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KANT'S PHILOSOPHY OF MATHEMATICS
SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY...
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KANT'S PHILOSOPHY OF MATHEMATICS
SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE [I
MODERN ESSAYS
I
1 Managing Editor:
Edited by
CARLI. POSY JAAKKO HINTIKKA, Boston University
Duke University, North Carolina, U_SA
Editors: DONALD DAVIDSON, University of California, Berkeley GABRIEL NUCHELMANS, University ofLeyden WESLEY C. SALMON, University of Pittsburgh
KLUWER ACADEMIC PUBLISHERS VOLUME 219
DORDRECHT I BOSTON I LONDON
Library of Congress Cataloging-in-Publication Data
TABLE OF CONTENTS Kant's philosophy of mathemat1cs modern essays I edtted by Carl J. Posy. p. cm. -- (Synthese library; v. 219) ISBN 0-7923-1495-6 (HB : aCid-free paper) 1. Mathematics--Phllosophy. 2. Kant. Immanuel. 1724-1804. I. Posy. Carl J. II. Series. QAB.4.K36 1992 91-86207 501--dc20
ISBN 0-7923-1495-6
PREFACE
vii
ACKNOWLEDGEMENTS
ix
CARL J. POSY -
Introduction: Mathematics in Kant's Critique of Pure Reason
I
I. CLASSIC PAPERS OF THE 1960'S AND 1970'S
Published by Kluwer Academic Publishers P.O. BOX 17, 3300 AA Dordrecht. The Netherlands. I I
, I
I
Kluwer Academic Publishers incorporates the publishing programmes ofD. Reidel, Martinus Nijhoff. Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publisher. 101 Philip Drive. Norwell, MA 02061, U.S.A. In all other countries, sold and distributed
by Kluwer Academic Publishers Group, P.O. BOX 322, 3300 AH Dordrecht, The Netherlands. Printed on acid-free paper
All Rights Reserved © 1992 by Kluwer Academic Publishers
No part of the material protected by this copyright notice may be reproduced or utilized in any fonn or by any means, electronic or mechanical, including photocopying, recording or by any infonnation storage and retrieval system, without written permission from the copyright owner.
JAAKKO HINTIKKA - Kant on the Mathematical Method CHARLES PARSONS - Kant's Philosophy of Arithmetic - Postscript MANLEY THOMPSON - Singular Tenns and Inmitions in Kant's Epistemology - Postscript PHILIP KITCHER - Kant and the Foundations of Mathematics
21/ 43
69 81
102 109
II. RECENT WORK CHARLES PARSONS - Arithmetic and the Categories J. MICHAEL YOUNG - Construction, Schematism and Imagination MICHAEL FRIEDMAN - Kant's Theory of Geometry STEPHEN BARKER - Kant's View of Geometry: a Partial Defense ARTHUR MELNICK - The Geometry of a Fonn of Inmition WILLIAM HARPER - Kant on Space, Empirical Realism, and the Foundations of Geometry CARL J. POSY - Kant's Mathematical Realism GORDON G. BRITIAN, JR. - Algebra and Inmition JAAKKO HINTIKKA - Kant's Transcendental Method and His Theory of Mathematics
Printed in the Netherlands
v
135 v 159 v177
221 245 v 257 293 315 341
Ii vi
TABLE OF CONTENTS
CONTRIBUTORS
361
INDEX
363
"'111
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PREFACE
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For the bulk of the twentieth century, Kant's philosophy of mathematics fell into low disrepute. His views suffered partly from the nineteenth and twentieth century challenges to the hegemony of Euclidean geometry. And their reputation declined partly because the great thinkers - Russell, Hilbert, Brouwer and others - who shaped the twentieth century's philosophical approach to mathematics each took aim in his own way at central Kantian doctrines. To be sure, the early years of the twentieth century saw lively debates about Kant's mathematical views. (The famous interchanges between Russell and Couterat are but one instance of this.) But the generally negative assessment of Kant's philosophy of mathematics led naturally to general neglect. So, with scant exception, the middle of the century saw little creative work on this topic. The one main exception is the late Gottfried Martin's work. Martin's Arithmetik und Kombinatoric bei Kant appeared in 1938. Though the book's positive thesis - that Kant viewed arithmetic as a fonnal axiomatic science is still highly controversial, its scholarship and its treasure trove of insights about Kant's way of thinking remain unsurpassed. It is indeed still the best starting place for anyone who wishes to do scholarly work on reactions to Kant's philosophy of mathematics. But for a long time Martin's work had sadly small impact on the wider philosophical community, and exploration of Kant's views about mathematics remained a philosophical backwater. All of that changed in the mid-1960's with the publication of a lively debate about issues in Kantian mathematics between Charles Parsons and J aakko Hintikka. The debate, which was fascinating in its own right, - each of these authors is an accomplished logician and a solid Kantian scholar had the added effect of reigniting philosophical interest in Kant's );houghts about mathematics. A trickle of fascinating studies - inspired by the issues that Parsons and Hintikka raised - followed in the 1970's. Then in the next decade the movement grew into a full-fledged renaissance of interest in Kant's philosophy of mathematics. This was further sparked by a 1983 conference on the topic at Duke University. vii Carl J. Posy (ed.), Kant's Philosophy of Mathematics, vii- viii.
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The present anthology is designed first of all to chronicle and publicize this renaissance. But hopefully it will also show the importance of Kantian thinking for general issues in the philosophy of mathematics. Part I includes a pair of the seminal papers by Hintikka and Parsons. It also includes influential papers by Manley Thompson and Philip Kitcher which followed up on this initial work and which appeared in the ensuing decade. Parsons and Thompson have added more recent postscripts to their papers. Part II contains a selection of papers that represent the flowering of interest in Kant's philosophy of mathematics during the 1980's and the beginning of the 1990's. It includes new work by both Hintikka and Parsons. I hope that the essays in this volume will demonstrate that Kant's views about mathematics are interesting, that they are defensible (or at least respectable in the light of our current knowledge) and that they are inseparable from his overall philosophical system. Indeed, a leisurely thumb through the volume will readily testify to the fascination that many modem Kant scholars and philosophers of mathematics have for Kant's mathematical views. As for defensibility, you will find in the pages that follow quite a few attempts to vindicate or reconstruct Kantian doctrines and arguments from a
I
modem perspective. In fact, far from viewing Kant's work on mathematics as dead, the volume contains several attempts to connect his views with quite recent findings in physical science, cognitive science and mathematical logic. Finally, you will find the volume rich and diverse with suggestions connecting Kant's mathematical views with his overall philosophy. This collection contains a mixture of scholarship and specUlation, reconS!rUction and close textual analysis.1t represents it fair sample of the current state of thinking on Kant's philosophy of mathematics. To be sure, you will find no unanimity on any single point; and it is hard to tell which, if any, of the perspectives and interpretations presented here will prevail. This very
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diversity and vigorous debate, however, are themselves the signs of a vibrant
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ACKNOWLEDGMENTS
Part of the editorial and clerical costs for this anthology were underwritten by a generous grant from the Duke University Research Council. The editor is also indebted to Tod Davis and Marianne Grochowski for editorial assistance. The editor gratefully acknowledges the permission given by the authors, editors and copyright holders of the following publications to reproduce them in original or revised form in this anthology: "Kant on the Mathematical Method" was first published in The Monist, volume 51, 1967, pages 352-375. It is copyright © 1967 The Monist, and is reproduced here by kind permission of Jaakko Hintikka and the publishers of The Monist, La Salle, Illinois, 61301. "Kant's Philosophy of Arithmetic" was first published in Sidney Morgenbesser, Patrick Suppes, and Morton White, eds., Philosophy, Science and Method: Essays in Honor of Ernest Nagel, 1969, St. Martin's Press. It is reproduced here by kind permission of Charles Parsons and the editors. The Postscript to "Kant's Philosophy of Arithmetic" was first published in Charles Parsons, Mathematics in Philosophy: Selected Essays, copyright 1983, Cornell University Press. It is reproduced here by kind permission of· Charles Parsons and Cornell University Press.
field of study.
Durham, 1991
"Singnlar Terms and Intuitions in Kant's Epistemology" was originally published in the Review of Metaphysics, Volume 26, copyright 1972, it is repnnted here by kind permission of Manley Thompson and the Review of Metaphysics. "Kant and the Foundations of Mathematics" was first published in The Philosophical Review, Volume 84, copyright 1975. It is reprinted here by kind permission of Philip Kitcher and The Philosophical Review.
ix Carl J. Posy (ed.), Kant's Philosophy of Mathematics. ix-x.
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ACKNOWLEDGEMENTS
"Arithmetic and the Categories" was first published in Topoi volume 3, copyright 1984, It is reprinted here by kind permission of Charles Parsons and of Topoi. "Construction, Schematism, and Imagination" was first published in Topoi volume 3, copyright 1984. It is reprinted here by kind permission of J. Michael Young and of Topoi. "Kant's Theory of Geometry," was first published in The Philosophical Review, Volume 94, copyright 1985. It is reproduced here by kind permission of Michael Friedman and of The Philosophical Review. "The Geometry of a Form of Intuition" was first published in Topoi volume 3, copyright 1984. It is reprinted here by kind permission of Arthur Melnick and of Topoi. "Kant on Space, Empirical Realism and the Foundations of Geometry" was first published in Topoi volume 3, copyright 1984. It is reprinted here by kind permission of William Harper and of Topoi. "Kant's Mathematical Realism," was first published in The Monist, volume 67, pages 115-134. It is copyright © 1984 The Monist, and is reproduced here by kind permission of the publishers of The Monist, La Salle, minois, 61301. "Kant's Transcendental Method and His Theory of Mathematics," was first published in Topoi volume 3, copyright 1984. It is reprinted here by kind permission of Jaakko Hintikka and of Topoi.
CARL
J.
POSY
INTRODUCTION: MATHEMATICS IN KANT'S CRITIQUE OF PURE REASON
Kant's views about mathematics have had eloquent and influential detractors among the very thinkers who shaped our present philosophical world view: Frege ridiculed Kant's claims about the role of pure intuition in arithmetic.! Russel denounced the Kantian theory of geometry.2 And Brouwer, who built his intuitionistic mathematics on a Kantian premise about the a priority of time, explicity rejected the a priority of space.3 Moreover these critics follow generations of "mainstream" dissenters going back at least to Eberhard.4 So even sympathetic readers of Kant are tempted to isolate his philosophy of mathematics as a small, separable, outmoded and embarrasing imperfection in the full critical framework. 5 But nothing could be further from the truth. The essays in this volume will, I believe, serve to affirm that Kant's views about mathematics are interesting, and that where they may be ntistaken the mistakes are explainable. Indeed, though the authors of these essays are not mindlessly uncritical of Kant, you will find in the pages that follow quite a few attempts to vindicate or reconstruct Kant's mathematical views from a modem perspective and to connect those views with quite recent work in empirical science. in mathematics and in logic. In this introduction, however, I want to address the question of separability. References to and even sustained discussions of mathematics abound in all of Kant's philosophy from the early writings through the great Critiques and their collateral works. But Kant's pivotal book is clearly the Critique of Pure Reason; so in this essay I shall introduce the papers that follow by showing how some of the issues that they raise bear on the central themes of that work. My point is not merely that the first Critique touches on mathematical issues. Rather, in introducing these papers I want to show you that these mathematical issues are bound intimately into the fabric of epistemology, metaphysics and philosophy of science that is woven by the Critique. I would like you to see that the effort extended in the present collection of essays to understand Kant's views about mathematics will also necessarily illuminate his most basic philosophical doctrines. I Carl J. Posy (ed.), Kant's Philosophy afMathematics, 1-17. tel 1QQ?' T(/lIwpr Academic Publishers. Printed in the Netherlands.
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CARL J. POSY
I have fu-ranged these introductory remarks according to the major divisions of the Critique of Pure Reason. And for readers who may not be familiar with its general outline, I have also included brief sketches of the strocture of the Critique and of its main sections. I. THE" AESTHETIC"
1. The Structure of the first Critique and of the "Aesthetic" Kant divided the Critique of Pure Reason into a "Doctrine of Elements" and a "Doctrine of Method." The Doctrine of Elements studies the scope and limitations of empirical knowledge. It focuses on the grounds of a priori knowledge (specifically on the possibility of synthetic a priori knowledge, which is Kant's response to Hume's skepticism) and argues that we can discover these possibilities by examining the presuppositions of unproblematic empirical knowledge. The "Transcendental Aesthetic," the first main division of the Doctrine of Elements, isolates the faculty of sensibility and explores its contribution to a priori knowledge. The second main division, the "Translendental Logic," (which concentrates on t"e intellect) is itself divided into two parts: The "Analytic" concerns the role of the intellect in empirically (and therefore sensorily) based knowledge. (The intellect thns limited is what Kant calls the faculty of understanding.) The "Dialectic" demonstrates that this faculty cannot be employed to provide nontrivial knowledge which is not sensorily based. ("Reason" is Kant's name for the intellect when it is not limited to sensorily based knowledge.) Central to Kant's arguments in all three of these divisions is his so called "critical" move (his analysis of the distinct epistemological, semalltic and ontological contributions of these separate faculties) and his doctrine of transcendental idealism (the view that sensibility and understanding do provide their own fully adequate theory of truth and their own ontology, quite separate from the semantics and ontology of reason)~ After having established these points to his own satisfaction, Kant devotes the "Doctrine of Method" to comparing the methodological techniques appropriate to the practice of empirical science, philosophy and mathematics. Turning to the "Aesthetic," three of the issues raised by this part of the Critique play central roles in several of the papers in this anthology: (a) Kant's notion of intuitions and of their distinctive epistemological roles; (b) Kant's exposition of space as an a priori form of intuition; "transcendentally ideal" and (e) Kant's claims that space and time "empirically real."
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INTRODUCTION
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Each of these themes actually also reverberates throughout the rest of the Critique. I shall discuss them in order.
2. The Nature of Intuitions According to Kant all synthetic judgements (and hence all nontrivial elements of knowledge) rest ultimately on intuitions. Iiltuition is thus for him a fundamental epistetuic notion. (The German word is "Anschauung.") Nevertheless, there is no clear consensus about what Kant means by Anschauung, and the opening essays by Hintikka and Parsons contain a debate about this basic notion. The debate is relevant to the philosophy of mathematics because Kant insists that mathematics contains synthetic knowledge. (Mathematics is, in fact, Kant's paradigm of synthetic a priori knowledge.) And so he must define "intuition" in 'a way that includes the evidence for mathematical jndgments. Indeed, Kant's argument for the intuitivity of geometric space provides one clear criterion for intuitions. This is the criterion of singularity. He says (at A25!B39) that "we can represent to ourselves only one space; and if we speak of diverse spaces, we mean thereby only parts of one and the same unique space."6 The opening passage of the "Aesthetic" appears, however, to add a second criterion, immediacy. "In whatever manner and by whatever means a mode of knowledge may relate to objects, intuition is that through which it is in immediate relation to them .. ."(AI9!B33) These dual characteristics, singularity and immediacy, are further reinforced at A320!B376-77, and they form the basis of Parsons' view in "Kant's Philosophy of Arithmetic." Thongh immediacy entails singularity, the reverse, says Parsons, need not hold. There are singular concepts, whose relation to their objects is presumably mediated by the more general concepts of which they are composed. Parsons also assumes that "immediate" awareness must be quite literally perceptual. This makes Kant's claim that geometry is synthetic quite plausible. For Kantian geometry, on Parsons' interpretation, is about ordinary spatial objects; and geometric reasoning nses standardly perceivable objects for its constructions. This reasoning is a priori because it abstracts from the accidental features of the particular object used and treats the object as a representative for all relevantly similar spatial figures. These same considerations, however, open a hard question about arithmetic. For numbers and numerical concepts are not concretely instantiated
the way geometric concepts are.
CARL J. POSY
INTRODUCTION
That is a question which Hintikka's interpretation is well suited to address. Hintikka claims that immediacy is not a separate criterion for intuitions, but rather a consequence of singularity. Moreover, Hintikka (building on E. W. Beth's work) suggests that the syntheticity of a mathematical judgment derives not from perceptual evidence for the judgment but rather from the role of singular representations in the proof of the judgment. Hintikka agrees that geometric proofs are a priori because of the arbitrariness of their figures. But he claims that this a priority is best expressed by the restrictions on the free variables used in the instances of universal quantifier introdnction and existential quantifier elimination that occur in formalizing these proofs. This approach clearly applies to number theoretic arguments as well. Indeed, existentialquantifier elimination is especially useful in arithmetic proofs (e.g., proofs by mathematical induction). That, he says, is the source of pure intuition for arithmetic. In "Kant on the Mathematical Method" Hintikka goes on to suggest that this reduction of intuitivity to the use of singular terms in formal proofs corresponds to the Euclidian notion of eethesis. That in tum, says Hintikka, is the model for the centrality of "construction" in Kant's theory of mathematics. Parsons, however, insists, as I said, that intuitivity requires something phenomenologically like perception. So to account for the syntheticity of arithmetic he finds its intuitive base in the system of numerals, a system which is indispensible for arithmetic calculation. He claims indeed that calculation plays the role arithmetic that spatial constructions play in geometry: Calculation goes beyond conceptual analysis, engages perceivable individuals and thus grounds arithmetic truths in intuition. Finally, Mauley Thompson, in "Singular Terms and Intuitions in Kant's EpistemOlOgy" builds on this dispute between Hintikka and Parsons to provide bis own intermediate view. He agrees with Hintikka that singularity and imuiediacy always coincide for Kant. But unlike Hintikka, he denies that linguistic singular terms generally represent Kantian intuitions. Intuitions, according to Thompson, almost never have direct linguistic, counterparts. Indeed, he says, Kant's philosophy is virtually without singular terms under-
3. Space as A Priori
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stood in the modem sense. "Space," "time," and numerals, to be sure, come
close to being singular terms. But (citing A7l9/B747) Thompson insists that the mathematical objects they denote do not exist. He accepts Parsons' point that mathematical construction - be it "ostensive" as in geometry or "symbolic" as in algebra-demonstrates possibility and not existence.
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Kant's "Metaphysical" Exposition" of the concept of space provides four arguments, the first two of which are designed to show that the representation of space is an a priori representation. That is to say, judgments whose justifications rest solely on this representation (especially the judgments of geometry) are judgments wbich can be known a priori. This claim of the a priority of geometry is perhaps the most frequently attacked claim of the Critique of Pure Reason. For it involves two subsidiary claims that have not fared well in modem science. It follows first of all from Kant's notion of a priority that the substantive claims of geometry are directly about empirical objects. There is thus no legitimate division in the Kantian scheme between abstract and applied geometry. Secondly, there is the !,nfamous Kantian belief that that these substantive a priori geometric judgments are just the theorems of old-fashioned Euclidean geometry. Objections to Kant's conflation of abstract and applied geometry (which stem from formalism in modern pure mathematics) strike at the heart of
Kant's transcendental idealism and his theory of empirical objects. I will discuss them fully when treating Kant's general notion of "objective Validity" in section 10 below. But for now let me tum to Stephen Barker's "Kant's View of Geometry: A Partial Defense," which addresses both sets of complaints in the setting of empirical science. In our own century, Barker notes, the conception of space that was prompted by relativity theory and then philosophically developed by thinkers like Reichenbach, Camap, Hempel and Nagel, has led to the separation of physical from purely mathematical geometry. In particular, physical science has established tests for the concept of straight line and for other geometric concepts. It is now a matter of empirical testing to determine which (if any) general laws are true of these concepts. And as a matter of fact, these laws are not those predicted by Euclidean geometry. So geometry is neither a priori nor Euclidean. 7 Barker also notes, however, that there has been a tradition of dissent from this standard modern view, a tradition wbich goes back at least to Poincare. Barker himself suggests that Euclidean geometry is consistent with our "ordinary conception of space." And he argues (along with Poincare) that the mode of thought wbich finds Euclidean geometry in conflict with empirical theories depends itself on some controversial assumptions about the criteria
for accepting scientific theories.
6
INTRODUCTION
Barker doesn't intend these considerations as a vindication of Kant. Indeed he points out that to be defensible Kant's geometric claims must be interpreted modestly. But he sees this fact itself as a ''partial defense," which shows that the attack on Kant's Euclideanism is itself dependent on sweeping and problematic theses in general philosophy of science.
Melnick contrasts this active spatializing behavior with the passive (indeed "reactive'') behavior associated with sensory stimuli. He uses this contrast to interpret Kant's matter/form distinction and to explain how space can be an intUition in its own right. (Spatial activity can, after all, be directed without attending to any accompanying sensations.) This sort of operational view of space gives poiguancy to problems about how subjectively generated activity (in this case spatializing activity) can dictate the forms of physical objects. Problems of this sort are problems of "objective validity," in Kant's terms; and Melnick responds to these problems by suggesting an operational account of Kant's general metaphysics. In particular he interprets Kant's transcendental idealism as an operational semantics for empirical language, and contrasts that with a purely descriptive language favored by transcendental realism. 8 The second ~argument for the claim that space is an intuition and not a concept rests on the fact that spatial regions extend infinitely. Kant points out that though a concept can have infinitely many things which "fall under" it (i.e., an infinite extension), no concept can comprehend infinitely many things "within it." Michael Friedman analyzes this argument by adapting Bertrand Russell's criticism that Kant's philosophy of mathematics rests on the outmoded constraints of Aristotelian logic. Without the Aristotelian limitation to monadic predicates, says Friedman, Kant would have been able to express the infinity of space in a purely conceptiIal fashion. For while a sentence in the language of monadic predicate logic ntight have (per accidens) an infinite model, no such sentence can force its model to be infinite. Thus, in particular, if the language is purely monadic, no simple collection of axioms for the concept of space can force the infinity of space. RusselJ employs this sort of logical criticism to undermine not only Kant's view of geometry but his entire theory of synthetic a prion· judgments. But Friedman, by contrast, simply uses our historical hindsight to "deepen our understanding of the difficult logical problems" with which Kant's theory of space is struggling. In particular, Friedman emphasizes certain topological and analytic properties of the Kantian conception of space (e.g., denseness and continnity) whose expression we now know requires a polyadic relational language and multiple quantifiers. Since, according to Friedman, Kant cannot form concepts which force these properties to hold of space, he must rest the
4. Space as an Intuih·on The latter two arguments of the "Metaphysical Exposition of Space" are each designed to shows that the original representation of space is an intuition and not a concept. That conclusion is, of course, needed in order to establish the syntheticity of geometry. The first of these paragraphs argues, as we have seen, that the representation of space is necessarily singular. The representation of space, says Kant, encompasses individual spatial regions as parts and not as subsidiary elements which fall under a more general concept. The second paragraph attempts to prove the intuitivity of space by pointing to its infinite extent and claiming that no concept could by itself establish the infinity of space.
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CARL L POSY
These arguments provide the material respectively for Arthur Melnick's "The Geometry of a Form of IntUition" and Michael Friedman's "Kant's Theory of Geometry." I wilJ discuss them in that order. At the heart of Kant's first argument for the intuitivity of space in his observation that an individual spatial region is derived from the original representation of total space by the imposition of lintitations. This is designed to show that the representation of total space could not be derived by abstraction from inferior representations in the way a general concept is derived from the elements of its extension. Melnick, however, elegantly develops the positive riotion that a spatial region is actively delimited by a specifically "spatial" behavior. He does this by outlining "spatial" behavior. He does this by outlin~ ing an "operational" interpretation of Kantian geometry. Melnick's central notion is that of "ostending or delimiting". a spatial region. The subject ostends a region in his inunediate vicinity by pointing and tracing out its boundaries. The ostension can be an active finger pointing; it can be a sweep of the gaze, or even a simple mental shift of attention. Melnick's interpretation captUres the global aspect of space because this regional ostending is a special case of the larger spatializing activity which includes moving one's entire body (e.g., by walking) and delimiting regions along the way.
properties on a non-concepttIal basis: pure intuition and construction in
intuition.
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CARL J. POSY
INTRODUCTION
5. Space as Empirically Real
The "Metaphysical Deduction" ptovides an elaborate derivation of Kant's list of these "categories" from a specially constructed table of the forms of judgment. And the "Transcendental Deduction" attempts to prove the objective validity of the categories in a most abstract way. This is then followed by the "Schematism" chapter (which uses the faculty of imagination to connect the abstract versions of the categories to concrete empirical concepts), the fourpart "Analytic of Principles" (which shows how these empirical versions of the categories actually structure our scientific knowledge), and a pair of sections relating Kant's views to more traditional historical and metaphysical issues. A number of the papers in this anthology touch on the role of the categories of quantity in the "Analytic", both in their abstract, transcendentaldeduction form and in their spatio-temponJ.! versions in the "Schematism" and in the "Analytic of Principles." The anthology also deals in some detail with the genenJ.! metaphysical issue of objective validity.
If we assume that the syntheticity and a priority of geometry both stem from the special nature of the reptesentation of space, then we must inevitably wonder how we can be certain that geometry holds true of physical objects. Kant's answer is the doctrine of transcendental idealism. I mentioned -Melnick's interpretation of this doctrine in Section (3) above, and I will discuss some other interpretations below. But there is a concomitant doctrine which deserves independent mention here. That is Kant's doctrine of "empirical realism." which is the subject of WJ.!liam Harper's "Kant on Space, Empirical Realism and the Foundations of Geometry." Empirical realism is the doctrine that space and the objects in space (though "transcendentally ideal") are perfectly real when viewed from the perspective of empiricru science. Harper interprets this doctrine in a strong way. Kant is arguing, he says. for the conclusion that the representation of spatial objects presupposes the presence of non-mental objects. Thus he views empirical realism as Kant's attempt to refute Berkeley's phenomenalism. The heart of Kant's argument, as Harper sees it, is the observation that ordinary perception must contain in it more than unadorned sensation. Perception must, in particular, contain a counterfactual element, a projection of "what would happen if ... ,. This element will include the constraints of geometry, constraints which detennine, for instance, how the object will appear when viewed from other positions. But counterfactuals of this sort are the rock on which a Berkeleyan phenomenalism must founder. For no phenomenalism can ultimately support such counterfactuals. Indeed, they rest, Harper insists, on the presupposition that there exists a non-mental object which is an "inexhaustably rich source of additional perceptible features". Thus you can see that the mathematical issues taken up by these papers issues conceming the syntheticity of arithmetic and geometry and of their physical applicability - are inseparable from themes like transcendental idealism and empirical realism which are at the heart of Kant's metaphysics, and general questions about the nature of intuitions and other basic elements of Kant's philosophy of mind. Let's turn next to the "Analytic." II. THE" AN ALYTIC"
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6. The Structure of the "AnoJytic" The "Analytic" investigates the grounds for and the objective validity of synthetic a priori judgments based on the formal concepts of the understanding.
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7. The Categories of Quantity Charles Parsons, in "Arithmetic and the Categories," points out that the categories of quantity (unity, plurality and totality) depend more on the part/whole relation than on their official derivation from universal, particular and singular judgments. He laments this because modem quantification theory (together with set theory, some of which may already be present in the Critique) can link the official derivation of these categories to the concept of number. This, in turn, would have provided a direct path to the principle of extensive magnitude stated in the "Axioms of Intuition." Drawing on sevenJ.! published and unpublished texts, Parsons then shows that Kant was struggling with two very delicate issues about number and quantity. Kant was first of all concerned with the distinction between continu_ ous and discretely ordered aggregates. He handled this, as Parsons demonstrates, by distinguishing between those aggregates whose "unifying concept" dictates a unique method for distinguishing its parts and those whose concept leaves the nature of the parts undetermined. The former are discrete (and therefore numberable), the latter are continuous. Parsons observes that this Kantian distinction does not require any element of intuition, even for the notion. of number. However, the second Kantian concern is precisely whether or not the notion of number per se must have an intuitive element. The "Schematism" clearly implies that "number" is an intuition based concept. But Parsons also brings passages that tell the other way.
CARL 1. POSY
INTRODUCTION
He suggests that we can adapt the modem distinction between structural and constructive treatments of the numbers to interpret the Kantian alternative positions. But he also suggests that Kant himself may never have come to a full conclusion on this.
9. The "Axioms of Intuition" and the "First Analogy"
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8. The "Schematism"
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The schema of a concept is a rule which guides the imagination to produce a paradigmatic representation of those objects which fall under the concept. The categories, of course, are so general that their schemata could not be actual images. So Kant argues that their schemata are general rules for the temporal ordering of appearances by the imagination. These schemata are importantly like mathematical objects: Both exemplify the general structure of intuition with particular concrete objects. To fully appreciate this similarity, and indeed to understand Kant's point in the "Schematism" chapter, we must understand Kant's notion of imagination. This is among the main tasks of J. M. Young's "Construction, Schematism and Imagination." The "construction" in Young's title refers to "ostensive construction" (Le., the activity of intuitively exemplifying mathematical concepts), an activity which lies at the heart of Kant's theory of mathematical proof. Like Parsons (in "Kant's Philosophy of Arithmetic") Young insists that the exemplifying intuition must be sensible. Indeed, again with Parsons, he suggests that a numeral system provides the best exemplifying instances for the concepts of arithemetic. Specifically, a numeral for n indicates the number n precisely because it is "the last thing that would be generated in a standard collection of n sensible things." This sort of collection (e.g., the first n numerals in some numeral system) is a concrete instance of the concept corresponding to the number n, and its "standardness" provides the generality needed for mathematical thought. But here is where imagination comes in. Using a numeral or a standardized collectiou to represent a particular number requires that we "construe" that collection as falling under the corresponding numerical concept. Kantian imagination, says Young, is the central element of the process of "construing." For construing an empirical object as falling under a concept involves imagining how that thing might present itself in other contexts. That in turn, Young claims, requires imaginative construction.
11
William Harper, expanding his analysis of geometry, argues that the "Axioms of Intuition" support a sophisticated empirical realism by requiring every dimension of any empirical object to have a precise real-valued maguitude. Though Harper does not do this, we can view his argument as responding to a criticism made in Philip Kitcher's "Kant and the Foundations of Mathematics. "9 Kitcher's complaint rests on the fact that we humans have perceptual thresholds and aims to assail the Kantian claim that space is infiuitely divisible (a claim crucial to the assigument of real-valued maguitudes). The usual argument for infinite divisibility rests on the possibility of iterating the division of any given spatial region. Kitcher points out that this possibility must not be purely logical (or conceptual), for then the claim of infinite divisibility would be anaJytic. However, he insists, we can have no observational ground for this possibility. For there will be observable magnitudes whose bisection we cannot observe. These will be maguitudes at or near the minimal threshold. And we caunot assume that the divisibility of more familiar spatial regions is a general feature of space. Harper responds to this by claiming once again that Kant's empirical realism supplies principles which outstrip our ordinary perceptual abilities. Thus, in particular, he points out that the result of any measurement can be located within observable intervals. Then he suggests that, even though an exact real number caunot be determined by human measurement, a device like van Fraassen's supervaluations can assure the empirical truth of the claim asserting the existence of a precise real number for the measurement. Harper argues that this approach will allow Kant a full empirical realism for mathematics. Indeed, he goes on to claim that the "First Analogy" (in which Kant establishes the permanence of physical substance) similarly establishes a general empirical realism even for unobservable physical objects. 10. Objective Validity At the end of the "Analytic" (at A239-40/B298-99) Kant suggests that were it not for the fact that space and time are the forms of empirical objects, mathematics would be a "mere play of imagination or of understanding." This observation has two contemporary upshots. The first is that - uulike the modern study of formal systems - Kant caunot
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CARL J. POSY
INTRODUCTION
allow the mathematical importance of a pure, uuinterpreted calculus. Modem abstract treatments of geometry, for instance, allow multiple interpretations of the axioms. And, similarly, alternative standard and non-standard realizations of Peano's axioms for arithmetic abound. Kant, however, seems to miss this altogether in his monolithic attachment to observable applications. This upshot is controversial. On the one hand nothiog Kant has said precludes the logical possibility of constructing abstract nninterpreted systems. Indeed, Stephen Barker points out in his essay that Kant explicitly allowed the consistency of non-Euclidean geometries. If we follow Barker's view, we can read the passage just quoted from the end of the "Analytic" as considering the possibility that mathematics be practiced abstractly without application to empirical objects. The main thmst of the passage is then that mathematics, thus considered, would not count as empirical knowledge. Michael Friedman, on the other hand, rejects the plausibility that Kant held any such a proto-modem view. Kant, he argues, "has no notion of possibility on which both Enclidean and non-Euclidean geometries are possible." The second upshot concerns the general ontology of mathematics. For it appears from the quoted Kantian remarks that the proper objects of mathematics are just the full-blooded objects of the empirical world. These, at any rate, are the only objects of which mathematics (or any science) can provide knowledge. But we must then ask whether Kant would admit a separable realm of purely mathematical objects. Here too the commentators will differ. Certaiuly Kant's remarks about numbers indicated that he thought of these as legitimate objects. On the other hand his remark at A719!B747 that in mathematics "there is no question of existence at all" seems to indicate that purely mathematical objects do not exist. Manley Thompson, in his essay, understands this last passage in that way. He suggests as a consequence that there are no objects with which to instantiate the quantified statements of a purely mathematical language. And thus he proposes to interpret generality in such a language ouly by free variables. Charles Parsons suggests in his "Kant's Philosophy of Arithmetic" a less austere reading accordiiig to which the quantifiers would be interpreted in a modal sense. Thus, in particular, existence in a mathematical language would connote the possibility of constructing an appropriate empirical object. These then are some of the central questions of the "Analytic": the ways in which concepts unify the representations of objects, the nature of imagination and categorial schemata, and the whole issue of objective Validity. Once again it should be clear that as you read these essays about quantity,
numerals, arithmetic and geometry, you will inevitably encounter the broader Kantian questions. Let us ruin finally to the essays which touch on the "Dialectic" and the "Doctrine of Method."
12
III. THE "DIALECTIC" AND THE "DOl;TRINE OF METHOD"
11. The Structure of the "Dialectic" Following a well established practice, Kant inserts a division of the Critique to diagnose the misuses of the concepts and principles detailed in the "Analytic." He arranges these "mistakes" systematically: They alI derive from natural but improper syllogistic uses of the categorial principles. HO'Yever, the real thrust of the "Dialectic" is Kant's sustained attack on traditional metaphysics (specifically, rational psychology, cosmology and theology). The "Dialectic" also contains his most forceful attack on transcendental realism. For the realist, according to Kant, is trapped into the natural mistakes that lead to the problems of traditional metaphysics. The transcendental idealist, by contrast, allegedly has the tools to avoid these mistakes.
12. The "First Antinomy" Kant's attack on transcendental realism is especially potent in the "Antinomy" section of the "Dialectic", where he attempts to show that the realist (hut not the transcendental idealist) is committed to four pairs of mutually contradictory propositions. The initial two antinomies are of special interest, since these are directly mathematical: The first commits the realist to both the finite and infinite extent of the world in space and in past time. The second commits him to the infinite divisibility of matter and to the opposite atomism as welL The first of these is the focus of my own essay on "Kant's Mathematical Realism." The essay interprets Kant's transcendental idealism as if it were the modem linguistic anti-realism (or "assertabilism") that has come to be associated with intuitionistic logic. Transcendental realism, by contrast, maintains the standard "classical" logic. This allows me to illuminate several features of the "Antinomy" argument. Thus, for instance, the conflicting claims about the age of the universe come as a pair of statements which are classically contradictory. If the realist must accept them both (as Kant claims) then transcendental realism is indeed inconsistent. These statements are not intuitionistically contradictory - I
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CARL J. POSY
INTRODUCTION
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display a Kripke model which falsifies them both - and so transcendental idealism escapes the reductio. Indeed, that same Kripke structure models Kant's claim that, for the idealist, the series of past moments may be continued indefinitely but not infinitely. I show, moreover, how to modify the philosophical side of assertabilism in order to explain the actual arguments that Kant. attributes to the realist and idealist. Finally I suggest that all of this offers an insight into Kant's view of the logic of mathematics. The idealist rejects classical logic for empirical judgments because there will always be observationally undecidable empirical claims. Undecidability plus assertabilism yield a non-classical logic. But, I point out, the "purity" of mathematical intuition leads Kant to believe that all mathematical judgments are decidable. So I conclude that though Kant advo-
only spatio-temporal intuition can ground the complete determination of mathematical objects. In particnlar Brittan uses Kant's views on algebra to dispute those who rest the intuitive nature of mathematics on the central role of calculation. The difference between algebra and arithmetic, he notes, is a difference of generality and not a difference in kind or in subject matter. And so, elaborating a point derived from Gottfried Martin, Brittan insists that algebraic reasoning is just as synthetic and intuitive as the other more specific branches of mathematics. Even in algebra, according to Brittan, this intuitivity rests on the possibility of complete detennination. But, in a bold proposal, Brittan suggests that it is relational structures whose properties are fully detertuinable by algebraic reasoning and not individual mathematical objects.
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cates an intuitionistic logic for empirical discourse, he favors a classical logic
for mathematics.
15. Mathematics and Transcendental Philosophy 13. The "Doctrine of Method"
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This last logical theme is echoed in the "Doctrine of Method," where Kant observes (at A792/B820) that "apagogical" (i.e., indirect) proofs have their "'true place" in mathematics but not in empirical science. For these proofs are classically but not intuitionistically valid. But the "Doctrine of Method" compares mathematics with empirical science (and indeed with philosophy) on more than just the question of their respective proof strategies. It analyzes the roles of axioms, hypotheses and definitions in each of these disciplines as well; and it diagnoses the possibility and methodological sources of synthetic knowledge in each of them. 14. Kant on Algebra
The "Doctrine of Method" contains Kant's only sustained discussion of algebra in the Critique, and this is the focus of Gordon Brittan's "Algebra and
Intuition." Brittan - who, unlike Thompson, does allow Kant a separable realm of mathematical objects - associates Kant's claim that mathematics requires
intuition with the view that these objects are "fully detennined." For Brittan the full detennination of a mathematical object means that all singular mathematical propositions which refer to that object are semantically bivalent. He notes that conceptual analysis alone cannot provide this bivalence, and so
In a long series of papers Jaakko Hintikka has produced perhaps the most comprehensive discussion of Kant's views on mathematical method. And indeed he has generally made great strides towards integrating considerations of mathematical method with the large systematic themes in Kant's speculative philosophY. This is apparent, for instance, in his "Kant on the Mathematical Method," which I discussed in section 2. In "Kant's Transcendental Method and His Theory of Mathematics" Hintikka connects his interpretation of Kant's views on mathematical method with the broadest of Kantian themes, the question of the objects and procedures of "transcendental philosophy." Specifically, according to Hintikka, Kant's notion of "transcendental knowledge" is not confined to knowledge about one particular conceptual system, but rather it refers to knowledge about the human activities which "create and maintain" that conceptual system. Kant's view of mathematical knowledge, says Hintikka, gives a prime illustration of this point. Recall that, according to Hintikka, Kantian mathematical method rests on the introduction of instantiating singnlar tenns in mathematical proofs. These tenns, which play the role of pure intuitions, ordinarily come without ties to specific empirical objects. So Kant is faced with the problem of how intuitions like these - nnconnected as they are to actual objects - can yield empirical knowledge a priori. And his answer, according to Hintikka, is that the method of instantiation anticipates the properties which we ourselves introduce "in the processes through which we come to know individuals (particulars)." Of course, those processes themselves are, for Kant, the processes involved in sense perception.
