since Hume’s Principle is a definition, we can come to know its consequences a priori in the same manner (or, at least, in a suitably similar manner) in which we obtain a priori knowledge of the consequences of more pedestrian definitions. Frege’s Theorem insures that all of second-order Peano arithmetic follows from Hume’s Principle plus standard second-order logic, so (since presumably second-order logic is a priori knowable and second-order consequence preserves a priori knowledge) it follows that we can, using the abstractionist recipe, obtain a priori knowledge of all of second-order arithmetic. (The question of analyticity is strictly speaking separate from that of aprioricity, and

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has been less of a focus for the abstractionists than it was for Frege himself, although Bob Hale has rekindled interest in this issue.) There are, unsurprisingly, deep questions regarding how Hume’s Principle and Frege’s Theorem accomplish this epistemological feat. In particular, there are deep worries regarding the connection between our reconstruction of arithmetic within the abstractionist framework and actual arithmetic practice: How do we know that the knowledge gained from Frege’s Theorem is, in fact, knowledge about the ordinary natural numbers (and not some isomorphic surrogate)? And how do we determine whether our (supposed) a priori knowledge of the former allows for an explanation of the a priori status of everyday mathematics? William Demopoulos’ “On the Philosophical Interest of Frege Arithmetic” [2003] (reprinted below as chapter 7) develops a sustained examination and critique of this aspect of the project (although the reader is encouraged to consult Fraser MacBride’s two contributions to this volume as well). Of course, even if the view in question is not, really, a version of logicism, the above sketch makes it clear that logic plays a crucial role in the abstractionist account of mathematical truth and mathematical knowledge. Defending the claim that second-order logic preserves the relevant epistemological properties is one outstanding lacuna in the abstractionist literature, although it is not one they are unaware of. The most sustained discussion of the issues is to be found in Stewart Shapiro and Alan Weir’s “Neo-Logicist Logic Is Not Epistemically Innocent” [2000], reprinted below as chapter 8. Setting the role of logic aside, however, there is much of interest to be said regarding: (a) the notion of implicit definition required for such an abstractionist project, (b) the more general abstractionist accounts of meaning and reference which might allow for such implicit definitions to succeed, and (c) the metaphysical account of abstract objects that would allow for our epistemological access to them to proceed via such stipulations. Although all of these and more are touched on (and often discussed in some depth) in the essays that follow, they are not the primary focus of this volume or the papers included in it. Instead, we are here interested in those philosophical problems that stem from mathematical issues arising within the abstractionist project. Thus, we shall move on to examine those aspects of abstractionism that are of a more technical nature (the reader interested in more straightforwardly philosophical aspects of the abstractionist project, such as the topics mentioned at the top of this paragraph, can do no better than to consult Fraser MacBride’s “Speaking with Shadows: A Study of Neo-logicism” [2003], although the first two essays in this volume, both titled “Is Hume’s Principle Analytic?”, by George Boolos [1997] and Crispin Wright [1999], also provide much useful philosophical background material). Arithmetic, it would seem, is, in one sense, the big success story of abstractionism, since the technical results, at least, seem to be for the most part settled – all that remains is sorting out the philosophical problems and issues

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that result from this abstractionist reconstruction. As we shift our attention to abstractionist accounts of other mathematical theories, however, we shall see that things are not always so successful even within the purely technical aspects of the project. Thus, our next task is to quickly survey the extension of this project to set theory and real analysis.

3.

Abstractionist real numbers

One of the two obvious test cases for extending any philosophy of mathematics past an initial account of the natural numbers is to attempt to reconstruct the continuum (the other test case is to provide an adequate account of sets or something like them, the subject of the next section). Abstractionism is no exception here, and it did not take long for both believers and critics to wonder what shape an abstractionist account of real analysis might take. Although various accounts differ in the details (and this difference tends to depend on varying attitudes towards Frege’s Constraint, see below), Bob Hale’s initial reconstruction (as found in his “Reals By Abstraction” [2000], reprinted as chapter 11 below) and those that follow are similar to the following, at least from a mathematical perspective. The first step in an abstractionist account of the real numbers is to note that we are already provided with the natural numbers via Hume’s Principle. We can obtain the integers from these by adding an additional abstraction principle to our theory – something like the following Difference Abstraction Principle (the universal quantifiers here are restricted to natural numbers and the arithmetical operators on the right-hand side of the biconditional are the standard operations on the natural numbers): DAP : (∀x)(∀y)(∀z)(∀w)(DIFF(x, y) = DIFF(z, w) ↔ x + w = y + z) This principle provides us with an object corresponding to the difference between two natural numbers – in other words, DAP provides us with (a priori access to) the integers. With the integers in hand, we can obtain the rational numbers from these by adding another abstraction principle to our theory – the following Quotient Abstraction Principle will do the trick (here, the initial universal quantifiers are restricted to integers, i.e. objects that are in the range of the DIFF operator, and the arithmetic operators on the right-hand side of the biconditional are the standard operations on the integers – these are definable in terms of DIFF and second-order logic): QAP : (∀x)(∀y)(∀z)(∀w)(QUO(x, y) = QUO(z, w) ↔ ((y = 0 ∧ w = 0) ∨ (y = 0 ∧ w = 0 ∧ x × w = y × z))) Note that the Quotient Abstraction Principle provides us, not only with the rational numbers, but also with an extra, ‘bad’ object: QUO(a, 0) for any integer a. This object results from the fact that we assume that our abstraction

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Introduction

operators are total functions, and thus certain unintended instances (such as division by zero in the present instance) nevertheless result in abstracts. The presence and role of ‘bad’ objects will be discussed in section 6 of this introduction. Now that we have the integers, we can obtain the reals by applying an abstraction principle that simulates Dedekind-style cuts on the rationals, such as the following Cut Abstraction Principle (here the universal quantifiers are restricted to non-empty bounded concepts holding only of non-‘bad’ rationals, i.e. objects in the range of the QUO operator other than the ‘bad’ object): CAP : (∀P)(∀Q)(REAL(P) = REAL(Q) ↔ (∀x)((∀y)(P(y) → y < x) ↔ (∀y)(Q(y) → y < x))) It is possible (although non-trivial) to prove that the objects provided by CAP are a complete ordered field, i.e. that they are isomorphic to the standard classical continuum (see Shapiro’s “Frege Meets Dedekind: A Neologicist Treatment of Real Analysis” [2000], reprinted as chapter 14 below, for details). Thus, the abstractionist position can account for not only the natural numbers, but the classical theory of the real numbers as well (from a technical perspective, at least). As a technical note, there seems to be no reason why, at this last step, we could not have applied, instead of CAP which encodes Dedekind’s notion of cut within the abstractionist framework, an abstraction principle that, when applied to sequences of rationals, provides the real numbers along the lines of Cauchy’s methodology. There is no formal reason why we could not formulate such a principle (e.g. let the abstraction principle in question map functions from the naturals to the rationals onto objects). The possibility of such alternate constructions raises a host of philosophical issues, however. Not least among them are the following questions: If it turns out that both CAP and an appropriate Cauchy-sequence principle are legitimate abstraction principles, then how are we to determine whether they provide us with access to the same objects? If not, then which one delivers the genuine real numbers (as opposed to merely an isomorphic copy)? Such questions are intimately tied up both with Frege’s Constraint and with the Caesar Problem, both of which will be discussed below.

4.

Abstractionist sets

The second natural extension of a foundational account of mathematics is to produce some account of set theory (or, at the very least, to provide some other theory that can do the work for which we normally invoke the theory of sets). The most notable attempt to provide such an account within the abstractionist framework is due to George Boolos, one of the most outspoken critics of the abstractionist view itself.

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Introduction

In “Iteration Again” [1989], Boolos compared and contrasted the iterative and limitation-of-size conceptions of sets. The former proposes to solve the problem posed by Russell’s paradox by claiming that sets must be formed in an infinitary step-by-step process, while the latter avoids paradoxes by claiming that only collections that are (in some sense) not too ‘big’ determine sets. One version of the limitation-of-size conception (the one Boolos used) can be formulated by defining ‘X is too big’ as ‘there is a bijection between X and the entire domain’. Boolos formulated an abstractionist version of the limitation-of-size conception of set along these lines. Letting “Big(P)” abbreviate the second-order formula asserting that there is an onto function from P to the entire domain, Boolos’ abstraction principle for extensions, called New V, is: New V : (∀P)(∀Q)(EXT(P) = EXT(Q) ↔ ((∀x)(Px ↔ Qx) ∨ (Big(P) ∧ Big(Q)))) New V provides a distinct object (an extension, or, more loosely, set) for each collection of objects provided that collection is smaller than the entire domain – concepts that hold of as many objects as there are in the domain, however, all receive the same abstract, the ‘Bad’ object (again, see below for discussion of ‘bad’ objects). Given New V, we can define a set to be the extension of a small concept: Set(x) =df (∃P)(x = EXT(P) ∧ ¬Big(P)) One object is the member of another object if and only if the second object is the extension of a concept which holds of the first object, or, in symbols: x ∈ y =df (∃P)(y = EXT(P) ∧ P(x)) (Note that ‘Bad’ objects can have, and be, members.) Given these definitions, New V entails many of the standard set theoretic axioms – extensionality, empty set, pairing, separation, replacement, and choice all follow (the union axiom does not follow on the above definitions, since the union of the singleton of the ‘bad’ object is not a set. Slight reformulations of this axiom do follow, however – for details see the chapters in section IV of this volume). In addition, the axiom of foundation holds if restricted to the pure sets (i.e. those sets that can be ‘built up’ from the empty set – see Gabriel Uzquiano and Ignacio Jané’s “Well- and Non-Well-Founded Extensions” [2004], reprinted as chapter 16 of this volume, for an in-depth examination of non-well-founded sets within the abstractionist framework). Thus, the only axioms that fail to follow, in some sense or another, are the powerset axiom and the axiom of infinity. It is easy to see why the axiom of infinity fails – if we take as our domain the hereditarily finite sets built from a single urelement (to serve as the ‘bad’ object), then the resulting model satisfies New V (since all ‘small’, i.e. finite, concepts receive extensions – the corresponding sets – while all ‘big’, i.e. infinite, concepts can be mapped onto our single urelement). In other words,

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although NewV entails that there must be infinitely many objects, it does not entail that there need be any non-‘Bad’ concept that holds of infinitely many objects. The proof that powerset fails is non-trivial, however – readers are encouraged to consult chapter 15 of this volume, “New V, ZF, and Abstraction” [1999] by Stewart Shapiro and Alan Weir, for the technical details. Given that New V does not allow us to reconstruct all of standard Zermelo Fraenkel set theory, work has been done exploring other abstractionist routes to set theory. Among these are Roy T. Cook’s “Iteration One More Time” [2004], which formulates an abstractionist version of the iterative conception of set based on an abstraction principle called Newer V. Newer V entails the extensionality, empty set, pairing, separation, powerset, and choice axioms, but fails to imply both the axiom of infinity and the replacement axiom. Other approaches include Bob Hale’s “Abstraction and Set Theory” [2000], which formulates an alternative version of the limitation of size conception, and Stewart Shapiro’s “Prolegomenon to Any Future Neo-Logicist Set Theory: Abstraction and Indefinite Extensibility” [2003], which examines the general conditions under which a restricted version of Basic Law V (such as New V) will entail various set-theoretic principles (these three papers are reprinted below as chapters 20, 17, and 18 respectively). Although one could debate how much we should worry about abstraction principles failing to imply powerset or replacement, there is no ignoring the fact that the failure of natural abstractionist accounts of set theory to provide a proof of the axiom of infinity is just that, a failure. Presumably, a successful defense of abstractionism will require a development of a set theory (or surrogate for it) that is stronger than any of the existing proposals, since any set theory which fails to guarantee the existence of any infinite sets is unlikely to be adequate to our needs. To be fair, there are abstraction principles that imply all the axioms of second-order Zermelo–Fraenkel set theory (Alan Weir considers such principles in his “Neo-Fregeanism: An Embarrassment of Riches” [2004], reprinted here as chapter 19). Unlike New V or even Newer V, however, these principles do not seem to codify plausible ‘definitions’ of the notion of set or collection – the sort of conception that could underlie successful a priori introduction of the notion of set or extension into our discourse. Instead, these principles seem tailor made to provide all of the set theoretic axioms, and it is unlikely that anyone could have conceived of them without extensive prior knowledge of advanced set theoretic methods (e.g. formulation of typical ‘distractions’ requires an understanding of notions such as strong inaccessible cardinal). Thus, unlike the case of arithmetic and real analysis, there seems to be much more work needed of a purely technical nature before the abstractionist can make any claim to have explicated the aprioricity and analyticity of set theory. The main technical problem is to find an appropriate abstraction principle for extensions that is satisfied only on uncountable domains of the right sort

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(presumably, something like an inaccessible rank). As of the time of writing this introduction, there does not seem to be any plausible abstraction principle that will do the job, although there is interesting work leading in this direction (e.g. see Shapiro’s “Prolegomenon to Any Future Neo-Logicist Set Theory. . . ” and the later sections of Cook’s “Iteration One More Time”).

5.

The first problem: too many abstraction principles

The first general problem plaguing the abstractionist project is that there seem to be too many abstraction principles. What is required, and what we, at present, fail to have, is some general criteria for distinguishing between acceptable and unacceptable abstraction principles. Clearly, Basic Law V, being inconsistent, is on the unacceptable side of the field, while Hume’s Principle, the pride and joy of abstractionism, is (it is hoped) on the acceptable side (if not, then presumably some suitably modification of it is, such as Finite Hume, discussed in the next section). The problem, however, is that mere consistency is not enough for acceptability, and as a result, we need some further guide to distinguishing the good from the bad. The initial formulation of this problem is (as is almost always the case in these debates) due to George Boolos (in Boolos [1990a]), who pointed out that there are abstraction principles that are consistent, but which are nevertheless incompatible with Hume’s Principle (or, in fact, with any abstraction principle guaranteeing the existence of infinitely many objects). Assuming that Hume’s Principle is acceptable if anything is, it follows that inconsistency, while sufficient for rejecting an abstraction principle as unacceptable, is not necessary. Crispin Wright [1997] provided perhaps the most well-known example of such an abstraction principle: his aptly-named Nuisance Principle (here FSD(P,Q) abbreviates the second-order formula asserting that the symmetric difference of P and Q, that is, the collection of objects that are either P-andnot-Q or are Q-and-not-P, is finite): NP : (∀P)(∀Q)[NUI(P) = NUI(Q) ↔ FSD(P, Q)] The Nuisance Principle can be satisfied on domains of any finite cardinality (in which case all objects receive the same nuisance), but can be satisfied on no infinite domain. Thus, the Nuisance Principle, although consistent, is as unacceptable an abstraction principle as is Basic Law V. The reason for the unacceptability is different, however. At first glance, the Nuisance Principle appears to derive its unacceptability, not solely in terms of its own formal properties, but rather in terms of its interaction with other principles (such as Hume’s Principle). The existence both of inconsistent abstraction principles, and of pairs of individually consistent but incompatible abstraction principles, has given rise to a collection of problems that have been labeled The Bad Company Objection. Chief amongst the concerns falling under this heading are:

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(1) The Existential Challenge: Given the existence of problematic principles of the same general form as Hume’s Principle (such as Basic Law V and the Nuisance Principle), what reason do we have for thinking that there are any good abstraction principles (including Hume) which have the privileged status the abstractionist claims for them? (2) The Epistemological Challenge: Even if one is convinced that there are good abstraction principles that can play a foundational role such as the one envisioned for Hume’s Principle, in general how do we tell the good principles from the bad?

The epistemological challenge, although clearly important, will be sidestepped here, since we are interested in those problems that are intimately connected to the mathematics of abstractionism. The existential challenge, however, is, or at least can be easily approached as, a logical/mathematical issue – i.e. what proof- or model-theoretic features will guarantee that an abstraction principle is acceptable? One common response to this version of the Bad Company Objection (one first put forward by Wright’s “Is Hume’s Principle Analytic?” [1999] and finessed by Shapiro and Weir in “New V, ZF, and Abstraction” [1999], both reprinted below) is to require that an abstraction principle be conservative in a certain sense. The intuitive philosophical idea is this: An acceptable abstraction principle is meant to be a definition of the abstracts that it introduces, but it is also meant to be no more than this. As a result, the principle in question should have no substantial consequences for those objects in the domain that are not abstracts. Put simply, Hume’s Principle might entail all sorts of interesting claims about numbers, and even interesting claims regarding the numbers corresponding to certain collections of cats, but Hume’s Principle should not imply any substantial non-numerical claim about cats (a numerical claim would be one containing at least one occurrence of the NUM operator). Hume’s Principle can be proven to be conservative, as we would expect. On the other hand, the Nuisance Principle turns out to be non-conservative, as we would hope (since it entails, for example, that there must be only finitely many cats). So at first glance the conservativeness constraint would seem to be doing the job that it was designed to do. There are problems, of course. For one, New V, the most promising abstractionist reconstruction of set theory so far (even if far from fully satisfactory), is non-conservative. The technical details can be found in Shapiro and Weir’s [1999] paper, but the informal idea is easy to grasp. Within the language of New V we can define the ordinals in the usual way – ordinals are just transitive pure sets, well-ordered by membership. By the familiar reasoning of the Burali-Forti paradox, we can conclude as usual that there is no set of all ordinals. Within the context of New V, however, this means that the collection of ordinals is ‘Big’ – i.e. there is an onto function from the ordinals to the entire universe. But, since the ordinals are well-ordered by membership, this

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imposes a well-ordering on the entire universe. So New V is not conservative, since it implies that the universe can be well-ordered (and, within second-order logic, we can express this claim using no set-theoretic terminology). It is worth noting that Cook’s [2004] iterative variant of abstractionist set theory fares no better on this score. The problems with the conservativeness requirement do not stop with the fact that it would seem to rule out principles (such as New V) that we might otherwise have wished to be acceptable, In addition, it turns out, as Alan Weir shows in his “Neo-Fregeanism: An Embarrassment of Riches” [2004], that there are consistent yet incompatible abstraction principles that pass the conservativeness constraint. Weir calls such principles distractions, and he shows, further, that if we try to strengthen the conservativeness constraint in various natural ways in order to avoid such pairs of distractions, analogous problems arise in the meta-theory. There are a number of other criteria that have been proposed for narrowing down the list of potentially good abstraction principles. One suggestion is that the equivalence relation on the right-hand side of the biconditional accurately reflect the mathematical content mastered when we actually first learn the mathematical theory in question. In other words, the criterion for identity of the mathematical objects in question, provided by the abstraction principle, should clearly reflect the criterion by which we actually learned to identify and distinguish the objects in question. (The various abstractionist set theories are of particular relevance here, since there does not seem to be one single notion of set underlying our mathematical practice, but a number of competing notions, which are reflected in the competing reconstructions such as New V and Newer V.) Critics of this approach, however, have suggested that such requirements confuse something like the order of discovery with the order of explanation (e.g. see MacBride’s “On Finite Hume” [2002] and “Could Nothing Matter” [2003], reprinted as chapters 5 and 6 below). According to this line of thought, abstraction principles are intended to provide a story about how we might come to know mathematical truths a priori, but there is no reason to think that the actual route that we took in first coming to know these truths is necessarily anything like the privileged route provided by the abstraction (since the initial knowledge could even have been a posteriori!).

6.

The second problem: Too many objects

One of the main (supposed) advantages of abstractionism is that abstraction principles imply the existence of more objects than we would expect from logic and definitions alone. Some (including, of course, Boolos) have objected to this, on the grounds that logic (or analytic statements, or a priori knowledge more generally) should not imply the existence of all (or most) of the objects studied by working mathematicians:

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Introduction . . . It was a central tenet of logical positivism that the truths of mathematics were analytic. Positivism was dead by 1960 and the more traditional view, that analytic truths cannot entail the existence either of particular objects or of too many objects, has held sway ever since. (Boolos [1997], pp. 249–250)

Nevertheless, abstractionism is hopeless without the assumption that at least some existential claims are analytic, or a priori knowable, or something similar – the position in question is (on one reading) nothing more than a detailed philosophical account of how such is possible. So for our purposes here we will ignore such general worries regarding ontological excess. This ontological success seems to come at a price, however, since the very abstraction principles (or, sometimes, natural generalizations of them) that provide us with the ontology of standard mathematics have a tendency to imply the existence of more objects than are strictly needed for the reconstruction of the mathematical theory in question. Unfortunately, these additional objects are often unwanted or inconvenient. The first such unwanted object is ‘anti-zero’. Hume’s Principle implies that, in addition to the countable infinity of finite numbers, at least one other number exists, namely the number of the universal concept denoted by “x = x” – this ‘number’ is anti-zero. The standard account of cardinal numbers as developed in ZFC implies that there is no largest cardinal number, however. Thus, as was first pointed out by George Boolos, the theory of cardinals derived from Hume’s Principle seems to contradict the spirit, if not the letter, of the standard theory of cardinality as derived in Zermelo-Fraenkel set theory, where there can be no cardinal number of all objects. It is important to note that there is no formal contradiction here. One can easily construct a model which satisfies both Hume’s Principle and the (second-order) axioms of Zermelo–Fraenkel set theory – just take any settheoretic model of second-order ZFC, and interpret the numerical operator in Hume’s Principle as mapping each concept onto the appropriate ZFC cardinal, if the concept’s extension is set-sized, and mapping all other concepts onto some other object. Boolos’ point, rather, must be that there is no model of Hume’s Principle plus second-order ZFC where the ZFC cardinal numbers are exactly the cardinal numbers as defined by Hume’s Principle (under the same ordering). There are a number of obvious moves one could make here, although each has its problems. Among them are: (a) We might deny that ZFC provides an account of all the cardinal numbers, arguing instead that through this means we only get a model of the cardinal numbers corresponding to setsized concepts (while Hume’s Principle provides us with a theory of all the cardinal numbers). While attractive, this option seems to challenge the idea that set theory (however it is formalized) can play the foundational role traditionally ascribed to it (a role that abstractionists presumably would prefer it to retain, hence the interest in set-theoretic abstraction principles such as New V).

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(b) We might adopt a (positive) free logic, so that some instances of the numerical operator fail to designate objects (such as the instance that purports to refer to anti-zero). This strategy, however, seems open to two problems. First, given the abstractionist’s rather lenient criteria for when a term refers (that it occur in a true statement of the appropriate sort), this response seems somewhat ad hoc. Second, if abstractionists make it part of their official view that some numerical terms can fail to refer, then this allows the critic of abstraction to ask why it is not possible that all numerical terms fail to refer (or to argue that since some numerical terms fail to refer, then it seems unlikely that we can know a priori that other numerical terms do refer). For a detailed discussion of free logic within the abstractionist context, see Shapiro and Weir’s “NeoLogicist Logic Is Not Epistemically Innocent” [2000], reprinted below. A final strategy, however, is to replace Hume’s Principle with some suitably modified version, such as Finite Hume (here “Inf(P)” abbreviates the secondorder claim that there are infinitely many P’s): FHP : HP : (∀P)(∀Q)(NUM(P) = NUM(Q) ↔ ((Px ≈ Q) ∨ (Inf(P) ∧ Inf(Q)))) Finite Hume’s Principle provides a cardinal number for each finite concept, but maps any concept with an infinite number of instances onto the same, ‘Bad’ object (assuming that the logic is not free). Frege’s Theorem still holds for Finite Hume’s Principle (since the finite cardinals, i.e. the natural numbers, behave just as they do in the case of Hume’s Principle). There is no largest cardinal number, however, since the ‘Bad’ object that is the value “NUM” assigns to any infinite concept cannot be interpreted coherently as a number at all (to see this, it is enough to note that Finite Hume maps concepts of differing cardinalities onto the ‘Bad’ object in any uncountable model). So Finite Hume does not entail the existence of any strange cardinal numbers, such as anti-zero. It does, however, provide us with a generic ‘Bad’ object, just as the Quotient Principle QAP and New V were seen to do earlier. While such an additional object does not, like anti-zero, seem to violate any intuitions regarding the order-type of the cardinal numbers, they do bring with them different, yet equally serious problems of their own. In examining such ‘Bad’ objects, however, let us return our attention to the ‘Bad’ object provided by New V and similar extensions-forming principles, as it is this object that has attracted the most attention in the literature. Now, all existent abstraction principles that purport to provide us with something like extensions or sets also provide at least one unwanted, ‘Bad’ object – in fact, if the extensions-forming operator EXT is a total function then they must, since the claim that each concept receives a unique extension is contradictory. Typically (as is the case with New V) all of the concepts which are too ‘badly behaved’ to determine sets get mapped on to a single object, the ‘Bad’ extension, and in the case of New V, the ‘Bad’ object is the extension of any concept that is equinumerous to the entire domain.

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Now, unlike anti-zero, where the jury is still out regarding whether or not it is a genuine number, the ‘Bad’ extension is clearly not a genuine extension or set at all. It is merely an artifact of the particular abstractionist means for obtaining the things that we do want, i.e. the other extensions. Since treating the ‘Bad’ extension as some novel, until now unrecognized, yet real set seems implausible, the other option would be to treat it exactly as just described – as an artifact of the fact that we are treating our abstraction operators as defining total functions. The first problem is, of course, the seeming unavoidability of such ‘Bad’ objects in the first place. Why should our account of set theory (or rational numbers, or perhaps natural or cardinal numbers) seem to require the existence of an additional, and unwanted, object in the first place? Shouldn’t it be possible to provide a foundational account of any mathematical theory that entails the existence of all, and crucially, only, the objects required by that theory? This sort of question, while important and intuitively quite troubling, is also rather loosely formulated. There are other problems associated with the existence of ‘Bad’ objects (in particular, the ‘Bad’ extension) that are a good bit more precise, however. If the ‘Bad’ extension is merely an artifact of our theory, and not a ‘real’ extension in some sense, then presumably any proof of a crucial set-theoretic result based on New V should not depend on the existence of the ‘Bad’ extension. In other words, any set theoretic axiom that turns out to be true (given New V) should be true solely in virtue of the settheoretic ‘behavior’ of the genuine extensions (and thus should not depend for its truth on the existence or ‘behavior’ of the ‘Bad’ extension). This requirement seems reasonable. Unfortunately, at least for the present attempts at reconstructing set theory within an abstractionist framework, it seems like a requirement that cannot be met. The problem is that, if we are not allowed to make use of the existence of the ‘Bad’ extension, we lose the proof that there are infinitely many sets. Boolos provides the following proof that New V entails the existence of at least two objects (this is the initial part of his proof that New V entails the existence of infinitely many objects): Let Ø be the concept [x : x = x] . . . since there is at least one object (e.g. EXT (x = x) or EXT (x = x)), Ø is small, Ø = V, and EXT(x = x) = EXT(x = x). ([1989], p. 90, notation modified to fit that used here)

Notice that the proof makes explicit reference to the ‘Bad’ extension (i.e. EXT (x = x)). This is not accidental – the result depends on the existence of the ‘Bad’ extension in order to guarantee that finite concepts are not ‘Big’. There does not exist any proof from New V to the pairing axiom that does not, in some way, make use of the ‘Bad’ extension. To see why, consider New V interpreted in a free logic that allows EXT to fail to take on a value when applied to ‘Big’ concepts. In such a logic, there will be

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a one-element model of New V – just let the domain contain (for example) the empty set. Then there are two concepts – the empty one, and the one holding of the empty set. Since the latter is ‘Big’ (it is, in fact, the entire universe), it need not receive an extension, so we can just map the empty concept onto the empty set, and we have our model. Thus, there seems to be a real problem in the way that abstraction principles for extensions such as New V behave. On the one hand, they seem to imply the existence of unwanted objects – in particular, the ‘Bad’ object. On the other hand, this object seems necessary in order to ‘bootstrap’ our way up to a proof that there are infinitely many sets. No satisfactory solution to this dilemma has been presented as of yet. Although the reader might be forgiven for thinking that this is, already, more than enough problems, a look at abstractionist reconstructions of real analysis is in order. As noted above, the abstraction principle generating quotients of integers also generated a ‘Bad’ object, but this seems less worrisome than anti-zero or the ‘Bad’ extension, since the traditional theory of the rationals was already plagued with a similar problem (i.e. the ill-definedness of 1/0 . More troubling, however, is the use of cut abstraction in the final step of the construction. Now, applying this particular abstraction principle to the domain does not, at first glance, seem to present any problems – we obtain, in fact, exactly the real numbers (or something isomorphic to them) and nothing else. The problem possibly arises, however, when we ask the following question: If we can use abstraction to take cuts on the rationals, as we did to obtain the reals, then is it permissible to take, as objects, the cuts on any linear order, by applying an appropriate abstraction principle? If the answer is “No”, the we seem faced with another particularly difficult instance of the Bad Company Objection – how are we to determine when we can, and when we cannot, apply an abstraction principle to a linear order to take cuts as objects? If the answer is “Yes”, however, then we are besieged by another worry: Generalizing such cut abstraction to any linear ordering whatsoever generates a large ontology (in the worse case, proper class sized). This is, at best, extremely surprising in a view that emphasizes its epistemic conservativeness. Additionally, some of the more powerful versions of generalized cut abstraction are incompatible with other, somewhat attractive abstraction principles, such as New V and variants of it (for a discussion of generalized versions of cut abstraction see Cook’s “The State of the Economy: Neo-Logicism and Inflation” [2002], reprinted as chapter 12 below, and criticism of it in Bob Hale’s “Reals by Abstraction”, [2000] chapter 11). Thus, in a number of ways the ontological power of abstractionism seems to backfire – the very strength of the view, the fact that it purports to provide us with an account of how we can have a priori knowledge of the existence and properties of those abstract objects studied by mathematicians, also is one of its weaknesses, since it also seems to provide us with a priori knowledge of the existence of until now unrecognized and, once recognized, unwanted

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objects such as anti-zero and the ‘Bad’ extension. Like the Bad Company objection before it, a satisfactory solution to this problem (or, better, this family of problems) would seem to be a matter of determining where to draw certain lines: How do abstractionists determine which principles (and which formulations of certain principles) will provide them with access to the objects required for mathematics without also entailing the existence of additional objects that are both unnecessary and, at times, inconvenient?

7.

The third problem: What objects?

The final major problem of interest here is the notorious Caesar Problem. Frege first points out the problem in the Grundlagen, where he considers an abstraction principle introducing directions (here the initial quantifiers range over lines, and “//” is the relation of parallelism): (∀a)(∀b)(DIR(a) = DIR(b) ↔ a//b) After pointing out that this definition provides us with the means for identifying directions, and distinguishing distinct directions from one another, he points out that: . . . this means does not provide for all cases. It will not, for instance, decide for us whether England is the same as the direction of the earth’s axis – if I may be forgiven an example which looks nonsensical. Naturally no one is going to confuse England with the direction of the Earth’s axis; but that is no thanks to our definition of direction. (Frege, [1974], §66, pp. 77–78)

Looking at this from a technical perspective, we can see the problem as follows: Abstraction principles, such as Hume’s Principle and New V, whose right-hand side can be expressed in purely logical vocabulary, place no constraints on which object in a particular domain plays the role, say, of seven, or the empty set (things are slightly more complicated in the case of abstraction principles, such as those used to construct the reals, where the equivelence relation on the right contains other abstraction operators). All that determines whether a particular set can serve as the domain of a model of either of these principles is the cardinality of the set – if the set is the right size, then any object in the set can be any number or set (the only requirement is that each object can play the role of at most one number, or one set). Much has been written on the Caesar Problem, but approaches to it generally take one of three routes: First, we can deny it is a problem, adopting a sort of structuralist approach to abstractionism where it does not matter whether Caesar turns out to be the number two, as long as we are guaranteed that some object plays this role. (Although the Caesar problem is not his main target, Crispin Wright’s “Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege’s Constraint” [2000], chapter 13 below, draws connections between ante rem structuralism and abstractionism, and is particularly relevant

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here.) Second, we can attempt to reformulate our abstraction principles in more complicated ways (e.g. by inserting modal operators in appropriate places or the like) so that the reference of numerical terms is more determinate. Third, we might argue that although abstraction principles alone do not determine which object, in particular, is picked out by a certain numeral, abstraction principles plus other background constraints do determine numerical reference uniquely. Which of these approaches is most promising has yet to be determined. In fact, as the literature grows, new variations on the Caesar Problem seem to sprout up at least as fast as attempts to solve them. Most important among these are: The Counter-Caesar Problem: How do we guarantee that particular Fregean numerals denote the same object as their natural language counterparts (e.g. does NUM(x = x) denote the same thing as the English locution “zero”)? The Julio Cesar Problem: How do we guarantee that the cardinal numbers provided by Hume’s Principle denote the same kind of objects as are denoted by mathematical terms occurring in natural language (e.g. does NUM(x = x) denote the same kind of thing as the English locution “zero”)? The C-R Problem: How do we determine whether abstracts provided by distinct abstraction principles are identical or distinct (e.g. is the complex number 0, provided by the appropriate abstraction principle, identical to the real number 0, provided by a different abstraction principle)? Although the Caesar Problem (and its cousins) results from certain formal characteristics of abstraction principles, responses to it tend to be less technical. Nevertheless, a number of the chapters included below contain extended discussions of it. (The reader is also encouraged to consult MacBride [2005] and Cook and Ebert [2005] for further discussion of variants of the Caesar Problem.)

8.

Indefinite extensibility

As the literature on these problems and other issues has grown, the notion of indefinite extensibility has become more and more central to purported solutions. One promising line of attack on both the ‘too-many-abstraction’ principles class of problems and the ‘too-many-objects’ class of problems has been to suggest that we restrict our attention to those abstraction principles that provide abstracts only for concepts which are not indefinitely extensible. Of course, this does little to help us until we know what indefinite extensibility is. Bertrand Russell seems to be the first person to discuss this notion when considering the cause of the various set-theoretic paradoxes:

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Introduction The contradictions result from the fact that . . . there are what we may call selfreproductive processes and classes. That is, there are some properties such that, given any class of terms all having such a property, we can always define a new terms also having the property in question. Hence we can never collect all of the terms having the said property into a whole; because, whenever we hope we have them all, the collection which we have immediately proceeds to generate a new term also having the said property. ([1906], p. 144)

The term “indefinite extensibility” is due to Michael Dummett, however, who extended Russell’s idea as follows: An indefinitely extensible concept is one such that, if we can form a definite conception of a totality all of whose members fall under the concept, we can, by reference to that totality, characterize a larger totality all of whose members fall under it. ([1993], p. 441)

It has become standard to use the term ‘definite’ for those concepts that are not indefinitely extensible. The ordinal numbers provide perhaps the clearest example of an indefinitely extensible collection. Consider any definite collection of ordinals (i.e. a set of ordinals). Given such a collection, we can immediately form a conception of an ordinal not in that collection (i.e. the ‘next’ ordinal, (i.e. either the successor of the greatest ordinal in the collection in question, or the supremum of the collection in question). As a result, there seems to be a sense in which we can never collect together all of the ordinals into a definite totality, since we could repeat this reasoning on such a collection to obtain an ordinal that is not in such a collection of all ordinals – contradiction (this is essentially just the reasoning behind the Burali-Forti paradox). An indefinitely extensible concept is thus one which allows for a certain sort of iteration – any time we have collected together some definite sub-collection of things falling under that concept, we can find a new object that is not in that collection. In fact, the ordinals are not only a clear example of the notion in question, but their structure seems to be fundamental to indefinite extensibility itself, since this iterability suggests that any indefinitely extensible collection will contain a structure isomorphic to the ordinals (it is worth noting, however, that Dummett would reject this Russellian characterization of indefinite extensibility). Thus, one way of characterizing indefinitely extensible concepts is “those concepts that are like the ordinals in relevant ways”. As Peter Clark points out in his contribution to this volume (“Frege, Neo-logicism and Applied Mathematics” [2004], chapter 3 below), another way of picking out the indefinitely extensible concepts is to just note that they are the ones whose extensions do not form sets. But neither of these suggestions, intuitively helpful as they are, do the abstractionist any real good. The abstractionist, remember, wishes to use the notion of indefinite extensibility in order to formulate a restricted version of Basic Law V (and other abstraction principles) which will provide an adequate

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set theory. As a result, no characterization of indefinite extensibility (such as those above) which uses set-theoretic notions can be of use, since using set theoretic notions to formulate one’s implicit definition of set would introduce a rather vicious circle into the picture. Thus, the abstractionist needs some neutral formulation of the notion in question. As of yet, no completely adequate account of indefinite extensibility has been found, at least none that is of the sort that could be mobilized by the abstractionist wishing to use it in formulating various abstraction principles. This is not to say, of course, that no work of interest has been carried out – on the contrary, at least half of the papers in the present volume make at least passing reference to the importance of this problem, and almost all of the papers in the last section, on set theory, contain detailed discussion of the issue. Of particular interest is Stewart Shapiro’s “Prolegomenon to Any Future Neo-Logicist Set Theory. . . ” (chapter 18 below), which contains both a detailed examination of indefinite extensibility as discussed by philosophers such as Dummett and Russell, as well as a sustained technical examination of what formal characteristics a successful abstractionist account of the notion requires.

9.

One last thing

Although the bulk of the literature on abstraction and its mathematics, and the majority of the papers to follow, focus either on the actual formalization of arithmetic, analysis, and set theory, or on the three major sorts of problem just outlined, there are of course many other crucial questions regarding abstraction to be answered and many other avenues to be explored. While space considerations preclude detailed discussion of them here, at least one of them deserves brief mention before moving on to the papers themselves. The question in question is this: In what ways can the abstractionist’s formal results be adopted or adapted by their philosophical opponents? For example, in “Frege’s Unofficial Arithmetic” [2002] (chapter 10 below) Agustin Rayo utilizes Frege’s Theorem (and corollaries of it) to provide a distinctly non-Fregean (in fact, somewhat Quinean) account of arithmetic (and, in particular, applied arithmetic). While Rayo suggests that the account he provides is at least inspired by Frege’s own views (views Frege held after he abandoned logicism), the project he sketches is worked out against a philosophical background quite different from the one assumed by most abstractionists. The point, to put it bluntly, is this: even if abstraction principles are not definitions in the sense the abstractionist suggests, they might nevertheless play some crucial role in our epistemological account of mathematics. It is thus hoped that this collection will serve as a repository of work on the technical aspects of abstraction principles which can be utilized by both the abstractionist himself and also by adherents of different, competing philosophical

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accounts of mathematics (even if the majority of the actual papers are working within the standard Fregean abstractionist picture).

References Black, M. [1965], Philosophy in America, Ithaca, Cornell University Press. Boolos, G. [1989], “Iteration Again”, Philosophical Topics 17: 5–21. Boolos, G. [1990a], “The Standard of Equality of Numbers”, in Boolos [1990b]: 3–20. Boolos, G. (ed.) [1990b], Meaning and Method: Essays in Honor of Hilary Putnam, Cambridge, Cambridge University Press. Boolos, G. [1997], “Is Hume’s Principle Analytic?”, in Heck [1997b]: 245–261, reprinted below as chapter 1. Boolos, G. [1998], Logic, Logic, and Logic, Cambridge, MA, Harvard University Press. Boolos, G. and Heck, R. [1998] “Die Grundlagen der Arithmetik §82–83”, in Boolos [1998]: 315–338. Burgess, J. [1984], Review of Wright [1983], Philosophical Review 93: 638–640. Clark, P. [2004], “Frege, Neo-logicism and Applied Mathematics”, in Stadler [2004]: 169–183, reprinted below as chapter 3. Cook, R. [2002], “The State of the Economy: Neologicism and Inflation”, Philosophia Mathematica 10: 43–66, reprinted below as chapter 12. Cook, R. [2003], “Aristotelian Logic, Axioms, and Abstraction”, Philosophia Mathematica 11: 195–202, reprinted below as chapter 9. Cook, R. [2004], “Iteration One More Time”, Notre Dame Journal of Formal Logic 44: 63–92, reprinted below as chapter 20. Cook, R. & P. Ebert [2005], “Abstraction and Identity”, Dialectica 59: 121–139. Demopoulos, W. [2003], “On the Philosophical Interest of Frege Arithmetic” Philosophical Books 44: 220–228, reprinted below as chapter 7. Fine, K. [2002], The Limits of Abstraction, Oxford, Clarendon Press. Frege, G. [1974], Die Grundlagen Der Arithmetic, J.L. Austin (trans.), Oxford, Basil Blackwell. Frege, G. [forthcoming], Grundgesetze der Arithmetik, C. Wright et al. (trans.), Oxford, Oxford University Press. Hale, R. [2000], “Reals by Abstraction”, Philosophia Mathematica 8: 100–123, reprinted below as chapter 11. Hale, R. [2000], “Abstraction and Set Theory”, Notre Dame Journal of Formal Logic 41: 379– 398, reprinted below as chapter 17. Hale, B. C. Wright [2001], The Reason’s Proper Study. Oxford, Oxford University Press. Heck, R. [1993], “The Development of Arithmetic in Frege’s Grundgesetze der Arithmetik”, Journal of Symbolic Logic 10: 153–174. Heck, R. [1997a], “Finitude and Hume’s Principle”, Journal of Philosophical Logic 26: 589– 617, reprinted below as chapter 4. Heck, R. (ed.) [1997b], Language, Thought, and Logic, Oxford, Oxford University Press. Hodes, H. [1984], “Logicism and the ontological commitments of arithmetic”, The Journal of Philosophy 81: 123–149. MacBride, F. [2000], “On Finite Hume”, Philosophia Mathematica 8: 150–159, reprinted below as chapter 5. MacBride, F. [2002], “Could Nothing Matter?”, Analysis 62: 125–135, reprinted below as chapter 6. MacBride, F. [2003], “Speaking with Shadows: A Study of Neo-logicism”, British Journal for the Philosophy of Science 54: 103–163. MacBride, F. [2005], “The Julio César Problem”, Dialectica 59: 223–236. Parsons, C. [1965], “Frege’s Theory of Number”, in Black [1965]: 180–203. Rayo, A. [2002], “Frege’s Unofficial Arithmetic”, Journal of Symbolic Logic 67: 1623–1638, reprinted below as chapter 10. Russell, B. [1902], “Letter to Frege” in van Heijenoort [1967]: 124–125.

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Russell, B. [1906], “On Some Difficulties in the Theory of Transfinite Numbers and Order Types”, Proceedings of the London Mathematical Society 4: 29–53. Shapiro, S. [2000], “Frege Meets Dedekind: A Neologicist Treatment of Real Analysis”, Notre Dame Journal of Formal Logic 41: 335–364, reprinted below as chapter 14. Shapiro, S. [2003], “Prolegomenon to Any Future Neo-Logicist Set Theory: Abstraction and Indefinite Extensibility”, British Journal for the Philosophy of Science 54: 59–91, reprinted below as chapter 18. Shapiro, S. & A. Weir [1999], “New V, ZF and Abstraction”, Philosophia Mathematica 7: 293– 321, reprinted below as chapter 15. Shapiro, S. & Weir [2000], “Neo-Logicist Logic Is Not Epistemically Innocent”, Philosophia Mathematica 8, 160–189, reprinted below as chapter 8. Stadler, F. (ed.) [2004], Induction and Deduction in the Sciences, Dordrecht, Kluwer Academic Publishers. Uzquiano, G. & I. Jané [2004], Well- and Non-Well-Founded Extensions”, Journal of Philosophical Logic 33: 437–465, reprinted below as chapter 16. van Heijenoort, J., (ed.) [1967], From Frege to Gödel: A Sourcebook in Mathematical Logic, Cambridge, MA, Harvard University Press. Weir, A. [2004], “Neo-Fregeanism: An Embarassment of Riches”, Notre Dame Journal of Formal Logic 44: 13–48, reprinted below as chapter 19. Whitehead, A. N. & B. Russell [1910–1913], Principia Mathematica, 3 vols., Cambridge, Cambridge University Press. Wright, C. [1983], Frege’s Conception of Numbers as Objects, Aberdeen, Aberdeen University Press. Wright, C. [1997], “On the Philosophical Significance of Frege’s Theorem”, in Heck [1997b]: 201–244. Wright, C. [1999], “Is Hume’s Principle Analytic?”, Notre Dame Journal of Formal Logic 40: 6–30, reprinted below as chapter 2. Wright, C. [2000], “Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege’s Constraint”, Notre Dame Journal of Formal Logic 41: 317–334, reprinted below as chapter 13.

IS HUME’S PRINCIPLE ANALYTIC?1 George Boolos

The reduction, however, cuts both ways. It is not easy to see how Frege can avoid the seemingly frivolous argument that if his reduction is really successful, one who believes firmly in the synthetic character of arithmetic can conclude that Frege’s logic is thus proved to be synthetic rather than that arithmetic is proved to be analytic. Hao Wang 2

There are a number of issues on which Crispin Wright and I disagree, some of them substantive and some merely terminological. For example, we disagree over whether the term “analytic” can be suitably applied to HP and whether a derivation of arithmetic from HP would establish a doctrine appropriately called “logicism.” I also have certain reservations, which I shall set out later, about his notions of explanation and reconceptualization. However, I think the areas of agreement about the interest of Frege’s derivation of arithmetic are both wide-ranging and far more significant than those of disagreement. In particular I want to endorse Wright’s closing suggestion that “the problems and possibilities of a Fregean foundation for mathematics remain [wide?] open” and the remark made earlier in his paper that “The more extensive epistemological programme which Frege hoped to accomplish in the Grundgesetze is still a going concern.” I also want to emphasize that I consider Wright to have made a great scientific contribution in showing contemporary readers 1 This article first appeared in Richard G. Heck Jr., ed., Logic, Language, and Thought, Oxford, Oxford University Press (1997). Reprinted by kind permission of Oxford University Press. A version of this paper was presented to a 1994 American Philosophical Association symposium on the topic of logicism. Crispin Wright was the co-symposiast and Charles Parsons the commentator. Michael Dummett much dislikes the designation “Hume’s Principle” because the remark in Hume’s Treatise (I, III, I, para. 5) which Frege cited with approval and from which the name derives, presupposes the doctrine that a number is an item composed of units, a doctrine which Frege is presumed to have refuted. Since this paper first appeared in a Festschrift for Michael, I used the designation “HP” instead. Cf. Chomsky and “LF.” 2 Wang, Hao, “The Axiomatization of Arithmetic,” Journal of Symbolic Logic 22 (1957), pp. 145–158, reprinted in Wang, Hao, A Survey of Mathematical Logic, Peking, Science Press (1963), pp. 68–81. The quotation, together with other extremely interesting observations, appears on p. 80.

3 Roy T. Cook (ed.), The Arché Papers on the Mathematics of Abstraction, 3–15. c 2007 Springer.

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how the deduction of the Peano postulates from HP could be carried out and in formulating the conjecture, subsequently verified, that HP is consistent. 3 The first issue I want to take up is whether a derivation of arithmetic from HP vindicates logicism. My view is: no logic, no logicism. It is clear what has to be established in order to show the truth of something we can call logicism with a clear conscience. Arithmetic has to be shown to be provable from an extension by definitions of a theory that is logically true. In technical parlance, arithmetic has to be interpreted in a logically true theory. It cannot be, trivially: Arithmetic implies that there are two distinct numbers; were the relativization of this statement to the definitions of the predicate “number” provable by logic alone, logic would imply the existence of two distinct objects, which it fails to do (on any understanding of logic now available to us). Wright states that if it has to be made out that HP is a truth of logic, then “the prospects are unimproved,” the prospects, I take it, being those for establishing a species of logicism. I infer that he does not consider HP to be a truth of logic. Nor do I: the principle implies the existence of too many objects. So I do not conclude, as Wright does, that the proof of Frege’s theorem by itself establishes logicism. It only shows the beautiful, deep, and surprising result that arithmetic is interpretable in Frege arithmetic, a theory whose sole nonlogical axiom is HP. Wright argues, though, that since HP is analytic, the proof yields “an upshot still worth describing as logicism, albeit rather different from the conventional understanding of the term.” I might be prepared to agree that something describable as logicism in a different understanding of that term would have been established if HP had been shown to be analytic or akin to something properly called a definition. But I doubt that it can be. Having to discuss whether HP is analytic is rather like having to consider whether hydrogen sulfide is deflogisticated. One can certainly see reasons why one might be tempted to call H2 S dephlogisticated: but if I am right in thinking that to deflogisticate is to combine with oxygen, there are conclusive reasons for not doing so. The main reason why the notion of analyticity is all but useless is discussing propositions of mathematics like HP is that, although an analytic statement is supposed to be one that is true in virtue of the meanings of the terms contained in sentences expressing it (and syntactic features of those sentences), the phrase “true in virtue of meanings” leaves it indeterminate how much mathematics may be used to get from facts about meanings to the truth of the statement, or, more exactly, how much mathematics it is allowable to use 3 Wright, Crispin, Frege’s Conception of Numbers as Objects, Scots Philosophical Monographs, vol. 2, Aberdeen, Aberdeen University Press (1983). The derivation is on pp. 154–168. The discussion of numbertheoretic logicism III is on pp. 153–154.

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in deriving the statement (or the statement that that statement is true) from reports of meanings. In brief, we are not told how strong the mathematics is that “in virtue of” permits. The stronger the mathematics permitted, the greater the number of analytic mathematical truths, of course. The point, in essence, is due to Gödel and is different from the objection raised by the question “Why mathematics rather than geology?” In the interest of trying to get at what’s really at issue between Wright and myself, however, I shall ignore the standard difficulties presented by “analytic,” including the uncertainty what the interest or point of classifying a statement as analytic is and the worry that complex logical argumentation might itself create semantic content, 4 and suppose that I understand the concept sufficiently well, well enough at least to know what’s meant by calling “all vixens are foxes,” etc. analytic and by saying that there is a semantic connection between “vixen” and “fox.” At the outset, let me acknowledge that I have no knock-down argument that will persuade a diehard defender of the claim that HP is analytic to abandon the view. All I shall offer are what strike me as some rather, and perhaps sufficiently, weighty considerations against that position. At first glance, HP might certainly seem analytic. In its statement “number” means “cardinal number” and, one would naturally wonder, isn’t it a matter of the semantic connection between “cardinal number” and “one–one correspondence” that two concepts have the same cardinal number just when things falling under one of them can be put in one–one correspondence with those falling under the other? Isn’t the cardinal of x the same as that of y just when there’s a one–one correspondence between x and y, and that because of what “cardinal number” means? So isn’t the left-hand side of HP close enough in meaning to the right-hand side for it to count as analytic? Doesn’t the left-hand side have the same sense as the right? 5 Let me begin to respond to this argument by recalling two features that analytic statements have been traditionally supposed to enjoy: first, they are true; secondly and roughly speaking, they lack content, i.e., they make no significant or substantive claims or commitments about the way the world is; in particular, they do not entail the existence either of particular objects or of more than one object. (It may be held that some analytic statement might entail the existence of at least one object, as will be the case if every logical truth counts as analytic.) Some have been tempted by the idea of analytic statements that happen not to be true, e.g., “the present king of France is a royal.” On the view in question, the semantic connection between “king” and “royal” suffices to ensure the analyticity of the entire statement, despite the failure of its subject to denote. But analytic statements are, and (since we are playing along) are analytically, 4 This possibility is suggested by a remark of Frege about condensation in §23 of his Begriffsschrift. 5 Thanks here to Arthur Skidmore.

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analytic truths, and the view may be put aside. The example is worth noting, however, for, as I am going to suggest later, HP suffers from a defect similar to that of “the present king of France is a royal,” which would not be analytic even if there were presently a (unique) king of France, since, of course, it would not be analytic that there is one. The main significant worry for the defender of the analyticity of HP concerns the quite strong content that it appears to possess. HP has consequences having to do with certain features of the domain of objects over which its firstorder variables range, in particular with the number of those objects there are. Much of the most interesting work in mathematical logic in the last 20 years or so has dealt with comparisons of strength of various logical and mathematical statements, examining which well-known theorems of mathematics can be derived from which logical principles (and vice versa!) in which background theories. We now know that Frege arithmetic is equi-interpretable with full second-order arithmetic, “analysis,” and hence equi-consistent with it. Learning that HP is analytic would not help us in the slightest with the problem of assessing the strength of various theorems, fragments, and subtheories of analysis, all of which would, I suppose, have to count as analytic. The first part of my worry about content is that HP, when embedded into axiomatic second-order logic, yields an incredibly powerful mathematical theory. Wright will say: Hooray! Math is analytic after all. But we don’t know what follows from its being so and we will have to study the subanalytic to see what (logically) entails what just as hard as before. It is known that HP does not follow (a word I will not surrender) from the conjunction of two of its strongest consequences: the (interesting) statements that nothing precedes zero and that precedes is a one–one relation. If HP is analytic, then it is strictly stronger (another non-negotiable term) than some of its strong consequences. It is also known that arithmetic follows from these two statements alone, and that arithmetic is strictly weaker than even their disjunction. 6 Faced with these results, how can we really want to call HP analytic? Frege, for a lengthy stretch of his career, held that the existence of infinitely many objects could be seen to follow from a set of principles and definitions that could, by his lights, be counted as analytic. He abandoned the view in 1906, according to Dummett, when he realized that his attempted patch to Basic Law V would not work. It is doubtful that Russell could be considered a logicist in the full sense of the term while writing Principia, whose stated aim is to analyze the notions employed in mathematics, not to show arithmetic to be a branch of logic. Despite the Gödel incompleteness theorems and Russell’s protestations that the axiom of infinity was no logical truth, it was a central tenet of logical positivism that the truths of mathematics were analytic. Positivism was dead by 1960 and the more traditional view, that analytic truths 6 For proofs of these results, see my “On the Proof of Frege’s Theorem,” in Benacerraf and His Critics, Adam Morton and Stephen Stich, eds., Cambridge, MA, Blackwell (1996), pp. 143–159.

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cannot entail the existence either of particular objects or of too many objects, has held sway since. Wright wishes to overthrow the tradition, but it should be asked how a statement that cannot hold if there are only finitely many objects can possibly be thought to be analytic, a matter of meanings or “conceptual containment.” On the symbolization that I prefer, HP reads: ∀F∀G(#F = #G ↔ F ≈ G) where “F ≈ G” is an abbreviation for a second-order formula expressing that there is a one–one correspondence between the objects falling under the concept F and those falling under the concept G. We need not here write out the formula, but must remember that it contains some first-order quantifiers. We must also remember the grammatical category of “#,” “octothorpe”: it is a function-sign, which when attached to a monadic second-order variable like “F,” produces a term of the same type as individual variables that occur in “F ≈ G.” It is essential to the proof of Frege’s theorem that octothorpe be so construed. Thus octothorpe denotes a total function from concepts to objects. Logic, plus the convention that function signs like octothorpe denote total functions, will guarantee that ∀F∃!x #F = x is true. It will not guarantee that HP is. HP entails, as Wright has put it with exemplary force and Cartesian clarity, that there is a partition of concepts into equivalence classes, in which two concepts belong to the same class if and only if they are equinumerous. If there are only k objects, k a finite number, then, since there are k + 1 natural numbers ≤ k, there will be k + 1 equivalence classes, viz. a class containing each concept under which zero objects fall, a class containing each concept under which exactly one objects falls, . . . , and a class containing each concept under which all k objects fall. (We need not here assume that concepts are individuated extensionally.) Thus, if there are only k objects, there is no function mapping concepts to objects that takes non-equinumerous concepts to different objects, for there won’t be enough objects around to serve as the values of the function, since k + 1 are needed. So if HP holds—even if only the left–right direction (the same direction as in the fatal Basic Law V) holds— there must be infinitely many objects. One person’s tollens is another’s ponens, and Wright happily regards the existence of infinitely many objects, and indeed, that of a Dedekind infinite concept, as analytic, since they are logical consequences of what he takes to be an analytic truth. He would also regard the existential quantification of HP (over the positions occupied by octothorpe) as analytic. But what guarantee have we that there is such a function from concepts to objects as HP and its existential quantification claim there to be? I want to suggest that HP is to be likened to “the present king of France is a royal” in that we have no analytic guarantee that for every value of “F,” there is an object that the open definite singular description “the number

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belonging to F” denotes. I shall also suggest that there may be some analytic truths in the vicinity of HP with which it is being confused. I hope that the suggestions will do justice both to the thought that there is a strong semantic connection between “the number of . . . ” and “one–one correspondence” and to the traditional idea that analytic truths do not entail the existence of a lot of objects. Our present difficulty is this: just how do we know, what kind of guarantee do we have, why should we believe, that there is a function that maps concepts onto objects in the way that the denotation of octothorpe does if HP is true? If there is such a function then it is quite reasonable to think that whichever function octothorpe denotes, it maps non-equinumerous concepts to different objects and equinumerous ones to the same object, and this moreover because of the meaning of octothorpe, the number-of-sign or the phrase “the number of.” But do we have any analytic guarantee that there is a function that works in the appropriate manner? Which function octothorpe denotes and what the resolution is of the mystery how octothorpe gets to denote some one definite particular function that works as described are questions we would never dream of trying to answer. (Harold Hodes’ article “Logicism and the ontological commitments of arithmetic” 7 contains much wisdom about these mysteries of mathematical reference.) Nevertheless, it would seem that if there is such a function, then whichever function octothorpe does denote, it also does the trick. 8 Thus, I am moved to suggest, very tentatively and playing along, that the conditional whose consequent is HP and whose antecedent is its existential quantification might be regarded as analytic. The conditional will hold, by falsity of antecedent, in all finite domains. By the axiom of choice, the antecedent will be true in all infinite domains, but then, we may suppose, nothing will prevent the consequent from being true. I also find plausible the suggestion that the right-to-left half of HP, which states that if F and G are equinumerous, then their numbers are identical, is analytic. It is the left-to-right half, which states that if F and G are not equinumerous, then their numbers are distinct, that blows up the universe. (E.g, consider the concept non-self-identical; call its number zero. Now consider the concept identical with zero; call its number one. By the left-to-right half of HP: since the concepts are not equinumerous, zero is not one.) The analogy with Basic Law V is obvious. Frege divided Basic Law V into Va, the left-to-right half, and its converse Vb. It was the left-to-right half that gave rise to Russell’s paradox. Vb has considerable claim to being regarded as a logical truth: (a) it is valid under standard semantics, thanks to the axiom of extensionality; (b) if the Fs are the Gs, as the antecedent asserts, then whatever 7 Hodes, Harold “Logicism and the Ontological Commitments of Arithmetic,” Journal of Philosophy 81 (1984), pp. 123–149. 8 Hartry Field has made a similar suggestion in his review of Wright’s Frege’s Conception of Numbers as Objects, which is reprinted in Field, Hartry, Realism, Mathematics, and Modality, Oxford, Blackwell (1989), pp. 147–170.

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“extension” may mean, the extension of the Fs is the extension of the Gs; and (c) if the antecedent holds, then the concepts Fand G bear a relation to each other that Frege called the analogue of identity. Thus under each of three familiar systems of formula-evaluation, Vb can never turn out false. In the case of both HP and Basic Law V, we have a principle whose left-to-right half requires that there be a function from concepts to objects respecting certain non-equivalences of those concepts. Unless enough objects exist, these nonequivalences cannot be respected. All that the right-to-left halves demand is that the equivalences be respected, as they can be trivially, by mapping all concepts to one and the same object. ∀F∀G(∀x(Fx ↔ Gx) → #F = #G), which has the same form as Basic Law Vb, can equally justifiably be claimed to be a logical truth, and the stronger ∀F∀G(F ≈ G → #F = #G) much more plausibly thought analytic, in virtue of the meaning of “#,” than its converse. There is a further difficulty, or at any rate a further aspect of the same difficulty: If numbers belonging to concepts F and G are supposed to be identical if and only if F and G are equinumerous, then how do we know that, for every concept, there is such a thing as a number belonging to that concept? We should not be led astray by the concision, symmetry, and apparent familiarity and obviousness of #F = #G ↔ F ≈ G into ignoring the fact that octothorpe is a function sign (for a function of higher type). Like constants and the usual sort of function sign, it may help in concealing significant existential commitments. (Perhaps because of that danger, Quine, concerned with ontology and logic’s role in its study, almost entirely avoids constants and function signs in his textbook Methods of Logic.) An analogy may help: if volumes are supposed to be translation- and rotation-invariant, finitely additive, and non-trivial, with singletons and balls of radius r having volumes 0 and 4πr 3 /3, respectively, then, as the “paradoxical” Banach–Tarski theorem shows, not every bounded set of points in three-space has a volume. It would thus be illegitimate to introduce a sign for a totally defined function from bounded sets of points in three-space to real numbers and assume that the function was translation-invariant, etc. And one had therefore better not say: it is analytic that volume is translation-invariant, etc., and it is analytic that there is always such a thing as the volume of any bounded set of points in three-space, for the conjunction of the two statements claimed to be analytic is false. Similarly, if numbers are supposed to be identical if and only if the concepts they are numbers of are equinumerous, what guarantee have we that every concept has a number? 9 Or, if we take ourselves to know that with every concept there is functionally associated some object, then how do we know that the associated object is a number belonging to F? 9 Profound thanks here to Peter Clark.

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It will be useful here to formulate HP in a way that expressly brings out its existential commitments. Let Numbers be the statement: for every concept F, there is a unique object xsuch that for every concept G, x is a number belonging to G if and only if F is equinumerous with G. Is Numbers analytically true? I see no reason at all to believe that it is analytic that for every F, there is such a (unique) object x. To reply that it is, since Numbers follows from HP, and HP is analytic, would seem to beg a question that ought not to be begged. Even more strongly, I don’t see any reason to think that it’s analytic that objects can be so assigned to concepts that any two concepts are assigned the same object if and only if they are equinumerous. It is not only the existence of a function of higher type making such an assignment of objects to concepts that seems synthetic to me: the weaker modal claim that objects can be so assigned strikes me as synthetic as well. I repeat that one person’s ponens is another’s tollens and admit again that I don’t have a knock-down argument against Wright’s view. I now want to raise some objections to Wright’s notion of a reconceptualization and his use of the term “explanation.” Discussing Frege’s (more-or-less) analogous case of directions and parallelism, Wright says, “we have the option . . . of re-conceptualizing, as it were, the state of affairs which is described on the right. That state of affairs is initially given to us as the obtaining of a certain equivalence relation . . . ; but we have the option, by stipulating that the abstraction is to hold, of so reconceiving such states of affairs that they come to constitute the identity of a new kind of thing . . . of which, by this very stipulation, we introduce the concept.” Part of the problem with this suggestion is this: in HP, numbers belonging to concepts are themselves among the objects over which the first-order variables on the right-hand side range. Talk of reconceptualizing a state of affairs would be in order only if the objects supposedly introduced by stipulation were new, objects that had not been previously quantified over. Whether old objects can be chosen to be identical or not under the right conditions would not seem to be a matter that it could be up to us to decide. It is here that the analogy between directions and lines and numbers and concepts breaks down: no one supposes that directions are any sort of constituent of lines, but on the Fregean treatment of number, numbers quite definitely are objects that both fall under concepts and are associated with concepts, as their numbers. However, when the objects allegedly introduced by this sort of stipulation are already objects quantified over in the equivalence relation, unexpected, and sometimes unwelcome, results can occur when we attempt to identify certain of them. We can’t, for example, stipulate that old objects be assigned to concepts in such a way that if some old object falls under one concept but not another then the two concepts are to be assigned different objects.

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Wright says, “The concept of direction is thus so introduced that that two lines are parallel constitutes the identity of their direction. It is in no sense a further substantial claim that directions exist and are identical under the described circumstances. But nor is it the case that, by stipulating that the principle is to hold, we thereby forfeit the right to a face-value construal of its left-hand side and thereby to the type of existential generalization which a face-value construal would license.” All well and good for directions, maybe, but what if the objects introduced on the left are already among those discussed on the right? Could there not then be a danger that a “substantial further claim” about those very objects, taken together, would be entailed? And of course there is such a danger: the generalized biconditional, or the biconditional with its free variables, taken as an axiom, might then entail that, e.g., there are many, many objects, too many for it to be capable of being regarded any longer as analytic. One might think: but does that not automatically show that HP isn’t analytic? How can an analytic truth be false in certain domains, indeed false in all the finite domains? There is of course a reply that is ready to hand, viz. that it’s analytically false that the objects that exist constitute any one of those finite domains. The response strikes me as incredible, but again, I don’t have a knock-down argument against the analyticity of HP, only a bunch of considerations. (Heidegger would hardly have welcomed the response, “Because, analytically, there is always the number of things that there are; so there couldn’t have been nothing rather than something.”) One final remark on reconceptualization. How can one call the left-hand side of HP a reconceptualization of the right if it can’t always be made to hold whenever the right-hand side does? Of course if the variables range over a set, one can always pick some new objects to play the role of the numbers belonging to subsets of that set, but why is one so sure one can do this if there is no set of objects over which the variables range? Wright’s idea that the role of HP is that of an explanation also worries me. In Frege’s Conception of Numbers as Objects, Wright writes: “the fundamental truths of number theory would be revealed as consequences of an explanation: [note the colon] a statement whose role is to fix the character of a certain concept.” 10 In the present paper, Wright calls HP “a principle whose role is to explain, if not exactly to define, the general notion of cardinal number.” Wright is impressed by the form of HP: a biconditional whose right limb is a formula defining an equivalence relation between concepts F and G and whose left limb is a formula stating when the cardinal numbers of F and G are the same. Since the sign for cardinal numbers does not occur in the right 10 Wright (1983), p. 153.

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limb, can one not appropriately say that HP explains the concept of a cardinal number by saying what it is for two cardinal numbers, both referred to by expressions of the form “the number of . . . ” to be identical? 11 Certainly. HP states a necessary and sufficient condition for an identity #F= #G to hold. Moreover the formula defining this condition doesn’t contain #. So if one wants merely to sum up this state of affairs by saying that HP explains the concept of cardinal number, I would not object. However, it is hard to avoid the impression that more is meant, that Wright holds that to call a statement an explanation of a concept is to assign it an epistemological status importantly similar to the one it was once thought analytic judgments, including definitions, enjoy. It is to this further suggestion that I wish to demur. I can’t help suspecting that Wright is using “explanation,” at least in the phrase “explanation of a concept,” as a term of art, as a member of the same family circle as “analytic,” “definition,” or “conceptual truth,” that the only reason he does not call HP an “analytic definition” is that it is not of the form: Definiendum(x) ≡ Definiens(x), and that he supposes it to be a super-hard truth like “all bachelors are unmarried” or “all equivalence relations are transitive.” The phrase “whose role” occurs in both quotations and may suggest that Wright thinks that HP has one and only one [pre-eminent] role, for “whose” seems in both places to mean “of which the” rather than “of which a.” This thought seems to me to be incorrect. HP might be taken as an axiom, the sole (non-logical) axiom in some axiomatization of arithmetic. It might be a sentence we want to show to be needlessly strong for some purpose, e.g., deriving arithmetic. It might serve as something to be obtained from Basic Law V. It might be used as an example of a beautiful proposition. Etc. etc. But there’s no such thing as the [unique] role of HP. It is certainly true that one of the ways in which HP can be used is to fix the character of a certain concept. Here’s how: lay Hume down. Then the concept the number of . . . will have been fixed to be such that numbers belonging to concepts will be the same if and only if the objects falling under one of the concepts are in one–one correspondence with those falling under the other. But Hume is no different in this regard from any other statement that we might choose to take as an axiom. The axiom of choice fixes the concept of set in a similar manner. Laid down, it determines that for any set of disjoint nonempty sets, there is a set with exactly one member in common with each of those sets. The principle of mathematical induction fixes the character of the natural numbers. The statement that bananas are yellow fixes the character of the concept of a banana. So nothing is said when it is said that one of the roles of HP is to fix the character of the concept of cardinal number. And HP doesn’t have a unique role. 11 I am grateful to Wright and Richard Heck for helpful comments on the whole of this paper but am particularly grateful to them here.

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Let me now defend myself about the “bad company” argument. What I think I was doing was illustrating that what is called (unfortunately, as Wright has stated) “contextual definition” is not, in general, a permissible way of introducing a concept. I didn’t mean to be arguing that it never was and gave the example of the principle governing truth-values as another example of a legitimate contextual definition. Different examples had different purposes. I cited Hodes splendid observation that the relation-number principle (the relation-number belonging to R is identical with that belonging to S if and only if R and S are isomorphic relations) leads to the Burali-Forti paradox in order to point out that Basic Law V was not an isolated case and that HP might well be expected to be powerful if consistent (as it is). I gave the example of parities in order to show that one couldn’t say that a contextual definition is OK if only it is consistent. (I had thought of nuisances, but I seemed to recall actually having heard of the “parity” of a set, and the notion is in any case a natural one.) The example of a principle true iff there are no more than two members was designed to show that one didn’t need heavy involvement with set theory to find a contextual definition incompatible with HP. And did I ever say that it would be impossible to demarcate the good contextual definitions from the bad? I merely said that it would seem to be a problem we have no hope of solving at present. I have to reserve judgment on the question whether Wright has solved the problem, but I certainly hope he has. Wright says I was wrong to say that there is no notion that V**, my revision of Basic Law V, is analytic of; what is true, he says, is “that there is no prior, no intuitively entrenched notion, no notion given independently, which V** is analytic of.” I happily accept the correction. I now want to make a somewhat conciliatory remark. I have been aspersing, at great length, the idea that HP is an analytic truth, all the while taking “analytic” to bear something like the sense it has in current philosophical discourse, namely, “truth in or by virtue of meanings.” I think that is the sense in which Wright uses the term too. But there may be another notion of analyticity on which the analyticity of HP might well be more plausible. It is the idea of Gödel’s, as outlined in both his paper “Russell’s mathematical logic” and his 1951 Gibbs lecture to the American Mathematical Society, 12 according to which a proposition is analytic if it is true “owing to the nature of the concepts occurring therein.” 13 Concepts, he says in the Gibbs lecture, “form an objective reality of their own, which we cannot create or change, but only perceive and describe.” By reflection, which of course 12 See also George Boolos, “Introductory Note to Kurt Gödel’s ‘Some Basic Theorems on the Foundations of Mathematics and their Implications’ ,” in Kurt Gödel, Collected Works, vol. III, Unpublished Essays and Lectures, Solomon Feferman et al., eds., Oxford, Oxford University Press (1995), pp. 290–304. 13 Gödel also describes propositions as analytic if they are true in virtue of the meanings of the terms expressing them, but it should be understood that his notion of meaning is much broader than that of “linguistic” meaning. For example, Gödel held that it is a matter of the meanings of “set” and “∈” that the axioms of set theory hold. The difference between the sense he attached to “meaning” and “concept” would not seem to be particularly significant.

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includes philosophical or mathematical or other intellectual work, we can sometimes arrive at an understanding of the natures of certain concepts that is sufficient to enable us to see the truth of certain propositions in which they occur. With the passage of time, our understanding of those concepts may improve and the truth of ever more analytic propositions become evident to us. Perhaps, as Schoenfield has ironically suggested, the rejection of the “axiom” of constructibility is one example of improvement in our perception of the meaning (in Gödel’s sense) of “set” or of the nature of sets. The thought that understanding of abstract objects may be achieved through a sort of perception of them, which is crucial to Gödel’s conception of the analytic, will certainly strike many contemporary philosophers as unacceptably mystical and at any rate highly implausible. (Perhaps, paradoxically, there is even a tinge of materialism in the suggestion that our knowledge of abstract objects arises from “something like a perception” of them: could there not be ways in which we interact with abstracta that yield knowledge of them that are not at all like perception?) But if—IF—a Gödelian notion of analyticity could be made out, then HP might well be among the first candidates for this new sort of analytic truth. Perhaps by taking the thought in the right way, we can “see” that if nothing exists, then zero, at least, has to exist, for it is then the number of things there are, and therefore that something does exist after all, but then there have to exist two things, for . . . This Fregean argument may strike one, as it does me, as a good example of the kind of reflection Gödel might have thought showed that the proposition that there are infinitely many natural numbers is analytic, on his understanding of “analytic,” if not on that of most of us who use the word. Maybe in the end we would also thus “see” the truth of HP. But even on such a Gödelian view of the analytic, at least two difficulties would confront the view that HP is analytic. The first is that (it is not neurotic to think) we don’t know that second-order arithmetic, which is equi-consistent with Frege Arithmetic, is consistent. Do we really know that some hotshot Russell of the 23rd century won’t do for us what Russell did for Frege? The usual argument by which we think we can convince ourselves that analysis is consistent—“Consider the power set of the natural numbers . . . ”—is flagrantly circular. Moreover, although we may think Gentzen’s consistency proof for PA provides sufficient reason to think PA consistent, we have nothing like a similar proof for the whole of analysis, with full comprehension. We certainly don’t have a constructive consistency proof for ZF. And it would seem to be a genuine possibility that the discovery of an inconsistency in ZF might be refined into that of one for analysis. Saying exactly which theories are known to be consistent is a difficult problem made even more difficult when one hears of respected mathematicians telling of their failed attempts to prove Q inconsistent, but ZF and analysis, and therefore also Frege arithmetic, are theories that are surely in the black area, not the grey. While we may regret that these theories may well be consistent and that it

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would probably be wise to bet on their consistency, we must not despair: we do not know that they are and need not yet give up hope that someone will one day prove in one of them that 0 = 1. Uncertain as we are whether Frege arithmetic is consistent, how can we (dare to) call HP analytic? One final worry, perhaps the most serious of all, although one that may at first appear to be dismissible or silly or trivial: as there is a number, zero, of things that are non-self-identical, so, on the account of number we have been considering, there must be a number of things that are self-identical, i.e., the number of all the things that there are. Wright has usefully dubbed this number, #[x: x = x], anti-zero. On the definition of ≤, according to which m ≤ n iff ∃F∃G(m = #F ∧ n = #G∧ there is a one–one map of F into G), anti-zero would be a number greater than any other number. 14 Now the worry is this: is there such a number as anti-zero? According to Zermelo–Fraenkel set theory, there is no (cardinal) number that is the number of all the sets there are. The worry is that the theory of number we have been considering, Frege Arithmetic, is incompatible with Zermelo–Fraenkel set theory plus standard definitions, on the usual and natural readings of the non-logical expressions of both theories. To be sure, as Hodes once observed in conversation, if #α is taken to denote the cardinal number of α when α is a set and some favorite objects that is not a cardinal number when α is a proper class, then HP will be a theorem of von Neumann set theory. But on that definition of #, # will not be translatable as “the cardinal number of.” ZF and Frege arithmetic make incompatible assertions concerning what cardinal numbers there are. And of course, the response “Well, these are just formalisms; the question of their truth or falsity doesn’t arise or makes no sense” is hardly available to one claiming that HP is analytic, i.e., an analytic truth. So one who seriously believes that it is has to be bothered by the incompatibility of the consequence of Frege arithmetic that there is such a number as anti-zero with the claim made by ZF + standard definitions (on the natural reading of its primitives) that there is no such number. It is thus difficult to see how on any sense of the word “analytic,” the key axiom of a theory that we don’t know to be consistent and that contradicts our best-established theory of number (on the natural readings of its primitives) can be thought of as analytic.

14 By the Schröder–Bernstein theorem, which can be proved in second-order logic, ≤ is anti-symmetric: if m ≤ n ≤ m, then m = n.

IS HUME’S PRINCIPLE ANALYTIC? 1 Crispin Wright

1. It was George Boolos who, following Frege’s somewhat charitable lead at Grundlagen §63, first gave the name, “Hume’s Principle”, to the constitutive principle for identity of cardinal number: that the number of F’s is the same as the number of G’s just in case there exists a one-to-one correlation between the F’s and the G’s. The interest—if indeed any—of the question whether the principle is analytic is wholly consequential on what has come to be known as Frege’s Theorem: the proof, prefigured in Grundlagen §§82–3 and worked out in some detail in Wright [1983] 2 that second-order logic plus Hume’s Principle as sole additional axiom suffices for a derivation of second-order arithmetic— or, more cautiously, for the derivation of a theory which allows of interpretation as second-order arithmetic. (Actually I think the caution is unnecessary— more of that later.) Analyticity, whatever exactly it is, is presumably transmissible across logical consequence. So if second-order consequence is indeed a species of logical consequence, the analyticity of Hume’s Principle would ensure the analyticity of arithmetic—at least, provided it really is second-order arithmetic, and not just a theory which merely allows interpretation as such, which is a second-order consequence of Hume’s Principle. What significance that finding would have would then depend, of course, on the significance of the notion of analyticity itself. Later I shall suggest that the most important 1 This paper first appeared in the Notre Dame Journal of Formal Logic 40, [1999], pp. 6–30. Reprinted by kind permission of the editor and the University of Notre Dame. 2 At pp. 158–169. An outline of a proof of the Peano Axioms from Hume’s Principle is also given in the Appendix to Boolos [1990]. The derivability of Frege’s Theorem is first explicitly asserted in Charles Parsons [1964]; see remark at p. 194. My own ‘rediscovery’ of the theorem was independent. I do not know what form of proof Parsons had in mind, but the reconstruction of the theorem is trickier than Frege’s own somewhat telegraphic sketch suggests. For an excellent recent overview of the ins and outs of the matter— early on, they remark that

§§82–3 offer severe interpretative difficulties. Reluctantly and hesitantly, we have come to the conclusion that Frege was at least somewhat confused in these two sections and that he cannot be said to have outlined, or even to have intended, any correct proof there (p. 407) —see Boolos and Heck [1998].

17 Roy T. Cook (ed.), The Arché Papers on the Mathematics of Abstraction, 17–43. c 2007 Springer.

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issues here are ones which are formulable without recourse to the notion of analyticity at all—so that much of the debate between Boolos and me could have finessed the title question. Boolos wrote that “having to discuss whether Hume’s Principle is analytic is rather like having to consider whether hydrogen sulphide is dephlogisticated” 3 —a question formulated, I suppose he meant, in a discredited theoretical vocabulary. That would be consistent, of course, with there being a good question nearby of which that was merely a theoretically unfortunate expression; it would also be consistent with there being enough sense to the theoretically unfortunate question to allow of a negative answer in any case. I myself do not believe that when the dust settles on analytical philosophy’s first century, our successors will find that the notion of analyticity was discredited by any of the well-known assaults. In particular, the two core lines of attack in “Two Dogmas of Empiricism”, namely. that the notion resists all non-circular explanation and that no statement participating in general empirical theory can be immune to revision, set an impossible—Socratic—standard for conceptual integrity and confuse analyticity with indefeasible certainty, respectively. What is undeniable, though, is that the status and provenance of analytic truths, and the cognate class of a priori necessary truths, would have to be a lot clearer than philosophers have so far managed to make them before a positive answer to our title question could be justified and shown to have the sort of significance which early analytical philosophy would have accorded to it. Boolos thought the situation was of the second kind noted: that the question is theoretically flawed but allows of well-motivated—though less than “knockdown”—arguments for a negative answer. To the best of my knowledge— I’m drawing just on three of his papers 4 which are reprinted in the excellent Demopoulos collection, 5 plus his ipsonymous paper in Richard Heck’s volume for Michael Dummett 6 —he proffered exactly five such arguments. In what follows I shall briefly explore how a character I shall call the neo-Fregean might respond to each of these arguments. Each is interesting, some are very searching, but—if I’m right—none does irreparable damage.

2. 2.1

The ontological concern

The ontological concern is epitomized in the following passage: I want to suggest that Hume’s Principle is to be likened to ‘the present King of France is a royal’ in that we have no analytic guarantee that for every value of ‘F’, there is an object that the open definite singular description, ‘the 3 Boolos [1997], p. 247. 4 Boolos [1987, 1990, 1986].

5 Demopoulos [1995]. 6 Boolos [1997], Heck [1997].

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Is Hume’s Principle Analytic? number belonging to F’ denotes . . . Our present difficulty is this: just how do we know, what kind of guarantee do we have, why should we believe, that there is a function that maps concepts to objects in the way that the denotation of octothorpe [that is: ‘#’, Boolos’s symbol for the numerical operator] does if HP is true? . . . do we have any analytic guarantee that there is a function that works in the appropriate manner? 7

The basic thought is that Hume’s Principle says too much to be an analytic truth. As normally conceived, analytic truth must hold in any possible domain. On a (purportedly) more relaxed conception, some analytic truths are allowed to hold in any non-empty domain. But how can a principle which entails— indeed, is strictly stronger than is necessary to entail—that there are infinitely many objects—indeed infinitely many objects of a special sort—possibly count as analytic? Here is the neo-Fregean reply. There is, to be sure, a perfectly good sense in which whatever is entailed by certain principles together with truths of logic may be regarded as entailed by those principles alone. In this sense it is undeniable that Hume’s Principle does entail the existence of infinitely many objects—at least if second-order consequence is a species of entailment. But the manner of the entailment is important. Hume’s Principle is a second-order universally quantified biconditional. As such, we are not going to be able to elicit the existence of any objects at all out of it save by appropriate input into (instances of ) its right-hand side. Thus we get the number zero by taking the instance of Hume’s Principle: Nx : x = x = Nx : x = x ↔ x = x

1≈1

x = x

(1)

together with its right-hand side as a minor premise. Compare the fashion in which we derive the direction of the line a from an instantiation of Frege’s illustrative equivalence for directions: (DE)

Da = Da ↔ a//a

together with its right-hand side as a minor premise: the necessary truth, modulo the existence of line a, that that line is parallel to itself. Sure, in the case of zero the minor premise: x = x

1≈1

x = x

(2)

can be established in second-order logic. So the existence of zero follows from this truth of logic, together with Hume’s Principle. If, accordingly, the latter can be regarded as, in all relevant respects, having a status akin to that of a definition, then the existence of zero is a consequence of logic and definitions. But that was exactly the classical account of analyticity: the analytical truths were to be those which follow from logic and definitions. So the existence of zero would be an analytic truth. And now with that in the bag, as it were, 7 Boolos [1997], p. 251. See more generally pp. 248–254, ibid.; also Boolos [1987] at p. 231 and Boolos [1990] at pp. 246–8. (The latter references are to the pagination in Demopoulos [1995].)

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nothing stands in the way of regarding x = 0 1≈1

x =0

(3)

as also an analytic truth, since it follows in second-order logic given only that there is such a thing as zero. But that is the right-hand side for the application of Hume’s Principle which, following Frege, we use to obtain the number one. So its existence is also analytic. We may now proceed in similar fashion to obtain each of the finite cardinals from putatively analytic premises, in second-order logic. Our result is thus not quite—when done this way— that it is analytic that there is an infinity of finite cardinals, but rather that of each of the finite cardinals, it is analytic that it exists. Doubtless this will be equally offensive to the traditional understanding of analyticity—the (as nearly as possible) existentially neutral understanding of analyticity—called forth in the above quotation from Boolos. But my point now is simply that, for the reasons just sketched, that understanding of analyticity had to be in jeopardy all along provided there is a starting chance that Hume’s Principle has an epistemic status relevantly similar to that of a definition. In sum: on the classical account of analyticity, the analytical truths are those which follow from logic and definitions. So if the existence of zero, one, and so on. follows from logic plus Hume’s principle, then provided the latter has a status relevantly similar to that of a definition, it will be analytic, on the classical account, that n exists, for each finite cardinal n. The idea which standardly accompanies the classical conception, that—with perhaps a very few, modest exceptions—existential claims can never be analytically true, is thus potentially in tension with the classical conception. If Hume’s Principle has a status not relevantly different from that of a definition, then we learn that the classical conception will not marry with this standardly accompanying idea. The core of the neo-Fregean stance is that Hume’s Principle does have such a status: that it may be seen as an explanation of the concept of cardinal number in general, covering the finite cardinals as a special case. Boolos asks, “If numbers are supposed to be identical if and only if the concepts they are numbers of are equinumerous, what guarantee have we that every concept has a number?” 8 Earlier he suggested, in the passage quoted, that there is no such guarantee—or anyway no “analytic guarantee”—proposing a parallel between the principle and the statement, “The present King of France is a royal”— something which is analytically true, modulo its existential presupposition. This is also Hartry Field’s position in his critical notice of Wright [1983]. 9 But I think this seemingly sane and reserved position is unstable. Consider the case of direction again. How do we know there are any objects which behave in the way that the referents of direction terms ought to behave, 8 Boolos [1997], p. 253. 9 Field [1984].

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given their introduction by the direction equivalence (DE), that is, given that they are identical just in case the associated lines are parallel and distinct just in case they are not? Shouldn’t we just say that provided there are such things as directions in the first place, that will be the condition for their identity and distinctness? Well, if this were the right view of the matter, there could be no objection to making the presupposition explicit. The following principle would then count as absolutely analytic: that for any lines a and b, ((∃x)(∃y)(x = Da & y = Db)) → (Da = Db ↔ a//b)

(4)

But think: how are we to understand the antecedent of this? The condition for its truth must now incorporate some unreconstructed idea of what it is for contexts of the form, ‘ p = Da’ and ‘q = Db’ to be true—unreconstructed because Field and Boolos have just rejected the proposed sufficient conditions for the truth of such contexts, where ‘ p’ and ‘q’ are, respectively, direction terms, incorporated in DE. However, no other such sufficient condition has been proposed. So, if we side with Field and Boolos, we don’t have the slightest idea, actually, of what satisfaction of the antecedent of the supposedly more modest and reserved formulation could consist in. True, the reserved formulation could be made to raise an intelligible issue if relativised to an antecedently given domain of quantification—the issue would be whether any of the objects thereby already recognised, perhaps certain equivalence classes, are appropriately identified and distinguished in the light of relations of parallelism among lines. But Frege, remember, was trying to address the question how we come by and justify the conception of a domain of abstracta in the first place. If it is insisted that abstraction principles always stand in need of justification by reference to an antecedently given domain of entities, that’s just to presuppose—not argue—that they are useless in that project. And it is so far to offer no alternative conception of how the project might be accomplished. The neo-Fregean contention, by contrast, is that, under the right conditions, such principles are available to fix the truth-conditions of contexts of identity for a certain kind of thing and thereby—given appropriate input on their right-hand sides—to contribute towards determining that, and how it is possible for us to know that things of that kind exist. Boolos’s question “If numbers are supposed to be identical if and only if the concepts they are numbers of are equinumerous, what guarantee have we that every concept has a number?”, raises a doubt in the way he presumably wished to do only if it is granted that the existence of numbers is a further fact, something which the (mere) equinumerosity of concepts may leave unresolved. But the neo-Fregean’s intention in laying down Hume’s Principle as an explanation is so to fix the concept of cardinal number that the equinumerosity of concepts F and G is itself to be necessary and sufficient, without further ado, for the identity of the number of Fs with the number of Gs, so that nothing more is required for the existence of those numbers beyond the equinumerosity of the concepts. This idea is discussed more fully in the early sections of

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Wright [1997] and in Bob Hale’s [1997]. The key idea is that an instance of the left-hand side of an abstraction principle is meant to embody a reconceptualisation of the type of state of affairs depicted on the right. Here is not the place to pursue this crucial idea further. My point is merely that Boolos’s question either ignores this aspect of the neo-Fregean position or assumes it is ill-conceived.

2.2

The epistemological concern

A recurrent element in Boolos’s misgivings about Hume’s Principle concerns its proof-theoretic strength—more accurately, the strength of the system which results from its addition to axiomatic second-order logic. In part this concern relates to the ontological issues just reviewed. But there is a separate strand, nicely captured by a passage towards the end of “Is Hume’s Principle Analytic?” Boolos was the first to show that second-order logic plus Hume’s Principle is equi-interpretable with second-order arithmetic, and hence that each is consistent if the other is. 10 But he was not himself inclined to take that result as settling the question of the consistency of Hume’s Principle. He writes: . . . (it is not neurotic to think) we don’t know that second-order arithmetic . . . is consistent. Do we really know that some hotshot Russell of the 23rd Century won’t do for us what Russell did for Frege? The usual argument by which we think we can convince ourselves that analysis is consistent—“Consider the power set of the set of natural numbers . . . ”—is flagrantly circular . . . Uncertain as we are whether Frege arithmetic is consistent, how can we (dare to) call HP analytic? 11

Now, I do not myself know whether disclaiming knowledge of the consistency of Frege arithmetic is neurotic or not. But we must surely look askance at the presupposition of the concluding question, which arguably—as did Quine— confuses analyticity and certainty, or anyway insists that certainty is a precondition for warranted analyticity claims. That seems to me a great mistake. There is nothing incoherent in the idea that we can be defeasibly justified in believing or claiming to know that a proposition is true which, if true, is analytic. The neo-Fregean claim, remember, is that Hume’s Principle serves as an explanation of the concept of cardinal number. If it harbours some subtle inconsistency, then of course it fails as such an explanation—just as Basic Law V failed as an explanation of a coherent notion of set. But we can surely be fairly confident—though by all means with our eyes open—that Hume’s Principle is successful in that regard, and correspondingly confident that it enjoys the kind of truth possessed by any successful implicit definition—and hence is analytic in whatever may be the attendant sense. 10 Boolos [1987]. For a detailed proof, see the first appendix to Boolos and Heck [1998]. 11 Boolos [1997], pp. 259–60.

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2.3

23

The concern about the universal number

The construction of the finite cardinals on the basis of Hume’s Principle relies entirely on the legitimacy of applying the numerical operator to some necessarily empty concept at the first stage, the concept not self-identical being the standard choice. On the face of it there should accordingly be no obstacle to applying the operator to the complement of any such concept, so arriving at the universal number, anti-zero—the number of absolutely everything that there is. Certainly Hume’s Principle as standardly formulated poses no obstacle to such an application. As Boolos puts it, As there is a number, zero, of things that are non-self-identical, so, on the account of number we have been considering, there must be a number of things that are self-identical, i.e., the number of all the things that there are. 12

Now, Hume’s Principle can be no less dubious than any of its consequences, one of which is the claim then that there is such a number. But . . . the worry is this: is there such a number as anti-zero? According to [ZF] there is no cardinal number that is the number of all the sets there are. The worry is that the theory of number we have been considering, Frege arithmetic, is incompatible with Zermelo–Frankel set theory plus standard definitions . . . one who seriously believes that [HP is an analytic truth] has to be bothered by the incompatibility of the consequence of Frege arithmetic that there is such a number as anti-zero with the claim made by ZF plus standard definitions . . . that there is no such number. 13

This objection, Boolos wrote, although it “may at first appear to be dismissible as silly or trivial”, is “perhaps the most serious of all”. It’s certainly an arresting objection, about which there is a good deal to say. Clearly there would be great discomfort in regarding any principle as analytically true if the cost of doing so was regarding Zermelo–Frankel set theory as analytically false. A first rejoinder would be that any such upshot would depend on cross-identification of the referents of terms in Frege arithmetic and terms in Zermelo–Frankel set theory—the “standard definitions” to which Boolos alludes. Who said numbers like anti-zero had to be sets, after all? However the more general worry underlying Boolos’s point—the worry about the coherence of Hume’s Principle with standard set theory—need not depend on such cross-identification. Grant the plausible principle (to which I shall return below) that there is a determinate number of F’s just provided that the F’s compose a set. Zermelo–Frankel set theory implies that there is no set of all sets. So it would follow that there is no number of sets. Yet for all we have so far seen, the property, set, lies within the range of the secondorder quantifiers in Hume’s Principle and the usual proof, via the reflexivity of equinumerosity, should therefore serve to establish, to the contrary, that there 12 Boolos [ibid.], p. 260. 13 Ibid.

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is such a number. So there would seem to be a collision with Zermelo-Frankel set theory in any case, whether or not anti-zero is identified with a set. However I think there is good reason to expect a principled and satisfying response to this general trend of objection. Consider the direction equivalence, DE, again. The reflexivity of the relation, . . . is parallel to . . . , ensures in the presence of DE that a has a direction, no matter what straight line a may be. But the question arises: what of the implications of DE for the case where a and b fail to be parallel because they are not even lines, as for example my hat fails to be parallel to my shoe. We might have been tempted to allow that the D-operator is totally defined—to allow that every object, without restriction, has a direction: in the case of an object which fails to be parallel to anything else because it is merely not a line, this would then be a direction that nothing else has. But a moment’s reflection shows that is not an option: if the failure of parallelism between my hat and my shoe is down to the unsuitability of either object to be parallel to anything, then by the same token they are not self parallel, and DE provides no incentive to regard either as having a direction at all. Moral: just as not every object is suitable to determine a direction, so we should not assume without further ado that every concept—every entity an expression for which is an admissible substituen for the bound occurrences of the predicate letters in Hume’s Principle—is such as to determine a number. That’s only a first step, of course. What is wanted for the exorcism of antizero is nothing less than grounds for affirming that whereas the concept, not self-identical, or any other self-contradictory concept, is a suitable case for application of the numerical operator, its complement is not. Here are two, independent such lines of thought: The first line is directed specifically at anti-zero. To accept Frege’s insight that statements of number are higher-level—that they state things of concepts—is quite consistent with the familiar observation that a restriction is needed which he does not draw. The basic case in which the question, how many F’s are there? makes sense—or at least has a determinate answer— is that of a special class of substitutions for ‘F’: what are sometimes called ‘count nouns’, or expressions for ‘sortal concepts’. While it is by no means the work of a moment to make this notion sharp, the usual intuitive understanding is that a sortal concept is one associated both with a criterion of application— a distinction between the things to which it applies and those to which it does not —and a criterion of identity: some principle determining the truth values of contexts of the form, ‘X is the same F as Y ’. ‘Tree’, ‘person’, ‘city’, ‘river’, ‘number’, ‘set’, ‘time’, ‘place’, are all, in at least certain uses, sortal concepts in the intended sense. By contrast, ‘red’, ‘composed of gold’, ‘large’—in general, purely qualitative predicates, predicates of constitution, and attributive adjectives—although syntactically admissible substituents for occurrences of the predicate letters in higher-order logic, are not. Call the latter class of expressions: mere predicables. Where F is a mere predicable, then, the suggestion is that the question, how many F’s are there? is ceteris paribus

Is Hume’s Principle Analytic?

25

deficient in sense and ‘the number of F’s’, accordingly, has no determinate reference. It is easy to see that ‘is self-identical’ is a mere predicable. For reflect that—prescinding from any cases of vagueness—mere predicables do nevertheless subserve determinate questions of cardinal number when their scope is restricted to that of some specific sortal concept: thus there can be a determinate number of red apples in the bowl, of gold rings in the jeweller’s window, and of large women at the reception. So if ‘self-identical’ were a sortal concept, it should follow that there can be determinate numbers of red self-identicals in the bowl, golden self-identicals in the jeweller’s window, and large self-identicals at the reception. However since ‘F and self-identical’ is equivalent to ‘F’, it follows that there can be no such determinate number wherever there is no determinate number of Fs. So self-identity is not a sortal concept. If we take it that, save where F is assured an empty extension on purely logical grounds, 14 only sortal concepts, and concepts formed by restricting a mere predicable to a sortal concept, have cardinal numbers, it follows that there is no universal number. To be sure, this first consideration will of course not engage the question whether we may properly conceive of a number of all ordinals, or all cardinals, or all sets—in general, cases where we are concerned with the results of applying the numerical operator to concepts which are (presumably) sortal but “dangerously” big. And as we saw, a variant of Boolos’s objection, that there is a potential clash of Hume’s Principle with Zermelo–Frankel set theory, does equally arise in those cases. However, a principled objection to the idea that there should be determinate numbers associated with these concepts may be expected to issue from the second line of thought, which concerns the tantalising notion of indefinite extensibility. As noted a little while ago, it seems natural and well motivated to suppose that the Fs should have a determinate cardinal number just when they compose a set. But a long tradition in foundational studies would argue that set-hood cannot be the right way to conceive of Frege’s intentionally all-inclusive domain of objects: that Cantor’s paradox shows, in effect, that there can be no universal set—no absolutely all-embracing totality which is subject, for example, to the operations and principles that provide for the proof of Cantor’s theorem. That is not the same as saying that unrestricted first-order quantification is illegitimate—a concession which would, of course, be fatal to Frege’s 14 A plausible general principle (suggested to me by Bob Hale) of which this exception would be a special case would be this: that a non-sortal concept, F, may nevertheless have a determinate cardinal number if every sortal restriction of it has the same cardinal number. This would not, of course, legitimate anti-zero, since the cardinality of sortal restrictions of the form, Gx & x = x, will vary with that of G. But it would save the standard Fregean definition of zero. (Would there be any instances of this principle other than those mere predicables which are necessarily uninstantiated?) More generally, we might—indeed, ought to—allow that a non-sortal F may determine a number if we know that all and only F-things are G, where G is sortal and non-indefinitely extensible. (But again, are there any such cases?)

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whole project. The point is rather that the objects that lie in the range of such unrestricted quantification compose not a determinate totality but one that is, in the phrase coined by Michael Dummett, “indefinitely extensible” 15 —a totality of such a sort that any attempt to view it as a determinate collection of objects will merely subserve the specification of new objects which ought, intuitively, to lie within the totality but cannot, on pain of contradiction, be supposed to do so. I do not know how best to sharpen this idea, still less how its best account might show that Dummett is right, both to suggest that the proof-theory of quantification over indefinitely extensible totalities should be uniformly intuitionistic and that the fundamental classical mathematical domains, like those of the natural numbers, or the reals, should also be regarded as indefinitely extensible. But Dummett could be wrong about both those points and still be emphasising an important insight concerning certain very large totalities— ordinal number, cardinal number, set, and indeed “absolutely everything”. If there is anything at all in the notion of an indefinitely extensible totality—and there are signs that the issue is now being taken up in productive ways 16 — one principled restriction on Hume’s Principle will surely be that F and G not be associated with such totalities. So that is a second definite programme for understanding how, in particular, not self-identical might determine a cardinal number even though self-identical does not. Indeed, when the range of both individual and higher-order variables is unrestricted, the complement of any determinate finite concept is presumably always an indefinitely extensible totality.

2.4

The concern about surplus content

This is the objection I find it hardest to be sure I properly understand. Here is one of Boolos’s expressions of it: It is known that Hume’s Principle does not follow . . . from the conjunction of two of its strong consequences: . . . that nothing precedes zero and that precedes is a one–one relation. If HP is analytic, then it is strictly stronger . . . than some of its strong consequences. It’s also known that arithmetic follows from these two statements alone . . . faced with these results, how can we really want to call HP analytic? 17

The objection is developed and endorsed by Richard Heck in recent work, 18 and I shall rely on his interpretation of it. Heck emphasises that there is a long conceptual leap involved in advancing to the concept of cardinal number enshrined in Hume’s Principle in full generality for one whose previous 15 Dummett first introduced this notion—which of course ultimately derives from one strand in Russell’s

Vicious Circle Principle—in his [1963] (reprinted in M. Dummett, Truth and Other Enigmas, London: Duckworth 1978, pp. 186–201). It is central to the argument of the concluding chapter of Dummett [1991]. See also his “What is Mathematics About?” in Dummett [1993] at pp. 429–45. 16 See for instance Clark [1998], Oliver [1998], and Shapiro [1998]. 17 Boolos [1997], p. 249. 18 Heck [1997a].

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acquaintance with cardinal number—a pre-Cantorian as it were—is restricted to finite arithmetic and its applications. The length of the leap is reflected in the results about the proof-theoretic strength of various systems, including Fregean arithmetic—i.e., Hume’s Principle plus second-order logic—secondorder Peano arithmetic and certain intermediaries which, building on work of Boolos’s, Heck demonstrates. 19 Here is his conclusion: . . . HP, conceptual truth or not, cannot be what underlies our knowledge of arithmetic. And no amount of reflection on the nature of arithmetical thought could ever convince one of HP, nor even of the coherence of the concept of cardinality of which it is purportedly analytic. Granted, any rationalist project of this sort will have to invoke a distinction between the ‘order of discovery’ and the ‘order of justification’. But the objection is not that Hume’s Principle is not known by ordinary speakers, nor that there was a time when the truths of arithmetic were known, but HP was not. It is that, even if HP is thought of as ‘defining’ or ‘introducing’ or ‘explaining’ our present concept of cardinality, the conceptual resources required if one is so much as to recognise the coherence of this concept (let alone HP’s truth) vastly outstrip the conceptual resources employed in arithmetical reasoning. Wright’s version of logicism is therefore untenable. 20

Heck goes on to consider whether some form of Hume’s Principle restricted to finite concepts might be resistant to the particular objection, that is, whether such a principle might be appreciable as a correct digest of its constitutive principles by one possessed just of the conceptual resources deployed in finite cardinal arithmetic and its applications. That is an interesting question, on which he offers interesting formal and informal reflection. But I have a prior difficulty in seeing that the original objection, concerning the conceptual excess of Hume’s Principle over second-order Peano arithmetic, does any serious damage to any contention that the neo-Fregean should want to make. Grant that a recognition of the truth of Hume’s Principle cannot be based purely on analytical reflection upon the concepts and principles employed in finite arithmetic. The question, however, surely concerned the reverse direction of things: it was whether access to those concepts and validation of those principles could be achieved via Hume’s Principle, and whether Hume’s Principle might in its own right enjoy a kind of conceptual status that would make that result interesting. The latter is, in effect, exactly the question raised by our title. But no particular view of it can be motivated merely by the reflection that the conceptual resources involved in Hume’s Principle, insofar as an extension of the notion of cardinal number to the infinite case is involved, considerably exceed those involved in ordinary arithmetical competence. More: it is unclear how anyone wishing to demonstrate the analyticity of arithmetic could clear-headedly acquiesce in the rules of debate implicit 19 See Heck [1997a], Section 4. 20 Heck [1997a], pp. 597–8.

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in Heck’s discussion. Those rules require that one canvasses some principle which is supposedly analytic of ordinary arithmetical concepts in the precise sense that it could be recognised by reflection as systematising those ordinary concepts and their proof theory. But, of course, an axiom could, in that sense, be analytic of a thoroughly synthetic theory, and itself as synthetic as that theory. (There might be a single such axiom which could be reflectively recognised as systematising exactly Euclidean geometry.) To be sure, it is a necessary condition of the success of the neo-Fregean project that the relevant principle does more than generate a theory within which arithmetic can be interpreted—there has to be a tighter conceptual relationship than that. But it is no necessary condition for the satisfaction of this necessary condition that there be no conceptual surplus of the axiom over the theory. And it is no sufficient condition of the analyticity of such an axiom that there be none; for again, a reflectively correct digest of a synthetic theory will be itself synthetic.

2.5

The concern about bad company

Boolos’s final objection is perhaps the most interesting and challenging of all. It begins with the excellent observation that there are close analogues of Hume’s Principle, specifically, principles taking the form of second-order abstractions, linking the obtaining of a (second-order logically definable) equivalence relation on concepts to the identity condition for certain associated objects, which are self-consistent (that is, the systems consisting of secondorder logic plus one of these principles are, arguably, consistent) yet which are inconsistent with Hume’s Principle. A nice example is what I have elsewhere called the Nuisance Principle (NP). The nuisance associated with the concept F is the same as the nuisance associated with the concept G just in case the symmetric difference between F and G—the range of things which are either F or G but not both—is finite. Straightforward set-theoretic reasoning leads to the conclusion that any universe in which NP is satisfied must be a finite one. 21 But it is, apparently, a self-consistent principle—it does have finite models. If Hume’s Principle is analytic, then NP is analytically false. But with what right could we make that claim—isn’t the analogy between the two principles near enough perfect? This challenge—there dubbed the ‘Bad Company’ objection—is treated in some detail in Wright [1997] on which Boolos’s “Is Hume’s Principle Analytic?” was commentary. My suggestion in that paper was that the first step to disarming it is a deployment of (something very close to) Hartry Field’s notion of conservativeness. A principle, or set of principles, is conservative with respect to a given theory when, roughly, its addition to that theory results

21 For details see Wright [1997] at pp. 221–5.

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in no new theorems about the old ontology. 22 One could hope that Hume’s Principle will be conservative with respect to any theory for which secondorder Peano arithmetic is conservative (that is, one would hope, any theory whatever). By contrast, the consistent augmentation of any theory, T , by NP will result in a theory of which it is a consequence that all categories of the original ontology of T are at most finitely instantiated. No pure definition could permissibly have that effect. So no merely conceptual-explanatory principle— no principle whose role, as that of abstractions is supposed to be, is merely to fix the truth-conditions of a range of contexts featuring a new kind of singular-term forming operator and is otherwise to be as close as possible to that of a pure definition—can permissibly have it either. Since it has consequences for the size of extensions of concepts which are quite unrelated to that which it purportedly serves to introduce, NP thus cannot be viewed as such a conceptual-explanatory principle. Moreover, any abstraction principle which clashes with Hume’s Principle by requiring the finitude of any domain in which it is to hold will be in like case. And indeed any abstraction principle which places an upper bound, finite or infinite, on the size of the universe will be non-conservative with respect to some consistent theory of things other than the abstracts it concerns. The particular analogy is therefore broken: Hume’s Principle, there is undefeated reason to hope, is conservative with respect to every consistent theory concerning things other than its own special ontology—the cardinal numbers. (That is, note, a kind of weak analyticity: if there were a possible world in which Hume’s Principle failed, it would have to be by dint of its misrepresentation of the nature of the cardinals in that world.) NP and its kin, by contrast, come short by this constraint. An abstraction is acceptable only if it is conservative with respect to every consistent theory whose ontology does not include its proper abstracts. It is a logical abstraction just in case its abstractive relation is definable in higherorder logic. The company kept by Hume’s Principle is thus, we may presume, that of conservative, logical abstractions. But are these all Good Companions? 22 A tidied version of the characterisation offered in Wright [1997] (at note 49, p. 232) would be as follows. Let:

()

(∀αi )(∀α j ) ((αi ) = (α j ) ↔ αi ≈ α j ),

be any abstraction. Introduce a predicate, Sx, to be true of exactly the referents of the -terms and no other objects. Define the -restriction of a sentence T to be the result of restricting the range of each objectual quantifier in T to non-S items—thus each sub-formula of T of the form (∀x)Ax is replaced by one of the form (∀x)(¬ Sx → Ax) and each sub-formula of the form (∃x)Ax is replaced by one of the form (∃x)(¬ Sx & Ax). The -restriction of a theory θ is correspondingly the theory containing just the -restrictions of the theses of θ . Let θ be any theory with which is consistent. Then is conservative with respect to θ just in case, for any T expressible in the language of θ , the theory consisting of the union of () with the -restriction of θ entails the -restriction of T only if θ entails T. The requirement on acceptable abstractions is, then, that they be conservative with respect to any theory with which they are consistent. (The tidying referred to, for which I am indebted to Alan Weir, consists in having the reference to the -restriction of θ , rather than as originally one simply to θ , in the clause for ‘conservative with respect to θ ’.)

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Recent critics of neo-Fregeanism have observed that they are not, so that the fifth concern extends beyond the point that Boolos himself took it to. I pursue the matter in the first Appendix.

3. It should now be apparent why I suggested earlier that my debate with Boolos could as well have proceeded, near enough, without recourse to the notion of analyticity. The point is simply that each of Boolos’s objections is, in effect, independent of the problematical aspects of that notion: what was really bothering him was not whether Hume’s Principle is analytic, but whether it is true, and whether and how we might be warranted in regarding it as being so. Thus, without any really significant loss, the five points of concern might be formulated as: 1. With what right do we regard ourselves as warranted in accepting a principle with such rich ontological implications—how do we know that there is any function which behaves as the referent of octothorpe must? 2. What warrant do we have for confidence that the strong theory—Fregean arithmetic—to which Hume’s Principle gives rise is a consistent theory? 3. Is not its inconsistency with Zermelo–Frankel set theory (plus standard definitions) a strong ground for doubting the truth of Hume’s Principle? 4. What warrant is there for accepting a principle which is supposed to provide a foundation for arithmetic yet has so much surplus content over arithmetic? 5. With what right do we accept a principle which seems to be on all fours with other consistent principles which are inconsistent with it?

These are all good concerns, and I hope I have indicated, point-by-point, something of the direction in which the neo-Fregean should try to launch respective responses to them. The crucial point remains that the notion of analyticity is not required to formulate the concerns. What is really at stake, rather, is the nature of our entitlement to Hume’s Principle. A worked-out account of the notion of analyticity, in all its varieties, might well provide an answer to the question. But the answer the neo-Fregean wants to give is not hostage to the provision of such an account. Let me rapidly recapitulate that answer. The neo-Fregean thesis about arithmetic is that a knowledge of its fundamental laws (essentially, the Dedekind-Peano axioms)—and hence of the existence of a range of objects which satisfy them—may be based a priori on Hume’s Principle as an explanation of the concept of cardinal number in general, and finite cardinal number in particular. More specifically, the thesis involves four ingredient claims: 23 (i) that the vocabulary of higher-order logic plus the cardinality-operator, octothorpe or ‘Nx: . . . x . . . ’, provides a sufficient definitional basis for a statement of the basic laws of arithmetic; 23 I here rely again on formulations given in Wright [1998a].

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(ii) that when they are so stated, Hume’s Principle provides for a derivation of those laws within higher-order logic; (iii) that someone who understood a higher-order language to which the cardinality operator was to be added would learn, on being told that Hume’s Principle governs the meaning of that operator, all that it is necessary to know in order to construe any of the new statements that would then be formulable; (iv) finally and crucially, that Hume’s Principle may be laid down without significant epistemological obligation: that it may simply be stipulated as an explanation of the meaning of statements of numerical identity, and that—beyond the issue of the satisfaction of the truth-conditions it thereby lays down for such statements—no competent demand arises for an independent assurance that there are objects whose conditions of identity are as it stipulates.

The first and third of these claims concern the epistemology of the meaning of arithmetical statements, while the second and fourth concern the recognition of their truth. With which of them would Boolos disagree? Even with a qualification I will come to in a minute, I think he had no quarrel with the first; nor, of course, with the second, which is just the point proved by Frege’s Theorem. And to accept just these two claims, of course, is already to acknowledge a substantial Fregean achievement: the analytical reduction of the primitive vocabulary of arithmetic to a base that contains just one nonlogical expression, the cardinality operator; and a demonstration that, on that basis, the fundamental laws of arithmetic can be reduced to just one: Hume’s Principle itself. The qualification concerning the first claim concerns the interpretation of the phrase “sufficient definitional basis”. No question of course but that Frege shows how to define expressions which comport themselves like those for successor, zero, and the predicate ‘natural number’, thus enabling the formulation of a theory which allows of interpretation as Peano arithmetic. But—as we remarked right at the start—it is one thing to define expressions which, at least in pure arithmetical contexts, behave as though they express those various notions, another to define those notions themselves. And it is the latter point, of course, that is wanted if Hume’s Principle is to be recognised as sufficient for a theory which not merely allows of pure arithmetical interpretation but to all intents and purposes is pure arithmetic. How is the stronger point to be made good? Well, I imagine it will be granted that to define the distinctively arithmetical concepts is so to define a range of expressions that the use thereby laid down for those expressions is indistinguishable from that of expressions which do indeed express those concepts. The interpretability of Peano arithmetic within Fregean arithmetic ensures that has already been accomplished as far as all pure arithmetical uses are concerned. So any doubt on the point has to concern whether the definition of the arithmetical primitives which Frege

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offers, based on Hume’s Principle and logical notions, are adequate to the ordinary applications of arithmetic. Did Frege succeed in showing how the concepts of arithmetic, as understood both in their pure and applied uses, can be understood simply on the basis of second-order logic and the numerical operator, as constrained by Hume’s Principle, or could someone fully understand the entirety of the construction without having the slightest inkling of the ordinary meaning of arithmetical claims? The matter needs more detail than I will offer here, but I think it’s clear that Frege did succeed in the more ambitious task, and a crucial first step in seeing that he did so is to realise that Hume’s Principle provides for the proof of a very important principle, dubbed N q by Bob Hale, to the effect that for each numeral, ‘n f ’, defined in Frege’s way, we can establish that n f = Nx : Fx ↔ there are exactly n Fs where the second occurrence of ‘n’ is schematic for the occurrence of an arabic numeral as ordinarily understood. 24 It follows that each Fregean numeral has exactly the meaning in application which it ought to have. That seems to me sufficient to ensure that Hume’s Principle itself enforces the interpretation of Fregean arithmetic as genuine arithmetic, and not merely a theory which can be interpreted as such. If this is right, then the key philosophical issues must concern the third and fourth claims. The importance of the third claim derives from the consideration that Hume’s Principle is not, properly speaking, an eliminative definition— it allows the construction of uses of the numerical operator which it does not in turn provide the resources eliminatively to define. Its claim to serve as an explanatory basis for arithmetic must therefore depend on its ability somehow to explain such uses in a non-strictly definitional fashion. Arguing the point requires stratifying occurrences of the numerical operator in sentences of Fregean arithmetic according to the degree of complexity of the embedding context, and making a quasi-inductive case: first, that a certain range of basic uses are unproblematic, and second, that at every subsequent stage, the type of occurrence distinctive of that stage may be understood on the basis of an understanding of the mode of occurrence exemplified at the immediately preceding stage. There are some complications with this; I’ve tried to work through the point in some detail elsewhere, 25 and will not repeat the detail here. For what it’s worth, it is Michael Dummett, rather than Boolos, who has been the most vociferous opponent of the third neo-Fregean claim. 24 I reproduce in the second Appendix the proof of this claim given at pp. 366–8 of Wright [1998]. 25 See Section V of Wright [1998] and—for a supplementary consideration in response to an objection

of Dummett’s—Section VI of Wright [1998a].

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It is the fourth claim—the claim that Hume’s Principle can be laid down as an explanatory stipulation, without further epistemological obligation— which seems to me to be the heart of the issue. Boolos was indeed uncomfortable with this claim, suspecting that more had been smuggled into the notion of explanation in this setting than was consistent with the seeming modesty of the explanatory thesis. But I do not feel that I have understood his reservations terribly well. If nominalism is a misconception—if it is possible to know of abstract entities and their properties at all—then it has to be because we have so fixed the use of statements involving reference to and quantification over such entities as to bring the obtaining of their truthconditions somehow within our powers of recognition. And whatever this fixing consisted in, it has to have been something we did by way of determination of meaning, and it should therefore have involved no epistemological obligations which are not involved in the construction of concepts and the determination of meanings generally. I really do not see why the fashion in which Hume’s Principle—if it indeed succeeds in doing so—determines the truthconditions of statements which configure the cardinality operator with secondorder logical concepts, should be epistemologically any more problematical than any definition or other form of stipulation whose effect is to fix the truthconditions of statements containing a targeted (type of ) term. It is of course— always—another question whether those truth-conditions are satisfied: something which a definition, without supplementary considerations, is powerless to determine. But a good abstraction principle always determines very explicitly what those supplementary considerations are to be—you have only to look at its right-hand side. If there are good reservations about this way of looking at Hume’s Principle, I do not think that they have yet been compellingly formulated. Whatever the ultimate assessment of that issue may prove to be, it is my hope that the foregoing overview of Boolos’s misgivings about the analyticity of Hume’s Principle may serve as a reminder of two things: first (we owe it to Frege to recognise) that there is still an unresolved debate to be had about the viability of something that is, in all essential respects, a Fregean philosophy of arithmetic and real and functional analysis; 26 second, that the progress made in the modern debate is owing in very considerable measure to George’s brilliant and unique articles on the issues. 26 This is a point that Boolos enthusiastically accepted:

. . . I want to endorse Wright’s . . . suggestion that the problems and possibilities of a Fregean foundation for mathematics remain [wide?] open and [his] remark . . . that ‘the more extensive epistemological programme which Frege hoped to accomplish in Grundgesetze is still a going concern. (Boolos [1997], p. 246). For interesting preliminary steps towards the extension of the neo-Fregean programme to the classical theory of the reals, see Bob Hale [2000].

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Appendix A. Conservativeness and modesty In their [2000], Shapiro and Weir observe that there are pairs of abstractions which result by various kinds of selection for φ in (D)

(∀F)(∀G)( F = G ↔ (φ F & φG) ∨ (∀x)Fx ↔ Gx)27

which are jointly unsatisfiable yet which are presumably conservative in the germane sense. For instance, take φ respectively as ‘is the size of the universe and some limit inaccessible’ and ‘is the size of the universe and some successor inaccessible’. (The Neo-Fregean should resist any tendency to impatience at the rarefied character of the example. These notions are definable in higherorder logic.) Any instance of schema (D) entails that some F is φ. So the two indicated abstractions respectively entail that the universe is limit-inaccessible sized and that it is successor-inaccessible sized. It cannot be both. Yet neither implication places any overall bound on the size of the universe—so these abstractions do not involve the kind of non-conservativeness which NP entrained. Still, they cannot both be in good standing. And if either is not, then it seems that neither should be. But by what (well-motivated) principle might they be excluded? What virtue does Hume’s principle have which they lack? What is intuitively salient about any D-schematic abstraction (henceforward “Distraction” 28 ) is that, the entailment notwithstanding, it provides no motive to believe that there is a concept which falls under its particular selection for ‘φ’—the result is obtained merely by exploitation of the embedded antinomy. For on the assumption of (∀F)¬(φ F) any Distraction entails Basic Law V: (∀F)(∀G)( F = G ↔ (∀x)(Fx ↔ Gx)) and thereby Russell’s Paradox. Such abstractions thus have no more bearing on the truth of the relevant ‘(∃F)φ F’ than instances of the following schema have: (∀F)F is φ-terological ↔ F does not apply to itself or φ F which likewise, on the assumption of (∀F)¬φ F entail the well-known Heterological paradox: (∀F)F is heterological ↔ Fdoes not apply to itself. 27 This is schema (D) discussed in some detail in Wright [1997]; see pages 216 and following. 28 Alan Weir’s puckish term.

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Again, we can select for ‘φ F’ that F is the size of the universe and some limit inaccessible, or the size of the universe and some successor inaccessible, or that F applies to God, or the Devil . . . and proceed to infer that the universe is limit-inaccessible in cardinality, or successor inaccessible, or that God, or the Devil, exists. It is long familiar how Liar-family paradoxes can occur not merely in contexts of self-contained aporia but may be exploited to yield unmotivated a priori resolutions of intuitively unrelated issues. The Cretan and the Curry Paradox are the best known examples of the latter. The schema for φ-terologicality, and Distractions as a class, merely provide two more. This perspective offers the option of a ‘holding’ response to the Shapiro/ Weir objection: “You persuade me”, the neo-Fregean may say, “that the general idea that a concept may be defined by stipulation of its satisfactionconditions is somehow confounded by the possibility of pairwise incompatible yet consistent instances of the rubric for φ-terologicality and I will concede that the neo-Fregean conception of an abstraction principle is put in similar difficulties by conservative yet pairwise incompatible instances of (D).” This response is dialectically strong. Who would suppose that roguish cases like “heterological” and instances of φ-terologicality somehow show that we may no longer in good intellectual conscience regard the general run of definitions of the form: X is F if and only if . . . X . . . as successful in fixing concepts? But then someone who had no other objection to the claim of Fregean abstractions to play the role of truth-condition fixers for the kinds of context that feature on their left-hand sides should not be fazed by roguish instances of (D). 29 It is only a holding response, however. It refurbishes one’s confidence that it has to be possible to draw the distinction which the neo-Fregean needs, but it does not draw it. The fact remains that just as a general explanation is owing of which are the pukka definitions of satisfaction-conditions, and which may be dismissed as rogues, so we still need a characterisation of which are the good abstractions and which are the (conservative but still) bad Distractions. In Wright [1997], motivated in part by the desire to legitimate Boolos’s axiom New V: (∀F)(∀G)( F = G ↔ (Big(F) & Big(G) ∨ (∀x)Fx ↔ Gx)) (where F is Big just if it has a bijection with self-identity) I ventured an additional conservativeness constraint which would be tolerant of at least some instances of schema (D) but would reject the majority. Roughly, it was that those consequences of such an abstraction which follow by exploitation of its “paradoxical component” have to be in ‘independent good standing’. I shall here attempt briefly to clarify and assess this proposal. 29 Cf. Wright [1997], pp. 220–1.

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Distractions entail conditionals of the form: ¬(∃F)(φ F) → (∀F)(∀G)( F = G ↔ (∀x)(Fx ↔ Gx)) The immediate intent of the proposed constraint is that anything derivable by the reductio of the antecedent of such a conditional afforded by its paradoxical consequent should be in independent good standing. New V fares well by this proposal: that there is a concept which is Big should presumably be a result in ‘independent good standing’ however that idea is filled out—for that selfidentity itself is Big follows from the definition of ‘Big’ in second-order logic. Of course, any abstraction will entail some such conditional. So the proposed constraint is quite general. How does Hume’s principle fare by it? Well enough, presumably, though in a different way. We may, for instance, obtain a relevant conditional by selecting ‘at least countably infinite’ for φ. But this time the resources required to make good the consequences of the denial of the antecedent are afforded not just by second-order logic but by Hume’s Principle itself, via its independent proof of the infinity of the number series. Indeed, it is just because it independently entails that denial that we are able to show that Hume’s Principle entails the selected conditional in the first place. By contrast, the kinds of roguish Distraction illustrated presumably fail the test. The only resources they have to show, e.g. that the universe is limit-inaccessible, or successor inaccessible, or whatever, are those furnished by the inconsistency of Basic Law V and the consequent modus tollens on the relevant conditional. So: an abstraction is good only if any entailed conditional whose consequent is Basic Law V (or, therefore, any other inconsistency) is such that all further consequences which can be obtained by discharging the antecedent are in independent good standing, as may be attested by their derivation in pure higher-order logic (like the case of New V) or their independent derivability from the abstraction in question (like the case of Hume’s Principle). But this is unclear in a crucial respect: what is the relevant sense of ‘independent derivability’? Clearly it would not be in keeping with the intended constraint if there were merely some collateral derivation of just the same suspect kind. The ‘independent derivation’ must be bona fide, must not proceed by “paradoxexploitative” means, as I expressed the matter. But what does that mean? In particular, how might it be characterised so as not to outlaw any proof by reductio ad absurdum? One possible response—the one I offered in Wright [1997]—was that a relevantly narrow sense of “paradox-exploitative” may be captured by reinvoking the previous (Fieldian) notion of conservativeness in the following way: a derivation from a conservative abstraction is paradox-exploitative just if there is a representation of its form of which any instance is valid and of which some instance amounts to a proof of the non-conservativeness of another abstraction. For instance, the derivation of the successor-inaccessibility of the universe from the Distraction canvassed above is paradox-exploitative because it may be schematised under a valid form of which another instance is a

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derivation, from the appropriately corresponding Distraction, that the universe contains exactly 144 objects. The only Distractions which are good are those which are both conservative and such that any of their consequences which may validly be derived by paradox-exploitative means, in the stipulated sense, may also validly be derived by non-paradox-exploitative means. Otherwise put: the second conservativeness constraint is that the paradox-exploitative derivations from an abstraction have to be conservative with respect to the results obtainable from it non-paradox-exploitatively. That was the essence of my previous proposal. In practice, its application would work like this. We would be defeasibly entitled to accept any (presumably) conservative abstraction, A, from which we had so far been able to construct no paradox-exploitative derivation—no proof of a valid form of which another instance demonstrated the non-conservativeness of another abstraction. But once we had such a derivation, it would then be inadmissible to accept A until we had found another non-paradox-exploitative derivation from it of the same conclusion: a formally valid derivation of which, so far as we could tell, no other instance was a proof of the non-conservativeness of another abstraction. That is apt to seem uneasily complex and less clearly motivated than one would wish. And one might worry about its reliance on our ability to judge non-paradox-exploitative derivations. However the play with ‘paradoxexploitation’, and its characterisation in terms of non-conservativeness may now seem inessential. The basic idea was that some abstractions—the Distractions and some others—are at the service of non-cogent proofs. We can tolerate this in particular cases so long as such proofs are matched by cogent ones of the same things. The natural—surely correct—objection to the derivation of, say, the successor inaccessibility of the universe from the appropriate Distraction is that it is unconvincing because “You could just as well prove the opposite—or anything—like that”, where “like that” means: by laying down a different (presumably consistent) Distraction and reasoning in just the same way. So a natural thought would be that we should ban those distractions—or abstractions generally—some of whose consequences are such as to deserve that complaint. That would suggest the following stipulation: that an abstraction A is unacceptable, at least pro tempore, if every proof it has yielded of some consequence C is such that, schematised so that any instance of it is valid, some other (conservative) abstraction yields a proof of the same form of something inconsistent with C. But there are still a number of salient concerns. First, it is not clear that any purpose is served by the continuing insistence on derivations of a given valid form. Why not just say that pairwise incompatible but individually conservative abstractions are ruled out—however the incompatibility is demonstrated— and have done with it? For think: if each such pair can be shown to be incompatible by proofs of a given single form, then the more complex formulation of the constraint is unnecessary; but if some pair cannot—if no derivation of C

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from A is of a valid form shared by some derivation of not-C from A*—then there will still be pairwise incompatible but conservative abstractions which survive the new test. So there will still be Bad Company, for which some further treatment will need to be devised. Besides, how is the proposed constraint meant to be applied to semantical—model-theoretic—demonstrations of consequence—for which of course, in the case of higher-order abstractions, there need be no effectively locating a corresponding derivation in higher-order logic? This whole direction was stimulated by the desire to save some ‘good’ Distractions, par excellence New V. It is therefore germane that, as Shapiro has since observed, New V itself is in any case non-conservative!—specifically that it entails that the universe can be well-ordered, and hence that the nonabstracts can. 30 This result, to be sure, does not show that there is nothing to be gained from attempting to refine the second conservativeness constraint of my [1997]—that it has no point. But it should occasion a re-think of the motivation for the general direction. I think there is something else amiss with the rogue Distractions— something which the second proposed constraint may well indirectly approximate but does not bring out with sufficient clarity. Start from the point that definitions proper should be innocent of substantive implications for the universe over which they range. Abstractions cannot in general match that, since in conjunction with logically (or other forms of metaphysically) necessary input, they may carry substantive implications for the abstracts whose concept they serve to introduce and hence—since those abstracts will be viewed, at least by neo-Fregeans, as full-fledged participants in the universe—at least some substantive implications for the universe as a whole. But to the extent that it is proposed to regard them as meaning-constituting stipulations, and hence as approximating definitions as nearly as possible, the character and scope of such implications needs to be curtailed. In brief: the requirement has to be that the only implications they may permissibly carry for the, as it were, enlarged universe in which their own abstracts participate must originate in what they imply—whether proof- or model-theoretically—about the abstracts they specifically concern. Hume’s Principle, for instance, implies of any object whatever that it participates in an at least countably infinite universe; but it carries that implication only via its entailment of the infinity of the cardinal numbers. This is a different requirement to Field conservativeness. A non-Field conservative abstraction—one that, as we put it intuitively above, entails new results about the prior ontology—may of course violate it. But it is possible for an abstraction to be immodest—for it to carry implications for other objects in 30 See Weir and Shapiro [1999]. In rough outline: we can derive the Burali-Forti paradox on the assumption that the concept, Ordinal, is not Big; but if Ordinal is Big, then there is a 1–1 correlation between Ordinal and x = x. So x = x may be well ordered by that correlation.

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the universe which cannot be shown to originate in implications it carries for its own proper abstracts—without thereby being demonstrably non-conservative. Consider the Limit-inaccessible Distraction again. As noted, this entails that the universe is limit-inaccessibly sized. But because, unlike NP, it places no upper bound on the size of the universe, it is not non-conservative in the way that NP is—it limits the extension of no other concept. And if it is nonconservative in any other way, we have yet to see how. However it is immodest. For its requirement that the universe have a certain kind of cardinality does not originate in any requirement that it imposes on its own abstracts. It is easy to overlook the force of “originate” here. The Limit-inaccessible Distraction, for instance, entails that any finite concept is non-φ. So it will allow singleton concepts to generate ‘well-behaved’ abstracts—abstracts whose identity and distinctness is governed by ordinary extensionality—of which there should therefore be no fewer than there are objects in the universe. 31 Thus this particular Distraction will indeed entail that its own abstracts are limit-inaccessible in number, from which the limit inaccessibility of the universe follows. 32 But—this is the crux—the result about the abstracts is not needed for the proof of the limit-inaccessibility of the universe. The Distraction provides no way of recognising the limit-inaccessibility of the universe which goes via a prior recognition of what it entails about its own proper abstracts. Rather the inference is the other way about: the proof that the Distraction entails that result about the universe as a whole is needed in order to obtain the result about its own abstracts. That is immodesty. Conservativeness constrains the kind of consequences which an acceptable abstraction is allowed to have: it is not allowable that there be any claim exclusively concerning the non-abstracts which was previously unprovable but which the abstraction, coupled with previous theory—now explicitly restricted to the previous ontology—enables us to prove. Modesty, by contrast, constrains the kind of ground which an acceptable abstraction can provide for consequences, not per se non-conservative, about the ontology of a theory in which that abstraction participates: such consequences must be grounded in what it requires of its own proper abstracts. But although the two constraints may seem different in character in this way, they are aspects of a single point au fond. Remember that the role of a legitimate abstraction, as I have repeatedly stressed, is merely to fix the truth-conditions of a class of contexts featuring a novel term-forming operator. It cannot have more than that role and yet retain the epistemically undemanding character of a meaning-stipulation. Logical abstractions, to be sure, are so designed that, consistently with their playing just this role, logical resources may enable us to show that there are abstracts of the kind they concern and to establish things about them. But no abstraction can be deemed to discharge the intended limited role successfully 31 Assuming that there no fewer singleton concepts than there are objects. 32 On standard cardinal-arithmetical assumptions.

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if, in conjunction with some consistent theory, it carries implications for the combined ontology which cannot be shown to derive from implications it has for its own abstracts. Non-conservativeness is (normally) one graphic way of failing that test. But if even a conservative abstraction entails some conclusion about the combined ontology which cannot be justified by reference to what it entails about its own abstracts, then knowledge of the truth of the abstraction cannot be founded in stipulation. Such an abstraction implicitly claims something about the world which might—for all we have shown to the contrary—be justified by reference to what it entails about its own abstracts; that is why we cannot accuse it of non-conservativeness. But equally, so long as we have no such justification, we have no defence against the suggestion that the abstraction is known only if we know that the world must be that way in any case, whether or not the abstracts themselves make it so. That would seem to demand knowledge about how the world would be even if the abstracts did not make it so. And that in turn is a substantial piece of collateral information which, by being prerequisite if we are to claim to be justified in laying down the abstraction in the first place, gives the lie to any claim that the abstraction is justified merely as a meaning-stipulation. In sum: an abstraction is modest if its addition to any theory with which it is consistent results in no consequences (whether proof- or model-theoretically established) for the ontology of the combined theory which cannot be justified by reference to its consequences for its own abstracts. And again, justification is the crucial point: an abstraction may fail this constraint even though every consequence it has for the ontology of a combined theory may be seen to follow from things it entails about its proper abstracts; in particular, it will not count if, as in the case of the Limit-inaccessible Distraction, a consequence for the combined ontology is needed as a lemma in the proof that the abstracts have a property from which that very consequence follows. Further clarification is needed of several matters: what kinds of proof should count in favour of the modesty of an abstraction—what it is to show that an abstraction independently carries certain implications for its own abstracts; whether the modesty constraint is effective against the general run of pairwise incompatible but (presumptively) conservative abstractions illustrated by Shapiro and Weir; what other constraints on Good Companions may be properly motivated. At the time of writing, these are largely open issues.

B. B.1

Proof of the principle, Nq Stage-setting

We assume the standard recursive definitions of the numerically definite quantifiers: (∃0 x)Fx ↔ (∀x)¬Fx (∃n+1 x)Fx ↔ (∃x)(Fx & (∃n y)(Fy & y = x)),

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and let ‘n f ’ abbreviate Frege’s definiens for n. Define ‘Pxy’ (immediate predecession) as (∃F)(∃w)(Fw & y = Nv : Fv & x = Nz : [Fz & z = w]) Define ‘Nat(x)’ (x is a natural number) as x = 0 ∨ P ∗ 0x where ‘P*xy’ expresses ancestral predecession. Let ‘(∃R)(F 1–1 R G)’ express that there is a one–one correspondence between F and G. We take three lemmas from the proof of the Peano axioms from HP outlined in the concluding section of Frege’s Conception (numbering as there assigned): Lemma 51: (∀x)(Nat(x) → x = Ny : [Nat(y) & P ∗ yx]—every natural number is the number of its ancestral predecessors. Lemma 52: (∀x)(Nat(x) → ¬P ∗ xx)—no natural number ancestrally precedes itself. Lemma 5121: (∀x)(∀y)(Nat(x) & Nat(y) → (Pxy → (∀z)(Nat(z) & (P ∗ zx ∨ z = x) ↔ (Nat(z) & P ∗ zy)))—if one natural number immediately precedes another, then the natural numbers which ancestrally precede the second are precisely the first and those which ancestrally precede the first. Finally, recall that Frege’s 0 is Nx : x = x and that each successive n + 1 f is N x : [x = 0 ∨ · · · ∨ x = n f ]. Each of these objects qualifies as a natural number in the light of the above definition of ‘Nat(x)’. Proof: 0 f qualifies by stipulation; n + 1 f qualifies if n f does—take ‘F’ in the definition of ‘Pxy’ as ‘[x = 0 ∨ · · · ∨ x = n f ]’ and ‘w’ as ‘n f ’ to show that P(n f , n + 1 f ); then reflect that Pxy → P ∗ xy and that P ∗ xy is transitive. (Frege’s Conception, Lemmas 3 and 4, respectively.)

B.2

Proof of Nq for Frege’s natural numbers

Induction Base: To show Nx : Fx = 0 f ↔ (∃0 x)Fx, it suffices to reflect that the left-hand side holds just if (∃R)(Fx 1–1 R x = x), which in turn holds just if ¬ (∃x)Fx. 33 Induction Hypothesis: Suppose Nx : Fx = n f ↔ (∃n x)Fx. We need to show that it follows that Nx : Fx = (n + 1) f ↔ (∃n+1 x)Fx. 33 As George Boolos remarked to me, Frege himself observes, at Grundlagen §75 and §78, that he is in a position to obtain proofs of N q for 0 f and 1 f , respectively.

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(Left-to right) Consider any F such that Nx : Fx = (n + 1) f . By Lemma 51 and the reflection that Nat(n f ), n f = Nx : [Nat(x) & P ∗ xn f ]. So by the Hypothesis (∃n x)(Nat(x) & P ∗ xn f ). But by Lemma 52, ¬P ∗ n f , n f . So (∃n x)(Nat(x) & (P ∗ xn f ∨ x = n f ) & x = n f ). So (∃y)(Nat(y) & (P ∗ yn f ∨ y = n f ) & (∃n x)(Nat(x) & (P ∗ xn f ∨ x = n f ) & x = y)). So by the recursion for the quantifiers (∃n+1 x)(Nat(x) & (P ∗ xn f ∨ x = n f ). But by Lemma 5121 and since P(n f , n + 1 f ), we have that (∀x)(Nat(x) & (P ∗ xn f ∨ x = n f ) ↔ Nat(x) & P ∗ (x, n + 1 f )). So (∃n+1 x)(Nat(x) & P ∗ (x, n + 1 f )). That establishes the desired result for one concept of which (n + 1) f is the number. But by HP, any G such that (n + 1) f = Nx : Gx will admit a one-one correspondence with that concept. So a lemma to the following effect will now suffice: (∀F)(∀G)((∃R)(F 1−1 R G) → ((∃n+1 x)Fx ↔ (∃n+1 x)Gx) A proof by induction—strictly, at third-order—suggests itself: Base: It suffices to show (∀F)(∀G)((∃R)(F 1−1 R G) → ((∀x)¬Fx ↔ (∀x)¬Gx)) Hypothesis: Suppose (∀F)(∀G)((∃R)(F 1−1 R G) → ((∃n x)Fx ↔ (∃n x) Gx). Consider any H such that (∃n+1 x) Hx. Then (∃x)(Hx&(∃n y)(Hy&y = x)). Let a be such that Ha & (∃n y)(Hy & y = a). Let J be one–one correlated with H by R. Let b be such that Jb & Rab. Then R one–one correlates Hx & x = a with Jx & x = b. So, by the Hypothesis, (∃n x)(Jx & x = b). So (∃x)(Jx & (∃n x)(Jx & x = b)). So (∃n+1 x)Jx. (Right-to left) Consider any F such that (∃n+1 x)(Fx). Then there is some a such that Fa & (∃n y)(Fy & y = a). So by the Hypothesis Ny (Fy & y = a) = n f . So, by HP, there is an R such that (Fy & y = a) (1–1 R ) (Nat(x) & P ∗ xn f ). Let R # correlate (Fy & y = a) with (Nat(x) & P ∗ xn f ) in just the fashion of R, and let it also correlate a with n f . Then (Fy) 1–1 R # (Nat(x) & (P ∗ xn f ∨ x = n f )). But, as established above (∀x)(Nat(x) & (P ∗ xn f ∨ x = n f ) ↔ Nat(x) & P ∗ (x, n + 1 f )). So Nx : Fx = (n + 1) f .