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INTRODUCTION
To be sure, Hintikka criticizes Kant for basing knowledge of particulars on sense perception. (He calls this Kant's "Aristotelian premise.") For Hintikka views sense perception (even Kantian sense perception) as entirely passive. And he contrasts this with the most general process of gathering information abont individuals, a process which includes the activities of seeking and finding. However, when we correct Kant, and view knowledge gathering in this more active way, we produce, Hintikka believes, a more consistently Kantian ("transcendental") epistemology. We also get a revised Kantian view of logic. This will center on instantiation rules and will found these rules in tnm on a general theory of seeking and finding. Moreover, Hintikka argues, the revision itself will produce improved readings of Kant's notion of a thingin-itself and of his general theory of identity.
REFERENCES
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Duke University NOTES 1 See Frege's Foundations of Arithmetic, especially sections 12 ff. 2 Russell's most sustained criticism of Kant's philosophy of geometry is contained in his An
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Essay on the Foundations of Geometry. 3 See chapter II of Brouwer's dissertation, Over de grondslagen del" wiskunde, his lecture "Het wezen der meetkunde," and his essay "Intuitionism and Formalism." 4 See the articles by J. A. Eberhard in Philosophischen Magazin (1788-1792) and Philosophischen Archiv (1793-1794). An excellent summary of Eberhard's criticisms and those of others together with a translation of Kant's response in the essay "On a Discovery According to which any New Critique of Pure Reason has been made Superfluous by an Earlier One" is fOWld in H. Allison's The Kant Eberhard Controversy. 5 See for instance Norman Kemp Smith's A Commentary on Kant's Critique of Pure Reason, (especially his treatment of the "Introduction" and of the concept of space in the "Aesthetic") and Jonathan Bennett's Kant's Analytic. ~ Making the same point about time, Kant says a' bit later on, that ''the representation which can be give only through a single object is intuition." (A32/B47). 7 It is worth noting that Kant was an explicit foil for the early nineteenth century mathematicians who challenged both the a priority and Euclidean natu~ of space, though these challenges weren't fully accepted, until much later. See M. Greenberg's Euclidean and Non-Euclidean Geometries: Development and History, Chapter 6. g Melnick has expanded this theme into a full-blown interpretation of the central parts of the Critique of Pure Reason in his book, Space, Time and Thought in Kant. 9 Actually, Harper is responding to a point in Charles Parsons' "Infinity and Kant's Conception of the 'Possibility of Experience'," a paper which influenced Kitcher's criticism.
Allison, H.: 1973, The Kant-Eberhard Controversy, The Johns Hopkins University Press, Baltimore. Bermett, J.: 1966, Kant's Analytic, Cambridge University Press, Cambridge. Brouwer, L.EJ.: 1907, Over de Grondslagen der Wiskunde. (Dissertation, University of Amsterdam; translated as 'On the Foundations of Mathematics,'), in Brouwer (1975). Brouwer, L.E.J.: 1909, 'Het Wezen der Meetkunde', (Translated as The Nature of Geometry'), in Brouwer (1975). Brouwer, L.E.J.: 1913, 'Intuitionism and Formalism', in Bull. Amer. Math. Soc. 20; reprinted in Brouwer (1975). Brouwer, L.EJ.: 1975, Collected Works, v. I. (A. Heyting, ed.), North Holland, Amsterdam. Frege, G.: 1955, The Foundations of Arithmetic. 0. L. Austin. trants.). Oxford University Press. OxfQfd. "Greenberg. M.: 1974. Euclidean and Non-Euclidean Geometries: Development and History (Second Edition), W. H. Freeman and Co., New York. Melnick, A.: 1989, Space, Time and Thought in Kant, Kluwer Academic Publishers, Dordrecht. Farsons, c.: 1964, "Infinity and Kant's conception of 'the 'Possibility of Experience"', Philosophical Review, 73. Russell, B.: 1956, An Essay on the Foundations of Geometry, Dover, New York. Smith, N. K.: 1929, A Commentary on Karrt's Critique of Pure Reason, Macmillan, London.
JAAKKO HINTIKKA
KANT ON THE MATHEMATICAL METHOD
I. MATHEMATICAL METHOD TURNS ON CONSTRUCTIONS
According to Kant, "mathematical knowledge is the knowledge gained by reason from the construction of concepts," In this paper, I shall make a few suggestions as to how this characterization of the mathematical method is to be under,stood, The characterization is given at the end of the Critique of Pure Reason in the first chapter of the Transcendental Doctrine of Method (A 713 = B 741),1 In this chapter Kant proffers a number of further observations on the subject of the mathematical method. These remarks have not been examined very intensively by most students of Kant's writings. Usually they have been dealt
with as a sort of appendix to Kant's better-known views on space and time, presented in the Transcendental Aesthetic. In this paper, I also want to call attention to the fact that the relation of the two parts of the first Critique is to a considerable extent quite different from the usual conception of it. To come back to Kant's characterization: the first important term it contains is the word 'construction'. This term is explained by Kant by saying that to construct a concept is the same as to exhibit, a priori, an intuition which corresponds to the concept.2 Construction, in other words, is tantamount to
the transition from a general concept to an intuition which represents the concept, provided that this is done without recourse to experience. 2. A VULGAR INTERPRETATION OF KANTIAN CONSTRUCTIONS
How is this tenn 'construction' to be understood? It is not surprising to meet
it in a theory of mathematics, for it had in Kant's time an established use in at least one part of mathematics, viz. in geometry. It is therefore natural to assume that what Kant primarily has in mind in the passage just quoted are the constructions of geometers. And it may also seem plausible to say that the reference to intuition in the definition of construction is calculated to prepare
21 Carl J. Posy (ed.), Kant's Philosophy of Mathematics, 21-42.
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KANT ON THE MATHEMATICAL METHOD
the ground for the justification of the use of such constructions which Kant gives in the Transcendental Aesthetic. What guarantee. if any. is there to make sure that the geometrical constructions are always possible? Newton had seen the only foundation of geometrical constructions in what he called 'mechanical practice' (see the preface to Principia). But if this is so, then the certainty of geometry is no greater than the certainty of more or less crude 'mechanical practice'. It may seem natural that Kant's appeal to inmition is designed to fumish a better foundation to the geometrical constructions. There is no need to construct a figure on a piece of paper or on the blackboard, Kant may seem to be saying. All we have to do is to represent the required figure by means of imagination. This procedure would be justified by the outcome of the Transcendenral Aesthetic, if this can be accepted. For what is allegedly shown there is that all the geometrical relations are due to the structure of our sensibility (our percepmal apparams, if you prefer the term); for this reason they can be represented completely-in imagination without any help of senseimpressions. This interpretation is the basis of a frequent criticism of Kant's theory of mathematics. It is said, or taken for granted, that constructions in the geometrical sense of the word can be dispensed with in mathematics. All we have to do there is to carry out certain logical argnments which may be completely formalized in terms of modem logic. The only reason why Kant thought that mathematics is based on the use of constructions was that constructions were necessary in the elementary geometry of his day, derived in most cases almost directly from Euclid's Elementa. But this was only an accidental peculiarity of that system of geometry. It was dne to the fact that Euclid's set of axioms and postulates was incomplete. In order to prove all the theorems he wanted to prove, it was therefore not sufficient for Euclid to carry out a logical argnment. He had to set out a diagram or figure so that he could tacitly appeal to our geometrical intuition which in this way could supply the missing assumptions which he had omitted. Kant's theory of mathematics, it is thus alleged, arose by taking as an essential feamre of all mathematics something which only was a consequence of a defect in Euclid's parricnlar axiomatization of geometry.3 This interpretation, and the criticism based on it, is not without relevance as an objection to Kant's fnIl-fiedged theory of space, time, and mathematics as it appears in the Transcendental Aesthetic. It seems to me, however, that it does less than justice to the way in which Kant acmally arrived at this theory.
It does not take a sufficient accOlmt of Kant's precritical views on mathematics, and it even seems to fail to make sense of the arguments by means of which Kant tried to prove his theory. Therefore it does not give us a chance of expounding fully Kant's real arguments for his views on space, time and mathematics, or of criticizing them fairly. It is not so much false, however, as
23
too narrow. 3. KANT'S NOTION OF INTUITION
We begin to become aware of the insufficiency of the above interpretation when we examine the notion of construction somewhat more closely. The defiuition of this term makes use of the notion of intuition. We have to ask, therefore: What did Kant mean by the term 'inmition'? How did he define the teIID? What is the relation of his notion of intuition to what we are accustomed to associate with the term? The interpretation which I briefly sketched above assimilates Kant's notion 'I of an a priori intuition to what we may call mental pictures. Intuition is something you can put before your mind's eye, something you can visualize, \ something you c:an represent to yo.ur imagination. This is not at all .the basic meanmg Kant hImself wanted to gIve to the word, however. According to hlS"j definition, presented in the first paragnaph of his lectures on logic, every particnlar idea as distingnished from general concepts is an inmition. Everything, in other words, which in the human mind represents an individual is an intuition. There is, we may say, nothing 'intuitive' about intuitions so defined. Inmitivity means simply individuality.' Of course, it remains true that later in his system Kant came to make intuitions intuitive again, viz. by arguing that all our human intuitions are bound up with our sensibility, Le., with our faculty of sensible perception. But we have to keep in mind that this connection between intuitions and sensibility was never taken by Kant as a mere logical consequence of the definition of intuition: On the contrary, Kant insists all through the Critique of Pure Reason that it is not incomprehensible that other beings might have intuitions by means other,-than senses.5 The connection between sensibility and inmition was for Kant something to be proved, 'not something to be assumed.6 The proofs he gave for assuming the connection (in the case of human beings) are presented in the Transcendenral Aesthetic. Therefore, we are entitled to assume the connection between sensibility and intuitions only in those parts of Kant's system which are logically posterior to the Transcendenral Aesthetic.
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JAAKKO H1NTIKKA 4. THE SYSTEMATIC PRIMACY OF KANT'S THEORY OF THE MATHEMATICAL METHOD
My main suggestion towards an interpretation of Kant's theory of the mathematical method, as presented at the end ofthe first Critique, is that this theory is not posterior but rather systematically prior to the Transcendental Aesthetic. If so, it follows that, within this theory, the term 'intuition' should be taken in the 'unintuitive' sense which Kant gave to it in his definition of the notioTI. In particular, Kant's characterization of mathematics as based on the use of constructions has to be taken to mean merely that, in mathematics, one is all the time introducing particular representatives of general concepts
and carrying out arguments in terms of such particular representatives, arguments which cannot be carried out by the sole means of general concepts. For if Kant's methodology of mathematics is independent of his proofs for connecting intuitions and sensibility in the Aesthetic and even prior to it, then we have, within Kant's theory of the method of mathematics, no justification whatsoever for assuming such a connection, i.e., no justification for giving
the notion of intuition any meaning other than the one given to it by Kant's own definitions. There are, in fact, very good reasons for concluding that the discussion of
the mathematical method in the Doctrine of Method is prior to, and presnpposed by, Kant's typically critical discussion of space and time in the Transcendental Aesthetic. One of them should be enough: in the Prolegomena, in the work in which Kant wanted to make clear the structure
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of his argument, he explicitly appeals to his discussions of the methodology of mathematics at the end of the Critique of Pure Reason in the beginning and dnring the argument which corresponds to the Transcendental Aesthetic, thus making the dependence of the latter on the former explicit. This happens both when Kant discusses the syntheticity of mathematics (Academy edition of Kant's works, Vol. 4, p. 272) and when he discusses its intuitivity (ibid., p. 281; cf. p. 266). Another persuasive reason is that at critical junctures Kant in the Transcendental Aesthetic means by intuitions precisely what his own definitions tell us. For instance, he argues abnut space as follows: "Space is not a' ... general concept of relations of things in general, but a pure intuition. For ... we can represent to ourselves only one space.... Space is essentially one; the manifold in it, and therefore the general concept of spaces, depends solely on the introduction of lintitations. Hence it follows that an ... intuition underlies all' concepts of space" (A 24-25 = B 39). Here intuitivity is
KANT ON THE MATHEMATICAL METHOD
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inferred directly from individuality, and clearly means nothing more than the latter. 5. THE HISTORICAL PRIMACY OF KANT'S THEORY OF THE MATHEMATICAL METHOD
But I am afraid that, however excellent reasons there may be for reversing the order of Kant's exposition in the first Critique and for putting the discussion of mathematics in the Methodenlehre before the Transcendental Aesthetic, my readers are still likely to be incredulous. Could Kant really have meant nothing more than this by his characterization of the mathematical method? Could he have thought that it is an important peculiarity of the method of mathematicians as distinguished from the method of philosophers that the mathematicians make use of special cases of general concepts while philosophers do not? Isn't suggesting this to press Kant's definition of intuition too far? The answer to this is, I think, that there was a time when Kant did believe that one of the main peculiarities of the mathematical method is to consider particular representatives of general concepts.7 This view was presented in the precritical prize essay of the year 1764. Its interpretation is quite independent of the interpretation of Kant's critical writings. In particular, the formulation of this precritical theory of Kant's does not turn on the notion of intuition at all. It follows, therefore, that the idea of the mathematical method as being based on the use of general concepts in concreto, Le., in the fonn of individual instances, was the starting-point of Kant's more elaborate views on
mathematics. Whether or not my suggested reading of Kant's characterization of mathematics is exhaustive or not, that is, whether or not intuition there
means something more than a particular idea, in any case this reading is the one which we have to start from in trying to understand Kant's views on mathematics. It is useful to observe at this point that the reading of Kant which I am suggesting is not entirely incompatible with the other, more traditional, interpretation. On one hand, a fully concrete mental picture represents a particular, and therefore an intuition in the sense of the wider definition. On the other hand, particular instances of general concepts are usually much easier to deal with than general concepts themselves; they are much more intuitive in the ordinary sense of the word than. general concepts. The two interpretations therefore don't disagree as widely as may first seem. What really makes the difference between the two is whether Kant sometimes had in mind, in
26
KANT ON THE MATHEMATICAL METHOD
addition to 'usual' intuitions in the sense of mental pictures or images, some other individuals that are actually used in mathematical arguments. This, I
the use of intuitions, i.e., on the use of representatives of individuals as distinguished from general concepts. After all, the variables of elementary algebra range over numbers and don't take predicates of numbers as their substitution values as the variables of a formalized syllogistic may do. Then we can also understand what Kant had in mind when he called algebraic operations, such as addition, multiplication, and division, constructions. For what happens when we combine in algebra two letters, say a and b, with a functional sign, be this f or g or + or . Or:, obtaining an expression like f(a, b) or g(a, b) or a + b or a . b or a: b? These expressions, obviously, stand for indiVidUal) numbers or, more generally, for individual magnitudes, usually for individuals different from those for which a and b stood for. What has happened, therefore, is that we have introduced a representative for a new individual. And such an introduction of representatives for new individuals, i.e., new intuitions, was just what according to Kant's definition happens when we construct something. The new individuals may be said to represent the concepts 'the sum of a and b', 'the product of a and b', etc. Kant's remarks on algebra therefore receive a natural meaning under my interpretation, quite apart from the question whether this meaning is ultimately reconcilable with what Kant says in the Transcendental Aesthetic. We might say that the purpose of Kant's use of the term 'intuition' here is to say that algebra is nominalistic in Quine's sense: the only acceptable values of variables are individuals.
think, is something we must make an allowance for. 6. KANT ON ALGEBRA
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In fact, if we have a closer look at Kant's actual theory of mathematics as presented at the end of the Critique of Pure Reason, we shall see that many things become natural if we keep in mind the notion of intuition as a particular idea in contra-distinction to general concepts. Usually, people read Kant's theory of the mathematical method in the light of what he says in the Transcendental Aesthetic. In other words, they read 'intuition' as if it meant 'mental picture' or 'an image before our mind's eye' or something of that sort. But then it becomes very difficult to understand why Kant refers to algebra and to arithmetic as being based on the use of intuitions. The point of using algebraic symbols is certainly not to furnish ourselves with intuitions in the ordinary sense of the word, that is, its purpose is not to furnish ourselves with more vivid images or mental pictures. Scholars have tried to reconcile Kant's remarks on algebra and arithmetic with his critical doctrines as they are presented in the Transcendental Aesthetic. The outcome of these attempts is aptly summed up, I think, by Professor C. D. Broad in a well-known essay on 'Kant's Theory of Mathematical and Philosophical Reasoning', where he says that "Kant has provided no theory whatsoever of algebraic reasoning.'" This is in my opinion quite correct if we read Kant's descriprion of the mathematical method in the light of what he says in the Transcendental Aesthetic. But then Broad's view becomes, it seems to me, almost a reductio ad absurdum of the assumption that the Transcendental Aesthetic is, in Kant's mind, logically prior to the discussion of mathematics at the end of the first Critique. For on this assumption the statements Kant makes on arithmetic and algebra are not only deprived of their truth but also of their meaning. If the Transcendental Aesthetic were logically prior to Kant's methodology of mathematics, it would become entirely incomprehensible what on earth Kant could have meant by his remarks on arithmetic and algebra which so obviLously are at variance with his professed theories. On the other hand, if we assume' that by 'intuition' Kant only meant any representative of an individual when he commented on arithmetic and algebra, a number of things, although not necessarily everything, become natural. If we can assume that the symbols we use in algebra stand for individual numbers, then it becomes trivially true to say that algebra is based on
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7. KANT ON ARITHMETICAL EQUATIONS
Kant's remarks on arithmetic present a somewhat more complicated problem. I shall not deal with them fully here, although they can be shown to square with the view I am suggesting. There is only one point that I want to make here. In the case of the arithmetic of small numbers, such as 7, 5, and 12, the ordinary reading of Kant's remarks is not without plausibility. What Kant seems to' be saying is that in order to establish that 7 + 5 = 12 we have to visualize the numbers 7, 5, and 12 by means of points, fingers, or some other suitable illustrations so that we can immediately perceive the desired equation. He goes as far as to say that equations like 7 + 5 = 12 are immediate and indemonstrable (A 164 = B 204). This is not easy to reconcile with the fact that Kant nevertheless described a procedure which serves, whether we call it a proof or not, to establish the truth of the equation in question and that he said that his view is more natural as applied to large numbers (B 16). I hope
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to be able to show later what Kant meant by saying that equations like 7 + 5 = 12 are 'immediate' and 'indemonstrable'. He did not mean that the equation can be established without an argument which we are likely to call a proof. 'Immediate' and 'indemonstrable' did not serve to distinguish immediate perception from an articulated argument, but to distioguish a certain subclass of particularly straightforward arguments from other kinds of proofs. The ordinary interpretation of Kant's theory therefore fails here too. Of the correct view I shall try to give a glimpse later.
he used the term exposition for a process analogous to that of mathematical
28
8. EUCLID AS A PARADIGM FOR KANT
29
construction.
The setting-out or ecthesis is closely related to the following or third part of a Euclidean proposition, the auxiliary construction. This part was often called the preparation or machinery (Ka~acrKE1Jti). It consisted in stating that the figure constructed in the setting-out was to be completed by drawing certain additional lines, points, and circles. In our example, the preparation reads as follows: "For let BA be drawn through the point D, let DA be made equal to CA, and let DC be joined." The construction was followed by the apodeixis or proof proper (i'm60E1~1~). In the proof, no further constructions were carried out. What
One good way of coming to understand Kant's theory of mathematics is to ask: What were the paradigms on which this theory was modelled? The most obvious paradigm, and in fact a paradigm recognized by Kant himself, was Euclid's system of elementary geometry.9 In the beginning of this paper, we
happened was that a series of inferences were drawn concerning the figure
saw that a usual criticism of Kant's theory of mathematics is based on a comparison between Kant's theory and Euclid's system. It seems to me, however, that it is not enough to make a vague general comparison. It is much more
way in which it was constructed.
useful to ask exactly what fearores of Euclid's presentation Kant was thinking of in his theory. In view of the interpretation of Kant's notion of intuition that I have suggested, the question becomes: Is there anything particular in Euclid's procedure which encourages the idea that mathematics is based on the use of particular instances of general concepts? It is easy to see that there is. For what is the structure of a proposition in Euclid? Usually, a proposition consists of five (or sometimes six) partslO First, there is an enunciation of a general proposition. For instance, in proposition 20 of the Elementa he says: "In any triangle two sides taken together in any manner are greater than the remaining one." This part of the proposition was called the 1tp01:a<Jl~. But Euclid never does anything on the basis of the enunciation alone. In every proposition, he first applies the content of the enunciation to a particular figure which he assumes to be drawn. For instance, after having enunciated proposition 20, Euclid goes on to say: "For let ABC be a triangle. I say that in the triangle ABC two sides taken together in any manner are greater than the remaining one, namely, BA, AC greater than BC; AB, BC greater than CA; BC, CA greater than AB." This part of a Euclidean proposition was called the setting-out or ecthesis (IlK9E<Jl (3 x)(Fx V Gx)
2
2
5
Now suppose we had a universe U in which for any choice of extensions of 'F' and 'G' this schema came out true. Even according to our notions of logic, there is a possible case in which this happens, and in which (since (1) is valid) there is also no conflict with (1), namely in which U contains fewer than four elements. Io that event the antecedent of the above would always be false?4 If one considers the minimal existence axioms which would be needed to prove the categorical '2 + 2 = 4' in modern set theory, we find that again they require the universe to contain at least four elements, which can be identified with the numbers 1,2,3,4.
61
If we accept first order quantification theory with identity as a logical framework, then it seems that we can maintain the symmetry of arithmetic and geometry in a weak sense, that such propositions as '2 + 2 = 4' imply or presuppose existence assumptions which it is logically possible to deny. To draw the line at this point and to declare thus that set theory is not logic seems to me eminently reasonable; but I shall not argue for this now, particularly since I have done so elsewhere.25 I think the presence of existence propositions in mathematics one of the considerations at stake in Kant's views on mathematics, but it is not clearly differentiated from others. His general views on existence imply that existential propositions are synthetic, but he never applies this doctrine directly to the existence of abstract entities. In the letter to Schultz cited above, Kant says that arithmetic, although it does not have axioms, does have postulates. Postulates as to the possibility of certain constructions, for Kant constructions in intuition, played the role of existence assumptions in Euclidean geometry. Schultz states as a postulate in the Priifung essentially that addition is defined. This factor is also present in Kant's remarks about "construction of con-
cepts in pure intuition," which he regards as the distinguishing feature of mathematical method. If the geometer wants to prove that the sum of the angles of a triangle is two right angles, he begins by constructing a triangle (A 716 = B 744). This triangle, as we indicated above, can serve as a paradigm of all triangles; althongh it is itself an individual triangle, nothing is used about it in the proof which is not also true of all triangles. The proof consists of a sequence of constructions and operations on the triangle.
Kant's view was that it is by this construction that the concepts involved are developed and the existence of mathematical objects falling under them is shown. Although we need not regard this theorem as implying or presupposing that there are triangles, Kant regarded a general proposition as empty, as not genuine knowledge, if there are no objects to which it applies. In this instance only the construction of a triangle can assure us of this. Apart from that, further existeIlce assumptions are used in the course of the proof, in the example of A 716 = B 744 of extensions oflines and of parallels. The same factor is also suggested in the rather puzzling passage in which Kant says that the operation with variables, function symbols, and identity in ttaditional algebraic calculation involves "exhibiting in intuition" the operations involved, which he calls "symbolic construction." Io fact, such operation presupposes that the functions involved are defined for the arguments we permit ourselves to substitute for the variables. Moreover, the construction of an algebraic expression for an object to satisfy a certain condition is the very
KANT'S PHILOSOPHY OF ARITHMETIC
paradigm of a constructive proof of the existence of such an object. However, I think there is something else at stake in this passage, which I shall come to.
instead of the successive addition of "units" we have a timeless relation, for example, that one set is the union of two others; but also with the application of these notions within modem mathematics, in which arithmetical statements can be made about structures which are entirely timeless, and in reference to which any talk of "successive addition" is in on the face of it entirely metaphorical. In the lener to Schultz, Kant qualifies his position in a way which does more justice to this more general character of arithmetic:
VI
It is by no means obvious that the existence assumptions which must be made in the deductive development of mathematics have any connection with sensibility and its alleged form. Frege for one was quite convinced that they did not. What Kant says that bears on this point is not completely clear, partly because in the nature of the case it is bound up with some difficult notions in his philosophy, partly because again he did not disengage this issue from some others. As a preliminary remark, we must observe that Kant certainly did not regard arithmetic as a special theory of, say, time, in the sense in which he regarded geometry as a special theory of space. It does not tum up in this connection in the proofs of the apriority of time in either the Aesthetic or the corresponding discussion in the Inaugural Dissertation (§ 12, § 14 no. 5). Nevertheless it is clear that according to Kant, the dependence of arithmetic on the forms of our intuition is in the first instance only on time. I should venture to say that space enters the picture only through the general manner in which inner sense, and thus time, depends on outer sense, and thus space. We shall be clear about the intuitive character of arithmetic when we are clear about the manner in which it depends upon time. Whenever Kant speaks about this subject, he claims that number, and therefore arithmetic, involves succession in a crucial way. Thus in arguing that intuition is necessary to see that 7 + 5 = 12: For starting with the number 7. and for the number 5 calling in the aid of the fingers of my hand as intuition. I now add one by one to the number 7 the units which I previously took together to form the nwnber 5, and with the aid of that figure [the hand] see the number 12 come into being. [B 15-16; emphasis mine]
When he gives a general characterization of number in the Schematism, the reference to succession occurs essentially:
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CHARLES PARSONS
62
The pure image of all magnitudes (quantorum) for outer sense is space; that of all objects of the senses in general is time. But the pure schema of magnitude (quantitatis), as a concept of the understanding, is number, a representation which comprises the successive addition of homogeneous units. [A 142 = B 182]
As I said, this seems to conflict not only with the interpretation which number and addition acquire in such constructions as Frege' s, in which
Time, as you quite rightly remark, has no influence on the properties of numbers (as pure determinations of magnitude), as it does on the property of any alteration (as a quantum), which itself is possible only relative to a specific condition of inner sense and its fann (time); and the science of number, in spite of the succession, which every construction of magnitude [Grosse] requires, is a"pure intellectual synthesis which we represent to ourselves in our thoughts.
Earlier in the letter he writes: Arithmetic, to be sure, has no axioms, because it actually does not have a quantum, i.e., an object of intuition as magnitude, for its object, but merely quantity, i.e., a concept of a thing in general by detennination of magnitude.
Kant is here in fact reaffirming a position affirmed in the Dissertation: To these there is added a certain concept which, though itself indeed intellectual, yet demands for its actualization in the concrete the auxiliary notions of time and space (in the successive addition and simultaneous juxtaposition of a plurality), namely, the concept of number, treated of by arithmetic. [§12J
These remarks place arithmetic less on the intuitive and more on the conceptual side of our knowledge. If arithmetic had for its object "an object of intuition as magnitude," i.e., forms such as the points, lines, and figures of geometry, then it would refer quite directly to a form of intuition. But instead it refers to "a concept of a thing in general"; the science of number is a "pure intellectual synthesis." This laner phrase especially suggests that arithmetical notions ntight be definable in terms of the pure categories and thus be associated with logical forms which do not refer at all to conditions of sensibility. Such a view would seem to conflict with the statement of the Schematism that number is a schema. The refere,!ce to "a concept of a thing in general" is no doubt to be meant in the same sense as that in which the categories are said to specify the concept of an object in general, and the pure intellectual synthesis is no doubt that of the second edition transcendental deduction, which is the synthesis of a manifold of intuition in general, which is for us realized so as to yield
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KANT'S PHILOSOPHY OF ARITHMETIC
knowledge only in application to intuitions according to our forms of inruition. Thus the "concept of an object in general" could give rise to actual knowledge of objects only if these objects can be given according to our forms of intuition. But does this merely mean that objects in space and time provide the only concrete application of these concepts which we can know to exist, as one might expect from the absence of special reference to inruition? Whether it means this or something more drastic is, I think, a special case of the general dilemma about the understanding which I mentioned in the beginning. In either case, however, it would be a plausible interpretation of Kant to say that the forms of intuition must be appealed to in order to verify the existence
logic, for example where the iuitial element is '0' and the (n + \)st numeral is obtained by prefixing'S' to the nth numeral, have the further property that each numeral contains within itself all the previous ones so that the nth numeral is itself a model of the numbers from 0 to n). The basis for the use of a concrete perception of a sequence of n tenus in verifying general propositions is that, since it serves as a representative of a strucUlre, the same purpose could be served by any other instance of the same strucrure, that in any other perceptible sequence which can be placed in a one-one- correspondence with the given one so as to preserve the successor relation. This might justify us in calling such a perception a "formal inruition." We might note that the physical existence of the objects is not directly necessary, so that we can abstract also from that "material" factor. An empirical intuition functions. we might say. as a pure intuition if it is taken as a representative of an abstract strucrure. Such a perception provides the fullest possible realization before the mind of an abstract concept. One of the important questions about Kant's philosophy of arithmetic is whether a comparable realization exists beyond the limits of scale of concrete perception. Before we can enter into this question, let me point out another closely related reason in Kant's mind for regarding mathematics as dependent on intuition. This. comes out in particular in the concept of "symbolic construction." The algebraist, according to Kant, is getting resnlts by manipulating symbols according to certain rnIes, which he wonld not be able to get without an analogous inruitive representation of his concepts. The "symbolic construction" is essentially a construction with symbols as objects of inruition:
assumptions of mathematics. However, it is not very clear how to apply the general conceptions derived from the Aesthetic and the Transcendental Deduction to the case at hand. The direct existence propositions in pure mathematices are of abstract entities, and it is only in the geometric case that they can be said to be in space and time. I do think that the objects considered in arithmetic and predicative set theory can be construed as forms of spatiotemporal objects. Fnll set theory wonld of course not be accommodated in this way, but it is not reasonable to expect that from a Kantian point of view impredicative set theory should be intuitive knowledge or indeed genuine knowledge at all. It could legitimately be said to posruIate entities beyond the field of possible experience. 26 VII
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It is narural to think of the natural numbers as represented to the senses (and of course in space and time) by numerals. This does not mean mainly that numerals function as names of numbers, although of course they do, but that they provide instances of the strucrure of the naruraI numbers. In the algebraic sense, the set of numerals generated by some procedure is isomorphic to the narural numbers in that it has an iuitial element (e.g., '0') and a successor relation which the notion of naruraI number requires. In this sense, of course, the numerals are abstract mathematical objects; they can be taken as geometric figures. But of course concrete tokens of the first n numerals are likewise a model of the numbers from \ to n or from 0 to n - 1. A set of objects has n elements if it can be brought into one-to-one correspondence with the numbers from I to n; a standard way of doing this is by bringing them in some order into correspondence with certain numerals representing these numbers, that is by counting. (The numerals used in work in formal
65
Once it [mathematics] has adopted a notation for the general concept of magnitudes so far as their different relations are concerned, it exhibits in intuition, in accordance with certain universal rules, __ all the various operations through which the magnitudes are produced and modified. When, for instance, one magnitude is to be divided by another, their symbols are placed together, in accordance with the sign for division, and similarly in the other processes; and thus in algebra by means of a symbolic construction, just as in geometry by means of an ostensive construction (the geometrical construction of the objects themselves) we succeed in arriving at results which discursive knowledge could never have reached by means of mere concepts. [A 717 = B 745]
That· this is a source of the clarity and evidence of mathematics and provides a connection of mathematics with sensibility is indicated by the following remark: "This method, in addition to its heuristic advantages, secures all inferences against error by setring each one before our eyes" (A 734 ; B 762). A connection of mathematics and the senses by way of symbolic opera-
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tions is already claimed in Kant's prize essay of 1764, Untersuchung iiber die Deutlichkeit der Grundsiitze der natiirlichen Theologie und der Moral,2? which presents a prototype of the theory of mathematical and philosophical method of the Discipline of Pure Reason in its Dogmatic Employment For example, consider the statement of the latter: Thus philosophical knowledge considers the particular only in the universal, mathematical knowledge the universal in the particular, or even in the single instance, although still always a prioli and by means of reason. [A 714 = B 742]
This distinction corresponds in the prize essay to the following, where the distinctive role of signs in mathematics is explicitly emphasized: "Mathematics considers in its solutions proofs, and inferences the universal
in [unter] the signs in concreto, philosophy the universal through [durch] the signs in abstracto."2' The certainty of mathematics is connected with the fact that the signs are sensible: Since the signs of mathematics are sensible means of knowledge, one can know with the same confidence with which one is assured of what one sees with one's own eyes that one has not left any concept out of account, that every equation has been derived by easy rules. etc.; thereby attention is made much easier in that it must take account only of the signs as they are known individually, not the things as they are represented generally.29
The prize essay suggests a position incompatible with the Critique of Pure Reason, namely that since in mathematics signs are manipulated according to
Illi
rules which we have laid down (in contrast to philosophy, where the value of any definition turns on its having a certain degree of faithfulness to preanalytic usage), operation with signs according to the rules, without attention to what they signify, is in itself a sufficient gnarantee of correctness.30 . These passages show that a connection between sensibility and the intuitive character of mathematics existed in Kant's mind before he developed the theory of space and time of the Aesthetic. However, unlike in the later work, no inference is drawn at this stage from this connection to a limitation
of the application of mathematics to sensible objects. The general point behind the observations on symbolic construction can be put in the following way: In general, a mathematical proposition can be verified only on the basis of a proof or calcnlation, which is itself, a construction in intuition. But in view of the remarks about '7 + 5 = 12', a more special fact may have influenced Kant. Certain "symbolic constructions"
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KANT'S PHILOSOPHY OF ARITHMETIC
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'2 + (1 + 1)' and the two 1's as it were added on to the '2'. A corresponding proof of '7 + 5 = 12' wonld involve five such steps instead of two. A similar observation concerning the schema (1) has been made by a number of writers. Although the schema does not imply that the universe contains any elements or that any construction can be carried out, the proof of it involves writing down a group of two symbols representing the F's, another such group representing the G's, and putting them together to get four symbols. So that it is not at all clear that '2 + 2 = 4' interpreted as a proposition about the combinations of symbols is not more elementary than the logically valid schema (1). I have already suggested that the "symbolic" construction in generating numerals is already enough to settle the question of their reference. In the same way the actna! carrying out of the calculations shows the well-defined character for individual argnments of recursively defined functions. However, induction, which I have wanted to leave out of account here, is involved in seeing that they are defined for all argnments. Maybe Kant ought to have said that apart from intuition I do not even know that there is such a number as '7 + 5'. And it seems that one could not see by a particnlar construction that there is such a number without also seeing it to be 12. This is in agreement with Hintikka's statement that the sense of Kant's statement that numerical formulae are indemonstrable is that the construction required for their proof is already sufficient. The considerations about the role of symbolic operations apply equally to logic and therefore undermine Kant's apparent wish to distingnish them on this basis. This appears more forcefully in modem logic, where instead of a short list of forms of valid iuference one has an infinite list which must be specified by some inductive condition. In my opinion this is a consequence to
be accepted and is even in general accord with Kant's statements that synthesis underlies even the possibility of analytic judgments.3 ! The special connection of arithmetic and time can, I think, be explained as follows: If one constructs in some way, such as on paper or in one's head, such a sequence of symbols as the first n numerals, the structure is already represented in the sequence of operations and more generally in the succession of mental acts of running through a group of n objects, as in counting. Thus time enters in through the succession of acts involved in construction or in successive apprehension. This connects with Kant's remark about number
associated with propositions about number actually involve constructions iso-
in the Schematism. In the operations involved in representing a number to the
morphic to the numbers themselves and their relations, or at least an aspect of them. Thus in Leibniz's proof that 2 + 2 = 4, '2 + 2' must be written out as
Time provides a universal source of models for the numbers. In particnlar,
senses, we also generate a structure in time which represents the number.
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KANT'S PHILOSOPHY OF ARITHMETIC
Kant held that it is only by way of successively perceiving different aspects of a manifold and yet keeping them in mind as aspects of one intuition that we can have a clear conception of a plurality. For quite small numbers this seems doubtful although not for larger ones. Nonetheless the element of succession appears even for the smaller ones in the comparison involved in generating or perceiving them in order, and the order is certainly part of our concept of number. What would give time a special role in our concept of number which it does not have in general is not its necessity, since time is in
according to a rule. To speak of a peculiar kind of intuition in the second case seems qnite tuisleading. The mathematical knowledge involved has a highly complex relation to "intuition" in the more specifically Kantian case.
some way or other necessary for all concepts, nor an explicit reference to
time in numerical statements, which does not exist, but its sufficiency, because the temporal order provides a representative of the number which is present to our consciousness if any is present at all. Of course it is one thing to speak of representation in space and time and another to speak of representation to the senses. What is represented to the senses is presumably represented in space and time, but maybe not vice versa. To establish a link of these two Kant would appeal to his theory of space and time as forms of sensibility. The relevant part of this theory is that the structures which can be represented in space and time are structures of possible objects of perception. The kind of possibility at stake here must be essentially mathematical and go beyond "practical" or physical possibility.32 Consider once again a procedure for generating numerals, say by starting with '0' and prefixing occurrences of'S'. The actual use of these as symbols requires that they be perceptible objects. Nonetheless we say it is possible to iterate the procedure indefinltely and therefore to construct indefinitely many numerals. Thus it is clear that the numerals (numeral types)33 which it is in this sense possible to construct extend far beyond the numeral-tokens which have ever been produced in history or which conld in any concrete sense actually be used as symbols.34 This possibility of iteration is necessary for the constructibility of indefinitely many numerals and therefore for the infinity of natural numbers to be given by intuitive construction. Moreover, some insight into such iteration seems necessary for mathematical induction. Insofar as the appeal to pure intuition for the evidence of mathematical statements is supposed to be an analogy of mathematical and perceptual knowledge, it holds less well for propositions involving the concept of indefinite iteration, such as these proved by induction, than for propositions such as 2 + 2 = 4. There seem to be two independent types of insight into our forms of intuition which a Kantian view requires us to have, that which allows a particular perception to function as a "formal intuition" and that which we have into the possible progression of the generation of intuitions
The complexity must be in some way present in the "intuitions" of space
and time since space is an individual which is given, but its structure also detertuines the limits of possible experience and contains various infinite aspects. No doubt the plausibility of the idea that space is present in immediate experience made it more difficult for Kant to appreciate the differences of the kinds of evidences covered by his notions of pure intuition. I am sure that more could be done to explicate the Kantian view of their connection.
In our discussion of intuition, we have somewhat lost sight of the view of logic which at the start we attributed to Kant, which except for the question of existence resembles the modern views called Platonist. Although Kant's view of intuition fits better with the modem tendencies called constructivist or intuitionist, it seems certain that the concept of pure inruition was meant to go with this view of logic and not to replace it. Without using notion like "concepts" and "object" in a quite general way, it is probably not possible to describe it. It wonld be hasty for that reason to identify Kant's conception of intuition with that of Dutch intuitionists, although Brouwer's undoubtedly shows some affinity. It would also be hasty to regard Brouwer's critique of classical mathematics as altogether in accord with Kantianism. POSTSCRIPT
The remarks in this paper about Kant's conception of intuition provoked some controversy_ Hintikka's reply to my criticisms- of his views concentrated on this issue. 34 It is agreed that Kant's basic conception of an intuition
is ofa representation of an individual object that relates to its object inunediately. The dispute concerned what I called the "immediacy criterion" and how it is related to the individuality or singularity criterion. 35 According to Hintikka, an intuition is simply a singular representation, the analogue among Kantian Vorstellungen of a singular term. In reply to my question why he thought the immediacy criterion nonessential, Hintikka says that it is "simply a corollary of the individuality criterion" (p. 342). He cited the followiog well-known passage: "This knowledge is either intuition or concept (intuitus vel conceptus). The former relates immediately to the object and is singular [einzeln],36 the latter refers to it mediately by means of a feature which several things may have in common" (A 320 = B 376-7). What is not immediate about concepts, as Hintikka reads this passage, is that
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KANT'S PHILOSOPHY OF ARITHMETIC
they refer to their objects "only through the mediation of a characteristic which several objects may share" (p. 342). Thus the immediacy of intuitions consists in their not representing their objects by way of properties that they may share with other objects. This is, so far, a defensible reading of the passage. To support Hintikka's thesis, however, it would have to be stretched to say that no representation that is mediate in this sense could be singular. Hintikka does not explicitly argne for this consequence, and strong textual arguments against it were subsequently offered by Robert, Howell.37 An important difficulty is that it would make singnlar judgments an exception to Kant's view that a judgment is a combination of two concepts. This would make nonsense of Kant's assituilation of singular judgments to universal. 38 Kant does not, to be sure, say that
ceptual, content of the representation.44 Nonetheless, the controversy about the immediacy criterion convinced me that my interpretation of its meaning was not so evident as I had thought. Howell draws a reasonable line between what is plain from the text and what has the character of a reconstruction. Howell's own further interpretation is certainly an interesting line of reconstrucrion and may be fruitful. As an interpretation of Kant's intentions, it has the difficulty of relyiug on ideas developed much later in response to problems Kant did not consider. Howell's view, like Hintikka's, attempts to make the distinction between intuitions and concepts entirely within general logic. Kant followed a logiciu tradition that neglected the distinction between singular and general terms. Without breaking with this tradition more clearly than he did, Kant could not give a clear account of how singular representations could enter into propositions and inferences. My strategy was not even to try to do this on Kant's behalf. In understanding immediacy as some kind of direct presence, I was treating the concept of intuition as from the beginning epistemic. Even if Howell is right about the strict definition of the immediacy of intuition, there is undoubtedly an epistemic sense of immediacy in Kant's writings, at stake when he says that mathematical axioms are immediately certain (A 732 = B 760).45 I do not see how to get around regarding some link between the immediacy of intuition and this epistemic sense as an assumption of Kant's, whether or not it was directly embodied in the way he understood the word "immediate" in the definition of intuition. Without some such interpretative hypothesis, I did not see how to make sense of Kant"s theory of geometty. It gives a straightforward explanation of how Kant could think that mathematics depends on sensibility, and extends it from the easier case of geometty to the harder case of arithmetic. I should point out that I intended "present" in a phenomenological sense; in imagination as well as perception an object is "present" in the relevant sense. It follows that intuition does not necessarily involve the existence of the object intuited. It is not clear to me how the direct-reference view achieves the same result, which seems necessary for an account of nonveridical perception.46 At this point I should mention a misunderstanding to which my view can give rise, which is expressed most clearly in the following remarks by Gordon Brittan:
there are singular concepts. It is not concepts themselves but only their use
that can be classified as universal, particular, and singular. 39 But there is no indication that the singular use of a concept makes it an intuition.
A more principled difficulty with Hintikka's interpretation is how it could be, on his account, that all our intuitions are sensible. A definite description is a singular term that refers to its object by means of concepts; quite apart from the passage where he seems pretty clearly to say the contrary,40 I do not see how even after Kant's logical analysis of mathematics (however that is to be interpreted) and the argumeut of the Aesthetic, that can be taken to be a representation of sensibility. Rejection of Hintikka' s view of the immediacy criterion does not of itself
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imply acceptance of my own. A via media is proposed by Howell. He agrees with Hintikka that the above-cited passage gives the strict definition of immediacy; it is simply the absence of "mediation" by marks or characteristics. So, far from agreeing with Hintikka that it is merely a corollary of singularity, he goes on to nnderstand it as direct reference in something'like the sense of
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modem theories of names and demonstratives.41 Indeed, his view is that
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CHARLES PARSONS
empirical intuitions at least are analogues of demonstratives.42 Hintikka himself reads immediacy as "direct reference to objects" (p. 342) and may have adopted a view like Howell's, although then he surely should not continue to hold that immediacy is a consequence of singularity.43 Howell's view of the strict definition of immediacy has the great advantage of putting into relief the somewhat hypothetical character of all three of the developed views we are considering, Hintikka's, Howell's, and mine. In my view it also shows the necessity of some further assumption, since it makes the bare-bones notion of immediacy very uninfonnative; indeed, to one trained in modern logic, it appears circular, for what could mediation by marks or characteristics be but some predicative, in Kantian tertuinology con-
If we were to accept Parsons' interpretation - that it is part of the meaning of "intuition" that intuitions are quasi-perceptual (and thus that the "immediacy criterion"" is independent of the singularity criterion and has epistemological import) - then how would we be able to understand Kant's claim (at B 146) that "as the Aesthetic has shown, the only intuition possible to us is sen· sible"? On Parsons' interpretation, that (human) intuition is sensible follows as a trivial conse· quence of the definition and should not require the extended argument of the Aesthetic. 47
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KANT'S PHILOSOPHY OF ARI>HMETIC
It cannot be my view (or anyone else's) that it is a trivial consequence of the definition that intuition is sensible; as I made clear above (section I), an intellectual intuition would be a counterexample. I do not see why Brittan should think that the more restricted thesis that human intuition is sensible should be immediate from the definition as I read it, since it makes no explicit mention of distinctively human (or even fiuite) intellectual capacities. It may be that he takes it to be obvious from Kant's point of view that it is ouly by sensibility that individuals are immediately present to us. However, this could be true ouly if "individual" is taken to mean concrete object. Since clearly only logical singularity is at issue in the definition of intuition, one cannot derive the sensible character of all human intuition in this way. There is an underlying reason why Brittan's misunderstanding should be natural. I did say that in intuition the object "is in some way directly present to the mind" (p. 112), and that word "present" does suggest that the object is given in the sense that Kant regards as characteristic of the intuition of finite intelligences, that is, that the tuind is to some degree passive and is apprehending an object that is "there" prior to its apprehension. Perhaps we do have to cancel such a suggestion in understanding the idea of an intellectual intuition (to the extent that we can understand it). If so, this reflects the intrinsic difficulty of fortuing a conception of a mode of intuition different from
here expressing a common conception of intuition, not necessarily his own, and that in the end his view is that it fits ouly empirical intuition. In fact it is quite clear (especially from §9) that when Kant in this passage talks of "objects," he means actual concrete objects. A priori intuition of such objects has to be prior to their being given. This, he says in §9, is possible ouly "if my intuition contains nothing but the form of sensibility, antedating in my mind all the actual impressions through which I am affected by objects."49 Kant's problem in this passage is thus how intuition can be "of' an object not yet given. I am not sure why, in Hintikka's view, there should be a problem. Hintikka grants that on my view there is but holds it to be insoluble. I hold that the claim that a priori intuition "contains nothing but the form of sensibility" is the main idea of Kant's solution. In §9 of the Prolegomena, as in many other passages, his stress is on the thesis that this form's being a condition of my tuind on the intuition of objects makes a priori knowledge of such objects possible. The question how knowledge resting on the form of sensibility is intuitive is prior. My own writing on mathematical intuition undertakes to offer a model of how forms may be given in intuition which are yet the forms of objects not yet given. Kant, in giving geometrical examples, must have thought this tolerably clear. - An obstacle to complete clarity is the absence in Kant's philosophy of a theory of mathematical objects. Of course Kant's writing on mathematics abounds in what would ordinarily be read as references to mathematical objects. Sometimes he seems to commit himself to them in a more philosophical way, as when he says that '7 + 5 = 12' is a singular proposition (A 164 = B 205), and that we can "give it [the concept of a triangle1 an object wholly a priori, that is, construct it" (A 223 = B 271). In sections V and VI of this essay I assumed that Kant was concerned with mathematical objects of the usual kind. This was an incautious assumption. The concept of object in tenns of which construction gives the concept of triangle an object is not Kant's primary one, and indeed in that passage Kant partly takes away what he has just given in saying that the triangle is "only the form of an object." Kant never talks explicitly of the existence of mathematical objects; existence for hini ~eems to be concrete existence; this is quite explicit in its schematization as actuality. He seems to decline to attribute existence to mathematical objects at all.50 But what, then, are a priori intuitions, as singular representations, intuitions of! Mathematics contains a priori knowledge, which is knowledge of objects in the full-blooded sense, that is of the objects given in empirical intuition. Sometimes, as in Prolegomena §8, Kant talks as if these objects were the
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CHARLES PARSONS
Howell's view of Kant's conception of intuition would serve to save Hintikka's analysis of Kant's philosophy of mathematics as a whole. But my own criticisms in section IV above of the rest of Hintikka's analysis would not be affected by the assumption that Kant's basic conception of intuition is as Howell claims. However, Hintikka directly criticizes my statement that "intuition is thus a source, ultimately the only source, of immediate knowledge of objects" (p. 112 above). Appealing to remarks of Kant in §§8-9 of the Prolegomena, he argues (p. 343) that my view would lead to a perverse conception of Kant's problem concerning mathematics. His view appears to be that my reading of the immediacy condition would make a priori intuitions "tuisnomers." He relies here on the puzzle expressed in §8 of the Prolegomena: For the question now is, "How is it possible to intuit anything a priori?" An intuition is such a representation as would immediately depend upon the presence of the object. Hence it seems impossible to intuit spontaneously a priori, because intuition would in that event have to take place without either a fonner or a present object to refer to, and in consequence could not be intuition. 48
I should remark that the claim abont intuition made in the second sentence fits Hintikka's conception even less well than mine. But it is clear that Kant is
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CHARLES PARSONS
KANT'S PHILOSOPHY OF ARITHMETIC
objects of a priori intuition. But how can a priori representations have such reference and still be singular? This is a difficulty conunon to my own and to Hintikka's view of intuition. A picture common to us is of pure intuitions as analogous to free variables, with predicates attached to them representing the concepts they "construct" If we are not to import into Kant the "mathematical-objects picture," then it seems we have to take the range of these variables to be empirical objects. Then a mathematical argument cannot, strictly speaking, establish existence, What pays the role of mathematical existence in Kant is constructibility. The most plausible reconstruction of Kant would be, in my view, to take constructibility of a concept to be a kind of possible existence of a (nonabstract) object falling under the concept Kant's view would then be in line with the modal interpretation of quantifiers discussed elsewhere in this volume [i.e. Mathematics in Philosophy], in particular, in connection with intuition, in Essay 1, section III, However, I say "a kind of possible existence" because it cannot be possible existence in the precise sense Kant gives to those words, The difficulty is not with existence but with possibility, which for Kant is what we might call real possibility, The fact that the concept of triangle can be constructed makes it appear that we can see the possibility of a triangle "from its concept in itself' (A 223 = B 271). What makes the construction itself fall short of showing possibility is that possibility involves agreement with the formal conditions of experience with respect to intuition and concepts, that is, not ouly the forms of intuition but the categories. 51 Thus in order to see the possibility of a triangle, we have to observe that space is a formal condition of outer experience and that "the formative synthesis through which we construct a triangle in imagination is precisely the same 'as that which we exercise in the apprehension of an appearance, in making for ourselves an empirical concept of it" (A 224 = B 271). This, of course, repeats the considerations advanced in the Axioms of Intuition. The "objective reality" in the full sense of mathematical concepts seems to be a proposition not of mathematics but of philosophy, To return to the original issue about the immediacy criterion for being an intuition: I would say that Kant as I interpreted him is of interest to the philosopher today, My line of interpretation parallels Kant's most important influence on rwentieth-century foundational research, through Brouwer and Hilbert. A concept of intuition like that I attributed to Kant is of interest in its own right 52 Hintikka, too, could defend his interpretations partly on the grounds of philosophical interest I must also grant a certain justice to the closing remarks of his conunent on my paper (pp, 344-345). Indeed, a central
problem for Kant was "why the knowledge so obtained [in mathematics] can be applied to all experience a priori and with certainty." There is an important aspect of Kant's answer to that question that I hardly touched on, namely the argument in the Analytic for the claim that mathematics necessarily applies to the objects of empirical intuition. However, I do not find an analysis of that argument in Hintikka's writings either. I do not think that either of us has undertaken the task of constructing a truly Kantian explanation of the a priori character of mathematics. In my own case, I doubt that such an explanation could be given without appealing to Kantian transcendental psychology.53
75
Harvard University
NOTES 1 An earlier version of this paper was written while the author was George Santayana Fellow in Philosophy, Harvard University, and presented' in lectures in 1964 to the University of Amsterdam and the Netherlands Society for Logic and the Philosophy of Science. I am indebted to 1. J. de longh, J. F. Staal, and G. A. van der Wal for helpful comments. I am also grateful to -Jaakko Hintikka for sending me two unpublished papers on the subject of this paper. 2 I.e., 1st edition, p. 320, 2nd edition, pp. 376-377. All passages are quoted in the translation of Norman Kemp Smith (London, 1929) with slight modifications. Other translations from German are my own. Translations of Kant's Inaugural Dissertation are by John Handyside, in Kanf s Inaugural Dissertation and Early Writings on Space, Chicago and London, 1929. 3 Kants Gesammelte Schriften, ed. by the Prussian Academy of Sciences, Berlin, 1902-1956, IX, 91. This edition will be referred to as "Ale" 4 "Kant's 'new method of thought' and his theory of mathematics", p. 130. Hintikka argues in detail for this thesis in a paper, "On Kant's notion of intuition (Anschauwzg)," in Terence Penelhum and J. H. MacIntosh (eds), Kant's First Critique (Bemont, Calif.. 1969). The same idea seems to underlie the analysis of Kant's theory of mathematical proof in E. W. Beth, "Uber Lockes 'allgemeines Dreieck'" in Kant-Studien 48 (1956-1957), 361-380. 5 "It is a mere tautology to speak of general or common concepts" (Logic 1, Ak. IX 91). 6 One might attribute to Kant the view that there are no such representations. The classification Kant makes in A 320 = B 376 and Logic §1 is of Erkenntnisse, which Kemp Smith translates as "modes of knowledge" but which in many contexts would be more accurately though inelegantly translated as "pieces of knowledge." Then the relation of a representation to its object is that thrOQgh which one can know its object. and it might be held that intuition in the full sense is the only singular representation which can p~vide such knowledge. This view would have the perhaps embarrassing consequence that an object which is not in some way perceived is not really known as an individual. 7 Cf. the examples of "truths of reason" given by Leibniz, Nouveaux Essais, IV, ii, § 1. 8 Ibid. IV, vii, §IO. 9 Arirhmetlk und Kombinatorik bel Kant (Diss. Freiburg 1934), eniarged ed. Berlin 1972; Kant's
76 Metaphysics and Theory
CHARLES PARSONS
of Science.
Manchester, 1953. ch. i; Klassische Ontologie der Zahl,
Kant-Studien Erganzungsheft 70, KGln. 1956, § 12. 10 Neither Leibniz nor Schultz seems to mention the fact that in order to prove fonnulae involv-
ing multiplication, such as '2 . 3 = 6', one also needs instances of the distributive law. 11 1. Aus mehrern gegebenen gleichartigem Quantis dUTCh ihre successive Verknupfung den Begriff von einem Quanta zu erzeugen, d. i. sie in ein Ganzes zu verwandeln. 2. Ein jedes gegebenes Quantum, urn so viet, als man will, d.i. sie ins Unendliche zu vergrossern, und zu vermindern (Prufung, I, 221). 12 Ak, X 554-558. 13 Arithmetik und Kombinatorik bel Kant, p. 64-5. 14 C. J. Gerhardt (ed.), Leibnizens mathematissche Schriften, Halle, 1849-1963, vn 78. Leibniz gives a definition of addition from which he claims commutativity follows immediately. One could read his argument as deriving the commutativity of addition from the commutativity of settheoretic union. 15 "UberLockes 'allgemeines Dreieck'." 16 "Kant's 'new method of thought'," ''On Kant's concept of intuition," also "Are logical truths analytic?" Philosophical Review 74 (1965), 178-203, "Kant on the mathematical method," This volume, 21-42. J7 It ought to be remarked that while no doubt the distinction which Kant makes between axioms and postulates derives historically from that of "common notions" and postulates in Euclid, Kant's distinction does not correspond exactly to Euclid's. Euclid's division is between more general principles and specifically geometrical ones. For Kant postulates are "immediately certain practical judgments," the action involved is construction, and their purport is that a construction of a certain kind can be carned out. The role they play is thus that of existence axioms. Euclid's common notions are all of a type which Kant asserted to be analytic propositions (A 164 = B 204, B 17), while axioms proper must be synthetic. 18 op. cit. p. 365. 19 Cf. W. V. Quine, Methods of Logic, revised ed., New York, 1959, §28. 20 "Kant's 'new method of thought' ," p. 130, also "Kant on the mathematical method." The texts are A 717-B 745, A 734-B 762. 21 In "Are logical truths analytic?" Hintikka develops a distinction between analytic and synthetic according to which some logical truths are synthetic. He suggests that the logical truths which are analytic according to this criterion are roughly those which Kant would have regarded as analytic. It follows, however, that in some of the arguments which according to Beth and Hintikka involve for Kant an appeal to intuition, the conditional of their premises and conclusion is analytic. In particular, this is true of the example that Beth works out in detail in Uber Lockes 'allgemeines Dreieck'." §7. In order to be applied to mathematical examples like Kant's, Hintikka's criterion would have to be extended to languages containing function symbols. The way: 'of doing this which seems to me most in the spirit of Hintikka's definition has some anomalous consequences. See also Logic. Language-Games. and Information, Oxford, 1973, chs. 6-9. 22 Beth, op. cit. p. 363. 23 In fact, (1) is analytic according to the criterion of "Are logical truths analytic?" (see note 21 above). However, according to another criterion which might be more in the spirit of Kant, to consider as synthetic a conditional whose proof involves formulae of degree higher than its antecedent, (1) is synthetic. Hintikka takes account of this in "Are logical truths tautologies?" by making an additional distinction between analytic and synthetic arguments, such that in the rele-
KANT'S PHILOSOPHY OF ARITHMETIC
77
vant sense the argument from the conjuncts of the antecedent of (1) as premises to its consequent as conclusion is synthetic. 24 Cf. Hao Wang, "Process and existence in mathematics," in Y. Bar-Hillel, E. I. J. Poznanski, M. O. Rabin, A. Robinson (eds.), Essays in the Foundations of Mathematics, dedicated to A. A. FraenkeI. Jerusalem, 1961,328-351, p. 335. 25 "Frege's theory of number" (1965), in Mathematics in Philosophy, N.Y. 1983. 26 An interesting intermediate case is how constructive proofs as the object of intuitionist mathematics could be interpreted from a Kantian point of view. According to Kant as I interpret him, certain empirical constructions can function as paradigms so as to establish necessary truths because of the intention or meaning associated with them. Intuitionism would require that our insight into th~se meanings be sufficient not only to establish laws directly relating to objects in space and tme but also to establish laws concerning the intentions as ''mental constructions." I leave open the question of whether this is possible from Kant's point of view or not. 27 Ak.1I272-301. 28 Die Mathematik betrachtet in ihren Aufiosungen, Beweisen, und Folgerungen das Allgemeine unter den Zeichen in concreto, die Weltweisheit das Allgemeine durch die Zeichen in abstracto (Erste Betrachtung, §2. heading, Ak. IT 278). 29 Denn da die Zeichen der Mathematik sinliche Erkenntnismittel sind, so kann man mit derselben Zuversicht, wie man dessen, was man mit Augen sieht, versichert ist, auch wissen, class man keinen Begriff aus der Acht gelassen, class erne jede einzelne Vergleichung nach leichten Regeln geschehen sei U.S.w. Wobei die Aufmerk-samkeit dadurch sehr erleichterr wird, dass sie nicht die Sachen selbst in ihrer allgemeine Vorstellung, sondern die Zeichen in ihrer einzelnen Erkenntnis, die da sinnlich ist, zu gedenken hat. (Dritte Betrachtung, § I, Ak. II 291). 30 But cf. the following: in der Geometrie, wo die Zeichen mit den bezeichneten Sachen iiberden eine Ahulichkeit haben, ist daher diese Evidenz noch grosser, obgleich in der Buchstabenrechnung die Gewissheit evenso zuwerliissig ist. (Ibid., 292). 3'1 [This "general accord" now seems to me quite tenuous, and Manley Thompson is probably right in saying that the synthesis required for analytic judgments is clearly distinguishable from that in mathematical judgments ("Singular Tenns and Intuitions in Kant's Epistemology,» p. 342, n, 23). Nonetheless, a reply to the main point, that logic is not entirely independent of intuitive construction, would demand a lot of Kant's distinction between intuitive and discursive proofs, as is clear from Thompson's interesting discussion Qfthis distinction (ibid., pp. 340-342). His interpretation implies rather extreme limits on the role of logic in mathematics. This raises a doubt whether Kant's distinction is in the end tenable.] 32 One might say that it is possible to construct tokens. The sense of possibility in which this is possible is, however, derivative from the mathematical possibility of constructing types (or mathematical existence of the types). For we declare that the tokens are possible either directly on the basis of the mathematical construction, or physically on the basis of a theory in which a mathematical space which is in some way infinite is an ingredient. 33 Cf. my «Infinity and Kant's conception of the 'possibility of experience'" in Mathematics in Philosophy (1964). 34 This does not imply that there is an upper limit on the numbers which can be individually represented, once we admit notations for faster-growing functions than the successor function. This happens already in Arabic numeral notation. The number-"I,OOO,OOO,OOO,OOO, if written in '0' and'S' notation with four symbols per centimeter, would extend from the earth to the mOOIL That there is such an upper limit follows, of course, from the assumption that human history
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CHARLES PARSONS
must come to an end after a finite time.
35 "Kantian Intuitions", Inquiry 15 (1972) 341-345. In this Postscript this paper is cited merely by page number. 36 Particularly since the singularity criterion itself is not in dispute, I should emphasize that my own understanding of its importance owes much to Hintikka's writings and conversation. I should belatedly thank him for explaining his ideas to me some years before they were published in English.
37 Kemp Smith translates einzeln in this passage as "single." The translation "singular" fits its use in Logic, §l (see above, p. 112), where it is paired with the Latin singularis. Thompson suggests that in the Critique passage it may mean that an intuition is a single occurrence. ("Singular Tenns and Intuitions, in Kant's Epistemology" this volume. p. 105, n. 13.) If intuitions are thus in effect events. that would rule out Hintikka's interpretation (though not the view of Robert Howell discussed below). Though this seems to me to agree with Kant's characteristic way of speaking about institutions, the point is not so clear as to be a serious argument in the present dispute. 38 "Institution. Synthesis, and Individuation in the Critique of Pure Reason," NaUs 7 (1973), 207-232, p. 210. 39 A 71 = B 96; Logic, §21, n. I (Ak., IX, 102). 40 LogiC §l, Note 2 (Ak., IX, 91). This point is discussed at some length by Thompson, "Singular Terms and Intuitions," pp. 83-84. 41 In the discussion of "the black man" in a letter to J. S. Beck, July 3, 1792 (Ak., XI, 347); cf. Howell, "Intuition, Synthesis, and Individuation," p. 210. Other examples that can be given, such as the Idea of God, involve either Ideas of Reason or mathematics and might therefore be regarded as exceptional. 42 "Intuition, Synthesis, and Individuation," pp. 210-211. The distinction between what he takes to be the definition and his further interpretation is not explicit in the paper; here I rely on his clarification of his views in a recent letter. 43 Ibid., p. 215. A somewhat similar picture is presented in Thompson, "Singular Terms and Intuitions." However, Thompson rejects Howell's view that certain demonstratives can be the linguistic expression of intuitions; see Thompson, pp. 91-92, and Howell's reply, p. 232. Thompson holds that a Kantian "canonical language" would be virtually without singular tenns (ibid., p. 10 I). 44 In a discussion in March 1983, after this Postscript has been written; Hintikka stated that Howell's interpretation of immediacy was the view Hintikka had maintained all along. 45 But obviously one needs to look carefully at how Kant and his contemporaries actually viewed the relation of concepts and their "marks." This matter was explored by my student Alan Shamoon in his dissertation, "Kant's Logic," Columbia University 1979. 46 Cf. the contrast on the following page (A 733 = B 761) between discursive and intuitive principles. 47 lowe this last observation to Manley Thompson. However, the direct-reference view might itself suggest an assumption such as I attribute to Kant; compare the connection between direct reference and sense-perception in the philosophy of Bertrand Russell. 48 Kant's Theory of Science, p. 50, n. 15. Chapter 2 of this book is a very clear and instructive discussion of Kant's philosophy of mathematics, with expositions both of Hintikka's and my own views. Howell seems not to be free of the same misunderstanding; see "Intuition, Synthesis, and . Individuation," pp. 210-211.
KANT'S PHILOSOPHY OF ARITHMETIC
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49 Ak., IV, 281-282, translation from Beck's edition. 50 Ak., IV, 282, emphasis Kant's. 51 "But in mathematical problems there is no question of this Ithe conditions Wider which the perception of a thing can belong to possible experience], nor indeed of existence at all, but only of the properties of the objects in themselves, solely in so far as these properties are connected with the concept of the objects" (A 719 = B 747). This passage is instructively discussed by Thompson, "Singular Tenus and Intuitions," pp. 98-101. It is largely through this paper that I became aware of the difficulties faced by my own views about mathematical objects and existence in Kant. Brittan comments on the same passage (Kant's Theory of Science, p. 66), but he seems to me to misread it in saying that "mathematical problems" have to do with real possibility. That seems to me to neglect the clear statement of A 223-4 = B 271 that construction does not establish such possibility, and even the remark in the present passage that in mathematical problems there is "no question" of the conditions under which perception of a thing can belong to possible experience. The point is subtle because Kant holds that mathematics is about really possible objects and that this can be established. But it is not mathematics that establishes it. 52 A 218 = B 265. Cf. the whole discussion of possibility this statement introduces. Thompson (p. 106, n. 21) sees a difficulty with a modalist interpretation of how Kant might deal with reference to mathematical objects in Kant's distinction between demonstrations and discursive proofs. I am not sure I understand what the difficulty is, but it is not evident that the attenuated version of modalism suggested for Kant in the text is more exempt from it than the direct version. But cf. note 30 above. 5~ See my «Ontology and Mathematics" in Mathematics in Philosophy (Ithaca, Cornell University Press, 1983), section III; also "Mathematical Intuition, Proceedings of the Aristotelian Society N.S. (1979-1980)." 54 I wish to thank Robert Howell and Manley Thompson for their helpful comments on an earlier version of this Postscript.
MANLEY THOMPSON
SINGULAR TERMS AND INTUITIONS IN KANT'S EPISTEMOLOGY
As Kant explains his concept of intuition, it seems clear that a representation must satisfy two conditions in order to be an intuition: it must be singular and it must relate immediately to its object. Charles Parsons has referred to these as "the singularity condition" and "the immediacy condition," and has doubts that within Kant's philosophy they boil down to the same thing.! Jaakko Hintikka, on the contrary, maintains that in Kant the immediacy condition is only the singularity condition stated in another way, so that "Kant's notion of intuition is not vety far from what we would call a singular term."2 Both Parsons and Hintikka focus their attention primarily on Kant's philosophy of mathematics, and Parsons holds that "Hintikka's theory really stands or falls on the interpretation of the role of intuition in mathematics" (p. 46). In this paper I want to emphasize Kant's treattnent of empirical judgments and. the role that intuition plays in them, as I believe (and will tty to show in the course of my discussion) that this context was the primary one for Kant. As textual evidence for his theory, Hintikka cites a passage from the first Critique in the first section of the Dialectic where Kant gives a classification of representations . ... an objective perception is knowledge (cognitio). This is either intuition or concept (intuitus vel conceptus). The former relates immediately to the object and is single, the latter refers to
it
mediately by means of a feature which several things may have in common [A 320 B 376-77]3
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After quoting this passage along with a remark from the Prolegomena §8, "Intuition is a representation such as would depend directly on the presence of the object," Hintikka concludes: These quotations . .. show what for Kant was the alternative to an immediate relation to objects. It was a reference to objects by means of certain marks or characteristics which may be shared by several objects, i.e., a reference to objects by means of general concepts. Hence, another way of saying that Anschauungen have an immediate relation to their objects is to say that they are particular ideas or IOO,}
Yl.n Gy) is equivalent to 3xFx --> 3yGy.) Moreover, it is the dependence of one quantifier on another - specifically, of existential quantifiers on universal quantifiers - that enables us to capture the intuitive idea of an iterative process formally: any value x of the universal quantifier generates a value y of the existential quantifier, y can then be substituted for x generating a new value y', and so on. Hence, the existence of an infinity of objects can be deduced explicitly by logic alone. We .can now begin to see what Kant is getting at in his doctrine of construction in pure intuition. For Kant logic is of course syllogistic logic or (a fragment of) what we call monadic logic. 14 So, for Kant, one cannot represent or capture the idea of infinity formally or conceptually: one cannot represent the infinity of points on a line by a formal theory like 1-6 above. If logic is
185
Space is represented as an infinite given quantity [Grosse]. Now one must ~ertainly think eveI)l concept as a representation in which an infinite aggregate [Menge] of different possible representations are contained (as their common characteristic (MerkmalD, and it therefore contains these under itself. But no concept, as such, can be so thought as if it were to contain an infinite aggregate of representations in itself. Space is thoug.1it'in precisely this way, however (for all parts of space in ifl-finitum exist simultaneously). Therefore t..'1e original representation of space is an a priori intuition, and not a concept (B40).
At first sight, it is not at all clear what Kant means by the distinction between "containing representations under itself" and "containing representations in itself' here, but our above considerations supply us with a plausible reconstruction. Monadic concepts can, of course, have infinite extensions: an infinite number of objects can happen to fall under any given monadic fonnnla. But monadic concepts (unlike polyadic formnlas) cannot/orce their extensions to be infinite: they do not (and cannot) contain an infinity of objects i.."1 their very idea, as it were. Hence. since the idea of space does have this latter property, it cannot be a (monadic) concept. The notion of infinite divisibility or denseness, for example, cannot be represented by any such formnla as 6: this logical form simply does not exist. Rather, denseness is represented by a definite fact about my intuitive capacities: namely, whenever I can represent (construct) two distinct points a and b on a line, I can represent (construct) a third point c between them. Pure intuition - specifically, the interability of intuitive constructions 15 - provides a uniform method for instantiating the existential quantifiers we would use in formulas like 6; it therefore allows us to capture notions like denseness without actually using quantifier-dependence. Before the invention of polyadic quantification theory there simply is no alternative. Thus, in Euclid' s geometry there is no axiom corresponding to our denseness condition 6. Instead, we are given a uniform method for actually constructing the point bisecting any given finite line segment: it suffices to join C in the figure on page 181 with its "mirror image" below AB - the resulting straight line bisects AB (Prop. 1. 10). This operation, which is itself constructed by iterating the basic operations (i), (ii), and (iii), can then be iterated as many times as we wish, and infinite divisibility is thereby represented. So we do not derive new points between A and B from an existential axiom, we construct a bisection function from our basic operations and obtain the new points as the values of this function: 16 in short, we are given what modern logic calls a Skolem function for the existential quantifier in 6. 17 For Kant,
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this procedure of generating new points by the iterative application of constructive functions takes the place, as it were, of our use of intricate rules of quantification theory such as existential instantiation. Since the methods involved go far beyond the essentially monadic logic available to Kant, he
places are only parts of the same boundless space, related to one another through a certain position. nor can you conceive to yourself a cubic foot unless it be bounded on all sides by the surrounding space.
views the inferences in question as synthetic rather than analytic. I8 Finally, we should note that our modem distinction between pure and applied geometry, between an uninterpreted formal system and an interpretation that makes such a system true, cannot be drawn here, In particular, the only way to represent the theory of linear order 1-6 is to provide, in effect, an interpretation that makes it true. 19 The idea of infinite divisibility or denseness is not capturable by a formula or sentence, but only by an intuitive procedure that is itself dense in the appropriate respect. By the same token, the sense in which geometry is a priori for Kant is also clarified. Thus, the proposition that space is infinitely divisible is a priori, because its truth - the existence of an appropriate "model" - is a condition for its very possibility.2o One simply cannot separate the idea or representation of infinite divisibility from what we would now call a model or realization of that idea; and our notion of pure (or formal) geometry would have no meaning whatever for Kant. (In a monadic context a pure or uninterpreted "geometry" cannot be a geometry at all, for it cannot represent even the idea of an infinity of points.) II
The above considerations make a certain amount of sense out of Kant's theory, but one might very well have doubts about attributing them to Kant. After all, Kant certainly had no knowledge of the distinction between monadic and polyadic logic, nor of quantifier-dependence, Skolem functions, and so on. So using such ideas to explicate his theory may appear wildly anachronistic, and my reading of B40 may appear strained In particular, the . distinction I stressed between "containing representations under itself' and "containing representations in itself' appears capable of a much simpler interpretation.· Kant is merely drawing a contrast between the predication relation and the part-whole relation: the relation of space to spaces is .not one of general concept to its instances but one of (individual) whole to its parts. Therefore, space is a singular representation (intuition), not a general representation (concept). These doubts are reinforced by a glance at § IS.B of the Inaugural Dissertation (1770), where the point is made in exactly this way: The concept of space is a singular representation comprehending all things in itself, not an abstract and common notion containing them under itself. For what you speak of as several
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Here the distinction between the part-whole relation and the predication relation is quite clear and explicit. Further, there is no confusing - and apparently extraneous - reference to infinity. Yet, from the present point of view, it is precisely this difference between the Critique and the Inaugural Dissertation that is most striking. Kant apparently found it necessary to modify his arguments between 1770 and 1781 (and again between 1781 and 1787, since B40 is itself a revision of the first edition passage at A2S), and we should ask ourselves why. First of all, it is clear that Inaugural Dissertation, § \S.B is, by itself, quite insufficient for Kant's purposes. Kant wants to show that our cognitive grasp of the notion of space is intuitive rather than conceptual or discursive, and the mere contrast between the part-whale relation and the predication relation certainly does not establish this. For, as Kant explicitly recognizes in 1781, there is indeed a "general concept of space (which is common to both a foot and an ell alike)" (A2S), and this general concept of space - which we might represent by Ix is a space' - does bear the predication relation to spaces or parts of space. In other words, there is both the general concept IX is a spacel, which bears the predication relation to spaces, and the singular individual space, which bears the whole-part relation to spaces (and is in turn also related via predication to IX is a space'). The question is: why should the latter have cognitive priority over the former? Wby should our idea or representation of space be identified with the actual individual space rather than the general concept IX is a space'? Moreover, there is also a general concept corresponding to the part-whole relation: the concept Cx is a part ofyl-this would presnmably be an example of the kind of "general concent of relations of things as such" (or simply "concept of relations") to which Kant is alluding at A2S 21 Given this concept and the concept IX is a spacel we can apparently frame a discursive or conceptual definition of the individual space: namely, "that space of which all other spaces are parts." From this point of view, then, it begins to look as if our cognitive grasp of the general concept of space and spatial relations (for example, the part-whole relation) precedes our cognitive graSp of the individual space. Kant rejects this last possibility at A2S = B39: Space is not a discursive or, as one says. general concept of relations of things as such. but a pure intuition. For, first, one can represent to oneself only one space; and when one speaks of several spaces, one means thereby only parts of the same unique space. Nor can these parts precede the
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one all-embracing space, as being, as it were, its constituents (and making its composition possible); on the contrary, they can be thought only in it: Space is essentially one: the manifold in it,. and hence the general concept of spaces as such. depends solely on limitations. It follows that an a priori intuition (that is not empirical) underlies all concepts of space.