References George Boolos [1986] “Saving Frege from Contradiction” in Proceedings of the Aristotelian Society 87, pp. 137–51; reprinted in Demopoulos, ed. [1995], pp. 438–52. George Boolos [1987] “The Consistency of Frege’s Foundations of Arithmetic” in Judith Jarvis Thompson, ed. [1987], pp. 3–20; reprinted in Demopoulos, ed. [1995], pp. 211–33. George Boolos [1990] “The Standard of Equality of Numbers” in Boolos, ed. [1990a], pp. 261– 77; reprinted in Demopoulos, ed. [1995] pp. 234–54. George Boolos, ed. [1990a] Meaning and Method: Essays in Honor of Hilary Putnam, Cambridge: Cambridge University Press. George Boolos [1997] “Is Hume’s Principle Analytic?” in Heck, ed. [1997], pp. 245–61. George Boolos and Richard. G. Heck, Jr. [1998] “Die Grundlagen der Arithmetik §§82–3” in Schirn, ed. [1998], pp. 407–28.

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Peter Clark [1998] “Dummett’s Argument for the Indefinite Extensibility of Set and Real Number” in Grazer Philosophische Studien 55, New Essays on the Philosophy of Michael Dummett, eds. J. Brandl and P. Sullivan (Vienna: Rodopi), pp. 51–63. William Demopoulos, ed. [1995] Frege’s Philosophy of Mathematics, Cambridge, Mass.: Harvard University Press. Michael Dummett [1963] “The Philosophical Significance of Gödel’s theorem”, Ratio 5, pp. 140–55. Michael Dummett [1991] Frege: Philosophy of Mathematics, London: Duckworth. Michael Dummett [1993] The Seas of Language, Oxford: The Clarendon Press. Hartry Field [1984] Critical Notice of Crispin Wright Frege’s Conception of Numbers as Objects, Canadian Journal of Philosophy 14, pp. 637–62; reprinted as “Platonism for Cheap? Crispin Wright on Frege’s Context Principle” in Field [1989], pp. 147–70. Hartry Field [1989] Realism, Mathematics and Modality, Oxford: Basil Blackwell. Bob Hale [1997] “Grundlagen §64”, Proceedings of the Aristotelian Society XCV11, pp. 243– 61. Bob Hale [2000] “Reals by Abstraction”, Philosophia Mathematica 8, pp. 100–23. Richard G. Heck, Jr., ed. [1997] Language, Thought and Logic, Oxford: The Clarendon Press. Richard G. Heck, Jr., ed. [1997a] “Finitude and Hume’s Principle”, Journal of Philosophical Logic 26, pp. 589–617. Alex Oliver [1998] “Hazy Totalities and Indefinitely Extensible Concepts: An Exercise in the Interpretation of Dummett’s Philosophy of Mathematics” in Grazer Philosophische Studien 55, New Essays on the Philosophy of Michael Dummett, eds J. Brandl and P. Sullivan (Vienna: Rodopi), pp. 25–50. Charles Parsons [1964] “Frege’s Theory of Number” in Philosophy in America, ed. Max Black, London: Allen and Unwin, pp. 180–203; reprinted in Demopoulos, ed. [1995], pp. 182–210. Matthias Schirn, ed. [1998] Philosophy of Mathematics Today, Oxford: The Clarendon Press. Stewart Shapiro [1998] “Induction & Indefinite Extensibility: The Gödel Sentence is True but Did Someone Change the Subject”, Mind 107, pp. 597–624. Stewart Shapiro and Alan Weir [1999] “New V, ZF and Abstraction”, Philosophia Mathematica 7, pp. 293–321. Judith Jarvis Thompson, ed. [1987] On Being and Saying: Essays in Honor of Richard Cartwright, Cambridge, Mass.: MIT Press. Crispin Wright [1983] Frege’s Conception of Numbers as Objects, Aberdeen: Aberdeen University Press. Crispin Wright [1997] “On the Philosophical Significance of Frege’s Theorem” in Heck [1997], pp. 201–44. Crispin Wright [1998] “On the Harmless Impredicativity of N= (‘Hume’s Principle’)” in Schirn [1998], pp. 339–68. Crispin Wright [1998a] “Response to Dummett” in Schirn [1998], pp. 389–405.

FREGE, NEO-LOGICISM AND APPLIED MATHEMATICS 1 Peter Clark Philosophy Department, School of Philosophical and Anthropological Studies, University of St Andrews, St Andrews, Fife, Scotland, KY16 9AL, UK

1.

Introduction—logicism and neo-logicism

A little over one hundred years ago (the letter is dated July 28, 1902) Frege wrote to Russell in the following terms: 2 I myself was long reluctant to recognize ranges of values and hence classes; but I saw no other possibility of placing arithmetic on a logical foundation. But the question is how do we apprehend logical objects? And I have found no other answer to it than this, We apprehend them as extensions of concepts, or more generally, as ranges of values of functions. I have always been aware that there are difficulties connected with this, and your discovery of the contradiction has added to them; but what other way is there?

Frege here poses an extremely good question, a recent answer to which this paper is really devoted. Whatever view one may finally adopt about whether the new answer succeeds one has to recognise that it is surely a very remarkable fact that one hundred years after the discovery of the Zermelo–Russell contradiction which follows from Basic Law Five of Frege’s Grundgesetze (Frege (1893)) we should now be actively discussing not as a purely historical enterprise but as a viable possibility in the foundations of mathematics, the programme more or less explicitly laid out in the earlier work of Frege Die Grundlagen der Arithmetik (Frege (1884)). This is because recent research has highlighted three crucial facts. First and most importantly that in full conformity with the spirit of Frege’s programme the deduction of the axioms of (Second Order) Peano Arithmetic from principles of higher order logic and “definition” 3 does not require appeal to Basic Law Five. Second that 1 This paper first appeared in Induction and Deduction in the Sciences, F. Stadler (ed.) [2004], Dordrecht, Kluwer Academic Publishers. Reprinted by kind permission of Springer Academic Publishers. 2 Frege (1902), pp. 140–41. 3 Actually one has to be careful how this result is stated. Formally the central result is that if a formalisation of the key ‘definition’ is added as an axiom to standard axiomatic second order logic, second

45 Roy T. Cook (ed.), The Arché Papers on the Mathematics of Abstraction, 45–60. c 2007 Springer.

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the principles of logic and “definition” (this definition has become known as Hume’s Principle) which are employed are consistent, 4 and third that even the proofs of the axioms for Peano Arithmetic given by Frege himself in the Grundgesetze der Arithmetik do not depend essentially upon Basic Law Five. 5 These technical facts, of which more later, open the way for the revival of Frege’s programme, they make it possible but they do not determine its form. What was Frege’s programme and what is its revived form explicitly? Frege’s programme was the result of an answer to the famous question raised in Section 62 of the Grundlagen viz.: “how then are numbers given to us, if we cannot have any ideas or intuitions of them?” The Fregean answer was “by explaining the senses of identity statements in which number words occur”. That explanation was to be provided at least in part by what has come to called Hume’s Principle: the claim that the cardinal numbers corresponding to two concepts are identical if and only if the two concepts are equinumerous. I say in part by Hume’s Principle because as Frege had already argued in another context at Section 56 of the Grundlagen whatever the merits of Hume’s Principle it can’t explain the senses of identity statements in which number words occur of the form “the number of F’s is n”, where n is not given in the form of “the number of G’s”, for some G. Frege then adopted the explicit definition of number in terms of classes or extensions “the number of F’s is the class of all concepts G, equinumerous with F”. But this explicit definition together with Basic Law Five, the comprehension axiom for class existence entails Hume’s Principle. With Axiom Five in place it looked as if Frege’s programme could be carried out. It was now possible to show that second order logic (Frege (1879)) together with Basic Law Five entails the Peano–Dedekind axioms for arithmetic. As such the truths of arithmetic could be seen to be analytic, they could all be seen to be consequences of general logical laws together with suitable implicit definitions (like Basic Law Five which implicitly defines the notion of an extension). Further arithmetic could be seen as a body of truths about independently existing objects—the finite cardinals—which were logical objects, order arithmetic (arithmetic with the full second order induction axiom) can be interpreted in the resulting theory, often called Frege Arithmetic. Certainly the result seems to have been known to Geach in the forties, Dummett in the fifties and was first recently explicitly noted by Charles Parsons in his 1965 paper “Frege’s Theory of Number” (Parsons (1965) reprinted in Demopoulos (1995), pp. 182–210). A very closely related result was published by Timothy Smiley in 1981 (Smiley (1981)). The full significance of the result as well as a well developed proof was given by Wright (1983). The result has been systematically investigated by George Boolos (who discovered it independently in the early eighties and Richard Heck (see especially Boolos (1987a), (1998), papers 17, 18 and 19, Heck (1993). The most accessible proof can be found in the Appendix to Boolos (1990a), Boolos (1998) paper 13). 4 In his 1983 Wright conjectured that the system of axiomatic second order logic together with Hume’s Principle is consistent but did not establish it. Burgess (1984), Hazen and Hodes provided elementary consistency proofs with ω and ω + 1 as domains. George Boolos however established the central consistency result which is that the theory is equi-consistent with analysis (see his (1987b), (1990b) and Boolos and Heck, paper 20 of Boolos (1998)). 5 This was certainly known to Frege (see especially Heck (1995)), but in a letter to Russell he dismissed the possibility of basing his system on Hume’s Principle, saying only that it faced difficulties which were different from those facing the attempt to use Basic Law Five (Frege (1902), letter xxxvi/7, p. 141).

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logical in the sense that knowledge of which requires nothing beyond knowledge of logic and definitions. If this could be extended to the Real numbers and other parts of mathematics then a foundation for mathematics would have been established on which mathematics was presented as uncontaminated by empirical notions, presented as a body of truths in its full classical form and shown to be applicable to reality, since it can be shown to be in fact the more and more elaborate drawing of consequences from meaning postulates. But deduction applies to everything that can be thought. In fact it seems to me that the fundamental logicist thought can be put simply as follows: there can be no thought without representation, there can be no representation without concepts and there can be no concepts without number. Of course the serpent had already entered this Eden, with the introduction of Basic Law Five which says (∀F)(∀G)(Ext(F) = Ext(G) ↔ (∀x)(Fx ↔ Gx)) But by the Comprehension Principle for Second Order Logic there is a property corresponding to the formula of Second order logic (∃F)(Ext (F) = x &¬ Fx). Russell’s paradox immediately results from allowing this property to fall under the universal quantifier (∀F) in Basic Law Five. Another way of putting the same point is to note that the Russell reasoning shows that it is a theorem of Second order logic that there is no function from properties to objects such that distinct properties (i.e. non-coextensive properties) are associated with distinct objects. This is just what Basic Law Five read from left to right in contrapositive form asserts there is. 6 So much for logicism. What about neo-logicism, the revived form of Frege’s programme? We should let Wright and Hale, the main proponents of this view speak for themselves. They say: 7 Neo-Fregeanism holds that Frege need not have taken the step which lead to this unhappy conclusion [The appearance of the Russell contradiction]. At least as far as the theory of natural number goes, it is possible to accomplish Frege’s central mathematical and philosophical aims by basing the theory on Hume’s Principle, adjoined as a supplementary axiom to a suitable formulation of second order logic. Hume’s Principle cannot, to be sure, be taken as a definition in any strict sense—any sense requiring that it provide for the eliminative paraphrase of its definiendum (the numerical operator, “the number of . . . ”) in every admissible type of occurrence. But this does not preclude its being viewed as an implicit definition, effecting an introduction of a sortal concept of cardinal number and, accordingly, as being analytic of the concept—and this, the neo-Fregean contends, coupled with the fact that Hume’s Principle so conceived requires a prior understanding only of second order logical vocabulary, is enough to sustain an account of the foundations of Arithmetic that deserved to be viewed as a form of logicism which, whilst not quite logicism in the sense of a reduction of arithmetic to logic, preserves the essential core and content of Frege’s two fundamental theses. 6 This is Frege’s own generalisation of the lesson of Russell’s paradox. (See also Boolos (1993)). 7 Hale and Wright (2000), Introduction.

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So two conditions will have to hold for this position to viable: first what has come to be known as Frege’s theorem (which certainly does hold), that is the mathematical claim that the Dedekind–Peano axioms postulates for number theory in their second order form can be derived from a combination of Second order logic and Hume’s Principle and second that it will have to be shown that Hume’s Principle and other so called abstraction principles which share its form constitute a legitimate means of introducing the names of numbers (of abstract objects in general) by, in effect, stipulation by implicit definition. One needs to be careful to state the claim of neo-logicism properly, it is: (i) Hume’s Principle is a stipulation which gives the truth conditions of a restricted class of statements of numerical identity (ii) The resulting explanation of the concept of number is complete however, in that it suffices for the second order derivation of the basic laws of arithmetic (iii) The existence of numbers is something discovered and not stipulated (the Platonism of Frege’s original theory is preserved) (iv) Our (a priori) knowledge of number is derived from a principle whose truth is a matter of stipulation. 8

Abstraction principles of which Hume’s Principle is a paradigm example come in two types conceptual abstractions and objectual ones, but all have the following form. There is a domain of entities, denoted say, by α, β, etc., and a relation R defined over them. Then an abstraction principle has the form ((α) = (β)) ↔ R(α, β) Where R( , ) is an equivalence relation among the α and β’s. An abstraction principle may be called a logical abstraction when the relation R( , ) is definable in purely logical vocabulary, e.g. equinumerosity among concepts or ordinal similarity among binary relations. Under the classical canonical interpretation (α) is the equivalence class of α under the relation R and exists (where it does) in virtue of a set existence axiom. That is the existence and uniqueness of (α) has in effect to be guaranteed by a separate principle of set or class existence. Wright and Hale however argue that in certain cases logical abstraction principles can play the role of stipulations and if the relation on the right hand side of the iff is ever satisfied then no further question concerning the existence of the (α) need arise. Conceptual abstraction principles are those in which α’s are concepts (as in the case of Hume’s Principle) and objectual abstraction principles are those in which the field of the equivalence relation comprises objects. In both cases and this is crucially so the abstracta the (α) are objects, so in the case of conceptual abstractions acts as a type down operation, from concepts to objects. 8 See also Demopoulos (1998), (2000).

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Of course Wright and Hale do not argue that it is always legitimate to introduce abstracta in this way. Two examples of conceptual logical abstraction principles which fail to introduce abstracta are Basic Law Five and what might be called Ordinal Hume which is the claim that (∀ R)(∀ S)(Ord R = Ord S ↔ R is similar to S) This has the form of an abstraction principle since similarity is an equivalence relation among binary relations. But Ordinal Hume leads directly to the Burali–Forti paradox. 9 Hume’s Principle in logical abstraction form says: (∀ F)(∀ G)(NxFx = NxGx ↔ (∃ R)(F1 − 1R G)) where (∃R)(F1−1R G) is an abbreviation for the standard formulation in the vocabulary of second order logic of the formula expressing that there is a relation R which establishes a one to one correspondence between the things falling under F and those falling under G, that is ((∃R)(∀x)(Fx→ ∃!y)(Gy & R(x,y)) & ∀z(Gz→ ∃!w)(Fw & R(w,z))) and the operator Nx . . . x is a term forming operator. Wright has argued that there are general principles which can distinguish between good and bad abstraction principles and in any case as is well known there is no similar problem about Hume’s Principle, since it is known to be consistent. Like Basic Law Five Hume’s Principle asserts the existence of a function from concepts to objects but unlike Basic Law Five it asserts that merely non-equinumerous concepts (not non-co-extensive concepts) can be sent to distinct objects and this is possible provided that the domain is (Dedekind) infinite. For a domain of k objects there are k + 1 non-equinumerous concepts definable over it, so no finite domain can satisfy Hume’s Principle. The values of the second order variables for Frege are concepts and objects are denoted by terms that may appear on either side of the identity sign, so the terms like NxFx denote objects. As such Nx . . . x will have to be thought of as a term forming operator and our theory of second order logic plus Hume’s Principle must be sufficiently strong to have as a theorem (∀F)(∃!x)(x = NxFx). In general one would expect that we would have to have an axiom asserting that for each F, NxFx was a term. This fact seems to me to have the profoundest significance for the claim that Hume’s Principle and other abstraction principles can be thought of analytic stipulations introducing the names of numbers (or other abstract objects). Nor do I see any of the various equivalent methods of adding Hume’s Principle to second order logic in order to derive Frege’s theorem as avoiding this difficulty, for they will all have to guarantee that (∀ F)(∃!x)(x = NxFx) holds in some way or other. If it is guaranteed then we may proceed as follows: consider the concept non-self identical. It is a truth of logic that the concept non-self identical is equinumerous with the concept non-self identical, so 9 See Hodes (1984) and Fine (1998).

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the right hand side of Hume’s principle for F and G both non-self identical reduces to a logical truth. So we may detach and assert as a theorem Nx ∼ (x = x) = Nx ∼ (x = x) and so infer (∃!y)(Nx ∼ (x = x) = y) and so introduce the name “zero” to designate that unique object. But that hinges on us being able to read the stipulation, to understand the stipulation, as providing a context into which it is appropriate to quantify in. But what guarantees that is the correct reading of the stipulation? This is my first sceptical argument then: it essentially concerns how the underlying logic is to function. On the one hand to presume NxFx is a term because it appears on one side of the identity sign seems to me to beg the question, while if the question is not begged and say a free logic is employed then I fail to see how the required existential postulates will form any intended contrast with “mere axiomatic postulation”. 10 But I will not dwell further on this matter here.

2.

Anti-zero Following Boolos we can write Hume’s Principle in the form ∗

F = ∗ G ↔ (F1 − 1G)

and understand it as asserting that there exists a total function from concepts to objects, call it *, such that non-equinumerous concepts are assigned distinct objects (that is the contrapositive of Hume’s principle read from left to right). Adding this principle to Second Order logic allows us to prove in the Fregean way that, with the successor relation defined in the usual manner viz., S(n, m) ↔ (∃ F)(∃y)(Fy & ∗ F = n & ∗ (x : Fx & x = y) = m) (i) the successor relation is functional and one–one (ii) but that with zero defined as *(x:x = x), (∀x) ∼ S(0,x) and n is Finite iff n is zero or Sˆ(n,0) where Sˆ is the ancestral relation of S. But that is just to say that the natural numbers form a Dedekind infinite sequence (that is (∃f)[(∀x)(∀y)(fx = fy → x = y) & (∃x)(∀y)(fy = x)]). The key step is clearly to be able to prove that every number has a successor and Frege’s proof works precisely because n is an object that can be proved not to fall under the concept “being less than n”. This was, one of, Frege’s triumphs in the Grundlagen. It might therefore seem 10 Wright and Hale consistently draw attention to the difference in methodology they see between mere axiomatic stipulation and their proposed methodology, whereby abstracta are taken as stipulations. I have not dwelt on what I regard in this respect as a separate issue namely that of the very strong existential import of second order logic. This is most clearly seen in a point that George Boolos constantly emphasised that is the very strong existential commitments embodied in the Second Order Comprehension Principle (∃X)(∀x)(Xx↔A(x)) where A(x) is a formula of second order logic not containing X free. The issue of the existential presuppositions of the Hume’s principle has been addressed by Shapiro and Weir (2000) and by Demopoulos in his review of M. Schirn (1998) in the Journal of Symbolic Logic. Shapiro and Weir conclude that “the neologicist has no non-question begging account of how there could be an epistemologically innocent route to the demonstration of platonistically construed mathematical existence claims.” (p. 188).

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little more than trite to claim that H.P. is an axiom of infinity, it clearly is, since it forces the domain to be infinite. No doubt it is, but the principle has many other odd entailments too. Some of which are surprisingly and worryingly strong. Since the function * is total we can take it that it is a theorem of Frege Arithmetic that (∀F)(∃!x)(x = *F). Call this theorem (**). Now as Frege noted if we let F be “. . . is a member of the natural sequence of numbers”, it follows that there is a number of finite numbers, Frege’s ∞1 and that since it succeeds itself it is not a finite number (since it is a theorem of the formal system implicit in the Grundlagen that if Finite(n) then ∼S(n,n)). Again it follows immediately from (**) that if we take F to be the concept “is self identical” then (∃!x)(x = *(x = x)), that is, that there is a number of all things that there are. This latter looks like a very strong claim indeed. But we should note immediately, as Boolos pointed out, that we cannot show within the theory (Frege Arithmetic) that the two numbers are distinct. 11 In what we might call the standard model of Hume’s Principle, with domain N, the natural numbers in which all infinite concepts are assigned the object zero, while all finite concepts are assigned the cardinality of the corresponding subset of N plus one, the object associated with *Finite (n) is zero, as is the object associated with *(x = x), since neither are finite but the set of all ordinals less than or equal to Aleph one is also a model for the theory (that is for Frege Arithmetic) and in this case *Finite (n) is assigned Aleph zero while *(x = x) is assigned Aleph one. So we certainly cannot prove in Frege Arithmetic that *Finite (n) = *(x = x). But a worry still remains for as Boolos asked “is there such a number as anti-zero?” 12 That seems to me a very good point indeed. For that, if anything is, seems to be a substantial matter, which cannot be decided by stipulation. Indeed this issue seems to generalise as Boolos pointed out into a general issue about the compatibility of two conceptions of cardinal number. One derived from what one might call the pure theory of cardinal number based on Hume’s Principle and one derived from set theory as we shall see below. However it is possible to dissolve this worry and Wright has done so. He points out that it was a prime tenet of Frege’s view that numbers were the numbers of sortal concepts. Clearly the concept “self identical” is not a genuinely sortal concept so it is not an appropriate instance of the theorem above (**). Thus as Wright remarks in his reply to Boolos: Moral: just as not every object is suitable to determine a direction, so we should not assume without further ado that every concept—every entity an expression for which is an admissible substituted for the bound occurrences of the predicate letters in Hume’s Principle—is such as to determine a number. 11 Boolos (1987b), p. 197. Page reference to the reprint in Boolos (1998). 12 Boolos (1997), p. 314. Page reference to the reprint in Boolos (1998).

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He goes on to say: So self-identity is not a sortal concept. If we take it that, save where F is assured an empty extension on purely logical grounds, only sortal concepts, and concepts formed by restricting a mere predicable to a sortal concept, have cardinal numbers, it follows that there is no universal number. 13 So the function * is not to be regarded as having as its domain all concepts, rather there is to be natural and principled restriction on its domain of interpretation. It is to be confined to sortal concepts. However it is now unclear whether we can any longer regard Hume’s Principle as a purely logical abstraction principle since it is not clear that the notion of being a sortal concept can be expressed in purely logical vocabulary. But as Wright was quick to note if we do accept the need to restrict the domain of the function * to genuinely sortal concepts then a second difficulty looms and to some of us it looms very large.

3.

The good company objection

Let us follow Wright’s lead and consider the sortal concepts “set”, “ordinal” and “cardinal number”. According to Hume’s Principle these concepts too ought to have a number associated with them by (**). But then we immediately invite the observation that there is bound to be conflict with the notion of number as embodied in set theory, say ZFC. According to ZFC there are no numbers associated with the concepts “set”, “ordinal” and “cardinal number” precisely because the extensions of such concepts do not form sets and ZFC embodies the principle: no set no cardinal number. The collections we have been considering are proper classes, not sets, so there is no number associated by ZFC (or set theory) with them. Considerations such as these suggested to Boolos that there were two conflicting conceptions of number in play. He remarks: 14 Two thoughts about the concept of number are incompatible: that any zero or more things have a (cardinal) number, and that any zero or more things have a number (if and) only if they are the members of some one set. It is Russell’s paradox that shows the thoughts incompatible: the sets that are not members of themselves cannot be the members of any one set. The thought that any (zero or more) things have a number is Frege’s; the thought that things have a number only if they are the members of a set may be Cantor’s and is in any case a commonplace of the usual contemporary presentations of the set theory that originated with Cantor and has become ZFC.

In a similar vein he elaborated on the issue: 15 The worry is that the theory of number we have been considering, Frege Arithmetic, is incompatible with Zermelo–Fraenkel set theory plus standard 13 Wright (2000) “Is Hume’s principle Analytic?” in the Notre Dame Journal of Formal Logic.

14 Boolos (1995) “Frege’s Theorem and the Peano Postulates”, in Boolos (1998), p. 291. 15 Boolos (1997), p. 314. Page reference is to Boolos (1998).

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definitions, on the usual and natural readings of the non-logical expressions of both theories. To be sure, as Hodes once observed in conversation, if *a is taken to denote the cardinal number of a when a is a set and some favourite object that is not a cardinal number when a is a proper class, then HP will be a theorem of Von Neumann set theory. But on that definition of *, *will not be translatable as ‘the cardinal number of.’ ZF and Frege arithmetic make incompatible assertions concerning what cardinal numbers there are. And of course, the response ‘Well, these are just formalisms; the question of their truth or falsity doesn’t arise or makes no sense’ is hardly available to one claiming that HP is analytic, i.e., an analytic truth. So one who seriously believes that has to be bothered by the incompatibility of the consequence of Frege arithmetic that there is such a number as anti-zero with the claim made by ZF + standard definitions (on the natural reading of its primitives) that there is no such number.

Since to be in the company of set theory is to be in very good company indeed, let us call this objection to the stipulatory nature of Hume’s Principle, the Good Company objection. Now there is a clear response which can be made by someone who wishes to defend the idea that the truth of Hume’s principle can be simply stipulated and that is that the conflict alluded to above is illusory. 16 After all set theory assigns numbers to sets, but Frege arithmetic assigns numbers to concepts. Frege arithmetic assigns a cardinal number to the Russell concept “non-self membered set” but in virtue of the way cardinal numbers are introduced in ZFC, no cardinal number can be assigned to a nonset. (It is worth recalling that in ZFC if X is any set, then there is an ordinal number α and a bijection f: α →X. For any set X the cardinality of X is the least ordinal α such that there is a bijection f: α →X. A cardinal number is an ordinal number α such that for no β < α is there a bijection f: β → α.) The response to the Good Company objection in short then is Frege arithmetic assigns a number to proper classes, ZFC is silent. In any case certainly there is no conflict. But this dismissal of the good company objection is too swift. Let us go back to the first quotation from Boolos. He rightly points out that two conceptions of number are incompatible. The first conception is embodied in Hume’s Principle and says that every sortal concept has a number and the other says that every sortal concept has a number if and only if the extension of that concept is a set. Call these the Fregean and non-Fregean conceptions respectively. Clearly Russell’s paradox does indeed show that these two conceptions are incompatible. “Non-self membered set” is certainly a sortal concept and so has a number, by the Fregean conception. But it is provably the case that the concept “non-self membered set” has no set as its extension so on the second conception by the contrapositive of the only if clause there can be no number corresponding to the concept. Now imagine someone, we had better call him “Anti-Hero” or simply “Villain” (in this context) who believes this: the conception of number we have is revealed in mathematical practice and he holds to the non-Fregean concept of number. Is it conceivable that he holds an analytically false belief, 16 This objection was put to me by Stewart Shapiro and Fraser MacBride.

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or a belief that a stipulation may show to be false? Our grasp of number is what we do with it in mathematics. Now it is certainly true that the nonFregean concept of number outlined above is entirely adequate for the whole of mathematics since it realised by ZFC (though ZFC does not entail it). The extra content which Anti-Hero in fact believes over ZFC is the claim that the mathematical universe is exhausted by ZFC. That claim certainly seems believable. What it certainly does not seem to be is analytically false and it seems to have enormous evidence in favour of it. It may seem that Anti-Hero is merely legislating on the content of all possible mathematics and that can’t be right. But Anti-Hero doesn’t believe that his view is analytically true, what he certainly believes is that it is not analytically false however. Perhaps it was something like this that Boolos had in mind when he wrote of Anti-Hero’s view that: “It is in any case a commonplace of the usual contemporary presentations of the set theory that originated with Cantor and has become ZFC”. Clearly this matter must be connected with Boolos’s rejection of the notion of proper class. He took the view that they are in fact just “a manner of speaking” in the sense that they really play the role of abbreviations whose use can always be eliminated in any formal theory by replacing them by their defining formulas and this latter view I think is directly connected with his support for the iterative conception of set and the claim that ZFC exhausts that iterative hierarchy. Whatever may be the case about this it does seem to me that Boolos’s Good Company objection is a compelling one, for the set theoretic conception of number is a perfectly viable one and it surely cannot be rejected as analytically false. A possible response would be to say that there are two conceptions of cardinal number one for concepts and one for sets, but then what would have happened to the Fregean foundational programme that this was the correct ontological and epistemological account of the nature of and of our knowledge of the cardinal numbers? We would then seem to be left with an impenetrable problem about reference, we have an account of Frege numbers, we have an account of “set” numbers but neither is apparently related to each other or the numbers of ordinary arithmetic. That response would amount to abandoning the foundational programme and would hardly be acceptable to the neo-Fregean.

4.

Indefinitely extensible concepts again—proper classes

Perhaps it was some considerations such as these which lead Wright to endorse the principle at the heart of the non-Fregean concept of number as characterised above. He writes: 17 Grant the plausible principle that there is a determinate number of F’s just provided the F’s compose a set 17 Wright (2000), reprinted in Hale and Wright (2000), pp. 307–32. Reference is to p. 314.

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and later . . . it seems natural and well-motivated to suppose that the F’s should have a determinate cardinal number just when they compose a set

If this is the line to follow then what we are looking for is some further principled restriction on range of the quantifiers in Hume’s Principle over and above that of the restriction to sortal concepts. Wright proposes that the concepts falling under the range of the universal quantifiers in Hume’s principle should be restricted to those which are “definite”. Recall that a concept is definite just when it is not “indefinitely extensible”. One sense of indefinite extensibility goes back to Russell and Poincaré. Russell, in 1903 had thought that the contradiction derivable from Basic Law V of Frege’s Grundgesetze showed that not every property determines a class simpliciter. 18 The fundamental question as he saw it was then “to determine, which propositional functions define classes which are single terms as well as many, and which do not?”. By 1906, after reading Poincaré he had changed his view. He wrote “the contradictions result from the fact that, according to current logical assumptions, there are what we may call selfreproductive processes and classes. That is, there are some properties such that, given any class of terms all having such a property, we can always define a new term also having the property in question.” 19 In formulating the Vicious Circle Principle he made a very similar claim: “Thus all our contradictions have in common the assumption of a totality such that, if it were legitimate, it would at once be enlarged by new members defined in terms of itself”. 20 Both the Russell of this period and Poincaré would have agreed that the objects falling under such self-reproductive properties like ordinal or set form no totality. As is very well known Dummett has recently revived this idea. 21 As Wright remarks of the idea: 22 I do not, myself, know how best to sharpen this idea, still less how its best account might show that Dummett is right both to suggest that the proof-theory of quantification over indefinitely extensible totalities should be uniformly intuitionistic and that the fundamental classical mathematical domains, like those of the natural numbers, or the reals, should also be regarded as indefinitely extensible. But Dummett could be wrong about both those points and still be emphasizing an important insight concerning certain very large totalities—ordinal number, cardinal number, set, and indeed ‘absolutely everything’.

So the idea that Wright is proposing then is this I take it. We should restrict the range of the quantifiers in Hume’s principle to definite concepts, that is 18 Russell (1903). 19 Russell (1906), p. 144.

20 Russell (1908), p. 63. 21 Dummett (1991), (1963) and Clark (1998).

22 Wright (2000), p. 316 of Hale and Wright (2000).

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those that are not indefinitely extensible. But one has to be careful here for there are at least two conceptions of indefinite extensibility, if there are any, one which is very closely related to the Russell–Wright line of thought which sees indefinite extensibility associated with concepts generating or holding of very large totalities, with in effect, proper classes (in the presence of strong versions of the axiom of choice all the proper classes we have been considering are equinumerous with the Universe) and one which Dummett has developed in which both N and R are indefinitely extensible. Deploying Dummett’s conception will of course have essentially the same effect as deploying a suggestion due to Heck 23 to the effect that analytic core of Hume’s Principle is Finite Hume. This is the Principle that (∀F)(∀G)(Fin(F) & Fin(G) → (∗ F = ∗ G ↔ (F1 − 1R G)) where the notion of finite can be spelled out in purely logical vocabulary (as the negation of the second order sentence expressing that F is (Dedekind) infinite). This is of course not in the form of an abstraction principle and so is unacceptable as an implicit definition for the neo-Fregean. The closely related principle (∀F)(∀G)((∗ F = ∗ G ↔ Not Fin(F)V Not Fin(G) V (F1 − 1G)) although it is an abstraction principle (the right hand side being an equivalence relation) would not suffice either since it conflicts with the Cantorian conception of cardinal number by assigning the same number to all infinite concepts. However what would seem to capture Wright’s restriction is precisely the notion of set as opposed to proper class. I take it what we want to say is that N and R are definite while “set”, “ordinal” and “cardinal number” are not. Can such a notion of definite be made out? Well of course it can: the natural candidate is, a concept F is definite iff it has a set as its extension. This will do exactly what Wright wants. But it would make the understanding of Hume’s principle parasitic upon the notion of set and our grasp of the set theoretic Universe. It could hardly then be argued that the truth of Hume’s Principle was guaranteed by stipulation. We have already noted that in the presence of a very strong choice principle the concepts we wish to exclude from the domain of Hume’s Principle are equinumerous with the Universe. We could use this fact to independently give a justification of Wright’s New Hume which would have the form (∀F)(∀G)(∗ F = ∗ G ↔ (InDef (F) & InDef(G)) v (F1 − 1G)) where InDef (F) would be a condition stating that F is equinumerous with the Universe. 23 Heck (1997c).

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This would still be a logical abstraction principle, provided that the condition InDef (F) were expressed by the formula: (∃R)((∀x)(Fx → (∃!y)Rxy) & (∀y)(∃!x) (Fx & Rxy)) 24 However one proceeds here there looks to be a dilemma: either restrictions on the range of the quantifiers in Hume’s principle are actually motivated by set theoretic considerations or some notion of definiteness can really be made out, which itself does not rely upon set existence principles. These would serve as a principled restriction but in either case clearly something other than the notion of (sortal) concept and of second order logic is motivating our understanding of Hume’s Principle and thus of our knowledge of number. Whatever route is the correct one, it seems hardly possible to regard the laying down of such principles as guaranteeing their truth by stipulation.

5.

Frege and the application of arithmetic

It is certainly true that Frege put the application problem at the heart of his philosophy of mathematics. To the question: “why does arithmetic apply to reality?”, the logicist provides the clear answer because it applies to everything that can be thought. It is the most general science possible. The partial contextual definition, provided by Hume’s Principle and the fundamental thought that numerical concepts are second level concepts yield Frege’s account of the applicability of mathematics. In the simplest case for which the question arises—the application of the cardinal numbers—the solution is that arithmetic is applicable to reality because the concepts, under which things fall, themselves fall under numerical concepts. Thus it is possible to prove in second order logic that ∃n xFx − F falls under the numerical property expressed by the numerically definite quantifier ∃n x if and only if the Frege numeral introduced by the partial contextual definition (Hume’s Principle) is indeed n. In other words the theorem that ∃n xFx ≡ n = NxFx can be obtained, from Hume’s Principle in second order logic. But there is a real difficulty with Frege’s solution to the application problem and that is that we are provided by Hume’s principle with at best a partial contextual definition. The principle cannot settle the truth conditions of sentences of the form q = NxFx where q is not given in the form NxHx for some H, this of course is the famous Julius Caesar problem. In the case of pure arithmetic the Julius Caesar problem can 24 Boolos suggested that we might regard the principle.

(∀F)(∀G)(EXT(F) = EXT(G) ↔ (InDef(F) & InDef(G)) v (∀x) (Fx ↔ Gx)) as a repair of Basic Law Five. This might well then be used as a justification of set theory based upon the Principle of the Limitation of size. But like that justification however the power set axiom might well prove a difficulty when taken together with the separation schema. Knowing that for example N is not “too big” does not help us with the claim that P(N) is not too big. Similarly knowing that X is definite might tell us very little about or nothing at all about the definiteness of P(X). The effect of introducing New Five has been very carefully studied by Shapiro and Weir (see Shapiro and Weir (1999)).

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properly be regarded as irrelevant, since there would be no other singular terms in the language but once the language is extended to include empirical singular terms, as it would in the language of applied arithmetic, Hume’s Principle will no longer settle the sense of numerical identities and the solution to the application problem will fail. Of course this issue does not arise in the formal language of the Grundgesetze, which is a language of pure arithmetic, since in that language all the objects it is possible to refer to are already given as extensions (value ranges) the identity conditions for which are given by Basic Law Five. But as Michael potter has put it: 25 “a formal language in which Julius Caesar cannot be spoken of is one in which he cannot be counted, and in such a language the applicability of arithmetic remains unexplained. At some stage in the development we shall have to extend the formal language by adding some empirical vocabulary, and we shall then have to address the Julius Caesar problem just as before.” The question naturally arises as to whether the neo-logicist fares any better than Frege with respect to the application problem. This seems very unlikely since the neo-logicist 26 relies exclusively upon Hume’s Principle and therein lays the real difficulty, as Frege long ago knew. In the same letter which I quoted at the beginning of this paper he wrote to Russell about the idea of letting his programme rest on Hume’s Principle alone the following: 27 We can also try the following expedient, and I hinted at this in my Foundations of Arithmetic. If we have a relation (ξ, ζ ) for which the following propositions hold (i) from (a, b) we can infer (b, a) and (2) from (a, b) and (b, c) we can infer ; then this relation can be transformed into an equality (identity), and can be replaced by writing, e.g., “§a = §b”. If the relation is, e.g., that of geometrical similarity, then “a is similar to b” can be replaced by saying “the shape of a is the same as the shape of b”. This is perhaps what you call “definition by abstraction”. But the difficulties here are not the same as in transforming the generality of an identity into an identity of range of values.

References Boolos, G. (1987a), “Saving Frege from Contradiction” Proceedings of the Aristotelian Society 87 (1987), pp. 137–51. Boolos, G. (1987b), “The Consistency of Frege’s Foundations of Arithmetic” in Thomson (1987), pp. 3–20. Boolos, G. (1990a), “The Standard of Equality of Numbers” in Boolos (1990b), pp. 261–77. Boolos, G. (Ed.) (1990b), Meaning and Method: Essays in Honor of Hilary Putnam (Cambridge: Cambridge University Press, 1990). 25 Potter (2000), p. 108. 26 Hale and Wright treat the Julius Caeser problem very seriously indeed, they say of it that it is “one

of the hardest the neo-Fregean must solve” (Hale and Wright 2000, pp. 14–16). They devote the whole of essay 14 of their (2000), (pp. 335–396) to the topic. 27 Frege (1902), p. 141.

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Boolos, G. (1993), “Whence the Contradiction?” Proceedings of the Aristotelian Society, Supp. Vol 67 (1993), pp. 213–33. Boolos, G. (1997), “Is Hume’s Principle Analytic?”, in Heck (1997a), pp. 245–62. Boolos, G. (1998), Logic, Logic and Logic (Cambridge MA: Harvard University Press, 1998). Boolos, G. & Heck, R. (1998), “Die Grundlagen der Arithmetik §§82–3”, in Schirn (1998), pp. 407–28. Brandl, J. & Sullivan, P. (Eds.) (1998), New Essays on the Philosophy of Michael Dummett (Vienna: Rodopi, 1998). Burgess, J. P. (1984), Review of Wright [1983], Philosophical Review 93, 1984, pp. 638–40. Clark, P. (1998), “Dummett’s Argument for the Indefinite Extensibility of Set and Real Number” in Brandl & Sullivan (1998), pp. 51–63. Demopoulos, W. (Ed.) (1995), Frege’s Philosophy of Mathematics (Cambridge: Harvard University Press, 1995). Demopoulos, W. (1998), “The Philosophical Basis of our Knowledge of Number”, Noûs 32 (1998), pp. 481–503. Demopulos, W. (2000), “On the origin and Status of Our Conception of Number” Notre Dame Journal of Formal Logic, 41 (2000), pp. 210–26. Dummett, M. (1963), “The Philosophical Significance of Gödel’s theorem”, Ratio 5 (1963), pp. 140–55. Dummett, M. (1991), Frege: Philosophy of Mathematics (London: Duckworth, 1991). Fine, K. (1998), “The Limits of Abstraction” in Schirn (1998), pp. 503–629. Frege, G. (1879), Begriffsschrift (Halle: L. Nebert, 1879). Frege, G. (1884), Die Grundlagen der Arithmetik (Breslau: W. Koebner, 1884); reprinted with English translation by J. L. Austin as The Foundations of Arithmetic (Oxford: Blackwell, 1950). Frege, G. (1893), Die Grundgesetze der Arithmetik vol. 1 (Jena: H. Pohle, 1893), part translated into English by Montgomery Furth in The Basic Laws of Arithmetic (Berkeley: University of California Press, 1964). Frege, G. (1902), Letter XV7 [xxxvi/7] in Frege (1980), pp. 139–42. Frege, G. (1980), Philosophical and Mathematical Correspondence, ed. G. Gabriel, 1980. Hale, R. and Wright, C. (2000), The Reason’s Proper Study (Oxford: Oxford University Press, 2001). Heck, R. Jr. (1993),“The Development of Arithmetic in Frege’s Grundgesetze der Arithmetik”, Journal of Symbolic Logic 58 (1993), pp. 579–601. Heck, R. Jr. (1995), “Frege’s principle” in J. Hintikka (ed.), From Dedekind to Godel, 1995, pp. 119–42. Heck, R. Jr. (1997a), Language, Thought and Logic (Oxford: Oxford University Press, 1997). Heck, R. Jr. (1997b), “The Julius Caesar Objection”, in Heck (1997a), pp. 273–308 (a). Heck, R. Jr. (1997c), “Finitude and Hume’s Principle”, Journal of Philosophical Logic 26, (1997), pp. 589–617. Hodes, H. (1984), “Logicism and the Ontological Commitments of Arithmetic”, Journal of Philosophy 81 (1984), pp. 123–49. Parsons, C. (1965), “Frege’s Theory of Number” in Mathematics to Philosophy, pp. 150–75. Potter, M. (2000), Reason’s Nearest Kin (Oxford: Oxford University Press, 2000). Russell, B. (1903), The Principles of Mathematics (London, George Allen and Unwin, 1903). Russell, B. (1906), “On some difficulties in the theory of transfinite numbers and order types”, reprinted in D. Lackey ed. Bertrand Russell Essays in Analysis (London, George Allen and Unwin, 1973), pp. 135–64. Russell, B. (1908), “Mathematical logic as based on the theory of types”, reprinted in R. C. Marsh ed. Logic and Knowledge (London, George Allen and Unwin, 1956), pp. 59–102. Schirn, M. (1998), Philosophy of Mathematics Today (Oxford: Clarendon Press, 1998). Shapiro, S. and Weir, A. (1999), “New V, ZF and Abstraction”, Philosophia Mathematica 1999, pp. 293–321. Shapiro, S. and Weir, A. (2000), “Neo-logicist Logic is not epistemically innocent” Philosophia Mathematica, pp. 160–89. Smiley, T. (1981), “Frege and Russell”, Epistemologica, 1981, pp. 51–6.

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Wright, C. (1983), Frege’s Conception of Numbers as Objects (Aberdeen: Aberdeen University Press, 1983). Wright, C. (1997), “The Philosophical Significance of Frege’s Theorem” in Heck (1997), pp. 201–45. Wright, C. (1998a), “On the Harmless Impredicativity of N=”, in Schirn (1998), pp. 339–68. Wright, C. (1998b), “Response to Dummett” in Schirn (1998), pp. 389–406. Wright, C. (2000), “Is Hume’s Principle Analytic”, Notre Dame Journal of Formal Logic, 40 (2000), pp. 6.

FINITUDE AND HUME’S PRINCIPLE 1 Richard G. Heck, Jr Brown University, Providence RI, U.S.A. E-mail: [email protected]

Abstract The paper formulates and proves a strengthening of ‘Frege’s The-

orem’, which states that axioms for second-order arithmetic are derivable in second-order logic from Hume’s Principle, which itself says that the number of Fs is the same as the number of Gs just in case the Fs and Gs are equinumerous. The improvement consists in restricting this claim to finite concepts, so that nothing is claimed about the circumstances under which infinite concepts have the same number. ‘Finite Hume’s Principle’ also suffices for the derivation of axioms for arithmetic and, indeed, is equivalent to a version of them, in the presence of Frege’s definitions of the primitive expressions of the language of arithmetic. The philosophical significance of this result is also discussed.

1.

Opening

In recent work, 2 George Boolos has, with an eye towards philosophical issues I shall discuss in Section 3, investigated the relative strengths of two sorts of systems of second-order arithmetic. The more familiar of these originates with the work of Dedekind and Peano; the less familiar, with that of Frege. Dedekind–Peano systems characterize the natural numbers in terms of properties of the sequence of natural numbers; these systems may be thought of as axiomatizations of finite ordinal arithmetic. The Fregean systems, on the other hand, characterize the natural numbers as finite cardinals. 3 Fundamental to such systems is an axiom specifying the condition under 1 This paper first appeared in the Journal of Philosophical Logic 26, [1997], pp. 589–61. Reprinted by kind permission of the editor and Springer Academic Publishers. 2 G. Boolos, “On the Proof of Frege’s Theorem”, in A. Morton and S. Stich, eds. Benacerraf and His Critics (Oxford: Blackwells, 1996), pp. 143–59. 3 For further discussion of this difference, see my “The Finite and the Infinite in Frege’s Grundgesetze der Arithmetik”, in M. Schirn, ed., Philosophy of Mathematics Today (Oxford: Oxford University Press, 1998), §5, pp. 429–66.

61 Roy T. Cook (ed.), The Arché Papers on the Mathematics of Abstraction, 61–84. c 2007 Springer.

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which two concepts 4 have the same cardinal number, together with another specifying under what conditions a cardinal number is finite. What is perhaps the most familiar (second-order) Dedekind–Peano system is axiomatized as follows: 1. 2. 3. 4. 5. 6. 7.

N0 Nx & Pxy → Ny ∀x∀y∀z(Nx & Pxy & Pxz → y = z) ∀x∀y∀z(Nx & Ny & Pxz & Pyz → x = y) ¬∃x(Nx & Px0) ∀x(Nx → ∃y Pxy) ∀F[F0 & ∀x∀y(Fx & Pxy → Fy) → ∀x(Nx → Fx)]

Let us call this system PA2 (for second-order Peano arithmetic). I have here formulated its axioms using a relational expression ‘Pξ η’, rather than the more usual functional expression ‘Sξ ’, to facilitate comparison with Fregean systems. The most familiar Fregean system has but one ‘non-logical’ axiom, Hume’s Principle, which states that the number of Fs is the same as the number of Gs just in case the Fs and Gs are in one–one correspondence. Taking ‘Eqx (Fx;Gx)’ to abbreviate one of the (many equivalent) second-order formulae which define ‘the Fs correspond one–one with the Gs’ (or, in Frege’s terminology, ‘the Fs are equinumerous with the Gs’), Hume’s Principle (HP) is then: Nx : Fx = Nx : Gx ≡ Eqx (F x, Gx) The second-order theory whose sole non-logical axiom is HP is FA (for ‘Frege arithmetic’). Note that ‘Nx : x’ is a unary, second-level, term-forming operator: The result of substituting any formula (possibly containing further occurrences of ‘Nx : x’) for ‘x’ in ‘Nx : x’ is a term. The definition of finite or natural number can be given in different ways. In Frege’s work, 5 zero and the relation of predecession are defined and, famously, the finite numbers are defined as those to which zero stands in the weak ancestral of this relation. The necessary definitions are thus: 0 = Nx : x = x Pmn ≡ ∃F∃y[Fy & n = Nx : Fx & m = Nx : (Fx & x = y)] Frege defines the strong ancestral of a relation Rξ η as follows: R ∗ ab ≡ ∀F[∀z(Raz → Fz) & ∀x∀y(Fx & Rxy → Fy) → Fb] 4 I shall use this term to denote whatever are in the range of the second-order variables. Though my choice of terminology certainly suggests a view about what these are, my remarks here do not depend upon it. It is, of course, essential to the logicist project that second-order logic is logic, but this is not at issue among those whose positions we shall be discussing. 5 See, of course, G. Frege, The Foundations of Arithmetic, 2nd. ed., trans. by J. L. Austin (Evanston, IL: Northwestern University Press, 1953), §§74, 76, 83.

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And he defines the weak ancestral of Rξ η thus: R ∗= ab ≡ R∗ ab ∨ a = b Frege’s definition of natural number is then: n is a natural number just in case P ∗= 0n. There are other ways to proceed, however. In sections K and of Grundgesetze der Arithmetik, Frege formulates a purely second-order definition of finitude, to state which we need an additional definition: 6 Btwx y (Rxy; a; b)(n) ≡ ∀x∀y∀z(Rxy & Rxz → y = z) & ¬R∗ bb & R ∗= an & R ∗= nb Thus, n is between a and b in the R-series if, and only if, Rξ η is a functional relation, in whose strong ancestral b does not stand to itself (i.e., there is no ‘loop’ from b to b), such that a stands in the weak ancestral of Rξ η to n, which in turn stands in the weak ancestral of Rξ η to b. Frege’s definition of finitude is then: 7 Finitex (Fx) ≡ ∃R∃x∃y∀z[Fz ≡ Btw(R; x; y)(z)] That is: A concept is finite just in case the objects falling under it may be ordered in a certain way, namely, as the objects between x and y in the Rseries, for some R, x and y. That this definition is correct follows from the central theorems of sections K and of Grundgesetze der Arithmetik, which are Theorems 327 and 348 of Grundgesetze: (327) Finite(F) → P ∗= (0, Nx : Fx) (348)P ∗= (0, Nx : Fx) → Finite(F) Thus, a concept is finite, in Frege’s sense, just in case its number is a natural number. Frege’s definition of natural number could, therefore, be replaced by: N(n) ≡ ∃F[Finite(F) & n = Nx : Fx] Of course, this definition will be adequate only in a theory strong enough to prove Theorems 327 and 348. 8 Analogues of these theorems are the crucial lemmas in the proofs of the main result of this paper (see Lemmas 3.1, 3.11, and 3.21). As we shall see, given Frege’s definitions of ‘0’ and ‘Pξ η’, Theorem 327 becomes a theorem of second-order logic. The proof of Theorem 348, however, must rely upon additional assumptions, for without additional assumptions, it is consistent that 6 G. Frege, Grundgesetze der Arithmetik (Hildesheim: Georg Olms Verlandsbuchhandlung, 1966). The definition is given in §158 of volume I. I shall insert the bound variables, such as ‘x’ and ‘y’ on the left-hand side here, into the definitions, but will drop them when doing so causes no confusion. 7 Frege does not explicitly formulate any such definition, but it is clear from the theorems proven in sections K and that this is what he intends. For further discussion, see my “The Finite and the Infinite”, op. cit. 8 Frege proves it in the system FA + FD. As we shall see, it is also provable in FAF + FD (and so in PAF + FD).

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all concepts have the number 0 (and, of course, it is consistent that some of these are not finite). In investigating the relative strengths of Dedekind–Peano and Fregean systems, there are two sorts of questions one might raise. First, one might inquire about the relative consistency of such theories. To ask whether FA is consistent relative to PA2 is to ask whether the consistency of FA would follow from that of PA2. One familiar sort of proof that it would consists in a demonstration that FA can be relatively interpreted in PA2. Roughly speaking, to interpret FA in PA2 is to give definitions of the primitives of FA in terms of the primitives of PA2, which definitions, when added to PA2, allow one to prove relativizations of the axioms of FA in PA2: By a relativization of a formula is meant, as usual, the result of restricting quantifiers occurring in the formula by means of some formula of PA2. 9 If FA can be interpreted in PA2, it follows immediately that, if there is a proof of a contradiction in FA, that proof can be mimicked in PA2, so that, if PA2 is (syntactically) consistent, so is FA. As it turns out, FA and PA2 are equi-interpretable—each can be interpreted in the other—and so equi-consistent—an inconsistency in either would imply an inconsistency in the other. Still, one might wonder whether FA is not, in some other sense, a stronger theory than PA2. This question is more easily understood when we have two theories formulated in the same language. Consider, for example, the following Dedekind–Peano system, which we shall call PAS (for ‘Strong’ Peano arithmetic): 1. 2. 3. 4. 5. 6. 7.

N0 Nx & Pxy →Ny ∀x∀y∀z(Pxy & Pxz → y = z) ∀x∀y∀z(Pxz & Pyz → x = y) ¬∃xPx0 ∀x(Nx → ∃yPxy) ∀F[F0 & ∀x∀y(Fx & Pxy → Fy) → ∀x(Nx→ Fx)]

Clearly, every axiom of PA2 is a theorem of PAS, but the converse does not hold. As far as the axioms of PA2 are concerned, zero could have as its predecessor Julius Caesar, so long as Caesar is not a natural number. Thus, PAS is strictly stronger than PA2. This is perfectly compatible with the fact that PA2 and PAS are equi-interpretable. (To interpret PAS in PA2, no ‘definitions’ are needed: Just restrict all the quantifiers in the axioms of PAS to the natural numbers, i.e., by the formula ‘Nx’.) 9 Of course, in the context of second-order logic, one must restrict not only the first-order, but also the second-order, quantifiers. As we replace ‘∀x A(x)’ by ‘∀x[R(x) → A(x)]’, so we replace ‘∀F A(F)’ by ‘∀F{∀x[Fx → R(x)] → A(F)}’.

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The question under discussion here concerns the proof-theoretic strength of the two systems. This question is harder to raise when the theories under discussion are not formulated in the same language: Obviously, the axioms of FA are not going to be theorems of PA2, since the axioms of FA are not even sentences in the language of PA2. Nor would expanding the language of PA2 to include such formulae help. To consider the relative strength of theories formulated in different languages, what we require is a bridge theory which relates (the referents of) the primitives of PA2 to those of FA. We can then ask whether, with the aid of one or another bridge theory, the theorems of FA can be proven in PA2. 10 One might wonder what difference there is between the question whether FA can be relatively interpreted in PA2, and the question whether the theorems of FA can be proven in PA2, with the aid of some bridge theory. For, one might ask, if FA can be relatively interpreted in PA2, will that not itself guarantee that there is some bridge theory with the aid of which the theorems of FA can be proven in PA2? namely, that theory whose axioms are exactly the definitions used in interpreting FA in PA2? The answer to this question is “No”. One must not overlook the fact that, in relatively interpreting one theory in another, it may be essential to relativize the axioms of the former theory: The usual relative interpretation of FA in PA2, for example, requires that the quantifiers occurring in Hume’s Principle be restricted to the natural numbers. There is no necessity that there should be a way of mimicking this restriction in any bridge theory, and there is certainly no need that any particular bridge theory should impose such a restriction. Our chief interest here is in the relative strength of various Fregean systems and various Dedekind–Peano systems. We thus must make use of a bridge theory which relates (the referents of) their primitives. The bridge theory in which we shall be interested is that which has the following three axioms: 0 = Nx : x = x Pmn ≡ ∃F∃y[Fy & n = Nx : Fx & m = Nx : (Fx & x = y)] Nn ≡ P ∗= 0n This theory we shall call FD—for ‘Frege’s definitions’, since these are the definitions Frege uses in deriving axioms for arithmetic (in particular, those of PAS) in FA. 10 One might well wonder what such a bridge theory must be like, if the provability of the theorems

of one system from those of another is to have the kind of interest it is here taken to have. I do not know how this question should be answered. Surely, however, it is sufficient if the axioms of the bridge theory are definitions of the primitives of one of the two theories in terms of the primitives of the other. The bridge theories we shall employ below are of this sort. The question we are considering is thus one of interpretability, rather than relative interpretability.

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The systems

We will here investigate the relative strengths of five different systems of arithmetic. The Dedekind–Peano systems at which we shall look are PA2 and PAS, mentioned above, and a third system, PAF, whose axioms are those of PA2 plus: ∀x∀y(Nx & Pyx → Ny) This axiom, which we also call PAF (for ‘predecessors are finite’), states that any predecessor of a natural number is a natural number. 11 As we shall see, PAF is stronger than PA2 and weaker than PAS. Before discussing the variations on FA at which we shall look, let me make a remark about the background logic in which we shall be working. In the case of the Dedekind–Peano systems, the logic is usually taken to be standard (axiomatic) second-order logic, with full, impredicative comprehension. In discussing FA and its relations to the Dedekind–Peano systems, however, it is convenient to take the logic also to contain the axiom Boolos calls FE, for ‘functional equivalence’: ∀x(Fx ≡ Gx) → Nx : Fx = Nx : Gx This axiom is clearly valid on any extensional semantics for second-order logic and so should itself be regarded as a truth of (extensional, higher-order) logic. 12 The system whose axioms are those of second-order logic, plus FE, Boolos calls Log. We shall suppose our background logic, throughout, to be Log. The Fregean systems at which we shall look are FA and a variation on it, in which Hume’s Principle has been weakened by restricting its range of application. The axiom is HPF (for Finite Hume’s Principle): Finite(F) ∨ Finite(G) → [Nx : Fx = Nx : Gx ≡ Eqx (Fx; Gx)] Here, the formula ‘Finite(F)’ may be defined via any of the equivalent secondorder definitions of finitude: We shall take it to be defined as Frege defines it. HPF states that finite concepts have the same number if, and only if, they are equinumerous and that no infinite concept has the same number as any finite one—making no further claim about the conditions under which infinite concepts have the same number. (For all that HPF says, all infinite concepts could have the same number, so long as no finite concept also has that number.) Call the theory whose sole non-logical axiom is HPF, FAF (for finite Frege arithmetic). 11 G. Boolos remarked to me that, when presenting “On the Proof of Frege’s Theorem”, he has heard it objected that PAF—or, more precisely, its consequence NPZ, to be mentioned below—cannot be true, since −1 surely precedes 0. But the theories in which we are interested here are theories of cardinal or ordinal numbers, and ‘Pξ η’ is defined as a relation between such numbers. Negative numbers are neither ordinals nor cardinals. 12 As should the axiom schema: ∀x(Fx ≡ Gx) → A(F) ≡ A(G).

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The results of this paper may now be summarized in the following diagram: FA ⇒ PAS ⇒ {PAF (⇒ FAF)} ⇒ PA2 Here, ‘⇒’ means: Is strictly stronger than, relative to the bridge theory FD; that is, ‘A ⇒ B’ means that every theorem of B is a theorem of A + FD, but that not every theorem of A is a theorem of B + FD. That FAF and PAF occur together in the braces indicates that they are equivalent, relative to FD: Every theorem of PAF is a theorem of HPF + FD, and every theorem of FAF is a theorem of PAF + FD. What we need to prove are thus the following: 1. 2. 3. 4.

FA ⇒ PAS PAS ⇒ PAF PAF is equivalent to FAF (modulo FD) PAF ⇒ PA2

Some of the required proofs have been discussed in detail by Boolos: I shall merely indicate how those proofs go. The main work of the present paper consists in establishing Theorem 3.

3.

On the philosophical significance of these results

Before turning to the proofs, let me make a couple of remarks about the inspiration for the present investigation and about its philosophical implications. In Frege’s Conception of Numbers as Objects, Crispin Wright rediscovered Frege’s Theorem, which states that the axioms of PAS are provable in FA + FD, proved it in some detail, and conjectured that FA is consistent, which it turned out to be. 13 On the basis of this result, Wright not only revived Frege’s logicist project, but claimed that it was substantially vindicated by the proof of Frege’s Theorem. If logicism were to be vindicated completely, of course, HP would have to be shown to be a logical truth, which it certainly cannot be, given our contemporary understanding of ‘logical truth’. Nevertheless, Wright argued, HP is ‘analytic’, whence the truths of arithmetic are logical consequences of an analytic truth and so, presumably, are themselves analytic. The sense in which HP is analytic is that, “even if inadequate as a definition, it nevertheless succeeds as an explanation; . . . it contrives to fix the meaning of the sorts of occurrence of [‘Nx : x’] which it fails to eliminate”. 14 In the paper mentioned at the outset, Boolos shows that FA is strictly stronger than PAS (and so PA2), relative to the bridge theory FD. His purpose is not primarily technical: He intends this to be one consideration in favor 13 C. Wright, Frege’s Conception of Numbers as Objects (Aberdeen: Aberdeen University Press, 1983). See especially Ch. 4. The consistency of the system was noted by Burgess, Hazen, and Hodes. Boolos later showed that FA and PA2 are equiinterpretable. For a proof, see Boolos and Heck, Jr., “Die Grundgesetze der Arithmetik §§82–3”, in Schirn, ed., pp. 407–28. 14 Wright, Frege’s Conception, p. 140. See also the statement of Number-theoretic Logicism (III) on p. 153.

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of the view that, contra Wright, HP is neither ‘analytic’, nor a ‘conceptual truth’, nor any such thing. 15 Boolos does not explain in detail why his result should trouble Wright, but his point seems to be that, since FA is significantly stronger than PAS (which is itself a stronger theory even than PA2, which is itself a very strong theory), it is implausible to claim that HP is a conceptual truth. It is difficult, however, to evaluate the force of this consideration: As Boolos recognizes, Wright would likely reply that since his view is the view that arithmetic is analytic—and, indeed, that the general theory of cardinality which FA embodies is analytic—he is simply being accused of holding that very view. Still, there is a stronger consideration in the vicinity. For the additional proof-theoretic strength of FA, as compared to PA2, reflects a very real, and very large, conceptual gap between second-order arithmetic and the general theory of cardinality. W. W. Tait has pointed out that ‘Hume’s Principle’ is something of a misnomer: In the passage Frege cites when introducing it, Hume is speaking not of cardinality in general but only of the cardinality of finite concepts (or sets, or whatever). 16 As of course he was. Prior to Cantor’s work on transfinite numbers, the view that all equinumerous concepts have the same number, whether they are finite or infinite, was almost universally rejected, because it gives rise to antinomies: For example, it implies that the number of natural numbers is the same as the number of even numbers, and that can seem absurd, because there are lots of natural numbers which are not even— indeed, according to Cantor, as many numbers as there are natural numbers. Cantor’s realization that one can coherently suppose, even in the infinite case, that all and only equinumerous sets have the same cardinality constituted as enormous a conceptual advance as his introduction of transfinite numbers was a mathematical advance. It is easy to forget this, so at home are we initiates with Cantor’s ideas. But it is just as easy to be reminded of it: One has an opportunity every time a student wanders into one’s office puzzled about these very antinomies. Indeed, my own work on this very paper was fundamentally altered by just such an experience. A friend of mine—a professional philosopher, and so no fool— was telling me about an objection one of his students had raised in lecture. The student had insisted that there is only one ‘kind’ of infinity, and my friend had been tempted to reply (but wanted to check with me first) that of course there was more than one kind of infinity, since both the natural numbers and the even numbers are infinite, and the infinities in question certainly cannot be of the same kind. He was troubled by my response. Not just philosophically troubled, mind you, but really bothered: As I conveyed Cantor’s ideas to him, 15 See G. Boolos, “Is Hume’s Principle Analytic?”, in R. Heck, ed., Language, Thought and Logic: Essays in Honor of Michael Dummett (Oxford: Oxford University Press, 1997), pp. 245–62. 16 Private communication. The term ‘Hume’s Principle’ should not confuse, however: It came into common use not because anyone thought Hume scooped Cantor, but because Frege introduced HP in §63 of Grundlagen by quoting from Treatise I iii 1.

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he kept saying, “That’s very worrying”, over and over again. I had to do a lot of explaining before he was again at ease. He made the leap, but my experience served to remind me how great a conceptual leap he made at that point—and so how great a conceptual leap Cantor himself had made. I am not going to argue that HP isn’t a conceptual truth: On that question I regard myself, to steal a phrase of David Wiggins’s, as a militant agnostic. My point is that, once one has recognized just how great a conceptual advance is required if one is to acknowledge the truth of Hume’s Principle, one can no longer accept that Frege’s Theorem has the sort of epistemological interest Wright and others have wanted it to have. What is required if logicism is to be vindicated is not just that there is some conceptual truth or other from which what look like axioms for arithmetic follow, given certain definitions: That would not show that the truths of arithmetic, as we ordinarily understand them, are analytic, but only that arithmetic can be interpreted in some analytically true theory. 17 To put the point differently, if we are so much as to evaluate logicism, we must first uncover the ‘basic laws of arithmetic’, laws which are not just sufficient to allow us to prove translations of arithmetical truths, but laws from which arithmetical truths themselves can be proven. (The distinction is not a mathematical one, but a philosophical one.) But, if these ‘basic laws’ are to be the basic laws of arithmetic, they had better be ones upon which ordinary arithmetical reasoning relies. If Frege’s Theorem is to have the kind of interest Wright suggests, it must be possible to recognize the truth of HP by reflecting on fundamental features of arithmetical reasoning—by which I mean reasoning about, and with, finite numbers, since the epistemological status of arithmetic is what is at issue. For what the logicist must establish is something like this: That there is, implicit in the most basic features of arithmetical thought, a commitment to certain principles, the (tacit) recognition of whose truth is a necessary precondition of arithmetical reasoning, and from which all axioms of arithmetic follow. Having identified these basic laws, we will then be in a position to discuss the question whether they are analytic, or conceptual truths, or what have you. What used to be my favorite argument for the analyticity of HP went roughly like this: HP is a conceptual truth, because it is part of the very concept of cardinality that equinumerous concepts have the same cardinal number. 18 Perhaps, but the argument overlooks the fact that, though this may be true of our present concept of cardinality, ‘we’ did not even have this concept of cardinality until about a 120 years ago. A recognition of the very coherence 17 If analysis were analytic, as Frege thought it was, then Euclidean geometry would be interpretable in an analytically true theory, via Cartesian co-ordinates. Are we to conclude that Frege’s position was inconsistent, since he held that geometry is not analytic, but synthetic a priori? Surely not. 18 Compare C. Wright, “On the Philosophical Significance of Frege’s Theorem”, in Heck, ed, pp. 201– 44. A somewhat different version of this claim is that, even if HP is not analytic of any preexisting concept of cardinality, it is perfectly in order to introduce such a concept by means of HP. This version of the claim becomes important in certain contexts.

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of our present concept of cardinality requires the conceptual leap I discussed above, whence, even if HP is analytic of our present concept of cardinality, it is extremely odd to attempt to ground our knowledge of arithmetic, of all things, upon it. Moreover, there is demonstrably no way in which a recognition of the truth of HP can arise simply from reflection on the nature of ordinary arithmetical thought—not, that is, if the principles governing ‘ordinary arithmetical thought’ are captured by the axioms of PA2 (or even of PAS) and the outcome of ‘reflection’ is something that could be written down as a proof. That is what follows from the fact that HP is proof theoretically stronger than PAS (and so PAF and PA2). The disparity of strength parallels the conceptual disparity remarkably well—so well as to remind one why well-conceived technical investigations can be so philosophically fruitful. To summarize and emphasize: HP, conceptual truth or not, cannot be what underlies our knowledge of arithmetic. For no amount of reflection on the nature of arithmetical thought could ever convince one of HP, nor even of the coherence of the concept of cardinality of which it is purportedly analytic. Granted, any rationalist project of this sort will have to invoke a distinction between the ‘order of discovery’ and the ‘order of justification’. But the objection is not that Hume’s Principle is not known by ordinary speakers, nor that there was a time when the truths of arithmetic were known, but HP was not. It is that, even if HP is thought of as ‘defining’ or ‘introducing’ or ‘explaining’ our present concept of cardinality, the conceptual resources required if one is so much as to recognize the coherence of this concept (let alone HP’s truth) vastly outstrip the conceptual resources employed in arithmetical reasoning. Wright’s version of logicism is therefore untenable. Of course, this does not imply that no form of logicism is defensible. And careful examination of Boolos’s proofs itself reveals a way forward. The important observation is that the distinction between finitude and infinitude plays a major role in these proofs. Consider, for example, the sort of model Boolos uses to show that FA is stronger than PAS. Take the domain of the model to be the natural numbers, together with Caesar and Brutus. Given any term of the form ‘Nx : Fx’, assign it a value according to the following scheme: Caesar, if there are infinitely many Fs and infinitely many non-Fs Brutus, if there are infinitely many Fs, but only finitely many non-Fs n, if there are exactly n Fs, for some natural number n Interpret the primitives of PAS according to the ‘definitions’ of the bridge theory FD (thus guaranteeing that all axioms of the bridge theory are true in the model): Thus, ‘0’ denotes the number 0; ‘Nξ ’ is true of xiff x is a natural number; and ‘Pξ η’ is true of the pair < x, y > i just in case either x = y = Caesar, or x = y = Brutus, or y = x + 1. It should be clear that the axioms of PAS are all true in this model. But HP is not: For example, ‘Even(ξ )’ having

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been appropriately defined, ‘Nx : [Nx & Even(x)] = Nx : Nx’ will be false in the model—the former term denoting Caesar, the latter, Brutus—even though the evens are equinumerous with the natural numbers. The important point is that, in this model, Hume’s Principle fails only for infinite concepts. Indeed, as Boolos essentially observes, HPF holds in every model of PAS + FD: So, in any model for PAS + FD, Hume’s Principle will fail to hold, if it does, only because there are some equinumerous infinite concepts which are assigned different numbers. The natural technical question is then: Is there a reasonable Dedekind–Peano system which, in the presence of FD, is equivalent to FAF? The answer is that there is: Relative to FD, FAF is equivalent to PAF. Now, in Grundgesetze, Frege actually derives the axioms of PAS in FA + FD, and these proofs do exploit the full power of HP (since the axioms of PAS are not provable in FAF + FD). But Frege’s proofs can easily be adapted to yield proofs, in FAF + FD, of the axioms of PAF: One need only relativize certain of the formulae appearing in those proofs to the natural numbers. Frege’s development of arithmetic thus does not depend essentially upon (though it may have been psychologically impossible without) the conceptual advance of which I have been speaking. This is striking enough, but it is all the more so since my objections to Wright’s attempt to ground arithmetic on Hume’s Principle simply cannot be raised against an attempt to ground it on HPF. For HPF’s weakness, as compared to HP, reflects the conceptual distance between them, too. There are two points to be made here: First, that recognizing the truth of HPF does not require making the conceptual advance made by Cantor; and, secondly, that one can be convinced of the truth of HPF merely by reflection on ordinary arithmetical thought. To take the first point: Just as HP may be thought of as the sole axiom of a general theory of cardinal numbers, HPF may be thought of as the sole axiom of a theory of finite cardinals. And since HPF makes no claims whatsoever about the conditions under which infinite concepts have the same cardinality, 19 it will not give rise to any of the antinomies generated by HP, whence one does not need to make Cantor’s leap before one can accept the truth of HPF. Indeed, not only could HPF have been recognized as true prior to Cantor’s work, it almost universally was. Bolzano, who was famously skeptical about HP in the infinite case, accepted HPF, 20 as did just about everyone else who considered the matter. For all that HPF says is that, in the finite case, all and only equinumerous concepts have the 19 This would be all the more clear were HPF formulated in a logic which allowed partial functions, so that it was defined only for finite concepts. But working in such a logic would complicate matters quite unnecessarily. Such a formulation would also answer the objection that, in its present form, HPF does not have the form of a Fregean abstraction. 20 See B. Bolzano, Paradoxes of the Infinite, trans., by F. Prihonsky (London: Routledge and Kegan Paul, 1950), §§21–2. In §22, Bolzano gives an argument for HPF similar to the one to be given in the next two paragraphs.

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same number—and who knows what we should say about the infinite ones, other than that none of them have got the same number as any of the finite ones. The second point is that this claim really is implicit in arithmetical reasoning and that one can convince oneself of its truth, come to understand why it is true, by (and perhaps only by) reflecting on basic aspects of arithmetical thought. Now, it is not initially obvious to what notion of finitude we might appeal in reflecting on our arithmetical thought and investigating whether a commitment to HPF is implicit in it. Nor is it clear whether that notion is itself a logical one. But I submit that the intuitive notion of a finite concept is that of one the objects falling under which can be counted, i.e., enumerated by means of some process which eventually terminates. Frege’s definition of finitude directly reflects this intuitive notion: For what the definition says is precisely that a concept is finite if, and only if, the objects falling under it can be ordered as a discrete sequence which has a beginning and an end. 21 Our intuitive notion of finitude can thus be straightforwardly transcribed into second-order logic—thereby showing, modulo the status of second-order logic itself, that this intuitive notion is a logical one. How then can one convince oneself of the truth of HPF? It suffices to realize that the process of counting, which lies at the root of our assignment of numbers to finite concepts, already involves the notion of a one–one correspondence: As Frege frequently points out, 22 to count is to establish a one–one correspondence between certain objects and an initial segment of a sequence of numerals, starting with ‘1’; the process ends with a numeral which names the number of objects counted. By the transitivity of ‘is equinumerous with’, concepts the objects falling under which are themselves equinumerous must be equinumerous with the same initial segments; 23 conversely, any concepts the objects falling under which can be put in one–one correspondence with the same initial segment must be equinumerous. So any two concepts the objects falling under which can be counted—i.e., any two finite concepts— will be assigned the same numeral by the process of counting—i.e., will have the same number—if, and only if, they are equinumerous. And, of course, no infinite concept will get assigned any number by the process of counting. That is enough to establish HPF. 21 That Frege intended his definition to correspond to this intuitive notion is, furthermore, clear from the way he proves Theorems 327 and 348 of Grundgesetze. See my discussion of this point in “The Finite and the Infinite”, op. cit. 22 See, e.g., G. Frege, “Review of E. G. Husserl, Philosophie der Arithmetik I”, in his Collected Papers, ed. B. McGuinness, trans., by H. Kaal (Oxford: Blackwell, 1984), p. 199, original page 319. 23 That there is only one such initial segment will follow from the finitude of the segments themselves, given Frege’s definition. For we shall be able to show that no distinct initial segments are in one– one correspondence. The proof will depend upon certain claims about the numerals themselves, claims corresponding to the axioms of PAF. See here Frege’s discussion of counting in Grundgesetze, Vol. I, §108. I should emphasize that the argument being given here can be formalized: Indeed, the proof of Theorem 3.1, below, can be read as a very rough formalization of it.

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Of course, one might yet have all kinds of worries about the claim that HPF is a conceptual truth. There are two broad classes of such worries: Those which arise from its impredicativity, and those which rest upon the fact that it implies the existence of a lot of objects (infinitely many). I am not going to say anything here about questions of the former sort. 24 But, with regard to the latter, let me say that one needs to be very careful with such objections. Any principle sufficient to ‘ground’ arithmetic in the relevant sense obviously has to imply the existence of infinitely many objects: So one cannot object to someone who is trying to establish that the truths of arithmetic are conceptual truths, or logical consequences of such, by saying that the principle on which he proposes to base arithmetic cannot be a conceptual truth, because no conceptual truth can imply that there are infinitely many objects. One might as well object that the principle yields arithmetic, that his premises imply his conclusion, i.e., accuse him of holding his view. Or, better, one should just say, flatfootedly, that arithmetic can’t be ‘analytic’, in any reasonable sense, since it implies the existence of lots of objects. But that is not so much an objection as a refusal even to discuss the matter. For no one interested in the question whether arithmetic is ‘analytic’ is likely to be moved by that thought. But, having said all of that, let me emphasize that the importance of the question whether HPF is ‘analytic’, in the context of discussions of logicism, should not be allowed to obscure the fact that how we answer it does not affect the philosophical interest of the modification of Frege’s Theorem to be presented below. If HPF really is the ‘basic law of arithmetic’, in the relevant sense, that is philosophically important, whatever its epistemological status might turn out to be.

4.

The relative strengths of the systems

We turn now to the proofs of the four results mentioned at the end of Section 2. In this section, we prove Theorems 1, 2, and 4. We will prove Theorem 3 in the following section. Theorem 1: FA ⇒ PAS. Proof: In the last section, we saw a countermodel which establishes that HP is not a theorem of PAS + FD. That all the axioms of PAS are theorems of FA + FD is the content of Frege’s Theorem, first proven by Frege in Grundgesetze der Arithmetik (though very nearly proven in Die Grundlagen der Arithmetik). 24 I mean to include so-called ‘bad company’ objections. For discussion of these, see the papers of Wright and Boolos in Heck, ed., op. cit. For discussion of general concerns about the impredicativity of HP, see M. Dummett, Frege: Philosophy of Mathematics (Cambridge, MA: Harvard University Press, 1991), pp. 187–89 and 217–22; C. Wright, “The Harmless Impredicativity of Hume’s Principle”, in Schirn, ed., op. cit.; Dummett’s reply, in the same volume.

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As I have discussed Frege’s proof of Frege’s Theorem in detail elsewhere, and as adaptations of Frege’s proofs will be employed below, we need not dwell on it here. 25 Theorem 2: PAS ⇒ PAF. Proof: Clearly, every axiom of PAF other than PAF itself is a theorem of PAS + FD (indeed, of PAS by itself). That PAF is can be proven by induction. If a = 0, then all of its predecessors are finite, since it has none, by Axiom 5 of PAS. Suppose, then, that Na, that, if Na and Pya, then Ny, and that Pab. We must show that, if Pxb, then Nx. So suppose Pxb. By Axiom 4 of PAS, x = a, so Nx. Done. That not every theorem of PAS is a theorem of PAF + FD should be obvious: The axioms of PAF make no claims whatsoever about what the predecessors of objects which are not natural numbers might be, whereas the axioms of PAS state that predecession is one–one, not just on the natural numbers, but universally. Construction of a model is left to the reader. Theorem 4: PAF ⇒ PA2. Proof: Clearly, every theorem of PA2 is a theorem of PAF + FD. Again, that the converse (roughly speaking) is not true should be obvious: PA2 is completely silent on the question whether zero, or any other natural number, has predecessors which are not natural numbers. To construct a model, let the domain consist of the natural numbers and Julius Caesar. Assign denotations to terms of the form ‘Nx : Fx’ according to the following scheme: 0, if there are no Fs or if everything is F n, if there are nFs, for some finite n > 0 JC, if there are infinitely many Fs, but not everything is F Interpret the primitives of PA2 and PAF according to the axioms of FD. The axioms of PA2 may be verified, but PAF fails: The sentence ‘P (Nx : x = 0,0)’ is true in the model. Now, since, by Theorem 3, to be proven in the next section, PAF is equivalent to FAF (in the presence of FD), it follows that FAF is strictly stronger than PA2. The model just given also shows this directly. For the sentence ‘Finitex (x = x) & Nx : x = x = Nx : x = x & ¬Eqx (x = x; x = x)’ is true in the model, whence HPF is false in the model. 25 See my “The Development of Arithmetic in Frege’s Grundgesetze der Arithmetik”, Journal of Symbolic Logic 58 (1993), pp. 579–601, reprinted, with a Postscript, in W. Demopoulos, ed., Frege’s Philosophy of Mathematics (Cambridge, MA: Harvard University Press, 1995), pp. 257–94. See also, of course, Wright, Frege’s Conception, Ch. 4.

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Theorem 3: PAF is equivalent to FAF

In the proofs to be given below, we shall appeal frequently to the following easy consequence of the second axiom of FD: Fa → P[Nx : (Fx & x = a); Nx : Fx] This is Theorem 102 of Grundgesetze, and I shall cite it as such below. The equivalence of PAF and FAF is, in essence, a consequence of the fact that Theorems 327 and 348 of Grundgesetze, mentioned above, can be proven both in PAF and in FAF, with the aid of the bridge theory FD. That is to say, Finite(F) ≡ P ∗= (0, Nx : Fx) can be proven in both theories. This fact will allow us to work back and forth between the condition of finitude, as it appears in HPF, and the claims about the natural numbers made in the axioms of PAF. The proof to be given here of the left-to-right direction—which is Theorem 327 of Grundgesetze—shows it to be a consequence simply of Frege’s definitions and the axiom FE of Log. ‘The number of a finite concept is natural number’ may thus be added to the list of arithmetical facts which are, modulo the status of second-order logic itself, undeniably logical truths. (Others are ‘0P1’ and ‘1P2’, which Boolos shows to be provable in FD.) We begin by noting that we can, without loss of generality, assume the relation which orders a finite set to be one–one (not just functional), and such that no object follows itself in the relevant series. We define: Betw(Q; a; b)(n) ≡ ∀x∀y∀z∀w(Qxy & Qzw → x = z ≡ y = w)& ¬∃x Q ∗ xx & Q ∗= an & Q ∗= nb Proposition : Log ⊢ Finite(F) ≡ ∃Q∃a∃b∀x[Fx ≡ Betw(Q; a, b)(x)]. Proof: Right-to-left: Trivial, since if Betw(Q; a; b)(x), then Btw(Q; a,b)(x). Left-to-right: Assume that Fξ is finite, i.e., that for some Rξ η, a, and b: ∀x[Fx ≡ Btw(R; a, b)(x)]. If ¬∃xFx, let Qξ η be the universal relation; let a and b be whatever you like. Then for no x Betw(Q; a, b)(x). If ∃xFx, say x, then we have that Btw(R; a; b)(x), and so: ∀x∀y∀z(Rxy & Rxz → y = z) & ¬R∗ bb& ∀x[Fx ≡ R ∗= ax & R ∗= xb] Define: Qxy ≡ Rxy & Fx & Fy. Then ∀x[Fx ≡ Betw(Q; a, b)(x)]. The proof is straightforward; I shall not present the details here. 26 26 The plan of the proof is as follows. First, Qξ η is functional, since Rξ η is. Second, if Q ∗ xy, then R ∗ xy; so, if Q ∗= ax & Q ∗= xb, then R ∗= ax & R ∗= xb, so Fx. Third, if Q ∗ xx, then R ∗ xx and Fx, so R ∗= xb; so it is enough to prove that, if R ∗= xb, then ¬Q ∗ xx; do this by induction on the converse of Rξ η. Fourth, prove that the converse of Qξ η is functional, using Theorem 133 of Begriffsschrift, the roll-forward theorem, and the fact that ¬∃x Q ∗ xx. Then prove that, if Fx, then Q ∗= ax, by noting that, if Fx, then R ∗= ax, so it is enough to prove that, if R ∗= ax, then, if Fx, then Q ∗= ax, which can be done by induction. Finally, prove that, if Fx, then Q ∗= xb, similarly.

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Lemma 3.1 (Theorem 327): FD ⊢ Finite(F) → P ∗= (0, Nx : Fx). Proof: If ¬∃xFx, then Nx : Fx = 0 (by FE and the first axiom of FD), whence certainly P ∗= (0, Nx : Fx). So we suppose throughout that ∃xFx. Suppose F is finite; by the proposition, there are objects aand band a relation, Rξ η, which is one–one, in whose (strong) ancestral no object stands to itself, and which is such that x is F iff R ∗= ax and R ∗= xb. It will thus suffice to prove that P ∗= [0, Nx : (R ∗= ax & R ∗= xb)]. Since ∃xFx, for some x, R ∗= ax and R ∗= xb, so R ∗= ab. So it will be enough to prove that R ∗= ay → P ∗= [0, Nx : (R ∗= ax & R ∗= xy)] which we can prove by (logical) induction. For whenever R ∗= mn, we can prove that Fn by showing that Fm and: ∀x∀y(R ∗= mx & Fx & Rxy → Fy) (This fact is an easy consequence of the definition of the weak ancestral.) By comprehension, we may take F ξ to be: P ∗= [0, Nx : (R ∗= ax & R ∗= xξ )]. It will thus suffice to prove: (i) P ∗= [0, Nx : (R ∗= ax & R ∗= xa)] (ii) ∀y∀z{R ∗= ay & P ∗= [0, Nx : (R ∗= ax & R ∗= xy)] & Ryz → P ∗= [0, Nx : (R ∗= ax & R ∗= xz)]}

We are assuming, of course, that Rξ η satisfies the conditions mentioned above. For (i): By (102), P[Nx : (x = a & x = a), Nx : x = a]. Since x = a & x = a iff x = x, Nx : (x = a & x = a) = 0, by FE. Hence, P(0, Nx : x = a) and so P ∗= (0, Nx : x = a). So it will suffice to show that Nx : (R ∗= ax & R ∗= xa) = Nx : x = a, for which, by FE, it suffices to show that R ∗= ax & R ∗= xa ≡ x = a. From right-to-left, this is obvious. For the other direction, suppose that R ∗= ax and R ∗= xa and x = a. Then R∗ ax and R∗ xa, so, by the transitivity of the ancestral, R∗ aa. Contradiction. For (ii): Suppose the antecedent. By comprehension, we may take Fξ and a in (102) to be, respectively, R ∗= a ξ & R ∗= ξ z and z, whence: P{Nx : [(R ∗= ax & R ∗= xz) & x = z], Nx : (R ∗= ax & R ∗= xz)} Since P ∗= [0; Nx : (R ∗= ax & R ∗= xy)], it will be enough to show that Nx : (R ∗= ax & R ∗= xy) = Nx : [(R ∗= ax & R ∗= xz) & x = z] for which, by FE, it is enough to show that: ∀x[(R ∗= ax & R ∗= xy) ≡ (R ∗= ax & R ∗= xz) & x = z] Left-to-right: If R ∗= ax and R ∗= xy, R ∗= ay; since Ryz, then R ∗= az. And if x = z, then R ∗= zy and Ryz, so R∗ zz, contradiction. Right-to-left: Since R ∗= xz and

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x = z, R∗ xz. We then have the following theorem of second-order logic, the roll-back theorem: 27 Q ∗ x y → ∃z(Qzy & Q ∗= xz) By the roll-back theorem, there is some w such that Rwz and R ∗= xw. Since Rξ η is one–one and Ryz, w = y, so R ∗= xy. Remark: The following are theorems of PAF: 28 N P Z : ¬∃xPx0 P1M F : ∀x yz(P ∗= 0z & Pxz & Pyz → x = y) Z E : Nx : Fx = 0 ≡ ¬ ∃x Fx Proof: Zero has no predecessor which is a natural number and, by PAF, only natural numbers precede natural numbers. So since zero is a natural number, it can have no predecessor at all. P1MF will follow immediately from Axiom 4 of PAF if we can show that x and y are themselves natural numbers. But this follows from PAF, since z is a natural number and x and y both precede it. For ZE: If ¬∃xFx, then ∀x(Fx ≡ x = x). So, by FE, Nx : Fx = Nx : x = x = 0. Suppose, then, that Nx : Fx = 0 and ∃x Fx, say, a. By (102), P[Nx : (Fx & x = a), Nx : Fx], so, by NPZ, Nx : Fx = 0. Contradiction. Lemma 3.11 (Theorem 348): PAF + FD ⊢ P ∗= (0, Nx : Fx) → Finite(F). Proof: We prove the equivalent: P ∗= 0n → ∀F[n = Nx : Fx → Finite(F)] The proof is by (logical) induction. We must thus establish that: (i) ∀F[0 = Nx : Fx → Finite(F)] (ii) P ∗= 0n & ∀F[n = Nx : Fx → Finite(F)] & Pnm → ∀F[m = Nx : Fx → Finite(F)]

For (i): Suppose 0 = Nx : Fx. By ZE, ¬∃xFx, so F is finite. For (ii): Suppose the antecedent, and suppose further that m = Nx : Fx. We must show that F is finite. Suppose ¬∃xFx. Then, by ZE, m = Nx : Fx = 0, so 27 The roll-back theorem is proved by induction. We must show

(i) Qxw → ∃z(Qzw & Q ∗= xz) (ii) ∃z(Qzw & Q ∗= xz) & Qwv → ∃z(Qzv & Q ∗= xz) The proof of (i) is trivial: Take z to be x. For (ii), assume the antecedent. Take z in the consequent to be w. By hypothesis, Qwv. And since, by the antecedent, Q ∗= xz & Qzw, certainly Q ∗= xw. 28 In fact, PA2 + NPZ + P1MF is deductively equivalent to PAF. The proof of PAF given above, in the proof of Theorem 2, depends only upon NPZ and P1MF, and not on the full force of Axioms 4 and 5 of PAS.

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Pn0, contradicting NPZ. So ∃xFx, say a, and P[Nx : (Fx & x = a); Nx : Fx]. But also, by hypothesis, P(n, Nx : Fx), and, since P ∗= 0n; P ∗= (0, Nx : Fx). So, by P1MF, n = Nx : (Fx & x = a). Hence, by the induction hypothesis, Finitex (Fx & x = a). It is then a simple matter to show that F too must be finite. Corollary 3.12: PAF + FD ⊢ Finite(F) ≡ P ∗= (0, Nx : Fx). Theorem 3.1: PAF+ FD ⊢ HPF. Proof: By Corollary 3.12, it suffices to show that P ∗= (0; Nx : Fx) → Nx : Fx = Nx : Gx ≡ Eqx (Fx, Gx) We prove the equivalent: P ∗= 0n → ∀F{n = Nx : Fx → ∀G[Nx : Fx = Nx : Gx ≡ Eqx (Fx, Gx)]} The proof is by induction. We must show that: (i) 0 = Nx : Fx→ ∀G[Nx : Fx = Nx : Gx ≡ Eqx (Fx, Gx)] (ii) P ∗= 0n & ∀F{n = Nx : Fx → ∀G[Nx : Fx = Nx : Gx ≡ Eqx (Fx, Gx)]} & Pnm→ ∀F{m = Nx : Fx→∀G[Nx : Fx = Nx : Gx ≡ Eqx (Fx, Gx)]}

For (i): Suppose that 0 = Nx : Fx. By ZE, ¬∃xFx. Now, if 0 = Nx : Gx, by ZE, ¬∃xGx, so Eq(F,G). Conversely, if Eq(F; G), then ¬∃xGx, so, by FE, Nx : Fx = Nx : Gx. For (ii): Suppose the antecedent, and suppose further that m = Nx : Fx. We must show that, for every G, Nx : Fx = Nx : Gx iff Eqx (Fx, Gx). Since P ∗= 0n and Pnm, P ∗= 0m and so P ∗= (0, Nx : Fx). Left-to-right: Suppose Nx : Fx = Nx : Gx. Since P(n, Nx : Fx), Nx : Fx = 0, by NPZ, and so, by ZE, ∃xFx, say a; similarly, ∃xGx, say b. By (102): P[Nx : (Fx & x = a), Nx : Fx] P[Nx : (Gx & x = b), Nx : Gx] Since Nx : Fx = Nx : G, Nx : (Fx & x = a) = Nx : (Gx & x = b), by P1MF. Moreover, since P(n, Nx : Fx), by P1MF, again, n = Nx : (Fx & x = a). So, by the induction hypothesis: Eqx (Fx & x = a, Gx & x = b) But then Eqx (Fx, Gx), since Fa and Gb. Right-to-left: Suppose Eqx (Fx;Gx). Once again, ∃xFx, say a, and ∃xGx, say b, and: P[Nx : (Fx & x = a), Nx : Fx] P[Nx : (Gx & x = b), Nx : Gx]

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Since P(n; Nx : Fx), n = Nx : (Fx & x = a), by P1MF. But, if Eqx (Fx, Gx), Fa, and Gb, certainly: Eqx (Fx & x = a; Gx & x = b) So, by the induction hypothesis, Nx : (Fx & x = a) = Nx : (Gx & x = b). But then, since P[Nx : (Fx & x = a), Nx : Fx] and P[Nx : (Gx & x = b), Nx : Gx], Nx : Fx = Nx : Gx, by Axiom 3 of PAF. That, then, establishes that HPF is a theorem of PAF + FD. We now turn to the proof that all axioms of PAF are theorems of FAF + FD. Our plan is simply to mimic Frege’s proofs of the axioms of arithmetic, relativized in the appropriate way to the natural numbers. To make these proofs work, we need to establish an analogue of Corollary 3.12. From this it will follow that, when talking about natural numbers, we are dealing only with finite concepts, so HPF will do the work HP does in Frege’s proofs. We divide the proof of Theorem 3.2 into two parts: The proof that all axioms other than Axiom 6 hold is relatively easy, and we prove this first; the proof that Axiom 6 holds is of special interest and so will be considered separately. First, we establish the corollary, by establishing an analogue of Lemma 3.11. Lemma 3.21 (Theorem 348, again): FAF + FD ⊢ P ∗= (0, Nx : Fx) → Finite(F). Proof: It will suffice to show that (∗ )P ∗= 0n → ∃F[Finite(F) & n = Nx : Fx] For then, suppose that P ∗= (0, Nx : Fx). Then, for some finite G, Nx : Fx = Nx : Gx. By HPF, Eqx (Fx, Gx), so F is finite. The proof of (∗ ) itself is by induction. We must show that: (i) ∃F[Finite(F) & 0 = Nx : Fx] (ii) P ∗= 0n & ∃F[Finite(F) & n = Nx : Fx] & Pnm→ ∃F[Finite(F) & m = Nx : Fx]

For (i): 0 = Nx : x = x and Finitex (x = x). For (ii): Suppose the antecedent, so that Finite(F) and n = Nx : Fx. Since Pnm, P(Nx : Fx, m), so by Axiom 2 of FD, for some G and b: Gb & m = Nx : Gx & Nx : Fx = Nx : (Gx & x = b) Since Finite(F), by HPF, Eqx [Fx, Gx & x = b], so Finitex (Gx & x = b). But then G too is finite. Corollary 3.22: FAF + FD ⊢ Finite(F) ≡ P ∗= (0, Nx : Fx). Lemma 3.23: FAF + FD ⊢ All axioms of PAF other than Axiom 6.

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Proof: Axioms 1, 2, and 7 do not require any special attention: Each of them is an immediate consequence of FD itself—indeed, of just the third axiom of FD. We thus need to prove Axioms 3, 4, and 5 and the Axiom PAF itself. Axiom 5 is: ¬∃x(Nx & Px0). Suppose that Pn0. By the second axiom of FD, there are Fand ysuch that: Fy & 0 = Nx : Fx & n = Nx : (Fx & x = y) But since 0 = Nx : x = x and Finitex (x = x), HPF yields that Eqx (x = x, Fx). But ∃yFy. Contradiction. (Note that this actually establishes NPZ.) Axiom 3 is: ∀x∀y∀z(Nx & Pxy & Pxz → y = z). So suppose that Na, i.e., that P ∗= 0a, and that Pab and Pac. By the second axiom of FD, there are Fand G, and yand z, such that: Fy & b = Nx : Fx & a = Nx : (Fx & x = y) Gz & c = Nx : Gx & a = Nx : (Gx & x = z) Since P 0a and a = Nx : (Fx & x = y), by Corollary 3.22, Finitex (Fx & x = y). Since Nx : (Fx & x = y) = a = Nx : (Gx & x = z), by HPF, Eqx (Fx & x = y, Gx & x = z). But then Eqx (Fx, Gx), since Fy and Gz, and certainly Finite(F). So, by HPF again, Nx : Fx = Nx : Gx and so b = c. Axiom 4 is: ∀x∀y∀z(Nx & Ny & Pxz & Pyz → x = y). So suppose that Naand Nb, and that Pac and Pbc. Note that P ∗= 0c. We shall make no further appeal to the assumptions that Na and Nb. Once again, there are F and G, and y and z, such that: ∗=

Fy & c = Nx : Fx & a = Nx : (Fx & x = y) Gz & c = Nx : Gx & b = Nx : (Gx & x = z) Since c = Nx : Fx and P ∗= 0c, by Corollary 3.22, Finite(F). And Nx : Fx = c = Nx : Gx, so by HPF, Eqx (Fx, Gx). But then, since Fy and Gz, Eqx (Fx & x = y, Gx & x = z); these are finite, since F and G are, so by HPF again, Nx : (Fx & x = y) = Nx : (Gx & x = z) and so a = b. (Note that this actually establishes P1MF.) PAF is: Nx & Pyx → Ny. As noted parenthetically above, the proofs of Axioms 4 and 5 in fact suffice to prove NPZ and P1MF, from which PAF follows. But it can also be proven directly. Suppose that P ∗= 0n and that Pmn. Since Pmn, there are Fand a such that: Fa & n = Nx : Fx & m = Nx : (Fx & x = a) Since P ∗= (0, Nx : Fx), F is finite; so Finitex (Fx & x = a), and so P ∗= [0, Nx : (Fx & x = a)]. But then P ∗= 0m. In the paper mentioned earlier, Boolos proves the surprising result that, in the bridge theory FD, Axiom 6, that is, ∀x(Nx → ∃yPxy), of PA2 is redundant, since it follows from Axioms 3, 4, and 5. More recently, he has observed that, in FD, Axiom 6 in fact follows from Axiom 3 alone. Boolos’s original proof

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of this extraordinary result is somewhat indirect: 29 I shall take the opportunity to give a direct proof here. Of course, since FAF + FD ⊢ Axiom 3, it follows that FAF + FD ⊢ Axiom 6. To prove Axiom 6, Frege proves P ∗= 0n → P[n, Nx : (P ∗= 0x&P ∗= xn)] to establish which he needs the crucial lemma: P ∗= 0n → ¬P ∗ nn It is for the proof of this lemma that Axiom 5 is needed. Our proof will differ from his, in the first instance, in that we do not make use of this lemma, but instead pack the necessary condition into the antecedent and prove the weaker: P ∗= 0n & ¬P ∗ nn → P[n, Nx : (P ∗= 0x & P∗= xn)] We complete our argument by also proving P ∗= 0n & P∗ nn → ∃yPny whence Axiom 6 follows by dilemma. Our proof will also differ in another way. Frege appeals to Axiom 4 at a crucial point, but we shall see that the necessary inference does not require it, even in his proof. What we shall use instead is the logicized version of the Law of Trichotomy: P ∗= 0b & P∗= 0c → P ∗ bc ∨ P∗ cb ∨ b = c The Law follows from Axiom 3 and the following strengthening of the famous Proposition 133 of the Begriffsschrift: 30 ∀x ∀ y∀z[R ∗= ax & Rxy & Rxz → y = z] & R ∗= ab & R∗= ac → R ∗ bc ∨ R∗ cb ∨ b = c Instantiating ‘R’ with ‘P’, ‘a’ with ‘0’, and noting that the first conjunct then follows from Axiom 3, the Law of Trichotomy follows immediately. 29 We have that: FD, 3, 4, 5 ⊢ 6. Boolos then observed that also: FD, 3, ¬6 ⊢ 4 & 5. But then, by truthfunctional logic: FD, 3, ¬6 ⊢ 6. And so: FD, 3 ⊢ 6. For a proof of a related result, see G. Boolos, “Frege’s Theorem and the Peano Postulates”, Bulletin of Symbolic Logic 1 (1995), pp. 317–26. 30 The strengthening lies in our assuming not that Rξ η is functional, but just that it is functional on the members of the R-series beginning with a.

Proof: We assume that ∀x∀y∀z[R ∗= ax & Rxy & Rxz → y = z] and R ∗= ab and prove that R ∗= ac → R ∗ bc ∨ R ∗ cb ∨ b = c by induction on c. We must thus prove: (i)

R ∗= aa → R ∗ ba ∨ R ∗ ab ∨ b = a

(ii)

R ∗= ax & (R ∗= ax → R ∗ bx ∨R ∗ xb ∨b = x) & Rxy → (R ∗= ay → R ∗ by ∨R ∗ yb ∨ b = y)

Since R ∗= ab, (i) follows from the definition of the weak ancestral. So suppose the antecedent in (ii). Note that R ∗= ay. Moreover, R ∗ bx or R ∗ xb or b = x. If R ∗ bx, then since Rxy, R ∗ by. Moreover, if b = x, then Rby, so R ∗ by. So suppose that R ∗ xb. By the roll-forward theorem, to be mentioned shortly, for some z, Rxz and R ∗= zb. But then Rxz and Rxy: And since Rξ η is functional on the R-series beginning with a and R ∗= ax, we have z = y. So R ∗= yb, hence R ∗ yb or b = y.

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Lemma 3.24: FD + Axiom 3 ⊢ Axiom 6 of PAF. Proof: We must establish ∀x[Nx → ∃yPxy] for which, in light of FD, it will suffice to establish: P ∗= 0n → ∃yPny We proceed by dilemma, proving each of: P ∗= 0n & P ∗ nn → ∃yPny P ∗= 0n & ¬P ∗ nn → ∃yPny The former follows immediately from the following theorem of second-order logic, the roll-forward theorem: 31 R ∗ ab → ∃y(Ray & R ∗= yb) For, then, if P∗ nn, for some y, Pny & P ∗= yn, so certainly ∃yPny. For the latter, we prove: P ∗= 0n & ¬P ∗ nn → P[n, Nx : (P ∗= 0x & P ∗= xn)] The proof is by induction. We thus need to establish: (i) FD ⊢ ¬P ∗ 00 → P[0, Nx : (P ∗= 0x & P ∗= x0)] (ii) FD + Axiom 3 ⊢ P ∗= 0a & {¬P∗ aa → P[a, Nx : (P ∗= 0x & P ∗= xa)] & Pab → {¬P∗ bb → P[b, Nx : (P ∗= 0x & P ∗= xb)]}

For (i): Suppose ¬P ∗ 00. Since P ∗= 00, by (102): P[Nx : (P ∗= 0x & P ∗= x0 & x = 0), Nx : (P ∗= 0x & P ∗= x0)] Now suppose that P ∗= 0x & P ∗= x0 & x = 0. Then, since P ∗= x 0 & x = 0, P ∗ x0. But then P ∗ x0 and P ∗= 0x, so P ∗ 00, contradicting our supposition. Thus ¬∃x(P ∗= 0x & P ∗= x0 & x = 0). By FE, Nx : (P ∗= 0x & P ∗= x0 & x = 0) = 0 and so P[0, Nx : (P ∗= 0x & P ∗= x0)]. For (ii): Suppose the antecedent and suppose further that ¬P∗ bb. Suppose, for reductio, that P∗ aa. By the roll-forward theorem, for some y, Pay and P ∗= ya. Since Pab, Axiom 3 implies that y = b. But then P ∗= ba and Pab, so P∗ bb. Contradiction. Hence ¬P∗ aa. By the induction hypothesis, then, P[a, Nx : (P ∗= 0x & P ∗= xa)]. Now, we need to show that P[b, Nx : (P ∗= 0x & P ∗= xb)]. Since P ∗= 0b and P ∗= bb, by (102): P[Nx : (P ∗= 0x & P ∗= xb & x = b), Nx : (P ∗= 0x & P ∗= xb)] So it is enough to show that b = Nx : (P ∗= 0x & P ∗= xb & x = b). And since Pab and P[a, Nx : (P ∗= 0x & P ∗= xa)], we have, by Axiom 3, that b = 31 The proof of this result is similar to that of the roll-back theorem, mentioned earlier.

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Nx : (P ∗= 0x & P ∗= xa). So we need only show that Nx : (P ∗= 0x & P ∗= xa) = Nx : (P ∗= 0x & P ∗= xb & x = b). By FE, this will follow from: ∀x[(P ∗= 0x & P ∗= xa) ≡ (P ∗= 0x & P ∗= xb & x = b)] Left-to-right: Suppose P ∗= 0x and P ∗= xa. Then since Pab, certainly P ∗= xb and, further, P ∗= 0b. Suppose x = b. Then P ∗= ba and Pab, so P∗ bb. Contradiction. Right-to-left: Suppose P ∗= 0x & P ∗= xb & x = b. Then P∗ xb. By the roll-back theorem, for some y, P ∗= xy and Pyb. Up to this point, we have been following Frege’s proof closely. Here, he uses Axiom 4 to conclude that, since Pab, a = y, whence P ∗= xa, and he is done. But we can in fact establish that a = y without appeal to Axiom 4. We have that P ∗= xy and Pyb. Since P ∗= 0x, certainly P ∗= 0y. Since, by the initial hypotheses of the inductive step, P ∗= 0a, the Law of Trichotomy yields that either P∗ ay or P∗ ya or a = y. Suppose that P∗ ay. By the roll-forward theorem, for some z, Paz & P ∗= zy. But since Pab, Axiom 3 implies that z = b. So P ∗= by & Pyb, so P∗ bb, contradiction. Similarly, if P∗ ya, then for some z, Pyz & P ∗= za. But since Pyb, Axiom 3 implies that z = b, so P ∗= ba Pab, so P∗ bb, once again. Hence a = y, and we are done. Theorem 3.2: FAF ⊢ FD ⊢ All axioms of PAF. Proof: By Lemmas 3.23 and 3.24.

6.

Closing

We have thus seen that FAF is equivalent, in the presence of the bridge theory FD, to PAF. By Theorem 4, then, FAF is strictly stronger than PA2. The following two questions now raise themselves: Whether there is some further weakening of HP which is provable in PA2 + FD and, if so, whether some such principle is equivalent, in FD, to the conjunction of the axioms of PA2. A natural axiom at which to look would be WHP (for Weak Hume’s Principle): Finitex (Fx) & Finitex (Gx) → [Nx : Fx = Nx : Gx ≡ Eqx (Fx, Gx)] WHP states only that finite concepts have the same number if, and only if, they are equinumerous and makes no claim whatsoever about the conditions under which infinite concepts have the same number as any concept, finite or otherwise. (As far as WHP is concerned, some infinite concepts could have the number zero, others one, and so forth.) Call the theory whose sole nonlogical axiom is WHP, WHP. It can be shown that, though WHP is provable in PA2 + FD, none of the axioms of PA2 which are not already theorems of FD itself are theorems of WHP + FD: Not even the disjunction of these axioms— that is, of Axioms 3, 4, 5, and 6—is a theorem of WHP + FD.

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Still, it is easy to see that there are no finite models of WHP + FD. For, in any model, there must be a number Nx : x = x; there must be a number Ny : (y = Nx : x = x), which, by WHP, will differ from Nx : x = x; there will be a number Nz : [z = Nx : x = x ∨ z = Ny : (y = Nx : x = x)], which again must differ from the first two, and so forth. Indeed, it is not terribly difficult to prove that PA2 can be relatively interpreted in WHP, and so that PA2 and WHP are equi-interpretable and therefore equi-consistent. As we saw earlier, however, the fact that a theory A is interpretable in another B is no guarantee that there is any reasonable bridge theory by using which one can prove the axioms of A in B. One might therefore wonder whether, in this case, there is some bridge theory other than FD, by appeal to which one could prove the axioms of PA2 in WHP. In fact there is, the necessary modification to the axioms of FD not being drastic. The bridge theory FDF has the same first and third axioms as FD, but we change the second axiom to: ∃F[Finite(F) & m = Nx : Fx] ∨ ∃F[Finite(F) & n = Nx : Fx] → Pmn ≡ ∃F∃y[Finite(F) & Fy & n = Nx : Fx & m = Nx : (Fx & x = y)] This axiom now states nothing at all about when numbers which are not the numbers of finite concepts precede one another. It requires, however, that, if a number is the number of a finite concept, then it will precede or be preceded by another number only if there is some finite concept which does the trick. The axiom, though complicated as stated, seems intuitive enough and is certainly true, since it is a theorem of FAF + FD, as can easily be seen. It may thus be considered a partial definition of one of the primitives of PA2 in terms of those of WHP. And it can be shown that, relative to FDF, WHP, PA2, and PAF are all equivalent. What philosophical interest this result might have for a logicist is a question I shall not pursue. 32

32 Thanks here to Charles Parsons, Alison Simmons, Jason Stanley, Jamie Tappenden, and Crispin Wright for discussion. The paper also benefitted from the comments of the Journal of Philosophical Logic’s referees. I owe a special debt to George Boolos, discussions with whom led to my work on this topic, as many others. George was a teacher, a colleague, a mentor, and a source of inspiration and courage—but most of all, he was a friend. This paper is dedicated to his memory.

ON FINITE HUME 1 Fraser MacBride†

Neo-Fregeanism declares there to be an a priori route that we may follow (guided by proofs and definitions) from an understanding of analytic truths to a grasp of the fundamental laws of arithmetic (see Wright [1997], pp. 202– 11). The purportedly analytic principle from which the neo-Fregean claims we may set out is Hume’s Principle, a principle that specifies the conditions under which concepts have the same cardinal number: (HP)∀F∀G[(N x : F x = N x : Gx) ↔ F1 − 1G] When this principle is adjoined to second-order logic, the system that results is Frege arithmetic. What makes it plausible to suppose that Hume’s principle provides a departure point from which we may successfully go onto grasp arithmetic a priori is the result called Frege’s theorem. For according to Frege’s theorem, the Peano postulates can be interpreted in Frege arithmetic and their interpretations proved in that system (see Boolos [1996]). There are, however, alternative departure points, other purportedly analytic principles, from which we may just as plausibly set out. As Richard Heck has shown (Heck [1997]), finite Hume’s principle is one such principle, a principle that states the conditions under which finite concepts have the same cardinal number: (HPF) ∀F∀G((Finite(F)vFinite(G)) → [(Nx : Fx = Nx : Gx) ↔ F1 − 1G]) The system that results from adjoining this principle to second-order logic may be called finite Frege arithmetic. Heck demonstrates that the Peano postulates have provable interpretations in finite Frege arithmetic just as they do in Frege arithmetic. 1 This paper first appeared in Philosophia Mathematica 8, [2000], pp. 150–9. Reprinted by kind permission of the editor and Oxford University Press. † I wish to thank Patrick Greenough, Katherine Hawley, Richard Heck, Alex Oliver, Stewart Shapiro, Crispin Wright and, especially, Peter Clark for discussion of this paper. I am also grateful to the audience at an Arché conference on abstraction held at the University of St. Andrews for their helpful comments.

85 Roy T. Cook (ed.), The Arché Papers on the Mathematics of Abstraction, 85–93. c 2007 Springer.

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To these technical results Heck adds the philosophical contention that it is from an understanding of finite Hume’s principle—rather than Hume’s principle—that the neo-Fregean should guide us to an a priori grasp of arithmetic. I will argue that Heck’s philosophical arguments are flawed. They do not give the neo-Fregean reason to lose nerve. An abstraction is a principle of the form: ((α j ) = (αk )) ↔ α j ≈ αk Abstraction principles tie the conditions under which entities of one kind (s) are identical to the obtaining of an equivalence relation (≈) amongst another kind of entity (αs). Such a principle may be read as offering a substantial, synthetic claim to the effect that facts about one kind of entity are necessarily connected to facts about another entirely distinct kind. But the neo-Fregean counsels that we need not always read abstractions in this way (Wright [1997], pp. 205–08). He suggests instead that an abstraction principle may be read (in certain circumstances) as offering an analytic claim. For where an abstraction contains an occurrence of a novel term-forming operator (“”) on expressions (“α1 ” · · · “αk ” . . . ) that are already understood, the abstraction may be read as embodying a stipulation that introduces the novel term into language. It may be read as stipulating that the truth conditions of identity statements featuring the novel operator (“(α j ) = (αk )”) coincide with the truth conditions of another familiar form of statement (“α j ≈ αk ”). Since one way to be analytic is to be stipulated, the neoFregean concludes, that under these circumstances, abstraction principles are analytic. According to neo-Fregean doctrine, Hume’s principle is an analytic abstraction. It introduces a novel cardinality operator by stipulating that the truth conditions of identity statements concerning cardinal numbers coincide with the conditions under which an equivalence relation amongst concepts may be familiarly said to obtain. Imagine a faultlessly rational character—call him ‘Hero’—who has mastered second-order logic but has yet to be introduced to any characteristically mathematical notions (see Wright [1998], p. 359). Despite his mathematical ignorance, Hero is able to grasp Hume’s principle because the familiar vocabulary (bound concept variables, the notion of one–one correspondence) which the principle uses to introduce the cardinality operator is second-order expressible. Hero is therefore able to appreciate a priori the conditions under which—according to the stipulation Hume’s principle effects—cardinal numbers are identical. Having come to appreciate Hume’s principle, Hero is then able to appreciate the truth of Peano’s postulates. For as Frege’s theorem shows, these postulates may be interpreted and their interpretations proved in a system generated from Hume’s Principle and the second-order logic that Hero has already mastered. The possible case of Hero—the neo-Fregean

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claims—shows how it is possible for mathematical knowledge to be acquired a priori. There are many challenges the neo-Fregean must overcome in order to substantiate the claim that Frege’s theorem may be invested with such epistemological significance. Amongst the most basic of these challenges are two. First, the neo-Fregean must establish that Hume’s Principle is nothing more than a stipulation even though in conjunction with second-order logic it generates a theory—Frege arithmetic—committed to the existence of infinitely many objects. Second, the neo-Fregean must establish—contrary to Quinean suspicions—that an understanding of second-order logic does not presuppose prior knowledge of mathematics. Nevertheless, provided such challenges can be met, it appears the neo-Fregean may legitimately claim that it is possible to travel a priori from an understanding of Hume’s Principle to a grasp of arithmetic. But the legitimacy of this final claim remains open to question. For even if it is granted that Hume’s principle may be employed to access some array or other of a priori truths, it remains to be established that these really are arithmetical truths and that the a priori knowledge acquired by employing Hume’s principle is genuinely arithmetical in character. Frege’s theorem, on its own, does not secure this result. All that Frege’s theorem strictly shows—given the assumption that abstraction principles can be used to generate a priori knowledge—is that there is a system of a priori truths (Frege arithmetic) that is capable of modelling arithmetic. It does not show that any of these a priori truths are arithmetical in character. According to Heck, the a priori truths that flow from Hume’s principle can only be arithmetical if Hume’s principle itself is a genuinely arithmetical principle. To establish that Hume’s principle is arithmetical, Heck claims, it must be shown to be a basic law “upon which ordinary arithmetical reasoning relies” (Heck [1997], p. 596). But, Heck goes onto argue, Hume’s principle does not underlie ordinary arithmetical reasoning and so the a priori knowledge which a grasp of Hume’s principle grounds cannot be arithmetical. Heck concludes that Frege’s theorem fails to map an a priori route to arithmetical knowledge. Heck bases his argument that Hume’s principle does not inform ordinary arithmetical reasoning on the contention that “no amount of reflection on the nature of arithmetical thought could ever convince one of HP” (Heck [1997], p. 597). Hume’s Principle says that all and only equinumerous concepts have the same cardinality. It follows that the number of natural numbers and the number of even numbers are the same (since the concepts natural number and even number can be put in one–one correspondence). But no amount of reflection on ordinary arithmetical reasoning—that is, “reasoning about, and with, finite numbers”—could ever convince one that these infinite cardinals are the same. Indeed it comes as something of a conceptual shock to discover that the concepts natural number and even number have the same cardinality.

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Since the evens form only a portion of the naturals, our naive inclination is to say that there are fewer evens than naturals. That is why it took not just an ordinary arithmetical thinker, but an intellect of great genius—Georg Cantor— to make the conceptual leap required to recognise the truth of what Hume’s principle tells us, that the concepts in question share the same cardinal. That is why it cannot be Hume’s principle that is implicit in ordinary arithmetical reasoning. By contrast, Heck continues, it is plausible to hold that Finite Hume’s principle informs habitual arithmetical practice. Finite Hume’s principle does not make any claim concerning the conditions under which infinite cardinals are the same or different. It is possible to accept Finite Hume without taking the conceptual leap that Cantor made. In fact, Heck claims, prior to the receipt of Cantor’s work, reflection on numerical practice led almost all thinkers to endorse finite Hume’s principle. Heck cites Bolzano as an example of such a thinker, a thinker who was sceptical about Hume’s principle (since he thought it possible for there to be infinite totalities that, even though equinumerous, nevertheless differed in multiplicity) but who endorsed Finite Hume’s principle. Heck goes so far as to claim that Finite Hume “really is implicit in arithmetical reasoning and that one can convince oneself of its truth, come to understand why it is true, by (and perhaps only by) reflecting on basic aspects of arithmetical thought”. We count by establishing a one–one correspondence between an initial segment of the sequence of numerals and the objects counted: we begin with “1” and end with a numeral n that stands for the number of objects. Consequently, if two finite concepts are equinumerous, then, by the transitivity of equinumerosity, the objects falling under one of those concepts will be one– one correspondent with the same initial segment of numerals as the objects falling under the other concept. In other words, those two finite concepts will have the same number. Conversely, if the objects falling under one concept are one–one correspondent with the same initial segment of the numerals as the objects falling under another concept—that is, if those two finite concepts share the same number—then, by the transitivity of equinumerosity, those concepts will be equinumerous. Reflection on the nature of counting thereby establishes that finite concepts have the same number if, and only if, they are equinumerous. Indeed—Heck claims—it is just such an argument that convinced Bolzano of the truth of finite Hume. Heck concludes that if the neo-Fregean claim to provide an epistemology of arithmetic is to have any legitimacy, the neo-Fregean should adopt finite Hume’s principle rather than Hume’s principle. The neo-Fregean account of how a priori knowledge may be generated applies just as well to the former principle as the latter. Just like its less modest cousin, finite Hume’s principle may be understood as an analytic abstraction, only a restricted one, stipulating the conditions under which finite cardinal numbers are identical. The additional vocabulary finite Hume’s principle uses to fix these conditions

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(the extra notion of finite) is also expressible in second-order logic. And, furthermore, interpretations of the Peano postulates are—as Heck shows— provable in the system that results from uniting finite Hume’s principle and second-order logic. But, unlike Hume’s principle, finite Hume is implicit in ordinary arithmetical practice. So, Heck claims, there is every reason to suppose that the a priori knowledge that finite Hume delivers—by contrast to the knowledge that flows from Hume’s principle—is arithmetical. Heck’s arguments are, however, flawed. To begin with, Heck’s claim that prior to the receipt of Cantor’s work it was finite Hume, rather than Hume’s principle, that informed arithmetical thinking is highly contentious. Prior to the receipt of Cantor’s work it seemed paradoxical to think that the number of natural numbers was the same as the number of even numbers even though the latter constitute only a portion of the former. But in order for this thought to appear a paradox, it is necessary not only to have an intellectual inclination to deny that those numbers are distinct, it is also necessary to have an intellectual inclination to affirm that they are the same. Yet if—as Heck supposes— pre-Cantorian thinkers only endorsed finite Hume, then they would have had no reason to affirm the identity of the number of natural numbers and the number of even numbers. They would have had no reason because finite Hume’s principle is entirely silent about identities amongst infinite numbers. So, it would not have seemed—as it did seem to them—a paradox that those numbers were distinct. Indeed, such thinkers as Heck describes would have been quite unable to frame a thought about the identity and distinctness of infinite numbers. If, prior to the reception of Cantor’s work, ordinary arithmetical reasoning had only been informed by finite Hume then Bolzano’s Paradoxes of the Infinite would have been a shorter book. Heck sketches an historical Cantor whose conceptual contribution was— through his work on the infinite—to clear the way for the introduction of a novel criterion of numerical identity that applied not only to finite but also to infinite numbers. According to Heck history, we fail to register the significance of Cantor’s contribution if we suppose Hume’s principle already informed ordinary arithmetical reasoning. But Heck’s sketch makes no sense of the fact that prior to Cantor it appeared paradoxical to affirm that equinumerosity amongst concepts sufficed for the identity of infinite numbers. In fact it makes better sense of history to describe the significance of Cantor’s contribution in a quite different way. According to this historical reconstruction, ordinary arithmetical thinking, prior to Cantor, was guided not only by Hume’s principle but also by the intuition that the cardinality of a collection of entities is always greater than the cardinality of any of its proper parts. The clash of principle and intuition made the identities of infinite numbers appear paradoxical to earlier thinkers. Hume’s principle, according to this reconstruction, led these thinkers to affirm that the natural numbers and the even numbers had the same cardinality, whereas intuition led them to suppose their cardinalities were different since the even numbers form only

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a proper part of the collection of natural numbers. Cantor’s contribution was to recognise this intuition was founded only on a parochial acquaintance with finite wholes and parts, and to enable us to resolve the paradox by persuading us to abandon the intuition. Cantor’s contribution was not to clear the way for the introduction of a novel principle of numerical identity, Hume’s principle. Rather—like a great moral leader—he persuaded us to see that a familiar principle should be applied in an unfamiliar case. Heck’s argument that finite Hume’s principle does inform ordinary arithmetical practice (because it may be arrived at by reflection on the process of counting) must also be questioned. First, if any abstraction principle can be arrived at by reflection on the process of counting, then that principle is not finite Hume. Finite Hume presupposes the intelligibility of the notion of an infinite cardinal. Suppose that concepts F and G fail to be equinumerous because F is infinite whereas G is finite. Then, by finite Hume, there is an infinite number belonging to the concept F that is distinct from the number belonging to the concept G. But, if the identity conditions for cardinals flow from reflection on the process of counting, then the notion of an infinite cardinal cannot make sense. For an infinite cardinal is a number that belongs to a concept the objects falling under which, by definition, cannot be counted; there is no initial segment of the numerals with which the objects falling under such a concept can be put in one–one correspondence. So finite Hume cannot be arrived at by the reflective route recommended by Heck. This suggests that if any abstraction principle arises from simple reflection on the counting process, it is another principle—weak Hume’s principle—that, unlike finite Hume, concerns only the identity conditions of finite cardinals: (WHP)∀F∀G((Finite(F) & Finite(G)) → [(Nx : Fx = Nx : Gx) ↔ F1 − 1G]) The suggestion receives historical support. Heck mentions Bolzano as an example of a thinker who endorsed finite Hume on the basis of reflection on the counting process (Bolzano [1950], §22). In fact, in the passage cited by Heck, it is weak Hume, rather than finite Hume, which Bolzano sanctions on that basis. 2 Second, it is far from evident that there is any abstraction even resembling finite Hume that is implicit in ordinary arithmetical practice. A character can readily be imagined who is capable of counting yet lacks the conceptual wherewithal to grasp finite Hume. He might, for instance, lack the notion of a relation that would be required to grasp the second-order definition of one– one correspondence embodied in finite Hume. Or he might be unable to comprehend the notion of an arbitrary property required to grasp the significance 2 Bolzano [1950], pp. 98–9 endorses the following principle: “Whenever, in fact, two finite sets are constituted so that every object a in the one corresponds to another object b in the other which can be paired off with it, no object in either set being without a partner in the other, and no object occurring in more than one pair: then indeed are the two finite sets always equal in respect of multiplicity”.

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of its second-order quantifiers. Alternatively, instead of having any thought of numerical identity, he might have a concern only for the employment of numerals. More generally, it is contentious to suppose that any theoretical principle is implicit in ordinary arithmetical practice. For not only is the notion of a theoretical principle implicit in practice a very murky and difficult notion to apply—as Kripke’s rule following paradox makes clear—the more radical possibility remains that our arithmetical practice should not be described in theoretical terms at all. Perhaps ordinary arithmetical practice, rather than being informed by a ghostly inner theory, is better understood as the exercise of a repertory of arithmetical techniques. Of course, if ordinary arithmetical reasoning is not informed by a secondorder abstraction, then a fortiori it is not informed by Hume’s principle. Heck assumes that the neo-Fregean can only succeed in providing an epistemology of arithmetic if the basic laws employed for that purpose actually inform ordinary arithmetical reasoning. So if Heck is correct to make this assumption it appears the epistemological project the neo-Fregean undertakes cannot succeed. Unfortunately, Heck makes no attempt to justify the assumption that an arithmetical epistemology must be derived from principles that we can retrieve by reflection on ordinary arithmetical reasoning. Heck’s remarks suggest the following argument. It is constitutive of arithmetical truths that they are derived from the basic laws that actually inform ordinary arithmetic. We might call these laws the ‘canonical sources’ of arithmetical truths. Since Hume’s principle is not a canonical source, the a priori truths that may be derived from it cannot be arithmetical in character. But this argument is far from convincing. The premise that an arithmetical truth can only be derived from canonical sources is unmotivated, and generalised it leads to the absurd conclusion that a stronger principle cannot be employed—perhaps for reasons of elegance—to prove the consequences of a weaker principle. Heck’s objections to neo-Fregeanism reflect a presupposition common amongst its critics. According to this presupposition, the neo-Fregean project is a hermeneutic one: it aims to show that what we ‘had in mind’ all along, when we reasoned arithmetically, is a priori. It is now generally recognised that Frege had no concern to determine that ordinary arithmetical reasoning is a priori (see Benacerraf [1981], Weiner [1984], and Dummett [1991], pp. 176– 79). Regrettably, it is not generally recognised that neo-Fregeans need have no hermeneutic concern either. Benacerraf exhibits such a lack of recognition in his discussion of Hempel: According to Hempel the Frege-Russell definitions of number, successor, and related concepts have shown the propositions of arithmetic to be analytic because they follow by stipulative definitions from logical principles. What Hempel has in mind here is clearly that in a constructed formal system of logic one may introduce by stipulative definition the expressions ‘Number’, ‘Zero’. . . in such

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Having outlined Hempel’s logicist project, Benacerraf proceeds to remark (next sentence) that the project fails because Hempel does not establish that the arithmetical truths we express in ordinary language are analytic: He [Hempel] is not entitled to that conclusion. Nor would he be even if the theorems of logic in their primitive notations were themselves analytic. For the only things that have been shown to follow from theorems of logic by stipulation are the abbreviated theorems of the logistic system. To parlay that into an argument about the propositions of arithmetic, one needs an argument that the sentences of arithmetic, in their preanalytic senses, mean the same (or approximately the same) as their homonyms in the logicistic system. That requires a separate and longer argument. I bring this up here not to berate Hempel but to use his views as an illustration of the epistemological motivation that drives twentieth century logicists.

But Benacerraf’s observation is misplaced. For neither Hempel nor any other neo-Fregean ever claimed to be putting forward a thesis about the ordinary senses of arithmetical expressions. Hempel’s expresses the very different nature of his epistemological project with exemplary clarity: The assertion that the definitions given above state the “customary” meaning of arithmetical terms involved is to be understood in the logical, not the psychological sense of the term “meaning”. It would obviously be absurd to claim that the above definitions express “what everybody has in mind” when talking about numbers and the various operations that can be performed with them. What is achieved by those definitions is rather a “logical reconstruction” of the concepts of arithmetic in the sense that if the definitions are accepted, then those statements in science and everyday discourse which involve arithmetical terms can be interpreted coherently and systematically in such a manner that they are capable of objective validation. Hempel [1945], p. 387

The objections that Heck (and Benacerraf) voice are ineffective because they misconstrue the nature of the neo-Fregean project. That project never was to uncover a priori truth in what we ordinarily think, but to demonstrate how a priori truth could flow from a logical reconstruction of arithmetical practice. By failing to recognise the nature of the beast, Heck fails to articulate a convincing objection to the neo-Fregean doctrine that there is an a priori route from Hume’s principle to knowledge of arithmetic. 3 3 Of course, there are a host of other difficulties the neo-Fregean must confront in order to make good their claims, difficulties concerning the ability of Hume’s principle to introduce objects and the tenability of the minimalist metaphysic that principle assumes. I explore these issues further in MacBride [2003].

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References Benacerraf, P. [1981]: “Frege: The Last Logicist” in P. French et al. (eds) Midwest Studies in Philosophy VI, pp. 17–35. Minneapolis: University of Minnesota Press. Reprinted in Demopoulos [1995], pp. 41–67. Benacerraf, P. and Putnam, H. (eds) [1983]: The Philosophy of Mathematics: Selected Readings. Cambridge: Cambridge University Press. Bolzano, B. [1950]: Paradoxes of the Infinite trans. F. Prinhonsky. London: Routledge and Kegan Paul. Boolos, G. [1996]: “On the Proof of Frege’s Theorem” in A. Morton and P. Stich (eds), Benacerraf and His Critics, pp. 143–59.Oxford: Blackwells. Demopoulos, W. (ed.) [1995]: Frege’s Philosophy of Mathematics. Harvard: Harvard University Press. Dummett, M. [1991]: Frege: Philosophy of Mathematics. London: Duckworth. Frege, G. [1950]: Foundations of Arithmetic, trans. J.L. Austin. Oxford: Basil Blackwell. Hempel, C. [1945]: “On the Nature of Mathematical Truth”, American Mathematical Monthly, 52, pp. 543–56. Reprinted in Benacerraf and Putnam [1983], pp. 377–93. Heck, R. [1997]: “Finitude and Hume’s Principle”, Journal of Philosophical Logic, 26, pp. 598–617. MacBride, F. [2003]: “Speaking with Shadows: A Study of Neo-Fregeanism”, British Journal for the Philosophy of Science, 54, pp. 103–63. Weiner, J. [1984]: “The Philosopher behind the Last Logicist” in C. Wright (ed.) Frege: Tradition & Influence. Oxford: Basil Blackwell. Wright, C. [1997]: “On the Philosophical Significance of Frege’s Theorem” in R. Heck (ed.), Language, Thought and Logic: Essays in Honour of Michael Dummett, pp. 201–44. Oxford: Oxford University Press. Wright, C. [1998]: “On the Harmless Impredicativity of N= (‘Hume’s Principle’)” in M. Schirn (ed.), The Philosophy of Mathematics Today, pp. 339–68. Oxford: Clarendon Press.

COULD NOTHING MATTER? 1 Fraser MacBride

According to the neo-Fregean we may acquire a priori knowledge of arithmetic’s fundamental laws by reflecting upon the (recognisable) second-order logical consequences of an a priori principle (Hume’s Principle) that specifies identity conditions for cardinal numbers: (HP)(∀ F)(∀ G)(Nx : Fx = Nx : Gx ↔ F 1 − 1 G) This epistemological contention receives mathematical support from Frege’s Theorem, the result that Peano’s axioms can be interpreted and their interpretations proved in the system (Frege arithmetic) that results from adjoining (HP) to second-order logic (see Wright 1997: 202–11). 2 Black (2000) brands this contention “implausible” and argues that (HP) provides the wrong sort of reason for believing in the infinity of the number series. I will argue that a central argument in Black’s paper is ineffective, relying upon a popular misconception of the epistemological character of the neo-Fregean project (see Black 2000: 233–36 and also Heck 1997: 597–98 and Lowe 1998: 49–50). It is important that this misconception is corrected and the concern Black voices assigned its proper place. Otherwise an accurate assessment will continue to evade us of the relative merits and demerits of the neo-Fregean philosophy of mathematics compared to any other.

1.

Black’s thought experiment

Black asks us to imagine a tribe of arithmeticians whose basic notion of number is that of finite ordinal. They arrive at this notion by reflecting upon their practice of counting. The tribe counts a totally ordered collection of objects by linking its members one by one with some other objects (the ‘numbers’) taken in a privileged order. The last number so assigned is the 1 This paper first appeared in Analysis 62, [2002], pp. 125–135. Reprinted by kind permission of the editor and Blackwell Publishing. 2 All references to Wright, and Hale & Wright are to the reprints of their papers in Hale & Wright 2001.

95 Roy T. Cook (ed.), The Arché Papers on the Mathematics of Abstraction, 95–104. c 2007 Springer.

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ordinal number of the collection. Black also asks us to imagine that the tribe does not recognise the number zero. Let “W(R)” stand for the second-order statement that the relation R well-orders its field and let “R ∼ = S” stand for the statement that the orderings R and S are isomorphic. Then, according to Black (2000: 234), the notion of ordinal number that informs the tribe’s practice may be encapsulated by a principle (let’s call it the Tribe’s Principle) that specifies identity conditions for ordinal numbers: (TP)(∀ R)(∀ S)[(W(R) & W(S) & ∃ xRxx & ∃ xSxx) → (oR = oS ↔ R ∼ = S)] Continuing the fantasy, the tribe progress to the notion of cardinal number by establishing that all orderings of a given non-empty set have the same ordinal number. Their notion of cardinal number may then be captured by the restriction of (HP) to non-zero cardinals: (RHP)(∀ X)(∀ Y) [(∃yXy & ∃zYz) → (Nx : Xx = Nx : Yx ↔ X1 − 1 Y)] On this basis the tribe develops finitary arithmetic (or at least that portion of finite arithmetic required for counting the objects that the tribe actually encounters). But despite the considerable advances made by the tribe they fail to take Cantor’s leap and only envisage the application of (RHP) to finite totalities. Black’s thought experiment is designed to reveal that it is possible to have a “coherent understanding” of finite arithmetic and its applications which is not epistemologically founded on (HP) (2000: 235–36). For even though the tribe possesses such an understanding the principle for cardinal identity (RHP) they employ eschews—by contrast to (HP)—a commitment to zero. Moreover, (HP) is inconsistent with the principle of ordinal identity (TP) upon which the tribe’s understanding is ultimately founded. For whereas (HP) may be satisfied only in infinite domains, (TP) may be satisfied only in domains of finite size (inducing the Burali-Forti paradox otherwise) (Hodes 1984: fn 16). Since a coherent understanding of finite arithmetic can be achieved without recourse to (HP) Black concludes that (HP) cannot perform the foundational epistemological role the neo-Fregean proposes for it. There are a number of distinct issues here that require to be disentangled. First, Black’s assumption that (TP) is a mathematical principle—a principle of ordinal identity—may be questioned. For (TP) fails to exhibit an arguably constitutive feature of any genuinely mathematical principle, namely the feature of being conservative with respect to non-mathematical theories (cf. Field 1980: 9–16). Roughly, a mathematical principle should not allow the derivation of any non-mathematical conclusions from non-mathematical premises that could not have been drawn from those premises already. (TP) fails to be conservative (in this sense) because it cannot be satisfied in any infinite domain, even if the domain in question is composed solely of nonmathematical objects. Consequently (TP) allows us to draw a conclusion that we might not otherwise have drawn, the conclusion that there are only

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finitely many non-mathematical objects. 3 If it is indeed a characteristic mark of mathematical principles that they are conservative, it is correspondingly doubtful whether the understanding of the tribe that is based solely upon (TP) is genuinely mathematical in character. 4 Of course, Black may reject the general assumption that mathematical principles are (by constitution) conservative—although Black will then have to undertake the burden of explaining away the implausibility of supposing that a purely general mathematical proposition rules out the possibility of so many non-mathematical objects. Alternatively, he may reject the more specific assumption that (TP) need be labelled ‘mathematical’ in the first place— although he will then have to explain why this manoeuvre is anything more than ad hoc. But in either case it is open to the neo-Fregean to question whether (TP) is capable of serving as any sort of epistemological foundation, never mind an alternative one to (HP). For they may argue (to take one plausible example) that it is an open epistemic possibility that there are infinitely many spacetime points. Since the truth of (TP) is incompatible with any such infinitary hypothesis (therein lies (TP)’s failure to be conservative) it also an open epistemic possibility that (TP) is false. It then becomes a mystery how grasp of a principle that—without Cartesian excess—may legitimately be doubted could serve as a foundation for acquiring knowledge of arithmetical truths. These difficulties may be avoided by weakening (TP) so as to render it relevantly conservative. This result may be achieved by strengthening the antecedent of (TP) to characterise only concepts of finite extension: (WTP) (∀ R)(∀ S)[(W(R) & W(S) & ∃xRxx & ∃xSxx & Finite(S) & Finite(R)) → (oR = oS ↔ R ∼ = S)] It may (in any case) be argued that it is (WTP) rather than (TP) that underwrites the counting practice of the tribe in Black’s thought experiment. If, as Black suggests, the tribe is chary of the infinite then it is just as plausible to suppose that their practice is described by a version of (TP) restricted to the finite. But (WTP) may be satisfied even in infinite domains and so—by contrast to (TP)— appears not only conservative (in the relevant sense) but also consistent with (HP). 3 Less roughly, let T be any non-mathematical theory and, for any sentence A, let A* be the result of restricting the quantifiers in A to non-mathematical objects. Similarly, let T* be the result of restricting all the quantifiers in the theory T to non-mathematicals. Then a mathematical principle N is conservative iff for any sentence A, if T* + N implies A* then T implies A (cf. Wright 2000: 319). Now let S be the sentence stating that the non-mathematical universe is finite. Since (TP) implies S, (TP) + T* implies S even if T does not. Therefore, (TP) fails to be conservative. 4 By failing to be conservative (TP) also broaches a (plausible) constraint on abstraction principles that—like (HP)—are intended to introduce novel concepts (Wright 1997: 297). However, it is unclear from Black’s discussion whether this constraint should apply to (TP). For Black does not distinguish between the case where (TP) is laid down as a piece of conceptual innovation (in which case it may be obliged to be conservative) and the case where (TP) is intended as a codification of existing numerical practice (that may already fail to be conservative).