One cannot arrive at the individual space by way of the concepts rx is a space' and ex is a part of y'. On the contrary, these concepts are themselves only possible via the intuitive act of "cutting out" parts of space from the singular intuition space. But now the question becomes why should this be so: why should the singular intuition cognitively precede the general concept? In the sentence immediately following the above quotation Kant appeals to our knowledge of geometry: So too are all principles [Grundsa:tze] of geometry - for example. that in a triangle two sides
together are greater than the third - derived: never from general concepts of line and triangle, but only from intuition, and this indeed a priori, with apodeictic certainty.
We find a sintilar appeal in Jnaugural Dissertation, § I5.C: "That there are not given in space more than three dimensions, that between two points there is only one straight line, that from a given point on a plane surface a circle can be described with a given straight liue, etc. - these cannot be concluded from some general notion of space, but can only be seen, as it were, in space in concreto." We also find a reference to "incongruous counterparts" (which are not mentioned in the Critique) and the claim that "here the diversity, namely, the discongruity, can only be noticed by a kind of pure intuition." Finally, Kant appeals to geometrical demonstrations: "Geometry does not demonstrate its own general propositions by thinking an object by means of general concepts as happens with things rational, but by subjecting it to the eyes by means of a singular intuition as happens with things sensitive." In the end, therefore, Kant's claim of cognitive priority for the singular intuition space rests on our knowledge of geometry.22 Our cognitive grasp of the notion of space is manifested, above all, in our geometrical knowledge. Hence, if we can show that this knowledge is intuitive rather than conceptual, we will have shown the cognitive inadequacy of the general concept of space and the priority of the singular intuition. Continuing our line of questions, then, we must ask why, at bottom, is conceptual knowledge inadequate to geometry: why must intuition play an essential role? Surely, the mere assertion that geometrical principles "cannot be concluded from some general notion ... but can only be seen, as it were, in space in concreto" is not expected to convince those, like the Leibnizians and
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Wolffians. who maintain precisely L,e opposite. Now it is at this point that Kant puts forward t,'te "infinity argnment" in the Critique. In the first edition we have § 2.5: Space is represented as an infinite given magILitude. A general concept of space (which is common to both a foot and an ell alike) can detennine nothing in regard to magnitude [Grossel. Were there no limitlessness in the progression of intuition, no concept of relations could, by itself, supply a principle of their infinitude (A25).
Neither the general concept rx is a spaceJ nor any concept of spatial relations (for example, ex is a part of yl) can yield a "pdnciple of infinitude." Such a principle can only be found in the '"limitlessness in the progression" or nnbounded iterability of pure intuition. Hence, since the idea of infinity - in parricular, the idea of infinite divisibility - is all essential part of geometry and therefore of our idea of space, the latter must indeed be a pure intuition and not a concept. The second edition passage at B40 is clearer, for Kant is more explicit that the problem is not with th~ general concept rx is a spaceJ in particular but with all general concepts as such. In the first edition, the problem appears to be merely that rx is a spaceJ says nothing whatever about magnitUde, whether infinite or finite. We might then be tempted to try concepts such as rx is a cubic foot of spaceJ or even rx is a greater space than y'. The passage at B40 (along with the final sentence of A, § 2.5) gets to heart of the matter: without the unbounded iterability of pure intuition, no concept - not even a relational concept like rx is a greater space than y' - can force its extension to be infinite; although, to be ·sure, any concept may have an infinite (possible) extension. The same idea stands out clearly in § 12 of the Prologomena (1783): "That one can require a line to be drawn to infinity (in indeifinitum), or that a series of changes (for example, spaces traversed by motion) shall be infinitely continued, presupposes a representation of space and time that can only depend on intuition, namely, in so far as it in itself is bounded by nothing; for from concepts alone it could never be inferred." Once again, Kant's conception of infinity and infinite divisibility can be clarified by contrasting it with modem formulations. We, of course, can easily represent infinite divisibility by means of concepts like ex is greater than y l - as we did above in the theory of dense linear order. In such a theory the points on a line are taken as primitive, and the line itself is built up from them in just the way Kant says it cannot be: the points relate to the line as "its constituents (and making its composition possible):' Yet what makes this· representation itselC possible is precisely the quantifier-dependence of
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modern polyadic logic: the logical form V.. \13. In the absence of such logical forms - and in accordance with the actual procedure of Euclid's geometry the natural alternative is to represent infinite divisibility by an intuitive constructive procedure for "cutting out" a stualler line segment from any given one: for example, Euclid's construction for bisecting a line segment of Prop. 1.10. Thus, whereas we can represent infinity by ''v'x3y(x is greater than y)', Kant would formulate this proposition by 'x is greater than f B(x)', where fB(x) is the operation of bisection, say. And, in this representation, the idea of infinity is not conveyed by logical features of the relational concept 'x is greater than Y', but by the well-definedness and iterability of the function fB(x): our ability, for al'Y given line segment x, to construct (distinct) fB(x), fB(fB(x», ad infinitum. This, I suggest, is why Kant gives priority to the singular intuition space, from which all parts or spaces must be "cut out" by intuitive construction ("limitation"). Only the unbounded iterability of such constructive procedures makes the idea of infinity, and therefore all "general concepts of space," possible. And, of course, it is this very same constructive ilerability that underlies the proof-procedure of Euclid's geometry.
and eighteenth centuries; that does reqnire genuine continuity - "all" or "most" real numbers; whose modern, "rigorous" formulation requires full polyadic logic - much more intricate fonns of quantifier-dependence than V.. \13; and, finally, whose earlier, "non-rigorous" formulation made an essential appeal (in at least one tradition) to temporal or kinematic ideas _ to the intuitive idea of motion. This branch of mathematics is of course the calculus, or what we now call real analysis. It goes far beyond Euclidean geometry in considering "arbitrary" curves or figures - not merely those constructible with Euclidean tools - and in making extensive use of limit
III
Even if we are on the right track, however, we have still gone only part of the way towards understanding construction in pure intuition. We can bring out what is missing by three related observations. First, as we noted above, the notions of denseness, infinite divisibility, and (even) constructibility with straight-edge and compass do nol amount to full continuity. These notions all involve denumerable sets of points which are but small fragments of the set Ill. of real numbers. Hence, to understand how full continuity comes in we have to go beyond Euclidean geometry. Second, these notions do not (on a modem construal) exploit very much of polyadic logic: just the logical form V.. \13. If t..his is all that were required modern logic would hardly need to have been invented. But we would like to understand why modern logic was invented and, in particular, why it was invented when it was. Third, the procedure of construction with Euc.lidean tools - with straight-edge and compass - does not really exploit the kinematic element that is essential to Kant's conception of pure intuition: no appeal is made to the idea that lines, circles, and so on are generated by the motion of points. So why did Kant think that motion is so important? These three observations are in fact intimately related. For there exists a branch of mathematics which was just being developed in the seventeenth
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operations.
From a modem point of view the basic limit operation underlying the calculus is explained in teITIlS of the Cauchy-Balzano-Weierstrass notion of convergence. Moreover, we also appeal to this notion in explaining the distinction between denseness and genuine continuity, in precisely expressing the idea that there are "gaps" is a merely dense set like the rational nllll1bers. Thus, let s" s" ... be a seqnence of rational numbers that converges to n, say - that approaches n as its limit (for example, let s, = 3.1, s, = 3.14, and in general Sn= the decimal expansion of 1t carried out to n places). This sequence of rationals converges (to "something" as it were), but in the set Q of rational mimbei-s (and even in the expanded set Q* of Euclidean-constructible, numbers) there is no \intit point it converges to. Such limit points are "missing" from a merely dense set like the rationals. A truly continuous set contains "all" such limit pOints. More precisely, using Cauchy's criterion of 1829, we say that a sequence SI' S2" .. converges if 'v'E 3N 'v'm 'v'n [m, n > N ~ ISm - s,1 < Ej,
where £ is a positive rational number and N, m, n are natural numbers. A sequence SI, S2.··· converges to a limit r if "IE 3N "1m [m>N ~ Ism-rl <Ej,
The problem with a merely dense order is that the first can be true even when the second is not, whereas a continuous order satisfies the additional axiom of Cauchy completeness - whenever a sequence converges, it converges to a limit r - which clearly has the logical form:
\l3VV ~ 3r\l3V Note the additional logical complexity of this axiom: in particular, the use of the strong form of quantifier-dependence \13"1. The increase in logical strength that I find so striking here can be best
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brought out if we compare the way points are generated by a completeness axiom with the way they are generated by Euclidean constructions (or, from a modem point of view, by the weaker form of quantifier-dependence V.. '0'3). In the latter case, although the total number of points generated is of course infinite, each particular point is generated by a finite number of iterations: each point is determined by a finite number of previously constructed points. In generating or constructing points by a limit operation, on the other hand, we require an infinite sequence of previously given points: no finite number of iterations will suffice. So limit operations involve a. much stronger and
more problematic use of the notion of infinity than that involved in a shnple process of iterated construction. Let us now return to Kant and the late eighteenth century. We cannot of course represent the ideas of convergence and transition to the limit by complex quantificational fonus like \if 3V. But the idea of continuous motion appears to present us with a natural alternative. Thus, for example, we can easily "construe!" a line of length re by imagining a continuous process that takes one unit of time and is such that at t = -l- a line of length 3.1 is constructed, at t = %a line of length 3.14 is constructed, and in general at t = n/n + 1 a line of length s" is constructed, where s, again equals the decimal expansion of
1t
carried out to n places. Assuming this process in fact has a
tenninal outcome, at t = I we have consttucted a line of length re. In some sense, then, we can thereby "construct" any real number. 23
In this style of representation the notion of convergence or approach to the iimit is expressed by a temporal process: by the idea of one point moving or becoming closer and closer to a second. This intuitive process of becoming
does the work of our logical fonn 113V, as it were. That the limit of a convergent seqnence exists is expressed by the idea that any finite process of temporal generation has a terminal outcome. This idea does the work of our logical form 3113V. In particular, then, what we now call the continuity or completeness of the points on a line is expressed by the idea that any finite motion of a point beginning at a definite point on our line also stops at a definite point on our line,24 What t.~e modern definition of convergence does, in effect, is
replace this intuitive conception based on motion and becoming with a formal, algebraic, or "static" counterpart based on quantifier-dependence and order relations. Now a temporal conception of the limit operation is explicit in the basic lenuna Newton uses to justify the mathematical reasoning of Principia: Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer to each other than by any given difference, become ultimately equal (Book I, § I, Lemma D.25
This lenuna is perhaps best understood as a definition of what Newton means by «quantiiy": namely, an entity generated by a continuous temporal process (it clearly fails for discontinuous "quantities"). Newton's conception of "quantities" as temporally generated is even more
explicit, of course, in his method of fluxions, where all mathematical entities are thought of as fluents or "flowing quantities." For example: I don't here consider Mathematical Quantities as composed of Parts extremely small, but as generated by a continual motion. Lines are described, and by describing are generated, not by any apposition of Parts, but by a continual motion of Points. Surfaces are generated by the motion of Lines, Solids by the motion of Surfaces, Angles by the Rotation of their Legs, Time by a continual flux, and so in the rest. 26
Moreover, Kant appears to be echoing these ideas in an hnportant passage about continuity in the Anticipations of Perception: Space and time are quanta continua. because no part of them can be given without being enclosed between limits (points and instants), and therefore only in such fashion that t.~is part is itself again a space or time. Space consists only of spaces, time consists only of times. Points and iristants are only limits, that is, mere places [Stellen] of their limitation. But places always presuppose the intuitions which they limit or determine; and out of mere places, viewed as constit1-'ents capable of being given prior to space or time, neither space nor time can be composed [zusammengesetzt]. Such quantities [Grossen} may also be calledjfowing Iftiessendl, since the synthesis (of the productive imagination) in their generation [Erzeugung] is a progression in time, whose continuity is most properly designated by the expression of flowing (flowing away) (B211-212).
For Kant, like Newton, spatial quantities are not composed of points, but rather generated by the motion of points. I take Kant's choice of language to be especially significant here, for his "fliessende Grossen" is the standard German equivalent of Newton's "fluents. "27 This expression is used, for example, by the mathematician Abraham Kastner in his influential textbooks on analysis and mathematical physics." Kastner's analysis text attempts to develop the calculus from a "rigorous" standpoint that makes no appeal to infinitely small quantities. In this connection he develops a version of Newton's method of fluxions, and, what is more remarkable for a German author of this period, he argues that Newton's fluxions are in some respects clearer and more perspicuous than Leibniz's differentials. Further, he explicitly applauds Collin Maclaurin's attempt, in his monumental Treatise of Fluxions (1742), to develop the calculus on the basis of a kinematic conception of the limit operation. 29
Without going into detail,'" the most basic ideas of the fluxional calculus are as follows. We start with fluents or "flowing quantities" x, y, conceived as continuous functions of time. We can then fonn the fluxions or time-
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derivatives .t, y, because continuously changing quantities obviously have well-defined instantaneous velocities or rates of change. 31 If we are then given a curve or figure y = f(x) generated by independent motions in rectangular coordinates of the fluents x, y, the derivative (slope of the tangent line) will be dy/dx = y/i says that s converges tofixo). We define the differentiability affix) at a given pointxo by an expression of the fonn:
V3V [Conv(s', x o)], where s' is a second sequence defined algebraically from fix). By understanding these notions formally rather than intuitively, we can, for the first time, both clearly and precisely distinguish them and clearly and precisely explore their logical relations: differentiability logically implies continuity but not vice versa, for example.44 IV
The present approach to Kant's theory of geometry follows Russell in assuming that construction -in pure intuition is primarily intended to explain mathematical "proof or reasoning, a type of reasoning which is therefore distinct
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from logical or analytic reasoning. Again following Russell, we have sought an explanation for this idea in the difference between the essentially monadic logic available to Kant and the polyadic logic of modem quantification theory. Further, we have tried to link this conception of mathematical reasoning with the very possibility of thinking or representing mathematical concepts and propositions. Thus, for example, "r cannot think a line except by drawing it in thought" (B 154), because only this representation permits me to use the concept of line in mathematical reasoning (such as Euclid's or
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Newton's) where properties like denseness and continuity play an essential role. Yet RuselI's assumption has been vigorously debated. It has been maintained that Kant did not deny, and indeed may have even affirmed, that mathematical inference is logical or analytic; his primary concern, rather, is with the status of the premises or axioms of such inferences. Geometry is synthetic precisely because its underlying axioms are synthetic; the (synthetic) theorems of geometry follow purely logically or analytically. This anti-Russellian view is clearly and forcefully stated by Beck: The real dispute between Kant and his critics is not whether the theorems are analytic in the sense of being strictly [logically J deducible, and not whether they should be called analytic nOw when it is admitted that they are deducible from definitions, but whether there are any primitive propositions which are synthetic and intuitive. Kant is arguing that the axioms cannot be analytic ... because they must establish a connection that can be exhibited in intuition.45
As Beck indicates, this view is attractive because Kant will not be refuted, as Russell thought, by the mere invention of polyadic logic. For even modern formulations of Euclidean geometry like Hilbert's will contain priruitive propositions or axioms, and pure intuition can be called in to secure their truth (to provide a model, as it were).46 Indeed, from this point of view, the discovery of logically consistent systems of non-Euclidean geometry should be seen as a vindication of Kant's conception. The existence of such geometries shows conclusively that Euclid's axioms are not analytic and, therefore, that no analysis of the basic concepts of geometry could possibly explain their truth (as Leibniz apparently thought). Assuming that Euclid's axioms are true, then, there is no alternatiye.but to appeal to a synthetic source: hence pure intuition.47 On the Russell-inspired interpretation developed here, by contrast, there can be no question of non-Euclidean geometries for Kant. Non-Euclidean straight lines, if such were possible, would have to at least possess the order properties - denseness and continuity - common to all lines, straight or curved. And, on the present interpretation, the only way to represent (the order properties of) a line - straight or curved - is by drawing or generating it in the space (and time) of pure intuition. But this space, for Kant, is necessarily Euclidean (on both interpretations). It follows that there is no way to draw, and thus no way to represent, a non-Euclidean straight line; and the very idea of a non-Euclidean geometry is quite impossible." (Another way to see the point is to note that the anti-Russellian interpretation would reinstate precisely the modern distinction between pure and applied geometry argued above to be unavailable to Kant.)
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The anti-Russellian interpretation draws its primary support from B 14: For as it was found that all mathematical inferences proceed in accordance with the principle of contradiction [nach dem Satze des Widerspruchs fortgehen] (which the nature of all apodeictic certainty requires), it was supposed that the fundamental propositions [Grundsatze] could also be recognized from that principle [aus dem Satze des Widerspruchs erkannt wUrdenl. This is erroneous. For a synthetic proposition can indeed be comprehended [eingesehenJ in accordance with [nach] the principle of contradiction, but only if another synthetic proposition is presupposed from which it can be derived [gefolgert}, and never in itself.
Kant seems to be saying that because inference from axioms to theorems was (correctly) seen as analytic, the axioms themselves were (incorrectly) thought to be analytic. But these axioms are really synthetic; for this reason (and only for this reason), so are the theorems. Kant therefore agrees with Russell that the conditional [Axioms ..... Theorems] is a logical or analytic truth;49 his ~ point is simply that the antecedent of the conditional is synthetic. I do not think this reading of the passage is forced on us. First of all, Kant does not actually say that mathematical inference is analytic, nor that the theorems can be analytically derived. The first sentence may mean only that mathematical proofs necessarily involve logical or analytic steps - and, of course, no logical fallacies. 50 Similarly, the last sentence does not explicitly say that the derivation of one synthetic sentence (theorem) from another (axiom) is analytic; the possibility that this derivation is itself synthetic is at least left open. Second, it is assumed that by fundamental propositions [Grundslitze] Kant means axioms, and this is doubtful. Kant's own technical term for axioms is Axiomen (cf. B204-205, B760-762), and at A25 he calls the proposition that two sides of a triangle together exceed the third a fundamental proposition [Grundsatz]. This latter is of course not an axiom in Euclid, but a basic (and therefore fundamental) theorem (Prop. 1. 20). So the error Kant is diagnosing here may not be the (really rather ridiculous) mi~e of transferring analyticity from inference to preruise (axiom), but the more subtle supposition that because logic plays a central role in the proof of basic theorems it is sufficient for securing their truth. A more fundamental problem for the anti-Russellian reading of BI4 is posed by Kant's conception of arithmetic. Kant is supposed to have a moreor-less modem picture of mathematical theories as strict deductive systems. The synthetic character of mathematics depends solely on the synthetic character of the underlying axioms. But this is certainly not Kant's picture of arithmetic. According to Kant, arithmetic differs from geometry precisely in having no axioms, for there are no propositions that are both general and synthetic serving as preruises in arithmetical arguments (B204-206). Thus, our conception of arithmetic as based on the Peano axioms, say, is completely
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foreign to Kant, and one cannot use the model of an axiomatic system to explain why arit.~metic is synthetic: one cannot suppose that arithmetical reasoning proceeds purely logically or analytically from synthetic axioms as premises.:5i Yet arit.lnnetic is the very first example Kant uses (at BI5-16) to illustrate, "-,,d presumably illuminate, the general ideas of B 14. Arithmetical propositions like 7 + 5 = 12 are synt.;'etic, not because they are established by analytic derivation from synthetic axioms (as we would derive them from the Peano axioms, say), but because they are established by the successive addition of unit to unit. This procedure is synthetic, according to Kant, because it is necessarily temporal, involving "t..1-te successive progression from one moment to another" (AI63)." Thus, for example, only the general features of succession and iteration in time can guarantee the existence and uniqueness of the sum of 7 and 5, which, as far as logic and conceptual analysis are concerned, is so far merely possible (non-contradictory). Similarly, only the unboundedness of temporal succession can guarantee the infinity of the number series; and so on. For Kant, then, arit.ljmetical propositions are established by calculation, a procedure that is sharply distinguished from logical argument in being essentially temporal. This is why Kant says that the synthetic character of arithmetical propositions "becomes even more evident if we consider larger nwnbers, for it is then obvious that, however we might turn and twist our concepts, we could never by mere analysis unaided by intuition be able to find the sum" (B 16). The reference to larger numbers makes it clear that intuition is not being called in to secure the truth of basic propositions - such as 2 + 2 = 4, perhaps - by "seuing them before our eyes. "Rather, intuition underlies the step-by-step process of calculation which, in its entirety, may very well not be surveyable "at a glance."" We have now reached the heart of the matter, I think, for it is the idea of a sharp distinction between calculation and logical argument that is perhaps most basic to Kant's conception of the role of intuition in mathematics. Thus, at B762-764 Kant contrasts matllematical and philosophical reasoning. Only mathematical proofs are properly called demonstrations, while philosophy is restricted to logical or conceptual ("acroamatic" or discursive) proofs. The latter "must always consider the universal in abstracto (by means of concepts)," the fonner "can consider the universal in concreto (in the single intuition), and yet still through pure a priori representation whereby all errors are at once made visible [sichtbar]" (B763). That Kant has calculation centrally in mind here is indicated by his refer-
ence to the methods employed in solving algebraic equations (B762), a reference which recalls the even more explicit conception of calculation found in the Enquiry Concerning the Clarity of the Principles of Natural Theology and Ethics (1763). See, for example, the First Reflection, § 2, entitled "Mathematics in its methods of solution [Ausflosungen], proofs, and deductions [Folgerungen] examines the uuiversal under symbols in concreto; philosophy examines the universal through symhols in abstracto":
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I appeal first of all to arithmetic, both the general arithmetic of indeterminate magnitudes [algebra], as well as that of numbers, where the relation of magnitude to unity is determinate. In both symbols are first of all supp~~, instead of the things themselves, together with special notations [Bezeicbnungen] for their increase and decrease, their relations, etc. Afterwards, one proceeds with these signs, according to easy and secure rules, by means of substitution, combination or subtraction, and many kinds of transformations, so that the things symbolized are here completely ignored, until, at the end, the meaning of the symbolic deduction is finally deciphered [entziffertJ.
As the Third Reflection, § I explains, this "symbolic concreteness" of mathematical proof accounts for the difference between philosophical and mathematical certainty. Since philosophical argument is discursive or conceptual, ambignities and equivocations in the meanings of general concepts are always· possible. Mathematics, on the other hand, works with concrete or singular representations that allow us to be assured of the correctness of its substitutions and transformations "with the same confidence with which one is assured of what one sees before one's eyes." As Kant puts it in the Critique, the step-by-step application of the easy and secure rules of calculation "secures all inferences against error by setting each one before our eyes" (B762).54 From the present point of view, the point could perhaps be reconstructed as follows. Mathematical proof, unlike logical proof, operates not only with predicates like 'x is even' and 'x is a triangle', but first and foremost withfunciion-signslike ex + y" and ethe bisector of z.,. In calculation we form functional terms by inserting particular arguments into the function-signs, we set up equalities (and inequalities) between such functional terms, and we substitute one functional term for another in accordance with these equalities. Since hoth the arguments and the values of our function-signs are individuals," the procedure of substitution is to be sharply distinguished from the subsumption of individuals under general concepts characteristic of logical or discursive reasouing. In particular, the essence of the former procedure lies in its iterability: f(a) can be substituted inf(x) to form a distinct functional termfif(a», while it of course makes no sense at all to subsume the predication F(a) under the
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predicate F(x)." Thus, the essentially "extra-logical" form of inference required is that which takes us from one object a satisfying a condition ... a ... to a second object/Cal satisfying another condition _ _ lea) ~ and from there to a third object/(f(a)) satisfying---flj(a)) ---, and so on. Now this conception of the role of calculation and substitution in mathematical proof also applies, mutatis mutandis, to the case of geometry. In Euclidean geometry we start with an initial set of basic constructive functions: the operation fLex, y) taking two points x, y to the line segment between them, the operation fE(x, y) taking line segments x, y to the extended line segment of length x + y, and the operation fc(x, y) taking point x and line segment y to the circle with center x and radius equal to y. We also have a specifically geometrical equality relation (congruence) and, of course, definitions of the basic geometrical fignres (circle, triangle, and so on). Euclidean proof then proceeds somewhat as follows. Given a figure a satisfy-
Nevertheless, there is of course an important difference between the two cases. As already noted above, geometry has axioms whereas arithmetic does not; moreover, geometry uses "ostensive construction (of the objects [Gegenstiinde] themselves)" (B745) in additon to the "symbolic" or "characteristic" (B762) construction conunon to algebra and arithmetic. Thus, Kant's discussion of algebraic construction has a decidedly "formalistic" tone: we "abstract completely from the nature of the object" (B745) and, as the above passage from the Enquiry puts it, "symbols are first of all supposed, instead of the things themselves" and "the things symbolized are here completely ignored." In geometry, on the other hand, such "formalism" is quite inappropriate: geometrical construction operates with "the objects themselves" (lines, circles, and so on). This difference between arithmetical-algebraic construction and geometrical construction is perhaps most responsible for the confusion that has surrounded Kant's theory. For it begins to look as if geometrical inntition has not merely an inferential or calculational role, but also the more substantive role of providing a model, as it were, for one particular axiom system as opposed to others (Euclidean as opposed to non-Euclidean geometry). Intuition does this, presumably, by placing the objects themselves before our eyes, whereby their specific (Euclidean) structure can be somehow discerned. From the present point of view, of course, there can be no question of picking out Euclidean geometry from a wider class of possible geometries. Rather, the difference between geometrical and arithmetical-algebraic construction is understood as follows. Geometry, uulike arithmetic and algebra, operates with an initial set of specifically geometrical functions (the operations f L, fE' and fe) and a specifically geometrical eqnality relation (congruence). To do geometry, therefore, we require not only the general capacity to operate with functional terms via substitution and iteration (composition), we atso need to be "given" certain initial operations: that is, intuition assures us of the existence and uniqueness of the values of these operations for any given arguments. Thus, the axioms of Euclidean geometry tell us, for example, ''that between two points there is only one straight line, that from a given point on a plane surface a circle can be described with a given straight line" (Inaugural Dissertation, § I5.C), and they also link the specifically geometrical notion of equality (congruence) with the intuirive notion of superposition (Prolegomena, § 12).60 Now one might at first suppose that the case of arithmetic is precisely the same. After all, we need inntitive assurance that the successor function, say, is uniquely defined for all arguments. The point, I think, is that the successor
202
ing a condition ... a ... , we construct, by iteration of the basic operations, a new constructive function g yielding an expanded fignre g(a) satisfying a condition - - - g(a) - - - . From this last proposition we are then able to derive a new condition _ _ a _ _ on our original figure a. Whereas the inference from ---g(a)--- to _ _ a __ can be viewed as "essentially monadic," and is therefore analytic or logical for Kant, the inference from ... a .. . to - -g(a)---is not: it proceeds synthetically, by expanding the figure a as far as need be into space the around it, as it were. Since this procedure is grounded in the indefinite iterability of our basic constructive operations, geometry is
synthetic for much the same reasons as is aritlunetic; and, therefore, the case of arithmetic is primary.57 Confirmation is apparently provided by the discussion at B 15-17. For Kant illustrates BI4 at great length with the example of arithmetic and only then touches on geometry, almost as a corollary." Just as little,is any fundamental proposition [Grundsatz] of geometIy analytic. That the straight line between two points is the shortest is a synthetic proposition. For my concept of straight {GeradenJ contains nothing of quantity [Grosse], but only a quality. The concept of shortest is
entirely an addition, and cannot be derived by any analysis of the concept of straight line. The aid of intuition must therefore be brought in, by means of which alone the synthesis is possible (B 16-17).
As the discussion of arithmetic has shown, the general concept of magnitude [Grosse] requires an inntitive synthesis (the successive addition of unit to unit). But geometry requires this concept as well (for example, in connecting the notion of straight line with the notion of shortest line). Therefore, geometry, just as much as arithmetic, is a synthetic discipline."
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Kant is imagining any such process of "visual inspection." It is much more
general form cif succession or iteration common to all functional operations
plausible thai, in precise parallel to his discussion of the angle-sum property at B743-745, he is referring to the Euclidean proof of this proposition (Prop. 1.20): We consider a triangle ABC and prolong BA to point D such that DA is equal to CA:
whatsoever. So it is not necessary to postulate any specific initial functions in
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definedness of the successor function is guaranteed by the mere form of iteration in general (that is. time). Thus, in explaining why geometry has axioms while arithmetic does not at B205-206, Kant refers to the need in geometry for general "functions of the productive imagination" like our ability to construct a triangle from any three line segments such that, two together exceed the third (this functional operation is of course definable, in Euclidean geometry, from the operations f L , fE' and fc= Prop. l,22). The point, presumably, is that no such specific functional operations need be postulated in arithmetic.6l In any case, the idea that pure intuition plays the more substantive role of providing a model for one particular axiom system as opposed to others - as the anti-Russellian interpretation reqnires - is rather obviously untenable and definitely unKantian. The untenability of such a view has been clearly brought out in an iostructive article by Kitcher." Kitcher supposes that the primary role of pure intuition is to discern the metric and projective properties (the Euclidean structure) of space. We construct geometrical figures like triangles and somehow "see" that they are Euclidean: "[Kant's] picture presents the mind bringing forth its own creations and the naive eye of the mind scanning these creations and detecting their properties with absolute accu-
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function is not a specific function at all for Kant; rather, it expresses the
arithmetic: whatever initial functions there may be. the existence and well-
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KANT'S THEORY OF GEOMETRY
racy" (p. 129). It is then easy to show that pure intuition, conceived on this quasi-perceptual model, could not possibly perform such a role. Our capacity for visualizing figures has neither the generality nor the precision to make the reqnired distinctions. Thus, for example, Kant's appeal to the proposition that two sides of a triangle together exceed the third at A25 is considered, and "We now imagine ourselves coming to know [it] in the way Kant suggests. We draw a scalene triangle and see that this triangle has the side-sum property" (p. 125). But this idea quickly founders on Berkeley's generality problem: how are we supposed to conclude that all triangles have the side-sum property and not, say, that all triangles are scalene? (Actually, in this connection, a more relevant dimension of generality is size. In elliptic [positive curvature] space, "small" - relative to the dimensions of the space itself - triangles have the side-sum property while arbitrarily large triangles do not. So one caunot argue from the properties of small, visualizable triangles to the properties of arbitrary triangles.)" It is extremely unlikely, however, that in appealing to intuition at A25
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We then draw DC, and it follows that er coincidence results. Why, that is, does he think we can anticipate a priori the results of other constructions without having to actually carry them out? Looked at this way, the issue is over homogeneity principles. Although Kant mostly talks of size homogeneity (so that little constructions verify also big results), I shall talk of positional or translational homogeneity. Why should a proximate local construction tell us anything about that same construction as carried out elsewhere? Note, if positional homogeneity obtained, we could verify a priori the complete local structure of Space everywhere and therefore also verify the result of large-scale constructions. I carry out a local construction according to a pair of rules and L':Ie upshot is coincidence. Why think that ca..rrying out that same construction. after an intermediate linear construction (with one's feet say) would also result in coincidence? I tlunk Kant's answer is simply to reiterate that it is the very same construction that is being carried out in both cases, guided completely by the same rules or directions. If Space were something objective, rather than merely our own activity, theu perhaps its features could change from place to place. If space were either a receptacle entity or a system of configurations of empitical items, then the features of this entity or system might very well vary, even the feature of coincidence. Put crudely, maybe the receptacle or else the configuration bends or twists or dips from locale to locale. The point is, as long as we are dealing with something objective, there is no way to verify a priori what its features shall be. On the other hand, if the subject matter of geometry is pure or our own productions then there is nothing objective in the nature of the producing to change the results. The only thing different about the constructions is an intermediate linear construction which separates them. The &1SWer to this consideration is that homogeneity does not follow from purity or constructivity. For example, the
THE GEOMETRY OF A FORM OF INTUITION
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number series is not homogeneous with respect to the distribution of primes, though there is nothing between two segments of the series, other than an intervening counting or reciting. The answer to Kant is to say that the intervening line construction can ma..ke a difference to coincidence results, that the "unit' of construction may have to include the intervening construction, so that it is literally not the very same construction that is carried out twice. The purity, that is, of geometry seems compatible with positional inhomogeneity, allowing thus, for example, for geometries of variable curvature, and likewise precluding anticipative or a priori verification. Kant is still right, I claim, that verification in geometry pertains to constructions, even if he is wrong about its being a priori. Indeed if the subjectmatter of geometry is coincidence of constructions then of course the verification of geometrical statements 'pertains' to constructions, in the sense that what is being verified is a coincidence of constructions. However. not only isn't the verification of geometry a priori. it is not even pure. The trouble is the incompatibility of spatializing or spatia-temporizing construction. Pairs of constructions cannot be directly carried out together, whereas verification is always ultimately a local matter of having all the required information at once where and when one is. Thus, one needs physical markers or physical siguals to verify coincidence results, but once these are introduced one needs empirical hypotheses about how things move or how forces operate. The very sense of spatial statements includes incompatibility of actually carrying out pairs of constructions, and so the very sense of spatial statements forces a distinction between what is stated and how it can be verified. The purity of the subject-matter of geometry is not only compatible with the empirical nature of its verification, but once homogeneity is given up as an assured ptinciple, the subject matter demands this empirical nature of verification. Another way of stating this point is that Kant's spatial operationalism and the central insight behind it, that thought can do nothing more than guide behavior, demands a wedge between means of verification ana what is thereby verified. His view then that thought obtains real siguificance by legitimating or reguIalating activity is not derived from some verificationist principle, but is an autonomous alternative to verificationism.
University of Illinois.
'1 WILLIAM HARPER
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IIII
KANT ON SPACE, EMPIRICAL REALISM AND THE FOUNDATIONS OF GEOMETRY'
I.
SPACE: KANT'S ANSWER TO BERKELEY
1. Kant and Berkeley One of the first reviews (Garve-Feder, 1782) of the Critique of Pure Reason described Kant's system as a form of idealism of a piece with that of Berkeley. Kant (Letter to Garve, August 7, 1783) was not pleased with this comparison. In the Prolegomena (13) he explained that his system, far from agreeing with Berkeley, was the proper antidote to Berkeley's objectionable form of idealism. In an explicit response to the offending review (prolegomena Appendix) Kant claimed that when Berkeley made space a mere empirical representation he reduced all experience to sheer illusion. Kant continued to stress Berkeley's failure to do justice to the special role of space as SOUTce of a priori constraints on experience when he distinguished ·'his view from Berkeley's in the second edition of the Critique (B 69-72, B 274, Note on B xi of Preface). In spite of these protests, quite a number of subsequent writers have offered interpretations of transcendental idealism that would have Kant in basic agreement with Berkeley. Perhaps the most clearly stated example is to be found in Colin Turbayne's classic paper (1955), but any interpretation that construes the manifold in intuitions as sensations or appearances as subjective contents of experience will make Kant's position true to the spirit of Berkeley's point of view. I shall use Turbayne as an example; but, if the interpretation I propose is correct then the way Kant uses space to support his empirical realism makes his position quite different from Berkeley's or from any kind of phenomenalism or any empiricism based on subjective experiences. Turbayne claims that Kant's main argument against transcendental realism was anticipated by Berkeley's main argument against materialism. He breaks the argument down into six steps which I paraphrase roughly as follows: The transcendental realist supposes that external objects of perception have an existence by themselves independently of what we can perceive. (2) What we can be immediately aware of is only the contents of our own representations. (1)
257 Carl J. Posy (ed.), Kant's Philosophy of Mathematics. 257-291.
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(3) Therefore, it is impossible to understa"ld how we could arrive at knowledge of external
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objects and we are led to skepticism about therr existence. (4) We are alSO led to skeptical idealism - the doctrine that we can only know the contents of our own representations. (5) Skepticism about external objects can be avoided by giving up transcendental realism and adopting transcendental idealism - the doctrine that external objects are appearances and so are contents of representations. (6) This supports empirical realism - the doctPJle that we have immediate perception of external objects.
Something like this kind of argument against 't:Fal1scendental realism does seem to be an important part of Kant's Copernican revolution in philosophy.' Turbayne uses quotations from Berkeley and Kant to illustrate u'leir agreement at each step. The last two steps represent the official position which combines transcendental idealism with empirical realism.
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EMPIRICAL REALISM & THE FOUNDATIONS OF GEOMETRY 259
Space is one issue on which Kant and Berkeley clearly differ. Kant held that we have knowledge a priori about space while Berkeley held that all spatial concepts are merely empirical. According to Berkeley (Theory of Vision 153-154) even the three-dimensionality of space is something that must be inferred from experience by associating visual with tactual sensations. As we said above. Kant regarded Lhis rejection of a priori constraints on space as a fatal flaw in Berkeley's account of the difference between truth and error. According to Turbayne (p. 236), Berkeley based the distinction between truth and error on the coherence of our ideas with one another in experience and Kant is committed to the same kind of account. He suggests (pp. 243-244) that Kant's appeal to their differences over space was no more than an attempt to keep his readers from realizing that this basic position was essentially the same as that of u':!e infamous Berkeley.3
Fifth Step Kant: (Transcendental Idealism). External bodies are mere appearances, and are therefore nothing but a species of my ideas, the objects of which are something only through these ideas. Apart from them they are nothing (A 370, Cf. A 491, Prolegomena 13).
Berkeley: As to what is said of the absolute existence of lL."1.thinking things without any relation to t.qeir being perceived, that seems petfectly unintelligible. Their esse is percipi, nor is it possible they shouid have any existence, out of the minds or thinking things which perceive them. (Prin.3).
Sixth Step
Kant: (Empirical Realism). I leave things as we obtain them by the sense their reality (Proleg, 13). In order to arrive at the reality of outer objects, I have just as little need to resort to inference as I have in regard to the reality of the object of my inner sense ... For in both cases alike the objects are nothing but ideas, the immediate perception of which is at the same time a sufficient proof of their reality. (A 371) ... An empirical realist allows to matter, as appearance, a reality which does not permit of being inferred, but is immediately perceived. (A 37 N).
Berkeley: [am of the vulgar cast, simple enougIl to believe my senses and leave things as I find them (Hylas ill). I migh.t as well doubt of my own being, as of the being of those things I actually see and feel ... Those immediate objects of perception, which according to you, are only appearances of things, I take to be the real things themselves. ~ If by material substance is meant only sensible body, that which is seen and felt ... then I am more cert:aill. of matter's existence than you, or any other philosopher, pretends to be (Hylas III).
As these quotations show, Berkeley cerntinly did not describe his position as one which reduces all experience to iIlusion.2 He regarded his idealist account of bodies as the proper defense of common sense empirical realism against skepticism. This is exactly the virtue Kant claimed for his own transcendental idealism.
2. Sellars' objections to phenomenalism On Berkeley's version of the coherence account Macbeth's dagger is illusory because he cannot grasp it or cut with it - the sense data involved in his experience do not fit into the sort of coherent pattern with other sense data that constitutes seeing a real dagger. Unifonnities among our sense data let us coordinate sight with touch and make a host of specific correlations among our subjective experiences. \Vhen these uniformities break down we find that our judgments have been in error. When Macbeth sees the dagger apparition hover Ln the air, he has some grounds for judging that it is not a real daggerreal daggers don't appear to hover in the air without visible snpport. Upon attempting to grasp it he would have more evidence - real daggers resist when grasped. On this view such breakdowns among the unifonnities that constitute experience of real daggers are what make Macbeth's dagger count as illusory. In order to accurately reconstruct the common sense distinction between truth and error we must be able to account for cases where one person's experience is ground for judging correctly that another's judgment is in error, even if the other doesn't realize his errOr. When I am situated so as to see that you are standing in front of an empty facade, as you claim to be in front of a house, I can correctly judge that you are in error even if you don't think so. The uniformities that ground an adequate coherentist account must apply impersonally to the experiences of all of us. In addition to applying impersonally to different observers the unifonnities
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that account for the difference betwen truth and error must apply to possible as well as actual experieuces. Macbeth·s dagger is illusory even if neither he nor anyone alse ever actually tries to grasp it or cut with it. It is illusory because one would not be able to succeed if he were to make the attempt. Even if no one is situated to observe them a real house has other sides and
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insides. Were one to make the appropriate observations he would have the appropriate experiences. Any idealist who leaves everyday empirical things as they are must believe and have good reason to believe many counterfacmals of this sort. In his discussion of phenomenalism Wilfrid Sellars (1963. pp. 60-106) argues that such counterfacmals cannot be analysed into uniformities among acmal sense contents. He points out that in order to specify the appropriate antecedents for the counterfactuals in _question one needs to refer to external objects. One striking example is the need for antecedents such as looking from different perspectives. The counterfacmal arrangements of bodies in space that would re-position the observer with respect to h'1e object would, themselves, have to be formulated in terms of counterfacmal as well as acmal sense contents. Thus the very conditions that would be used to define these
possible sense contents would have to be based on other conditionals of the same sort. According to Sellars (1963, p. 80), a phenomenalist might reply by claiming that there are independent general laws about sense data that do not need to be formulated by reference to external bodies and which can be supported by induction based on actual sense data alone. Sellars' answer is that what the phenomenalist needs are generalizations which would apply impersonally, but the best that the phenomenalist can get are uniformities that are valid only for his own particular experience.4 For Kant the fundamental a priori constraint that space is three dimen-
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sional together with the a priori constraints on shape and perspective that can be established by geometrical constructions provide richer material for generating an empirical realism. I shall argue that the species of representation Kant uses to account for external bodies is the kind of objective perception exemplified by observations of those perceptible feamres presented by a three-dimensional object at a specific location and orientation with respect
to the observer. Kant's a priori constraints build in the assumption that such an object has another side even if only one side is being observed. They also require that the object has a determinate shape that is systematically related to an indefinitely large array of perspectives from which it could be observed.
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These spatial assumptions provide exactly what is needed to get around the objections we have considered. The antecedents of the required conterfacmal conditionals can be cashed out as specific alternative arrangements of the object and h':le observer's body in space. Grasping at the dagger is bringing one's hand into the appropriate location and orientation as one squeezes. Similarly, the antecedents relevant for observing other parts of the house are generated by specifications of locations and orientations for the body of an observer relative to the house. The various uniformities on shape and perspective that support the specific content of these counterfactuals are impersonal in just the way required. That
a quarter-shaped object will present a circular aspect to an observer who looks at it from a perspective orthogonal to and centered on its head's side and present an elliptical aspect to one who looks at it from an appropriately different angle is not something idiosyncratic to any particular observer. Such laws are part of what is to count as nonnal observation of shaped objects in
space.' To the extent that Berkeley's position is vulnerable to these objections to phenomenalism, while Kant's a priori constraints on space get around them, it is plausible to argue that Kant can be taken at his word when the claims that Berkeley reduced experience to illusion when he made space a merely empirical representation. 6
3. Refutation of idealism In addition to the objecrions we have considered Sellars (1963, pp. 83-84) also argues that a phenomenalist is committed to the external world of bodies in space and time when he refers to perceivers and their personal identities. This is one of Kant's own arguments. It is a major theme in the transcendental deducrion and the refutation of idealism. Several other writers, including notably Peter Strawson (1966), have argued that Kant is correct on this point because the path traced out by a person's body as he moves about over time through an enduring world of external bodies in space is all that provides for his ability to collect his various subjective episodes into an experience belonging to a single person. Margaret Wilson (1972, pp. 597-606) has suggested that the foregoing objecrion only shows that one must use external body concepts to describe the subjective contents of experience and does not show that judgments about external objects have to be known to be true. She argues that even if this undermines a position which attempts to reduce all experience to sensory
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EMPIRICAL REALISM & THE FOUNDATIONS OF GEOMETRY 263
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contents alone it need not be decisive against Descartes' more modest skeptical position. According to Wilson (ibid., p. 603) Descartes' only essential contentions are (1) Our most confident ordinary employment of physical object concepts is in a significant sense compatible with the non-existence of physical objects, and (2) Judgments which purport only to describe our experience, without claiming the actual existence of entities other than ourselves, are not similarly challengeable.
She uses the following example of Descartes' demon hypothesis in action to support the claim that these contentions are plausible: Consider, the Cartesian may say, the case of a man approaching an oasis across the desert. First he perceives only the tops of the palm trees. After a while he perceives the trunks. Although his perceptions of the trunks occur after his perceptions of the leafy tops, he will naturally take both to be perceptions of one set of stable objects, not of temporally successive sets of objects. As he gets nearer, he sees a bird in one of the trees. He sees the bird stretch open its beak, then close it, then fly off. Then he hears a shrill note. While he perceives the bird-flight before the bird-cry, he takes it to have occurred afterwards. In other words, he implicitly makes all the usual distinctions between subjective and objective time order, in complete conformity with the examples of the Second Analogy. Now let us suppose (1) that the oasis was a mirage; or (2) that the man was not awake; Or (3) that he was in the clu~ches of a deceitful demon or super-scientist, who was in some manner providing him with a fantastic series of perceptual experiences. Certainly, the Cartesian will continue, there is a sense in which this man not deceived about the character of his own perceptual experiences. Yet he certainly was deceived in taking them directly to represent an outer reality. Now, how can we ever be sure that our 'outer experience' is not deceptive in precisely this manner, etc .. .. ?
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The science fiction version of the demon hypothesis is especially compelling today. How do I know that I'm not just a brain in a vat. Perhaps my present experiences, and indeed my whole life's experiences, are nothing but responses by my brain to artificial inputs provided by ingenious super scientists. This kind of hypothesis seems to obviously a coherent possibility in principle, even if it cannot be achieved yet by today's scientists, that it has revived the demon hypothesis as an epistemological puzzle of concern to philosophers. 7 The two contentions Wilson acribes to the Cartesian correspond to the first two steps in the paralogism argument that made Kant look like Berkeley. 8 They have the effect that our judgments about the snbjective contents of our experiences are immediate, but that the existence or non-existence of external objects is independent of our judgments about them. The demon hypothesis challenges our ability to arrive at knowledge of external objects (step 3) and invites the skeptical conclusion that all my knowledge is limited to the subjective contents of my own experience.
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On Berkeley's version of it, the idealist move in step 5 gives up the Cartesian contention (I), but keeps contention (2). According to Berkeley external bodies are accounted for by visual and tactual ideas that contain nothing beyond what is immediately given to the senses. On his view the esse of these ideas is percipi so that my judgments about what I perceive immediately are incortigible9 Kant's skeptical idealist (step 4 in the paralogism argument) conld say that he gives up (1) and that he interprets each perception claim about external bodies as asserting no more than the subjective content of that particular perception. He could then say, as Berkeley does, that in his view, our perceptions of external objects are immediate; but, such a view would certainly not provide for an empirical realism. 10 The Berkeley that Turbayne shows us would interpret my judgments about external objects as asserting appropriate uniformities among the subjective conients of my experience." On the assumption that (contrary to what I have argued above) these uniformities can be made available within Berkeley's framework, this more realistic kind of subjective idealism does defuse the skeptical argument. On this idealistic assumption the demon hypothesis is incoherent because the truth of my claim that external bodies exist comes down to the sarne thing as having my subjective experiences satisfy the appropriate uniformities. On the interpretation I shall propose. there is an important difference between the ways Kant and Berkeley give up contention (I). According to transcendental idealism external bodies are accounted for by appearances and appearances can be immediately perceived by us. The difference is that the appearance I perceive now is correctly construed as an object the existence of which is independent of my perception of it. On this interpretation, appearances are objective rather than merely subjective contents of perception and my judgnients about them are not incortigible. Even though appearances are empirically real so that they are independent of anyone's actual perception of them they are not transcendentally real because they are not independent of what conld be perceived by observers like us. Kant still has available an idealistic answer to the demon hypothesis. On the interpretation I shall defend I can assume that my judgment abont an appearance I perceive is false only by assuming that it fails to cohere with a host of other claims about outer appearances which I assume to be true. There is, on this view, no way to coherently assume that all my judgments abont outer objects are false. By giving this objective account of appearances Kant has broken the connection between immediate perception and incortigibility. He holds both that my perception of the appearance presented to my senses now is immediate
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experiences are construed as supporting nothing beyond incorrigible claims about subjective contents then they cannot support any such objective time order. On the assumption that all my outer experience is hallucination I have given up any grounds on which I could know that the temporal order my experiences seem to have is the one they actually do have. The demon hypothesis is incoherent because on it there is no way to prevent my experience from collapsing into a solipsism of the present moment. 14
and that it includes an objective judgment that can in principle be mistaken. On this view the fact that my present judgment is corrigible does not mean that it is in any way doubtful. My perceptions of the outer appearances presented to me are not mediated by any more direct perceptions of tIle subjective contents of my experience. Even though they are corrigible they are as
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immediate and certain as any perceptions I can have. Kant's transcendental idealism gives up contention (2), as well as contention (I). On his view my judgments about the subjective contents of my own experience are no more imIllediate than my judgments about the outer appearances I am presented with. According to the refutation of idealism (B275-279), my having knowledge of the determinate temporal sequence of my subjective experiences depends upon my having determinate knowledge about outer things. If this is correct, then the need, pointed out by Sellars and Strawson, to appeal to my body's path through an objective world in order to know how my subjective experiences fit together in time requires knowing the truth of some judgments about outer things.I2 Though Kant and the skeptical idealist agree in treating my judgments about external objects and my judgments about my subjective experience as equally immediate they do so in opposite ways. Where the skeptical idealist would treat claims about external objects as incorrigible claims about subjective experience, Kant would treat my judgments about my subjective experiences as no less corrigible than my judgments about external objects. Recently Hilary Putnam (1981, pp. 1-20) has argued that the demon hypothesis '1 am a brain in a vat' is self refuting because if it were true I would be unable to use the word 'vat' to refer to actual vats in the world. Putnam (p. 62) points out affinities between his views, Kant's position on sensations, and Wittgenstein's private language argument. Other writers including Sellars (1963, Chapters 3 and 5; 1968, Chapters I and II) and Jonathan Bennett (1966, pp. 202-209) have also given interesting arguments in support of Kant's position that bring out affinities with Wittgenstein. 13 1 think that Kant's argument can be profitably interpreted along the lines these writers suggest. For Kant, just as for Putnam, the demon hypothesis about my own case gives up what is needed in order for me to make some objective reference it
requires me to make. Unlike Putnam, Kant focuses on the objective reference required to have knowledge of the temporal order of my own experiences. If my past tense judgments are to connect together in an appropriate way then I must be able to use 'now' to demonstratively refer to a location in an objec-
tive time order that defines a past for these judgments to refer to. If my
II.
KANT'S EMPIRICAL REALISM
1. Appearance: The undetermined object of an empirical intuition
The following taxinomy can help to explicate the species of representation 15 that count as intuitions:
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The genus is representation [Vorstellung] in general 'repreasentatio). Under it stands the representation with consciousness (perceptio). A perception which relates only to the subject as a modification afits state is sensation [Empfindung] (sensario),. an objective perception is cognition [Erkenntnis] (cognitio). This is either intuition [Anschauung] or concept [Begriff] (intuitus vel conceptus). The fonner relates immediately to the object and is single, the latter relates to it mediately by means of a feature [Merkmal] which several things may have in common. The concept is either an empirical or a pure concept; and the pure concept, so far as it has its origin only in the understanding (not in the pure image of sensibility), is called a notion (Notio). A concept formed from notions and transcending the bounds of experience [Erfahrung1 is the idea [Idee] or concept of reason [Vemunftbegriff]. (A 320/B 377).
This passage distinguishes intuitions from sensations on the one hand and
concepts on the other. I shall deal with the distinction between intuitions and concepts before dealing with the distinction between intuitions and sensations. An intuition is single and relates to its object immediately while a concept
relates to its object mediately by means of a feature which several things can have in common. An intuition is single in that it is a singular representation -
one that can have only one particular object - while a concept can be satisfied by many distinct instances. In this respect, the distinction corresponds roughly to that between an individual referring expression and a predicate expression in symbolic logic. Hintikka ,has argued (1969) that this sort of logical distinction between particular ideas and general concepts captures the essence of Kant's use of inmition. Other writers (Parsons, 1964; Sellars, 1968; Howell, 1973) have argued that, on Kant's view, a demonstrative element is essential to any intuition. The emphasis, in this passage, on the immediacy with which an inmition is in relation to its object may support
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t.'1ese writers. Our problem is to get clear about the sort of empirical intuitions t.~at Kant uses to account for external bodies. For these intuitions at least, a demonstrative reference to a specified actual instance is essential. Consider the empirical intuition I have as I observe three coins arranged on the desk before me. Presumably, Lhe object of this intuition is a complex of several individual things. It may be, as Kant sometimes suggests, that any such inmition of a complex is a complex of simpler intuitions; but, even if this were so, a complex intuition would still be an intuition. My intuition can still be singular in Lhat it unambiguously designates this one instauce of the arrangement of coins. The importaut singularity of intuitions, at least of empirical intuitions of the sort that concern us here, is to refer demonstratively to a single instauce. It does not matter whether the specified instauce turns out to be simple or complex. In the example under consideration, the object of my intuition is whatever is actually present now at the location I specify when I refer to the coins before me on my desk. Kant distinguishes intuitions from sensations in the following manner: an intuition is a cognition or objective perception while a sensation only relates to the subject as a modification of its state. Since Kant would regard Berkeley's ideas as mere sensations, this distinction between intuitions and sensations is vital to the difference between his transcendental idealism and Berkeley's subjective idealism. Adequate treatment of this difference will require explicating the role of sensation in empirical intuitions. This explication will benefit from a consideration of additional passages in which Kant distinguishes between empirical and pure intuitions. Among these passages the following paragraph deserves to be quoted in full because this will help set the stage for the explication to follow: Our knowledge [Erkenntnis] springs from two fundamental sources of the mind; the first is the receiving of representations (the receptivity for impressions), the second is the power to know Ierkennen] an object through these representations (spontaneity for concepts); through the first an object is given to us, through the second it is thought in relation to that representation (which is a mere detennination of the mind). Intuition and concepts, therefore, constitute the elements of all O!lr knowledge [Erkenntnis], so that neither concepts without an intuition in some way corresponding to them, nor intuition without concepts, can yield knowledge [Erkenntnis]. Both may be either pure or empirical. They are empirical when they contain sensation (which presupposes the actual presence of the object), and when there is no admixture of sensation with the representation they are pure. Sensation may be called the material of sensible knowledge. Pure intuition, therefore, contains only the fonn under which something is intuited, and pure concept only the form of the thought of an object in general. Only pure intuitions or concepts are possible a priori, empirical ones must be a posteriori (A 50/B 74-A 51/B 75).
According to this passage, an intuition is empirical when it contains sensa-
EMPIRICAL REALISM & THE FOUNDATIONS OF GEOMETRY 267
tion; moreover, sensation presupposes the actual presence of the object, and it is through receptivity for impressions that objects are given to us. The passage also implies that all my information which is not contributed by my own mental apparatus must come from the input of sensations to that apparatus. That there is a tight correspondence between such characteristics as the shape and hardness of a physical object, e.g. a rubber ball, and the kind of experience I have when I look at and handle it, is one of the faruiliar facts of iife. When I look at the ball, what I see depends on the ball, the perceptual circumstances, and my psychoiogical circumstances; but, given the specification of these contingencies it is independent of my decision. 16 I can decide to look or not to look but, if I look, what I see is not all up to me. According to Kant, sensations are essential to this independence. In this system they link up my mental machinery with the world: The effect [Wirkung] of an object on the faculty of representation, so far as we are affected by it, is sensation. That intuition which is in relation to the object through sensation is called empirical. (A 19-20/B 34)
Notice that Kant does not say that sensation is perception of what it corresponds to. On the contrary, in all three passages quoted above, he carefully restricts sensation to a modification of the state of the subject only, to that representation which is a mere determination of the mind, and to the effect of an object on the facnlty of represenation so far as we are affected by it. The sort of perception which is in relation to an object, through sensation, is empirical intuition - not sensation itself. How does an empirical intuition contain sensation? Consider the following proposal: Just as an instauce of a sign desigu can function as a token for a sentence in so far as it is subjected to rules that govern correct usage for that sentence, so also may an array of sensations function as the token for an empirical intuition.17 The connection between any token and the represenation it is used to token is provided by the rules that govern correct tokening of the type of representation in question. An empirical intuition is just a sensation episode that is subjected to rules appropriate for tokening that specific type of intuition. According to this proposal, seeing that there are coins on my desk is distinguished from merely having sensations of a certain kind in that seeing is subject to rules that govern judgments about objects of experience while mere sensations are not. This makes the distinction between an empirical intuition and mere sensation analogous to the distinction between asserting that there
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are coins over there and merely mouthing the words. On the other hand. also according to this proposal, seeing that there are coins over there is distinguished from merely asserting the corresponding judgment in that seeing required being in perceptnal circumstances appropriate to produce the corresponding sensations while asserting does not require any such immediate relation to the object of the judgment. The last passage quoted, where sensations and intuitions were characterized' continues with the following sentence, which characterizes appearance as the undetennined object of an empirical intuition:
other things, that coins be rigid enough to resist when touched and that they have appropriate boundaries on the sides not being observed. The additional content provided by the empirical concept of a coin connects the appearance presented to me now wit"!:l other appearances that are not now presented to me but would be presented to an appropriately located observer. Were! to pick up and exaruine one of the coins I would be presented with an appearance that included tactual as well as visual inforarntion. All of the directly presented features, the shape, texture and resistance presented to my fingers as well as the shape presented to my sight are located in one space relative to the location and orientation of my body.19 As I construe II1em here, Kant's appearances are just those objective properties of actual things in space that follow geometrically from those perceptible features that would be presented directly to the senses of an appropriately situated human observer. In many passages Kant tells us that imagination is the process by which sensations are worked up into empirical intuitions. According to this picture we can think of an outer appearance as that set of sensible features which I intuit in an object simply through the taking up of sensations into the imagination according to the general rules for having an outer intuition at all. The -appearance is the content of a minimally conceptualized intuition. The object of such an intuition is characterized as this something - qua having the perceptible features generated by the imagination under the guidance of only my present sensations and the pure concept of a spatial object. Even when the ascription of content to the object of my outer intuition is limited in this way, the general spatial rules require some definite connections between the appearance actually presented to me and further appearances that would be presented to appropriately located observers. For example, (as we remarked above, Note 5), the shape presented to me is systematically related to what would appear to other perspectives. This comntitment to further appearances makes the ascription of even these most directly perceptible features both objective and corrigible. I take this to be Kant's main point when he insists that empirical intuitions are objective perceptions and not mere sensations.
The undetennind object of an empirical intuition is called appearance (A 20/B 34)
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In so far as it is an object of an outer intuition an appearance must be subject to the general rules that characterize the pure concept of an object in space. These rules are generated by the role of space as the pure form of all outer intuition and require that the object have a location with respect to the observer in three-dimensional space as well as satisfy all the constraints on shape and perspective that can be established by geometrical constructions. The appearances Kant uses to account for external bodies are objects of outer intuitions; therefore, whatever may be undetermined about them, they must at least satisfy these general rules. I propose that the object of an outer empirical intuition is undetermined in so far as it is subjected to none but these general rules for objects in space. Consider again the empirical intuition I have as I observe the coins on my desk. I leave the object undetermined when I limit my judgment about it to just those perceptible features actually presented to me now, together with whatever these features imply according to the general rules governing objects in space. The appearance is simply something - qua presenting the aspect of a specific triangular array of three dime-shaped objects viewed from my relative location and perspective. The basic idea here is that the features which generate the content of an appearance are just those perceptible features that are actnally exposed to the appropriate senses of the observer. 18 The shapes on occluded sides of the coins are not part of the content of this appearance. The very sarne appearance could have been presented by rods embedded in the desk with exposed ends shaped like tops and edges of dimes; it could also have been presented by an appropriately focused hologram. An hallucination would not count as observing the Sarne appearance even though I might tuistake one for such an observation. When I judge that what I see are coins, I subject the object of my intuition to rules that require more of it than just this appearance. I require, among
2. Transcendental idealism and empirical truth Kant's most developed exposition of the way his transcendental idealism supports an empirical realism is to be found in the long paragraph (A 190-191; B235-236) which opened his first edition version of the second analogy and was retained unchanged as the third paragraph in his second edition version. I
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shall attempt to show how my explication of Kant's basic conception of appearance as the nndetermined object of an empitical intuition illuminates what I take to be the two centtal ideas Kant inttoduces in this celebrated passage.20 One of these is a transcendental sense of 'appearance' according to which even such a complex solid object as a house counts as an appearance. The other is an account of empitical truth that is objective and yet avoids the demon argument which plagues the transcendental realist conception of truth as correspondence with things as they are in themselves. Kant tell us that a house is not a thing in itself, but an appearance. He explicates this by glossing 'appearance' as "a representation the transcendental object of which is unknown". He then asks what we are to nnderstand by the connection of the manifold in the appearance itself, when an appearance is neverthelsss not anything in itself. Now, as soon as I unfold the transcendental meaning of my concepts of an object, I realize that the house is not a thing in itself but only an appearance, that is, a representation, the trimscendental object of which is unknown; therefore, what am I to understand by the question: how the manifold may be connected in the appearance itself (which is yet nothing in itself)?
When I judge that what I see before me is a house, I ascribe more to the object of my experience than just those directly perceptible features that are now afforded to my senses. These additional ascriptions go far beyond what the general concept of an object in space requires in order that the shape from other perspectives cohere geometrically with what I observe. Accordingly the house before me is not as undetermined an object of empitical intuition as the perspective-bound appearances I have been explicating. Lewis Beck (1978, pp. 143, 146) is surely correct that Kant uses a thicker notion of appearance when he applies it to such complex objects as houses. 2l He calls this Kant's transcendental sense of 'appearance' and identifies the more perspective relative notion I have been explicating with what he takes to be Kant's contrasting empirical sense of 'appearance'.22 Kant's rhetorical question, at the end of this passage, can be understood as asking how the manifold of a transcendental appearance can be independent even though it contains nothing beyong contents of representations. Kant's somewhat enigmatic answer is given in the next passage, which immediately follows his question in the text. That which lies in the successive apprehension is here viewed as representation, while the appearance which is given to me, notwithstanding that it is nothing but the swn of these representations, is viewed as their object; and my concept, which I derive from the representations of apprehension,. has to agree with it.
EMPIRICAL REALISM & THE FOUNDATIONS OF GEOMETRY 271
That which lies in the successive apprehension is presumably a manifold of empitical appearances. These appearances have a two-sided character. On the one hand each is the content of an empirical intnition and therefore can be viewed as a representation. On the other hand, as actual features of objects in space, they are connected with one another in a manner that is independent of anyone's apprehension of them. The empirical intnitions in my apprehension make demonstrative reference to a spatio-temporal vicinity. The independence of the object of my experience is provided by all the additional empitical appearances to be found in that viciuity. In this way a transcendental appearance tlIat contains nothing beyond contents of representations can, nevertheless, be viewed as the independent object that my concept has to agree with. My concept is derived from the representations of apprehension, in that the perceptible features actnally presented to me lead me to judge that what is before me is a house, rather than (say) a ship, tree, or an empty stage prop. If my judgment is correct, this concept has to agree with whatever turns out to be tl,e actual object of my experience. This demonstrative reference, rigidly denoting whatever is at a spatio-tem. pora! v.icinity, is the most important contribution of Kant' a priori requirement that the object of an outer empirical intuition have a determine location relative to the body of the observer in three-dimensional space. The specific geometrical constraints on the relation of three-dimensional shape to perspective also play an important role. They provide a framework which allows the identification of the house as an independent empitical object which underlies all the appearances in its manifold. This object is whatever affords the mereological sum of all the three-dimensional shaped surfaces revealed in these various empitical appearances. It is the empirical substance of which the various perceptible feamres revealed in these appearances are determinations. ..On this account when I look at a quarter from a perspective 45° from perpendicular to its head side, I directly see its non-occluded snrface as sometiling shaped like an appropriate part of a three-dimensional disk located and oriented in the way specified. There is none of that difficult business of seeing it as elliptical but judging it to be ronnd which plagued G. E. Moore. I think that allowing for direct perception of oriented shaped surfaces in three-dimensional space is fundamental to any Kantian account of how observers from different perspectives can see over-lapping parts of the sarne empitical substance. The following acconnt of empitical truth completes Kant's explication of how to construe appearance as the fonnal-being referred to by the representations in my apprehension:
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One soon sees that though truth is agreement of the cognition [Erkenntnis] wiL1. the object, only the fannal conditions of empirical truth can be in question here, and appearance in contrast with the representations of apprehension can be represented as an object distinct from them only if it [the appearance] stands under a rule, which distinguishes it [the apprehension of this appearance] from every other apprehension and makes necessary some one particular kind if connection of
Ill. PHAENOMENAo APPEARANCES THOUGHT ACCORDING TO THE UNITY OF THE CATEGORIES
the manifold. That in the appearance which contains the condition of this necessary rule of apprehension is the object (A 191/B 236).23
So far as we have explicated it, Kant's empirical realism supports the common sense realm of candlesticks, ships and houses against skeptical reductions to subjective contents of experience; but, our explication has been lintited to observables in a sense close to that advocated by va." Fraassen (1980) in his anti-realist constructive empiricism. It would be disappointing for some of those who see Kant as providing a foundation for scientific methodology to find that his empirical realism does not support existence claims about the non-observables posrulated by modem science. In a passage at (A 249) Kant tells IlS that:
Consider my observation of my own house as I stand before it. Let A be the proposition that what is before me now is that particular house. If it is empirically true that A obtains here-now, then the object of my experience must agree with my cognition - i. e., with my judgment that A is the case here now - and must also be representable as something independent of the representations in my apprehension of it. Therefore, whatever is before me over there must satisfy the condition of a rule that distinguishes it from any possible object of experience that fails to be an instance of A. This rule distinguishes my apprehension, qua an apprehension of an instance of A, from any ot.;'er
apprehension - i. e., from any apprehension of anything that fails to lle an instance of A. In this example the representations of my apprehension are my perceptions of the perceptible features actually presented to me. These would include the shape relative position and orientation of the facing surfaces, etc. The object
of my experience is whatever is present at the appropriate, spatio-temporal vicinity of the location to which I now refer demonstratively. There are many
!
more perceptible features there to be observed than the ones now presented to me. It is this demonstrative reference to an inexhaustibly rich source of addi-
tional perceptible features that gives the object of my experience its independence from the representations in my apprehension of it.
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I believe that this account of empirical truth is the heart of Ka."t's Copernican revolution in epistemology. In place of the transcendent notion of truth as correspondence with the way things really are in themselves, he gives us empirical truth as correspondence with what can count for us as the actual objects of our experience. This transcendental idealism avoids the Cartesian argument for skepticism at least as well as Berkeley's subjective idealism. The demon hypothesis cannot be empirically true because it assumes away the demonstrative reference to an independent object of experience required to provide the empirical content that could make it true. The
advantage over Berkeley is that it provides for the independence and objectivity required by our common sense empirical distinctions between truth and error.
I. The principle of extensive magnitudes
Appearances. so fur as they are t.1tought according to the unity of the categories. are called
Phaenomeoa
The various categorical principles Kant argues for impose additional constraints on the basic idea of apprearance as the undetermined object of an empirical intuition. I think that these constraints transform the account of empirical truth by adding comntitments that go beyond observables. I shall illustrate this point by considering some consequences of the axioms of intuition.
According to the Axioms of Intuition all appearances are extensive magnitudes. When Kant tells us that these magnitudes are determinate he is requiring that, for example, at any instant in time ratios of lengths along any specified dimensions of an object in space determine specific real numbers. It
is important to note t.lJ.at Kant takes this commitment to determinate extensive magnitudes as a constitutive condition on appearances. When explicating outer-appearance as the undetermined object of an empirical intuition I cbumed that the content of such apprearances is lintited to features to objects in space that are directly accessible to the senses of observers like us together with whatever these features imply according to the rules constituting the general concept of an object in space. We have seen that the qualitative geometrical constraints on perspective require that judgments about shapes directly presented to an observer at one perspective carry systematic comntitments to further aspects that would be presented to observers at other appropriately oriented perspectives. Now we see that the general concept of an object in space also carries commitment to determinate extensive magnitudes. Even when I limit my judgment about the object of my empirical intuition to
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the outer appearance presented to me ! must include in the content of my
judgment that, at any given instance, e. g. now, and relative to any appropriate specification of a standard, the length along each specifiable dUnension of the object has a detenninate value. Consider a spatial example where lengths are being compared. Let A 1••• An be some finite sequence of propositions such that each Ai asserts that the distance (relative to a specific meter stick) between the centers of mass of two quarters on my desk now falls in the i th of n adjoining tiny length segments. We are to make these segments small enough so that each Ai is below the threshold of human sensory detection, but large enough so that the disjunction of all the A;s is something we can observe to be true. This holds if we limit our observations to whatever comparisons of length we can establish with unaided sight and touch as we lay the meIer stick across the coins, and il continues to hold even if we allow what we observe to be enhanced by the best measuring instruments science can provide. The principle of extensive
magnitude makes commitments that go beyond the resolving powers of human observation even if these powers are extended by instruments. An appearance includes the specification that each spatial dUnension in it be a detenninate extensive magnitude. But, even given the specification of an appropriate standard and time, does it also include specification of what the exact value of each of these magnitudes is? On the account I have been proposing the answer to this question is no, because these exact values are
not implied by observable features even under the most lavish construal and application of mathematical rules constituting the general concept of an object in space. This has the effect that the disjunction A 1 V ..• v An will be empirically true even though none of its disjuncts is. Similar examples will show that an existential statement can be empirically true even when each of
its instances is empirically indeterminate. Indeed, it will be empirically true that each magnitude has some determinate value, even though for each magnitude it will be empirically indetenninate exactly what this value is. Another option would be to include the specification of the exact value of each magnitude as part of the empirical content itself. This would remove any empirical truth value gaps generated by the commitment to determinate extensive magnitudes, but it would lead to a problem pointed out by Charles Parsons (1964). On this option Kant is faced with a dilemma. Either he would have to claim that humans have the ability to, in principle, make infinitely fine discriminations of extensive magnitudes! or he would have to claim that what counts as empirical content is not determined by the discriminations that humans, even in principle, could make. Neither hom of this dilemma is very
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attractive. The first hom would seem to commit him to someLhing patently false. Certainly our best instruments now fall far short of the precision required, and, though we can expect improvements that allow c1oserapproximations. nothing in the way such improvements have been made in the past
makes it plausible to suggest that such approximations could even in principle culminate in exact values. The second horn of the dilemma, on the other hand, would seem to fiy in the face of the whole idea of Kant's Copemican revolution in epistemology. If empirical content is inaccessible to OUf senses, even augmented by the best instruments we could in principle, devise, then
how could we know empirical truth from error? On the option I propose this dilemma is avoided in a way that seems in keeping with Kant's Copernican revolution and with his specific account of the a priori as something we impose on nature. The principle of extensive magnitudes shows that the basic account of empirical truth carries commitment to a more detenninistic ideal in which the value of each magnitude is exactly seuled. Any coherent way of filling out the specifications required by this ideal that is left open by what is settled by the empirical content will act as an admissible valuation in a supervaluation semantics appropriate to the account of empirical truth. 24 Whatever holds according to every way of filling out the ideal by arbitrarily assigning these values in some coherent way will count as empirically true. Therefore, the principle of extensive magnitudes contributes considerable strength to the account of empirical truth, even if we allow that the exact values of these magnitudes are empirically indeterminate. This way that empirical truth carries commitment to a more determinate mathematical ideal turns it from a fairly restrictive observationalism to a possible foundation for scientific realism, without violating the spirit of Kant's Copernican revolution in Epistemology.
2. The principle of the first analogy Though I shall not argue the point here, I think that Kant's arguments for the principles of the analogies can best be understood as an attempt to show that extension of the empirical content that can be appealed to in the account of empirical truth beyond what is presented to the observer here-now to additional observables at other times and places reqnires commitment to enduring substances and causal laws. If, as seems to be the case, the ouly candidate for the enduring substances that underly many observable changes are nonobservables, then the first analogy, as well as the axioms of intuition, will carry commitment to the empirical reality of some non-observable entities.
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Kant tells us (B 233) that the principle of the First Analogy can be expressed by the requirement thai all change is alteration - a succession of opposite determinations of a substance which abides. This principle requires that any change which we might be inclined to describe as the destruction of an empirical thing must be a succession of opposite states of some underlying subslance which persists through the change. Consider one of those familiar white styrofoam coffee cups. Now destroy it by smashing it to pieces. If both the before and the after states have to be determinations of the same substance then that substance cannol be (he cup which was destroyed. What is available to both persist through and be releVal1! enough to ground this change? One obvious candidate is the mereologicai SU01 of all the little styrofoam particles. If the after state is these particles all jumbled about in disarray while the before slate was these same particles assembled togeLlter into the cup, the change can be a proper alteration. Now burn the little pieces of styrofoan•. What is available to count as a substance which underlies this change? Presumably, some postulated collection of non-observable entities - something like molecules or atoms. 25 Thus, the principle of the First Analogy, together with such familiar happenings as destructions by burning which break something up into parts smaller than we can observe, seems to carry commitment to just the sort of non-observable theoretical entities dear to the hefu-t of a scientific realist.
3. Indeterminacies We have seen how the principle of extensive magnitudes makes commitments
that generate empirical truth value gaps. The same would hold for commitments generated by the First Analogy. Even if some version of the kineticmolecular theory of gasses turned out to be empirically true of some specified volume of gas there would not be any specific assignment -of positions and momenta to the individual particles (at any given time) that would be singled out as the unique empirically true one. Only the existential proposition that there was some such distribution of momenta would be empirically true. There would be a very large range of possible assignments of these maguitudes that would be equally compatible with the apprearances - roughly all ones which afford average kinetic energy values within the observable tolerances. This possibility of truth value gaps is a feature wbich my account of empirical truth shares with Carl Posy's (1983, 1984) inmitionistic rendering of Kant's transcendental idealism. I think any account of Kant's position which
EMPIR!CAL REALISM & THE FOUNDATIONS OF GEOMETRY 277
takes seriously his relativization of empirical trut.h to possible objects of experience will have to allow for such indetenninacies. Posy's intuitionisti-
cally motivated aCCOU!1t is one way to do this. My account of empirical truth with its supervaluation way of dealing with indetenninacies is another. Posy
(1983) has done an admirable job showing how his proposal can illuminate Kant's difficult discussion in the First Antinomy. I think the supervaluation approach can offer a comparably illuminating analysis of this difficult passage, but the details will have to wait for another occasion. IV.
GEOMETRY
1. Kant's commitment to a priori constraints on space
We have noted (Section I) that, according to Kant, Berkeley reduced all experience to sheer illusion when he made space a merely empirical representa-
tion. This suggests that Kant's commitment to the claim h'tat geometry provides knowledge a priori of constraints on objects of outer sense is deeper than his desire 10 provide a philosophy of mathematics. It suggests that he thought these a priori spatial constraints are what prevent his apprearances from collapsing into. merely subjective contents of experience - that they.are what separate his empirical realism from the objectionable form of idealism he attributed to Berkeley. My account of Kant's empirical realism appeals to the constraint that any object of an outer intuition must bave a determinate location and orientation relative to the body of the observer, and to the numerous specific constraints
on shape and perspective that can be revealed in geometrical constructions. These spatial constraints provide a framework within which appearances can be construed as objective features of things our senses can carry immediate
information about. If this account is correct then Kant's defence of empirical realism is based on these constraints on space. So, Kant may have been justified if he thought that his defence of empirical realism would be threatened unless he could appeal to such knowledge a priori about space.26 The sort of constraints on shape and perspective that my account of Kant's empirical realism needs can be illustrated by Shimon Ullman's (1979) various structure from motion theorems. Ullman's basic theorem concerns
orthographic projections. 27 Given three distinct orthographic projections of four non-coplaner points in a rigid configuration the structure and motion compatible with these views is uniquely detennined (up to a reflection about the image plane).