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The modal-re-constructive character of neo-Fregean epistemology

The fact remains, however, that (HP) incorporates a commitment to zero not even (WTP) incurs. And, as Black remarks, it is not only fantastical tribes whose arithmetical understanding may be zero free. The Greeks did not recognise zero nor did Dedekind or Peano in their original formulations of the axioms of arithmetic. But, according to Black, if (HP) is to perform the foundational role the neo-Fregean intends then “we must say that the reason there are infinitely many numbers is that 0 counts as one of them”. Since none of the aforementioned thinkers would have endorsed such a claim Black reasons that (HP) “can no longer be regarded as making explicit the ideas which already underlay our use of the natural number system” (Black 2000: 236). Black concludes that—contrary to neo-Fregean intention—(HP) cannot perform any foundational role in the epistemology of arithmetic. Black’s criticism is an instance of a general form of objection to the neoFregean programme that bemoans (HP) for its strength. According to objections of this form, (HP) cannot provide an analysis of the concept number because it incorporates existential commitments that no ordinary arithmetical reasoner needs to countenance. Usually such objections focus upon the commitment of (HP) to infinite numbers, numbers that prior to Cantor went almost entirely unnoticed (Heck 1997: 597–98). Black’s criticism reveals that (HP) might also be faulted as an analysis because it is committed to zero and— although Black does not mention the possibility—perhaps other finite numbers too (after all (HP) is also committed to 1, another number that has not always been recognised as such). However, objections of this form are in general misguided. 5 This is because the epistemological success of the neo-Fregean programme need not rely upon the effectiveness of (HP) as an analysis of ordinary arithmetical notions. Of course, neo-Fregeans do sometimes speak of (HP) as an “analysis” of the ordinary notion of number or “analytic of” that concept (see, for example, Wright 1983: 106–07 and Hale 1997: 99). Nevertheless, an alternative epistemology may be gleaned (and extrapolated) from what the neo-Fregeans have to say that makes no relevant play with the notion of analysis and obviates Black’s criticism. Black’s criticisms fail to take proper account of the modal character of this epistemology. Neo-Fregean epistemology (so envisaged) offers an account of how it is possible to acquire knowledge of the fundamental laws of arithmetic (by deriving them from (HP)). It thereby undertakes to describe “an a priori route” (that goes via the recognition of zero) to knowledge of the laws of arithmetic (Wright 1997: 279–80). But is not thereby committed, as Black 5 There are, in addition, particular reasons why objections to (HP) that complain about the commitment of (HP) to infinite numbers are also misguided. I discuss some of these reasons elsewhere (MacBride 2000).

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assumes, to saying this route is “the” only one available. It is consistent with our coming to recognise arithmetical truths in one way that we could have, and perhaps do, come to recognise their truth by different means. So the mere fact that certain figures—historical or imaginary—could have achieved a coherent understanding of arithmetic from which a recognition of zero was absent does nothing to compromise the neo-Fregean contention that it is possible for (HP) to serve as a basis for acquiring arithmetical knowledge. Black’s criticisms also fail to take proper account of the re-constructive character of the neo-Fregean epistemology. According to Black, (HP) can only discharge a foundational role if it makes explicit the principles that actually underlie established arithmetical usage. But, according to the neo-Fregean epistemology, (HP) can only discharge its intended role because, in the first instance, it does not answer to existing usage. Rather, the neo-Fregean claims, (HP)—properly understood—is nothing more than a stipulation that serves to introduce a novel operator (“Nx”) into our language (Wright 1997: 278, Wright 2000: 317–18, Hale & Wright 2000: 142, Hale & Wright 2001: 14). The introduction is achieved by implicit definition: the meaning of the novel operator is fixed by stipulating that the truth conditions of identity statements (“Nx : Fx = Nx : Gx”) in which it occurs coincide with the conditions under which an equivalence relation amongst concepts may be said to obtain (“F 1 − 1 G”). And it is because (HP) is intended merely as a stipulation that the neoFregean feels able to legitimately claim that (HP) is a priori. Nevertheless, the neo-Fregean continues, (HP) provides a basis for grasping arithmetical truths a priori because (as Frege’s theorem demonstrates) the system that results from (HP) and second-order logic allows for a reconstruction of ordinary arithmetical practice in the following sense. It—Frege arithmetic—suffices for the interpretation of the laws of ordinary arithmetic and the proof of their interpretations. It is in virtue of the interpretative powers of the system (HP) engenders that the neo-Fregean takes himself to be retrospectively entitled to characterise (HP) an arithmetical principle, a principle of cardinal (in the usual sense) identity. To simply complain that (HP) fails to “make explicit the ideas which already underlay our use of the natural number system” (Black 2000: 236) is to fail to take into account the re-constructive character of the epistemology proposed and the crucial role Frege’s theorem performs in the envisaged reconstruction.

3.

The counter-Caesar problem

Whilst this characterisation of the neo-Fregean programme obviates the criticisms of Black (and others) it also brings into focus a potentially critical epistemological difficulty quite peculiar to that programme. For—strictly speaking—Frege’s theorem does not establish that the truths of ordinary arithmetic are themselves a priori. It only establishes that there is a system of truths (Frege arithmetic) capable of modelling (interpreting) the laws of arithmetic.

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Assume the best case scenario for the neo-Fregean and suppose the truths that comprise Frege arithmetic are a priori in character. An additional argument is still required to show that the truths of arithmetic inherit the epistemological status of their Fregean models (see Benacerraf 1981: 20, Heck 1997: 596). There appear to be at least two styles of strategy—re-constructive and hermeneutic—whereby the neo-Fregean might endeavour to address this issue. According to the re-constructive strategy, the neo-Fregean may accept that the truths expressed in Frege arithmetic concern an entirely novel subject matter and merely model the truths of ordinary arithmetic. Nevertheless, the neo-Fregean may still maintain that Frege’s theorem bears epistemological significance for ordinary arithmetic. For, the neo-Fregean may argue, the mappings that the theorem establishes a priori between the a priori truths of Frege arithmetic and ordinary arithmetic suffice to demonstrate the operational effectiveness (reliability), if not the truth, of ordinary arithmetical claims. 6 Alternatively, the neo-Fregean may adopt the hermeneutic strategy according to which all the truths of the ordinary arithmetic are expressed by truths of Frege arithmetic. Ordinary arithmetic will then automatically inherit the a priori status of the latter system. The difficulty attendant upon this second strategy is, in a sense, the reverse of the more familiar Caesar problem (or at least one member of that family of problems). The latter difficulty concerns our capacity (or lack of it) to establish that the terms and sentences occurring in two different theories (concerning, for example, persons and numbers respectively) are, as we might pre-theoretically suppose, terms for quite different entities (Julius Caesar, 2) and sentences expressive of very different truths. The former difficulty may appropriately be dubbed the ‘counter-Caesar’ problem. It concerns our capacity to establish that the terms and sentences figuring in two different theories (ordinary arithmetic, Frege arithmetic) are, as the neo-Fregean would have it, terms for the same entities (2, Nz:[z = Nx: x =x v z = Ny:(y = Nx: x = x)]) and sentences expressive of the same truths. In fact, the neo-Fregean appears to adopt the second strategy and is therefore obliged—if he is to realise the epistemological pretensions of his programme—to take on the counter-Caesar problem with all the seriousness usually reserved for the Caesar problem itself. The neo-Fregean proceeds upon the assumption “that to define the distinctively arithmetical concepts is to so define a range of expressions that the use thereby laid down for those expressions is indistinguishable from that of expressions which do indeed express those concepts” (Wright 2000: 322). He then claims that the stipulation of (HP) gives rise to a pattern of linguistic use that is ensured by the interpretability of Peano’s axioms in Frege arithmetic to be equivalent to ordinary arithmetic. 6 The neo-Fregean may also add that the existence of these mappings is in no way compromised by the commitment of Frege arithmetic to zero (and other numbers ordinary arithmetic fails to recognise). So, the neo-Fregean may conclude, a commitment to zero can hardly provide a reason for denying (HP) the status of an epistemological foundation.

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So, relying on the aforementioned assumption, the neo-Fregean concludes that regardless of the different underlying principles ((HP), Peano’s axioms or some other source) that gave rise to that same pattern of use the very same arithmetical truths are thereby expressed. 7 An analogy may help bring into relief the epistemological character now envisaged for the neo-Fregean project. According to Davidson, we may achieve insight into the nature of language by reflecting upon the possibility of a radical interpreter who (entirely ignorant of a given language L) constructs a theory knowledge of which suffices for interpreting a speaker of L (Davidson 1973). The radical interpreter does not, however, make any attempt to describe the inner psychological mechanisms that in fact account for the speaker’s mastery of L. Nonetheless, Davidson claims, the theory supplied by a radical interpreter provides insight into the character of the complex linguistic abilities displayed by ordinary speakers of L. This is because (very roughly) the theory is empirically adequate to their linguistic performances. In an analogous way, the neo-Fregean claims, we may achieve insight into the nature of arithmetical knowledge. This time we are asked to reflect upon the possibility of a character (call him “Hero”) who (entirely ignorant of arithmetic) seeks to construct a theory knowledge of which suffices for the competent use of ordinary arithmetical language. 8 Hero makes no efforts to uncover the psycho-genetic origins of our arithmetical abilities. Nonetheless, the neo-Fregean claims, the a priori theory provided by Hero provides insight into the character of arithmetical knowledge. This is because grasp of his theory provides Hero with the ability to engage in a practice of use equivalent to the arithmetical performance displayed by ordinary speakers. The a priori theory Hero supplies is Frege arithmetic. The assurance that knowledge of this theory engenders competence in arithmetic flows (very roughly) from an appreciation of Frege’s theorem.

4.

Meaning-theoretic foundations of neo-Fregeanism

How then should the neo-Fregean programme be assessed once it is liberated from the popular misconception that it was ever intended to characterise whatever principles in fact underlie ordinary arithmetical reasoning? If it is to carry conviction the neo-Fregean assumption—that ‘arithmetical’ systems which exhibit the same pattern of use refer to the same objects and express 7 More generally, the neo-Fregean must establish that the stipulation of (HP) provides for two distinct patterns of use that are respectively equivalent to the distinct uses of pure and applied arithmetical language. The neo-Fregean takes the former result to be established by demonstrating the intepretability of Peano’s axioms within Frege arithmetic. The latter result is secured by deriving from (HP) the principle (Nq) that relates (in the intuitive manner) pure occurrences of the numerals “nf ” of Frege arithmetic with appropriate applied occurrences of the numerals “n” of ordinary arithmetic (Wright 2000: 322, 330–32): (Nq) nf = Nx : Fx ↔ there are exactly n Fs. 8 The character Hero was introduced by Wright (1997: 247) for the heuristic purpose of showing how Frege’s definitions of zero and its successors might be grasped upon the basis of (HP) and second-order logic. Here I extend Hero’s role to show how ordinary numerals might similarly be grasped.

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the same truths about them-must be underwritten by a general and principled conception of how meaning and use relate. That conception must include a meaning-theoretic doctrine of (at least) the following strength: (MSU) Meaning (truth and reference) supervenes on use.

But by relying upon (MSU) the neo-Fregean undertakes a distinctive theoretical commitment that it is far from trivial. First, if the neo-Fregean employs (MSU) then he will confront a substantial explanatory challenge. On the one hand, the notion of ‘use’ figuring in (MSU) cannot be so behaviouristically conceived that it becomes implausible to suppose intentional facts could ever supervene on facts about use (a difficulty familiar to us from Kripke’s rule-following paradox). On the other hand, the relevant notion of ‘use’ cannot be so intensionally conceived that it becomes impossible for ordinary arithmetic and Frege arithmetic to exhibit the same pattern of use. To vindicate his stance it is incumbent upon the neoFregean to supply an account of use that navigates between these undesirable consequences. Second, a version of Black’s original concern resurfaces—this time in its proper place. For even if such a notion of ‘use’ can be supplied, the neoFregean faces the further difficulty that ordinary arithmetic and Frege arithmetic exhibit different patterns of use (in whatever sense of use that might turn out to be). This is because Frege arithmetic is a far richer language than ordinary arithmetic. It includes expressions putatively referring to numerical objects for which there may be no corresponding ordinary arithmetical terms: for example, the number of identical things, the number of natural numbers, the number of non-self-identical things, and so on. To accommodate this point, the neo-Fregean must claim that the meaning of a given expression does not supervene upon the global pattern of use associated with it, but rather upon some relevant local holism. More specifically, the neo-Fregean must claim that the meanings of expressions in Frege arithmetic that do correspond to terms in ordinary arithmetic are determined by a local pattern of use from which the more colourful, unfamiliar expressions of Frege arithmetic are excluded. The neo-Fregean will then seek to harness Frege’s theorem to show that ordinary arithmetical terms exhibit the very same local pattern of use, and so mean the same, as their Fregean counterparts. But to make good such claims the neo-Fregean is obliged to undertake a further explanatory task: to explain the significance of the notion ‘relevant local holism’ and to provide some principled account of why meaning should be taken to supervene on that sort of use. Part of this task will be to systematically specify—in the face of widespread Quinean scepticism that it cannot be done (see Quine 1976 and, more recenly, Fodor & LePore 1992: 163–83)—those aspects of the use of an expression that confer meaning (and belong to the relevant local holism) from those aspects that do not confer meaning (and

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belong only to a wider pattern of use). There is, however, reason to be sanguine about the neo-Fregean’s prospects of executing this task. After all, if some such contrast could not (at some level) be made out then the possibility of translating from a richer language into a more impoverished one, or acquiring greater knowledge of a language that we have already partially learnt, would appear to be foreclosed. Moreover, if it turns out that the explanatory task cannot be discharged it is always open to the neo-Fregean to avoid the counterCaesar problem by adopting the first strategy proposed and seeking only to model ordinary arithmetical usage. Black concludes with the suggestion that his real objection to neoFregeanism is not so much to do with the commitment of (HP) to zero (or the existence of some other object). He writes: “rather the problem is the way in which (HP) generates an infinity of numbers, generating new numbers to count the numbers already there with a tail biting circularity” (2000: 237). Black here raises an objection to the impredicative character of (HP), a form of objection familiar from Dummett’s critique of the neo-Fregean programme (Dummett 1991: 226–29). But this sort of objection must surely be distinguished from Black’s earlier concern that (HP) fails to make explicit the ideas that underlie ordinary arithmetic. For even supposing a 21st century Boolos were to uncover a predicative version of (HP) with sufficient strength to generate an ‘arithmetical’ system, the principle in question might still fail to underlie ordinary practice. Moreover, even if (HP) did underlie ordinary practice this would in no way address the objections to (HP) based upon its impredicative character. Finally, it is worth reflecting that impredicativity (specified in such general terms) appears to be a ubiquitous phenomenon. In the linguistic environment into which we are thrown the meaning of a given word can never—or so it seems—be determined independently of some significant portion of the sentential contexts in which that word occurs. It may well be that the terms introduced by (HP) are impredicative in a stronger and more objectionable sense than this. But until we have equipped ourselves with a more discerning means of saying just why this is so we—Black and Dummett included—should be more reticent to dismiss (HP) on such generic grounds. 9

References Benacerraf, P. 1981. Frege: The Last Logicist. In Midwest studies in philosophy VI, eds. P. French et al., 17–35. Minneapolis: University of Minnesota Press. Black, R. 2000. Nothing matters too much, or Wright is wrong. Analysis, 60: 229–37. Boolos, G. 1997. Is Hume’s Principle Analytic? In Heck: 1997a: 245–62. Davidson, D. 1973. Radical Interpretation. Dialectica 27: 313–28. 9 The neo-Fregean programme remains, however, beset by many other challenges, not least that of justifying the metaphysical assumption it presupposes of an intimate communion between language and reality. I explore these issues further in MacBride [2003]. For comments and discussion of the present paper thanks to Robert Black, Peter Clark, Roy Cook, Bill Demopoulos, Philip Ebert, Patrick Greenough, Jonathan Hesk, Stephanie Schlitt, Stewart Shapiro, Crispin Wright, and an anonymous referee for this journal.

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Dummett, M. 1991. Frege Philosophy of Mathematics. London: Gerald Duckworth & Co. Ltd. Field, H. 1980. Science without Numbers. Oxford: Basil Blackwell. Fodor, J. & LePore, E. 1992. Holism: A Shopper’s Guide. Oxford: Basil Blackwell. Hale, B. 1997. Grundlagen §64. Proceedings of the Aristotelian Society 97: 243–61. Reprinted in Hale & Wright 2001: 90–116. Hale, B. & Wright C., 2000. Implicit Definition and the a priori. In New Essays on the A Priori, ed. P. Boghossian and C. Peacocke, 286–319. Oxford: Clarendon Press. Reprinted in Hale & Wright 2001: 117–50. Hale, B. & Wright, C. 2001. The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics. Oxford: Clarendon Press. Heck, R. 1997. Finitude and Hume’s Principle. Journal of Philosophical Logic, 26: 598–617. Heck, R. 1997a. Language, Thought and Logic: Essays in Honour of Michael Dummett. Oxford: Oxford University Press. Hodes, H. 1984. Logicism and the Ontological Commitments of Arithmetic. Journal of Philosophy 81: 123–49. Lowe, E.J. 1998. The Possibility of Metaphysics: Substance, Identity and Time. Oxford: Clarendon Press. MacBride, F. 2000. On Finite Hume. Philosophia Mathematica, 8: 150–59. MacBride, F. 2003. Speaking with Shadows: A Study of Neo-Logicism. British Journal for the Philosophy of Science, 54: 103–63. Quine, W.V.O. 1976. Carnap and Logical Truth. In his Ways of Paradox and other essays, 107–32. Cambridge, Mass.: Harvard University Press. Wright, C. 1983. Frege’s Conception of Numbers as Objects. Aberdeen: Aberdeen University Press. Wright, C. 1997. On the Philosophical Significance of Frege’s Theorem. In Heck 1997a: 201–44. Reprinted in Hale & Wright 2001: 272–306. Wright, C. 1998. On the Harmless Impredicativity of N= (‘Hume’s Principle’). In M. Schirn (ed.), The Philosophy of Mathematics Today, ed. M. Schirn, 339–68. Oxford: Clarendon Press. Reprinted in Hale & Wright 2001: 229–55. Wright, C. 2000. Is Hume’s Principle Analytic? Notre Dame Journal of Formal Logic 40. Reprinted in Hale & Wright 2001: 307–22.

ON THE PHILOSOPHICAL INTEREST OF FREGE ARITHMETIC 1 William Demopoulos

1.

Fregean logicism: the laws of logic have an arithmetical content

Traditional “Fregean” logicism held that arithmetic could be shown free of any dependence on Kantian intuition if its basic laws were shown to follow from logic together with explicit definitions. It would then follow that our knowledge of arithmetic is knowledge of the same character as our knowledge of logic, since an extension of a theory (in this case the “theory” of secondorder logic) by mere definitions cannot have a different epistemic status from the theory of which it is an extension. If the original theory consists of analytic truths, so also must the extension; if our knowledge of the truths of the original theory is for this reason a priori, so also must be our knowledge of the truths of its definitional extension. The uncontroversial point for traditional formulations of the doctrine is that a reduction of this kind secures the sameness of the epistemic character of arithmetic and logic, while allowing for some flexibility as to the nature of that epistemic character. Thus, it is worth remembering that in Principles (p. 457), Russell concluded that a reduction of mathematics to logic would show, contrary to Kant, that logic is just as synthetic as mathematics. Nevertheless, the methodology underlying this approach to securing the aprioricity of arithmetic by a traditional logicist reduction has been challenged. For example, Paul Benacerraf, 2 who focuses on Hempel’s 3 classic exposition, tells us that 1 This paper first appeared in Philosophical Books 44, [2003], pp. 220–228. It is reprinted by kind permission of the editor and Blackwell Publishing. With the exception of section headings and a small number of minor stylistic changes, it is unaltered. A Postscript addresses Hale and Wright’s response to the original paper. 2 “Frege: The last logicist,” in William Demopoulos (ed.) Frege’s philosophy of mathematics (Harvard University Press: 1995), pp. 42 and 46. 3 C. G. Hempel, “On the nature of mathematical truth,” Hilary Putnam and Paul Benacerraf (eds.) The philosophy of mathematics: selected readings, second ed. (Cambridge University Press: 1983), 377–393.

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The Arché Papers on the Mathematics of Abstraction . . . logicism was . . . heralded by Carnap, Hempel . . . and others as the answer to Kant’s doctrine that the propositions of arithmetic were synthetic a priori . . . in reply to Kant, logicists claimed that these propositions are a priori because they are analytic—because they are true (or false) merely “in virtue of” the meanings of the terms in which they are cast. . . . According to Hempel, the Frege-Russell definitions . . . have shown the propositions of arithmetic to be analytic because they follow by stipulative definitions from logical principles. What Hempel has in mind here is clearly that in a constructed formal system of logic (set theory or second-order logic plus an axiom of infinity), one may introduce by stipulative definition the expressions ‘Number,’ ‘Zero,’ ‘Successor’ in such a way that sentences of such a formal system using these introduced abbreviations and which are formally the same as (i.e., spelled the same way as) certain sentences of arithmetic—e.g., ‘Zero is a Number’—appear as theorems of the system. He concludes . . . that these definitions show the theorems of arithmetic to be mere notational extensions of theorems of logic, and thus analytic. He is not entitled to that conclusion. Nor would he be even if the theorems of logic in their primitive notations were themselves analytic. For the only things that have been shown to follow from the theorems of logic by [stipulative definitions] are the abbreviated theorems of the logistic system. To parlay that into an argument about the propositions of arithmetic, one needs an argument that the sentences of arithmetic, in their preanalytic senses, mean the same (or approximately the same) as their homonyms in the logistic system. That requires a separate and longer argument.

Benacerraf is questioning whether the logicist can claim to have established any truth of arithmetic on the basis of a successful reduction. What is required according to Benacerraf, is a supplementary argument showing that the logicist theorems have the preanalytic meanings of their ordinary arithmetical analogues. But Benacerraf’s demand for a further argument is not justified. The philosophical interest elicited by traditional logicism derived from the fact that it was thought implausible that the concepts and laws of logic could have an “arithmetical content.” To have successfully dispelled this belief it would have been sufficient to have shown that the concepts of logic allow for the explicit definition of notions which, on the basis of logical laws alone, demonstrably satisfy the basic laws of arithmetic. The philosophical impact of the discovery that the concepts and laws of logic have an arithmetical content in this sense would not have been in any way diminished by the observation that the preanalytic meanings of the primitives of arithmetic were not the same as their logicist reconstructions. The sense in which the logicist thesis must be understood in order to be judged successful cannot therefore be the one for which Benacerraf claims Hempel must argue. Notice also that independently of one’s view of meaning and truth in virtue of meaning, it must be conceded that traditional logicism would have provided a viable answer to Kant had it succeeded in showing that arithmetical knowledge requires only an extension of logic by explicit definitions. Hempel’s appeal to these notions addresses a different issue: Frege left the

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problem of securing the epistemic basis of the laws of logic largely untouched. Benacerraf’s Hempel should be understood as proposing to fill this gap by suggesting that the laws of logic are true in virtue of the meanings of the logical constants they contain. Like Frege, Hempel seeks to secure the aprioricity of arithmetic by an argument that proceeds from its analyticity. But Hempel’s version of logicism differs from Frege’s, for whom “analytic” merely meant belonging to logic or a definitional extension of logic, by providing a justification for the analyticity of logical laws: logical laws are analytic, not by fiat as on Frege’s account, but because they are true in virtue of the meanings of the logical terms they contain. From this it would follow that if logical laws are true in virtue of meaning, so also is any proposition established solely on their basis, where “established solely on their basis” is intended to encompass the use of explicit definitions. The clarity of the thesis that the laws of logic are true in virtue of meaning is therefore central to Hempel’s presentation of the view. Also central is the substantive and further claim that the basic laws of arithmetic can be recovered within a definitional extension of logic. The implied criticism of Hempel’s appeal to truth in virtue of meaning gains its force from the difficulties that stand in the way of establishing the traditional logicist thesis that arithmetic is reducible to logic in the original sense of the doctrine. Certainly, the failure to sustain this thesis led to more ambitious applications of the notion of truth in virtue of meaning. But if the basic laws of arithmetic had been recovered as a part of logic—not merely shown to have analogues that are part of some formal system or other, but to be part of logic—what more would be needed to infer that they share the epistemic status of logical laws? Once Hempel is not represented as seeking to secure the truth of the basic laws of arithmetic by an appeal to the derivability of mere formal analogues or a blanket appeal to the notion of truth in virtue of meaning, it is clear that he simply doesn’t owe us the argument Benacerraf claims he does. The difficulties that attend traditional logicism are therefore not the methodological difficulties Benacerraf advances, but the simple failure to achieve the stated aim of showing arithmetic to be a definitional extension of logic. This point is obscured by Benacerraf’s suggestion that the reduction might proceed from second-order logic with an axiom of infinity or from some version of set theory. Neither theory supports the truth in virtue of meaning account that underlies Hempel’s formulation of logicism. A reduction to second-order logic with infinity would mean a reduction to a system augmented with an axiom like Whitehead and Russell’s; but no one ever thought such a system was true in virtue of meaning. As for a reduction to set theory, set theory is properly regarded as the arithmetic of the transfinite. Why should a reduction of the natural numbers to such a generalized arithmetic be regarded as a means of establishing its aprioricity on a less synthetic footing? The only coherent logicist methodology would therefore seem to be the one just outlined: to reduce arithmetic to a theory like Begriffsshrift’s. Unfortunately, such a theory is either too weak, or in the presence of Frege’s theory of classes, inconsistent.

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Wright and Hale’s neo-Fregean alternative

The renewed interest in logicism is based on the fact that the secondorder theory having Hume’s principle 4 as its only non-logical axiom—“Frege arithmetic”—has a definitional extension which contains the Dedekind–Peano axioms. The neo-Fregean program of Crispin Wright and Bob Hale seeks to imbed this logical discovery into a philosophically interesting account of our knowledge of arithmetic by subsuming Hume’s principle under a general method for introducing a concept by an “abstraction principle.” 5 This program explains the epistemological interest of the discovery that arithmetic is a part, not of second-order logic, but of Frege arithmetic, by the program’s account of concept introduction. The key to achieving this goal is the idea that abstraction principles have a distinguished status: they are a special kind of stipulation. Their stipulative character shows them to be importantly like explicit definitions even if their creativeness suggests an affinity with axioms; and it is a central tenet of neo-Fregean logicism that abstraction principles are sufficiently like definitions to yield an elegant explanation of why arithmetical knowledge is knowledge a priori. The neo-Fregean program has a methodological dimension that parallels the role of the theory of definition in traditional logicism. Frege accords a statement the status of a proper definition if it meets conditions of eliminability and conservativeness. The classical theory of definition is supplemented by the neo-Fregean methodology of good abstractions. Thus, the theory of definition mandates that a definitional extension must be conservative in the familiar sense of not allowing the proof of sentences formulated in the unextended vocabulary which are not already provable without the addition of the definitions which comprise the extension. But “extensions by abstraction” need not be conservative in this sense; indeed interesting extensions are interesting precisely because they are not conservative in the sense of the theory of definition. The neo-Fregean theory of good abstractions allows for classically non-conservative extensions—extensions which properly extend the class of provable sentences—while imposing a constraint on the consequences an extension by good abstraction principles can have for the ontology of the theory to which they are added. This methodology is constrained and principled, it is just not constrained in the same way as the classical theory of definition. We can, perhaps, put the difference by saying that the constraints on definition have a more purely epistemic motivation than do the constraints the neo-Fregean imposes on good abstractions. 4 Hume’s principle tells us that for any concepts F and G, the number of Fs is identical with the number of Gs if, and only if, the Fs and the Gs are in one–one correspondence. 5 By an abstraction principle, Hale and Wright mean the universal closure of an expression of the form (X) = (Y) ↔ X ᑬ Y, where ᑬ is an equivalence relation, the variables X and Y may be of any type, and the function may be of mixed type. In the case of Hume’s principle, the equivalence relation is the (second-order definable) relation on concepts of one–one correspondence, and the “cardinality function,” is a type-lowering map from Fregean concepts to objects.

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In my view, the reticence of the classical theory of definition to allow a mere definition to properly extend the theory to which it is added is wellfounded, and should also inform the epistemic basis of a principle as rich as Hume’s. My goal here is to consider whether the neo-Fregean account of Hume’s principle as a kind of stipulation can support the epistemological claim of neo-Fregeanism to have secured the aprioricity, if not the analyticity, in one or another traditional sense of the notion, of our arithmetical knowledge. The matter is taken up by Hale and Wright in their paper “Implicit definition and a priori knowledge”—and by Wright in his “Is Hume’s principle analytic?” which, notwithstanding its title, is not concerned to secure the analyticity of Hume’s principle but to address the question of its epistemic status within the neo-Fregean program and the light it sheds on our arithmetical knowledge. 6 Wright and Hale use the stipulative character of Hume’s principle as a premise in an argument for the aprioricity of our arithmetical knowledge. This becomes clear when we reflect on the fact that they are concerned to show that our knowledge of arithmetic can be represented as resting on a principle that introduces the concept of number. In acquiring the concept of number, we acquire a criterion of identity for number—a criterion for saying when the same number has been given to us in two different ways as the number of one or another concept. This criterion of identity—Hume’s principle—affords the only non-logical premise needed to derive the basic laws of arithmetic. Our arithmetical knowledge is secured, therefore, with our grasp of the concept of number and is based on nothing more than what we acquire when we are introduced to the concept. But since this knowledge rests on a stipulation, it is unproblematically knowledge a priori.

3.

Recovering the epistemic status of ordinary arithmetic by modeling

This is essentially the same account of the philosophical interest of Frege arithmetic that is elaborated by Fraser MacBride in two thoughtful papers 7 that address this issue. For MacBride the neo-Fregean explanation of the aprioricity of our knowledge of arithmetic runs as follows: We first stipulate a criterion of identity for a special kind of objects; call them cardinal numbers. That certain fundamental truths about these objects are established on the basis of a stipulation guarantees that our knowledge of those truths is knowledge a priori. This is to be contrasted with an account which would seek to infer the aprioricity of our knowledge of arithmetic from theses about meaning or truth in virtue of meaning. The neo-Fregean account does not depend on a traditional notion of analyticity: since neo-Fregeanism demands only 6 Both reprinted in their collection of their papers, The reason’s proper study (Oxford University Press: 2002), as chapters 5 and 13, respectively. 7 “Finite Hume,” Philosophia mathematica 8 (2000) 150–159, and “Can nothing matter?,” Analysis 62 (2002) 125–134.

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the relatively uncontentious concession of the aprioricity of a stipulation, it can claim that its explanation of the aprioricity of arithmetic need not address the difficulties associated with defending traditional conceptions of analyticity. The fact that the reduction to Frege arithmetic requires more than a merely definitional extension of second-order logic suggests an objection to neoFregean logicism that is closely related to the one we saw Benacerraf urge against to Hempel: How, one might ask, does our knowledge of the truths that hold of the objects the neo-Fregean has singled out—the Frege-numbers— bear on our knowledge of the numbers, on the subject matter of ordinary arithmetic? In so far as the epistemological issues are issues concerning ordinary arithmetic, have they even been addressed by the neo-Fregean? In this form, the objection presupposes only preservation of subject matter—a minimal requirement that it would be difficult to justify not meeting—and says nothing about preservation of meaning. The first of two neo-Fregean responses to this objection that I wish to review holds that it is because ordinary arithmetic can be “modeled” in Frege arithmetic that the epistemological status of the truths of Frege arithmetic is shared by the truths of ordinary arithmetic. Wright remarks (p. 322) that this answer is too weak. And although MacBride does not endorse this response, neither does he reject it as altogether unlikely. Nevertheless, I think it is worth recording exactly why such a straightforward answer, couched in terms of the relatively unproblematic relation of modeling, can’t be right. It is clearly possible to stipulate the conditions that must obtain for the properties of a purely hypothetical and imaginary “abstract” physical system to hold without in any way committing ourselves to the existence—or even the dynamical possibility—of such a system. Our knowledge that such abstract systems are configured in accordance with our stipulations is no less a priori than our knowledge that, for example, the four element Boolean algebra has a free set of generators of cardinality one. But it sometimes happens that abstract configurations “model” actual configurations, in the sense that there is a correspondence between the elements of the two systems that preserves fundamental properties. It is clear that in such circumstances we take ourselves to know more than that an imaginary example has the properties we stipulate it to have: if the example is properly constructed, we know the dynamical behavior of a part of the physical world. But of course the fact that an actual system is “modeled” by our imaginary system, together with the fact that our knowledge of the properties of our imaginary system is a priori knowledge because it depends only on our free stipulations, are completely compatible with the claim, obvious to preanalytic intuition, that our knowledge of the dynamical behavior of the actual system is a posteriori. Whatever role stipulation may have in fixing the properties of the abstract system by which the behavior of some real process is modeled, it lends no support to the idea—and

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would never be regarded as lending support to the idea—that our knowledge of the real process is knowledge a priori. There is a disanalogy between the number-theoretic case and our example that might seem to undermine its effectiveness as a criticism. In the numbertheoretic case the existence of the correspondence between ordinary numbers and the “Frege-numbers” that model them is known a priori. But the correspondence between the abstract system of our example and the actual system is not known a priori; it depends on the a posteriori knowledge that there are in reality configurations of particles having the postulated characteristics. This is of course entirely correct. However it is of no use to the neo-Fregean, since to know a priori that there is a mapping between the ordinary numbers and the Frege-numbers it is necessary to have a priori knowledge of the existence of the domain and co-domain of the mapping. To be of any use to the neoFregean, the fact that the Frege-numbers model the ordinary numbers therefore requires that our knowledge of the ordinary numbers be a priori. But if the modeling of the ordinary numbers by the Frege-numbers presupposes that our knowledge of the ordinary numbers is a priori, it cannot be part of a noncircular account of why ordinary arithmetic is known a priori. The general point may be put as follows: The fact that M models N , so that for any sentence s, s is true in M if and only if s is true in N , does not entitle us to infer that because the sentences true in M are known a priori, the sentences true in N are known a priori. Indeed it is perfectly possible that (with the obvious exception of the logical truths) our knowledge of sentences true in N is wholly a posteriori. So even if we grant that an assumption rich enough to secure an infinity of objects is correctly represented as a stipulation, it remains unclear how the neo-Fregean can use this fact to answer the question which motivates his account of arithmetical knowledge—it remains unclear how it yields an account of our knowledge of the numbers, knowledge that we have independently of the neo-Fregean analysis. Notice that this objection depends only on an observation about the modeling of one domain by another, and that, in particular, it does not require the resolution of various difficult issues in the theory of meaning.

4.

Recovering the epistemic status of ordinary arithmetic by preserving a pattern of use

There is an alternative to the response based on modeling. The idea is that since Frege arithmetic captures the “patterns of use” exhibited by our ordinary number-theoretic vocabulary, both in pure cases and in applications, we are justified in inferring not merely that the Frege-numbers model the ordinary numbers but that the Frege-numbers are the ordinary numbers. Showing that Frege arithmetic captures the patterns of use of our ordinary number-theoretic vocabulary constitutes a considerable strengthening of the claim that ordinary arithmetic is merely modeled by Frege arithmetic.

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Suppose we grant, both that Frege arithmetic captures the patterns of use of our ordinary number-theoretic vocabulary and that because of this, ordinary arithmetic and Frege arithmetic share the same subject matter. Vindicating the claim that ordinary arithmetic and Frege arithmetic share a subject matter is only one of the difficulties that the weaker understanding of the view in terms of modeling fails to address. If we are only modeling ordinary arithmetic, it is unproblematic to hold that as a statement of the modeling theory Hume’s principle is known a priori because it is a mere stipulation. The difficulty, as we saw, is that this fails to transfer to the aprioricity of the truths we are modeling—to the truths of ordinary arithmetic. Is this difficulty removed when the neo-Fregean account is extended to one which claims to capture the patterns of use of our ordinary number-theoretic vocabulary? And does it illuminate the epistemic status of the basic laws of arithmetic to observe that, in the neo-Fregean reconstruction of our patterns of use, Hume’s principle has the status of a stipulation? When neo-Fregeanism is understood to preserve our patterns of use, it becomes virtually indistinguishable from the traditional idea that the account of the numbers given by Frege arithmetic is analytic of the ordinary notion of number, so that a major burden of the account now falls on establishing the adequacy of an analysis in something very much like the traditional sense. This is a task the neo-Fregean had sought to avoid, since once the neoFregean has to defend the idea that the patterns of use of ordinary numerical expressions have been captured, the simplicity of urging the stipulational character of Hume’s principle, and then basing the aprioricity of arithmetic on this footing, has been lost: the principle no longer governs the introduction of a new concept but is constrained to capture an existing one. But let us grant both that sameness of pattern of use implies sameness of reference and that Frege arithmetic does in fact capture the pattern of use of our ordinary arithmetical vocabulary and, therefore, articulates a successful reconstruction of our arithmetical knowledge. Since we have given up the idea that Hume’s principle is being used simply to introduce a new concept, but forms part of an attempt to articulate principles that capture our numerical concepts as they are given by the patterns of use of our ordinary number-theoretic vocabulary, its justification does not consist merely in its being laid down as a stipulation. Rather, Frege arithmetic is justified because it captures the fundamental features of the judgements—pure and applied—that we make about the numbers. For the neo-Fregean, the reconstruction must not only capture an existing concept by recovering the patterns of use to which our arithmetical vocabulary conforms, it must also illuminate the epistemic status of our pure arithmetic knowledge. Having the status of a stipulation is not, of course, a characteristic of Hume’s principle that is recoverable from our use of our arithmetical vocabulary, but is something the reconstruction imposes on the principle in order to illuminate the basis for our knowledge of the propositions derivable

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from it. But it is unclear what is achieved if one has captured the pattern of use of an expression by a principle that—in the reconstruction of the knowledge claims in which that expression figures—is regarded as a stipulation. Does this confer the epistemological characteristics that the notion of a stipulation is supposed to enjoy on the knowledge claims that have been reconstructed? To establish that the epistemic basis for the knowledge these judgements express resides in the stipulative character the neo-Fregean analysis assigns to Hume’s principle, it is not enough to show that Frege arithmetic captures patterns of use. The essential point is not all that different from what we have already noted when discussing the response based on modeling, and it can be seen by an example that is not all that different from the one cited in that connection. Suppose the world were Newtonian. We could then give a reconstruction of our knowledge of the mechanical behavior of bodies by laying down Newton’s laws as stipulations governing our use of the concepts of force, mass and motion. 8 But the fact that in our reconstruction the Newtonian laws have the status of stipulations would never be taken to show that they are in any interesting sense examples of a priori knowledge. Why then should the fact that the neo-Fregean represents Hume’s principle as a stipulation be taken to show that arithmetic is known a priori? The neo-Fregean reconstruction of the patterns of use of expressions of arithmetic leaves the epistemic status of the basic laws of arithmetic as unsettled as it was on the suggestion that Frege arithmetic merely models ordinary arithmetic. Neither reconstruction supports the epistemological claim of the neo-Fregean to have accounted for the aprioricity of our knowledge of arithmetic. Whether that account is put forward as a theory within which ordinary arithmetic can be modeled, or whether it is said to capture the patterns of use of our number-theoretic vocabulary, it fails to have the direct bearing on the epistemic basis of our arithmetical knowledge that the neo-Fregean supposes it to have. Showing that Hume’s principle is correctly represented as a stipulation may be one route to securing it as a truth known a priori, but it is questionable whether, proceeding in this way, the task of revealing the proper basis for the aprioricity of arithmetic is made any easier than it would be by general reflection on why its basic laws are plausibly represented as known truths.

5.

Frege arithmetic and the analysis of number

Putting to one side the problem of establishing the aprioricity of arithmetic on a correct basis, a compelling argument that Frege arithmetic captures preanalytic intuitions about the numbers can be extracted from the neo-Fregean 8 There are of course well-known historical examples along these lines. Cp. Ernst Mach’s The science of mechanics (Open Court: 1960, sixth American edition, translated by Thomas J. McCormack), whose famous definition of mass (p. 266) even has the form of an abstraction principle. Thanks to Peter Clark for calling my attention to Mach’s rational reconstruction of Newtonian mechanics.

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corpus: Since the Dedekind–Peano axioms codify our pure arithmetical knowledge, their derivability constitutes a condition of adequacy which any account of our knowledge of number should fulfill. By Frege’s theorem, Frege arithmetic satisfies this condition of adequacy. But what makes Frege arithmetic an interesting analysis of the concept of number is that it not only yields the Dedekind–Peano axioms, but derives them from an account of the role of the numbers in our judgements of cardinality—from our foremost application of the numbers. As such, it is arguably a compelling philosophical analysis of the concept of number since, as Wright has observed, one can show that the Frege-number of Fs = n if, and only if, there are, in the intuitive sense of the numerically definite quantifier, exactly n Fs. 9 But once the project of securing a correct analysis is divorced from the project of securing a body of truths as analytic or a priori, neither the fact that Frege arithmetic satisfies our condition of adequacy nor the fact that it connects the pure theory of arithmetic with its applications—essential as each is to securing it as a correct analysis of number—addresses the question of the epistemic status of our knowledge of arithmetic. This conclusion is not particularly surprising. Both neo-Fregean strategies we have been considering are variants on the methodology of reconstruction associated with Carnap. For Carnap the thesis that arithmetical knowledge is non-factual, and therefore, a priori, was not in serious doubt. And since the aim of a reconstruction is simply to delimit more precisely the extension of a predicate, we should never have expected that a Carnapian reconstruction of arithmetical knowledge would in any way justify the claim that our knowledge of arithmetic is a species of a priori knowledge. It is precisely in respect of their epistemological significance that Fregean logicism and neo-Fregean logicism—reduction to logic by explicit definition vs. reconstruction by Frege arithmetic—come apart.

Postscript (added 2004) Hale and Wright have replied to the criticism raised in Section 4: Let ‘Newton’ denote the conjunction of Newton’s laws as ordinarily understood, and ‘NewStip’ denote the (perhaps typographically indistinguishable) conjunction of the corresponding stipulations taken as introducing certain concepts of force, mass and motion. Then Demopoulos’s claim is—or ought to be, if the parallel is to be damaging—that while we may, by laying down NewStip, acquire some a priori knowledge (in some sense, knowledge about (some things we are calling) force, mass and motion), we obviously do not thereby acquire a priori knowledge of Newton—as we ought to do, if we can, in just or essentially the same fashion, acquire a priori knowledge of truths of ordinary arithmetic by stipulating Hume’s Principle, etc. . . . [But clearly,] the mere possibility of regarding (the sentences which formulate) Newton’s laws as stipulations introducing concepts of force, mass and motion (as distinct from generalisations to which 9 See The reason’s proper study, p. 251 and pp. 330ff.

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bodies conform) does not, and cannot, by itself justify the claim that NewStip ‘captures a pattern of use’ exhibited by ‘ordinary’ statements of Newtonian dynamics. 10

But it was never claimed that regarding Newton’s laws as stipulations is what justifies the contention that NewStip captures a pattern of use. Rather, what justifies the contention that Frege arithmetic preserves a pattern of use is that it recovers the deductive structure of a body of pure and applied unreconstructed knowledge claims. The point at issue is whether, by representing certain principles as recoverable from a stipulation, a reconstruction sheds any light on their epistemic status. The comparison with the Newtonian case makes it transparent that from the fact that we can recover a pattern of use from a stipulation, nothing follows regarding the aprioricity or otherwise of our knowledge of the principles being reconstructed. The situation would, of course, be entirely different if, in accordance with the methodology of Fregean logicism, it had proved possible to recover arithmetic from logic plus explicit definitions.

10 “Responses to commentators,” Philosophical books 44 (2003) 245–263, pp. 248–249.

“NEO-LOGICIST” LOGIC IS NOT EPISTEMICALLY INNOCENT 1 Stewart Shapiro The Ohio State University E-mail: [email protected]

Alan Weir University of Glasgow E-mail: [email protected]

1. A number of philosophers in recent years, most notably Crispin Wright and Bob Hale, 2 have tried to revive something recognisably akin to the logicist programme of showing that mathematical truths are in some sense analytic, a priori or at any rate “epistemically innocent”. They do so in the face of widespread scepticism with respect to the “a priori” and especially with regard to the idea of a priori proofs of existence. Their neo-logicism seems to involve two main tenets: firstly that mathematical truths are not known a posteriori, in the way empirical truths are known, but neither are they known via some Kantian form of intuition; rather our knowledge of mathematics arises from our ability to derive mathematical truths from rules or principles which are “analytic” or “meaning-constitutive” or in some sense explanatory of key mathematical notions such as that of natural number. Secondly, the realist thesis that this mathematical knowledge is knowledge of a world which is in some sense mind-independent or objective. 1 This paper first appeared in Philosophia Mathematica 8, [2000], pp. 160–189. Reprinted by kind permission of the editor and Oxford University Press. 2 See for example Crispin Wright, Frege’s Conception of Numbers as Objects (Aberdeen University Press, 1983); ‘The Philosophical Significance of Frege’s Theorem’ in Richard Heck Jr. (ed.) Language, Thought and Logic (Oxford: Oxford University Press, 1997), pp. 201–244; ‘On the Harmless Impredicativity of N= ’ in Matthias Schirn (ed.) The Philosophy of Mathematics Today (Oxford: Clarendon, 1998), pp. 339–368. Bob Hale, ‘Dummett’s critique of Wright’s Attempt to Resuscitate Frege’, Philosophia Mathematica 2 (1994), pp. 122–147.

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Clearly neo-logicism is a very attractive position for anyone sympathetic to a fairly traditional view of mathematics as a body of objective truths knowable a priori but who is worried by the standard epistemological problems faced by platonistic mathematics: how could we gain knowledge of a world of causally inert abstract objects, and so forth? The neo-Fregean answers, roughly speaking: by virtue of our knowledge of what we mean when we use mathematical expressions. More fully, we understand mathematical concepts 3 by following, in some sense, rules and following these rules is constitutive of understanding the expressions which express those concepts. These rules are entities which, like mathematical objects, are not part of the concrete, physical world; by tracing out the consequences of these rules we can find out truths concerning abstract objects. Even those such as Quine hostile to the notion of analyticity as glossed by the positivists recognise that our knowledge of the truth of sentences such as ‘if John is tall then either John is tall or Mary is short’ and ‘ “John is tall and Mary is short” entails that “John is tall” ’ may well proceed not through ordinary empirical means, nor through any special faculty of intuition but arise rather out of our understanding of operators such as the conditional, disjunction and conjunction, understanding which Quine claims is encapsulated, in the latter two cases, in his verdict matrices for the sentential connectives. 4 We will call such knowledge “epistemically innocent”. The phrase is deliberately somewhat vague, pending fuller amplification by neo-logicists. Certainly if there are truths which are analytic in the positivists’ sense, these will count as epistemically innocent; conversely any truths which require empirical verification (including holistic verification in Quinean fashion) or verification via some sort of Kantian intuition do not count as epistemically innocent. But we allow that the term may apply even if the positivist account of analyticity as truth by virtue of meaning fails. The neo-logicist claim we take then to be that there are principles which are epistemically innocent in something like the same way in which “if John is tall then either John is tall or Mary is short” is but which are strong enough to generate at least a sizeable body of standard mathematics, enough for the needs of the physical sciences, perhaps. The simple principles involved in elementary fragments of propositional logic are of course insufficient for the derivation of, for example, arithmetic. Neo-logicists have appealed instead to second-order abstraction principles of the form: αx(ϕx) = αx(ψ x) ↔ (ϕ, ψ) where α is some term-forming variable-binding operator which forms singular terms from open sentences ({x: ϕx} and nxϕx − the class of ϕ’s and the number of ϕ’s—are classic examples) and is an equivalence relation over 3 We use “concept” in the everyday, non-Fregean sense, unless otherwise indicated. 4 W. V. O. Quine, Word and Object (Cambridge, Mass. MIT Press, 1960) §13 and The Roots of Reference

(La Salle, Illinois: Open Court, 1974) §§20–21.

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properties. For the first of those two cases, the corresponding abstraction principle is “Hume’s Principle”: 5 ∀F∀G(nxFx = nxGx ↔ F1 − 1G) with F1 − 1G the second-order sentence which expresses the existence of a one–one correspondence between the F’s and the G’s. From this principle (plus suitable “bridging” definitions) one can derive, in standard second-order logic, all the theorems—including of course theorems expressing the infinitude of the natural numbers—of the usual Peano-Dedekind formulation of the theory of second-order arithmetic. This result—the derivability of second-order Peano arithmetic from Hume’s Principle—has become known as “Frege’s Theorem”. 6 For the second case, abstraction using the class operators, we get, much more problematically of course, Frege’s notorious Axiom V, which for the extensions of (Fregean) concepts takes the form ∀F∀G({x : Fx} = {x : Gx} ↔ ∀z(Fz ↔ Gz)). In the present paper, we accept for the sake of argument that abstraction principles such as Hume’s Principle are, indeed, epistemically innocent, at least on some natural readings (but see ahead Section 5). 7 We accept furthermore that not only are simple logical principles, such as ∨I and & E, or the related conditional theorems, epistemically innocent; so, too, are certain at least of the quantifier rules, for instance the natural deduction rules of universal generalisation and existential elimination. Moreover we include here these rules applied to second-order variables; that is, whilst some such as Quine would block derivation of Frege’s Theorem at the outset by refusing to accept the legitimacy of second-order logic, we do not object to second-order logic per se. Our claim will be, nonetheless, that Frege’s Theorem requires 5 The term is George Boolos’, following Frege, Grundlagen §63. See George Boolos, ‘The Standard of Equality of Numbers’ in George Boolos (ed.) Meaning and Method: essays in honour of Hilary Putnam (Cambridge Eng.: Cambridge University Press, 1990), pp. 261–277, see p. 267; the article is reprinted in the collection of Boolos’ papers Logic, Logic and Logic (Cambridge, Mass.: Harvard University Press, 1998), pp. 202–219. For the Grundlagen see The Foundations of Arithmetic translated by J. L. Austin, second edition (Oxford: Blackwell, 1980), p. 73. Frege’s rather honorific reference to the Treatise Book I, III.i is garnered from Baumann’s Die Lehren von Raum, Zeit und Mathematik (Berlin, 1868). 6 The phrase is Wright’s from a suggestion by Boolos—see ‘On The Philosophical Significance of Frege’s Theorem’, p. 203. 7 Boolos, Field, and Dummett have argued that the fact that Axiom V is classically inconsistent but is formally very similar to, e.g. Hume’s Principle, irredeemably vitiates the claim of abstraction principles to be analytic truths or at any rate to be epistemically unproblematic. See George Boolos, ‘The Standard Equality of Numbers’, p. 273, ‘Whence the Contradiction?’ in Proceedings of the Aristotelian Society, Supplementary Volume LXVII (1993), pp. 213–233 (reprinted in Logic, Logic and Logic, pp. 220–236), Hartry Field, Realism, Mathematics and Modality (Oxford: Blackwell, 1989), p. 158, Michael Dummett, Frege Philosophy of Mathematics (London: Duckworth, 1991), pp. 188–189, p. 208. For a more nuanced development (we claim!) of this objection see Stewart Shapiro and Alan Weir, ‘New V, ZF and Abstraction’, Philosophia Mathematica, 7 (1999), pp. 293–321. Wright’s ‘The Philosophical Significance of Frege’s Theorem’ is in large part a response to such objections. One radical response is to reject the logic (significantly weaker than classical logic) used in the derivation of antinomies and hold that Axiom V is not in fact inconsistent or at any rate not trivial. Cf. Alan Weir ‘Naïve Set Theory is Innocent!’, Mind 107 (1998), pp. 763–798.

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use of first- and second-order logical principles which are not epistemically innocent. More exactly, certain of the logical principles which are essential to the derivation of a theorem of infinity, when this is construed as expressing the existence of infinitely many mind-independent entities, are at least as problematic epistemologically as axioms of infinity laid down simply as postulates. Our supposed knowledge of these principles is, we will argue, every bit as mysterious as Kantian intuition of an infinity of numbers. We will look at two main cases in turn: the second-order axiom of comprehension applied to non-instantiated properties (Sections 2 and 3) and the first-order existential instantiation and universal elimination principles as applied in standard nonfree classical logic (Sections 4 and 5). We finish in the sixth section with a summary of our overall conclusions.

2. The standard second-order logic needed in the derivation of Frege’s Theorem includes straightforward generalisation of the first-order quantifier rules plus an Axiom Scheme of Comprehension. 8 Thus, for example, universal elimination and existential introduction become ∀FF F/ P F/ P ∃FF where P is any simple predicate constant or parameter and is any open sentence with F free. The (impredicative) Comprehension Scheme consists in all instances of: ∃R∀x 1 , . . . , ∀xn (Rx1 , . . . , xn ↔ ϕx1 , . . . , xn ) where R is an n-place relation variable and ϕ is any formula of the language in which R does not occur free. Finally we add a Substitution rule permitting substitution of co-extensional predicates: 9 ∀x(ϕx ↔ ψ x), θ . θ[ϕ/ψ] Now on the face of it, our neo-logicist, in taking second-order logic as characterised above as a body of epistemically innocent truths, is committed by the Axiom Scheme of Comprehension to a strong realism about properties, committed moreover to the a priori demonstrability of this strong realism. 8 It would be neater to dispense with Comprehension in favour of use of λ terms and λ conversion. But since most treatments of second-order logic use the comprehension scheme rather than λ abstracts we will follow suit. The points we will make in connection with Comprehension can be rephrased to apply to the λ term version of second-order logic. 9 This rule is a derivable rule in pure second-order logic but we will be concerned with expansions of the language to include complex singular terms formed by binding open formulae in one free variable x with operators such as the numerical operator nx(. . . x . . . ) or the class operator {x: . . . x . . . } and to handle these we need either a device such as λ abstraction or else a rule of the above sort. Since it could be dispensed with in a treatment with λ abstracts we will not question the epistemic status of this rule.

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At any rate, the neo-logicist seems committed to it being demonstrable that whatever it is that second-order variables range over exists, indeed exists in a mind-independent fashion. For the neo-logicist views mathematical theories such as number theory as arising (at least on a “rational reconstruction”) from the extension of an “empirical” or non-mathematical second-order language by the addition of new operators such as nxPx, thereby generating existential assertions with the same (or similar) degree of objectivity as pertains to the original “empirical” language. This has the advantage of treating the mathematical and empirical subfragments of language as semantically homogenous thereby easing, it is hoped, the problem of explaining the applicability of the allegedly “epistemically innocent” realm of pure mathematics to the epistemologically guilt-stained empirical realm. The downside to this for the neologicist is that the impredicative comprehension axiom must be assumed not only to be true in the same objective sense as other sentences of the original empirical sector, we must also be able to know or verify its truth in an epistemically innocent fashion. If comprehension is known via intuition or known a posteriori then no demonstration of a mathematical truth can be innocent if it relies essentially on comprehension just as no demonstration is innocent if it relies on an axiom of set theory conjecturally justified holistically in terms of the fruitfulness of its empirical consequences. We have agreed that our knowledge of the validity of simple rules such as &E and ∨I is epistemically innocent; we may well expect that there can be other less simple cases of epistemic innocence which do not share all the features of &E and ∨I. But one cannot help but notice the enormous leap from simple rules of the above type to complex principles such as: ∃R∀x 1 , . . . , ∀xn (Rx1 , . . . , xn ↔ ϕx1 , . . . , xn ) (especially when one reflects on those impredicative cases where ϕ can contain bound second-order variables). We can give a fairly plausible account of how we know &E is sound, grounded perhaps in nothing much more than the Quinean verdict matrices for &; the neo-logicist needs to come up with something along similar lines (more complex no doubt) which will explain how we know, of each instance of the above, that it is true. A Quinean might dismiss neo-logicism on these grounds alone: if one assumes that it is a priori (or innocently) demonstrable that a realm of properties exists independently of the mind, such a person may say, then maybe you can derive the existence of abstract objects. But anyone sympathetic to the “anti-Anselmian” intuition that one cannot derive existence from concepts alone will refuse to accept the above axiom scheme of comprehension as epistemically innocent. For each instance entails the existence of a property and, (in)famously, Quine argued that second-order logic is not logic at all, but is set theory in disguise. As early as 1941, he claimed that properties are too

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obscure to serve logic, and should be replaced with items like classes. 10 Once we invoke classes, however, we have crossed the border out of logic and into mathematics proper. Later, Quine wrote: 11 Set theory’s staggering existential assumptions are. . . . hidden. . . . in the tacit shift from schematic predicate letter to quantifiable set variable.

For Quine, then, second-order logic is a wolf in sheep’s clothing. It is set theory made to look like logic, by having variables ranging over properties/sets. But it is important to note that the Quinean is only the most extreme of the opponents the neo-logicist faces and even if the neo-logicist can confute the Quinean position, he or she is far from home and dry. To see this, let us assume for the sake of argument that properties exist in as mind-independent a fashion as scientific entities do but assume as little else as we can about properties so that we can be neutral as to what exactly the nature of properties is (perhaps they are just classes of objects, for example). Consider now a philosopher—Macari, let us say—who accepts second-order logic as correct subject to one, rather minor looking amendment. She accepts only the following form of the Comprehension Schema as logically valid: ∃x1 , . . . , ∃xn ϕx1 , . . . , xn → ∃R∀x1 , . . . , ∀xn (Rx1 , . . . , xn ↔ ϕx1 , . . . , xn ). That is, focusing on one-place open sentences for simplicity, Macari agrees that it is a logical truth that to every such sentence which is instantiated by something or other, there corresponds a co-extensional property. But she refuses to accept that logic alone tells us that there are uninstantiated properties, so refuses to conclude that to predicates such as x = x there corresponds a property. Macari’s attitude is not all that odd or lacking in motivation. Macari may have what one might loosely call “Aristotelian” reasons for being sceptical about uninstantiated properties. 12 Macari accepts, let us suppose, that it is just as sound to infer wisdom exists, because Socrates is wise, as it is to infer that Socrates exists, because that same proposition is true. But it is a far more contentious step to assume that if a sentence involving a predicate P is true, then a mind-independent property corresponds to P, even when P is the predicate x = x. For Macari, this is as substantive an assumption as the 10 W. V. O. Quine, ‘Whitehead and the rise of modern logic’, in P. A. Schilpp, The Philosophy of Alfred North Whitehead (New York: Tudor Publishing Company, 1941), pp. 127–163. 11 W. V. O. Quine, Philosophy of Logic (Englewood Cliffs, New Jersey: Prentice-Hall, 1970), p. 68. 12 There seems to be enough textual support for the idea that Aristotle was opposed to the idea of ‘ante rem’ universals to justify use of his name; but since Macari’s view of properties is doubtless not all that close to Aristotle’s we will refer to the position as “aristotelian”, with lower case “a” in analogy with “platonism” in philosophy of mathematics. The key tenet of Macari’s view is only that there is no epistemically innocent proof that uninstantiated properties exist; she need not believe, for example, in ‘concrete universals’ which are somehow both properties and extended physical objects but which, unlike mereological fusions, are ‘wholly present’—whatever that means—wherever their instances are present.

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assumption of the existence of an abstract object. It is certainly hard to see how this assumption can be known in as innocent a fashion as the soundness of &E and ∨I can be known. Macari’s views, we submit, err, if anything, on the side of generosity to neo-logicism, in allowing that to any arbitrary instantiated predicate there corresponds a property. Neo-logicists, if they are to convince Macari that abstract objects can be shown to exist in something like an a priori fashion must therefore do so by adding abstraction principles to a logic which does not already embody the assumption that uninstantiated properties exist, i.e. a logic which is no stronger than Macari’s “aristotelian” second-order logic (let’s call it “A2L”), that is standard second-order logic but with Comprehension restricted as above. But here they face a huge problem: arithmetic, in particular a theorem of infinity, is not derivable from Hume’s Principle in A2L. The stumbling block is the number zero, defined by the neo-logicist as nxPx for some predicate P x with ⊢ ∀x ∼ P x, e.g. with x = x for P x. On the face of it, Macari ought to hold that nx(x = x) is, or could in some situations be, an empty term standing for nothing rather than standing for something, namely the number zero. For the intended interpretation of the numerical operator nxϕx is as a function which maps properties to objects in a certain way; and where ϕx is x = x there is no property available as argument for the function. But in order to keep separate a number of distinct problems with neo-logicism we will leave discussion of free logics till later (Sections 4 and 5) and assume that all singular terms, simple or complex, have a reference. We will therefore ensure that there is at least a “dummy” referent assigned to all numerical terms of the form nxϕx, even where nothing satisfies ϕx. The aristotelian interpretation of the numerical operator is as follows. For simplicity let properties, the range of the monadic second-order quantifiers, just be non-empty subsets of the domain of individuals. In each model, the semantics, via the recursion theorem, will ensure that each open sentence ends up getting assigned a subset (empty or not) of the domain as its extension. To do this for the language containing the numerical operator, we first partition all the non-empty subsets into equivalence classes under the relation of equinumerosity and map these equivalence classes via a one:one function f into the domain of individuals. We then select some arbitrary member d of the individual domain as the image of the equivalence class whose sole member is the empty extension Ø and thus extend f to a function N by mapping {Ø} to d. Since d is arbitrary, there is no requirement that d not occur in the range of f; it might, for example, turn out to be the image of f applied to the equivalence class A of all unit sets. The numerical operator is then interpreted as a function which maps the extension of E each open sentence ϕxgo the N-image of the equivalence class in our partition to which E belongs. In the model envisaged, then, N(A), that is d, is the object which is, in the model, the number one so that d may number both the empty set and every unit set.

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On this extension of A2L to the language including the numerical operator, we can, of course, prove that zero—nx(x = x) on the Fregean definition— exists. For ∃x(x = nx(x = x)) is a simple consequence of the appropriate instance of the axiom schema of identity which holds in A2L. 13 But the rub is that we cannot prove the standard facts about zero which we need in order to prove a theorem of infinity. In particular, we cannot prove ∃x Suc(x, 0), where Suc(x, y) abbreviates the standard Fregean definition of x succeeds y: Suc(x, y) ≡ df. ∃F∃G(x = nw Fw & y = nwGw & ∃z(F z & nw(Fw & w = z) = nwGw)); and we cannot rule out the possibility that for some positive number j, 0 = S j 0, where S j 0 is the standard Fregean numeral for the number j. 14 In particular, as in the case set out above, we can have as true in an aristotelian model 0 = S0, i.e. 0 = 1. The assumption 0 = S j 0 does not lead to conflict with Hume’s Principle since in A2L, with its restriction on the Comprehension scheme, we cannot derive: nx(x = x) = nx J x < − > 1 − 1((x = x), J x) from Hume’s Principle, where J x is (x = 0 ∨ · · · ∨x= S j−1 0) (x= 0 in the example given). Since Jx is provably instantiated, we will be able to prove from the relevant instance of the Comprehension scheme: ∃G∀x(Gx ↔ J x)

(i)

If we had available full Comprehension we would also have for the instance (x = x): ∃F∀x(F x ↔ x = x)

(ii)

from which, after excising the initial second-order existential quantifiers for application of ∃E (or else after existential instantiation), we get: ∀x(Gx ↔ J x)

(iii)

∀x(F x ↔ x = x).

(iv)

∀E on Hume’s Principle yields nx F x = nx Gx ↔ F1 − 1G

(v)

S j 0 abbreviates nx J x so that two applications of the Substitution rule takes us (iii), (iv), and (v) to: 0 = S j 0 ↔ 1 − 1((x = x), J x). 13 If t = u is defined as ∀F(Ft → Fu) (equivalently ∀F(Ft ↔ Fu)) the derivation of each instance is fairly trivial. If it is primitive, we add each instance of t = t as an axiom. 14 i.e. takes the form nx(x = 0 ∨ . . . x = S j−1 0).

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Since we can disprove the right-hand side, (ii) along with the rest of our logical apparatus would yield a reductio ad absurdum of 0 = S j 0. But since (ii) is not forthcoming in aristotelian second-order logic, neither is this standard proof. As a consequence, the Fregean proof of infinity fails in A2L because we cannot prove there are n + 1 numbers less than or equal to n; for all the aristotelian knows, zero happens to be identical to some number Sk 0 between zero and n. As we have seen, it is possible for 0 = 1 to hold in a model in the aristotelian framework in which case the number of numbers less than or equal to one is just one. Nor can the neo-logicist hope that it is merely the standard proofs of the theorem of infinity that fail, that the aristotelian may be able to find some more complex ways of arriving at infinity. For there are aristotelian models of any finite size of Hume’s Principle. To see this, take any finite domain of individuals D = {d1 , . . . , dn }. As above, we interpret the numerical operator by partitioning the subsets of D into classes of equinumerous classes:– there will be n + 1 of these including the class containing the single zero-sized subset of D, namely Ø and with the range of the second-order quantifiers the non-empty subsets of D. 15 Interpret nxϕx by the map N which takes X ⊆ D to dk , where X is the extension of ϕ and is of size k = 0 and which takes the empty extension to d j for some j which is doubling up as “zero”. Then Hume’s Principle: ∀F∀G(nx F x = nx Gx ↔ F1 − 1G) is satisfied because for any non-empty subsets S1 and S2 of D assigned to F and G (remember assignment of Ø is not possible in this model) the left-hand side of the biconditional is true iff M assigns to F and G the same element di iff F and G belong to the same partition class under the equinumerosity equivalence relation (since F and G are non-empty) iff the right-hand side is true. 16 In the model given, all the other principles of second-order logic, including all the instances of aristotelian Comprehension and the Substitution rule are sound. Hence Hume’s Principle (HP) does not semantically entail, in models sound for A2L, a theorem of infinity and so such theorems are not derivable in A2L from HP. The neo-logicist cannot claim, then, that the existence of infinitely many numbers is demonstrable in an epistemically innocent fashion unless he or she can show that it is demonstrable, in such an epistemically innocent fashion, 15 Though the argument goes through just as well for a property realist who distinguishes the extension of a predicate from the property it stands for, so long as they allow some interpretations in which no property has an empty extension. 16 The aristotelian interpretation also yields a (not very exciting!) one-element model of Axiom V. If d is the sole individual, there are two subsets of domain D: Ø and {d} and so only one non-empty subset falling in the range of the predicate variables. We assign {d} to d and Axiom V comes out true for every assignment of predicate extensions to the two variables F and G of the embedded biconditional, since every assignment assigns {d} to both variables. This, however, is the only aristotelian model (up to isomorphism) of Axiom V .