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Ullman (pp. 134-6) discusses a neat experiment in which visual information processing suppor.able by this theorem can be observed. Moving points are projected onto a screen with motions compatible with ortbographic projections of points on the surfaces of two rotating transparent coaxial cylinders. As you look you cannot help but see them as points on the rotating rigid three dirnensional cylinders. This suggests that we visualize as though we operated with a wired-in program which first looks for some possible rigid body in relative motion interpretation of the sensory input. We can see UIlman's
theorem as providing constraints on what can count as a rigid body in relative motion interpretation. mlman's shape from motion theorem tells us that a rigid configuration that afforded this ortbographic projection
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straints that apply to anything that can be an outer object of experience for us. On this view geometrical constructions provide us with a way of making the pure form of our outer intuition transparent to ourselves. 2. Kant's account of geometrical constructions Perhaps the most salient example of geometrical construction in Kant's writing (A 716-17; B 744--5) is the one used in Euclid's proof of Proposition 32 (in the Elements) - that the sum of the interior angles of a plane triangle equals a straight angle (180°). Euclid's proof of this proposition appeals to proposition 29 about various equal angles made when a straight line falls on two parallel lines. The following diagram is a construction which shows that all the marked angles must be equal.
to one perspective, and this ortbographic projection
to a perspective corresponding to a 45° rotation to the right could not afford this ortbographic projection
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I think that, according to Kant, anyone who properly understands this diagram cannot help but be compelled to see that all the marked angles must be equal - and that this would hold for any straight line falling on two parallei lines. The heart of Euclid's proof of proposition 32 is the following construction.
D
to a perspective corresponding to the opposite 45° rotation. I shall call this my salient illustration. According to Kant such constraints on shape and perspective are built into the structure of space which is the pure form of outer intuition. He holds that geometrical constructions offer us a priori knowledge of these structural con-
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From proposition 29, or by inspection of the present diagram, we see that angle ABC equals angle DCE and angle BAC equals angle ACD. From the way we constructed auxiliary lines CE and CD we can now see immediately that ACB plus ACD plus DCE equals BCE, since these three angles together just are the straight angle BCE. I think this proof shows the iotuitive force that geometrical constructions provide. It is very hard to reason through this diagram without feeliog compelled to accept Euclid's general proposition that the sum of the ioterior angles of a plane triangle equals a straight angle (180'). Kant attempts to justify this compUlsion by his account of geometrical constructions. The following passage gives Kant's explanation of the role of the triangle diagram as a constructive definition of the concept of a planetriangle.
schema of the triangle can exist nowhere but in thought. It is a rule of synthesis of the imagination, in respect to pure figures in space_ (A 141/B 180)
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To construct a concept means to exhibit a priori the intuition which corresponds to the concept_ For the construction of a concept we therefore need a non-empirical intuition. The latter must, as intuition, be a single ebject, and yet none the less, as the construction of a concept (a universal representation), it must in its representation express universal validity for all possible intuitions which fall under the same concept_ Thus I construct a triangle by representing the object which corresponds to this concept either by imagination alone, in pure intuition, or in accordance therewith also on paper, in empirical intuition - in both cases completely a priori, without having porrowed the pattern from any experience_ The single figure which we draw is empirical, and yet it serves to express the concept, without impairing its universality.
The single empirical figure I draw functions as the pure intuition which underwrites a real definition of the geometrical concept of a plane triangle. As a real definition it displays sure marks by which to identify any figure that is to count as a plane triangle and it also provides an actual instance which shows that this concept is not empty. (parsons, '1969; Beck, 1956)28 Kant goes on to tell us more about how it is that this siogle empirical figure can serve to express a pure geometrical concept without impairing the generality of that concept.
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For in this empirical intuition we consider only the act whereby we construct the concept, and abstract from the many determinations (for instance, the magnitude of the sides and of the angles), which are quite indifferent, as not altering the concept 'triangle' _(A 714/B 742)
These remarks can be usefully amplified by the followiog passage from the schematism. No image could ever by adequate to the concept of a triangle in generaL It would never attain that universality of the concept which renders it valid of all triangles, whether right-angled, obtuse-angled, or acute-angled; it would always be limited to a part only of this sphere_ The
281
The diagram can express the pure iotuition of the geometrical concept of a plane triangle in so far as my reasoning about it appeals only to the schema of this concept. The schema is a rule for the synthesis of imagination reqnired to construct any ostensive representation of a plane triangle. The following passage contrasts what Kant calls the ostensive character of this sort of geometrical construction with the symbolic constructions to be found in algebra. thus in algebra by means of symbolic construction. just as in geometry by means of an ostensive construction (the geometrical construction of the objects themselves), we succeed in arriving at results which discursive knowledge could never have reached by means of mere concepts. (A 717/B 745)
It also illustrates Kant's commitment to the claim that Euclid's proposition 32 does not follow analytically from the mere concept of a plane triangle. The result depends essentially on the additional content provided by the construction. The schema for this pure concept is embedded in and shows us constraints on our framework for ostensively recogniziog figures in space. Thus, this pure intuition reveals a general constraint on space as the pure form of outer sense. For Kant, as I understand him, what I see immediately when I recognize a plane figure as a triangle is gnided by the very sarne rules I would use to construct an image of a triangle 10 imagination or to draw my own diagram of a triangle on paper. These are also the sarne rules I would follow to trace out a plane triangle with my finger or with the path of my whole body as I walked out a triangular pattern on a football field. The plane on which I conctruct or recognize the triangle must be oriented relative to my body in three-dimensional space. Even 10 the imagination, I think, Kant would claim, the plane on whiCh a plane geometry construction is carried out is imagined as oriented in a three-dimensional space relative to a point of view. 29 I think for Kant these general rules for recognizing or constructing any plane triangle support the auxiliary construction of lioes CD and CE in the same plane. They also underwrite the intuitive reasoniog whereby I am compelled to recognize that the sum of the ioterior angles equals the straight angle BCE. Sioce this construction and intuitive reasoning is supportable from the schema for the general concept of a plane triangle, the conclusion I reach for the figure under consideration must hold for any fignre I could ostensively recognize to be a plane triangle.
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3. What about the non-Euclidean geometry of modern physics?
I used Euclid's constructive proof that the sum of the interior angles of a plane Triangle equals 180° to explicate Kant's account of geomeTrical constructions as an endorsement for the intuitive compulsion a proof like this provides. The fact that Kant cites this as a paradigm example of geomeTrical construction gives some snpport to those who claim that he was comntitted to the a priori application of Euclidean geometry to the space in which we can apprehend outer objects of experience. It is now well known that the sum of the interior angles of a plane Triangle is a key mark discriminating between Euclid's geometry and the various non-Euclidean geomeTries of constant curvature. GeomeTries of positive curvature make the sum of the interior angles greater than 180°. This is clearly illustrated in Poincare's model of plane Riemannian geometry in which the plane is identified with the outer surface of a Euclidean sphere. GeomeTries of negative curvature all make the sum less than 180°. These include the classic hyperbolic geomeTries of Lobachevsky and Bolyai. In a geometry with variable curvature the sum of the interior angles of a plane Triangle marks the local curvature of the plane on which the Triangle is constructed. If the sum is 180 0 then the space is locally Euclidean. Kant has some good company if he was committed to Euclidean geometry _ even among mathematicians who were aware of non-Euclidean geometries. I think it was an appreciation of just the sort of intuitive compulsion Kant's theory of constructions attempts to explicate that led Frege (1959) to claim that only Euclidean geometry fits our intuition and led Poincare (1898) to suggest that Euclidean geometry ought to be retained even at considerable cost in additional complexity to physical theory. Nevertheless, I think that today most of us, children of the relativistic age as we are, would regard it as hopelessly Quixotic to continue to claim that Euclidean geometry is the correct geometry of the physical objects we meet in space-time. The weight of evidence is too solidly lined up behind modern physical theory. Does this not show, therefore, that the very foundation of Kant's empirical realism has been overturned by modern physics? Strawson's (1966) attempt to save something of Kant's account of spaceby making it apply to a merely visual geometry - will not do. This attempt and others like it (e.g. Walker's, 1978) which remove the clash with physical theory by giving up comntitment to objective constraints on physical things, will not preserve the fundamental role of space as a framework within which appearances can generate an empirical realism. 3o I think Melnick (this
EMPIRICAL REALISM & THE FOUNDATIONS OF GEOMETRY 283
volume) is absolutely correct in his identification of the subject matter of Kant's account of geometry with the structure of our framework for meeting things outside us. Kant's space is the physical space we move our bodies around in. His straight lines correspond to rigid rods that can be rotated inside their boundaries and to paths of light rays. I think one can keep this physical interpretation of the subject matter of geometry and keep enough of Kant's account of geomeTrical verification to give the constraints his empirical realism needs without flying in the face of modern pbysics. The key idea has been put forward by James Hopkins (1973) in an interesting paper attacking Strawson. It is this: What we can establish by geometrical constructions is limited by our perceptual capacities.
Hopkins (p. 24, 25) points out that we could not take in any diagram that accurately represented the relative sizes and distances between two stars. Either the dots representing the stars would be too small to see 01: the distances would be so great we could not survey the diagram. I think what Hopkins is pointing out is correct and important. The limitations on what we can use diagrams to represent are quite significant. Even two parallel lines one centimeter apart - each say O. 5 mm thick and 150 meters long could not be taken in by us. If we got far enough back to take in the end points· we would be so far back that we would not be able to resolve the separate lines. This shows that Euclid's parallel's postulate could not be established by any geomeTrical construction we could carry out. If Kant had claimed to be able to establish this postulate by constructions he would have violated his own basic injunction about extending concepts ouly valid for objects of experience beyond the limits of what we can experience. Any specification of what happens as parallel lines are extended indefinitely would correspond to an ideal of pure reason not to a principle constitutive of possible objects of experience. If we use the supervaluation method of representing commitment to the possibility of some such idealization then Kant's account of geomeTrical construction would comntit him only to the envelope corresponding to a whole family of geomeTries each of which captured local constraints on threedimensional shape and perspective up to tolerances provided by our perceptual capacities.
When you or I carry out our construction for the sum of the interior angles of a plane Triangle the inmitive compulsion our result carries is not misleading, so long as we recognize that what we establish only holds up to tolerances provided by our perceptual capacities. Similarly, we really can constructively establish Ullman's various shape from motion theorems up to
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such tolerances. Even though the constraints we get this way are vague and have vague limits on their vagueness. they have considerable bite. Lots of hypotheses are clearly beyond the tolerances allowed. For example, there could not be any rigid object, about as wide as my head, that would generate, at about arms length, the three orthographic projections specified in my salient illustration (Section IV, I) of what Ullman's theorem rules out. Such constraints capture a good deal of what Kant wanted from his account of geometry as a source of a priori knowledge. They depend on pervasive and accessible features of the actual capacities our sensory systems have and of the environment in which they have evolved to operate. These are very broad and deep features of what Wittgenstein called our form of life. They are not something we could change by adopting new social conventions. Nor could we find out tomorrow, on the basis of some new theory, that we have been wrong about these things all along. This does not imply that these constraints could never possibly change, only that to change mem we would have to undergo rather gross changes in our bodies or their local environments. If the behaviour of light rays and measurably rigid rods were to change so dramatically that they become unreliable as indicators of shortest paths between macroscopic local locations, then perhaps the constraints would change. I believe Kant would say that such a change would be impos. sible.31 Rather than just follow him in this. I want to point out the difference between having such physical changes actually begin to happen (which would be rather noticeable) and changing our theories about what has been happening all along. This difference points out an important sense in which geometrical constructions give us theory independent constraints on observation. It suggests to me that a Kantian alternative to some of the excesses of the last twenty years is to rush away from the idea of an observation theory distinction.
NOTES
I •
* The earliest ancestor of this paper was a talk I gave at the Canadian Society for the History and Philosophy of Science in 1974. In 1978 I commented on Colin Turbayne at the Rochester Conference honouring Lewis Beck's retirement. This led me to develop the argument in Section I. The first written draft was in May 1982 and its first public presentation was at a conference at the University of Western Ontario in Spring of 1982. A version was delivered as a lecture in my graduate seminar as Visiting Professor at Princeton in spring of 1983. Versions were also presented at a Duke Conference and at a Columbia University Philosophy Department colloquium. I am grateful for the insightful questions and comments received from many of the people who heard one or another of these presentations.
EMPIRICAL REALISM & THE FOUNDATIONS OF GEOMETRY 285 Special thanks are due to Robert Butts, Dan Garber, Ralf Meerbote, Calvin Nonnore. George Pappas, Margaret Wilson and Bas van Fraassen. Section n benefited from acute stylistic criticism generously provided by Paul Kirchner. I Most (but not all) of the passages Turbayne uses to support Kant's commitment to this argument come from the fourth Paralogism (A 367-380) which was dropped from the second edition of the Critique. The new Refutation of Idealism (B 275-279) in the second edition uses a different argument (see Section 3). One can also, perhaps, quibble over some of the steps and the way they are arranged to bring out the affinity to Berkeley. Nevertheless, I think it is fair to say that Kant remained committed to something like this argument. RaJf Meerbote (correspondence with me) disputes this. I think he is correct unless special care is taken with step 2. See Note 8 below for a suggested interpretation of step 2 under which the argument is compatible with the refutation of idealism. It is under this interpretation (which differs from Turbayne's Berkleyean construal of (2) that I hold it plausible be assume that some such argument is important to Kant's Copernican revolution in philosophy. 2 Several writers, e. g. N. K. Smith (Commentary, p. 156) have taken Kant's accusations of illusiONism as evidence that he misunderstood Berkeley's position. What George Miller (1973, pp. 316-322) has called the traditional view of the relation between Kant and Berkeley would explain these apparent misunderstandings on the hypothesis that Kant only knew Berkeley's work through distorted second hand sources. Turbayne (pp. 225-227), Miller (op. cit.) and Henry Allison (1973. pp. 43-45) have made it plausible to assume that Kant had far more access to Berkeley's work than the traditional view would allow. In particular they point out that a Gennan translation of Berkeley's dialogues was readily accessible to Kant. The hypothesis that Kant actually read the dialogues allows one to entertain the view that Kant's reference to the 'good Berkeley' in his B 70 passage we cannot blame the good Berkeley for degrading bodies to mere illusion: which Turbayne finds evidence of animus and Allison of condescension is really only irony obtained by applying to Berkeley the very same rhetorical device he applies to Hylas (the defender of common sense realism) in the dialogues-
Phil: "Have patience, good Hylas, and tell me once more whether there is anything immediately perceived by the senses expect sensible qualities. I know you asserted there was not; but I would now be infonned whether you still persist in the same opinion. " 3 Turbayne, Margaret Wilson (1971), Goerge Miller (1973) and Henry Allison (1973) all point out that in Kant's day Berkeley's position was regarded very unsympathetically. 4 George Pappas brought to my attention James Cornman's (1973) defence of the idea there can be laws COlUlecting sense data. I believe this defence will not work if sense data are to be construed as incorrigible subjective contents of experience (see Note 14). 5 It is not swpising that Sellars, the author of the objections to phenomenalism I have been considering, should take such rules governing what he called point-of-viewish aspect of perception as the key to an interpretation of Kant's transcendental idealism. In 'Kant's Transcendental Idealism' 1975 and in 'The Role of Imagination in Kant's Theory of Experience' 1978, Sellars proposes what I take to be just the sort of account I shall defend here. Indeed, this paper can be well constnied as an attempt to make some of the details of this kind of account more explicit and to document more extensively its textual support in Kant's writing. 6 Margaret Wilson (1971), George Miller (1973) and Henry Allison (1973) all argue impressively against Turbayne that it was reasonable for Kant to draw the conclusion that Berkeley's
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qeatment of space renders his position unable to support an empirical realism. I agree with most of what these writers have to say and offer these additional arguments in support of the view that Kant's conclusion about Berkeley's position is true as well as having been reasonable for him to
draw. 7 John Pollock (1974) and Hilary Putnam (1981) are but two of many recent examples. 8 It is well known that Kant explicitly addressed this paralogism against Descartes' position (A 368), so it is not too surprising that the assumptions of the demon hypothesis should correspond to the assumptions leading to the skeptical nadir of the transition from transcendental realism to transcendental idealism. Some care must be taken with step 2. For one thing, it will tum out that for Kant, unlike Berkeley, immediate awareness need not mean unchallengeable. For another, Kant will distinguish between subjective and objective contents of representations. A Berkelean construal of step 2 on which the content in question is subjective and inunediacy implies not open to challenge is what corresponds to Wilson's contention 2. On this construal, I shall argue, Kant's transcendental idealism gives up contention (2) as well as contention (1). If step two is interpreted so as to include objective content and immediate awareness so as to allow for corrigibility then Kant does not give up step 2. Under this interpretation (in which step (2) is not the same as Wilson's contention (2» Kant's Fourth Paralogism argument, in which step (2) is retained, is quite compatible with his refutation of idealism, which I shall argue rejects Wilson's contention (2). 9 Berkeley's subjective idealism is a salient example in a long tradition of phenomenalistic empiricism that is characterized by the attempt to ground all acceptable knowledge claims is sensations or some kind of incorrigible data base in experience. This tradition, which includes Hwne, Mill, Russell at some stages, the Camap of the Aujbau, and C. I. Lewis, is still alive today in the work of R. M. Chisholm and John Pollock. All of these writers including the most sophisticated agree with Berkeley in holding to contention (2) and most of them give up contention(1) in some way or other. In recent years, perhaps to a great extent due to the influence of Wittgenstein, this idea of a secure data base incorrigible claims about subjective contents of experience has lost power. As this has happened more and more Kant scholars have opted for objective rather than subjective readings of transcendental idealism. If my interpretation is correct then this has been a good trend, for Kant' position always was distinctively different from Berkeley's in that his appearances should never have been construed as incorrigible data. 10 Turbayne (pp. 232-3) considers this skeptical idealism (step 4) to be the first stage of the solution to the skeptical problem. I think Kant considered this position as no better than what Turbayne calls the deepest skepticism of step 3. 11 Note that, on Turbayne's version of it, Berkeley's position would also have to allow thatjudgments about external objects could be mistaken. Even if my judgment about the subjective content I now have were incorrigible my judgment that uniformities appropriate to the claim that there is a real dagger there obtain is subject to error. 12 Of the two interpretations of Kant's refutation of idealism argument suggested by WIlson (1972, pp. 604-605) this is the one that she grants would make trouble for her. It is also the one that best coheres with what Kant says the argument proves (B 275). 13 Barry Stroud (1968) also makes interesting comparisons between Kant's refutation of idealism argwnent and Wittgenstein's private language argumenL Unlike Sellars and Bennett, however, he did not actually propose an argument for Kant's conclusion. 14 This collapse into the solipsism of the present moment provides an additional compelling argument against the hypothesis that there are laws about sense data construed as incorrigible
EMPIRICAL REALISM & THE FOUNDATIONS OF GEOMETRY 287 subjective reports that could be learned empirically from observed regularities in one's subjective experience (See Note 4. ) I explore connections between the foregoing interpretation of the Refutation of Idealism and the Second Analogy Passages on the distinction between subjective and objective succession in Harper (1984). That paper also uses the nice passage from Wilson as a paradigm of the Demon argument and the Cartesian asswnptions it requires. I first saw this kind of interpretation of Kant's Refutation of Idealism, where the key argument is the failure of subjective idealism to support objective truth conditions about past subjective experiences, in Jonathan Bennett (1966). Paul Guyer (1983) has recently provided an extensive discussion of the origin and interpretation of Kant's Refutation of Idealism which also makes the core of the argument depend on these considerations. 15 It is worth remarking that Kant uses 'representation', and each sub-heading on this list, ambiguously as a token term to refer to particular mental episodes (representings) and also as a type term to specify a kind of representation qua - what it represents and how it represents it. The important use is the type use. I shall attempt to restrict my use of 'representation' to it and reserve representing for the token use. When, for example. I speak of my empirical intuition of this coin I shall be speaking of a kind of representing which can be speCified as a representing of this coin under certain perceptual circwnstances. According to this usage the same intuition could have tokened by another person or have failed to be tokened at all should it have been that someone else Or no one at all had satisfied the relevant perceptual circumstances. 16 Van Fraassen (Lecture at University of Western Ontario, fall tenn 1981) had recently suggested that one's inability to believe there is no ball in his hand when he is confronted with it does nouhow that belief is not voluntary any more than the fact that one cannot steal when he is in his own bathtub surrounded by only his own possessions shows that stealing is not voluniary. According to van Fraassen circumstances can sometimes constrain belief. Presumably, one of the most important kinds of constraint is provided by perceptual circumstances. It is to our relative inability to ovenide these circwnstances that I point when I speak of the independence of percep_ tion.
17 Sellars (e.g. 1963a) has emphasized the idea that entities of various kinds could play in thought a role suitably analogous to the role placed by a sentence, e. g. This is a quarter-shaped object before me, i'1 English. My proposal here is designed to be in the spirit of his general use of dot quotes and his own accounts of Kant's intuitions (e. g. 1968, Chapt. L 1975,1978). 18 This account of appearances is very close to Sellar's account of what we see of an object (e.g. 1981, Sections 15-24). It is also very close to Gibson's account of 'affordances' for human per_ ception (Gibson, 1966, 1979). I hope to explore some of the connections with Gibson's work in a later paper. Indeed, I expect that some of Kant's use of geomeny to ground his acCOunt of appearances as undetermined objects of empirical intuition can help answer some of the objections (e.g. in J. A. Fodor and Z. W. Pylyshyn, M. I. T. Occasional paper # 12) that have been raised against Gibson. 19 One of the salient differences between colors and such geometrical properties as shape is that the same geometrical property can often be presented to touch as well as to sighL This legitimate and important integration of sense realms, noted as early as Aristotle, provided motivation for the infamous distinction between primary and secondary qualities. According to Berkeley (Theory of Vision, 121-146) vision is only presented with colour, and colour is never presented to touch. so that these two sense realms are entirely distinct. He insists
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that the shapes presented to touch and the shapes presented to sight are entirely different species related milch as the combinations of writer's letters are related to the sounds of speech. In 'The Perception of Shape' (1983a) David Sanford has offered an excellent exploration of and defence for the claim that the shapes we see are the same as the shapes we touch. This is the best discussion of the issuef know of. 20 In 'Kant's Empirical Realism and the Distinction Between Subjective and Objective Succession' (Harper, 1984) I presented a line by line interpretation of this paragraph. This section of the present paper is, mostly, a summary of the main points in that interpretation; however, it does contain some additional remarks I hadn't thought to make before. 21 Beck's paper contains an admirable brief gloss of Kant's entire third paragraph in B. A good deal of my longer interpretation was cast in the fonn of commentary and expansion on Beck's gloss. 22 Beck distinguishes three distinct versions of Kant's empirical sense of 'appearance'. See (Harper, 1984) for my exposition of their relation to my explication of appearance as undetermined object of an empirical intuition. 23 This gloss of the two occurences of 'it' in the translation was suggested to me in correspondence by Lewis Beck. It is a salient part of the reading he provides in Beck (1978),
pp. 144-146).
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24 See van Fraassen (1966) for the basic account of supervaluation. See Thomason (1973) and Hans Kamp (1981) for applications of supervaluations to problems of indeterminacies generated by vagueness. 25 The Metaphysical FoWldations of Natural SCience suggests that Kant himself would opt for a plenum account based on postulated centers of inverse square attractive forces and inverse cube repulsive forces. 26 I have only argued that Kant's own defence of empirical realism is based on his appeal to a priori spatial constraints. I have not attempted the ambitious task of showing that an empirical realism that does not presuppose such constraints is impossible. Nevertheless, I hope the arguments of this section will make it implausible to suppose that one can recover an empirical realism without presupposing some such constraints. More importantly, I hope to have shown that the sort of constraints Kant's defence requires are not SO very implausible to presuppose. 27 Orthographic projection is not an accurate representation of the visual infonnation afforded to an ob$erver surveying a relatively large object before her. Ullman explores several other projection schemes. One is a perspective projection scheme according to which three views of five elements are mostly sufficient to uniquely specify the configuration and relative motion. This scheme is better at discriminating reflections in the image plane, but not so efficient at ruling out other alternative configurations and motions. The most sophisticated model of a projection scheme for human vision he considers is a polar-parallel projection scheme. Points corresponding to local texture patterns on a large surface are treated as approximately parallel projections to give detailed information about the local surface shape, while the larger structure is pinned down by polar-projection of these various local textured areas. This scheme apparently provides a fair approximation to the strengths and weaknesses of actual human visual discrimination. 28 Beck (1956) and Parsons (1969) have made a convincing case that Kant interprets such constructions as real definition of a geometrical concept. Real definitions are not analytic nor are they to be merely conventional stipulations. I have tried to make it plausible that geometrical constructions can play such an exalted role. 29 David Sanford (1983b) has an interesting discussion of the commitment to orientation relative to a point of view of a visual field. [think Kant would agree with this and would extend the point to imagination as well as perception.
I
EMPIRICAL REALISM & THE FOUNDATIONS OF GEOMETRY 289 30 Reid's Geometry of Visibles has been reswrected as one of a number of proposals for construing visual geometry as non-Euclidean (Angell, 1974; Daniels 1972). Strawson's attempt to save Euclidean geometry by dividing viSUal geometry from physical interpretations plays right into the hands of these advocates for non-Euclidean geometries of visual experience. 31 I do think that it is rational now to proceed as though such changes could not happen. Indeed I think that we all really do proceed this way. We cannot help but make them into conceptual commitments for us. I also think that conceptual commitments can be rationally changed (Harper,
1978).
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43-63. Angell, R. B.: 1974, 'The Geometry ofVisibles'.Nous 8 87-117. Beck. L w.: 1956, 'Kant's Theory of Definition', Philosophical Review 65, 179-19l. Beck, L. W.: (ed.): 1972, Proceedings of the Third International Kant Congress, Reidel, Dordrecht. Beck, L. W.: 1978, Essays on Kant and Hume, Yale University Press, New Haven. Bennett, 1.: 1966, Kanfs Analytic, Cambridge University Press, Cambridge. Berkeley, G.: 1954, Three Dialogues Between Hylas and Philonous ed. by C. M. Turbayne, Bobbs Merri1l, Indianapolis. Berkeley, G.: 1965, A Treatise Concerning the Principles of Human Knowledge, ed. by C. M. Turbayne, Bobbs Merrill, Indianapolis. Berkeley, G.: 1969, An Essay Towards a New Theory of Vision. ed. and introducted by A. D. Lindsay, New York. Cornman, 1. W.: 1973, 'Theoretical Phenomenalism', NOlls 7, 120-138. Daniels, N.: 1972, 'Thomas Reid's Discovery of a Non-Euclidean Geometry', Philosophy of Science 39, 219-234. Euclid: 1956, Elements, Trans. by T. C. Heath, Dover, New York. Fodor, 1. A. and Pylyshyn, Z. W.: 'How Direct is ViSUal Perception?', MIT Occasional Paper # 28. Frege, G.: 1959, The Foundations ofArithmetic, trans. by 1. Austin, Blackwell, Oxford. Gibson, 1. 1.: 1966, The Senses Considered as Perceptual Systems, Houghton Mifflin, Boston. Gibson, 1. 1.: 1979.An Ecological Approach to Visual Perception, Houghton Mifflin, Boston. Guyer, P.: 1983, 'Kant's Intentions in the Refutation of Idealism" Philosophical Review 92,
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pp.145-155. Miller, G.: 1973, 'Kant and Berkeley', Kant Studien 64, 315-335. Mohanty, I. N. and Shehan, R. W. (eds.): 1982, Essays on Kanfs Critique of Pure Reason. University of Oklahoma Press, Norman, Oklahoma. Parsons, C.: 1964-, 'Infinity and Kant's Conception of the "Possibility of Experience"', Philosophical Review 73,183-97. Parsons, C.: 1969, 'Kant's Philosophy of Arithmetic', in Morgenbesser et al. (eds.), Philosophy. Science, and Methodology: Essays in Honor of Ernest Nagel, St. Martin's Press, New York; reprinted in this volume pp. 43-79. Penelhum and Macintosh (eds.): 1969. The First Critique, Wadsworth, Belmont, California. Poincare, H.: 1898, 'On the Foundations of Geometry', The Monist 9, 1-43. Pollock, J.: 1974, Knowledge and Justification, Princeton University Press, Princeton. Posy, C.: 1983, 'Dancing to the Antinomy: A Proposal for Transcendental Idealism', American Philosophical Quarterly 20, 81-94. Posy, C.: 1984, 'Kant's Mathematical Realism', The Monist 67 (1984); reprinted in this volume pp.293-313. Putnam, H.: 1981, Reason, Truth and History, Cambridge University Press, Cambridge. Sanford, D.: 1983a, 'The Perception of Shape', in C. Genet and S. Shoemaker (eds.), Knowledge and Mind: Philosophical Essays. Oxford, pp. 130-158. Sanford. D.: 1983b, 'Impartial Perception', Philosophy 58, 392-395. Sellars, W.: 1963, Science, Perception and Reality, Humanities Press, New York. Sellars. W.: 1963a, 'Abstract Entities', Review of Metaphysics 16, 627-671. Sellars, W.: 1968, Science and Metaphysics, Humanities Press, New York. Sellars, W.: 1975, 'Kant's Transcendental Idealism', in Laberge et al. (eds.), Ottawa Congress on Kant in the Anglo-American and Continental Traditions, Editions d'Universite d'Ottawa, Onawa. Sellars, W.: 1978, 'The Role of Imagination in Kant's Theory of Experience'. in Johnstone (eds.), Categories: a Colloquium, Penn. State Press, pp. 231-245. Sellars, W.: 1981, 'Foundations for a Metaphysics of Pure Process', The Monist 64, 1-88. Smith, N. K: 1929, A Commentary to Kant's Critique of Pure Reason, Macmillan, London. Strawson, P.: 1966. The Bounds of Sense, Methuen, London.
EMPIRICAL REALISM & THE FOUNDATiONS OF GEOMETRY 291 Stroud, B.: 1968, 'Transcendental Arguments', Journal ofPhiiosophy6S, 241-256. Thomason, R.: 1973, 'Supervaluations, The Bald Man and The Lottery', Unpublished Mimeo, University of Pittsburgh. Torretti, R.: 1978, The Philosophy o/Geometry from Riemann to Poincare, Reidel, Dordrecht. Turbayne, c.: 1955, 'Kant's Refutation of Dogmatic Idealism', The Philosophical Quarterly 20, 225-244. Ullman, S.: 1979, The Interpretation of Visual Motion, MIT Press, Cambridge (Mass.). van Fniassen, B.: 1966, 'Singular Terms, Truth Value Gaps, and Free Logic', Journal of Philosophy 63, 481-495. van Fraassen. B.: 1980, The Scientific Image, Oxford University Press, Oxford. Walker, R. C. S.: 1978, Kant, Routledge and Kegan Paul, London. Wllson, M.: 1971, 'Kant and the Dogmatic Idealism of Berkeley', Journal of the History of Philosophy 9, 459-476. Wllson, M.: 1972, 'On Kant and the Refutation of Subjectivism'. in L. W. Beck (ed.), (1972).
CARL J. POSY
KANT'S MATHEMATICAL REALISM
Though my title speaks of Kant's mathematical realism, I want in this essay to explore Kant's relation to a famous mathematical anti-realist. Specifically, I want to discuss Kant's influence on L. E. J. Brouwer, the 2Oth-cenrury Dutch mathematician who built a contemporary philosophy of mathematics on, constructivist themes which were quite explicitly Kantian. I Brouwer's
theory (called intuitionism) is perhaps most notable for its belief that constructivism (whatever that means) requires us to abandon the traditional (classical) logic of mathematical reasoning in favor of a different canon of reasoning, called intuitionistic logic. Brouwer thought that classical logic is intrinsically bound up with a nonconstructive (or "realistic") view of mathematics~ This means that, according to Brouwer, when we do mathematics we must give up bivalence (the principle that a given sentence either is true or is
detenninately false), we must no longer use such familiar logical laws as excluded middle, and we must sometimes forebear from the classic method
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of reductio ad absurdum. All of these are intuitionistically invalid classical principles. Now scholars generally agree that Brouwer did borrow Kantian themes. But there is very little consensus about which details he borrowed. And there is certainly little agreement about how this same set of doctrines could foster such radically different programs as intuitionism and Hilbert's formalism. (Hilbert too was an avowed Kantian.) So I have set myself the task of trying to give a precise - though perhaps anachronistic - account of those Kantian themes Brouwer might be developing in his critique of classical logic. I will, indeed, consider two rival accounts of Brouwer's debt to Kant. This task is especially interesting to me, because I am fond of using intuitionistic techniques to interpret and vindicate some Kantian arguments - not so much arguments about mathematics but mainly ones about empirical science. These are arguments in which Kant defends his famous transcendental idealism, TI, (the view that empirical objects are mere appearances) from its rival 'transcendental realism, TR, (the view that these same objects are things in themselves). I think of this project as exploring Kant's debt to 293 Carl J. Posy (ed.J, Kant's Philosophy of Mathematics, 293-313.
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Brouwer. Any philosopher owes a debt (even posthumously) to the progenitor of techniques that can vindicate some of his reasoning. But the point is that I can't just go about attributing intuitionistic logic to Kant simply because it salvages this or that argument. To make this project plausible, I'll have to find in TI a philosophical ground to support these logical moves. This ground can be the theme Brouwer borrowed from Kant. As I said, I'll suggesttwo rival interpretations of Tl to do that job: the first, a fairly standard ontological one, unfortunately fails, both as a reading of Kant and of Brouwer. The second, which applies some currently popular notions from the philosophy of language, does work for Kant (and perhaps for Brouwer). But my main thesis now is that this second interpretation has a most remarkable consequence. When these linguistic notions are clarified and applied to Kant, they show him to be an intuitionist for empirical science, but a realist, or at least an advocate of classical logic, for mathematics! So my remarks will fall under four main topics:
beginning is ruled out, because the event of creation ex nihilo could have no observable cause. [A427/B455].) The idealist escapes this dilemma. For, according to Kant, TI's world need be neither finite nor infinite. (A504/B532). In fact, it is neither finite nor infinite. (A520/B548). Passages containing this attack on TR have themselves suffered much critical attack. Jonathan Bennett says that Kant's discnssion of measurability simply distorts the properties of discretely ordered series.3 Bertrand Russell tells us that Kant is unable to express clearly the infinity of a class (in this case the class of past durations), and that Kant's basic move is undermined by the discovery of infinite numbers.4 There's a logical question as well: Why is Tl free from the disjunction (finite vs. infinite past) while TR is not? Lastly, there is a simple question of consistency: How can Kant square his claim that TI's world is neither finite nor infinite with his subsequent claim just a page later that the idealist's world does indeed extend infinitely into the past? (A521/B549). (Perhaps Kant is distinguishing, in the later passage, between an actual and potential infinity. But if that is so why isn't the realist entitled to the same distinction? Realism about physical objects is perfectly compatible with denying actual infinity.) [2] Here is where I think some contemporary insights and techniques will help. Observe first that Bennett is right in one respect: Kant is indeed Concerned with a discrete linearly ordered series, a sequence of equal temporal durations (say, days) stemming backwards from some fixed moment. Indeed we can say (without anachronism) that the elements are ordered by a relation satisfying the axioms of a linear order. Now suppose we introduce a very simple formal first order language, Lr (T for temporality), whose variables range over these equal durations. And suppose Lr has a single, nonlogical, binary symbol, 'B', which denotes that discrete ordering of durations. B (x, y) means that interval x began at a moment distinguishably prior to the beginning of y. In this language we can symbolize the claim about a finite beginning by
I. Display Kant's debt to Brouwer. II. Describe the standard interpretation, and why it fails. (I will do these first two very briefiy.J2 III. Describe the linguistic reading, and clarify some of its oft rnis~d details. IV. Apply this reading to Kant, and use it to support the snrprising conclusions I mentioned. I will conclude with some general remarks about how all of this interacts with some other themes in the Critique of Pure Reason. and with a discussion of the sort of anachronism that mayor may not be lurking in all this. 1. KANT'S DEBT TO BROUWER,
[I] Let me tum to my first task and qnickly describe an instance where current intuitionism may help us interpret a controversial Kantian defense of TI. I have in mind Kant's attempt, in the famous "Antinomy" chapter, to produce an internal contradiction in the realist's empirical world view - in particular in his view about the age of the universe. According to Kant, the transcendental realist is committed to the view that the world either had a finite beginning in time or extends infinitely into the past. Kant then proves that neither alternative is possible, and thus reduces TR ad absurdum. (An infinite past is impossible, he says, because it is impossible to measure such a long temporal stretch. [A 426/B454]. And a finite
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(I) 3xVy"B(x,y).
The claim about an infinite past becomes (2) V:i3Y#xB(y, x).
Let me step out of my announced order for a minute to say that I think this simple translation into 4, which seems so natural today (languages like LT compose our modern lingua franca), this simple move is probably the most
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anachronistic move I will make. I'll discuss that a bit later. But I'll also point out that it's no less foreign to Brouwer (perhaps it is even more so) than it is to Kant. But if, for the moment, we do allow this barbatism, and we do assume that (I) and (2) are fair representations of Kant's alternatives, then the first group of objections melts away (by force, to be sure). There are no ordinal irregulatities hidden by (I) and (2); each is perfectly consistent with the linear order of time. (2) is precisely the expression of infinity that Russell ultimately advocates, and the existence of infinite numbers is (sttictly speaking) irrelevant to this pair offonnulas. As for the second problem, the logical one, that dissolves as well, if we associate TR with classical logic and TI with the logic that has been fonnulated for intuitionism. For the result of disjoining (I) and (2) is a classical logical truth, but not an intuitionistic one. As for the final problem - the one about the consistency of Kant's denial that the world is infinite with his later claim that the world does reach infinitely into the past -let me explain that latter remark. Let's assume for the moment that TI does link physical existence with experience. Still, according to Kant, we can never be satisfied with partial scientific results. We never rest easy with just those facts and objects which we have experienced. This is dictated by what he calls the "faculty of reason", the faculty that governs theoryfonnation and our general inferential activity. In particular this part of our personalities demands that we must always consider how any series of cosmological discoveries can be extended to discoveries documenting ever more remote past durations.