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that there is an uninstantiated property, unless, in other words, they can show that aristotelianism on properties can be shown to be wrong by an epistemically innocent argument. The “aristotelian neo-logicist” will, it is true, be able to prove (∃F∀y ∼ Fy) → Inf where Inf is a theorem of infinity; that is, if an empty property exists then there are infinitely many numbers. Even if that conditional is demonstrable in an epistemically innocent fashion, this is of little use to the neo-logicist if our knowledge of the existence of an empty property is no better placed or explicable than our knowledge of the existence of infinitely many numbers. Certainly the neo-logicist cannot argue that the hypothesis that there is an empty property is holistically confirmed by the fact that it enables us to develop the mathematics needed by science. If we take this Quinean course, we may as well help ourselves to the standard axiom systems of number theory, analysis etc. Perhaps it will be felt that to credit us with an intuitive knowledge of the existence of one empty property is more plausible than crediting us with an intuitive grasp of infinitely many numbers, or even an intuitive grasp of the fact that infinitely many numbers exist. But the neo-Kantian need not appeal to direct intuition of every number, nor does there seem any relevant difference between admitting direct intuition of one non-concrete entity and admitting direct intuition of many; 17 any such appeal to intuition represents abandonment of neo-logicism. One patch the neo-logicist might try out is to add to Hume’s Principle each instance of the following “Zero” axiom scheme: ∼ ∃xϕx ↔ (nxϕx = 0) This ensures that if ϕ and ψ are both unsatisfied then nxϕx = nxψ x = 0, whereas if θ is satisfied then nxθ x = 0. Call the theory consisting of HP plus all instances of the above schema HPP · HPP entails that the universe is infinite. Model-theoretically we can show this as follows: if there are only finitely many individuals k in a domain then there are k equivalence classes under equinumerosity on the domain of non-empty sets so we need all k distinct individuals to number the non-empty properties so as to satisfy HP. But we also need an individual which is the referent of nxθ x, where θ has an empty extension and this individual, the referent of 0, is constrained by the Zero axiom scheme to be different from the k individuals which number the instantiated properties; hence there can be no finite model of HPP. A salient point to note here is that we run into problems if we replace the schematic variable ϕ with a second-order variable because in our aristotelian theory such 17 Compare Russell’s view that if one accepts that one universal—of resemblance—exists, one may as well accept a plurality: (Bertrand Russell, The Problems of Philosophy, London: Williams and Norgate, 1912, pp. 150–151).

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variables range over non-empty properties and the case we wish to capture here is one in which ϕ is uninstantiated. This brings to the fore how ungainly the theory HPP is; how different in form it is from an abstraction principle, formulable as a single sentence, such as HP, not to mention simple inference rules such as &E. What reason is there to suppose that the infinitary theory HPP is epistemically innocent? Since HPP contains infinitely many sentences, there is no way we can think of it as grounded in inferential practice in the way that &E is grounded in the verdict matrices for &. However, there is a single formula related to the theory HPP which will yield, even in aristotelian second-order logic, Frege’s theorem and that is HPP∗ : ∀F∀G((nx F x = nx Gx ↔ F1 − 1G) & (∼ ∃x F x ↔ (nx F x = 0))) In other words, we conjoin to HP an instance of the Zero axiom scheme but with an ordinary variable in place of a schematic one, a variable bound by one of our initial quantifiers. It follows from the right conjunct of HPP* that no property has number zero; since all properties are instantiated, in our aristotelian models, all falsify the left-hand side ∼ ∃ xFx of the right conjunct. Hence, as with HPP, there must be infinitely many numbers. Starting with our definition of zero as nx(x = x) we can then prove the Peano Postulates using the other “bridging” definitions of successor or predecessor and so on. What we cannot prove, though, and here is a key difference with HPP, is that there is exactly one “zero”, in the sense of one number which all uninstantiated properties have. This can be seen by adding to our interpretations “virtual” properties in addition to the “real” properties which exhaust the range of the second-order quantifiers and letting formulae in one individual variable whose extension is empty be assigned an arbitrary such virtual property. The numerical operator is then to be interpreted as a function on properties, real or virtual in which equinumerous real properties are assigned the same value but distinct virtual properties can be assigned distinct values. Suppose, for example, that we can prove ∼ ∃xϕx. The formula ϕx might, for example, be x < 0. We cannot go on to prove that nx(x < 0) = 0 because we cannot apply Hume’s Principle to empty properties. If we had full Comprehension we could prove ∃F∀x(Fx ↔ x < 0) and so assume for existential elimination (i) ∀x(Fx ↔ x < 0). We instantiate HPP* and derive the right conjunct by & E. Our generalised substitution rule yields, from (i) (∼ ∃xϕx ↔ (nxϕx = 0)).

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Having already proved that ∼ ∃x x < 0, we would then have been able to conclude nx(x < 0) = 0. But we do not have full Comprehension! We only have, for the instance in question: ∃x x < 0 → ∃F∀x(Fx ↔ x < 0) and we cannot discharge the antecedent. This plurality of “zeros” is a problem for the neo-logicist, even though Frege’s Theorem follows from HPP∗ . For PA is only one possible formulation of arithmetic (albeit a very important one) where we understand by “arithmetic” a theory investigated (in natural language plus some notation) by mathematicians; moreover mathematicians do not solely (or even mainly) deal with formalised theories of arithmetic, but also use arithmetic principles in order to prove theorems in various different areas, pure and applied. It is by their ability to account for and explain real mathematical theories that philosophies of mathematics, and the formal systems they utilise, must be judged. In “real arithmetic” we say things like “the number of numbers less than zero is itself zero”; such applications of “the number of Ps is zero” are also very commonplace in applications of arithmetic in real situations. If a philosophy of mathematics cannot account for the truth of such assertions, that is a serious defect. All this on the assumption that HPP∗ (and the aristotelian version of the Comprehension scheme) are epistemically innocent. But once again we can ask the question what grounds are there for thinking that HPP* is innocent? It is not even an abstraction principle. We will raise this question again with respect to a related principle which arises when neo-logicism is set in the framework of plural quantification.

3. George Boolos, the most influential advocate of plural quantification, proposed that one of its values is that we can use it to interpret second-order quantification without invoking either properties or classes. 18 Perhaps the neologicist can use this to bypass the issues concerning which properties exist. If Hume’s Principle can also be interpreted “pluralistically” in such a way as to retain its innocence (we have granted it innocence on the usual way of reading it) then the neo-logicist would seem to be home and dry. Boolos suggested that a monadic, existential second-order quantifier be considered a counterpart of a plural quantifier, “there are (objects)”, in natural language. The following illustration is called the Geach-Kaplan sentence: Some critics admire only one another. 18 George Boolos, ‘To Be is to Be a Value of a Variable (or to Be Some Values of Some Variables)’, Journal of Philosophy 81 (1984), pp. 430–449; ‘Nominalist Platonism’, Philosophical Review 94 (1985), pp. 327–344; ‘Reading the Begriffschrift’, Mind 94 (1985), pp. 331–344. These essays are reprinted in Logic, Logic and Logic at pages 54–72, 73–87, and 155–170, respectively.

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It has a (more or less) straightforward second-order rendering, taking the class of critics to be the domain of discourse: ∃F(∃xFx & ∀x∀y((Fx & Axy) → (x = y & Fy))). According to the usual reading, the formula would correspond to “there is a non-empty class (or a non-empty property) F of critics such that for any x in F and any y, if x admires y, then x = y and y is in F”. But this implies the existence of a class or property, while the original “some critics admire only one another” does not, at least prima facie. Boolos developed a rigorous, model-theoretic semantics for monadic, second-order quantification in these terms. Some philosophers with nominalist tendencies have invoked the Boolos semantics in order to obtain the benefits of second-order quantification without encumbering oneself with a second-order ontology. A good deal. Our neo-logicist might attempt a similar maneuver, in order to make the logic more tractable. In this case, however, there are troubles at every turn. Second-order logic enters into Hume’s Principle (and Frege’s theorem) in two places. Like any second-order abstraction, Hume’s Principle has prenex universal quantifiers binding monadic property variables. The Boolos plural construction originally was limited to existential second-order quantifiers but Boolos extended it to universal quantifiers. He glosses a second-order universal quantification of the form ∀F(F) along the lines: no matter which things the Fs are, holds of the Fs.

We will give an example later illustrating how he glosses the schematic phrase:– holds of the Fs. However, on the right-hand side of Hume’s Principle we find the notion of equinumerosity and the definition of equinumerosity invokes a variable over (binary) relations on the domain: two (monadic) properties are equinumerous if there is a one-to-one relation from the extension of one of them onto the other. So our first problem is that the Boolos plural construction is limited to monadic property variables, while the second-order definition of equinumerosity has a binary relation variable. However, if there is a (definable) pair function in the language, then relations can be introduced in the usual manner. Variables over binary relations are replaced with monadic variables ranging over pairs: ∃R is rendered ∃F′ , where ′ is obtained from by replacing each occurrence of Rtu with F< t, u>, where < t, u > is the ordered pair of t and u. This move is not available here, at least not without begging the crucial question. There can be no pair function on a finite domain with more than one member. If the domain has size n, then there are n 2 ordered pairs. So our neo-logicist cannot introduce a pair function until she has shown that the universe has either at most one object in it or is infinite. This throws a monkey wrench into the works. The neo-logicist wants to use Hume’s Principle and

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therewith Frege’s Theorem to establish that the universe is infinite, by showing that the natural numbers exist (and are distinct). But on the present plan, she cannot even formulate Hume’s Principle (via plural quantification with pairing) without first showing that the universe is either non-plural or infinite. Since we non-Hegelians know the first disjunct is false, this means showing that the universe is infinite. The plan is frustrated before it can even get started. The existence of a pair function on each infinite domain is equivalent to the full axiom of choice, since it amounts to κ 2 = κ, for each cardinality κ. How can our “pluralist” neo-logicist claim that the existence of pairs is epistemically innocent? This amounts to the epistemic innocence of full choice. Perhaps we can be helpful here. The neo-logicist might introduce pairs via an abstraction principle: ∀x∀y∀z∀w( (x, y) = (z, w) ≡ (x = z & y = w)). Call this the Pair Principle. Strictly speaking, it is not in the same form as other abstraction principles, since the right-hand side (symbolised as (x = z & y = w)) represents a four-place relation and so not an equivalence relation. But it has at least the flavor of an equivalence relation (symmetry, transitivity, reflexivity), and so is in the spirit of other abstraction principles. 19 The Pair Principle lies between first-order abstractions and second-order abstractions. If our neo-logicist can maintain that it is epistemically innocent, then an acceptable “plural” formulation of equinumerosity can be produced. The antecedent represents quite a big “if” since the principle will in effect give the neo-logicist an infinite universe. But for the rest of this section, we will make that concession and assume that the right-hand side of Hume’s Principle is kosher. What, next, of the left-hand side of the principle in which we find, crucially, the abstraction operator itself—“the number of”. On the standard reading, this is a function from properties (or classes) to objects, but we are considering here a philosopher who is trying to get by without acknowledging properties or classes at all. We thus cannot think of the abstraction term as denoting a function, for there is (or may be) nothing for it to operate on. This is not an insuperable stumbling block: English and other natural languages with a definite article construction allow that construction with respect to plurals. We speak, for example, of “the dogs in the room” or “the numbers less than 12”. So it seems to make sense to use the plural definite article with a second-order variable, to produce the locution, the F’s or, here, the number of F’s 19 Reflexivity: (x = x & y = y); symmetry: (x = z & y = w) entails (z = x & w = y); transitivity: (x = z & y = w); and (z = t & w = u) entail (x = t & y = u). Bob Hale invokes a pair abstraction in “Reals by abstraction” (Philosophia Mathematica (3) 8, 100–123; reprinted in Hale and Wright [2001], 399–420). Frege’s theorem only needs to invoke instances of Hume’s principle applied to finite properties with finite extensions and so it may be possible to formulate a “predicative” version of the Pair Principle in which no occurrences of occur (at least untyped) inside terms of the form (x, y). The (set-theoretic) statement that the Pair Principle is satisfiable on every infinite domain is equivalent to choice. However, as far as we know, the Pair Principle does not entail the axiom of choice, since it only produces a pair function on the universe.

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(we will attempt to be as neutral as possible on the right way or ways to approach the syntax and semantics of such locutions). Certainly Boolos found application of the number operator to plural phrases perfectly sensible. 20 Here is one of his glosses of Hume’s Principle, illustrating also his way of reading second-order universal quantifiers plurally: Hume’s Principle is the statement that no matter which things the Fs and Gs may be, the number of Fs is the same as the number of Gs just in case the Fs and Gs are in one–one correspondence. 21

So we finally have a plural-quantifier version of Hume’s Principle. Call this Plural Hume. Let us assume that, as written, Plural Hume is epistemically innocent. What does it say? In particular, what of the initial plural quantifiers? Macari, it will be recalled, agrees that instantiated properties exist, and so instantiated properties can be objects in the range of second-order variables; but Macari denies that empty properties exist (or at least that we can prove innocently that they do). We saw in the previous section that under Macari’s assumption, Frege’s Theorem does not go through using Hume’s Principle alone. In Macari’s system, Hume’s Principle is satisfied on finite domains. But if the neo-logicist goes the route of plurals, and formulates Plural Hume, then Frege’s Theorem does not go through for the same reason:—if our neo-logicist relies on plurals, she has played into Macari’s hands. The reason for this is fairly simple. The ordinary locution, there are F’s such that . . . entails that there is at least one F. Indeed, this is what the locution says. 22 In traditional terms, the plural quantifier declares that a certain predicate is instantiated. For example, I cannot announce that there are vicious elephants in our backyard, unless I have reason to believe that there are some. The empty property (if there is one) cannot instantiate a plural existential quantifier. Similarly if I say that no matter what things the Fs are, if everything which is an F is a G, then Hamish is an F, i.e. no matter what the F’s are: (∀x(Fx → Gx) → Fh) then I do not seem to be committed, absurdly, to Hamish being non-selfidentical and what I say seems perfectly coherent if, for example, only Hamish is a G. 23 Thus the plural quantifiers are analogues only of Macari’s aristotelian quantifiers, not the standard second-order quantifiers. If, therefore, the Boolos program is successful, it eliminates a commitment to instantiated properties 20 See, for example, Boolos [1993]. 21 Op. cit., p. 223. Perhaps “no matter which things the Fs and Gs are” is better than “no matter which

things the Fs and Gs may be” as the latter may seem to import a modal element foreign to straightforward universal quantification. 22 Boolos notes that the locution there are F’s might entail that there are at least two F’s, but one F is enough for present purposes. See, for example, Boolos [1984], p. 443. 23 If there is no way of reading universal second-order quantification without commitment to properties or classes, including empty ones, then someone sticking to Boolos’ original motivation for plural quantification would have to drop ∀F as a primitive and use the ∼ ∃F ∼ translation.

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or to non-empty sets. For that reason we cannot validate, in the way required by neo-logicists, the full axiom scheme of Comprehension; only the restricted form available to Macari is sound for plural quantifiers. And so Frege’s Theorem is blocked from the start. By reasoning parallel to that in the Macari case we can show that there are models in which the numbers form a finite subset of the infinite domain, with 0 = S n 0, for some number n. Indeed there will be a model of size one, one in which the pairing axiom holds. Plural Hume does not entail the Peano postulates. However, Boolos’ pluralist agenda included the program of rendering standard “non-aristotelian” second-order languages into the language of plural quantification. (This language is akin to lawyer’s English, with indexed pronouns and related constructions playing a role similar to “the party of the third part” and so on.) He does this by including in his translations a clause to handle explicitly what would be empty properties. Let (F) be a formula with the monadic, second-order variable F free. Let ∗ be the result of replacing each occurrence of Ft in (F) with t = t. In standard terms, ∗ states that holds of the empty property. Then Boolos renders the second-order ∃F(F) as something like: Either there are some F’s such that (F), or ∗

For example, he “translates” the second-order set-theoretic truth ∃F∀x(Fx ≡ x∈ / x) as (after some simplification) “Either there are some sets that are such that every set is one of them iff it is not a member of itself or every set is a member of itself” (ibid.). 24 In this case, the second disjunct does no work (being provably false), but it is part of the “translation”. Applying the same idea to ∀F(F) we get: no matter what the Fs are, (F) and ∗

with conjunction in place of disjunction. Since plural English can get rather complex we will carry out some translations into a formalised version of it but in order that the plural reading be borne in mind we will use (F) for no matter what the F’s are and (EF) for there are some F’s such that. So we translate, e.g. the sentence ∀F Ft, which is false in standard second-order logic, into: ((F)Ft & t = t) which is also false even in a one-element universe in which no uninstantiated property exists. Can we amend Hume’s Principle along those lines? The matter is complicated a bit in the present context. Since we have introduced a higherorder operator the number of F’s , we cannot just replace the “F” with something like t = t to handle the uninstantiated case. The phrase The 24 ∗ (F) here is ∀x(x = x ↔ x ∈ / x) which is provably equivalent to ∀x(x ∈ x).

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number of t = t is not grammatical. One way round this is to introduce λ terms and thereby a translation *λ in which [∀F(F)]∗λ = ((F)(F)) & (λx(x = x)) with a dual clause for ∃F; thus ∀F Ft becomes (F)Ft & λx(x = x)t. But since we can always eliminate λ terms by λ conversion, starting from λ terms with narrowest scope and working outwards substituting equivalents for equivalents, we can in this way arrive at a translation ∗∗ which is the result of applying λ conversion to the ∗ λ translation. Hence applying ∗ λ to ∃F(nxFx = nx(x = x)) (i.e. there is some non-zero number) gives us: (E F)(nxFx = nx(x = x)) ∨ n(λx(x = x) = n(λx(x = x)) 25 and then λ conversion yields the ∗∗ translation (E F)(nxFx = nx(x = x)) ∨ nx(x = x) = nx(x = x) in which the second disjunct is a logical falsehood. Monadic Comprehension: ∃F∀y(Fy ↔ ϕy)(whereFdoes not occur in ϕ) becomes, after λ conversion: (E F)(∀y)(Fy ↔ ϕy) ∨ (∀y)(y = y ↔ ϕy) which is equivalent to (∃y)ϕy → (E F)(∀y)(Fy ↔ ϕy), i.e. If there is a ϕ then there are some Fs such that anything is an F if it is a ϕ and is a ϕ if it is an F.

We submit that if this is formalised so as to be put to use in a derivational system, it must be formalised as Macari’s aristotelian (monadic) axiom of Comprehension, with the antecedent requiring the existence of a ϕ; 26 it ought not to be formalised as standard comprehension. In one sense, the Boolos program validates full second-order comprehension: each instance is translated into a truth of the plural quantification framework. But never a truth with which Macari would quibble. For example, where ϕ in the Comprehension scheme is instantiated by y = y, the translation yields (∃y)y = y → (E F)(∀y)(Fy ↔ y = y). What happens to Plural Hume under the Boolos translation? Here things are complicated by the fact that we have two initial universal quantifiers to 25 If we have λ terms, we can do all variable-binding using them, formalise the numerical operator as n(λxϕx) and abbreviate the latter as nxϕx. 26 As remarked above, we leave aside the complication that the plural reading is indeed plural, that is, seems to require more than one ϕ; even ‘at least two’ is arguably not quite right as a reading either.

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work on. Plural Hume, using our conventions about “pluralese” (and leaving unpacked the definition of one:one correspondence) is: (F)(G)(nx F x = nx Gx ↔ F1 − 1 G). Applying ∗∗ to (F) we get: ((F)[(G)(nxFx = nxGx ↔ F1 − 1G)]) & [(G)(nx(x = x) = nxGx ↔ (x = x)1 − 1G)]. We must now apply ∗∗ to the two (G) formulae in square brackets. For the first, it yields (using the standard definition of 0 as nx(x = x)): ((G)(nxFx = nx Gx ↔ F1 − 1G)) & (nx F x = 0 ↔ F1 − 1x = x). For the second we get ((G)(0 = nx Gx ↔ (x = x)1 − 1G)) & (0 = 0 ↔ (x = x)1 − 1(x = x)). Putting all this together the result is ((F)((G)(nx F x = nx Gx ↔ F1 − 1G) & (nx F x = 0 ↔ F1 − 1x = x)))& (G)((0 = nx Gx ↔ (x = x)1 − 1G) & (0 = 0 ↔ (x = x)1 − 1(x = x))). Call this Amended Plural Hume. Now the last conjunct is a logical truth and the second and third are symmetrical so that we can delete one of these which gives, with some further simplification: (F)(G)((nx F x = nx Gx ↔ F1 − 1G) & (∼ ∃x F x ↔ nx F x = 0)). That is: No matter what the Fs are and no matter what the Gs are, the number of Fs = the number of Gs just in case there is a one–one correspondence between them and there are no Fs iff the number of Fs = zero.

But this is none other than the non-schematic HPP∗ . We conjoin to Plural Hume a clause specifying that the number of Fs is zero iff there are no Fs. HPP∗ yields, we have seen, Frege’s theorem but fails to capture many applications of the numerical operator by allowing a plethora of zeros. This a form of plurality which supporters of plural quantification will be less keen on. In sum, the plural version of HP does not give us Frege’s theorem. Plural HPP∗ does but does not give us the applications we want. Moreover we can ask again why think plural HPP∗ , which is not an abstraction principle, is innocently true. After all, none of the formal versions of HP can be presumed to be explicitly part of the concept of number possessed by everyday speakers (or even most mathematicians); indeed it is doubtful whether any are even implicitly held by hoi polloi. The best case for the epistemic innocence of HP is surely that something like:

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the number of apples equals the number of oranges iff the apples and oranges can be paired off one:one

is part of competent users’ tacit understanding of number (and so on for other sortal concepts). If Boolos’ plural reading of quantifiers is on the right lines, the most faithful reading of these “platitudes” which jointly are constitutive of our notion of number (compare the conceptual functionalists’ idea that our folk psychological notions arise from psychological platitudes) is the original Plural Hume not Amended Plural Hume, and the original is certainly too weak to do the work the logicist requires of it. The neo-logicist might abandon the project of reconstructing arithmetic using Hume’s Principle and consider other abstraction principles, perhaps principles which are not constitutive of any ready-to-hand notion. 27 But given the equivalence in power of the plural quantification and aristotelian second-order logic, and the existence of oneelement models of Axiom V, this strategy will not work either. To conclude these two sections and the examination of logicism’s use of second-order logic: we have considered the effect of weakening second-order logic by allowing as epistemically innocent only the assumption of arbitrary instantiated properties, or equivalently, allowing only pluralist interpretations of second-order quantification. But the result, combined with Hume’s Principle, does not yield Frege’s Theorem. Moreover thus far we have been very generous to the neo-logicist. Many who are realist about properties, for example, assume only a “sparse” rather than an “abundant” theory of properties: that is, they do not assume that to each arbitrary predicate—(x is an electron or x is a baseball or x is a pleasant dream)—there corresponds a property. (And even those who favour an abundant theory do not always claim they know a priori that it is true.) This suggests that no form of the axiom scheme of Comprehension is epistemically innocent. But if we drop it completely, then it is easy to produce finite models of HP (and indeed of Axiom V), models 28 in which any number can be a “rogue” number in the same way that zero is a rogue number in the aristotelian version of Fregean arithmetic. Take any finite set of D individuals and any subset S of the power set of D such that there are at most n equivalence classes over S under the equivalence relation of equinumerosity. There is thus a one:one function f from this set of equivalence classes into D. We let S be the range of the monadic predicate quantifiers and select some member d of D as our rogue number; we then interpret the numerical operator as follows: where the extension s of ϕx belongs to S, the referent of nxϕx is f (E(s)), where E(s) is the equivalence class s belongs to; otherwise the referent of nxϕx is d. As before, this gives us a model of 27 Cf. Wright’s response to Boolos on the question of what notion “New V”, an abstraction principle which is a weakened form of Axiom V, is constitutive of:—“The Philosophical Significance of Frege’s Theorem”, pp. 239–240. 28 Non in standard ‘unfaithful Henkin’ models, of course: cf. Cf. Stewart Shapiro, Foundations without Foundationalism (Oxford: Clarendon, 1991), p. 89.

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second-order logic (minus Comprehension) plus Hume’s Principle, this time one in which (∃F∀y ∼ Fy) → Inf can fail, if the empty set belongs to S. Our overall conclusion, then, is that the neo-logicist program, as standardly developed, requires the use of the strongest form of the impredicative axiom scheme of Comprehension if it is to achieve its goals (e.g. the derivation of Frege’s theorem) but that this principle is on a par with outright stipulation of the Peano-Dedekind or ZF axioms in terms of epistemic innocence.

4. We turn now from the presuppositions the neo-logicist makes with respect to second-order logic to those made with respect to first-order. If one accepts that standard classical first-order logic is epistemically innocent then one already accepts that one can prove existence claims innocently, for one can prove ∃x(Fx ∨ ∼ Fx) or, in first-order logic plus identity, 29 ∃x(x = x). Even though this is not to prove very much about whatever it is which exists, the neo-logicist might take these standard theorems as revealing at least an ad hominem problem for those opponents who reject the idea of a priori or epistemically innocent existence proofs but nonetheless accept first-order logic. However, we conjecture that most contemporary logicians would respond by denying that standard non-free logic is epistemically innocent. The theorem ∃x(x = x) is harmless, they might say, since we know some things exist. And the quantifier rules ∀E and ∃I: ∀xϕx ϕx/t

ϕx/t ∃xϕx

are harmless if applied in a language in which one can reasonably assume or suppose that no singular term is empty, that is non-denoting. Making these assumptions certainly simplifies the proof theory and model theory. However, if one was interested in what could be derived by pure logic alone, or if one was working in a language containing complex singular terms which may not denote—for example, description terms or terms constructed using function terms standing for partial functions—then the above simple rules must be rejected in favour of more complex rules such as ∀xϕx, E(t) ϕx/t

ϕx/t, E(t) ∃xϕx

where E(t) represents some way of expressing the claim that t exists. Whether or not this would be the usual response of an “anti-Anselmian” logician, we submit it is the right response to make to the neo-logicist. Moreover Wright himself seems to presuppose some sort of free logic background at 29 From now on, include the standard theory of identity in “first-order logic”.

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least in the sense of allowing for empty domains. 30 Furthermore, to hold both that numerical terms such as nx(x = x) are genuine singular terms and that ∃I is unrestrictedly valid for all complex terms gives the neo-logicist a very easy victory but a Pyrrhic one since the key question is so obviously begged. Let us look, then, at Hume’s Principle and theorems of infinity against a free logic background. The restrictions on second-order ∀E and ∃I which blocked the proof of infinity in an unfree first-order background will, of course, a fortiori, do so in a free logic background. In order to isolate the problem with free logic, we grant the neo-logicist for these purposes standard second-order rules and axiom schemes, our sole amendment being altering first-order ∀E and ∃I as above, that is to ∀xϕx, E(t) ϕx/t

ϕx/t, E(t) ∃xϕx

Different policies on, and semantics for, “E” (it may, for example, be some complex formula containing t) will yield different frameworks for free logic. The problem for the neo-logicist is that in some of these the crucial proof of the infinity of the natural numbers is not forthcoming from Hume’s Principle. We cannot look at all possible free logics of course; rather we will look at one framework under which it is plausible that Hume’s Principle is epistemically innocent but under which Frege’s Theorem fails and at a rather different one on which the latter holds but it is very implausible that Hume’s Principle is epistemically innocent. It is then is up to the neo-logicist to refute our contention that these two approaches are representative in the sense that any free logic will fall on one or other horn of the dilemma. For the first example, we take the inner domain/outer domain framework— cf. Read Thinking about Logic (Oxford: Oxford University Press, 1994), pp. 134–7, 146–7. One divides the domain of individuals, of referents of individual constants, into two classes, the inner domain of “real” individuals, this domain being the range of the individual quantifiers, and an outer domain of “dummy” or “virtual” individuals, which do not belong in that range, so intuitively do not “exist”, though they can be assigned to nonvariable terms as referents. One can then adopt a number of policies for the value of an atomic sentence Pt in an interpretation in which t is assigned a merely virtual referent. One can always set such sentences as gappy, neither determinately true nor determinately false; or else always set them false, or (this option is rather unmotivated) always true, or let them vary freely as for sentences in which there is no failure of “real” reference. Predicate extensions then are assigned positive and negative extensions, each being divided into real and virtual components which together exhaust the particular extension. 30 Op. cit., pp. 235–6 where he writes, concerning the question whether the property λx(x = x) is too big, “But it will, presumably, be too big, however exactly that notion is defined, if the universe is empty”.

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Since we are working in a second-order framework, additional complications can arise if we want to mark failure of predicates to stand for properties. In a general treatment we would have domains of individuals, of properties and of n-ary relations. We then ask whether, e.g. the assignment to P in Pt of a merely virtual property induces truth-gaps or not; the semantics will proceed by assigning extensions: either of individual (positive and negative) satisfiers to properties or, dually, to each individual a set of properties which the individual has, and a negative set of properties which it lacks. To keep things as close as possible to the standard case, let us assume bivalence (so we can drop negative extensions) and confine the free logic component to the first-order case alone. So the domain of individuals D divides into exclusive and exhaustive subdomains I and O, the inner and outer domains; let us also take identity to be a primitive, its extension being all pairs α, α, for α in D, i.e. α virtual or real. So if terms t and u stand for the same individual, real or virtual, t = u is true, if they stand for distinct individuals, real or virtual, the sentence is false. This yields pretty much the standard theory of identity: t = t is valid for all t, Leibniz’s law—from ϕx/t and t = u conclude ϕx/u—is sound and so these two can be added as axiom scheme and inference rule respectively. This yields the standard theorems constraining identity to be an equivalence relation: ∀x x = x ∀x∀y(x = y → y = x) ∀x∀y∀z((x = y & y = z) → x = z). The main difference is that, since we are working in a free logic background, we cannot move from universal to existential generalisations as freely as before, i.e. cannot conclude ∃x x = x from ∀x x = x. As remarked, we are assuming the second-order quantifiers are interpreted as closely as possible to the standard way. Hence we can take the range of these quantifiers to be the power set of D thus allowing the “platonic” case of “properties” uninstantiated by a real individual. In fact, if the virtual domain is not empty we get a number of distinct empty properties even where we simply identify (which we do here, purely for convenience) properties with subsets of the domain of individuals; if v is in O, then Ø and {v} are two distinct properties both satisfying ∼ ∃yFy. Now in this framework we cannot prove, in any system sound for that semantics, ∃x(x = x), or indeed ∃x(ϕx) for any ϕ, since there are interpretations in which I is the empty set and all individuals are virtual (though we will be able to prove ∃F∀y ∼ Fy). This framework, then, does not build the neo-logicist assumption that there are a priori existence proofs, at least of the existence of individuals, into the underlying logic. Can we then show that by augmenting it with abstraction principles we can nonetheless prove, from

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epistemically innocent principles alone, that some things exist, indeed that an infinity of different numbers exist? The answer is no, for reasons analogous to the aristotelian case. There are finite, indeed empty models of Hume’s Principle in this framework, that is models in which I is finite or empty. For the latter case, take a “universally free” model in which D = O = {v} and I = Ø and so not even ∃x(x = x) true, far less a theorem of infinity is true. The Fregean proof of the infinity of the natural numbers breaks down right at the outset: we do not have ∃x(x= nx(x = x)). To see what happens to Hume’s Principle in this model, note that there are only two properties: {v} and Ø, both of them empty, that is both have empty real subextensions. Both properties, therefore, are equinumerous: with {v} assigned to F and Ø to G the formula which expresses the condition that a one:one correspondence holds between F and G, namely: ∃R∀x((Fx → ∃!y(Rxy & Gy))&(Gx → ∃ ! y(Ryx & Fy))) is true in the model. For remember that the quantifiers ∀x and the uniqueness quantifier ∃!y 31 on the right-hand side both range over the inner domain I, a domain which is empty in this case. Hence assigning to R any relation over D one likes satisfies the right-hand side of the biconditional. Every assignment to x trivially renders Fx → ∃!y(Rxy & Gy) true; for none renders it false, since there are no assignments from I to x; likewise for the second conditional. By assigning v as the number of each property with an empty real extension, thus to nxϕx, for every ϕ, we ensure the left-hand side is true so that the instance: nx F x = nx Gx ↔ ∃R∀x((F x → ∃ !y(Rx y & Gy)) &(Gx → ∃ !y(Ryx & F y))) of Hume’s Principle is true in the model; and the same holds for the other three possible assignments of pairs of properties to F and G. More generally, ensure there is at least one virtual individual and partition the properties of the universe, the subsets of D, into equivalence classes under equinumerosity (of the real extensions of the properties). If the inner domain is of size k, there will be k + 1 of these; 32 since D is of size at least k + 1, there is a function N which maps these equivalence classes into the total domain, inner plus outer. If we then interpret nxϕx via N, i.e. assign to it the value of N applied to the element of the partition to which the extension of ϕx belongs, then it is easy to see Hume’s Principle comes out as true in the model. Note that we could add as an additional requirement on admissible models that all simple singular terms have real referents. If there is some finite number n of simple singular terms, then will be able to construct finite models of Hume’s Principle in which standard ∀E and ∃I are sound where instantiation is for simple terms, finite models which can be of any size ≥ n. 31 i.e. ∃!yϕy is (∃yϕy & ∀x(ϕx → x = y)). 32 Since k can be infinite, we assume the Axiom of Choice in the metatheory.

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One might also note that Axiom V is satisfiable in this framework too; in fact, there are models of every cardinality of Axiom V and indeed of any abstraction principle. 33 This tolerance of arbitrary sets of abstraction principles affords no comfort to the neo-logicist since we still cannot prove any set, any non-proper class, exists. The outer domain of “virtual” objects, in other words, is best seen as a technical device. It gives us a semantics in which the free logic versions of first-order ∀E and ∃I are sound (with E(t) true just when t has a referent in the inner domain). And even if we were prepared to tolerate a baroque ontological slum in which virtual objects have some type of being, this will still not help the neo-logicist since there are models where the total domain D, including virtual individuals as well, is finite.

5. However, there are other frameworks for free logic than the inner/outer framework above. For instance, there are free logics in which the existence premiss E(t) is taken to be self-identity: t = t is true only if the term t refers. 34 This framework seems better suited to the neo-logicist position. The neologicist can hold that t = t entails ∃x x = t but deny that t = t is valid for every t. In some special cases, however, such an identity may be provable. For example, nx(x = x) = nx(x = x) is provable from the instance of Hume’s Principle in which we instantiate both predicate variables with x = x, for the right-hand side of this instance is a logical truth. Let us take E(t), then, to be t = t. But in order to prevent confusion with the identity relation in which t is identical with t is always true, we will retain “=” with the interpretation given above and expand our language to include in addition this second “existential” identity relation which we will express as “t ∼ = t”, evaluating such sentences as true in a model iff the referent of t belongs to the inner domain of the model. 35 The amendments to standard rules needed are then: ∀xϕx, t ∼ =t = t ϕx/t, t ∼ . ϕx/t ∃xϕx 33 Where I is of cardinality κ, let O be a disjoint set of cardinality 2κ . There is thus a one:one function B from the power set of I into D. Every property contains a real subextension (possibly empty). Interpret {x: ϕx} by a map which takes property P to B(X), where X ⊆ I is the real subextension of P. Given our interpretation of =, this is easily seen to satisfy Axiom V (for courses of values):

∀F∀G(({x : Fx} = {x : Gx}) ↔ ∀z(Fz ↔ Gz)). For, given any assignment of properties P1 and P2 to F and G, the left-hand side of the embedded biconditional is true iff B(rP1 ) = B(rP2 )—rPi being the real subextension of P—iff the real subextensions of P1 and P2 are identical iff (since ∀z ranges only over I), ∀z(Fx ↔ Gz) is true. The argument generalises to arbitrary abstraction principles. 34 See Dana Scott et al., Notes on the Formalization of Logic, Part II, Sub-Faculty of Philosophy Study Notes, University of Oxford. 35 Though when we leave wide identity = out of consideration, it does not matter whether we divide the individual domain into real and virtual components or simply let empty terms denote nothing at all.

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We can also lay down in the semantics that individual parameters can only be assigned real referents in I. 36 This ensures the soundness of standard ∀I and ∃E and also the logical truth of a ∼ = a,where a is any parameter (though t ∼ = ∼ t is not true in general). Since the = version of Leibniz’ law is also sound in this semantics—the premiss t ∼ = u can only be true when both t and u are nonempty and stand for the same referent—reflexivity, symmetry and transitivity of identity still hold in the generalised form above (i.e. ∀x x ∼ = x holds, even though t ∼ t may fail). = So we still have a fairly standard-looking theory of identity. The neo-logicist can then argue that the mere introduction of numerical and class terms alone does not prove that numbers and classes exist so that there is no questionbegging assumption from the outset that “cheap” proofs of existence theorems are available. However, it is still true, in this framework, that the identity relation is a one:one map from non-self-identity onto itself, reading identity as “wide” non-existential identity, i.e. (using λ abstracts for clarity here) (λx(x = x) 1 − 1 λx(x = x)) holds by dint of the relation λx(λy(x = y)). More fully this is: ∀x((x = x → ∃ !y(x = y & y = y)) & (x = x → ∃!y(y = x & y = y))). ∼ x) 1–1∗ λx ∼ (x = ∼ x)), where the ∗ indicates Indeed we also have λx ∼ (x = one:one functions expressed in terms of ∼ = rather than =. For although we do not have ⊢ t ∼ = t, for arbitrary t, we do have ⊢ a ∼ = a, where a is a parameter; hence we have both ⊢ a = a → ∃!y(a = y & y = y) and ⊢∼ (a ∼ =y&y∼ = y) 37 = a) → ∃ ! ∗ y(a ∼ and in general can prove anything from a = a (e.g. ∃!*y(a ∼ = y & y∼ = y)) ∼ and anything from ∼ (a = a) (e.g. ∃!y(a = y & y = y)) by ex contradictione quodlibet. Thus (λx ∼ (x ∼ = x)) also holds by dint of = x) 1 − 1∗ λx ∼ (x ∼ the relation λx(λy(x ∼ = y)). Now that we have two different notions of identity we have a number of different versions of Hume’s Principle, for instance HP∗ : ∀F∀G (nx F x ∼ = nx Gx ↔ ∃R∀x((F x → ∃!∗ y(Rx y & Gy)) &(Gx → ∃!∗ y(Ryx & F y)))) where ∃!∗ y(Rx y & Gy) is ∃y((Rx y & Gy) & ∀z((Rx z & Gz) → z ∼ = y)). 36 In models with empty I, atomic formulae containing parameters are all to be interpreted as true since there are no admissible assignments under which they are false. 37 With ∃!∗ ϕx abbreviating (∃yϕy & ∀x(ϕx → x ∼ = y)).

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Our original HP but with = interpreted in the wide non-existential fashion in our free logic setting is, for comparison. 38 ∀F∀G(nxFx = nxGx ↔ ∃R∀x((Fx → ∃!y(Rxy & Gy)) & (Gx → ∃!y(Ryx&Fy)))) ∃ !y(Rx y & Gy) being ∃y((Rxy & Gy) & ∀z((Rxz & Gz) → z = y)). Focusing on HP*, we can prove by instantiating F and G by λx(∼ (x ∼ = x)), (and using the Substitutivity Rule) the following: N x ∼ (x ∼ = x) ∼ = nx ∼ (x ∼ = x) ↔ ∃R∀x((∼ (x ∼ = x) → ∃ !∗ y(Rx y & ∼ (y ∼ = y))) & (∼ (x ∼ = x) → ∃ !∗ y(Ryx & ∼ (y ∼ = y)))). Since the right-hand side of the biconditional is, as we have seen, provable, we can deduce (nx ∼ (x ∼ = x) ∼ = nx ∼ (x ∼ = x)) (i.e. 0 ∼ = 0 with zero defined ∼ ∼ ∼ using =) and so prove ∃x(x = nx ∼ (x = x)) using our latest version of free ∃I where E(t) = df. T ∼ = t. And from here the Fregean proof of the infinity of the natural numbers can proceed as before. So HP∗ amounts to an axiom of infinity even in the context of free logic with existential identity. However, this is another place at which we must depart from our policy of not contesting the neo-logicist claim that abstraction principles such as Hume’s Principle are epistemically innocent. For the plea of innocence is more plausible with respect to HP than HP∗ . One key notion which neo-logicists have used to argue for the innocence of Hume’s Principle and against the idea that they beg any questions in teasing out ontological commitments from such abstraction principles is the idea of “reconceptualisation”. Taking as their text Grundlagen §64, the neo-logicists do not claim that in general one can generate objects from concepts; rather one shows how to “reconceptualise” some thoughts so as to generate new concepts which carve up the “state of affairs” represented in the thoughts in a different way (as involving an identity between directions rather than a parallelism between lines, for instance). More generally, the neo-logicist argues that one can “recarve” the concepts involved in thinking of a state of affairs in which an equivalence relation obtains in such a way as to generate new concepts, pertaining to abstracts. Moreover there is to be, using Frege’s directions example: 39 absolutely no gap between the existence of directions and the instantiation of properties and relations among lines. 38 And of course there are a number of other of versions of Hume’s Principle using permutations of = and ∼ =; the further variants produce no philosophically relevant new cases. 39 C. Wright, “On the philosophical significance of Frege’s theorem”, Language, thought, and logic, edited by Richard Heck, Jr., Oxford, Oxford University Press, 201–244.

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But what Wright must mean here is that there is absolutely no epistemic gap between the existence of directions (or the truth of identities between abstracts) and the instantiation of properties and relations among the original domain of objects. Reconceptualisation cannot do the job it is intended to do unless in addition to referring to a process of conceptual innovation or creation it also carries an epistemic punch. For suppose numbers do exist of necessity. Then “0 exists” and “0 ∼ = 0” will be semantically equivalent to (various readings of) “there is a one:one map from non-self-identity onto itself”. Indeed “0 exists” will be semantically equivalent to P → P , for any P. But the neo-logicist needs more than this, if neo-logicism is to answer the epistemological worries usually directed against platonism. The neo-logicist needs there to be an epistemic equivalence so that we can know that 0 ∼ = 0, know the truth of the left-hand side of the relevant instance of HP*, on the basis of our knowledge of the truth of the right-hand side (though we do not know it on the basis of our knowledge that P → P). So to say that we “reconceptualise” some “state of affairs” described in terms of relations among properties into one involving objects such as numbers is just to assert that we can know the objects exist on the basis of our knowledge of the relations among properties. It is not to show how we can know this. In the case of wide identities such as 0 = 0, the epistemic gap may indeed be small. But x = x is not, of course, an existence predicate and in knowing the truth of 0 = 0 we are not coming to know anything about the existence of objects. The identity t ∼ = t, on the other hand, has the sense or informational content of [t exists] or a sense very close to it: competent speakers, after all, are to use such identities as the existential premisses of the ∃I and ∀E rules. So in this case we are being asked to accept that the state of affairs consisting of there being a one:one mapping of the existential identity relation λx(x ∼ = x) onto ∼ nx(∼ x∼ x) itself can be reconceptualised into the thought [nx(∼ x ∼ = x)] = = in such a way that there is not even the slightest epistemic gap between the latter proposition, which is tantamount to “the number zero exists” and the former which is held to be a (second-order) logical truth. But this is to ask us to accept right at the outset that some epistemically innocent truths 40 are equivalent to existence claims. This is precisely the point on which we remain to be convinced:– it is no clearer how we can know zero exists on the basis of our knowledge of the one:one map on non-self-identity than we can know that it exists on the basis of our knowledge that if the moon is made of green cheese then it is made of green cheese. It may seem that there is a closer link between “0 ∼ = 0” and “there is a one:one map from non-self-identity onto itself” than there is between “0 ∼ = 0” 40 Let us for the sake of argument grant epistemic innocence to (λx ∼ (x ∼ x) 1 − 1∗ λx ∼ (x ∼ x)) = = though it is far from innocent on the aristotelian understanding of second-order logic.

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and P → P. The existence of zero is an “ingredient” of the state of affairs of the one:one map on the property, and not on the conditional state of affairs, if there is such a thing. Stripping away the metaphor, the claim of an especially close epistemic link between the two sides of the biconditional double instantiation of Hume’s Principle surely rests on the idea that the principle is analytic or in some sense constitutive of our notion of number. While this may be plausible for HP, applied to HP∗ it entails the claim that there is a purely conceptual link between a truth of (standard) second-order logic and an existence claim, i.e. it entails that one can get existence out of meanings alone, the very thing which has to be demonstrated not taken as a premiss. Far from there being no epistemic gap between the left and right sides of instances of HP∗ , there is a large chasm bridged, according to the neo-logicist, by “reconceptualisation”. But to anyone not already convinced that meaningconstitutive principles can generate mind-independent existence claims, this is a bridge too far.

6. Overall we have seen that the neo-logicist needs to show not only that some second-order principles are epistemically innocent, if the full neo-logicist programme is to be successfully accomplished; neo-logicism requires the full axiom scheme of Comprehension which, we argued, embodies substantive non-innocent ontological commitments (at least if the semantics for the pure mathematical sector is taken to be homogeneous with a broadly realist semantics for the non-mathematical sector). Moreover if the neo-logicist assumes the innocence of standard non-free first-order logic then he or she begs the question against opponents of neo-logicism. If not, then if identity does not have existential import, Frege’s Theorem fails whereas if it does have existential import, then Frege’s Theorem holds but the interpretation of the required abstraction principles, such as HP∗ , will beg the question in much the same way. Our conclusion, therefore, is that the neo-logicist has no non-questionbegging account of how there could be an epistemically innocent route to the demonstration of platonistically construed mathematical existence claims. 41

41 Thanks to the participants at the Abstraction Day conference, St. Andrews, Scotland, 14th November 1998 for discussion and to Michelle Friend for comments on an earlier draft.

ARISTOTELIAN LOGIC, AXIOMS, AND ABSTRACTION 1 Roy T. Cook

1.

Introduction

Neo-logicism is the view that various branches of mathematics can be reformulated in terms of abstraction principles that we can stipulate, and thus come to know the truth of, a priori. The main success story of neo-logicism so far is the derivation of arithmetic from Hume’s Principle: HP: (∀P)(∀Q)[Num(P) = Num(Q) ↔ P ≈ Q] where P ≈ Q is the second-order formula asserting that there is a one-toone correspondence between the P’s and the Q’s. Recently Stewart Shapiro and Alan Weir have criticized this view, arguing in ‘Neo-logicist logic is not epistemically innocent’ [2000] that abstraction principles do not provide us with a priori access to the objects necessary for mathematics: Frege’s theorem requires use of first- and second-order logical principles which are not epistemically innocent. More exactly, certain of the logical principles which are essential to the derivation of a theorem of infinity, when this is construed as expressing the existence of infinitely many mind-independent entities, are at least as problematic epistemologically as axioms of infinity laid down simply as postulates. Our supposed knowledge of these principles is, we will argue, every bit as mysterious as Kantian intuition of an infinity of numbers. (p. 162)

Shapiro and Weir suggest that someone skeptical of the strength of full secondorder logic could accept that: . . . it is a logical truth that to every . . . sentence which is instantiated by something or other, there corresponds a co-extensional property. But she refuses to accept that logic alone tells us that there are uninstantiated properties, so refuses to conclude that to predicates such as x = x there corresponds a property. (p. 165) 1 This paper first appeared in Philosophia Mathematica 11, [2003], pp. 195–202. Reprinted by kind permission of the editor and Oxford University Press.

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To avoid such problematic assumptions, Shapiro and Weir suggest that the neo-logicist abandon standard second-order logic with the full comprehension schema and replace it with the following more restricted version of comprehension, guaranteeing the existence of a property for every instantiated predicate ( any (possibly complex) predicate): AristComp : (∃x1 , x2 , . . . , xn )((x1 , x2 , . . . , xn )) → (∃R)(∀x1 , x2 , . . . , xn ) (R(x1 , x2 , . . . , xn ) ↔ (x1 , x2 , . . . , xn )). Shapiro and Weir call the resulting logic Aristotelian. For the neo-logicism, the important difference between Aristotelian logic and standard second-order logic is that one cannot derive the existence of infinitely many numbers from HP in Aristotelian logic: . . . the Fregean proof of infinity fails . . . because we cannot prove that there are n + 1 numbers less than or equal to n; for all the Aristotelian knows, zero happens to be identical to some number Sk 0 between zero and n. As we have seen, it is possible for 0 = 1 to hold in a model in the Aristotelian framework in which case the number of numbers less than or equal to one is just one. (p. 168)

Shapiro and Weir’s result generalizes to any abstraction principle as long as the equivalence relation on the right-hand side of the biconditional is formulated in purely logical terminology—any such neo-logicist abstraction principle (including Frege’s notorious Basic Law V) will have a one-element model. Given an abstraction principle AP: AP: (∀P)(∀Q)[@(P) = @(Q) ↔ E(P, Q)] We can construct an Aristotelian model of AP by letting the domain consist of a single object, call it a. @ is then the function that maps any property or predicate onto a. Since there is only one property, {a}, AP is trivially satisfied. Thus, on an Aristotelian conception of logic, every second-order abstraction principle is consistent, but no second-order abstraction principle will imply that there are infinitely many objects. Shapiro and Weir conclude from all this that: . . . the neo-logicist has no non-question-begging account of how there could be an epistemically innocent route to the demonstration of platonistically construed mathematical existence claims. (p. 188)

Just because the neo-logicist project does not meet one of its goals (or even perhaps its most important goal) does not mean that it meets none of its goals, however. Even on the Aristotelian conception of logic, neo-logicism does provide us with a great deal, as the following two case studies demonstrate.

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Case study 1: Hume’s Principle

The first step in deriving second-order arithmetic from HP is the definition of the finite (or ‘natural’) numbers, the constant ‘0’, the binary successor relation ‘S’, and the ternary relations addition ‘+’ and multiplication ‘×’. In standard second-order logic HP implies that the definitions of S, + , and × define total functions, i.e. given these definitions, all of the Peano axioms for arithmetic follow from HP (for details see Wright [1983]). On the Aristotelian conception of logic we do not get all of this, but even so HP implies a significant chunk of Peano arithmetic. The successor and addition relations need not be total, but we can prove that they are partial functions. Of the seven Peano axioms, four are not implied by HP trivially since they contain ‘0’, the name of the number of the empty property, which does not (or might not) exist on the Aristotelian conception of logic. 2 The other three do follow. Of course, just because 0 does not exist does not mean that no numbers exist. We can define the number 1 as: 3 1 =df Num(x is a finite number ∧ (∀y)(S(x, y) ↔ + (x, x, y))) Assuming that we do not countenance empty models, 4 the existence and uniqueness of 1 follow from HP in Aristotelian logic. As a result we can formulate alternatives to each of the problematic Peano axioms in terms of ‘1’. For example, the problematic successor axiom can be expressed as: 5 (∃x)(x = 1) ∧ (∃x)(∀y)(¬(Sy, x)) Along similar lines, we can replace the axioms for addition, multiplication, and induction containing ‘0’ with: (∀x)(∀y)(S(x, y) → + (y, 1, x)) (∀x)(×(x, 1, x)) (∀P)[((∀w)((∀z)(¬S(z, w)) → P(w))) ∧ (∀x)(∀y)(S(x, y) → (P(x) → P(y)))) → (∀x)P(x)] Each of these follows from HP on the Aristotelian conception of logic. Interestingly, on the Aristotelian picture of logic, although we cannot prove that there are infinitely many numbers, we do get a substantial description of the behavior of the numbers that do exist. There are two main differences 2 Actually, as Shapiro and Weir point out, 0 is in a sense guaranteed to exist, since ‘Num(x = x)’ is guaranteed to have a referent. If there is no property P such that (Px ↔ x = x), however, then HP does not apply to 0. As a result, 0 could be identical to any object, including any other number. Thus, the real problem here is that, in Aristotelian logic, 0 may be very badly behaved. 3 In this formula and those below the quantifiers should be understood to be restricted to finite numbers. 4 In later sections of their paper Shapiro and Weir challenge this very assumption, examining the prospects for neo-logicist arithmetic in a free logic. Here, however, our concern is with Aristotelian logic and its motivations, such as Boolos’ plural reading of second-order quantification. As a result, assuming that the domain is non-empty seems both unproblematic and, since there might be no property to pick out the supposed empty domain, entirely natural. 5 Note that the number without a predecessor need not be identical to 1.

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between the consequences of HP in standard second-order logic and its consequences on the Aristotelian conception. First, the relations of successor and addition can turn out to be partial functions on Aristotelian logic. Second, we were forced to reformulate some of the axioms so that they did not rely explicitly on the existence (or, more accurately, on the good behavior) of 0. The axioms we obtain in Aristotelian second-order logic, however, are relatively natural 6 axioms for the non-zero natural numbers, and are defective only in the sense that we are not assured that successor, addition, and multiplication are total functions.

3.

Case study 2: NewV

A second case study is useful to convince us that the phenomenon in question is quite general. In this section we will investigate the neo-logicist treatment of set theory from the perspective of Aristotelian logic, based on Boolos’ [1989] NewV: 7 NewV: (∀P)(∀Q)[Ext(P) = Ext(Q) ↔ ((P is ‘Big’ ∧ Q is ‘Big’) ∨ (∀x)(Px ↔ Qx))] where ‘P is ‘Big’ is an abbreviation for the second-order formula asserting that the P’s are equinumerous with the entire domain. In standard secondorder logic, with ‘is a set’ and ‘∈’ defined in the standard way (see Boolos [1989] for details), NewV entails the extensionality, empty set, pairing, union, separation, and replacement axioms, but not the powerset axiom or the axiom of infinity. On the Aristotelian account of second-order logical consequence, however, NewV implies extensionality, pairing, union, and replacement, but fails to imply empty set or separation. The fact that the empty-set axiom does not follow on the Aristotelian conception of logic is unsurprising, since the empty-set axiom is equivalent to the claim that there is a property that is not ‘Big’ and has no instances. The failure of the axiom of replacement seems more surprising until one realizes that it (plus the claim that some set exists) implies the empty-set axiom. A revised version of separation that does not imply the existence of an empty set does follow from NewV in Aristotelian logic: Arist.Separation: (∀P)(∀x)((x is a set ∧ (∃w)(w ∈ x ∧ Pw)) → (∃y)(y is a set ∧ (∀z)(z ∈ y ↔ (z ∈ x ∧ Pz)))) 6 Historically, of course, 0 was not accepted as a legitimate number by many groups that were otherwise mathematically quite sophisticated, including the Greeks. In addition, both Peano and Dedekind formulated their original arithmetical axioms with 1 as the initial number. Thus, there is some reason for thinking that the axioms that do follow from HP on Aristotelian logic capture the (or an) intuitive conception of the natural numbers. Thanks are owed to Fraser MacBride for pointing this out. 7 I am ignoring the well-documented problems with NewV (see Shapiro and Weir [1999] and Boolos [1989]). Even if NewV is an inadequate foundation for a neo-logicist theory of sets, it nevertheless provides us with another nice example of how abstraction principles behave in Aristotelian contexts.

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Thus, with suitable reformulations, the only axiom that does not follow from NewV on the Aristotelian conception of logic that does follow on the standard conception is the empty-set axiom. Again, we see a division between existential principles such as the empty-set axiom and principles that just govern the behavior of and interactions between sets without implying the existence of any particular sets. On the Aristotelian conception of logic NewV does not guarantee that any sets exist (since it has a one-element model) but it does guarantee that any sets that do exist behave exactly as we would expect them to behave, i.e. they satisfy (many of) the standard axioms of set theory.

4.

What does neo-Fregean abstraction actually give us?

There are two initial reactions that one might have to these results. The first, optimistic, reading was suggested at the beginning of this essay. Even if the Aristotelian challenge is correct, the neo-logicist can retrench, arguing that he has still given us something useful. First, even in the Aristotelian context, HP and NewV provide us with enough for many of the basic applications of arithmetic and set theory. In Aristotelian logic HP plus the claim that there are n distinct objects implies that there are (at least) n distinct numbers, and HP plus the claim that there is a non-numerical object implies that there are infinitely many finite numbers. 8 Similarly, NewV plus the claim that there are two distinct objects implies that there are infinitely many sets. Thus, adding HP or NewV to a suitably robust physical theory (i.e. one that contained one or more of these additional claims) would surely allow us to carry out much of the mathematics necessary for science. Second, and perhaps more importantly, abstraction principles such as HP and NewV imply strong constraints on the behavior of the concepts that they purport to define, even if they do not (on their own) imply the existence of any (or many) of the objects supposedly falling under the scope of these concepts. If the neo-logicists can still defend the claim that Hume’s Principle in some sense defines the concept of cardinal number (or NewV defines the concept of set), independent of any ontological implications, then they will have provided the philosophy of mathematics with something valuable. Replacing a collection of axioms haphazardly compiled over decades or centuries with a single principle that tells us exactly what a mathematical concept means and how the objects falling under it must behave is certainly a step in the right direction. Shapiro and Weir’s objections do not in any way affect this part of the neo-logicist project, since it is perfectly conceivable that we could provide a suitable account of the meaning of a concept without thereby judging one way or another what objects fall under this definition, if in fact any do. 8 None of this is of help to the neo-logicist, however, unless ‘there are n objects’ or ‘there is a nonnumerical object’ is knowable a priori.

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It is this last point, however, that brings us to the pessimistic reading. On this interpretation, neo-logicism was doomed from the start, since we should not expect any definition of a concept, even an implicit one, to imply the existence of infinitely many objects. George Boolos has expressed something like this worry: Despite the Godel incompleteness theorems and Russell’s protestations that the axiom of infinity was no logical truth, it was a central tenet of logical positivism that the truths of arithmetic were analytic. Positivism was dead by 1960 and the more traditional view, that analytic truths cannot entail the existence of either particular objects or of too many objects, has held sway since . . . it should be asked how a statement that cannot hold if there are only finitely many objects can possibly be thought to be analytic, a matter of meanings or ‘conceptual containment’. ([1997], pp. 249–250)

According to this line of thinking, what we should expect from our definitions, at best, is constraints on when the concept being defined is applicable. Thus, there must have been something wrong with the original formulation of neologicism, and the results of Shapiro and Weir’s paper (and of the present essay) finally show us exactly what that flaw was. The pessimistic reading, however, seems a bit harsh. If we accept from the beginning the idea that logic and definitions cannot have existential consequences, then neo-logicism is a non-starter. What the pessimist has got right, however, is emphasizing that, if neo-logicism is to be successful in explaining how we come to know of the existence of infinitely many mathematical objects from definitions and logic alone, then the neo-logicists owe us an explanation of the role of logic and, more specifically, a justification of their particular choice of logic. The results of Shapiro and Weir demonstrate that the choice of logic is more crucial than one might initially think. Neo-logicism follows not merely from the conjunction of views about the nature of definition and stipulation alone, but follows from these claims plus a view about what the correct account of logic is. Wright and Hale have explicitly pointed out the importance logic has to play in the neo-logicist project and the relative lack of attention it has received: The logicist theory about a particular mathematical theory is that its fundamental laws are obtainable on the basis just of definitions and logic. It would at the time of writing be a justifiable complaint that while much attention has been paid by neo-Fregeans, and their critics, to the first component in the recipe— issues to do with abstractions in general and Hume’s Principle in particular— comparatively little has been given to the second component: the demands, technical and philosophical, to be made on the logical system which is to provide the medium for the proofs the neo-Fregeans need. ([2001], p. 429, emphasis added)

Shapiro and Weir’s paper represents an important first step in fleshing out the second of these issues, the requirements on the logic of neo-logicism, and

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the present discussion has, it is hoped, helped to sharpen their insights even further. While they are right to point out that: . . . the neo-logicist has no non-question-begging account of how there could be an epistemically innocent route to the demonstration of platonistically construed mathematical existence claims ([2000], p. 188, emphasis added) 9

it would be premature to conclude that no such non-question-begging account is possible. Rather, their arguments serve to point out that work remains to be done, and the direction that this work needs to take. 10

References Boolos, G. [1989], “Iteration Again”, Philosophical Topics 17: 5–21, reprinted in Boolos [1998], pp. 88–104. Boolos, G. [1997], “Is Hume’s Principle Analytic?”, in Heck [1997]: 245–261, reprinted in Boolos [1998], pp. 301–314. Boolos, G. [1998], Logic, Logic, and Logic, Cambridge Mass, Harvard University Press. Heck, R. [1997], Language, Thought, and Logic, Oxford, Clarendon Press. Shapiro, S. and A. Weir [1999], “NewV, ZF, and Abstraction”, Philosophia Mathematica 7: 293–321. Shapiro, S. and A. Weir [2000], “Neo-logicist Logic Is Not Epistemically Innocent”, Philosophia Mathematica 8: 160–189. Wright, C. [1983], Frege’s Conception of Numbers as Objects, Scots Philosophical Monographs, vol. 2, Aberdeen, Aberdeen University Press. Wright, C. and R. Hale [2001], The Reason’s Proper Study, Oxford, Oxford University Press.

9 Actually, they are not quite correct here, since, even in the Aristotelian context, HP implies ‘(∃x)(x = 1)’, which is, one would think, a ‘platonistically construed mathematical existence claim’. 10 A version of this note was presented to members of Arché: The Centre for the Philosophy of Logic, Language, Mathematics, and Mind at the University of St Andrews and benefited considerably from the resulting discussion. Thanks are also owed to Peter Clark, Philip Ebert, Fraser MacBride, Graham Priest, Agustín Rayo, Stewart Shapiro, Crispin Wright, and an anonymous referee for helpful comments and criticism.

FREGE’S UNOFFICIAL ARITHMETIC 1 A. Rayo

In The Foundations of Arithmetic and The Basic Laws of Arithmetic, Frege held the view that number-terms refer to objects. 2 Later in his life, however, he seems to have been open to other possibilities: Since a statement of number based on counting contains an assertion about a concept, in a logically perfect language a sentence used to make such a statement must contain two parts, first a sign for the concept about which the statement is made, and secondly a sign for a second-order concept. These second-order concepts form a series and there is a rule in accordance with which, if one of these concepts is given, we can specify the next. But still we do not have in them the numbers of arithmetic; we do not have objects, but concepts. How can we get from these concepts to the numbers of arithmetic in a way that cannot be faulted? Or are there simply no numbers in arithmetic? Could the numbers help to form signs for these second-order concepts, and yet not be signs in their own right? 3

To illustrate Frege’s point, let us consider the number–statement ‘there are three cats’. It might be paraphrased in a first-order language as: 4 (∃3 x)[C AT(x)].

(1)

If its logical form is to be taken at face value, (1) can be divided into two main logical components: first, the predicate ‘C AT(. . . )’, which for Frege refers to the (first-order) concept cat; and, second, the quantifier-expression ‘(∃3 x)[ . . . (x)]’, which for Frege refers to a second-order concept (specifically, the second-order concept which is true of the first-order concepts under which

1 This paper first appeared in Journal of Symbolic Logic 67, [2002], pp. 1623–1638. Reprinted by kind

permission of the editor and the Association for Symbolic Logic. 2 This is reflected in his definition of number. See, for instance Frege (1884) §67. 3 Notes for Ludwig Darmstaedter, pp. 366–7. I have substituted ‘second-order’ for ‘second-level’. 4 As usual, ‘(∃ x)[φ (x)]’ is defined as ‘∃x(φ (x) ∧ ∀y(φ (y) → x = y))’, and (for n > 1) ‘(∃ x)[φ (x)]’ n 1 is defined as ‘∃x(φ (x)∧ (∃n−1 y)[φ (y)∧ y = x])’.

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precisely three objects fall). 5 Significantly, Frege would regard neither of these components as referring to an object. Let us now consider a close cousin of ‘there are three cats’, namely, ‘the number of the cats is three’. This sentence might be paraphrased as: the number of the cats = 3.

(2)

If its logical form is to be taken at face value, (2) cannot be divided into a predicate and a quantifier-expression, like (1). Instead, Frege would take ‘the number of the cats’ and ‘3’ to be names, referring to numbers (which he regarded as objects). Frege saw a deep connection between sentences like (1)—in which something is predicated of a concept—and sentences like (2)—in which something is predicated of the number associated with that concept. An effort to account for this connection was a main theme in his philosophy of arithmetic. But, after the discovery that Basic Law V leads to inconsistency, he found much reason for dissatisfaction with his original proposal. As evidenced by the quoted passage, he no longer felt confident about the possibility of getting from concepts to their numbers ‘in a way that cannot be faulted’. Towards the end of the passage, Frege considers an alternative: the view that there really are no numbers in arithmetic, and that—appearances to the contrary—numerals are not names of objects. They do not even instantiate a legitimate logical category, they are merely orthographic components of expressions standing for second-order concepts. The grammatical form of a sentence like (2) is therefore not indicative of its logical form. Presumably, ‘the number of the cats = 3’ is to be divided into two main logical components. First, the expression ‘. . . cats’, which refers to the (first-order) concept cat; and, second, the expression ‘the number of the . . . = 3’, which refers to a secondorder concept (specifically, the second-order concept which is true of the firstorder concepts under which precisely three objects fall). The numeral ‘3’ is merely an orthographic component of ‘the number of the . . . = 3’, in much the same way that ‘cat’ is an orthographic component of ‘caterpillar’. The outermost logical form of (2) is therefore identical to that of (1). If, in addition, it turns out that the logical form of ‘the number of the . . . = 3’ corresponds to that of ‘(∃3 x)[ . . . (x)]’, then the logical form of (1) is identical to that of (2). It is unfortunate that Frege never spelled out his unofficial proposal (as we shall call it) in any detail. In particular, he said nothing about how first-order arithmetic might be understood. Luckily, Harold Hodes has developed and defended a version of the Unofficial Proposal. 6 On Hodes’s reconstruction, a 5 For Frege, a first-order concept is a concept that takes objects as arguments, and an (n + 1)th-order concept is a concept that takes nth-order concepts as arguments. See Frege (1831903), §21. Unless otherwise noted, we shall use ‘concept’ to mean ‘first-order concept’. 6 See Hodes (1984). See also Wright (1983) pp. 36–40 and Bostock (1979), volume II chapter 1.

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sentence ‘F(n)’ of the language of first-order arithmetic is to be regarded as abbreviating a higher-order sentence ‘(FX )((∃n x)[Xx])’, where ‘(∃n x)[ . . . x]’ refers to a second-order concept, and ‘(FX )( . . . X . . . )’ refers to a third-order concept. For instance, the first-order sentence ‘PRIME(19)’ abbreviates a certain higher-order sentence ‘(PrimeX )((∃19 x)[Xx])’. On Hodes’s version of the Unofficial Proposal, quantified sentences involve quantification over second-order concepts. More specifically, they involve quantification over finite cardinality object-quantifiers: the referents of quantifier-expressions of the form ‘(∃n x)[ . . . x]’. 7 Thus, the first-order ‘∃zP RIME (z)’ would abbreviate the result of replacing the position occupied by ‘(∃19 x)[ . . . x]’ in ‘(PrimeX )((∃19 x)[Xx])’ by a variable ranging over finite cardinality object-quantifiers, and binding the new variable with an initial existential quantifier. Hodes’s account of first-order arithmetic therefore requires third-order quantification. And the obvious extension to nth-order arithmetic (for n ≥ 2) would call for (n + 2)th-order quantification. Such logical resources are increasingly problematic. 8 Here we shall see that more modest resources will do. We will develop a version of the Unofficial Proposal within a second-order language, and show that it can be used to account for nth order arithmetic (for any finite n). This, in itself, is a surprising result. But it is especially important in light of the fact that, although the use of higher-order languages is often considered problematic, recent work has done much to assuage concerns about certain second-order resources. 9 We will also see that the Unofficial Proposal has important applications in the philosophy of mathematics.

1.