,
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always to push for further extensions of any sequence of discoveries, combined with the belief that such extensions will be found but the refusal to act as if they have been found. 5 There is a simple Kripke model which shows the compatibility of (3) with the axioms of linear order. The nodes of the model are artanged as a single infinite chain {a,} ,. The domain of a, is the set {I, 2, 3, ... n. And we interpret B(j, k) as j > k. (Numerical increase represents increasing remoteness into the past.) Now for-
mulas (2) and (3) are not inmitionalistically equivalent. (Indeed, (I) and (2) are both false in t1tis same Kripke model.) That explains Kant's dual attitudes toward infinity. The world is not infinite in the sense of fonnula (2), but is in the sense of (3). Classically, (2) and (3) are logically indistingnishable: so the realist is indeed barred from this resolution of the antinomy. That solves all the problems I raised above. [3] As it turns out t1tis same device is quite useful in interpreting several other passages in the Critique where the regulative effect of the faculty of reason is invoked. Thus, for instance, the principle of causality which is defended in the second analogy, needs something like this to reconcile it with the arguments of the first and third antinomies. 6 And the model structures that logicians use to study intnitionism provide handy devices for depicting the notions of "objectivity" and "unity" that figure prominently in Kant's Transcendental Deduction.7 There are other applications as well.
So there are exegetical advantages to assuming that the transcendental realism/idealism debate includes this modern debate about the logic of empir-
If (as the idealist requires) this feature of the mind has a role in forming
ical discourse. But as I said at the start, even if we ignore the inherent
our scientific picture of the empirical world, then that science must satisfy our
drive for "completeness" of the series. But, at the same time, pure reason
anachronism, still all of this becomes philosophically interesting only if we understand TI in a way which ma.kes it reasonable to link it with intnitionism.
alone cannot provide us with the observations and experiences we haven't yet
Let me tofu to that question.
had. (Observations and experiences require the faculty of "intuition.") Kant encapsulates all this in the slogan that the faculty ofreason posits the infinity of the world as a regulative ideal. Butnaming it isn't explaining it. Once again intuitionism comes to the rescue. We can symbolize the effects
of all of these seemingly diverse pulls in fonnula (3): (3) Vx - - 3yox B(y, x)
The intuitionistic double negation here means that given any duration, x, we cannot help but eventually discover a distinct y which is prior to x. This
gives a sympathetic reading to Kant's notion of a regulative idea: the drive
2. ONTOLOGICAL IDEALISM
[4] There is a more or less standard interpretation of TI which at first sight might do the job. This is the view which makes TI into a literal physical constructivism - called phenomeualism - and according to which material objects are entirely human constructions.
This interpretation is standard because it works within the widely accepted correspondence theory of truth and preserves the familiar truth tabular meanings of the logical connectives.
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And indeed it does fit many Kantian texts. Physical objects are often described as constructed ("synthesized") from the material of intuition. 8 Mathematical objects are thought of as similarly constructed, though out of different, more rarified material.9 This reading is consistent with the general assumption that Kant meant intuitions (both empirical and mathematical intuitions) to apply directly to their objects in a de re relation. 10 And since it views mathematical objects as reified versions of the principles governing the construction of empirical objects, it does a good job of explaining Kant's views on applied mathematics. I should add that this picture of the scientist and mathematician somehow "constructing" their objects seems to fit a number of Brouwerian passages as well." [5] Nevertheless this picture of constructivism won't work for Brouwer or for Kant. Though it does invalidate bivalence,12 still it fails for Brouwer because it does not generate the full intuitionistic logic. It will not for instance validate - - (Pt V - Pt) and other tautologies that figure prominently in many Brouwerian arguments. So this ontological constructivism - even if it is a Kantian theme - can't be one that Brouwer borrowed in order to ground his logical beliefs. 13 But in fact it fails as a reading of Kant too, and for three reasons. First, this interpretation makes it impossible to understand the realist's reasoning in the antinomy. If physical events and objects are (for the realist) mind - independent things in themselves, why should their measurability or immeasurability in any way affect the truth of judgments about their size? Secondly, if objects are constructed out of the material of sensation, what hope is there for the existence of objects too faint, too far, or too small to be perceived? Yet Kant clearly held that things of this sort (distant stars, miniscrile particles and the like) most definitely are physical objects, and do exist. l ' Finally, this reading blurs any appreciable difference between Kant and Berkeley. The two will share the sarne theory of truth and the sarne phenomenalist conception of empirical objects. This is bad on its own (phenomenalism for material objects is an abhorent doctrine); but it's even worse, because Kant explicitly separates himself from Berkeley on several occasions. l5 So, all in all, we have to look for some other philosophical ground generating intuitionistic logic - both in mathematics, and (for Kant) in empirical science. Michael Dummett, building on suggestions stemming from Heyting (and really aimed at mathematics) has provided a philosophical argument which (if sound) will invalidate bivalence and will generate an intuitionistic logic. It is
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the argument from the so called assertability theory of meaning. 3. ASSERTABILISM REFINED
[6] The main thrust of this argument is really qnite general, and can easily apply to empirical as well as to mathematical language. It goes like this: Given some fragment of language (say Lro or the language of number theory), the main premise of the argument claims that we must replace the standard referential notion of truth for sentences of that fragment with a notion of "sufficient evidence" (warranted assertability). Instead of speaking about what must be the case in the world (or in some model) for a given senteriCe to be true, we speak about what sort of evidence will suffice to warrant the speech act of asserting the sentence. Indeed we must replace the classical notion of a model (or possible state of affairs) with the notion of an evidential state: a collection not of objects, but of bits of evidence sufficient to support or refute some set of sentences. Whatever logical role standard truth in a model used to play in the language this notion of assertability at an evidential situation now plays. In particular it is assertability that is now to be preserved by valid inferences, and it is assertability that figures into the meanings of the logical particles. These facts will then fix the corresponding logic for that language. And that will be intuitionistic logic, because, as it tums out, the orily statements guaranteed to be assertable at every evidential state are precisely the intuitionistic tautologies. Or so the argument goes. Now this argument - or at least its assertabilist premise - has gained a certain notoriety; mainly, I think because of Durnmett's spirited defence of thit main premise by use of arguments about the tearnability of language, and the possibility of communication. But I also think that some confusions have snuck in under all this hullabaloo over the replacement of truth by proof. For one thing. because sufficient evidence is now the criterion of semantic success (i.e., of truth), some people have slipped into the habit of saying that propositions (understood assertabilistically) are about units of evidence. (That the objects of mathematics are proofs, and the objects of empirical science are bits of empirical evidence). This is a confusion. Assertabilism per se says nothing about the objects of discourse. Indeed, the assumption that a theory of truth involves a theory of objects is a correspondentist assumption to start with. You can be a perfectly good assertabilist and simultaneously hold that objects of one sort or another are totally mind-independent.
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Secondly, all this jwnping and shouting about the assertabilist premise has obscured the fact that assertabilism alone will not suffice to generate an intu-
(A4) (A -7' B) is assertable in a situation iff, if A is assertable in the situation B is. (AS) VxA is assertable in a situation iff Aa is thus assertable for every a spoken about by the given language. (A6) 3xA is assertable in a situation iff for some such a, Aa is thus assertable.
itionistic logic. In order to go from assertabilism to intuitionism we have to
settle four additional matters. I call these additional considerations (i) epistemic myopia, (ii) assertability conditions, (iii) constructive optimism, and (iv) the long vs, the short semantic view. Let me briefly explain them in order, [7] First, these evidential states that replace models must be limited in a certain way. That is, we must conceive of these states in such a way that it is always possible that at least some proposition or other being evaluated at a given epistemic situation is undecided at that point by the ''relevant evidence."
- This is certainly true of our ordinary notion of empirical evidence (at some point in time), Thus I may learn tomorrow that there is oil under all that worthless clay in my backyard. But right now, I don't have enough evidence to confirm that claim; so I can't assert it. And that is a fact about my current epistemic state. Now of course for rigor we have to be more general, and speak of epistemology in the abstract, or of the general theory of epistemic states. But if there's to be any hope of scotching bivalence, we had better see to it that our epistemology does impose some sllch limits on its notion of evidential state. I call this the condition of "epistemic myopia." For it rules out the infamous "God's eye" view of knowledge. It admits a notion of gradual evidential growth, and says that sometimes that growth takes you beyond the evidence you happen to see within a given evidential state. (Formally, you might think of the nodes in a Kripke or Beth model as a progression of evidential states.) [8] The second thing we have to do is agree upon some general conditions which (once we set ground-rules for asserting atomic claims) will tell us about the compound propositions assertable in a given evidential state. As it turns out, that's less trivial than you might think. In particular it's not clear that we can simply substitute "epistemic situation" for "model" and plug in "assertability" for "truth" in the standard truth conditions derived from Tarski. Table I shows what this would look like. Table I (AI) (A & B) is assertable in an epistemlc situation if and only if A and B are both tl;tus assertable. (A2) -A is assertable in a situation iff A is "negatively assertable." [Ordinarily this is interpreted as A ~~, -1 some known falsebood.] (A3) (A V B) is assertable in a situation iff either A is thus assertable or B is.
This approach falters though because it is unenforceable in practice. Clause (AS) for instance is impractical when we are talking about very large collec-
tions of objects. And (A6) - which is a version of the requirement for "constructive existence proofs" - simply clashes with common practice and with almost all learned opinion. No one - not even a constructivist - always sits ~_d waits to actually prove an instance before asserting an existence claim
3xPx. Quite the same holds for disjunction in (A3). The problem is that these clauses aren't merely myopic about assertability - they're just plain blind. Only things that are concretely touched or bwnped into in an evidential state COlmt as evidence at that state.
Now mathematical assertabilists often solve this problem by importing what they call an "effective procedure." Four our purposes we can think of this as a definable operation which takes pieces of evidence into pieces of
evidence and which guarantees to produce an output for any given input (within a finite time). If you admit effective procedures (or better the recognition of effective procedures) as relevant to assertability then you get the assertability conditions I've sketched in Table II. Table II (A3), (A V B) is assertable at a situation iff that situation includes (recognition of) an effective procedure which results in evidence confirming A or evidence confirming B. (A4), (A ---+ B) is assertable in a situation iff that situation contains (recognition of) an effective procedure converting evidence for A into evidence for B. (AS)' (Ax) A is assertable in a situation iff the situation contains (recognition of) an effective procedure taking evidence that a exists into evidence for Aa. (A6), (3x) A is thus assertable iff the situation contains (recognition of) an effective procedure which will produce evidence for Aa for some a.
This is still a constructivism. It still requires that existentials be wituessed and that the assertability of disjunctions entail the assertability of one of the disjuncts. But it's a mild constructivism - it allows me to be satisfied with a guarantee of eventual success instead of the "blind" requirements of the stricter view.
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On this milder view I'm entitled to assert right now that some large number [say lOw + I] is either prime or composite - because I know a method to figure that out; and I know that once I start I'm guaranteed success. On the strict view I have no such rights. So. even if I adopt a basic assertabilist view of truth, still I will have to choose between the strict constructivism of Table I, or the milder version of Table 2. That's the second matter to be settled. [9] My third point is that this difference - between ntild and strict constructivism - is a difference that makes a difference for our concern with intuitionistic logic. For suppose I do espouse a basic assertabilism and equate truth with knowledge, and suppose I do admit that knowledge is limited at any given point. But now suppose I adopt a mild constructivism, and I couple all of this with a sort of brazen optimism. Suppose I claim that every proposition (of the language in question) will (or at least can) be constructively decided. (I.e., - suppose I am prepared to assert that given a meaningful proposition, and given enough time, I will be able either to prove it constructively or refute in constructively.) This isn't so outlandish a belief as one might think. It's precisely Hilbert's view about large parts ofmathematics. 6 But this Hilbertian optimism, which can be expressed in formula 4, combines with my ntild constructivism to validate excluded middle, and in fact the full classical logic.
Peircian assertabilism which defines truth as eventual assertability (Le., assertability in some eventually realised epistentic situation). The Peircian picture is one in which mankind progresses from epistemic situation to situation, learning (and asserting) new things, and adding them to an ever growing stock of recoguized truths. To be true is to have been put in this stockpile of assertable truths. Thongh tied to assertability, truth is now a tenseless notion. (Thus if I strike oil tomorrow it is just true outright that there is oil under my yard.) Now Dummett disparages this picture. Essentially it loosens truth from the grip of immediate assertability, and he thinks that's a step towards realism. But he's wrong. Indeed, his own linguistic support of basic assertabilism automatically supports this tenseless variety as welL 17 [To be sure, this Peircian long view faces some problems about corrigibility and convergence. But as things tum out these ills plague the local version as well.] The fact is that this Peircian version is a legitimate assertabilism. Recoguizing this fact is the final connection I mentioned. For when we combine this view with the Hilbert style constructive optintism, then the result is not intuitionism at all. It is a bivalent language - no matter which style of constructivism we adopt! All told, with these three latter issues (style of constructivism, optimism, long vs. short semantic view) to be settled, I've distinguished eight different assertabilist positions. Some will indeed invalidate bivalence, some will yield intuitionistic logic for any language to which they are applied. But some will not. I've tabulated the possibilities on Table Ill. So clearly, the move from assertabilism to intuitionism is far from guaranteed.
(4) Va3p [Pr(p, a) v Pr(p, lXl] (where a ranges over the propositions of our language, p over pieces of evidence (proofs) and Pr is a proof predicate.) The reason: Well think about any formula of the form (A v -A) at some epistentic situation. I ntight not be in a position to assert either disjunct but I am assuming that I can assert formula (4) with A for a, and this itself provides the _guarantee I need in order to assert the disjunction. So - unless the assertabilist can refute this optintism or constrain it - he must either ,adopt the stingy strict constructivism of Table r or tread the primrose path right back to classical logic. [10] Finally, as for bivalence: Notice that up till now I have been speaking as if our assertabilist premise simply equates truth with assertability-at-anepistentic situation. (So those propositions which are actually assertable right now, are the only ones which are actually true, and those not currently assertable are either false or without truth value.) Certainly that's the way Dunnnett looks at things. But by putting things this way I am automatically excluding the prominent
TableID Strict Constructivism: Short Semantic View (i) Non-optimistic
(1) non-bivalent
(2) no classical logic (ii) Optimistic
(b) Long View (i) Non-optimistic
(I) non-bivalent (2) no classical logic
(1) non-bivalent
(2) no classical logic
(ii) Optimistic
Mild Constructivism: (a) Short View
(I) bivalent (2) No classicallogic. 18
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(1) non-bivalent (2) no cIassicallogic (1) non-bivalent
(2) classical logic (b) Long View: (i) Non-optimistic
(1) non-bivalent
(2) no c1assica11ogic (li) Optimistic
(1) Bivalent
(2) classical logic
4. KANT THE ASSERTABILIST
[II] Now did Brouwer use an assertabilism to motivate his rejection of classicallogic? Two remarks are in order: (a) Brouwer would not accept Dummett's linguistic arguments supporting the general assertabilist premise for mathematics. These rest on considerations that are foreign to Brouwer's professed outloOk. 19 (b) On the other hand there are other places (eg. his proof of the bar theorem and in some posthumously found notes) where he endorsed the assertabilist definition of some logical particles. 20 And clearly he did identify Hilbert's constructive optimism with classical logic and bivalence. 21 So the evidence is mixed. But if a general assertability reading does rehabilitate Brouwer (perhaps by force) then it can do the same for Kant. There are powerful reasons to accept this reading. If Tl can be some version of assertabilism which does yield an intuitionistic logic, that would account for the idealist's moves in the antinomy. Since assertability is ontologically neutral, there would be no problem with phenomenalism. Moreover we could begin to make sense of the realist's arguments in the antinomy. He would be a constructive optimist - the sort who demands constructive verification for his judgment but who confidently assumes that he can always get it. That will by why he adoprs classical logic, but also connects measurability to truth. If we do read Kant this way, then we will have to understand intuitions (empirical or mathematical) as units of evidence and not as the building blocks of objects. Indeed, intuitions would be the special sort of evidence demanded by synthetic atomic (or at least singular) judgments. [Pt is
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assertable in an evidential state, just in case that state contains an intuition of t which is also a P -experience.] So we would have to speak of intuitability de dicto and to think of Kant's elaborate apparatus of faculties as a theory about different varieties of evidence. 22 But, once again, even granting that Kant's transcendental idealism is an assertabilism the main thing we have to do is find some internal cues to place it at a position on Table III which generates an intuitionistic logic. I think there are some textual hints which do this job. Thus, for instance on the question· of constructivism, Kant comes down clearly on the mild side: The postulate bearing on the knowledge of things as actual, does not indeed demand immediate perception (therefore sensation of which we are conscious) of the object whose existence is to be mown. What we do, however, require is the connection of the object with some actual perception in accordance with the analogies of experience ... (A255/B272)
The "actuality" Kant is talking about here is simply the existence of empirical objects. He is admitting that even if I don't strike oil, still I tuight have enough geological knowledge to read signs which will allow me to assert right now that there's oil under my yard. Much the same holds for unobservable empirical objects (distant stars. small particles, etc.) about which we have enough scientific evidence to assert their existence. I think on the question of the long (peircian) vs. short (Dummettian) semantic view, Kant is equally clear: To call an appearance a real thing prior to our perceiving it, either means that in the advance of experience we must meet with such a perception, or it means nothing at all ... (A493/B521)
Here he is admitting nonverified truths, but ouly on the ground that they are eventually verified. He says similar things a bit later about historical truths as well. This is a Peircian position. Now, consulting Table III, we see that if our Kantian idealist is indeed a mildly constructive Peircian, then all our logical questions come down to the issue of constructive optimism. If he is an optimist (and believes that all empirical questions can ultimately be answered), then it's classical logic and bivalence for him. If not, then not. Here too Kant's view (qua transcendental idealist) is loud and clearly announced (at least for empirical languages): In the explanation of natural appearances much must remain uncertain. and many questions insoluble, because what we know of nature is by no means sufficient. in all cases, to account for what has to be explained... (A477 /B505)
There's no optimism here. So for Lr it is intuitionistic logic and irs atrendant complications. Thus, if we accept the assertabilist reading of Kant, and if
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Dummett and Heyting are right about Brouwer, then we do find in Kant a philosophical theme regarding empirical discourse, whose logical consequences >1-TO developed by the inmitionists for mathematical discourse. That, I think, is the better theory of what Brouwer borrowed from Kant. But there is one important difference. On this interpretation the application of Kant's general views to the case of mathematics will not Jead him to be a
These are not isolated passages. They represent a general Kantian theme: Proper rational sciences - disciplines concerned ultimately with the nature of our representations - are legitimately optimistic. The issue comes down in the end to a matter of control. In empirical science the material out of which we fashion our evidence is sensory. And that means we must depend on passively receiving it. We may try to search for the needed intuitions and put ourselves in position to receive them. But there is no guarantee that such a search will succeed or will even terminate. In the rational sciences the material is within us. So. as Kant says, "it is impossible to plead unavoidable ignorance." (A477/B505). We must take care to note that it is only the internal (spontaneous) origin of .its evidence that justifies, for Kant, this sanguine attitude towards the rational sciences. Forgetting that is precisely the realist's error in the antinomy: He wants to combine the optimism of pure reason with a constructivism based on empirical inmitions. That is the combination which Kant drives ab absurdum. But retwning to our case of mathematics we can now see that when we combine Kant's generally far-sighted (peircian) assertabilism, and his overall mild constructivism, with the optimism generated by mathematical intuition, we have the urunistakable recipe for a bivalent language and a full classical logic! I would conclude that while he might be an intuitionist for empirical science, Kant would simultaneously advocate classical logic for mathematical science. [12] That's the conclusion I announced at the outset. But if you look at it for a moment - it may seem to you that I've defeated my own purpose. Think all the way back to my solution of the antinomy about the age of the universe. I said that Kant advocates the "regulative" infinity of the universe in the sense of formula (3). This means in our current terminology that at an evidential state - at any point in which we have documented some finite part of the past - we are enjoined to search further for even more remote past durations. Moreover, we are guaranteed that this search will be fruitful. We will be able to document that past. (That is, the Kripke model will have further nodes.) This guarantee is provided by "transcendenral philosophy" - it follows from the conditions of the possibility of experience. So it is legitimate. But then, we ought to combine that guarantee with Kant's mild constructivism to warrant the claim right now that these remote (not yet documented) durations do exist. I know this reasoning is quite general so I find myself in the following simation: I can imagine myself given a duration x (at some evidential state) - and
proto ~ intuitionist in that field. II
Two facts are clear about Kant' views on the "relevant evidence" for mathematical statements. (a) He does say that we construct our mathematical inmitions, and (b) mathematical activity is, for him, purely intellecmal. (It belongs to the faculty of reason.) There's considerable controversy about how to interpret Kant's claims that the mathematician "constructs his objects in pure intuition". All I need say now is that this is not the "strict constructivism" described before. (Kant's not saying that I have to have constructed an object with the property P before I can assert 3xPx.) I think what we have here is simply a combination of his general view that synthetic atomic claims must rest on intuitions, with his special theory that mathematical (pure) intuitions are wholly the product of the mind. Whether or not pure intuitions are the sort of things we can experience, and whether or not this experience has some sensory element, one thing is certain. Pure intuitions are not passive or "receptive." They are, as Kant sometimes says, "spontaneous" mental representations. They stem from the mind alone. That above all distinguishes them from the empirical intuitions
needed for ordinary science. Other disciplines which share this penchant for internally, or spontaneously generated bits of evidence include ethics and the so called "transcendental philosophy," which studies the conditions of possible experience. Now according to Kant, these disciplines enjoy a certain epistemic advantage. Since their subject matter is internal to 'the mind, it is, he says, more readily known, . .. every question arising within their domain should be completely answerable in terms of what is known, inasmuch as the answer must issue from the same sources from which the question proceeds. (A476/B504) 1~1l
II",·
You couldn't ask for a clearer statement of the optimistic attitude fostered by these disciplines. But if you do, Kant himself provides it: It is not so extraordinary as at first seems the case that a science should be in a position to demand and expect none but assured answers to all the questions within its domain, although up to the present they have perhaps not been found. In addition to transcendental philosophy there are two pure rational sciences ... , pure mathematics and pure ethics. (A480!B508)
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I ask about the existence of a prior state (i.e., I ask about 3y BIy, x)). But then recalling the reasoning just completed - I confidently assert 3y BIy, x). Is this not a general procedure? So can I not say Vx3y BIy, x)? Precisely the outlawed infinity of formula (2). I've brought back the the dirty bath water with the baby! The answer in brief is that this bit of reasoning will not suffice to justify the existential claims in question. Look back at the passage at (A255/B272) I quoted from the Postulates in order to support my initial claim that Kant is a mild constructivist in general. Notice that it is causal information (an empirical causal law) that counts as a guarantee of existence here. My point: That sort of evidence can count, but evidence from pure reason can't count. What we have here is what Kant (in , the "Amphiboly") calls the "Transcendental Location of Concepts" (A268/B234), the association of concepts with specific faculties. I would interpret this as a classification of concepts according to the faculty which may be invoked to provide evidence for jndgments in which that concept occurs. The notion that I have symbolized by "B" is an empirical one. It requires empirical evidence, empirical intuitions, provided by the faculty of sensibility. To be sure the faculty of reason, does govern some concepts and judgments which have empirical import. It does for instance structure the arena for empirical searches 24 (In the jargon I employed above we can say that it does build the model structures which we use to depict our empirical evidential situations.) Thus is guides our empirical searches for actual objects. But that alone doesn't allow it to get inside and provide the intuitions or connections which ground that "actually." This, I think is the heart of Kant's famous "crit-
pretations can have no solid ground to stand on. This accusation, in the present context, really has three prongs: It is addressed against
- ical" turn. 5. ANACHRONISM
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[13] If we do read Kant in this general assertabilist way, and if Dummett is right about the origins of Brouwer's intuitionism, then there certainly is a profound debt there. (Indeed, we would also have a theory about the origins of some of Hilbert's doctrines as well.) To be sure many details have to be filled in and questions answered before this becomes a full fledged theory of Kant. But let me close by addressing (briefly) just one of them. The question of anachronism, that I promised at the start. The point of this accusation is that Brouwer was writing in a scientific climate so foreign to Kant that, as history, my proposed Brouwerian inter-
(al My use of formal langnages (like L T ) to express empirical and philosophical claims for Kant. (b) My use in particular of a full first order language with its quantifiers and multiple quantifiers; and (c) The general assertabilist premise as a reading of Kant's notion of truth. [l4] I'll speak to these in order. Regarding the first, clearly there is no issue here of LT as a formal syntax. That device is as benign as using English to 'describe Kant's views - so long as we don't falsely ascribe any implicit meanings or attitudes hidden in the formalism. The real question is the autonomy of logic as a topic neutral science. It is ultimately to emphasize that notion that formal languages came into their own. Two comments are in order. First, this question of autonomy was itself a matter of heated dispute between Brouwer and Hilbert themselves. Hilbert was quite taken with the autonomy of logic. Brouwer denied the autonomy of logic, and as a consequence he disparaged the widespread formalization of mathematical theories. But this Brouwerian view has not deterred his students; and the formal study of intuitionism is the order of the day, among intuitionists and nonintuitionists alike. And so at the very least we might say that whatever success these methods have in expressing Brouwer's ideas they can have for Kant as well. But in fact (and this is my second comment), on this dispute about the autonomy of logic Kant stands closer to Hilbert. Kant was perfectly at home with the idea of a theory of judgments which is topic neutral. The unschematized categories and his whole notion of general logic are two levels of abstraction which fit into this general way of talking. Kant wasn't as clear on this as Russell and Hilbert, but there is no great distortion in using their devices to illuminate his thinking. [15] Tuming to the second accusation: Really the force behind this accusation is Russell. And really the force behind Russell's objection is his conception of infinity. Russell was very much concerned with the question of how our finite powers of thought and imagination can properly grasp the mathematical infinite. He came to believe that the V3 quantifier combination does this job, and captures the notion of infinity in a way that Kant (who had no quantifiers) could never achieve.
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Russell, I think, was quite right that his work does crystallize a notion of infinity which is thoroughly unKantian, a notion based ultimately on the idea of an arbitrary (perhaps even undescribable) function, This is essentially a set theoretic notion, and it should be contrasted with the idea which underlies Kant's conception of infinity. That is the idea of a continued iteration of some simple process, like describing or dividing a line segment, or like generating a sequence of events and their predecessors. But Russell was wrong in his implication that this is an illegitimate (or contradictory) notion of infinity, It is in fact Brouwer's notion of infinity, (It was captured at first in his claims about the "primary intuition of the continuum," and ultimately refined in his theory of infinitely proceeding sequences.)25 Brouwer and Kant were concerned with precisely the same question as Russell, how can a finite intelligence deal meaningfully with the infinite. And they came up with the idea of a sequential process underlying both temporal (and for Kant at least) spacial infinity.26 Moreover, Brouwer and some of his followers have shown that many parts of mathematics can be built on this more dynamic conception of infinity. And Russell was also wrong in thinking that his static infinite has sale claim on the \13 quantifier sign design. To whatever extent we succeed in formalizing intuitionistic mathematics using the machinery of quantified logic to that extent we have expressed the Kantian' infinity in a first order language. If this is a reconstruction (or classical interpretation), then again if it works for Brouwer, it can work for Kant as welL Another way to put this: Brouwer differed with Hilbert not only on the autonomy of logic and language, but also on the notion of infinity. Hilbert did hold a more Russellian notion of infinity. Kant, we saw, looks more like a proto-Hilbertian on the first question. But on the second question, the dynamic vs. the static conception of infinity, Kant seems to be a forerunner of Brouwer. This alone, however, will not lead him to chuck classical logic for mathematics; because, we have seen, of his Hilbert-like optimism. [16] Finally, on the question of assertabilism for Kant, let me say first of all that it is no more anachronistic to attribute an assertability theory of truth to Kant than a referential theory. Both the correspondence and the assertabilist notions of truth that we have today are expressed in terms unknown to Kant. Insofar as our project is to recreate Kant's actual state of mind, we might want to say that he was struggling (sometimes successfully, sometimes inexpertly) to express a view which we today have more clearly stated. Kant clearly thought that TI had "logical" upshots. And the best way to generate these sorts of logical consequences is by means of what we today would call a theory of truth. There is no anachronism in assuming that he had an inkling
of connections and consequences which we today can establish with formal rigor. But I myself prefer a less temporally chauvinistic attitude. The fact is that we are still concerned with many of the same questions that engaged Kant, in particular the questions surrounding infinity. But we are still far from the goal of full understanding. We don't yet have a satisfactory idealistic theory of truth, or even a fully satisfactory understanding of quantification. And so it seems to me that we can't help but profit from testing our ideas on Kant's problems.
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NOTES I See for instance Brouwer's remarks in Chapter 11 of his dissertation. Over de grondslagen der wiskunde. (Translated on pages 11-101 in L E. J. Brouwer, Collected Works v.I. North Holland,
Amsterdam, 1975. Hereafter I will make page references to this anthology when citing Brouwer's works. I will abbreviate it CW.) See also Brouwer's suggestion in "Intuitionism and Formalism" (1913, CW page 123) that his own position accepts Kant's views on the a priority of time, and rejects Kant's notions about space. 2 I have discussed these matters a bit more extensively in "Dancing to the Antinomy", "Transcendental Idealism and Causality," and "The Language of Appearances and Things in Themselves". "Dancing to the Antinomy" American Philosophical Quarterly, vol. 20, (1983), pages 81-94. 3 J. Bennett, Kant's Dialectic, Cambridge University Press, London, 1974, section 40. 4 B. Russell, Principles of Mathematics, Norton. See especially section 435. 5 It is tempting to confuse this notion of regulativity with Kant's notion of continuation in indeftnitum (A51O-11/B538-9). I fell into this mistake in "Dancing to the Antinomy" as did Bennet in Kant's Dialectic section 46. But the in infinitum/in indefinitum distinction concerns only the initial conditions on a given regressive series. Thus for instance the series of divisions discussed in the second antinomy is given regulatively, but nevertheless is continuable in
infinitum. 6 See my 'l'ranscendental Idealism and Causality" in Kant on Causality, Freedom and Objectivity, edited by R. Meerbote and W. Harper. University of Minnesota Press, Minneapolis, 1984. I have discussed some of these notions in ''Transcendental Idealism and Causality". 8 This is especially prominent in the A-Deduction, A 103 ff. 7
To know anything in space (for instance a line) I must draw it. and thus synthetically bring into being a detenninate combination of the given manifold.... Actually this is a theme that goes back in Kant's writing at least as far as the Prize Essay of 1763: A cone may signify elsewhere what it will: in mathematics it originates from the arbitrary representation of a right angled triangle rotated on one of its sides. The explanation obviously originates here, and in all other cases through synthesis. (AK II, 296)
CARL J. POSY
KANT'S MATHEMATICAL REALISM
\0 See-C. Parsons "Mathematical Intuition", Proceedings of the Aristotelian Society, vol. 80 (1979-80), pages 145-68.
22 I have treated this in more detail in ''The Language of Appearances and Things in Themselves," Synthese, vol. 47, (19S1), pages 313-352. 23 "For the natural appearances are objects which are given to us independently of our concepts, and the key to them lies not in us and our pure thinking, but outside us; and therefore in many cases, since the key is not to be found, an assured solution is not to be expected."
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... there are iterative complexes of sensations whose elements are permutable in point of time. Some of them are completely estranged from the subject. They are called things. For instance individuals, i.e., human bodies, the home body of the subject included. are things, ... Mathematics wmes into being when the two-ity created by a move of time is divested of all quality by the subject, and when the remaining empty form of the common sub-stratum of all
two-ities, as basic intuition of mathematics, is left to an unlimited unfolding creating new mathematical entities .... ("Consciousness, Philosophy and Mathematics" 1948, CW, p. 480.) See also Brouwer's remarks about the construction of mathematical objects at the end of Chapter I in his dissertation. 12 That's because some constructions may leave some properties of the constructed objects eternally undecided. 13 In general _ under the minimal deviation from the truth tabular meanings - if A is truthvalueless so too are -A and thus -A. Now take A as (Pt V -Pt) with Pt undecided. 14
Thus from the perception of the attracted iron filings we know of the existence of a magnetic matter pervading all bodies, although the constitution of our organs cuts us off from all immediate perception of this medium ... The grossn~ss of our senses does not in any way decide the form of possible experience in general. (A226/B273) (See also A52-2/B550). 15 See Bxl(n), B70-71, B274 and the "Appendix" to the Prolegemena. 16 See D. Hilbert, "Mathematische Probleme",Arch, d. Math. u. Phys. (3), 1901; and "UberDas Unendliche". Math., Ann., (96), 1926. 17 Consider for instance Dwnmett's argument from language learning (i.e., from the premise that assertability conditions are inevitably the novice's first contact with meaningful complete sentences; see Truth and Other Enigmas, Harvard, University Press, Cambridge MA:, 1978, pages 217ff.) TIris argument certainly encompasses the Peircian assertabilism. For the novice can't escape the preponderance of deferred justifications in our everyday discourse. Quite similarly, Dwnmett's Wittgensteinian argument (as found in Truth and Other Enigmas pages 216-27 and Elements of Intuitionism, Oxford, 1977, pages 360-89) can be adapted to the Peircian case as well. The Peu'cian, like the Dwnmettian, ties "understanding" a sentence to a grasp of its assertability conditions, and thus does link understanding with actual behavior. 18 It might seem strange that the optimist should reject classical logic, and even stranger that he should do this while advocating the long semantic view. But you must recall that logic per se is linked with asserting behavior rather than with beliefs about the future. Under the strict constructivist view we cannot use these guarantees to assert as yet unwitnessed existentials or undecided
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disjunctions. 19 See for instance the criticism of Hilbert towards the end of Chapter ill in Brouwer's dissertation, and the similar criticism almost fifty years later in "Historical Background, Principles and Methods of Intuitionism" (1952), CW pages 50S-15. 20 See Ober Definitionsbereiche von Funktionen" (1927). CW pages 390-405, and the editor's note (4) (CW page 603) to "The non-equivalence of the constructive and the negative order relation on the continuum" (1949) CW pages 495-96. 21 See for instance "Intuitionistiche Betrachtungen iiber den Formalismus" (192S) CW pages
409-14.
313
(A480-1/8508-9). 24 This is how I Wlderstand the Appendix to the "Dialectic", entitled ''The RegUlative
Employment of the Ideas of Pure Reason." See in particular A547-8TB674-6. 2S Compare the dissertation description of the "basic intuition" of continuity with the presentation of the continuum as a "spread" in subsequent writings, e.g., Begrundung der Mengenlehere unabhangig yom logischen San yom ausgeschlossenan Dritten, CW 150-221. 26 I w!luld distinguish here between notions like this iterative conception of infinity and notions which depend intrinsically on multiple quantifier shifts (e.g., the arithmetical hierarchy). The latter, I think, are wholly language bound, and do not correspond to any specific Kantian ideas. 27 Since the first appearance of this paper I have published four additional pieces which are relevant to the topics I have covered here. "Autonomy, Omniscience and the Ethical hnagination" (in Y. Yovel, ed. Kant's Practical Philosophy Reconsidered, Kluwer, 1989) explores the parallels in Kant's practical philosophy to the view that I have here called his "mathematical realism." "Where Have All the Objects GoneT' (The Southern Journal of Philosophy (1986) XXV, Supplement) touches on Kant's views about the nature of mathematical objects. This ontological theme as well as some phenomenological and semantic issues are taken up in "Mathematical as a Transcendental Science." (in D. FlZillesdal et. al. eds., Phenomenology and the Formal Sciences, University Presses of America, 1990). That paper also expands on the remarks in note (5) above concerning the "Second Antinomy." Finally, "Kant and ConceptuaJ Semantics" (Topoi, 10, 1991) qualifies my attribution to Kant of a modem assertabilism and considers his anti-realism in the context of his Leibnizian background.
GORDON G. BRITTAN
ALGEBRA AND INTUITION
A great deal of excellent work on Kant's philosophy of mathematics has been done in the recent past, much of it facilitated by a deep knowledge of the traditional criticisms made of his philosophy and by contemporary developments in logic and set theory. In particular, I have in mind papers by Michael Friedman, Jaakko Hintikka, Charles Parsons, Carl Posy, Manley Thompson, and J. Michael Young.' Any new discussion of the topics they address should be able to presuppose some acquaintance with them. Their net effect has been to make Kant's position, in its general outlines, philosophically credible as well as historically interesting, perhaps especially as concerns arithmetic. Moreover, with the exception of Posy, whose commentary on Kant takes place against a background provided by Brouwer and the Dutch Inmitionists, there is a surprising degree of consensus among them. There remain important differences, of course, and a great deal of disagreement concerning the details. But in this paper I will be more interested in the similarities, and will in any case spend an inadequate amount of time on the details. My approach will be to isolate one or two of the main problems that Kant's philosophy of mathematics is designed to solve, relate them to a central theme, that there are deep difficulties with the "inferenrial" interpretation shared by almost all of the commentators mentioned, and draw more attention than is usual to his various remarks about algebra These remarks serve to clarify the concept of "consttuctibility" on which his position turns, and to underline some of the difficulties to which its analysis leads. They also serve to undennine the twin tendencies among most of the above-mentioned commentators to ascribe "symbolic consttuctions" to arithmetic, identifying arithmetic with algebra in the process, and to logic (variously understood), assimilating logic to mathematics. Both of these tendencies, as I hope to show, are mistaken. In my view, Kant is working his way to a rather modern and abstract conception of algebra as consisting of sets on which certain iterable operations are defined, a fact that demonstrates his originality and insight while it explains his obscurity.2 315 Carll. Posy (ed.), Kant's Philosophy o/Mathematics, 315-339. rr"l laO? KIr""'Ar-
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There are several themes running through all of Kant's work. A main one is that a merely conceptual determination of the objects of experience is never adequate; what he calls "intuitions" (Anschauungen) are also required. That the objects of our experience are determinable in a variety of ways is a fact. That they must be determined in certain of these ways follows from the argument of the Transcendental Deduction: application of the concept of the self as subject of experience necessitates, among other things, that the objects of that experience are spatial, enduring, and causally connected. Since such determination of objects is both "intuitive" and necessary, it is, in Kant's own vocabulary, synthetic a priori. What Kant adds by way of explanation is that this is possible only if the determination involves both a passive content, contributed "by the world," and an active form, contributed "by us." Thus the first half of the Critique of Pure Reason. In the second half, Kant shows why and in what ways the adequate determination of other types of so-called "transcendent" objects is not possible, and hence that we cannot have anything amounting to knowledge of them. Now as I understand Kant's strategy in this connection, it is to begin with what might appear to be the most problematic case for his thesis, the case of mathematics. For the objects of mathemarics, in his view, are completely determined; we have knowledge about them, indeed a priori knowledge of the most secure kind. What he needs to show is that even in the case.of mathematics a merely conceptual determination is not complete. Even mathematics requires (sensible) intuitions for an adequate determination of its objects, hence even mathematics is synthetic (as well as a priorz). Once convinced of this, we should more readily accept the general thesis that there is no (adequate) determination without (sensible) intuitions. Let me try to spel! this theme out in more detail, adding qualifications where important. First, Kant often talks about the determination of objects. But he also, and apparently interchangeably, talks about the verification and falsification of propositions. Thus, to determine an object (or type of object) a with respect to a property P is simply to verify or falsify the proposition "a is P." We can always replace talk about objects and properties with talk about the trnthvalues of propositions. This is iruportant for the following reason. It is unclear whether Kant has a view concerning the role and status of "mathematical objects" per se, still less whether they exist in any sense of the word. 3 It is arguably the case that he thinks mathematical propositions are true of concrete and not merely abstract objects, if they are trne at aiL But he does think
mathematical propositions are trne and false and hence in this minimai sense we can talk about "mathematical objects" and their determination. To talk in this guarded way, it should be clear, is not to quantify over such "objects:" With this qualification well in mind, I will continue to talk, as does Kant, about "mathematical objects" (numbers, plane figures, et. al.) and their determination; among other things, it allows for their convenient comparison with "empirical objects" and ''transcendent objects." Second, Kant claims that every well-formed mathematical proposition is decidable; in this sense, all "mathematical objects" can be completely determined. Mathematics, he says, may ... demand and expect none but assured answers to all the questions within its domain, although up to the present they have perhaps not been found (CPR, A480JB508).