Transformation

We will see that there is a general method for ‘nominalizing’ arithmetical formulas as second-order formulas containing no mathematical vocabulary. As an example, consider ‘The number of the cats is the number of the dogs’. This sentence might be nominalized as ‘The cats are just as many as the dogs’, or: ˆ OG(x)],10 x[C ˆ AT(x)] ≈ x[D where ‘≈’ expresses one–one correspondence. 11 7 See Hodes (1990) §3. 8 Hodes (1990), observation 5, offers a nominalization of second-order arithmetic which does not

exceed the resources of second-order logic. But it proceeds by encoding Ramsey sentences, and is therefore not a version of Frege’s Unofficial Proposal. 9 See Boolos (1984, 1985a, 1985b), McGee (2000), and Rayo and Yablo (2001). 10 Syntactically, an expression of the form ‘ x[φ ˆ (x)]’ takes the place of a monadic second-order variable. But the result of substituting ‘x[φ(x)]’ ˆ for ‘Y ’ in a formula ‘(Y )’ is to be understood as shorthand for: ∀W (∀x(W x ↔ φ (x)) → (W)). 11 That is, ‘X ≈ Y ’ abbreviates ∃R[∀w(Xw → ∃!v(Yv ∧ Rwv)) ∧ ∀w(Yw → ∃!v(Xv ∧ Rvw))]

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Consider now the sentence ‘the number of the cats is 3’. It can be nominalized as: ˆ AT(x]); 3 f (x[C where numeral-predicates are defined in the obvious way:

r 0 f (X ) ≡d f ∀ v¬X (v); r 1 f (X ) ≡d f ∃W ∃v(0 f (W ) ∧ ¬W (v) ∧ ∀ w(X (w) ↔ (W (w) ∨ w = v))); r 2 f (X ) ≡d f ∃W ∃v(1 f (W ) ∧ ¬W (v) ∧ ∀ w(X (w) ↔ (W (w) ∨ w = v))); r etc. This sort of nominalization can easily be generalized. In order to do so, we work within a two-sorted second-order language L containing the following variables: first-order arithmetical variables, ‘m 1 ’, ‘m 2 ’, . . . , monadic secondorder arithmetical variables ‘M1 ’, ‘M2 ’, . . . , first-order general variables, ‘x1 ’, ‘x2 ’, . . . , and, for n a positive integer, n-place second-order general variables X 1n , X 2 , . . . . 12 We assume that L has been enriched with a single higher-level predicate ‘N’ taking a monadic second-order general variable in its first argument-place and a first-order arithmetical variable in its second argument-place. 13 The well-formed formulas of L are defined in the usual way, with the proviso that an atomic formula can contain arithmetical variables only if it is of the form m i = m j , Mi m j or N(X i1 , m j ). 14 On the intended interpretation, arithmetical variables are taken to range over the natural numbers, and general variables are taken to have an unrestricted 12 As a precaution against variable clashes, we divide monadic second-order general variables in two: the 1 —which we abbreviate Z —will be paired with first-order arithmetical variables; the X 1 X 2i i 2i+1 — which we abbreviate X i —will be used for more general purposes. Also to avoid variable clashes, we 2 —which we abbreviate R —will be divide dyadic second-order general variables in two: the X 2i i 2 —which we abbreviate Ri2 —will be used paired with second-order arithmetical variables; the X 2i+1 3 — for more general purposes. Finally, we divide triadic second-order general variables in two: the X 2i 3 —which which we abbreviate Si —will be paired with third-order arithmetical variables; the X 2i+1 we abbreviate Ri3 —will be used for more general purposes. For n > 3, we use Rin as a terminological variant of X in . We will sometimes appeal to the introduction of unused variables. We employ ‘m’ as an unused first-order arithmetical variable, ‘w’, ‘v’, and ‘u’ as unused first-order general variables, ‘M’ as an unused second-order arithmetical variable, ‘W ’, ‘V ,’ and ‘U ’ as unused monadic second-order general variables, and, for each n > 1 (to be determined by context), we employ ‘R’ as an unused n-place secondorder general variable. (It is worth noting that appeal to unused variables could be avoided by renumbering subscripts.) It will often be convenient regard ‘x’, ‘y’, and ‘z’ as arbitrary first-order general variables and ‘X ’, ‘Y ’, and ‘Z ’ as arbitrary (monadic) second-order general variables. 13 For a discussion of higher-order predicates see Rayo, A. “Word and Objects.” Noûs 36, 436–464 (2002). 14 Formally, the well-formed formulas of L can be characterized as follows: (a) N(X 1 ,m ) and m = j i i m j are formulas; (b) for any n-place atomic predicate P other than ‘N’, P(xi 1 , . . . , xi n ) is a formula; (c) Mi m j and X in (x ji , . . . , x jn ) are formulas; (d) if φ and ψ are formulas, then ¬φ, (φ ∧ψ), ∃m i φ, ∃Mi φ, ∃xi φ, and ∃X in φ are formulas; and (e) nothing else is a formula.

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range. 15 In addition, ‘N(X i1 , m j )’ is true just in case the number of the X i1 is m j . Consider ‘The number of the cats is three’ as an example. It can be formalized in L as: ∃ m 1 (N(xˆ1 [C AT(x1 )], m 1 ) ∧ 3(m 1 ));

(3)

where, again, the number predicates are defined in the obvious way:

r 0(m) ≡d f ∃W (0 f (W ) ∧ N(W , m)); r 1(m) ≡d f ∃W (1 f (W ) ∧ N(W , m)); r 2(m) ≡d f ∃W (2 f (W ) ∧ N(W , m)); r etc. 16 Arithmetical predicates such as ‘S UCCESSOR’, ‘S UM’ and ‘P RODUCT’ can easily be defined in terms of ‘N’ and purely logical vocabulary. 17 So, without appealing to arithmetical primitives beyond ‘N’, the whole of pure and applied second-order arithmetic can be expressed within L. It will be convenient to introduce the following definitions, which are couched in purely logical vocabulary: Definition 1: F(X ) ≡d f ¬∃W (∃ w(¬Ww ∧ ∀ v(Xv ↔ (Wv ∨ v = w))) ∧ W ≈ X ) (there are at most finitely many Xs). Definition 2: ∃ f X φ (X ) ≡d f ∃ X (F(X ) ∧ φ(X )). 15 More precisely, first-order arithmetical variables are taken to range over the natural numbers, and first-order general variables are taken to have an unrestricted range. The range of the second-order variables is to be characterized accordingly. For instance, on a Fregean interpretation of second-order quantification, second-order arithmetical variables are taken to range over first-order concepts under which natural numbers fall, and second-order general variables are taken to range over first-order concepts under which arbitrary objects fall. 16 We use number-predicates rather than numerals for the sake of simplicity, but it is worth noting that our nominalization could be carried out even if L was extended to contain numerals. To see this, note that—using standard techniques—any formula φ of the extended language can be transformed into an equivalent formula φ ∗ of the original language in which numerals have been eliminated in favor of corresponding number-predicates (defined as above). One can then identify the nominalization of φ with that of φ ∗ . 17 The definitions run as follows:

S UCCESSOR(m i , m j ) ≡d f ∀V ∀U [(N(V , m i )∧ N(U , m j )) → ∃u(Uu ∧ wˆ [Uw ∧w = u] ≈ V )]; S UM(m i , m j , m k ) ≡d f ∀V ∀U ∀W [(N(V , m i )∧ N(U , m j ) ∧ N(W , m k ) ∧ ∀w(V w→ ¬Uw)) → w[V ˆ w ∨ Uw] ≈ W ]; P RODUCT(m i , m j , m k ) ≡d f ∀V ∀U ∀w[(N(V , m i )∧N(U , m j )∧N(W , m k )) → ∃R[∀v∀u((V v∧ Uu) → ∃!w(Ww ∧ Rvuw)) ∧ ∀w(Ww → ∃!v∃!u(V v∧ Uu ∧ Rvuw))]]:

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Our nominalization method can now be generalized to encompass the whole of first-order arithmetic by way of the following transformation: 18

r Tr(∃ m i (φ)) = ∃ f Z i ∧ Tr(φ); r Tr( m i = m j ) = Zi ≈Z j ; r Tr( N(Xi , m j )) = X i ≈ Z j . Intuitively, the transformation works by replacing talk of the number of the Fs by talk of the Fs themselves. As an example, let us return to ‘the number of the cats is three’. It can be formalized in L as: ∃ m 1 (N(xˆ1 [C AT(x1 )], m 1 ) ∧ 3(m 1 )); which Tr converts to: ∃ f Z 1 (xˆ1 [C AT(x1 )] ≈ Z 1 ∧ 3 f (Z 1 )); or, equivalently: 3 f (xˆ1 [C AT(x1 )]). For further illustration, note that ‘the number of the cats is the number of the dogs’ can be formalized in L as: ∃ m 1 [N(xˆ1 [C AT(x1 )], m 1 ) ∧ (N(xˆ1 [DOG(x1 )], m 1 )]. which Tr converts to: ∃ f Z 1 [xˆ1 [C AT(x1 )] ≈ Z 1 ∧ xˆ1 [DOG(x1 )] ≈ Z 1 ], or, equivalently: xˆ1 [C AT(x1 )] ≈ xˆ1 [DOG(x1 )]. It is worth emphasizing that mixed identity statements such as ‘m i = x j ’ are not well-formed formulas of L, so our transformation has not been defined for them. Intuitively, this means that the transformation is undefined for sentences along the lines of ‘The number 2 is Julius Caesar’, which do not express internal properties of a mathematical structure. We call such sentences Caesar sentences. This is as it should be. The view that numbers are objects led Frege to the uncomfortable question of whether the number belonging to the concept cat is, for instance, Julius Caesar. But in the context of our nominalizations, 18 The remaining clauses are trivial:

r r r r r r r r r

Tr(¬φ) = ‘¬’ ∧ Tr(φ); Tr(φ∧ψ) = ‘(’ ⌢ Tr(φ) ⌢ ‘⌢’ ⌢ Tr(ψ) ⌢ ‘)’; Tr(∃xi (φ)) = ∃xi ⌢ (Tr(φ)); Tr(∃X i (φ)) = ∃X i ∧ (Tr(φ)); Tr(X i x j ) = X i x j ; Tr(∃Rin (φ)) = ∃Rin ⌢(Tr(φ)); Tr(Rin (x j1 , . . . , x jn )) = Rin (x j1 , . . . , x jn ); Tr(xi = x j ) = xi = x j ; Tr(Pnj (xi 1 , . . . . , xi n )) = Pnj (xi 1 . . . , xi n ).

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such questions never arise, because number-terms do not refer to objects. ‘The number belonging to the concept cat is the number belonging to the concept dog’ is nominalized as ‘the objects falling under the concept cat are in one– one correspondence with the objects falling under the concept dog’, and ‘the number belonging to the concept cat is 3’ is nominalized as ‘there are three objects falling under the concept cat’. The question whether Julius Caesar is the number belonging to the concept cat isn’t only uncomfortable because it appears to be non-sensical. It also underscores a problem Paul Benacerraf made famous, that if mathematical terms refer to objects, then nothing in our mathematical practice determines which objects they refer to. 19 A remarkable feature of the Unofficial Proposal is that it avoids Benacerraf’s Problem altogether. It would, however, be a mistake to conclude from this that the Unofficial Proposal is the last word on Benacerraf’s Problem, since the inscrutability of reference pervades far beyond arithmetic.

2.

Second-order arithmetic

On the assumption that there are infinitely many objects in the range of the general variables of L, a certain kind of coding can be used extend Tr so that it encompasses second-order arithmetic (thanks here to . . . ). Intuitively, the coding works by representing each arithmetical concept Mi by a dyadic relation Ri . Specifically, we represent the fact that a number m j falls under Mi by having it be the case that some concept W under which precisely m j objects fall be such that some individual v bears Ri to all and only the individuals falling under W . 20 We implement the coding by enriching our transformation with the following two clauses: 21

r Tr(∃ Mi (φ)) = ∃ Ri ⌢ Tr(φ); r Tr( Mi m j ) = ∃ v(F(u[R ˆ i (v, u)]) ∧ Z j ≈ u[R ˆ i (v, u)]).

3.

Higher-order arithmetic

It is possible to express any (non-Caesar) formula in the language of n-th order arithmetic as a formula of L for which Tr is defined, provided that the range of the general variables contains at least גn−2 many objects. 19 See Benacerraf (1965). 20 We represent the fact that the number zero falls under M by having it be the case that some object i

bears Ri to nothing. Thus, in order to represent the fact that zero does not fall under Mi we must have it be the case that every object bears Ri either to n objects for some n > 0 falling under Mi , or to infinitely many objects. 21 Polyadic second-order quantification can be defined as monadic second-order quantification over sequences, which can be simulated within first-order arithmetic.

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Consider the case of third-order arithmetic. Intuitively, we proceed by pairing each second-order concept αi with a triadic relation Si in such a way that a set of numbers M j falls under αi just in case there is some object x with the following property: (*) For any number n, M j n holds just in case there is some object y such that there are exactly n vs. satisfying Si (x, y, v). 22 So that the ‘empty’ second-order concept (i.e. the second-order concept under which no first-order concept falls) may be represented, we let Si represent the fact that M j falls under αi only if there is an object x such that it is both the case that (*) is satisfied, and that there is no y such that Si (x, y, x). The ‘empty’ second-order concept can then be represented by any relation Si such that for every x there is some y such that Si (x, y, x). Formally, if ‘αi ’ is a monadic third-order variable restricted to the natural numbers, 23 we define a transformation C as follows: 24

r C(∃ αi φ ) = ∃ Si ⌢ C(φ); r C(αi (M j )) = ∃ x[∀ y(¬Si (x, y, x)) ∧ ∀ m(M j m ↔ ∃ y(N(v[S ˆ i (x, y, v)], m)))]

On the assumption that the range of the general variables contains least continuum many objects, it is easy to verify that, for any formula of third-order arithmetic, φ, on which C is defined, φ ↔ C(φ). By using n-adic relations instead of triadic ones, this procedure can be extended to n-th order arithmetic. And, on the assumption that the range of the general variables contains at least גn−2 objects, it will be the case that, for any formula of n-th order arithmetic, φ on which C is defined, φ ↔ C(φ).

4.

Numbering numbers

One would like to be able to number cats. But one would also like to be able to number numbers. One would like to say, for example, that the number of primes smaller than ten is four. And, unfortunately, an expression such as ‘N(mˆ i [P RIME-LESS - THAN-10(m i )], m j )’ is not well-formed formula of L because ‘N’ can only admit of a general variable in its first argument-place. 25 22 We represent the fact that the number zero falls under M by having it be the case that some object y j is such that there are no vs satisfying Si (x, y, v). Thus, in order to represent the fact that zero does not fall under M j we must have it be the case that every object y is either such that that there are n vs satisfying Si (x, y, v) for some n > 0 falling under M j , or such that there are infinitely many vs satisfying Si (x, y, v). 23 For instance, on a Fregean interpretation of third-order quantification, ‘α ’ ranges over second-order i concepts under which fall first-order concepts under which fall natural numbers. 24 The remaining clauses are trivial. 25 In analogy with the above, we let the result of substituting ‘m ˆ i [φ (m i )]’ for ‘M j ’ in a formula ‘ψ(M j )’ be shorthand for

∀M(∀m i (Mm i ↔ φ(m i )) → (M)).

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To remedy the situation, we may define a predicate ‘NN(Mi , m j )’, by appealing to the same sort of coding as before. Informally, ‘NN(Mi , m j )’ is to abbreviate a formula of L to the effect that there is a binary relation R with the following properties:

r for any number n, Mi n holds just in case some member of the domain of R is paired with exactly n+ 1 objects; 26

r every member of the domain of R is paired with finitely many objects; r for any x and y in the domain of R, if the objects paired with x are as many as the objects paired with y, then x = y;

r the domain of R contains exactly m j objects. 27 The new predicate allows us to say that the number of primes smaller than ten is four. It also allows us to say that the number of primes smaller than three is the number of objects falling under the concept cat: ∃ m 2 (NN(mˆ 1 [P RIME - LESS - THAN-6(m 1 )], m 2 ) ∧ N(xˆ1 [C AT(x1 )], m 2 )). 28 And, as desired, our any expression of the form NN(Mi , m j ) is definitionally equivalent to a well-formed formula of L.

5.

Formulas of L and their transformations

Our nominalization method is now complete. 29 Caesar sentences aside, any formula in the language of n-th order applied arithmetic can be expressed as a formula of L for which Tr is defined. And the result of applying Tr is always a formula with no mathematical vocabulary. We may now give a general characterization of the relationship between a formula and its transformation. In order to do so, consider the following five principles, all of which hold on the intended interpretation of L: 1. ∀ X (∃ m(N(X , m)) → ∃ !m(N(X , m))) (If m is a number of the Xs, then m is the number of the Xs.) 26 We require that a member of the domain of R be paired with n + 1 objects rather than n objects in order to accommodate the fact that the number zero might fall under Mi , since every member of the domain of R must be paired with at least one object. 27 More precisely, ‘NN(M , m )’ is to abbreviate: i j

∃R[∀m k (Mi m k ↔ ∃w∃W ∃u(Rwu ∧ ∀v(Wv ↔ (Rwv ∧ v = u)) ∧ N(W , m k ))) ∧ ∀w∀v(Rwv → ∃ f W ∀u(Wu ↔ Rwu)) ∧ ∀w∀v∀W ∀V ((∃u(Rwu) ∧ ∀u(Wu ↔Rwu) ∧ ∀u(V u↔Rvu) ∧ W ≈ V ) → w = v)∧ ∃W (∀v(Wv ↔ ∃u(Rvu)) ∧ N(W , m j )) ]; for m k an unused variable. 28 Whereas ‘C AT (. . . )’ may be regarded as an atomic predicate, ‘P RIME - LESS - THAN-6(. . . )’ abbreviates a complex formula constructed using the arithmetical predicates defined in Footnote 17. 29 So far we have only been concerned with the arithmetic of finite cardinals. But it is worth noting that a similar transformation could be applied to the language of infinite cardinal arithmetic.

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2. ∀ m∃ X N(X , m) (Given any number m, there are some objects such that m belongs to those objects.) 3. ∀ X (∃ m(N(X , m)) ↔ F(X )) (A number belongs to the Xs just in case they are at most finite in number.) 4. ∀ X ∀ Y [∀ m(N(X , m) → (Y , m)) ↔ X ≈ Y )]. (A number belonging to the Xs is also a number belonging to the Ys just in case the Xs are in one–one correspondence with the Ys.) 5. ∃ X ¬F(X ) (There are infinitely many things in the range of the general variables.)

Let Ꮽ be the conjunction of these five principles, and let φ T r be a notational variant for Tr(φ) . It is possible to show that, for any sentence φ of L, 30 Ꮽ ⊢ φ ↔ φT r

where ‘⊢’ expresses derivability in a standard second-order deductive system. In order to prove this result, a few preliminaries are necessary. ˆ Ri (v, u)], m))): Definition 3: N(Ri , m j ) ≡d f ∀ m(M j m ↔ ∃ v(N (u[ Definition 4: If mi1 , . . . , m ik , M j1 , . . . , M jl are arithmetical variables, we let ______________________ m i1 , . . . , m ik , M j1 , . . . , M jl abbreviate the following: (N(Z i1 , m i1 ) ∧ · · · ∧ N(Z ik , m ik )∧ N(R j , M j1 ) ∧ · · · ∧ N(R jl , M jl )). Definition 5: If φ is a formula of L, with free arithmetical variables mi1 , . . . , m ik , M j1 , . . . , M jl , we let φ ↔* φ T r abbreviate the universal closure of the following: ______________________ m i1 , . . . ,m ik ,M j1 , . . . ,M jl → (φ ↔ φ T r ). If φ contains no free arithmetical variables, we let φ ↔* φ T r be φ ↔ φ T r . Finally, we proceed to our main result: Theorem 1: If φ is a well-formed formula of L, then Ꮽ ⊢ φ ↔* φ T r. See appendix for proof. [An interesting feature of the proof is that the fifth conjunct of Ꮽ is required only to ensure the adequacy of the coding for second-order variables set forth in Section 2. In particular, the fifth conjunct is not required to prove a version of the theorem restricted to first-order 30 Here and in what follows I assume that, as a precaution against variable clashes, φ contains no variables for the form Z i , Ri or Si .

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arithmetic. On the other hand, without its fifth conjunct—or, alternatively, without a principle guaranteeing the existence of infinitely objects in the range of the arithmetical variables—the standard arithmetical axioms do not follow from Ꮽ.] Corollary 1: (Completeness of Ꮽ with respect to applied arithmetic) If φ is a sentence of L and T is the set of true sentences of L which do not contain ‘N’, then either A ∪T ⊢ φ or Ꮽ ∪T ⊢ ¬φ. Proof: Let φ be a sentence of L. It is easy to verify that φ T r does not contain ‘N’. Therefore, either T ⊢ φ T r or T ⊢ ¬φ T r , since either φ T r ∈ T or ¬φ T r ∈ T . But, since φ contains no free variables, it follows from our theorem that Ꮽ ⊢φ ↔ φ T r . So, either Ꮽ ∪ T ⊢ φ or Ꮽ ∪ T ⊢ ¬φ. Corollary 2: Suppose Ꮽ holds when ‘N(X , m)’ is interpreted as ‘the number of the Xs is m’. Let φ (m i ) be a well-formed formula of L, and let ψ(Zi ) be Tr(φ (m i )). If there are at most finitely many Fs, then φ(m i ) is true of the number of the Fs just in case ψ(Z i ) is true of the Fs. 31 Proof: Immediate from theorem.

6.

Interpreting second-order languages

We have taken care to ensure that the outputs of our transformation are always second-order formulas. So an interpretation for second-order quantifiers is all we need to make sense of our nominalizations. Frege took secondorder quantifiers to range over concepts, but Fregean concepts might be considered problematic on the grounds that they constitute ‘items’ which are not objects. Not any alternative will do. On Quine’s interpretation, second-order logic is ‘set-theory in sheep’s clothing’. So we would have succeeded in eliminating number-terms from arithmetic only by making use of set-terms. And, from the perspective of the Unofficial Proposal, set-terms are presumably no less problematic than number-terms. Nor is any progress made by interpreting second-order logic as Boolos has suggested. 32 Some of our definitions make essential use of polyadic second-order quantifiers, which Boolos treats as ranging (plurally) over ordered n-tuples. And, again, from the perspective of the Unofficial Proposal, ordered-pair-terms are presumably no less problematic than numbers-terms. 31 In fact, the result is slightly more general. Suppose φ (m , . . . , m ) is a formula of L and let ψ(Z , i1 in i1 . . . , Z i n ) be Tr(φ (m i 1 , . . . , m i n )); suppose, moreover, that there are at most finitely many F1 s, at most finitely many F2 s, . . . , and at most finitely many Fn s. Then φ(m i 1 , . . . , m i n ) is true when m i 1 is the number of the F1 s, m i 2 is the number of the F2 s, . . . , and m i n is the number of the Fn s just in case ψ(Z i 1 , . . . , Z i n ) is true when the Z i 1 s are the F1 s, the Z i 2 s are the F2 s, . . . , and the Z i n s are the Fn s. 32 See Boolos (1984) and Boolos (1985a).

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Some deviousness is needed to avoid Fregean concepts without betraying the spirit of the Unofficial Proposal. One way of doing so is by defining second-order quantifiers implicitly, in terms of an open-ended schema, as in McGee’s ‘Everything’. Another is by interpreting second-order logic as in Rayo and Yablo’s ‘Nominalism through De-Nominalization’. Alternatively, one might argue that genuine second-order quantification is to be accepted as a primitive.

7.

Applications

Frege’s Unofficial Proposal—the view that number–statements are to be eliminated in favor of their transformations—can take several different forms, depending on the sort of elimination one has in mind. On an approach like Hodes’s, number–statements are taken to abbreviate their transformations. As a result, number-terms do not refer to objects, and there is room for rejecting the existence of numbers altogether. The Unofficial Proposal might therefore provide a basis for a nominalist philosophy of arithmetic. It should be noted, however, that unless the universe is infinite, φ T r will not always have the truth-value that φ receives on its standard interpretation. In order to avoid infinity assumptions, a nominalist might claim that a number– statement φ abbreviates ‘necessarily, (ξ → φ T r )’, where ‘ξ ’ is a sentence stating that there are infinitely many objects, such as ‘∃ X ¬F(X )’. On the plausible condition that it is possible for the universe to infinite, ‘necessarily, (ξ →φ T r )’ is true if and only if φ is true on its standard interpretation. 33 A different approach towards the Unofficial Proposal might serve the purposes of the Neo-Fregean Program, championed by Bob Hale and Crispin Wright. Neo-Fregeans believe that Hume’s Principle allows us to reconceptualize the state of affairs which is described by saying that the Fs are as many as the Gs, and that, on the reconceptualization, that same state of affairs is rightly described by saying that the number of the Fs is the number of the Gs. 34 A version of the Unofficial Proposal might allow Neo-Fregeans to make the more general claim that every number–statement φ describes—on the appropriate reconceptualization—the state of affairs which is otherwise described by φ T r . Even if the Unofficial Proposal is to be abandoned altogether, it would be a mistake to neglect the connection between number–statements and their transformations described in Section 5. For non-nominalist accounts of mathematics must yield the result that there is no special mystery about how one might come to know what the truth-values of mathematical sentences are. But, on the assumption that A can be known to be true, our theorem ensures 33 For more on modal strategies, see part II of Burgess and Rosen (1997). Hodes discusses a modal strategy in Section III of Hodes (1984). 34 See Wright (1997), Section I, and Hale (1997).

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that this goal can be achieved for the case of pure and applied arithmetic. Let φ be an arithmetical sentence of L. When Ꮽ is known, it follows from our theorem that one is in a position to derive φ ↔ φ T r . So, insofar as one is in a position to know the truth of φ T r , which contains no arithmetical vocabulary, one is also in a position to know the truth of φ. 35 (Of course, one may not be in a position to know the truth of φ T r . In that case one is not, for all that has been said, in a position to know φ. But that cannot be used as an objection against a non-nominalist account of mathematical knowledge. Such an account is required to show that mathematical knowledge is no more mysterious than non-mathematical knowledge, not that all knowledge is unproblematic.)

8.

Logicism

Our theorem provides us with a partial vindication of Logicism. For whenever φ is a sentence of pure arithmetic (appropriately expressed in L), φ T r is a sentence of pure second-order logic. Moreover, Tr allows us to express formulas of pure arithmetic as formulas of pure second-order logic in a way which preserves compositionality. 36 This would constitute a complete vindication of Logicism if it were true as a matter of pure logic that, for every appropriate φ, Tr(φ) has the truth-value that φ would receive on its standard interpretation. Unfortunately, the general equivalence in truth-value holds only if the universe is big enough, and the size of the universe is not a matter of pure logic. Tr doesn’t reduce arithmetic to logic—but it comes close.

Appendix The theorem is proved by induction on the complexity of φ. Trivial cases are omitted.

r Assume φ = N(X i , m j ). Then φ ↔* φ T r is the universal closure of N(Z j , m j ) → (N(X i , m j ) ↔ X i ≈ Z j ), which is an immediate consequence of Ꮽ (first and fourth conjuncts). 35 For a more detailed treatment of this issue see Rayo, A. (2004) “ Frege’s Correlation.” Analysis 64, 119–122. It is worth noting that the completeness of the second-order Dedekind-Peano axioms yields a similar result for the case of pure second-order arithmetic, and that the quasi-categoricity result in McGee (1997) yields a similar result for the case of pure set-theory. 36 Unlike nominalization in terms of Ramsey sentences, Tr respects the logical connectives and quantifiers:

r r r r

Tr(¬φ) = ‘¬’ ⌢ Tr(φ), Tr(φ ∧ ψ) = Tr(φ) ⌢‘∧’ ⌢ Tr (ψ), Tr(∃m i φ) = ∃ f Zi ⌢ Tr(φ), Tr(∃Mi φ) = ∃Ri ⌢ Tr(φ).

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r Assume φ = m i = m j . Then φ ↔* φ T r is the universal closure of (N(Z i , m i ) ∧ N(Z j , m j )) → (m i = m j ↔ Z i ≈ Z j ), which is an immediate consequence of Ꮽ (first and fourth conjuncts).

r Assume φ = M j m i . Then φ ↔* φ T r is the universal closure of m i , M j → (M j m i ↔ ∃ v(F(u[R ˆ j (v, u)]) ∧ Z i ≈ u[R ˆ j (v, u])).

We make the following two assumptions: N(Z i , m i ),

(1)

N(R j , M j ).

(2)

Recall that (2) is shorthand for ˆ j (v, u)], m))), ∀ m(M j m ↔ ∃ v(N(u[R

(3)

from which it follows immediately that (M j m i ↔ ∃ v(N(u[R ˆ j (v, u)], m i ))).

(4)

From (1) and (4), together with Ꮽ (first and fourth conjuncts), it follows that M j m i ↔ ∃ v(Z i .u[R ˆ j (v, u)]),

(5)

And from (1) and (5), together with Ꮽ (first, third and fourth conjuncts), it follows that M j m i ↔ ∃ v(F(u[R ˆ j (v, u)]) ∧ Z i ≈ u[R ˆ j (v, u)]) :

(6)

Discharging assumptions (1) and (2) we get: m i , M j → (M j m i ↔ ∃ v(F(u[R ˆ j (v, u)]) ∧ Z i ≈ u[R ˆ j (v, u)])).

(7)

And the desired result follows from (7) by universal generalization.

r Assume φ = ∃ m i ψ(m i ). Let ψ have free arithmetical variables m i , . . . , 1 m ik ,M j1 , . . . , M jl distinct from m i . 37 Then φ ↔* φ T r is the universal closure of: m i1 , . . ., m ik , M j1 , . . ., M jl → (∃ m i ψ(m i ) ↔ ∃ f Z i ψ T r (Z i )).

By inductive hypothesis, the following is provable from HP: m i , m i1 , . . . , m ik , M j1 , . . . , M jl → (ψ(m i ) ↔ ψ T r (Z i )).

(1)

We make the following two assumptions: m i1 , . . . , m ik , M j1 , . . . , M jl ,

(2)

∃ m i ψ(m i ).

(3)

37 The case where ψ has no free arithmetical variables distinct from m , and the case where ψ does not i contain m i free require trivial differences in terminology. We ignore them for the sake of brevity.

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By Ꮽ (second and third conjuncts), it follows from (3) that ∃ m i ∃ W (F(W ) ∧ N(W, m i ) ∧ ψ(m i )).

(4)

So, by existential instantiation, F(C) ∧ N(C, c) ∧ ψ(c).

(5)

c, m i1 , . . ., m ik , M j1 , . . ., M jl → (ψ(c) ↔ ψ T r (C)).

(6)

But by (1) we have: And from (2), (5), and (6) we may conclude ψ T r (C).

(7)

∃ f Z i ψ T r (Z i ),

(8)

Thus, making again use of (5), and, discharging assumption (3), ∃ m i ψ(m i ) → ∃ f Z i ψ T r (Z i ).

(9)

∃ f Z i ψ T r (Z i ).

(10)

F(C) ∧ ψ T r (C).

(11)

Conversely, assume By existential instantiation: It is a consequence of (10) and Ꮽ (third conjunct) that ∃ mN(C, m).

(12)

From (12) we obtain the following, by existential instantiation: N(C, c).

(13)

c, m i1 , . . ., m ik , M j1 , . . ., M jl → (ψ(c) ↔ ψ T r (C)).

(14)

But by (1) we have: And from (2), the second conjunct of (11), (13), and (14) we may conclude ψ(c).

(15)

∃ m i ψ(m i ),

(16)

Thus, and, discharging assumption (10), ∃ f Z i ψ T r (Z i ) → ∃ m i ψ(m i ).

(17)

Finally, we combine (9) and (17), and discharge assumption (2): m i1 , . . . , m ik , M j1 , . . ., M jl → (∃ n m i ψ(m i ) ↔ ∃ f Z i ψ T r (Z i )). (18) The desired result is then obtained by universal generalization.

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r Assume φ = ∃ M j ψ (M j ). Let ψ have free arithmetical variables m i , . . . , 1 m ik , M j1 , . . . , M jl distinct from M j . 38 Then φ ↔* φ T r is the universal closure of: m i1 , . . ., m ik , M j1 , . . ., M jl → (∃ M j ψ(M j ) ↔ ∃ R j ψ T r (R j )).

By inductive hypothesis, the following is provable from HP: M j , m i1 , . . ., m ik , M j1 , . . ., M jl → (ψ(M j ) ↔ ψ T r (R j )).

(1)

We make the following two assumptions: m i1 , . . ., m ik , M j1 , . . ., M jl ,

(2)

∃ M j ψ(M j ).

(3)

By Ꮽ (second, third and fifth conjuncts), it follows from (3) that ∃ M j (∃ R(N(R, M j )) ∧ ψ T r (M j )).

(4)

So, by existential instantiation, N(P, C) ∧ ψ(C).

(5)

C, m i1 , . . ., m ik , M j1 , . . ., M jl → (ψ(C) ↔ ψ T r (P)).

(6)

But by (1) we have: And from (2), (5) and (6) we may conclude ψ T r (P).

(7)

∃ R j ψ T r (R j ),

(8)

Thus, and, discharging assumption (3), ∃ M j ψ T r (M j ) → ∃ R j ψ T r (R j ).

(9)

∃ R j ψ T r (R j ).

(10)

ψ T r (P).

(11)

∃ M∀ m(Mm ↔ ∃ v(N(u[P(v, ˆ u)], m))).

(12)

Conversely, assume By existential instantiation, The following is a logical truth: But (12) is definitionally equivalent to ∃ MN(P, M).

(13)

N(P, C).

(14)

So, by existential instantiation,

38 The case where ψ has no free arithmetical variables distinct from M , and the case where ψ does not j contain M j free require trivial differences in terminology. We ignore them for the sake of brevity.

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But by (1) we have: C, m i1 , . . ., m ik , M j1 , . . ., M jl → (ψ(C) ↔ ψ T r (P))

(15)

And from (2), (11), (14), and (15) we may conclude ψ(C).

(16)

∃ M j ψ(M j ),

(17)

Thus, and, discharging assumption (10), ∃ R j ψ T r (R j ) → ∃ M j ψ(M j ).

(18)

Finally, we combine (10) and (18), and discharge assumption (2): m i1 , . . ., m ik , M j1 , . . ., M jl → (∃ M j ψ(M j ) ↔ ∃ R j ψ T r (R j )).

(19)

The desired result is then obtained by universal generalization.

References Beaney, M., ed. (1997) The Frege Reader, Blackwell, Oxford. Benacerraf, P. (1965) “What Numbers Could not Be,” The Philosophical Review 74, 47–73. Reprinted in Paul Benacerraf and Hilary Putnam, Philosophy of Mathematics. Benacerraf, P., and H. Putnam, eds. (1983) Philosophy of Mathematics, Cambridge University Press, Cambridge, second edition. Boolos, G. (1984) “To Be is to Be a Value of a Variable (or to be Some Values of Some Variables),” The Journal of Philosophy 81, 430–49. Reprinted in George Boolos, Logic, Logic and Logic. Boolos, G. (1985a) “Nominalist Platonism,” Philosophical Review 94, 327–44. Reprinted in George Boolos, Logic, Logic and Logic. Boolos, G. (1985b) “Reading the Begriffsschrift,” Mind 94, 331–34. Reprinted in George Boolos, Logic, Logic and Logic. Boolos, G. (1998) Logic, Logic and Logic, Harvard, Cambridge, Massachusetts. Bostock, D. (1979) Logic and Arithmetic, Clarendon Press, Oxford. Burgess, J., and G. Rosen (1997) A Subject With No Object, Oxford University Press, New York. Frege, G. (1884) Die Grundlagen der Arithmetik. English Translation by J.L. Austin, The Foundations of Arithmetic, Northwestern University Press, Evanston, IL, 1980. Frege, G. (1893/1903) Grundgesetze der Arithmetik. Vol. 1 (1893), Vol. 2 (1903). English Translation by Montgomery Furth, The Basic Laws of Arithmetic, University of California Press, Berkeley and Los Angeles, 1964. Frege, G. (1919) “Notes for Ludwig Darmstaedter.” Reprinted in Michael Beaney, The Frege Reader. Hale, B. (1997) “Grundlagen x64,” Proceedings of the Aristotelian Society 97, 243–61. Reprinted in Bob Hale and Crispin Wright, The Reason’s Proper Study. Hale, B., and C. Wright (2001) The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics, Clarendon Press. Heck, R., ed. (1997) Language, Thought and Logic, Clarendon Press, Oxford. Hodes, H. T. (1984) “Logicism and the Ontological Commitments of Arithmetic,” Journal of Philosophy 81:3, 123–49. Hodes, H. T. (1990) “Where do Natural Numbers Come From?” Synthese 84, 347–407. McGee, V. (1997) “How We Learn Mathematical Language,” Philosophical Review 106:1, 35– 68. McGee, V. (2000) “Everything.” In Gila Sher and Richard Tieszen, Between Logic and Intuition.

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Rayo, A. (2002) “Word and Objects.” Noûs 36, 436–64. Rayo, A. (2004) “Frege’s Correlation.” Analysis 64, 119–22. Rayo, A., and S. Yablo (2001) “Nominalism Through De-Nominalization,” Noûs 35:1. Sher, G., and R. Tieszen, eds. (2000) Between Logic and Intuition, Cambridge University Press, New York and Cambridge. Wright, C. (1983) Frege’s Conception of Numbers as Objects, Aberdeen University Press, Aberdeen. Wright, C. (1997) “The Significance of Frege’s Theorem.” In Richard Heck, Language Thought and Logic.

REALS BY ABSTRACTION 1 Bob Hale

1.

General aim and basic ideas

1.1

Abstraction

A Fregean abstraction principle is now usually taken to be a principle of the general form: ∀α∀β(§α = §β ↔ α ≈ β) where ≈ is an equivalence relation on entities denoted by expressions of the type of α and β and § is an operator which forms singular terms when applied to constant expressions of the same type. The most prominent examples in Frege’s own writings are the Direction equivalence: the direction of line a = the direction of line b iff lines a and b are parallel together with what is now often called Hume’s principle: the number of Fs = the number of Gs iff the Fs and the Gs are 1–1 correlated and his ill-fated Basic Law V: the extension of F = the extension of G iff F and G are co-extensive In general, an abstraction principle seeks to give necessary and sufficient conditions for the identity of objects mentioned on its left-hand side in terms of the holding of a suitable equivalence relation between entities of some other sort. The Direction equivalence is a first-order abstraction, because its equivalence relation is a first-level relation on objects, whereas Hume’s principle and Basic Law V are second-order, their equivalence relations being second-level relations on concepts.

1.2

Frege’s logicism

Frege discusses at Grundlagen §§60–67 the suggestion that number might be contextually defined by means of Hume’s principle, but rejects it because 1 This paper first appeared in Philosophia Mathematica 8, [2000], pp. 100–123. Reprinted by kind permission of the editor and Oxford University Press.

175 Roy T. Cook (ed.), The Arché Papers on the Mathematics of Abstraction, 175–196. c 2007 Springer.

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he can see no way to solve what is now called the Caesar problem. The problem is that while Hume’s principle provides the means to settle, at least in principle, the truth-values of identity-statements linking terms for numbers when those terms are of the form ‘the number of Fs’(or definitional abbreviations of such terms), it appears not to enable us to answer questions of numerical identity, when one of the terms is not of that form, such as whether the number of Jupiter’s moons = Julius Caesar. Frege then immediately switches to his well-known explicit definition of number in terms of extensions (or classes): the number of Fs = the class of concepts 1–1 correlated with F. This requires him to provide a theory of extensions or classes, which he does by means of Basic Law V. As is well known, Basic Law V is inconsistent. Frege’s own attempt to arrive at a restricted axiom on classes which is both consistent and able to serve in its place as the basis for his hoped-for derivation of arithmetic from logic was unsuccessful and he eventually abandoned his belief that arithmetic could be provided with a purely logical foundation. Further, whilst we now know—or at least think we know—how to formulate a consistent theory of sets, this affords no comfort to anyone in sympathy with Frege’s logicist project, for two reasons. One is that this theory—Zermelo–Fraenkel set theory, say—is not plausibly viewed as a purely logical theory, owing to the very substantial existence assumptions it involves. The other is that Frege’s definition of number cannot be consistently embedded in the theory, because the objects with which it identifies cardinal numbers are too big to be treated as sets.

1.3

Neo-Fregean logicism

As far as elementary arithmetic goes, Frege’s only indispensable appeal, in Grundlagen and in Grundgesetze 2 to his explicit definition of number (and thence to Basic Law V) is in proving Hume’s principle from it. That is, once Hume’s principle has been established as a theorem, no further appeal need be made, either to the explicit definition or to Basic Law V, in deriving as theorems what are, near enough, the Dedekind–Peano axioms for arithmetic. These include, crucially, the axiom asserting that every natural number has another natural number as its successor, which amounts (in the presence of the others) to the assertion that there are infinitely many natural numbers. This fact is now, following a suggestion of the late George Boolos, 3 referred to as Frege’s Theorem. What Frege’s Theorem asserts, in effect, is that if Hume’s principle is added to a standard formulation of second-order logic as a further axiom, the resulting system suffices for the derivation of elementary 2 As far as Grundlagen goes, this is quite clear from a reading of §§68–83 and is emphasised by Crispin Wright in his Frege’s Conception of Numbers as Objects (Aberdeen University Press 1983). That the same is true of Grundgesetze is shown by Richard G. Heck, Jr., in “The Development of Arithmetic in Frege’s Grundgesetze der Arithmetik” The Journal of Symbolic Logic 58 (1993), pp. 579–601. 3 cf. “The Standard of Equality of Numbers” in George Boolos (ed.) Meaning and Method: Essays in Honor of Hilary Putnam (Cambridge University Press 1990), pp. 261–77.

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arithmetic. It is known that this system is consistent—or at least, that it is so, if second-order arithmetic is. Whether this fact supports any kind of logicism about arithmetic depends, of course, on the status of Hume’s principle. Boolos, along with many others, denies—plausibly, in my view—that it can be regarded as a truth of logic. Further, Hume’s principle cannot be taken as a definition, in any strict sense, because it does not permit the elimination of numerical terms in all contexts. This does not settle the issue, however, since it may be claimed that the principle is analytic, or a conceptual truth, in some sense broader than: either a truth of logic or reducible to one by means of definitions. That it can be so regarded is the view—now often called neo-Fregean logicism—of Crispin Wright and myself. 4 I do not intend, here, to defend this view of arithmetic against the many objections to our claim that Hume’s principle is a conceptual truth about numbers. Nor shall I offer a solution to the Julius Caesar problem 5 —though this must be (and we believe can be) done, if our view is to be viable. Nor, finally, shall I offer a general philosophical defence of the idea—which is again central to our view—that abstraction principles (provided they are consistent and perhaps meet certain other constraints) provide a legitimate means of introducing concepts of various kinds of abstract object in such a way that the existence of those objects depends only upon there being true instances of their right-hand sides. 6 Instead, what I want to do is explain one way in which I think it may be possible to extend our view beyond elementary arithmetic, to encompass the theory of real numbers. I say ‘one way’ because there are, on the face of it, several different ways in which one might try to do this.

1.4

Reals via Fregean set theory

In some ways, the most obvious approach—the one which has probably received most attention in recent work 7 —is a set-theoretic one. This would involve formulating a consistent Fregean axiom for sets to replace Basic Law V—an axiom which could form the basis of a theory of sets powerful enough 4 cf. Wright Frege’s Conception . . . , “The Philosophical Significance of Frege’s Theorem” in Richard G. Heck, Jr. (ed.) Language, Thought, and Logic: Essays in Honour of Michael Dummett (Oxford 1997) and “Is Hume’s principle analytic?” in Bob Hale & Crispin Wright The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics, Oxford: Clarendon Press 2001; Hale Abstract Objects (Blackwell 1987), “Dummett’s critique of Wright’s attempt to resuscitate Frege” Philosophia Mathematica (3) Vol. 2 (1994), pp. 122–47 and “Grundlagen §64” Aristotelian Society Proceedings 1997, pp. 243–62. For Boolos’s opposed view, see “Is Hume’s principle analytic?” in Heck (ed) op cit. 5 qv works cited in fn 4. 6 qv works cited in fn 4. 7 cf. George Boolos “Iteration Again” Philosophical Topics XVII, 2 (1989) pp. 5–21, also “Saving Frege from Contradiction” Aristotelian Society Proceedings 1987, pp. 137–51 and “Basic Law V” Aristotelian Society Supplementary Volume 67 (1993), pp. 213–34; Crispin Wright “The Philosophical Signficance of Frege’s Theorem”, Stewart Shapiro and Alan Weir “New V, ZF and Abstraction”, Philosophia Mathematica (3), vol. 7, 293–321.

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to support one or other of the usual set-theoretic constructions (Dedekind’s or Cantor’s) of the reals. The most obvious way to do this is by means of a suitably restricted version of Basic Law V, and a good deal of work has been done on one particular axiom of this sort, which builds in a restriction on the ‘size’ of concepts which are permitted to have sets corresponding to them which obey the principle of extensionality. 8 I shall not discuss this work here, save to remark that some of it seems to me to show that the prospects for obtaining a satisfactory treatment of the reals along this line are uncertain at best. In particular, as Boolos observed, 9 a theory based on secondorder logic plus this axiom alone, without further comprehension or existence assumptions, will not enable us to prove either an axiom of infinity or a power set axiom. So it will not yield sets large enough for the construction of the reals. This is not conclusive evidence against a broadly set-theoretic approach, of course, since it may be possible to formulate some other more powerful but still consistent Fregean axiom for sets which will give us large enough sets. Or again, it may be possible to justify supplementing this particular restricted version of Basic Law V with other principles to obtain a strong enough theory. I take no stand on that question here. 10 Instead, I want to pursue a quite different approach, which is in some respects much more like that taken by Frege in his incomplete treatment of the reals in Grundgesetze, although it differs from Frege’s in at least one quite fundamental way. This approach can roughly be described by saying that it tries (i) to minimise reliance on set theory and (ii) to obtain the reals very directly by means of abstraction principles, without any form of set-abstraction. In these respects, I think my approach may be seen as the most direct and natural way of extending the neo-Fregean position to the reals. Just as basing elementary arithmetic on Hume’s principle minimises (and, indeed, eliminates) reliance on set theory by avoiding a definition of cardinal numbers as certain equivalence classes, introducing them instead via a specifically numerical abstraction—so my approach to the arithmetic of real numbers will minimise (and indeed eliminate) reliance on set theory by avoiding a definition of reals as sets of one kind or another, introducing them instead via abstraction principles which— even if not happily described as purely numerical—are not distinctively set-theoretical. 8 The axiom (New V) is: ∀F∀G[*F= *G ↔ ((Small(F)∨ Small(G)) → ∀x(Fx↔ Gx))], where a concept is Small if fewer objects fall under it than fall under the universal concept ξ = ξ , and *F is what Boolos calls the ‘subtension’ of F (the subtensions of Small concepts being sets)—see Boolos “Saving Frege from contradiction”, Proceedings of the Aristotelian Society 87 (1986/87), pp. 137–51; also below p. 190ff. 9 cf. “Iteration again”. 10 For a brief discussion of this possibility, see Crispin Wright “On the Philosophical Significance of Frege’s Theorem”, section XI.

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1.5

179

Reals as ratios of quantities

Frege’s actual (incomplete) treatment of the reals in Grundgesetze Pt III 11 is, of course, unsatisfactory—if only because it relies, as does his theory of cardinal numbers, on an inconsistent theory of extensions, and cannot be simply relocated within any standard (and plausibly consistent) theory of sets such as ZF or NBG because the objects with which he proposes to identify the reals are too big to be treated as sets. In any case, such a relocation would obviously betray Frege’s philosophical aims, since it would leave our entitlement to the substantial existential commitments of the theory quite unaccounted for. From a philosophical standpoint, the most striking and most important features of Frege’s treatment of the reals are two: (i) the real numbers are to be defined as ratios of quantities [§§73,157] and (ii) in regard to the analysis of the notion of quantity, the fundamental question requiring to be answered is not: What properties must an object have, if it is to be a quantity? but: What properties must a concept have, if the objects falling under it are to constitute quantities of a single kind? [§§160–61]. Briefly and roughly, his insistence that reals be defined as ratios of quantities derives from his belief that the application of reals as measures of quantities is essential to their very nature, and so should be built into an adequate definition of them. It is this, more perhaps than any other single consideration, which underlies his dissatisfaction with the theories of Cantor and Dedekind, on which the applicability of the reals appears, in Frege’s view, merely as an incidental extra. As regards the second point, it is obvious to anyone that there are many different kinds of quantity (lengths, masses, volumes, angles, etc.) and that addition and comparison (as greater or less) make sense only as applied to quantities of the same kind. Since we may not simply take the notion of a kind of quantity for granted, as already understood and itself in no need of analysis, we cannot explain what a quantity is by saying that it is something which can be added to, or be greater or less than, (other) quantities of the same kind. If an explanation of quantity is not to be vitiated by circularity in this way, Frege thinks, it must take as its target the notion of a kind of quantity, and say what characteristics a collection of entities must, as a whole, possess if it is to form what he calls a quantitative domain [ein Grössengebiet]. When that has been done, what it is to be a quantity can be easily stated—an object is a quantity if it belongs, together with other objects, to a quantitative domain. I believe Frege was substantially right on both points. Here I shall simply assume as much, without argument. Where I disagree with him is over the analysis of what he calls quantitative domains. For reasons which I shall not go into, Frege decides that the elements of a quantitative domain should themselves be relations and—heavily influenced by a passage from Gauss [quoted 11 For expositions see Michael Dummett Frege Philosophy of Mathematics (Duckworth, 1991), ch. 22 and Peter Simons “Frege’s Theory of Real Numbers” History and philosophy of logic 8 (1987), pp. 25–44.

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in Grundgesetze §162]—analyses such a domain as an ordered group of permutations on an underlying set, with composition as its additive operation. Since quantities themselves are, on his approach, relations of a certain sort, real numbers, when defined as ratios of quantities, turn out to be relations of relations. One advantage of Frege’s approach is that it provides very easily for negative as well as positive real numbers. I do not have space to discuss Frege’s view properly here. Whilst there is justice in his criticism of earlier writers who simply help themselves to the notion of quantities being of the same kind, I think that the notions of addition and quantitative comparability are central and fundamental to the general notion of quantity in a way Frege fails to acknowledge. Accordingly, I shall propose a different account of quantitative domains—one which gives a central role to the idea that the elements of such a domain may always be added to yield further elements.

2. 2.1

Quantities and reals Types of quantitative domain

I distinguish between the entities (usually concrete objects) which may stand in various quantitative relations to one another—such as being longer than, or being as long as—and quantities themselves, which I take to be abstract objects introduced by abstraction on quantitative equivalence relations—for example: the length of a = the length of b ↔ a is as long as b This way of introducing (terms for) quantities makes no explicit mention of addition. However, a full analysis of the notion of a quantitative relation would, I claim, show that the notion of addition is nevertheless central to that of quantity. I do not have space to go into details here, but the essential idea is this. Among quantitative relations, we may distinguish—as conceptually basic— what may be called relations of simple quantitative comparison (e.g. longer than/as long as, heavier than/as heavy as, etc.) from relations of numerically definite or determinate comparison (e.g. twice as long as, 2.4 kg heavier than, etc.). A necessary condition for φ to denote a kind of quantity is that it be associated with a pair of relations of simple quantitative comparison: more φ than and as φ as. In virtue of this, things which are φ may be partially ordered with respect to φ-ness. However, the existence of an associated pair of such relations—a strict partial ordering relation and a cognate equivalence relation—is insufficient for φ-ness to be a kind of quantity. There are enormously many adjectives in ordinary use which may be substituted without violence to sense or syntax in the schemas: more φ than and as φ as—‘sweet’, ‘elegant’, ‘graceful’, ‘pretty’, ‘clumsy’, ‘ambitious’, ‘impatient’, ‘irrascible’, ‘probable’, . . . is clearly no more than the start of a potentially very long list. But in the case of only relatively few of them is it remotely plausible that they denote something properly describable as a quantity. It is therefore

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necessary to enquire what further condition needs to be satisfied, if such a pair of relations are properly to be viewed as quantitative. I contend that what makes the difference between quantitative ordering relations and others is that in the case of a quantitative ordering relation, but not otherwise, the entities which can significantly be asserted to stand in the relation can (at least in principle) be combined in such a way that compounds must come later in the relevant ordering than their components. In other words, for more φ than to be a quantitative ordering relation, there must be an operation of combination c on items lying in the field of more φ than, analogous to addition, such that c is more φ than a and ab c is more φ for any a,b in more φ than’s field, ab than b. 12 Quantitative domains are composed of (abstract) quantities. My aim in this section is to provide an informal axiomatic characterisation of such domains, on the basis of which it will be possible to introduce real numbers by means of an appropriate abstraction principle. Instead of simply laying down a single set of axioms for something to be a quantitative domain, I shall distinguish several—successively richer—types of quantitative domain. This will be helpful later, when I come to consider questions about the existence of quantitative domains. 1.1 A minimal q-domain is a non-empty collection Q of entities closed under an additive operation ⊕, which commutes, associates and satisfies the strong trichotomy law that for any a,b∈Q we have exactly one of: ∃c(a = b ⊕ c), ∃c(b = a ⊕ c) or a = b. Any minimal q-domain is strictly totally ordered by < , defined by: a < b ↔ ∃c(a ⊕ c = b). Multiplication of elements of Q by positive integers is easily defined—inductively—in terms of ⊕. 1.2 A normal q-domain is any minimal q-domain meeting the [Archimedean] comparability condition: ∀a,b∈Q ∃ (ma > b). Here and subsequently (unless explicitly indicated), m (and later n as well) ranges over positive integers. This requires quantities to be finite, in the sense that no quantity is infinitely greater (or smaller) than any other—it rules out infinitesimal 12 The basic idea is of course not new. It is, in particular, central to the theory of measurement advanced by N.R. Campbell in a number of works first published in the 1920s, the most important of them being Physics: the Elements (originally published by Cambridge University Press, 1919, and subsequently republished as Foundations of Science (Dover 1957)—see Part 2) and Measurement and Calculation (Longmans, Green & Co, London 1928). A briefer popular statement of his theory is given in What is Science? (Methuen, London 1921—see ch. VI). Whilst there is much in Campbell’s overall theory which I think we neither can nor need accept, I believe that Campbell was right, pace critics such as Brian Ellis (see Basic Concepts of Measurement, Cambridge University Press 1966, ch. IV), to insist upon the importance of a physical analogue of addition, and right too (at least in essentials) in taking there to be an important distinction between fundamental and derived measurement. More recent treatments of measurement—see, for example, the comprehensive text of Krantz, D.H., Luce, R.D., Suppes, P. and Tversky, A. (Foundations of Measurement New York and London: Academic Press 1971 (vol 1), 1989 (vols 2,3))—have not looked kindly on these distinctive features of Campbell’s approach. I need hardly emphasise that the very rough and dogmatic statement of my view, both here and in the text, requires both considerable qualification and further explanation, as well as defence.

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quantities. With his eye on Euclid’s Def.4 of Elements Bk.V, Howard Stein 13 describes it as the condition necessary and sufficient for a and b to have a ratio. It might be compared, in status, to the requirement on concepts presupposed by Hume’s principle, that the concepts through which it quantifies be sortal— which might be described as the condition for a concept to have a (cardinal) number. Where Q, Q* are any normal q-domains, not necessarily distinct, we introduce ratios of quantities by the abstraction principle: EM

∀a, bεQ∀c, dεQ∗ [a : b = c : d ↔ ∀m, n(ma < => nb ↔ mc< => nd)]

That is, ratios a:b and c:d are the same just if equimultiples of their numerators stand in the same order relations to equimultiples of their denominators. 14 The condition for identity of ratios is framed so as to allow that one and the same ratio may be at the same time a ratio of pairs of quantities of different kinds— belonging to different domains—such as masses and lengths. The operation in terms of which comparability is ultimately defined (i.e. addition of quantities) is, of course, domain specific—no sense is given to adding a length and a mass, for instance. But this does not preclude the introduction of ratios so that the same ratio may be found among, say, both masses and lengths. 1.3 A normal q-domain Q is full if ∀a,b,c∈Q∃q∈Q(a:b = q:c). This condition, which is a restricted form of the ancient postulate of ‘fourth proportionals’, ensures that, given a pair of ratios a:b and c:d, there is a quantity c′ such that c′ :b = c:d, so that we may always, without loss of generality, restrict attention to ratios with common denominators. I shall refer to it as CD. It is easy to see that CD ensures that there is no smallest quantity. 15 1.4 A full q-domain may be incomplete, in the sense that it may include only quantities which are rationally measurable; in consequence, the set of all ratios on a full domain is not guaranteed to include ratios corresponding

13 “Eudoxos and Dedekind: On the Ancient Greek Theory of Ratios and its Relation to Modern Mathematics” Synthese 84 (1990), pp. 163–82. Whilst the approach I pursue here differs quite radically from anything suggested by Stein, I have derived much benefit from this excellent paper. 14 This is, of course, the central principle in the ancient theory of proportion presented in Euclid’s Elements Book V (cf. Def.5) and standardly attributed to Eudoxos. I should perhaps emphasise that EM is not an abstraction principle of the form characterised at the outset. On the other hand, it should be clear that it is intended to work in essentially the same way as paradigm abstractions like the Direction equivalence and Hume’s principle and that it is reasonable to regard it as one. We might bring EM into line with the characterisation of abstraction principles with which I began by first defining an equivalence relation on ordered pairs of quantities: E[(a,b), (c,d)] ↔ ∀m,n (ma < => nb ↔ mc ⇔ nd), and then setting: Ratio(a,b) = Ratio(c,d) ↔ E[(a,b), (c,d)]. Alternatively, if it were felt desirable to avoid reliance on the notion of an ordered pair, we could introduce an extension of the notion of an equivalence relation so as to allow relations of arity greater than 2 to qualify as equivalence relations. Later we shall meet another abstraction principle which does not, as it stands, conform to the usual characterisation, but which may readily be brought into line in one or other of these ways. 15 Although I am not identifying quantities, as such, with numbers of any kind, it should be fairly clear that a full domain, and likewise the domain of ratios on it, is dense, and that we can develop an ‘arithmetic’ of ratios structurally analogous to that of the positive rationals.

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to any, much less all, (positive) irrational numbers. 16 If ratio-abstraction is to yield all the positive reals, we require a complete domain. Indulging—for convenience, but avoidably—in set-theoretic language, we say that a subset S of quantities belonging to a q-domain Q is bounded above by b iff for every quantity a in S, a ≤ b. A quantity b ∈ Q is a least upper bound of S ⊆ Q iff b bounds S above & ∀c(c bounds S above → b ≤ c), and finally that a q-domain Q is complete iff Q is full and every bounded above non-empty S ⊆ Q has a least upper bound.

2.2

Real numbers

We may straightforwardly define ‘bounded above’ ‘lub’ and ‘ordercomplete’ for ratios in a way that parallels our definitions of these notions for quantities and then prove, as an easy consequence of the completeness of the underlying domain, that where Q is any complete q-domain, the set RQ of ratios on Q is order-complete. 17 It can be shown that if Q and Q∗ are any ∗ complete q-domains, they are isomorphic, so that R Q = R Q , i.e. the set of ratios on Q is identical with the set of ratios on Q∗ . Thus provided there exists at least one complete q-domain, we can introduce the positive real numbers, by abstraction, as the ratios on that domain. In standard constructions of the various number systems, negative numbers make their entry at an early stage. The method by which this is accomplished—introducing a new, enlarged domain including negative numbers as certain ordered pairs (difference pairs) of numbers belonging to an underlying domain—is, however, perfectly general, in the sense that it is quite inessential to it that the numbers in the underlying domain should be natural numbers. Of course, we must start with the natural numbers if we want to get just the integers—but in general, all that is required for the application of the method itself is that the objects belonging to the underlying domain have the requisite arithmetic properties. There is, so far as I can see, no reason, either technical or philosophical, why this step may not just as well be taken at a (much) later stage. In particular, essentially the same construction can be used to get negative reals, starting from positive ones, as difference pairs of positive reals. Letting x, y, z, . . . range over, and ⊕ stand for addition of, positive reals, we obtain difference pairs of positive reals by the abstraction: D (x, y) = (z, w) ↔ x ⊕ w = y ⊕ z 16 Of course, since quantitative domains, as I have characterised them, do not include either a zero quantity or negative quantities, the ratios on such domains will not, in any case, have elements corresponding to all the reals. 17 Proof : Let S be any bounded above subset of R Q . By CD, each ratio in S can be expressed with a single common denominator, so that the members of S are: a1 :b, a2 :b, . . . , ai:b , . . . The set of numerators of these ratios is a non-empty subset of Q, and so—by the completeness of Q—have a least upper bound a◦ . Since every ai = a◦ , ai :b = a◦ :b for every ratio ai :b in S. And if some ratio p:q is less than a◦ :b, it follows [by CD] that p:q = p′ :b for some p′ , with p′ < a◦ . But then by the completeness of Q, there is some ak among the numerators of the ratios ai :b so that p′ < ak , and hence a ratio ak :b in S such that p′ :b < ak :b. So a◦ :b is a least upper bound of S.

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Defining < , > , addition, subtraction and multiplication and zero for dpairs in the obvious way, it can be shown that the collection R of d-pairs forms a field with the operations + and ×. Further, there is a subset P of R, namely the set of all pairs (x, y) such that (z, z) < (x, y), meeting the conditions: (i) if (x, y), (z, w) ∈ P then (x, y) + (z, w) ∈ P ∧(x, y) × (z, w) ∈ P and (ii) if (x, y) ∈ R, then exactly one of (x, y) ∈ P, (y, x) ∈ P or (x, y) = (z, z) holds

Thus R is an ordered field. There is an obvious isomorphism between the strictly positive subset P of R and the positive reals as previously defined. Using this, it can be shown without too much difficulty that R is complete.

3.

The existence of quantitative domains

Our result thus far is conditional: real numbers may be obtained by abstraction on quantities, if there exists at least one complete q-domain. Even if this were the best result that could be obtained, it is not completely obvious that this would signal the collapse of the neo-Fregean abstractionist approach to foundations. It might be possible to provide principled reasons for adopting different attitudes towards the question of the existence of reals and that of the natural numbers, holding that while the latter admits of resolution, a priori, in the affirmative, the existence of the reals is a matter on which no similar a priori assurance is to be expected. According to such a view, the existence of (at least) finite cardinal numbers would be a matter of necessity— whatever the universe might be like, its ingredient objects would be assignable to distinguishable sorts or kinds; there would be some sortal concepts or other, under which the objects fell, so that for various concepts F and cardinal numbers n, there would be facts of the form: the number of Fs = n. More importantly, for any such sortal concept F, there will be a sortal—F-andnot-F—logically guaranteed to have no objects falling under it, in terms of which 0 may be defined, thus giving the necessary toe-hold for a Fregean proof of the existence of an infinite collection of finite cardinals. But there can be no similar a priori guarantee that the physical universe comprises quantities which are real-valued—it is perfectly conceivable, even if in fact false, that the physical world should be discontinuous. So a result which says, in effect, that if it does exhibit continuity, the real numbers are available to measure it, might not appear utterly outrageous. Defending this position would, naturally, require speaking to the contrary intuition, that while it may be in some way an empirical question whether the physical universe is continuous, and so an empirical question whether the reals have ‘objective’ application [in the sense that there actually are real-valued quantities—contrast the idea that using the reals simply affords a useful simplification of applied mathematics], the existence of the reals should not itself be an empirical, a posteriori matter.

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Clearly, however, it is important to enquire whether a neo-Fregean can secure a stronger result. Evidently, the question of greatest interest is whether there can be proved to exist a complete q-domain. But it is worth emphasising that the question arises, not only for the case of complete q-domains, but equally for q-domains of the more modest kinds described—thus far, nothing has been done to establish the existence of a full q-domain, or even that of a normal, or even minimal, one. Even the question of the existence of a minimal domain is anything but trivial. A minimal domain is, by definition, non-empty. Since such a domain is closed under its addition operation and satisfies the additive trichotomy condition, it must comprise arbitrarily large quantities, and thus be at least countably infinite. To anyone who thinks of quantities as physical entities of some sort, the existence of such a domain must, for this reason, appear open to serious question. On my own view, quantities such as lengths, masses, angles, etc., should not be thought of a physical entities; they are, rather, abstract objects, ‘introduced’ via abstraction principles employing appropriate equivalence relations on the concrete objects whose lengths, masses, etc., they are. But this makes no essential difference, so far as the present question is concerned. At least, it will make no difference if the existence of a given length, say, is taken to be contingent upon the existence of a suitable concrete entity of which it is the length; for in that case, the ground for doubt about the existence of arbitrarily large quantities of any given kind remains. Clearly there must be an analogous doubt about the existence of arbitrarily small quantities, and hence about the existence of a full q-domain. However, it seems to me that these doubts may be assuaged and that we can actually prove the existence of at least one domain of each of the kinds I have distinguished, including complete domains. The crucial point here is to notice that whilst quantities as such are not identified, in my approach, with numbers, nothing in the characterisation of qdomains precludes such domains being composed of numbers. As previously remarked, Hume’s principle suffices for a derivation of the Dedekind–Peano axioms for elementary arithmetic, and hence for a proof of the existence of an infinite sequence of natural numbers—0, 1, 2, . . . . Omitting 0 to obtain the strictly positive naturals, N+ , and adjusting the usual recursive definitions of + and × to suit, we can easily show that N+ constitutes a minimal—and indeed a normal—q-domain. It is clear that N+ is not itself a full domain, i.e. it does not satisfy CD. + However, the collection R N of ratios on N+ does constitute a full domain. To see this, note first that since N+ is normal, there exists a ratio a:b for every a and b in N+ . Let a, b, c, d, e, f be any elements of N+ . Then what we must show is that there is a ratio g:h such that [a:b]:[c:d] = [g:h]:[e:f]. It is quite straightforward to verify that [a:b]:[c:d] = ad:bc = ade:bce = [ade:bcf]:[bce:bcf] = [ade:bcf]:[e:f]

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so that [ade:bcf] is our required ratio. 18 In the presence of CD, satisfaction + by R N of the minimality and normality conditions follows easily from their + satisfaction by the underlying domain N+ . Thus R N is a full domain. What we have, in effect, is a quite natural way of obtaining the positive rationals by abstraction on the positive natural numbers—each and every positive rational is simply a ratio positive natural numbers. Thus 3/4 just is the ratio 3:4. Of course, it is also the ratio 6:8 and the ratio 9:12, etc., but that is no problem, since these are all simply one and the same ratio in our sense (i.e. by the lights of EM). + It is clear that iteration of the abstractive procedure which yields R N from N+ will not yield any new kind of q-domain. The crucial point emerges above, in the observation that [a:b]:[c:d] = ad:bc. This holds quite generally—any ratio of ratios of positive natural numbers are simply ratios of positive natural numbers. In the same way, ratios of ratios of ratios of positive natural numbers collapse to ratios of positive natural numbers. Iteration of the abstraction to ratios of higher order thus merely gives us the positive rationals all over again. Thus the operation by which we obtained a full domain from an underlying normal one cannot, when re-applied to a full domain, yield a complete one. This is a special case of a quite general fact about first-order abstraction: no first-order abstraction on an infinite domain can generate a ‘new’ domain of greater cardinal size than that abstracted on. It follows that if a complete domain is to be obtained by abstraction, we must invoke a second-order abstraction. In this way—and only in this way—we may advance from a domain of objects of given cardinality to a strictly larger domain of abstracts. Given an initial domain comprising κ objects, there will be 2κ properties of those objects. By taking these properties, rather than the objects which have them, as our underlying domain for an abstraction, we may obtain a strictly larger collection of abstracts—up to (but not more than) 2κ of them. 19 We take as our initial domain the (at least countably infinite) full domain N+ R of ratios on N+ . Our goal is to obtain a complete domain Q# by cutabstraction, so-called because of its obvious correspondence to Dedekind’s + construction. 20 As anticipated, cut-abstraction operates, not directly upon R N itself, but upon properties of a certain kind defined over its elements, which I shall call cut-properties. These are defined by reference to the ordering on + R N . Informally, a cut-property is a non-empty property whose extension is 18 Recall that a, b, c, d, e, f are all positive integers. A ratio is unchanged by multiplying its numerator and denominator by the same positive integer. Hence a:b = ad:bd. Similarly, c:d = bc:bd. But the ratio to one another of ratios with a common denominator is simply the ratio of their numerators, so [ad:bd]:[bc:bd] = ad:bc, whence [a:b]:[c:d] = ad:bc. Further e:f = bce:bcf and ad:bc = ade:bce. Hence, since [ade:bcf]:[bce:bcf] = ade:bce, we have: [a:b]:[c:d] = ad:bc = ade:bce = [ade:bcf]:[bce:bcf] = [ade:bcf]:[e:f] 19 If κ is infinite and CH holds, then we shall, of course, get more than κ abstracts only if we get exactly 2κ of them; but I am not assuming CH, much less GCH. 20 cf. Richard Dedekind Stetigkeit und Irrationale Zahlen (1872), translated by Wooster Woodruff Beman as “Continuity and Irrational Numbers” in Richard Dedekind Essays on the theory of numbers, New York: reprint, Dover Publications (1963), pp. 1–27.

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a proper subset of R N and which is downwards closed [i.e. ∀a∀b(Fa→(b < a → Fb)) 21 ] and has no greatest instance [i.e. ∀a(Fa→ ∃b(b > a ∧ Fb))]. We now introduce objects—cuts—corresponding to cut-properties by the abstraction principle: Cut: #F = #G ↔ ∀a(Fa ↔ Ga) where F,G are any cut-properties on R N + and a ranges over R N .

+

+

Q# is the collection of all cuts, #F, for cut-properties F on R N . It may be shown that Q# constitutes a complete domain, in the sense previously explained. Obviously the main thing here is to verify that Q# has the least upper bound property, i.e. where φ varies over proper+ ties of cuts on R N , and bounds above and lub are defined in an obvious way, that if ∃Fφ(#F) and φ is bounded above then φ has a least upper bound. This can be done, mimicking the usual proof, by defining the property H by: H a ↔ ∃F(φ(#F) ∧ Fa)—we can then show that H is a cut-property and that #H is a lub of φ. We may define #F+ #G to be #H , where H a ↔ ∃b∃c(Fb ∧ Gc ∧ a= b⊕c), and #F × #G to be #P, where Pa ↔ ∃b∃c(Fb ∧ Gc ∧ a = b⊗c). With the aid of these and some supplementary definitions, it can then be proved that Q# is full, i.e. that it is a minimal q-domain which also meets the normality and common denominator conditions.

4.

Safe abstractions and safe sets

Are the abstraction principles which I have employed all in good standing? The question is urgent, since we know that not all abstraction principles are acceptable, if only because some—Basic Law V being the obvious example— are inconsistent. And there may be other constraints, besides consistency, with which good abstractions must comply. A thorough examination of the question lies well beyond the scope of this paper, but I should like to conclude by saying a little about it. Of the abstraction principles I have used, two—ratio-abstraction (EM) and difference abstraction—are first-order, while the other two – Hume’s principle and Cut—are second-order. In the case of first-order abstraction, we abstract upon a domain of objects of some kind, and thereby come to recognise objects of another kind; with a second-order abstraction, by contrast, we abstract upon a domain of concepts, themselves defined on some underlying domain of objects, and come to recognise ‘new’ objects, i.e. objects of a kind other than those belonging to this underlying domain. I shall call the field of an abstraction’s equivalence relation the domain for the abstraction, and in the case where this is a domain of (first-level) concepts, I shall call the domain of objects on which these concepts are defined the underlying domain. 21 Here and subsequently a,b, . . . range over elements of R N + .

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In the case of second-order abstractions, the underlying domain—if it has a determinate size at all—is much smaller than the domain for the abstraction; if the underlying domain has cardinality κ, then the domain for the abstraction (assuming it to comprise all the concepts defined on the underlying domain, and assuming concepts to be individuated extensionally) has cardinality 2κ . In consequence, the abstraction may ‘generate’ up to 2κ abstracts—and so many more abstracts than there are objects in the underlying domain. It is this feature of second-order abstractions which has led some writers to think that it is these abstractions—in contrast with first-order abstractions—which pose the greatest worry, as far as the risk of inconsistency is concerned. I think that is correct, and I shall therefore focus on the second-order abstractions. In fact, since Hume’s principle is known to be consistent, I shall concentrate upon the other second-order abstraction I have used—cut-abstraction. Cut—in contrast with Hume’s principle and Basic Law V—is a restricted abstraction principle, in the sense that the domain for the abstraction comprises only cut-properties on a certain specified underlying domain of objects. It is obvious that if the side constraints on it are ignored, Cut is just a notational variant on Basic Law V. Clearly, then, from unrestricted Cut, we could derive Russell’s contradiction. If we define a Russell property R by: Rx ↔ ∃F(x = #F ∧ ¬Fx), then by unrestricted Cut we have: #R = #R ↔ ∀x(Rx ↔ Rx), whence: #R = #R—so #R exists, and we may proceed: 1 1 3 3 3 3 3 3 1 12 12 12 12

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

R(#R) ∃F(#R = #F ∧ ¬F(#R)) #R = #F ∧ ¬F(#R) #R = #F #R = #F ↔ ∀x(Rx ↔ Fx) ∀x(Rx ↔ Fx) R(#R) ↔ F(#R) ¬F(#R) ¬R(#R) ¬R(#R) R(#R) → ¬R(#R)

assn 1, Def R assn 3∧E (unrestricted) Cut 4,5 ↔E 6 ∀E 3∧E 7,8 ↔ E 2, 3, 9∃E 1,10 →I

(12) (13) (14) (15) (16) (17) (18)

¬R(#R) #R = #R #R = #R ∧ ¬R(#R) ∃F(#R = #F ∧ ¬F(#R)) R(#R) ¬R(#R) → R(#R) R(#R) ↔ ¬R(#R)

assn =I 12, 13 ∧I 14 ∃I 15 Def R 12,16 →I 11,17 ↔I

With the constraints on Cut in place, however, this derivation will not go through without two further assumptions: to establish the existence of #R, and

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to justify the (second-order) ∀E step involved at line (5), we must assume that + R is a cut-property on R N ; and for the application of ∀E at line (7), we must + further assume that #R is in R N . Since the contradiction at line (18) depends upon these further assumptions, we may apply reductio to infer that either R + + isn’t a cut-property on R N , or #R is not an element of R N . Does that settle the matter? Well, no. The particular cut-abstraction principle I’ve used may be viewed as a special case of a general schema which runs: (#)#F = #G ↔ ∀a(Fa ↔ Ga) where F, G are any cut-properties on a suitable domain Q and a ranges over Q. A suitable domain Q here will be any domain with an at least dense linear ordering, with respect to which cut-properties are definable. Two obvious questions which may be raised about this general schema are: Are all its instances safe? If not, what distinguishes those which are from those which are not? I’ll venture a few somewhat tentative thoughts about these questions. Perhaps the first thing I should say is that I am not, so far as I can see, committed to endorsing all instances of (#)—i.e. to defending its universal closure with respect to Q—though I would think that, should it prove that some of its instances are either prone to Russell trouble or otherwise unsafe, it should be possible to provide some principled characterisation/explanation of the limitations here. It is clear that so long as the underlying domain Q for an instance of (#) is not inclusive of all objects whatever, any derivation of Russell’s contradiction can be seen, not as showing the inconsistency of that instance (#), but as a demonstration that either the Russell property R cannot be a cut-property on Q or the Russell cut #R cannot be an element of Q. If the universe of all objects whatever constitutes an admissible underlying domain for cut-abstraction, then the Russell cut, if there is such an object at all, must belong to that domain—so the second option lapses. But the first remains open. There will be such an object as the Russell cut only if the Russell property is a cut-property on the universe. But, at least in the absence of any compelling independent reason to think (#) defective, a derivation of the Russell contradiction would seem to give us ample reason to think that the Russell property cannot be a cut-property on the universe. If what I have said is right, it is possible to block Russell trouble without challenging the assumption that the universe constitutes an admissible underlying domain for cut-abstraction. The point is, however, somewhat academic since there are other worries—having more to do with Cantor’s paradox than with Russell’s—which are, I think, best answered by rejecting that assumption. Briefly, cut-abstraction, for all I have said thus far, may be applied to any domain on which cut-properties are definable – that is, any domain with an at least dense linear ordering. If the chosen domain is strictly dense (i.e. dense— like the rationals—but not complete—like the reals), then an instance of cutabstraction will inflate, in the sense that there are more abstracts ‘generated’

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than there are objects in the underlying domain (i.e. the domain on which the cut-properties are defined). 22 If it is dense but complete, then there will be no inflation—the collection of abstracts will be isomorphic to the underlying object domain. If the universe of all objects whatever admits of a strictly dense linear ordering and can be taken as a domain for cut-abstraction, we shall wind up with more abstracts (and so more objects) than there are objects altogether! How should we avoid this disastrous conclusion? The answer I shall tentatively commend makes crucial play with the contrast I drew previously between unrestricted abstractions, such as Hume’s principle, and restricted ones, such as cut-abstraction. In the case of Hume’s principle, it is essential that the first-order quantifiers on its right-hand side be allowed to range unrestrictedly over all objects whatever, including—crucially—the numbers themselves. In this sense, the first-order quantifiers in Hume’s principle must be understood impredicatively. If instead those quantifiers were restricted so as to range only over objects other than numbers, we could not prove the infinity of the sequence of finite numbers—at least, not without the additional assumption that there exist infinitely many objects of some other kind. With cut-abstraction, by contrast, it is unnecessary—in order to ensure that the abstraction delivers all the abstracts we require—to construe its first-order quantifier impredicatively in this way. Moreover, if we do allow that—in particular, if we allow an instance of the cut-schema whose first-order quantifier ranges over all objects whatever—then we will (provided the universe admits of a strictly dense ordering) run into Cantor-type trouble. But we do not have to allow this. As I have explained, cut-abstraction is—in contrast with Hume’s principle, and Basic Law V—a restricted abstraction, in the sense that each instance of the cut-schema (#) involves a restriction to a specified underlying domain, over which its first-order quantifier ranges. All I have said thus far about what constitutes a suitable underlying domain is that it shall be some densely ordered collection of objects. But as far as I can see, nothing stands in the way of imposing a further restriction which will preclude application of cut-abstraction to the universe as a whole. It may seem that the most obvious way to do this would be to incorporate a ‘limitation of size’ requirement in the conditions for a suitable domain for cut-abstraction—the idea would be to require that any suitable domain Q for cut-abstraction be smaller than the universe. This would bring cut-abstraction much closer to the modified version of Basic Law V which George Boolos dubbed New V. Following Boolos, say that a concept F is a subconcept of a concept G iff ∀x(Fx→ Gx), and that F goes into G iff F ≈ H for some subconcept H of G. Let V be the concept [x: x= x], and say that F is small iff V does not go into F. Define F to be similar to G iff (F is small ∨G is 22 As an anonymous referee, Stewart Shapiro and his student Roy Cook (independently) pointed out to me, cut-abstraction inflates at every cardinality, in the sense that, for every cardinal κ, there is a domain of size κ with a strictly dense linear order on it, so that cut-abstraction applies to yield a ‘new’ domain of size 2κ .

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small → ∀x(Fx↔ Gx)). Similarity is an equivalence relation. New V is then the abstraction: New V *F = *G ↔ F is similar to G If we agree—as I think we should—that the numbers may only properly be assigned to genuine sortal concepts—that is, roughly, concepts F with which are associated not only criteria of application but also criteria of identity—then we should be happy with this modification (of either cut-abstraction or Basic Law V) only if we are persuaded that self-identity is a genuine sortal. For if a concept F can have a number only if F is sortal, then, assuming Hume’s principle, F can be equinumerous with itself only if it is sortal. And if it can’t be equinumerous with itself, it can scarcely be equinumerous with any other concept. Since small is defined so that F is small iff self-identity doesn’t go into F, New V is a real restriction of Basic Law V only if self-identity is a genuine sortal. I do not think it is. A simple argument due to Crispin Wright shows, in effect, that if self-identity were a genuine sortal, many concepts which are plainly not sortal would qualify as such. The argument turns on the point that whenever a concept G is genuinely sortal, its restriction by any other (even merely adjectival) concept F—i.e. the conjunctive concept: F-and-G—will likewise be sortal. For example, since horse is, presumably, genuinely sortal, so is white horse, for all that the restricting concept white is no sortal. Thus if self-identical were a genuine sortal, so would be any restriction of it, such as white-and-self-identical. However, since white-and-self-identical is equivalent to white, it would follow that white is after all a sortal concept. Since white (or white thing) is not a genuine sortal, neither can self-identical be one. For the same reason, clearly, no concept which applies universally can be a genuine sortal concept. 23 If this is right, some other means of formulating the needed restriction is required. There is an obvious next thought. Why should we not simply stipulate that a predicate Q determines a suitable domain for cut-abstraction only if Q is genuinely sortal? Since neither self-identity, nor any other predicate (such as ‘F ∨ ¬F’) which is guaranteed application to all objects whatever, is a genuine sortal, this will ensure that the universe of objects as a whole— even if it admits of a strictly dense ordering—is not an admissible domain for cut-abstraction. 23 cf. Wright “Is Hume’s principle analytic?”. Wright formulates the argument slightly differently, as follows: “Call a concept that is not sortal a mere predicable. Where F is a mere predicable, the question: “How many F’s are there?”, is deficient in sense and “the number of F’s” has no determinate reference. However, attaching a mere predicable to a genuine sortal, G, produces a complex, restricted sortal, F-and-G, such that there can be, and normally will be, a determinate number of objects falling under it. Thus if Fis any mere predicable, and self-identity is a genuine sortal, there will be a determinate number of objects which are F and self-identical. But since F and self-identical is equivalent to F, it follows that there can be no such determinate number wherever there is no determinate number of F’s—i.e. wherever F is a mere predicable. So self-identity is not a sortal concept”.

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A thorough defence of this proposal requires more space than I have here. To conclude, I should like to comment briefly on three points. (i) It might be observed that a restriction of admissible domains to those specifiable by sortal concepts will not, on the face of it, exclude certain very large domains such as those comprising all ordinals, or all cardinals, or all sets (since the relevant concepts appear to qualify as genuinely sortal)—giving rise to concern that paradox may still be derivable from cut-abstraction by taking one or other of these collections as underlying domain. I think this might be met in either of two ways. First, any attempt to generate paradox from (#) by taking the ordinals, say, as domain will—so far as I can see—rely on the idea that the collection of all ordinals is universe-sized. That requires the assumption that the concept ordinal number is equinumerous with some concept under which every object—whether an ordinal number or not—falls. But if what I have already said is right, concepts can be equinumerous only if both are sortal, and there can be no universal sortal concept, so that this assumption can be rejected, and there will be no need to strengthen the restriction on cut-abstraction to preclude taking the ordinals, etc., as domains. But second, even if it should prove necessary to exclude the ordinals, etc., as admissible domains for cutabstraction, there is a quite natural way to do this. Instead of requiring simply that an admissible domain be given by a sortal concept, we might require that such a domain should have a determinate cardinal size. Since being the extension of a sortal concept is at least a necessary condition for a collection to have a determinate size, this restriction would encompass the one already proposed. If this necessary condition is not sufficient—i.e. if certain sortal concepts fail to have determinately-sized extensions – then those concepts will be excluded by the revised restriction. In particular, what Michael Dummett has called indefinitely extensible concepts, such as ordinal, cardinal and set itself, will be excluded. (ii) It may be objected that restricting admissible domains for cutabstraction in either of the ways suggested is arbitrary or ad hoc. And the objection might be thought to draw strength from the neo-Fregean’s willingness (and, indeed, need) to employ unrestricted abstractions such as Hume’s principle. I shall make just two quick points in reply, leaving—no doubt— much more to be said. First, as should by now be clear, it is in fact false that Hume’s principle is a completely unrestricted abstraction—although its first-order quantifiers are unrestricted, its initial second-order quantifiers are— crucially—restricted to sortal concepts. Second, my proposed restriction(s) on cut-abstraction appear to be no more arbitrary or ad hoc than the restriction which New V seeks to build into Basic Law V. It is true that the manner in which the restriction is imposed on (#) differs, formally, from what happens with New V—where what is done is not to restrict the range of any quantifier, but to complicate the equivalence relation—with the effect that when F and G are not small, *F and *G exist, but are identified irrespective of whether their

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concepts are co-extensive. But I think this difference is superficial. Provided that the conditions for a first-level concept to be sortal can be expressed (using only logical vocabulary) in a second- (or perhaps third-) order language, I can see no reason why (#) should not be recast in essentially the same mould as New V. And if they cannot be so expressed, that is bad news (if it really is bad) not only for (#) but for New V too, for reasons already mentioned. But I am not persuaded that it would be bad news—since I see no ground for assuming that every philosophically important concept must be capable of definitive expression in the purely logical vocabulary of a second- or thirdorder language. (iii) Finally, a quick word about the state of the economy. Some recent writers 24 have claimed—plausibly, in view of the obvious risk of some form of Cantor’s paradox—that acceptable abstractions should be, in some sense, non-inflationary. Is cut-abstraction inflationary, in any objectionable sense? Some care needs to be exercised in characterising the relevant notion of inflationariness, since a great part of the point and interest of abstractions lies in the fact that they ‘generate’ objects which are ‘new’, and so, in a certain sense, ‘expand’ the underlying domain. So that in one way, inflation—or at least domain-expansion—is just what the neo-Fregean wants. Of course, this way of putting the matter is potentially very misleading, since it gives the entirely false impression of ontological prestidigitation—in which abstraction creates objects out of nothing, as it were, much as a practised conjurer appears to pull pigeons out of thin air. The neo-Fregean can, and should, insist upon a more sober description of what is going on. What an abstraction does, if all goes well, is to set up a concept—of direction, or cardinal number, or whatever—by supplying necessary and sufficient conditions for the truth of identity-statements linking terms which purport reference to objects falling under it. It draws our attention to the possibility of redescribing—or reconceptualising—the state of affairs which consists in line a being parallel to line b, for example, in terms of the holding of the relation of identity between certain objects, the direction of a and the direction of b. 25 Accepting the proposed reconceptualisation does not—in and of itself—involve acknowledging the existence of these objects. What it involves, rather, is accepting that the question whether there are such objects reduces to the question whether suitable instances of the right-hand side of the abstraction principle are indeed true. So what an abstraction does is not to ‘create’ objects, but to equip us to recognise, identify and distinguish objects which we could not recognise, identify and distinguish before—i.e. in advance of grasping the concept which the abstraction introduces. 24 See Kit Fine “The limits of abstraction”, in M.Schirn ed. The Philosophy of Mathematics Today, Oxford University Press (1998), pp. 503–629. 25 For fuller discussion of this idea, see Wright “On the Philosophical Significance of Frege’s Theorem”, §I; Hale “Dummett’s critique of Wright’s attempt to resuscitate Frege”, §2; and Hale “Grundlagen §64” passim.

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If inflation of this kind is acceptable, what kind might not be? Kit Fine writes: Two necessary conditions for the truth of an abstraction principle hold as matter of logic. . . . In the first place, it follows from the truth of an abstraction principle that its underlying criterion of identity on concepts should be an equivalence relation . . . Secondly, it follows from the truth of an abstraction principle that the identity criterion should not be inflationary, the number of equivalence classes must not outstrip the number of objects. There must, that is to say, be a one–one correspondence between all of the equivalence classes, or their representatives, on the one hand, and some or all of the objects, on the other. It is, of course, on this score that Law V proves unacceptable; for where there are n objects, it demands that there be 2n abstracts. 26

There is, I think, some ambiguity or vagueness in these remarks which we need to resolve if avoidable confusion is to be avoided. Let us say that an abstraction A inflates on an underlying domain D if A’s equivalence relation partitions D into more equivalence classes than D has elements. Then one might say that an abstraction is weakly inflationary if there is some domain on which it inflates, and strongly inflationary if it inflates on every domain (or perhaps—a little less exiguously—on some domain of cardinality κ, for every cardinal κ). 27 To require of an acceptable abstraction that it should not be (even) weakly inflationary would stop the neo-Fregean project dead in its tracks, before it even got moving (as it were). It will be clear that I think there is no good ground to impose such a requirement, and I shall not discuss it further. It is much more plausible to require that acceptable abstractions should not be strongly inflationary. 28 Some of the neo-Fregean’s key abstractions, including the other crucial second-order abstraction, Hume’s principle, satisfy this requirement. 29 But whilst the requirement that abstractions not be strongly inflationary is 26 “The limits of abstraction”, p. 506. 27 This characterisation of weak and strong inflation applies directly only to abstractions—like Hume’s

principle and Basic Law V—which are not restricted abstractions in the sense previously explained, i.e. are not such that their formulation already involves a specification of a particular domain as the underlying domain for the abstraction. Since any particular cut-abstraction, such as Cut, is restricted in this sense, there can be no question of its being strongly inflationary. We can, however, properly ask of the corresponding general schema—(#) in the case of Cut—whether it is strongly inflationary. 28 More plausible, because it might seem that strong inflation is bound to give rise to a version of Cantor’s paradox. It might also be thought that if an abstraction is strongly inflationary, then there could be no hope of showing that it is satisfiable, i.e. has a model—for let D be any domain, of cardinality κ, say. Then any strong abstraction inflates on D, i.e. its equivalence relation partitions D into more than κ equivalence classes, and so ‘generates’ more than κ abstracts. Thus D cannot be a model for the abstraction. But D was any domain whatever, so our abstraction can have no models. On reflection, it should be apparent that this short argument involves an unstated assumption—that the domain of any putative model for an abstraction must be the underlying domain for the abstraction. As against this, I cannot see why, in setting up model for a restricted abstraction—such as cut-abstraction—we should not choose as the domain of the model some larger collection which properly includes the collection which is to play the rôle of the underlying domain for the abstraction. 29 Hume’s principle inflates, of course, on any finite domain, but can be shown—assuming Choice, but without assuming CH or GCH—that it does not inflate on any infinite domain.

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more plausible, I can see no compelling reason to accept it in full generality— that is, as applying both to unrestricted abstractions and restricted ones. It may be necessary to insist that no unrestricted abstraction can be strongly inflationary. But, as I have tried to make plausible, it is unnecessary to require this of restricted abstractions. The cut-schema, in particular, is strongly inflationary in the sense that for every cardinality κ, there is an admissible domain of cardinality κ on which an instance of (#) inflates. But that, so far as I can see, does no harm, provided admissible domains are restricted to those given by genuine sortal concepts (or perhaps, those of determinate cardinal size).

5.

Summary and concluding remarks

My aim in this paper has been to set forth one plausible way in which a neo-Fregean account of arithmetic may be extended to encompass the real numbers. I have followed Frege himself in suggesting that the reals should be introduced as ratios of quantities. This approach, as Frege perceived, demands a prior analysis of the notion of quantity. I have agreed with Frege, too, in thinking that this should be done by providing a general characterisation of what he called quantitative domains, but have offered a somewhat different account of them from that given in Grundgesetze. Ratios of quantities are introduced by an abstraction principle based on the ancient theory of proportion which comes down to us from Eudoxos. The positive reals are then obtainable as ratios of quantities in a complete quantitative domain, and zero and the negative reals by essentially the move by which the integers are standardly constructed as difference-pairs of natural numbers. My construction, taken by itself, establishes only a conditional result: if there exists a complete quantitative domain, then the reals may be introduced as ratios of quantities on it. However, as I argue in the second half of the paper, there is a route by which a neo-Fregean may establish the existence of at least one complete domain, starting with the natural numbers (as given by Hume’s principle), by successively applying ratio-abstraction to obtain a full domain and a suitably adapted version Dedekind’s method of cuts to obtain from this a complete domain. Two points deserve emphasis: first, quantities, though (on my account) abstract objects which are sharply to be distinguished from the concrete entities which stand in various quantitative relations to one another, are not themselves to be identified with numbers; and second, although I use a version of Dedekind’s method in proving the existence of a complete domain, there is no question, on the present approach, of defining the reals as or in terms of Dedekind cuts. Here is not the place to elaborate upon the significance of these points. The first is, I believe, integral to the defence of my approach against several more or less familiar objections to older attempts to treat real numbers as directly abstracted from quantitative relations among concrete entities— but that defence is best conducted in the context of a more searching analysis

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of the notion of quantity than I have had space for here. Such an analysis would also do much to motivate the axiomatic characterisation of quantitative domains which I have been obliged to state somewhat dogmatically, without the philosophical defence it surely requires. The second is essential to the claim of the present approach to respect Frege’s belief—I would say, insight— that a satisfying foundational account of the real numbers should introduce them in a way which expressly provides for their applications. 30, 31

30 If one disregards this constraint—as I think one should not—then it would, of course, be possible to obtain the reals by Fregean abstraction in a much simpler and more direct way than I have described. One might, for example, start with the natural numbers as given by Hume’s principle, obtain rationals by some form of ratio-abstraction (such as that employed here, but there are obviously other ways in which this might be done) and then directly introduce the reals as cuts by cut-abstraction (either as explained here, or in some similar way). 31 I am indebted to an anonymous referee for this journal, and to Roy Cook, Jim Edwards, Gary Kemp, Pierluigi Miraglia, Philip Percival, Stewart Shapiro, Neil Tennant and Crispin Wright for helpful discussion of earlier versions of this material, as well as to my audiences at presentations of parts of it in Cambridge, Columbus OH, Glasgow, L’Institut d’Histoire et Philosophie des Sciences et des Techniques in Paris and St. Andrews. Very special thanks are due to my colleague Adam Rieger. Work on this paper was carried out during my tenure of a British Academy Research Readership—I am most grateful to the Academy for its generous support.

THE STATE OF THE ECONOMY: NEO-LOGICISM AND INFLATION 1,2 Roy T. Cook

1.

Introduction

In recent years there has been a resurgence of interest in logicism as a viable philosophy of mathematics, stemming in great part from Crispin Wright’s Frege’s Conception of Numbers as Objects [1984] and the formal and philosophical work of George Boolos. Before this work it was generally accepted that Frege’s project of reducing mathematics to pure logic was devastated by Russell’s detection of a paradox produced by Frege’s notorious Basic Law V. Frege’s project has recently been reborn, with some modifications. In this paper I explore some of the landscape surrounding this project, concentrating on the prospects for a successful neo-logicist reconstruction of the real numbers. I focus on Bob Hale’s “Reals by Abstraction” [2000] and his use of a cut abstraction principle, as this approach seems to be the one most likely to be generalizable to complex analysis, functional analysis, etc. There is a serious problem that plagues Hale’s project. Natural generalizations of the sort of principle needed to construct the reals imply that there are far more objects than one would expect from a position that stresses its epistemological conservativeness. In other words, the sort of abstraction needed to obtain a theory of the reals is rampantly inflationary. After arguing for this claim with respect to Hale’s treatment, I will indicate briefly why this problem is likely to reappear in any neo-logicist reconstruction of real analysis. 1 This paper first appeared in Philosophia Mathematica 10[2002], pp. 43–66. Reprinted by kind permission of the editor and Oxford University Press. 2 This title is taken from a phrase used by Bob Hale [2000] in his own discussion of neo-logicist abstraction principles and domain inflation.

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Abstraction principles An abstraction principle 3 is any second-order formula 4 of the form: (∀P)(∀Q)[@(P) = @(Q) ↔ E(P, Q)]

“@” here is a function from properties (or relations) to objects, and E is an equivalence relation on the properties (or relations). Abstraction principles allow us to take, as objects, characteristics that the properties or relations have in common. Frege’s Basic Law V is: BLV : (∀P)(∀Q)[EXT(P) = EXT(Q) ↔ (∀x)(Px ↔ Qx)] Frege derives all of arithmetic from BLV plus second-order logic, but Russell’s discovery that BLV is inconsistent with the second-order comprehension axiom renders this result less noteworthy. The resurrection of logicism stems for the observation that Frege’s only ineliminable use of BLV occurs in his derivation of Hume’s Principle: HP : (∀P)(∀Q)[NUM(P) = NUM(Q) ↔ P ≈ Q] [P ≈ Q is the second-order formula asserting that there is a one-to-one correspondence between the P’s and the Q’s] 5 The “NUM” operator is, in effect, a number generating function, mapping properties onto the number corresponding to the cardinality of the extension of the property. Unlike BLV above, HP is consistent. It can be added to any theory that has an infinite model, and the new theory will have (infinite) models of the same cardinality 6 as the original theory. Frege’s derivation of arithmetic in the Grundgesetze can be reconstructed from second-order logic plus HP, thereby avoiding the troublesome BLV. 7 This result, quite remarkable as a mathematical fact independent of any philosophical implications, has come to be called Frege’s Theorem. 8 Of course, HP, with its explicit reference to numbers via the “NUM” function, is not a logical truth. Thus, the neo-logicist must abandon the hope that 3 I am ignoring “objectual” abstraction principles where the abstraction operator “@” maps objects onto objects, as the phenomena that interest us here involve only “conceptual” abstraction, where properties or relations are mapped onto objects. 4 I assume standard set theoretic semantics for second-order logic, where the second-order predicate variables range over the full powerset of the domain, and which therefore satisfies the comprehension scheme:

(∃R)(∀x1 , x2 , . . . , xn )(R(x1 , x2 , . . . , xn ) ↔ ) for each formula not containing R free. For details see Shapiro [1991]. 5 Although I often phrase the equivalence relation for an abstraction principle in everyday English, every abstraction principle considered in this paper can be expressed using only the resources of second-order logic (plus, in some cases, previously defined abstraction operators). 6 This result depends on the axiom of choice. 7 See Boolos [1987] and Heck [1993]. 8 Another abstraction principle which will be used in examples later in the paper is a size-restricted version of Basic Law V: NewV : (∀P)(∀Q)[EXT(P) = EXT(Q) ↔ ((Pis “Big” ∧ Qis “Big”) ∨ (∀x)(Px ↔ Qx))]

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one can reduce all of mathematics to truths of pure logic, but this is not surprising. The discovery of Russell’s paradox, coupled with the failure of Russell and Whitehead’s subsequent logicist attempt in the Principia Mathematica [1913], suffice to render the original logicist project implausible. In addition, Boolos argues that this sort of reduction of mathematics to pure logic is in principle impossible: mathematics has ontological commitments, while on the contemporary conception logic does not. 9 One can argue, however, that the crucial aspect of Frege’s logicism is not the reduction of all of mathematics to truths of logic. Instead, Frege’s main goal was to demonstrate the analyticity of mathematics, saving it from Kant’s charge of a priori yet synthetic (see Coffa [1991]): The problem becomes, in fact, that of finding the proof of the proposition, and of following it up right back to the primitive truths. If, in carrying out this process, we come only on logical laws and on definitions, then the truth is an analytic one . . . If however, it is impossible to give the proof without making use of truths which are not of a general logical nature, but belong to the sphere of some special science, then the proposition is a synthetic one. (Frege [1884], p. 4, emphasis added)

According to Frege, the aprioricity of mathematics is a direct consequence of its analyticity: For a truth to be a posteriori, it must be impossible to construct a proof of it without including an appeal to facts, i.e. to truths which cannot be proved and are not general. But if, on the contrary, its proof can be derived exclusively from general laws, which themselves neither need nor admit of proof, then the truth is a priori. ([1884], p. 4)

Thus, the reduction of mathematics to logic was just the particular strategy Frege adopted to secure the analyticity 10 and apriority of mathematics. Although Frege abandoned his project, Wright has revived it, stressing that the part of Frege’s project that is of interest is not the reduction of mathematics to logic but rather a demonstration of the analyticity, or at least a prioricity, of (much of) mathematics: Frege’s Theorem will still ensure . . . that the fundamental laws of arithmetic can be derived within a system of second-order logic augmented by a principle whose role is to explain, if not exactly to define, the general notion of identity of cardinal number, and that this explanation proceeds in terms of a notion which can be defined in terms of the concepts of second-order logic. If such an explanatory [where “P is Big” is an abbreviation for the second-order formula asserting that the P’s are equinumerous with the entire domain]. NewV is consistent, and satisfied by the hereditarily finite sets Vω . Many of the standard axioms of ZFC (but not infinity or powerset) can be reconstructed using NewV. 9 Except for (∃x)(x = x), the claim that there is at least one object. Even this, however, is only accepted for convenience. We could easily formulate a logic that countenanced the empty model. 10 The assumption that logic is analytic if anything is seems unproblematic. Even Quine admits something like this in “Two Dogmas of Empiricism” [1951]!

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The Arché Papers on the Mathematics of Abstraction principle . . . can be regarded as analytic, then that should suffice . . . to demonstrate the analyticity of arithmetic. Even if that term is found troubling, as for instance by George Boolos, it will remain that Hume’s principle – like any principle serving implicitly to define a certain concept – will be available without significant epistemological presupposition . . . So one clear a priori route to the recognition of the truth of . . . the fundamental laws of arithmetic will have been made out. And if in addition [Hume’s principle] may be viewed as a complete explanation – as showing how the concept of cardinal number may be fully understood on a purely logical basis – then arithmetic will have been shown up by Hume’s principle . . . as transcending logic only to the extent that it makes use of a logical abstraction principle – one [that] deploys only logical notions. So, always provided that concept formation by abstraction is accepted, there will be an a priori route from mastery of second-order logic to a full understanding and grasp of the truth of the fundamental laws of arithmetic. Such an epistemological route . . . would be an outcome still worth describing as logicism. ([1997], pp. 210–211, emphasis added)

Although the neo-logicists are a bit vague regarding exactly what the special status of abstraction principles is, the general idea seems to be something along the following lines: Acceptable abstraction principles provide something akin to an implicit definition of the abstracts generated by the principle, providing an explanation, although not necessarily a complete 11 explanation, of what it is to be an abstract of the relevant sort. This explanation provides us with a method by which we can come to know truths about these abstracts a priori. 12 Thus, the abstraction principles are meant, among other things, to provide some sort of epistemological advantage – the idea being that we can get all of arithmetic, for example, from the epistemologically unproblematic HP. Finally, although abstraction principles are not logical truths, the fact that they invoke only logical terminology on the right-hand side of the biconditional in giving the truth conditions of the identity on the left supports the claim that neo-logicism provides “an outcome still worth describing as logicism”. Of course, as is well known, most 13 of modern mathematics can be reconstructed quite nicely in Zermelo–Fraenkel set theory. In addition, some philosophers, such as Gödel [1947], argue that the axioms of set theory are a priori knowable. 14 The interest of the neo-logicist project, then, depends on the extent to which it can be argued that the necessary abstraction principles 11 Abstraction principles, according to many critics (and some defenders) of neo-logicism, notoriously fail to solve the “Caesar Problem”. 12 This emphasis on how we come to know the truths of mathematics seems to be what is crucial in the passages from Frege’s Grundlagen quoted above. 13 One standard textbook on set theory, Kunen [1980], contains the following as an exercise:

Verify that within ZC [ZFC minus replacement] one may develop at least 99% of modern mathematics. (p. 147) 14 Unlike the neo-logicists, Gödel would not have claimed that the axioms of set theory are analytic, as their truth depends not only on the meaning of the terms involved but also (in some manner) on our direct intuition of the set theoretic universe (see Gödel [1947]).

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are epistemically “cheap” in a way in which the axioms of ZFC are not. A first step towards this goal is an account of which abstraction principles are neo-logicistically acceptable. Mere consistency is not enough for an abstraction principle to be acceptable. Presumably if two abstraction principles are both acceptable, then their conjunction should be as well, yet we can formulate consistent abstraction principles that are only satisfiable on finite domains. 15 Such a principle and HP are not jointly satisfiable, so we need more stringent requirements on which abstraction principles are acceptable. The point of this paper is to examine one such constraint. Before moving on, a technical fact relevant to the satisfiability of abstraction principles needs to be noted: If the right-hand side of the biconditional in an abstraction principles contains no non-logical vocabulary, then the abstraction principle will be satisfiable on a domain of size κ if and only if it is satisfiable on any domain of size κ. A proof 16 of this result can be found in Fine [1998], although the reasoning behind it should be clear once one realizes that an abstraction principle only requires that there be a distinct object for each equivalence class of properties (or relations). It implies nothing regarding which object is associated with which class.

3.

Inflation

A number of requirements on abstraction principles have been proposed. The constraint that concerns us here is the idea that suitable abstraction principles should be non-inflationary. Informally this is just a requirement that the abstraction principles should not imply the existence of too many objects, reflecting the intuition, due to von Neumann, that the way to avoid the set theoretic paradoxes is by avoiding collections that are too large, i.e. what are now known as proper classes. 17 Consider BLV. The source of its inconsistency can be traced, at least in part, to the fact that it assigns a distinct object (an extension) to each collection of objects in the domain, violating Cantor’s theorem. To avoid this sort of contradiction, it is a good start to require that acceptable abstraction principles do not involve equivalence relations that partition the domain into more collections than there are objects. Kit Fine argues that: 15 The following, much discussed abstraction principle has come to be called the Nuisance Principle:

NP : (∀P)(∀Q)[NUI(P) = NUI(Q) ↔ (P, Q)] [where (P, Q) abbreviates the second-order formula asserting that the collection of objects that are either P-and-not-Q or are Q-and-not-P is finite]. NP is satisfiable on any finite domain, but on no infinite one. 16 Fine’s argument can be generalized to universal generalizations of logical abstraction principles. 17 I use the term “proper class” in the technical sense, referring to collections of sets that are too big to themselves be sets. Intuitively, at least, there are in fact collections that are (or at least might be) bigger than proper classes, such as various large collections of proper classes.

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Along similar lines, Wright writes that: The cells into which the relevant equivalence relation partitions the universe of Concepts must not outrun the population of objects which constitute the range of the first-order variables in the abstraction principle. ([1997], p. 222)

This way of phrasing the prohibition on domain inflation begs the question against the neo-logicist reconstruction of the reals, since the strategy is to add a suitable abstraction principle to a theory satisfied by a countable domain to get a theory that is only satisfied by an uncountable domain. The neo-logicist approach depends on some abstraction principles being at least somewhat inflationary. We can get at the spirit of the ban on inflationary abstraction principles with something akin to Boolos’ take on the matter: . . . it was a central tenet of logical positivism that the truths of mathematics were analytic. Positivism was dead by 1960 and the more traditional view, that analytic truths cannot entail the existence either of particular objects or of too many objects, has held sway ever since. ([1997], pp. 249–250, emphasis added)

Ignoring the issue of whether analytic principles should entail the existence of particular objects, we can assume for the sake of argument that some acceptable abstraction principles might be inflationary, i.e. their addition to a theory with models of size κ might result in a theory whose models all have domains larger than κ. This domain inflation should not be too rampant, however. Acceptable abstraction principles should not imply the existence of too many objects, at least not if they are to be “epistemologically cheap”. There may be no way to delineate exactly what “too many” means in the previous sentence. On the contrary, like the vague predicate “red”, there might be no sharp line marking off where the extension of “too many” begins. Even so, we can lay down a number of precise ways in which an abstraction principle might be inflationary, even if we cannot determine with certainty which of them are neo-logicistically acceptable and which are not. Given an abstraction principle AP and a set of object S, the restriction of AP to S is the result of replacing every first-order quantifier “∀x” (“∃x”) with “∀x ∈ S” (“∃x ∈ S”) and replacing every second-order quantifier “∀X ” (“∃X ”) with “∀X ⊆ S” (“∃X ⊆ S”). An abstraction principle AP generates κ objects when applied to the domain S iff, for every domain D such that S ⊆ D and D satisfies 18 the restriction of AP to S, the cardinality of D– S ≥ κ. (Here, 18 Since all that is relevant to the satisfaction of a logical abstraction principle is the cardinality of the domain, I say that a set D satisfies an abstraction principle AP if there is some model with D as domain that satisfies AP.

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and below, κ, γ , and λ are infinite cardinals.) 19 We now define the notion of κ-inflationary: 20 An abstraction principle AP is κ-inflationary if, for any domain S of cardinality κ, the application of AP to S generates γ objects where γ > κ. 21

Using the notion of κ-inflation, we can now define some more general senses in which an abstraction principle can be inflationary: Strictly Non-inflationary:

Locally Inflationary:

Boundedly Inflationary:

Unboundedly Inflationary: Universally Inflationary:21

An abstraction principle AP is strictly non-inflationary if there is no κ for which AP is κ-inflationary. An abstraction principle AP is locally inflationary if there are (only) finitely many k’s such that AP is κ – inflationary. An abstraction principle AP is boundedly inflationary if there are infinitely many κ‘s such that AP is κ – inflationary but there is some γ such that, for all λ > γ , AP is not λ-inflationary. An abstraction principle AP is unboundedly inflationary if, for every κ, there is a γ > κ, such that AP is γ – inflationary. An abstraction principle AP is universally inflationary if, for every κ, AP is κ-inflationary.22

HP is strictly non-inflationary, since it can be added to any theory with an infinite model and the result will have a model of the same cardinality. Shapiro and Weir [1999] show that, if the Generalized Continuum Hypothesis holds, then NewV (see note 7) is unboundedly inflationary, since on this assumption it is satisfied at every successor cardinal but at no singular cardinal. 24 Finally, it is clear that any abstraction principle used to obtain the real numbers must 19 Notice that, for any abstraction principle AP containing only logical vocabulary on the right-hand side of the biconditional and any set S of cardinality κ, if AP applied to S generates γ objects, then γ > κ. 20 I am ignoring cases where abstraction principles inflate on finite domains since they are irrelevant to the case at hand. Hume’s Principle inflates on finite domains, yet this inflation has rarely been the target of serious criticism. In addition this sort of inflation is critical to the success of the neo-logicist project given the possibility that there are only finitely many non-abstract objects in the world. 21 Since the satisfaction of an abstraction principle depends solely on the cardinality of the domain, we could have phrased this as:

An abstraction principle AP is k-inflationary if there is some domain S of cardinality κ such that the application of AP to S generates γ objects where γ > κ. 22 Locally inflationary and universally inflationary are equivalent (roughly) to Hale’s weakly and strongly inflationary, respectively (see Hale [2000], p. 121). 23 Universal inflation is one way of formalizing the intuition that some mathematical concepts given by abstraction principles are indefinitely extensible (see Dummett [1963] and note 47 below). 24 Shapiro and Weir also prove that it is consistent with ZFC that NewV has no uncountable models at all.

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be at least locally inflationary, since the point of the abstraction is to take us from a countable domain to an uncountable one where we can formulate real analysis. The question thus becomes: Where should we draw the line with respect to domain inflation? As has already been pointed out, abstraction principles that are locally inflationary must be acceptable. In addition, we can treat locally inflationary principles and boundedly inflationary principles as roughly on a par, since in both cases we are confronted with cases where the abstraction may blow up our ontology, but only so much. At some point we reach an upper limit beyond which the abstraction principle does not inflate. Along similar lines, we can think of unboundedly inflationary and universally inflationary principles as equally problematic, since in both cases the problem, if any, has to do with the fact that its (possibly repeated) application might multiply the underlying ontology without limit. Thus, we need to determine whether unboundedly or universally inflationary abstraction principles are neo-logicistically acceptable. A number of considerations can be brought to bear against unboundedly and universally inflationary abstraction principles, in addition to the points already canvassed against inflation more generally. I will give a different argument against each sort of inflation, although unbounded and universal inflation are sufficiently similar that problems with one are likely to indicate problems with the other. The neo-logicist is claiming that the abstraction principles implicitly define, or at least ground our use of mathematical concepts and theories. Definitions of the abstract objects of mathematics, even implicit ones, ought to determine a unique group of objects which necessarily fall under the definition. If this “defining” abstraction principle is unboundedly inflationary, however, then the neo-logicist has failed in his task. Assume that we have some unboundedly inflationary abstraction principle AP and there are κ objects in the universe 25 (including the abstracts guaranteed to exist by AP). Let γ be the least cardinal > κ such that AP is γ -inflationary. Then, had there been γ objects in the universe, there would have, by AP been more than γ (and thus more than κ) abstracts. But then the original abstracts are not all of the objects whose identity conditions are given by AP. This process can be repeated indefinitely (and transfinitely), so we never have all the objects that fall under the purview of AP. In other words, if AP is unboundedly inflationary then it fails to secure a definite collection of objects as the domain of its abstraction operator, but instead gives us different abstracts relative to how many objects exist. 26 25 This way of setting things up implies that AP is not κ-inflationary.

26 There is a tempting response at this point. One might point out that, in addition to the abstracts falling

under the principle AP, other sorts of abstract objects such as real numbers, sets, and groups will also exist and will exist necessarily. Therefore, the cardinality of the actual world is the same as the cardinality of every possible world, namely, however many objects could possibly exist. In other words, even if AP alone does not secure a definite extension for the abstraction operator, this unique extension might be secured by AP plus the fact that all of the sets of ZFC exist. While this is true, it does not save unboundedly inflationary

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This argument against unbounded inflation is quite compelling, especially if we think that abstraction principles should generate abstract mathematical objects that exist necessarily instead of providing an undetermined multitude of objects whose existence depends on the number of non-abstract objects present in the universe. The case against universal inflation is a bit different, but equally worrying. In moving from an abstraction principle that is unboundedly inflationary to one that is universally inflationary we have replaced one problem with another. With unbounded inflation there was no unique collection of abstracts generated by the abstraction. In the case of universal inflation, there might be a unique collection of objects generated by the abstraction principle, but if so, then it is an extremely badly behaved collection. In other words, if an abstraction principle is universally inflationary, then it will be satisfied (if satisfied at all) only by a structure that is at least the size of the smallest proper class. Assume that AP has a set-sized model M. Then there is some κ such that the domain of M has cardinality κ. If AP is universally inflationary, then application of AP to a domain of size κ produces γ objects for some γ > κ. M satisfies AP, so the domain of M contains at least γ objects, but then the cardinality of the domain is greater than κ. Contradiction. This sort of issue seems to be what Hale has in mind where he writes that, although boundedly inflationary abstraction principles are neo-logicistically acceptable: It is much more plausible to require that acceptable abstraction principles not be strongly inflationary [equivalent to my universally inflationary]. Some of the neo-Fregean’s key abstractions, including the other crucial second-order abstraction, Hume’s Principle, satisfy this requirement. ([2000], p. 120)

He adds in a footnote that: . . . it might seem that strong inflation is bound to give rise to a version of Cantor’s paradox. It might also be thought that if an abstraction is strongly inflationary, then there could be no hope of showing that it is satisfiable, i.e. has a model. (p. 120)

Hale touches on two main worries here, each of which deserve closer scrutiny. First, there is the claim that universally inflationary abstraction principles are likely to be susceptible to set-theoretic paradoxes such as Cantor’s paradox (or Russell’s or Burali-Forti’s). The idea is simple: If, for any κ-sized collection, the universally inflationary principle AP generates, say, 2κ new objects, then it seems plausible that when applied to a proper class, or any other sort of structure, it would also inflate. This is one way of explaining what goes wrong with Frege’s Basic Law V. The reasoning is not general, however. There could be abstraction principles that inflated on all sets but did not inflate on proper classes. For example, principle AP might inflate on any collection that can be abstraction principles from the force of the objection, because an adequate definition should determine a unique extension independently of the existence of any other objects.

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well-ordered, but on no structure that cannot. If this is the case, then, as long as there are proper classes too large to be well ordered, AP might be satisfiable even though it is universally inflationary. We will see a potential candidate for such a satisfiable yet universally inflationary abstraction principle below. Hale’s second worry is that we might be faced with insuperable difficulties when attempting to prove the consistency/satisfiability of universally inflationary abstraction principles. The standard definition states that a sentence is satisfiable if and only if there is a set theoretic model (read: set as domain plus appropriate assignments to various bits of language) such that the sentence is true in that model. Any universally inflationary abstraction principle fails to be satisfiable in this sense, yet it is still possible that some structure (such as a proper class) might make the sentence true. This is a serious problem for the neo-logicist. As we have seen, some abstraction principles are consistent while others that resemble the former a great deal are not. Thus, one of the most important parts of defending a neo-logicist abstraction principle as acceptable is to demonstrate its satisfiability. This will prove difficult, if not impossible, for universally inflationary abstraction principles since our methods for studying and manipulating proper classes are less powerful and less secure than our set theoretic machinery. There is a disturbing historical irony here. The notion of proper class was introduced as a result of, among other things, reflection on what exactly went wrong in Frege’s Grundgesetze. The idea was to draw a distinction between the logically safe sets and the problematic proper classes, which were in some sense too large to be safely manipulated like sets. If the neo-logicist reconstruction of Frege’s project pushes us once again into the realm of proper classes, historical sensitivity (or perhaps merely superstition) should cause some worry. 27 Thus, the neo-logicist should be extremely wary of unboundedly and universally inflationary abstraction principles. While these sorts of abstractions are not necessarily susceptible to the sorts of paradoxes usually associated with “bad” abstraction principles, and are even (possibly) satisfiable, they nevertheless take us far from the epistemically innocent implicit definitions that the neo-logicists argue acceptable abstraction ought to provide.

4.

Hale’s reconstruction of the reals

I take it as a desideratum of a successful philosophy of mathematics that it must account for enough mathematics to handle scientific applications. It follows that the neo-logicists need, at a minimum, to be able to reconstruct the theory of the real numbers. In fact, the success or failure of the neo-logicist project seems to hinge on their successful treatment of the real numbers. If this difficult case can be dealt with, then it is plausible that most other areas of 27 This point was brought to my attention by Jon Cogburn.

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contemporary mathematics can be handled by relatively unproblematic neologicist constructions building on the continuum. On the other hand, if the neo-logicist is unable to account for the reals, then the project has failed to provide a foundation 28 for mathematics. Reconstructing arithmetic from unproblematic abstraction principles might be interesting, and even mathematically important, but arithmetic is too simple a theory to allow us to conclude anything interesting about mathematics as a whole. It is at this point that the work of Bob Hale [2000] becomes relevant. Although others, including Simons [1987] and Dummett [1991] have written on Frege’s treatment of the reals, Hale is the first to attempt a full-scale neologicist account. Thus Hale’s work is of independent interest in an investigation of the prospects for a neo-logicist reconstruction of analysis. More importantly, given our purposes here, Hale’s account provides us with a useful case study of inflationary abstraction principles. Hale puts much stock in the fact that the reals are not just any sort of mathematical object but are, like the natural numbers and rational numbers, quantities: The most striking and most important features of Frege’s treatment of the reals are two: (i) the real numbers are to be defined as ratios of quantities . . . and (ii) in regard to the analysis of the notion of quantity, the fundamental question requiring to be answered is not: What properties must an object have if it is to be a quantity? but: What properties must a concept have, if the objects falling under it are to constitute quantities of a single kind? ([2000], p. 104)

Hale’s strategy is to, first, set up a general theory of quantity; second, to argue that if a certain sort of quantity exists, then ratios on those quantities can serve as the reals; and third, to prove, with the use of a novel abstraction principle, that the requisite sort of quantities exist. Hale begins by giving definitions of various sorts of “Quantitative Domain”. The series of definitions he proposes are intended to flesh out the second feature of Frege’s treatment of the reals – determining what properties a concept must have if the objects falling under that concept are quantities. A minimal q(uantitative)-domain is: . . . a non-empty collection Q of entities closed under an additive operation ⊕ which commutes, associates, and satisfies the strong trichotomy law that for any a, b ∈ Q, we have exactly one of: ∃c (a = b ⊕ c), ∃c (b = a ⊕ c), or 28 This is not to say that the work stemming from neo-logicism does not have other interesting philosophical applications. For example, Harold Hodes [1984] considers the following Order Type Abstraction Principle:

OTA : (∀P)(∀Q)[ORD(P, nd))] The new structure resulting from the application of EM to a normal q-domain Q is called R Q . 29 Finally, Hale defines a full q-domain to be a normal qdomain where we have: ∀a, b, c ∈ Q∃ q ∈ Q(RAT(a, b) =

RAT (q, c))(p. 107)

This can be reworded to avoid the reliance on abstraction. A normal q-domain is full iff: ∀a, b, c ∈ Q∃q ∈ Q(∀m, n(ma = nb ↔ mq = nc) ∧ ∀m, n(ma < nb ↔ mq < nc) ∧ ∀m, n(ma > nb ↔ mq > nc)) Finally, a q-domain Q is said to be complete iff: . . . every bounded above non-empty S ⊆ Q has a least upper bound. (p. 108)

This completes Hale’s first task – specifying what criteria a concept Q must meet for the objects falling under Q to be quantities of various varieties. Hale next points out that any two complete q-domains are isomorphic. 30 Applying the abstraction principle EM above, we get the result that, for any two complete domains Q and Q*, R Q = R Q∗ . Thus, according to Hale, the reals can be obtained as the ratios of any complete domain, as long as some such domain exists. All that remains is the third step in Hale’s argument – the proof that there is a complete q-domain. Hale’s argument is relatively straightforward: The 29 The principle EM, which is equivalent to a sort of pairing axiom, is strictly non-inflationary. 30 This fact depends on the explicit use of second-order quantifiers in the definition of complete q-

domains and their implicit use (in securing the fact that we are talking about the standard natural numbers and not some non-standard model of them) in the definition of normal q-domains.

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neo-logicist already has access to the positive natural numbers N + via HP. The natural numbers constitute a normal q-domain, but not a full one. An application of the abstraction principle EM to N + , however, gives us the ratios on the positive naturals R N + , which is a full domain (although not complete) and an obvious candidate to serve as the positive rational numbers. The next move is to apply the following Cut Abstraction Principle to R N + (reworded to fit the notation used here): C A : (∀P)(∀Q)[CUT(P) = ((∀x)((x ∈ R

N+

CUT (Q)

↔

∧ P and Q are cut properties30 on R N + ) → (Px ↔ Qx)))]31

This gives us the required complete q-domain, and we need only apply EM once more to obtain the reals. So far the neo-logicist project looks good, as long as HP, EM, and CA are acceptable. We will assume that HP and EM are acceptable, and concentrate on CA, since CA is responsible both for guaranteeing that there are uncountably many quantities and for producing the complete q-domain.

5.

Abstraction principles and abstraction processes

We have seen how CA generates a complete, uncountable q-domain from (something like) the rationals. There seems to be no principled reason for restricting this procedure, however. We should, prima facie, be able to apply cut abstraction to any linear ordering guaranteed to exist by previously accepted principles. To paraphrase Georg Kreisel, 33 one can argue that the evidence for the applicability of CA derives from the more general idea that cuts can be taken on any linear order whatsoever. In other words, we need to distinguish between abstraction principles (such as CA) and the more general 31 Hale defines a cut-property on R N + as follows:

. . . a cut property is a non-empty property whose extension is a proper subset of R N + and which is downwards closed [i.e., (∀a) (∀b) (Fa → (b < a → Fb)] and has no greatest instance [i.e., (∀a) (Fa → (∃b) (b > a ∧ Fb)]. (p. 112) We can generalize this to any linear ordering by replacing R N + with a name for the order in question. Hale’s definition, however, is a bit unwieldy, since it implies that the only linear orders with non-trivial cuts are dense. Thus, we can use a slightly modified version of Hale’s definition [for a linear order (Q, b, or a = b holds.

Notice that, in addition to its obvious similarity to Hale’s original definition of minimal q-domain, this definition respects Hale’s intuition that:

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. . . what makes the difference between quantitative ordering relations and others is that in the case of a quantitative ordering relation, but not otherwise, the entities which can significantly be asserted to stand in the relation can (at least in principle) be combined in such a way that compounds must come later in the relevant ordering than their components. In other words, for more than to be a quantitative ordering relation, there must be an operation of combination ⊕ on items lying in the field of more than, analogous to addition, such that for any a, b in more than’s field, a ⊕ b is more than a and a ⊕ b is more than b (p. 106)

In other words, the sum of two members of a quantitative domain must be larger than either member. 44 With this definition of minimal q*-domain in place, we get the following result: Theorem 3: (AC): Given an infinite cardinal κ, there is a minimal q*-domain (A, ⊕) such that: |A| ≤ κ and | Comp (A, ⊕)| > κ. Proof: Given an infinite cardinal κ, let λ be the least cardinal ≤ κ such that 2λ > κ. Let A be the subset of functions from λ (as an ordinal) into the set of positive rationals Q + such that f ∈ A iff there is an ordinal γ < λ such that for all ordinals αβ ≥ γ , f (α) = f (β). For f , g ∈ A, let h = f ⊕ g iff, for every α ∈ λ h(α) = f (α) + g(α). f < g iff at the least α such that (α) = g(α), f (α) < g(α). It is easy to verify that A is a minimal q-domain. The proof that |A| ≤ κ and |Comp(A, ⊕)| > κ is similar to the proof of Theorem 1. If we define normal q*-domain by substituting “minimal q*-domain” for “minimal q-domain” in Hale’s definition of normal q-domain, and similarly define full q*-domain by replacing “normal q-domain” with “normal q*domain”, then analogues of Theorem 3 hold for normal and full q*-domains. Given an f , g, h in the q*-domain A constructed in the proof of Theorem 3, let q be defined as: for all α < λ, q(α) = [ f (α) × h(α)] ÷ g(α). It follows that q ∈ A and f : g = q : h, so A is a full q*-domain. Universal inflation has returned. If the neo-logicist wishes to apply of cut abstraction to full q-domains but prohibit taking cuts on full q*-domains, then some relevant difference between the two sorts of structure needs to be explained. Neither definition appears, prima facie, to be more natural or intuitive than the other as an explication of our pre-formal notion of quantity, and the technical differences between the two definitions are subtle. 44 The main difference between a minimal q-domain and a minimal q*-domain is that a minimal q*domain does not have to be closed under subtraction. This does not seem absurd, at least if ordinary usage of language is our guide. It seems natural to say that “Cleopatra is as beautiful as Athena and Helen combined”, yet it is much less natural to say “Athena is as beautiful as Cleopatra is more beautiful than Helen” or “Athena’s beauty is equal to Cleopatra’s beauty minus Helen’s beauty”.

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There is one final move the neo-logicist might try in order to avoid the problems hovering around cut abstraction. Since the original Fregean idea of taking extensions as objects was salvaged via NewV by making use of the notion of “Big”-ness (see note 7), perhaps the neo-logicist can restrict cut abstraction to linear orders that are not “Big”, formulating something like the following Size-restricted Cut Abstraction Principle: SCA : (∀P)(∀Q)(∀H )(∀ λ, A has a model of size κ. So if A is unboundedly inflationary then for every cardinal λ there is a κ > λ such that A has no model of size κ. Say that A is unboundedly satisfiable if, for every cardinal λ, there is a κ > λ such that A has a model of size κ. Notice that if A is unboundedly satisfiable, then (assuming choice) we can turn any set into a model of A by adding more elements: for every set d, there is a model of A whose domain contains d. In the best cases, the “new” elements will be the new abstracts. Shapiro and Weir [1999] show that if the generalized continuum hypothesis is true, then New V is satisfiable at all regular cardinals, and so it is unboundedly satisfiable. However, it is independent of Zermelo–Fraenkel set theory (with choice) whether New V is actually unboundedly satisfiable. It might not have any uncountable models at all. So, again, how much inflation is too much? Cook [2001] argues that only strictly non-inflationary and boundedly inflationary abstraction principles should be acceptable to a neo-logicist. Let us examine the arguments, since they go to the heart of the goals of neo-logicism. 10 The usefulness of this notion turns on the axiom of choice, since in that case, for any distinct cardinals κ, κ ′ , either κ < κ ′ or κ ′ < κ.

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Concerning unbounded inflation, Cook writes: The neo-logicist is claiming that the abstraction principles implicitly define, or at least ground our use of mathematical concepts and theories. Definitions of the abstract objects of mathematics, even implicit ones, ought to determine a unique group of objects which necessarily fall under the definition. If this ‘defining’ abstraction principle is unboundedly inflationary, however, then the neo-logicist has failed in his task.

Assume, for example, that an abstraction principle A is unboundedly inflationary, and suppose that M is a model of both A and the background theory T . Let κ be the cardinality of the domain of M, and let γ be the smallest cardinal greater than κ such that A is γ -inflationary. Cook continues: . . . had there been γ objects in the universe, there would have, by [ A], been more than γ (and thus more than κ) abstracts. But then the original abstracts are not all of the objects whose identity conditions are given by [A]. This process can be repeated indefinitely (and transfinitely), so that we never have all the objects that fall under the purview of [ A]. In other words, if [A] is unboundedly inflationary then it fails to secure a definite collection of objects as the domain of its abstraction operator, but instead gives us different abstracts relative to how many objects exist.

In a note, Cook adds that “an adequate definition should determine a unique extension independently of the existence of any other objects”. Some abstraction principles do characterize a unique domain of objects, at least up to isomorphism. The present cut abstraction principle (CP), for example, yields all and only the real numbers, plus two extra abstracts. Another example would be the restriction of Hume’s principle to finite concepts (see Heck [1997]). This yields all and only (an isomorphic copy of) the natural numbers. Cook is correct that if an abstraction principle A is intended to characterize a unique structure (such as the natural numbers or the real numbers), then it should not be unboundedly inflationary. In this case, A should yield the required objects and no others. It should not inflate on any domain that is as large as or larger than the requisite structure. However, it is not true that every legitimate abstraction principle determines “a definite collection of objects” as the range of the defined operator. Some principles do yield “different abstracts relative to how many objects exist”. Consider Hume’s principle. Thanks to Frege’s theorem, it implies the existence of the natural numbers and the cardinality of the natural numbers (i.e., ℵ0 ). But what of other cardinalities? Since Hume’s principle has countable models, it does not, by itself, entail that the cardinality of the continuum exists. But Hume’s principle does entail that if there is a property that holds of continuummany objects, then the cardinality of the continuum exists. So, for example, Hume’s principle and (CP) together entail that the cardinality of the continuum exists. In general, which cardinal numbers exist depends on how many objects

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there are. I, at least, do not see this as a problem with Hume’s principle as an abstraction. Some acceptable abstractions are open-ended, in the sense that the abstracts they yield depend on the ontology of the background theory. Increasing the ontology might increase the abstracts. A different problem with unboundedly inflationary abstraction principles is that they may conflict with each other. Weir [2000] formulates a pair of “distraction” principles B, B ′ , such that B and B ′ are each unboundedly satisfiable, but are mutually inconsistent. Suppose that the background theory has a model of size κ0 . To extend this to satisfy B, we add κ1 > κ0 abstracts. But this new model does not satisfy B′ . To satisfy B′ , we add κ2 > κ1 abstracts. But once we add these abstracts to satisfy B ′ , we no longer satisfy B. To (re-)satisfy B, we have to add κ3 > κ2 more abstracts. But then we no longer satisfy B′ . In short, the conjunction B&B ′ is universally inflationary. Faced with such a pair of abstractions, the neo-logicist must find a principled way to choose among them. Or else she can play it safe and reject any unboundedly inflationary abstraction principle, and require that all acceptable abstractions be boundedly inflationary. Then, once we are satisfied that the universe is sufficiently large, the abstraction will be satisfied no matter how much larger we go on to recognize the universe to be. Let us turn to Cook’s treatment of universally inflationary principles—those that inflate on every cardinality. Of course, if an abstraction A is inconsistent, then it is unacceptable. Suppose that A is consistent, but universally inflationary. Let b be a set and κ = |b|. Since A is κ-inflationary, A cannot be satisfied on b. Since b is arbitrary, A has no models whose domain is a set (with a cardinality). As Cook puts it, A “will be satisfied (if satisfied at all) only by a structure that is at least the size of proper class”. This, he says, is problematic, since proper classes are “extremely badly behaved”. The idea is that if A can be satisfied only on a proper class, then it yields a proper class of abstracts (so to speak). Thus, the abstraction takes “us far from the epistemically innocent implicit definitions that the neo-logicists argue acceptable abstraction ought to provide”. In sum, Cook’s claim is that the “generation” of a proper class of abstracts is incompatible with the epistemic goals of neo-logicism. Notice that this judgement comes from the external perspective. Internally, the neo-logicist claims that we can come to know about the existence of some objects through deduction from principles in the neighborhood of implicit definitions or analytic truths. Externally, we use the set-theoretic meta-theory, accepted already by the established mathematician, to show that a certain abstraction principle yields more objects than there are members of any element of the set-theoretic hierarchy. Cook seems to hold that a proper class of abstracts is indeed “too many” objects to obtain this way. As noted above, for neo-logicism to have a chance, we have to temper the widely-held view that definitions, or principles much like definitions, have no ontological consequences. Cook’s claim, in effect, is that enough is enough. He presupposes that from the external

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perspective of an advocate of Zermelo–Fraenkel set theory, the abstracts must constitute (or be equinumerous with) a set. But the fact is that the objects of mathematics do not constitute a set, for well-known reasons. So Cook’s thesis entails that neo-logicism must fall short of its grand goal of providing an epistemic foundation for all of mathematics. A neo-logicist set theory and a neo-logicist theory of ordinals and cardinals is out of the question. Thus, the neo-logicist must rest content with an account of arithmetic, real and complex analysis, and perhaps a little more. The main (external) question that remains is just how big the neo-logicist’s ontology can be. What is the cardinality of the objects yielded by all acceptable abstractions together? Presumably, it will be ℵα , for some ordinal α. If the neo-logicist wants to avoid demanding revisions to established mathematics, she must provide some other epistemic foundation for those branches of mathematics— such as set theory, ordinal theory, and cardinal theory—whose ontology is not a set. For what it is worth, I believe that Cook’s view begs the question against the neo-logicist quest. So far as I know, no argument has been given that the objects yielded by an abstraction principle must constitute a definite, set-sized totality. The neo-logicist thesis is that an acceptable abstraction is akin to an implicit definition, providing an epistemic foundation of the theory of the objects it yields. There is no requirement that the objects be limited in any way, or that they constitute a definite totality. Perhaps further discussion of this should await either specific arguments concerning the limits of abstraction principles or the presentation of particular candidate principles that do yield a proper class of abstracts. We briefly revisit the issue at the end of the next section.

6.

The acceptability of cut abstraction

I now turn to the inflation and satisfiability of the abstraction principles presented here: (DIF), (QUOT), and (CP). Unlike Basic Law V and Hume’s principle, the quantifiers in all three of these principles are restricted. Since the right hand side of the difference principle (DIF) explicitly invokes addition on the natural numbers, (DIF) entails the existence of a difference-abstract for each pair of natural numbers, but that is all. It says nothing about “differences” of (pairs of) other objects, and in particular, it does not yield differenceabstracts for (pairs of) difference-abstracts. Similarly, the quotient principle (QUOT) yields a ratio for each pair of integers, but nothing else. And (CP) yields a cut for each property of rational numbers, but nothing else. Since there are only countably many integers, the difference principle is satisfiable on any domain that contains the natural numbers (using standard coding techniques if necessary). In a sense, (DIF) is universally satisfiable in that it is satisfiable on any domain over which it is defined. And so it does not inflate on any such domain. Since there are only countably many

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rational numbers, the quotient principle (QUOT) is satisfiable on any domain that contains the integers, i.e., on any domain on which it is defined. Since there are continuum-many distinct cuts, (CP) inflates on the rational numbers, but that is the end of its inflation. So (CP) is boundedly inflationary, in that it is satisfiable on any domain that is at least the size of the continuum and contains the rational numbers. Concerning inflation and satisfiability, the neo-logicist cannot do any better than this. If she hopes to recapture real analysis, she will need principles that yield continuum-many abstracts. The cut principle does that, and no more. Perhaps we should not be sanguine. The main reason why (CP) does not inflate beyond the real numbers is that it only defines cuts for properties of rational numbers. But we can mimic the development of (CP) for any linear order 11 “< ” defined on a set or class h. Let P be a property of items in h, and suppose that r ∈ h. Say that r is an upper bound of P, written P ≤ r , if for any s ∈ h, if Ps then either s < r or s = r . In other words, P ≤ r if r is greater than or equal to any object that P applies to (under the given linear order). Consider the following abstraction principle: (h,

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