So far as I can see, Kant gives no argument for this claim and we, chastened by the Godel results, know that it is false. Yet it seems to figure as an important premise in his argument that methematical determination requires intuitions. I am not at all sure why he thought that it was trne (over and above the fact that everyone up to the present century thought so). Perhaps he believed that it follows from the doctrine of the spontaneity of ''pure intuitions" that we can always actively geuerate the ''pure'' intuitions required· for the further determination of mathematical objects, and do not have to wait passively for the "empirical" intuitions which might otherwise be unavail5 able, but this use of the doctrine, with a qualification to be noted in the third paragraph hence, would seem to have no support other than the claim that every well-formed mathematical proposition is decidable. Of course, Kant
invokes the doctrine of "pure intuition" for a number of reasons; but none of the other reasons, however good, guarantees the ready availability of the intuitions demanded. We use the metaphor of "maker's knowledge"6 in transcendental philosophy and pure ethics as well as in pure mathematics,7 to explain what has already been assumed, that every proposition within these particular domains is decidable. The situation with regard to mathematics in this respect contrasts sharply with the situation in the natoral sciences. In the explanation of natural appearances much must remain uncertain, and many questions
insoluble, because what we know of nature is by no means sufficient, in all cases, to account for wh~lt has to be explained (CPR, A477/B505).
The, surrounding context implies that what is at stake here is the receptivity of the "empitical" intuitions on the basis of which empirical objects are determined. Just as the spontaneity of "pure" intuitions guarantees the decidability
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of mathematical propositions, so the receptivity of "empirical" intuitions, and the corollary that we must wait for "the world" to provide them, precludes our knowing in advance when and if they will occur. 8 At the same time, the complete determination of empirical objects, the objects of experience, is not required by the argument of the Transcendental Deduction, although it
remains an ideal of reason. But Kant also suggests a rather different line of argument for the contrast between mathematics and the natural sciences as concerns their respective decidability.9 Our experience of an object is never completely determinate
because to know a thing completely, we must know every possible [predicate], and must detennine it thereby, either affirmatively or negatively_ The complete determination is thus a concept. which, in its totality. can never be exhibited in concreto (CPR, A573/B610).
An empirical object can never be completely determined. The question is, why not? It is an aspect of Kant's general thesis that we can know in abstracto of any object that it has one or the other of two contradictory predicates or properties, but at least in a great number of cases we can kuow which of these two it has in fact only when the object is given in concreto, in intuition. Only then can we decide, on the basis of an "empirical" intuition, whether it is P or not-Po Now according to Kant, an object given in concreto is always given as a spatial-temporal particular. The further determination of it is thus always with respect to the forms of space and time. But then an object cannot be detennined with respect to every possible predicate, because at least some predicates (in Thompson's example, having an immortal soul) do not apply to spatial-temporal particulars. A completely determinate object would be possible only if some of our intuitions were "intellectual," but again according to Kant, such intuitions are, at least for us humans, impossible. iO This line of argument suggests a way in which Kant's claim concetuing the decidability of mathematical propositions might be supported. If we held that the incompleteness of empirical objects resulted solely from the fact that some among their possible properties could not be had by spatial-temporal particulars, then mathematical objects might be held to be completely detenninate in the sense that all of their possible properties were, appropriately, spatial temporal.ll This suggestion receives some small additional support from the fact that in the passage quoted earlier at A480/B508, Kant is careful to say that pure mathematics can answer all questions within its domain; within its domain, possible properties do not outrun those which in principle it is possible to ascribe on the basis of spatial-temporal intuitions, "pure" or otherwise.
Having clarified, if not also secured, its first premise, we can now outline Kant's master argument for the claim that the determination of mathematical objects (the decidability of mathematical propositions) requires intuitions. 1. Mathematical objects are completely detenninable (the decidability thesis).12 2. Concepts do not completely detennine mathematical objects (the conceptual under-detennination thesis).l3 3. Either concepts or intuitions determine objects (the framework thesis). 4. Therefore, intuitions are required to complete the detennination of mathematical objects. Of course, Kant does not quite leave it at this. He makes two important additions to the argument, both having to do with the elaboration of its third premise. First, there is a subsidiary argument that goes roughly as follows: (i) All intuitions are either sensible or intellectual (spatial-temporal arnot). (ii) But we humans (i.e., beings endowed with our perceptual abilities and conceptual capacities) are not capable of intellectual intuitions. (iii) Therefore, all our intuitions are sensible. (iv) Therefore, the intuitions required to complete the determination of mathematical objects are sensible. This argument, set out in the Transcendental Aesthetic, has nothing specifically to do with mathematics, although it compels Kant to provide an elaboration of its corollary. Thus, second, although it is one of Kant's framework principles that all representations are either concepts or intuitions, and hence that there is no other way in which objects can be determined, he also thinks that one must provide an explanation of just how sensible intuitions, in the cases of arithmetic and geometry, are required, of just how they serve to complete the determination of matllematical objects. We might call this the "intuitivity thesis." It is that sensible intuitions are required, in ways that can be spelled oul in detail, to determine mathematical objects. 'JYpically it is supported by adducing paradigmatic examples. There are two separate parts to Kant's case. On the one hand, the conceptual under-determination thesis mnst be established. On the other hand, the intuitivity thesis, and the kinds of explanations it involves, must be established. Any account of Kant's position which fails to do both is itself seriously incomplete, although it might be
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claimed (as certain commentators do implicitly) that either suffices to estab.lish the conclnsion of the master atgument, and the synthetic chatacter of
denied. IS Systems of non-Euclidean geometry ate logically consistent. But if the truth of the axioms of Euclidean geometry cannot be determined conceptually, then, on the assumptions that they ate true and that the concept/intuition distinction is exhaustive, an appeal to intuition must be made to detennine their truth. Intuition alone serves to pick out one atnong the various self-consistent geometries. There ate two further variations on this line of interpretation that should be mentioned. On one, it is pointed out that the axioms or first principles of a given mathematical theory typically under-detennine its objects. Consider the axioms for an elementary Euclidean geometry.19 These axioms chatacterize sets of objects, the various set-theoretical structures that satisfy them. But the axioms do not succeed in completely characterizing these sets; atnong other things, the sets that satisfy the axioms most closely resembling those that Kant had in mind differ in catdinality. But again, if the axioms do not succeed in completely characterizing the mathematical objects to which they apply, and all mathematical objects can be characterized completely, then appeal must be made to intuition. This variation has a straightforwatd application to arithmetic as well. We can take the Peano Axioms as characterizing the basic elements of arithmetic, numbers. That they under-detennine them, in the appropriate sense, follows from the well-known fact that very differeni set-theoretical structures satisfy them. On the other variation on this line of interpretation, it is pointed out that on Euclid's fonnulation of them, the axioms for elementary geometry ate incomplete; certain theorems cannot be proved on their basis. In the case of arithmetic, an even stronger result can be demonstrated: elementary number theory is essentially incomplete in the sense that not all true mathematical propositions can be proved using a fiuite set of axioms (and finitary methods ofproof).2o There is much to be said for this line of interpretation, on either of its variations, patticulatly from a contemporary point of view. There is a cleat sense in which Euclid's axioms for geometry don't, and any set of axioms for arithmetic can't, completely determine the objects within their domains. On the assumption of the decidability and framework theses, appeal to intuition must be made. But there is also a great deal to be said against this line of interpretation.21 In the first place, Kant does not have in mind anything like the modem picture of mathematical theories as deductive systems, theorems derived from well-defined sets of axioms. Although he does refer to the axioms (more
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It is not possible to survey all of the attempts to establish one or the other of the conceptual under-detennination and intuitivity theses. I will look at two already well established in the literature, more to clarify the theses and the sense of "detennination" at stake than anything else, and the "calculational" variation on one of them. We should then be in a position to exatnine Kant's various rematks on algebra and to detennine their bearing. We can begin by effecting a preliminary classification of attempts to establish the two theses on the assumption that, for Kant, mathematical proposi15 tions can be decided on the basis of something like a deductive atgument. Eventually this assumption will be given up. Thus some commentators focus attention on the premises of such arguments - the axioms, basic propositions, or principles of arithmetic or geometry.16 Other commentators focus instead on the logic invoked to derive their conciusionsP On one line of interpretation, the axioms, basic propositions or principles do not serve to detennine completely the objects which they chatacterize. On the other line of interpretation, inferences from the~e as premises. using the monadic or Aristotelian quantification theory available to Kant, ate not capable of deciding all mathematical propositions within their domain, although using general quantification theory or other specifically mathematical fonns of reasoning would. On both, further detennination or decision requires intuitions (in senses of that tenn that have not yet been specified). The first line of interpretation is suggested most directly by the following well-known passage: For as it was found that all mat.i.ematical inferences proceed in accordance with the principle of contradiction (which the nature of all apodeictic certainty requires), it was supposed that the fundamental propositions [Grundsatzel could also be recognized from that principle. This is erroneouS. For a synthetic proposition can indeed be comprehended in accordance with the principle of contradiction, but only if another synthetic proposition is presupposed from which it can be derived, and never in itself (CPR, B14).
The appatent implication is that those (Leibniz and his followers) who correctly saw that the inference from axioms to theorems is logically valid, mistakeuly held that the axioms themselves ate conceptual truths. That they ate not conceptual truths follows from the fact that they can be consistently
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precisely, the postulates) of Euclidean geometry, he expressly denies that arithmetic has axioms,22 and suggests rather (as well shall see shortly) that arithmetical propositions are established on the basis of calculation, not argumentation. Nor does he make a sharp distinction between the syntactically characterized axioms and the set-theoretical structures that satisfy them on which this line of interpretation depends. This is to say that for all of its independent philosophical plausibility, this line of interpretation is seriously anachronistic. In the second place, and much more serious, the appeal to intuition on this line of interpretation is completely vacuous. It is part of Kant's master argument that an explanation be provided of precisely how intuitions further determine the basic objects of mathematics, numbers for example, and plane figures. But there is no such explanation forthcoming on this line. To precisely what is appeal made in picking out, e.g., elementaty Euclidean geometry from the set of logically consistent geometries? There is nothing in our visual experience, still less in the character of drawn figures, that would allow us to discriminate between them. The difficulty is even more marked in the case of arithmetic: how does "intuition" (whatever it ntight be) serve to decide for or against particular set-theoretical constructions of the natural numbers? Thus, while a good, albeit seriously anachronistic argument can be made on this first line of interpretation for the conceptual under-determination thesis, it does not provide us with the materials for understanding, let alone the reasons for accepting, the intuitivity thesis. It remains, indeed, something of a mystery on this line of interpretation how the further requisite deterntination of mathematical objects is to be carried out. The other interpretation focusses not on the axioms and first principles, but on the inferential procedures used. There are, in fact, a number of passages in which Kant emphasizes the character of the reasoning used in mathematics and connects it directly to the synthetic a priori status of mathematical propositions. For example, in the Critique of Pure Reason at B744-745:
line parallel to the opposite side of the triangle. and observes that he has thus obtained an external adjacent angle which is equal to an internal angle - and so 00. In this fashion, through a chain of inferences guided throughous by intuition. he arrives at a solution of the problem that is simultaneously fully evident and general (my italics).
Suppose a philosopher be given the concept of a triangle and he be left to find out, in his own way, what relation the sum of its angles bears to a right angle. He has nothing but the concept of a figure enclosed by three straight lines, along with the concept of just as many angles. However long he meditates on these concepts, he will never produce anything new. He can analyse and clarify the concept of a straight line or of an angle or of the number three, but he can never arrive at any properties not already contained in these concepts. Now let the geometer take up the question. He at once begins by constructing a triangle. Since he knows that the sum of two right angles is exactly equal to the sum of all the adjacent angles which can be constructed from a single point on a straight line, he prolongs one side of the triangle and obtains two adjacent angles which together equal two right angles. He then divides the external angle by drawing a
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Here it seems explicit that it is the inference, and not the axioms, that requires intuition. 23 Now what is it about mathematical reasoning that is intuitive? This line of interpretation proceeds somewhat as follows. 24 Kant's conception of "logic" was thoroughly Aristotelian, which is to say approximately eqnivalent to monadic quantitification theory. Now the striking thing about this theory is that monadic formulas always have finite realizations Or models, a fact which is closely linked with their decidability. Polyadic formulas, on the other hand, often have only denumerable models, as, for example, when existential are dependent on universal quantifiers, as in (x)(EyJ FxY" But mathematical reasoning involves, in a variety of ways, the introduction of an unlintited (or infinite) number of new individuals, in guaranteeing the closure of the basic arithmetical operations, for example, or in proofs which require the density (or continuity) of the Euclidean straight line. Io its essentially "finite" character, monadic quantification theory does not harbor the resources to prove all of the mathematical theorems that polyadic quantification theory does. Identifying conceptual determination with monadic provability, Kant saw that many theorems reqnired, in a sense to be indicated momentarily, an appeal to intuition. TIris is to put a complex view too simply. But the main point should be clear: monadic quantificatioo theory, together with the various axioms and basic propositions of geometry and arithmetic, does not serve to determine completely the objects at stake, and this in large part because the determining procedures needed are not invariably finitaty. Polyadic quantification theory, on tltis line of interpretation, thus belongs to mathematics. What is particularly appealing about this line is that, unlike the first, it ties the conceptual under-determination thesis to the intuitivity thesis in a very plausible way. The existence of the requisite number of points cannot be demonstrated from the traditional axioms of Euclidean geometry if we have no more than monadic quantification theory at our disposal; monadic formulas cannot ''force'' models that have enough objects in them. We must tum, instead, to their "construction,"25 which in this case involves the continued bifurcation of a line segment originally given in concreto, in intuition, as a spatial o~ect. 26 Similarly, the determination of particular numbers, as the result of employing the basic arithmetical operations, is a calculational
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procedure which presupposes the continued iterability of these operations. This calculational procedure, in so far as it takes time, is in a fundamentally Kantian sense "intuitive." Monadic methods do not fully determine mathematical objects; their complete determination requires an appeal to spatial objects and temporal activities, i.e., to what is apprehended under the forms of intuition. There are, I believe, a great variety of things wrong with this interpretation, despite its considerable merits.27 But we should confine our attention to those that deal most directly with the theme of detennination.28
procedures involved in the second line of interpretation. This emphasis, quite apart from the monadic/polyadic distinction to which Friedman ties it (for reasons to be spelled out in the next paragraph) is worthy of closer examination, particularly since it brings out certain crucial features of mathematical reasoning and sets the stage for an exaruination of Kant's remarks on algebra which, in my view, are incompatible with it. There are three corollaries to the "calculational" interpretation of Kant's position. One, arithmetic is primary, and a better case can be made for its "intuitive" character than for geometry.32 Two, calculational procedures are in a fundamental sense "symbolic" and thus there is, from Kant's point of view, no reason to distinguish very sharply between arithmetic and algebra.33 Three, since the procedures of quantification theory, monadic or polyadic, are in generally this same sense "symbolic," there is, again from Kant's point of view, no reason to distinguish very sharply between "logic" and mathematics, despite his intentions to the contrary.34 All three of these corollaries are, in my view, mistaken, in large part because they misconstrue the role of algebra and the allied concept of a "symbolic construction" in Kant's thought. Such at least is one burden of this paper. To the extent that the calculational view implies them, it must be rejected, and with it the most promising way in which the second line of interpretation of Kant's position has been put. What exactly is the calculational view? Emphases vary among the commentators listed, but this is perhaps the core of the view.35 Mathematics is, Kant says, the science of quantity or magnitude. Maguitudes are determined, by way of a construction, through the application of calculational procedures. The case of arithmetic is paradigm. To determine some sum, for example, is to calculate it, to successively iterate the operation of addition, a unit at a time, until the sum is reached. The procedure, successfully performed, both guarantees the "existence" of the required sum and verifies that it is correct. The possibility that a given magnitude can be constructed is thus no more than the possibility that a given operation can be repeated, and that the rules governing it be followed. There are two "intuitive" aspects to this sort of activity. One is that it takes time; on this interpretation, the metaphor of "construction" is taken rather literally, the determination of maguitude is sometlting one does. Whether the result is to be taken as a "making" or a "finding," it involves a progressive enumeration that is temporal, hence intuitive. The other aspect is that traditional monadic logic does not have the conceptual resources to represent it. 36 To represent a progression. in fact, requires formulating a set of axioms for arithmetic or using set-theoretical concepts to define the natural numbers. Friedman emphasizes that mathematics requires, notably
In the first place, there is no reason in Kant's text (over and above the presumed correctuess of this line of interpretation) to think that he had some insight into the fact that any consistent monadic formula has a fiuite model; this result was first demonstrated, after all, in the 20th century. Nor is there any suggestion in the more mathematically sophisticated Leibniz, so far as I can see, that the use of "transfinite" methods goes hand in hand with the adoption of a polyadic notation; in common with a long tradition, he thought (incorrectly) that polyadic formulas were reducible to monadic ones. Nor, from a contemporary point of view, does Kant's presumed assimilation of polyadic quantification theory to mathematics do him much credit, for he assumes (as we have seen) that mathematics is generally decidable, whereas we know that polyadic quantification theory is not. This is obviously not intended as a criticism of Kant; the point is simply that if he had the kind of insight with which he is credited into the model-theoretic character of Aristotelian logic, then there is perhaps some reason to think that he would have understood that polyadic or general quantification theory is undecidable (since there is no effective way to generate counter-examples in models that contain at least a denumerable number of objects).29 In the second place, the distinction between monadic and polyadic quantification theory does not square very well with the general characterization Kant gives of "logic." There is, of course, no doubt that by "logic" he intended the traditional Aristotelian, essentially monadic, theory. But his characterization of logic as "empty," "the mere form of knowledge," etc.,30 includes (universally free) polyadic quantification theory plus identity as well. The latter theory is in the same precise sense "empty" in that its theorems hold in or of all models, including the model that contains no objects. 31 Although we can, trivially, prove a great many more propositions using a polyadic notation, and in this sense polyadic procedures, the objects that realize formulas in this notation are not more precisely "determined." These criticisms aside, I want to follow up the emphasis on calculational
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in the detennination of irrational magnitudes, the unlimited iteration of particular operations and the indefinite extendibility of the number series on which it depends. Now construction on the calculational view is "symbolic" in this sense.37 Numbers are represented by numerals, sensible tokens which are in this sense -themselves "intuitive." The numerals function as names or "symbolic constructions" of numbers, but they also, and more importantly, model the structure of the number series.
mentary, rejecting the majority tradition, also tends to suggest that a better case can be made for the synthetic a priori status of arithmetical than of geometrical propositions.
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In the system using Arabic nwnerals and base ten, for example, the nwneral '12' serves as a name of the number twelve, but the sequence of numerals from' 1' through' 12' also provides a model of what we take to be the structure of the corresponding numbers, since it has an initial element and a successor relation.... If we produce sensible tokens of the numerals' I' through '12' on paper or chalkboard, we have a model of what we take to be the structure of the nwnbers one through twelve. Since the perceptible tokens can be used merely as representations of an abstract structure, replaceable by any other instance of that structure, we can use them to verify them, ... , to verify arithmetical propositions.38
In fact, of course, we operate by and large with numerals when we construct magnitudes, particularly when the magnitudes are so large that we run out of fingers and toes, relying on familiar properties of the base ten system of representing nwnbers. This calculation with numerals, which are no more than symbolic constructions of numbers, does not simply abridge the otherwise laborious process of adding, subtracting, etc., numbers unit by unit; it affords a verification of its result because, once again, there is a fundamental isomorphism between the numerals and the structure of the number series. Arithmetic thus conceived consists not so much of a body of general truths or axioms as of calculational techniques, methods for finding magnitudes or solving equations, although these tecltniques rely on the general properties of both numbers and numerals already mentioned. Arithmetic is therefore prior to geometry in at least two senses. In the first place, it is more general. It has to do with the calculation of any magnitudes and does not have to do, in particular, with spatial (or, for that matter, temporal) objects. This generality derives from the fact that arithmetical constructions are purely symbolic, whereas geometrical constructions are invariably ostensive. In the second place, the iteration of what are basically arithmetic operations on which calculation depends is presupposed by geometry as well, for example in gnaranteeing both the indefinite extendibility and bifurcation of given line segments. Geometry, one might say, is applied arithmetic, although as already indicated, on the present line of interpretation no very sharp line can be drawn between pure and applied mathematics. Recent com-
4
As mentioned, I think that Kant's views on algebra tbrow a great deal of light on his philosophy of mathematics and his use of the concept of constructibility. I also think that his views on algebra are incompatible with certain aspects of the "calculational" interpretation of his position, although in other respects they support and elaborate it. I begin with a much-commented passage at A716/B744 of the Critique of Pure Reason: But mathematics does not only construct magnitudes (quanta) as in geometry; it also constructs magnitude as such (quantitas), as in algebra. In this it abstracts completely from the properties of the object that is to be thought in tenns of such a concept of magnitude. It then chooses a certain notation for all constructions of magnitude as such (numbers). that is, for addition, subtraction, extraction of roots, etc, Once it has adopted a notation for the general concept of magnitudes so far as their different relations are com:erned, it exhibits in intuition, in accordance with certain universal rules, all the various operations through which the magnitudes are produced and modified. When, for instance, one magnitude is to ~ divided -by another, their symbols are placed together in accordance with the sign for division, and similarly in other processes; and thus in algebra, by means of a symbolic construction, just as in geometry by means of an ostensive construction (the geometrical construction of the objects themselves), we succeed in arriving at results which discursive knowledge could never have reached by means of mere concepts (my italics).
This passage is problematic in several respects. But it is a key to understanding Kant's position. As is invariably the case with Kant, it turns on distinctions, here between magnitudes (quanta) and magnitude as such (quantitas) and between ostensive and symbolic constructions. The distinctions are clearly and quite closely related. The paradigm of an ostensive construction, at least in this passage, is provided by geometry, where the postulates and definitions afford the "synthetic" means to carry out the construction of particular figures which as such are "ostended" and the eventual objects of (visual) perception. But it is important to note that for Kant arithmetic as well is "ostensive."39 As much is indicated by the passages at B 15-16 and elsewhere that indicate how numbers can be "ostended" by fingers, strokes on a page, and so on, all of them spatial representations. In both cases, arithmetical as well as geometrical, the ostension is taken to provide us with a kind of evidence on the basis of which the
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corresponding claim (that 7 + 5 = 12, that the sum of the interior angles of a triangle is equal to two right angles) may be verified. The ostension, we might say, controls the claim.40 Now it is also true that Kant distinguishes between arithmetic and general arithmetic and couples the latter with algebra.41 The evident parallel between the two types of arithmetic is that they involve the same combinatorial operations. 42 The difference between them, as Kant puts it in #2 of the First Reflection of the Enquiry Concerning the Principles of Natural Theology and Ethics is that general arithmetic has to do with indeterminate magnitudes and arithmetic proper with numbers ("where the ratio of the magnitude to unity is determinate"). Michael Friedman suggests that "the arithmetic of numbers is concerned only with rational magnitudes, whereas general arithmetic or algebra is also concerned with irrational or incommensurable magnitudes."43 This is fine as far as it goes, but it does not go far enough. First, the reason why arithmetic proper is concerned only with rational magnitudes is that only rational magnitudes can, in the appropriate arithmetical sense, be "ostended." There are no determinate magnitudes, no homogeneous combinations of ostensive representations of numbers, corresponding to the irrationals. Second, general arithmetic has to do not simply with the "construction" of numbers, whether rational or not, but with "magnitude" per se, including the definition of complex curves. Kant has a much more general concept of the "indeterminate" than Friedman allows. For him, algebra is a logic of types, not of individuals, and applies as much to geometry as to arithmetic.44 Third, all "determinate" mathematical objects are in certain respects qualitative, "... for instance, the difference between lines and surfaces, as spaces of different quality, and with the continuity of extension as one of its qualities ..." (CPR, A715/B 743). But algebra considers these objects in their merely quantitative guise, as magnitudes as such, quantitas rather than quanta.45 Moreover, the qualitative/quantitative contrast which is here used to characterize algebra would seem to have little to do with the rational{rrrational contrast that Friedman wants to take as primary. But the most important and immediate point is simply this: arithmetic proper has an obviously "intuitive" foundation, ostensive constructions on the basis of which sample arithmetical claims may be verified, algebra or general arithmetic does not. 46 The subject matter, in the sense already introduced, "controls" the adequacy of its descriptions. In the case of algebra, on the other hand, the "subject matter" does not exist unless and until various combinatorial operations have been performed, i.e., the quantities involved are reached by way of an "analytical" characterization.47 Given that they cannot in general be "ostended,"
what other sorts of controls might there be on their admission? Or, in the absence of appropriate ostensions, do we simply dismiss them as "empty" and "unreal"?48 This is a problem for Kant, which the tendency to conflate arithmetic and algebra as part of the calculational interpretation simply sidesteps. There is another closely connected problem: given that algebra is, from one point of view, a system of rules (and not a set of determinate objects) licensing various combinatorial operations, what in tum licenses or legitiruizes the rules? These same points can be reinforced from a different direction and in an alternate vocabulary. Ostensive constructions can be identified with the use of "synthetic" methods, symbolic constructions can be identified with the use of "analytic" methods, as these terms were commonly understood in the geometrical tradition.49 The question then is this: does every analytic ("symbolic") construction have to be backed, as Descartes, for example, insisted, by a synthetic ("ostensive") proof, and, if not, what guarantees the adequacy of analytic constructions? Kant responds directly to the first of these questions in the controversy with Eberhard. The situation is straightforward with respect to the tradition of Greek geometry, he indicates, for on that tradition every proof given is synthetic and hence "constructive." Thus, one of Kant's favorite examples. . Apollonius first constructs the concept of a cone, i.e., he exhibits it a priori in intuition (this is the first operation by means of which the geometer presents in advance the objective reality of the concept). He cuts it according to a certain rule, e.g., parallel with a side of the triangle which cuts the base of the cone at right angles by its summit, and establishes a priori in intuition the attributes of the curved line produced by this cut on the surface of the cone. Thus, he extracts a concept of the relations in which its ordinates stand to the parameter, which concept, in this case, the parabola, is thereby given a priori in intuition. Consequently the objective reality of this concept, Le., the possibility of the thing with these properties, can be proved in no other way than by providing the corresponding intuition. 5o
But Kant is also very much aware that since the 16th century and the work of Vieta and Descartes, and particularly in the work of Newton, analytic methods had been generally adopted.51 So with this fact in mind, he continues: One could ... address to the modern geometers a reproach of the following nature: not that they derive the properties of a curved line without first being assured of the possibility of its object (for they are fully conscious of this together with the pure, merely schematic cOnStruction, and ~ they also bring in mechanical construction afterwards if it is necessary), but that they arbitrarily think for themselves such a line (e.g., the parabola through the fonnula ax = y2), and do not, according to the example of the ancient geometers, first bring it forth in a conic section. This would be more in accord with the elegance of geometry, an elegance in the name of which we are
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often advised not to completely forsake the synthetic method of the ancients for the analytic
have a special status. The point here, rather, seems to be that intuition is needed to verify the correct application of these rules, that these rules are used to construct maguitudes, and that therefore algebra, like geometry and arithmetic, is synthetic. 56 From this point of view, I think the case of algebra, much more clearly than the other two, demonstrates the fundamental correctness of the line of interpretation that emphasizes the "intuitive" character of the reasoning involved in mathematics, although it would be a tuistake, we have seen, to isolate its temporal character. Algebraic objects are conceptually under-determined in a much more obvious way than numbers or plane figures. It is the reasoning about them that is guided throughout by intuition,
method which is so rich in inventions. 52
With some Wlcertainty, I read this passage as follows. Contemporary geometers, despite their nse of analytic methods, are always in a position to demonstrate the objective reality (real possibility) of the curves with which they work. But this does not require that every analytic construction be backed by a synthetic proof. That is, it would be more "elegant" to bring it forth in a classically geometrical construction, and we should not "completely forsake" the synthetic method of the ancients, but these are not required by the possibility of 1:.1e Curve analytically characterized. Undoubtedly Kant was aware of the fact that many of the curves thus characterized could not be brought forth by the conic sections or any of the other constructive methods available to the Greeks, or to Descartes for that matter. How, then, is the "possibility" of such curves to be established? The passage originally quoted at A7l6/B744 of the first Critique is, I think, even more explicit that geometrical diagrams do not have to be supplied for algebraic results in its sharp separation between ostensive and symbolic constructions. For the requirement that every analytic demonstration be backed by a traditional synthetic proof just is the requirement that all constructions be ostensive. At the same time, Kant wants .to provide some sort of founda~ tion for these algebraic results. He does it by introducing the notion of a symbolic construction that is in some sense "intuitive" and therefore "synthetic," in the process going well beyond what Descartes, and a fortiori the Greeks, had meant by "construction,'" "intuit," and "synthetic."S3 What we "intuit" are
the operations performed on certain otherwise unspecified "objects" by way of their symbolic representation. It is just because the results of the combinatorial operations in algebra cannot invariably be ostended that the control is placed not on the exhibition of objects but on the carrying out of rulegoverned operations. This leaves questions concerning the combinatorial operations themselves,
or the rules which license them, still open. That is to say, the symbols are manipulated according to certain "universal rules" and, I take it, the application of these rules is guided, or "controlled," by our intuition of the symbols (in the paradigm case, marks on paper) manipulated.54 But it is never made clear what the status of these "rules" - for addition, subtraction, multiplication, division, and the extraction of roots - is. He denies that arithmetic or
algebra have axioms in severa! places,55 nor would it help to solve the question of status if the universal rules were "axioms" for Kant for he never really says why the Wldoubted axioms (read "postulates") of Euclidean geometry
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the reasoning being in some sense constitutive. For the rest, whatever else
might be said about them, the rules and combinatorial operations they define are, at least for Kant, clearly a priori, although how intuitions, pure or otherwise, might guarantee their status is, so far as I can see, never indicated. But all the master argument requires is that their application be "intuitive" in some general sense.
A great deal more needs to be said, of course, about the notion of a "symbolic construction." For the moment, however, I want to underline two of its very genera! implications. The first is that whereas in geometry and arithmetic the objects (numbers and figures) ostended "control" the resulting claims about them, in algebra the claims, in the form of the symbolic results of the various combinatorial operations we perform, "control" the objects. An object is adntissible on this latter tack if it is constructed in the right sort of way. We have no independent access to the object, no ostension to which we can appeal: from one point of view this is just the difference, Kant rightly saw, between a variable and a constant. The other, related implication is that Kant is working his way towards the concept of a relational structure. What a symbolic equation "pictures," so to speak, is a relation between magnitudes
(as such, and otherwise unspecified). The "objects" are simply whatever satisfies the relational structure. Thus "algebra abstracts completely from the properties of the object that is to be thought in terms of such a concept of magnitude." But to say this is to say that the problem is no longer the "determination" of particular mathematical objects, but the determination of general relational structures. Algebraic objects remain relatively indeterminate. 57 Relational structures, in turn, are "determined" by being "bnilt up" in certain ways, that is, by being "constructed." The construction makes use of symbols, sensible tokens, and presupposes the iteration of the basic operations; to this extent, it is "intuitive." But it is maiuly guided by the combinatorial rules of construction. In moving to a more modem conception of
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algebra, Kant strains his doctrines of "determination through iutuition" to the limit.58
great deal of controversy iu the 17th and 18th centuries concerniug algebra and its foundation. 59 The general tendency was to say that its procedures, barely systematized at the time, were merely "heuristic;" they could be justified ouly on the basis of the fact that they "worked." On my suggestion, Kant provides something like this answer, although he prepares the way to it much more carefully by showing why a reduction of it to either arithmetic or geometry would be iuappropriate. Here as elsewhere his position is thoroughly "anti-reductionist." Second, Kant appears to waver on a central issue,
332
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In fact, I think that iu moviug to a more modem conception of algebra, Kant was forced to go beyond the doctrine of "determiuation through intuition" as it is illustrated and understood in the cases of geometry and arithmetic. Although this is pure speculation on my part, I suggest that reflection on the character of algebraic methods, and iu particular on the "indeterminate" character of algebraic variables was one source of his central philosophical problem, how to secure the "objectivity" of at least some of our judgments about experience.
The specifically algebraic problem, as I reconstructed it, was to "determine" the relational structures to which application of the combinatorial rules gives rise, in the absence of ostensive coustructions to which appeal might otherwise be made. Kant's solution, certaiuly nowhere explicit iu the text, is that these relational structures are "determined" by way of a direct application to the objects of our experience, that is, the objects that have been fully conceptualized in accord with the requiremeU(s of the argnment of tJie Transcendental Deduction. Algebra and algebraic results cannot be justified piecemeal. Rather, they are to be justified in so far as they are generally true of the objects of experience. Or, to put it much too simply, what guarantees the objective reality (real possibility) is not ostensive construction; what guarantees it is its wholesale application, the fact that it gives empirically verifiable results, that it "work." Algebra, iu this respect uulike arithmetic and geometry, is legitituized iu practice. Kant generalized this presumed reflection very roughly as follows. Our scientific claims about the world constitute a relational structure or series of
such structures. The "objects" postulated or presupposed by the truth of these claims are iu this sense under-determined. Their future determiuation, which as we saw earlier is completable but never complete, is by way of a rather global fitting of this structure to our experience (itself made possible by the application of the Categories). Algebra gives us the outline of a form of the world which is validated by its empirical applications. Here are two small pieces of evidence, not for the generalization, which is admittedly vague and intended as no more than suggestive, but for the claim about algebra. that demonstrating the objective reality of its "objects" requires the machinery of the transcendental philosophy. First, there was a
whether the mathematician can or cannot demonstrate the "real possibility"
of the objects with which he or she deals. In the Critique of Pure Reason, A223-4/B271, Kant writes: It does, indeed. seems as if the possibility of a triangle could be known from its'concept in and by itself ... , for we can, as a matter of fact, give it an object completely a priori, that is, we can construct it. But since this is only the fann of an object, it would remain a mere product of the imagination, and the possibility of its object would still be doubtful. To detennine its .possibility, something more is required, namely, that such a figure be thought under no conditions save those
upon which all objects of experience rest.
But iu a passage already quoted from his polemic against Eberhard, iu the course of which he announces that he wants to clarify his earlier discussion of "construction of concepts" in the Critique, K;mt writes: Consequently, the objective reality of this concept [of a parabola} Le., the possibility of a thing with these properties, can be proven in no other way than by providing the corresponding intuition ... ,60
a view very much reiuforced iu his letter to Reiuhold of May 19, 1789. The second passage suggests that the mathematician can demonstrate real possibility, by way of an ostensive construction of his or her concepts, while the first passage appears to deny it. I am not sure how these passages, and others like them, are to be reconciled. But it might be the case that in the former Kant has mathematics as generalized to iuclude algebra in mind. For algebra, unlike arithmetic and geometry, gives us the mere form of an object, quantity without quality. Quality, iu the form of experience, is hence needed to secure "reality." In the latter passage, Kant has geometry or arithmetic proper iu mind, for in these cases ostensive construction already includes the sort of possibility of perceptual experience that assures the "real possibility" of their objects. There is, however, no direct textual evidence for this reconciliation, and the passage at A223-4/B271 specifically mentions an ostensively constructible triangle. Fiually, despite the similar abstractness of the two discipliues, and the
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same sort of emphasis on combinatorial and transformational rules, algebra should be distinguished from "logic." For algebra has the kind of empirically verifiable applications to the objects of experience that logic, at least as Kant conceives it, can never have. 6J
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I See Friedman, "Kant's Theory of Geometry," reprinted in this volume, pp. 177-219, and "Kant on Concepts and Intuitions in the Mathematical Sciences," Synthese 84 (1990), pp. 213-257; Hintikka, the various essays on Kant reprinted in Logic. Language-Games, and Information (Clarendon Press, 1973) and Knowledge and the Known (Reidel, 1974), "On Kant's Notion of Intuition," in Penelhum and Macintosh. eds.• The First Critique (Wadsworth. 1969), "Kant's Theory of Mathematics Revisited," in Mohanty and Shehan, eds." Essays on Kant's Critique of Pure Reason (University of Oklahoma Press, 1982), and "Kant's Transcendental Method and His Theory of Mathematics." in this volume pp. 341-359; Parsons, "Kant's Philosophy of Arithmetic," reprinted with a Postscript in this volume pp. 43-79 and "Arithmetic and the Categories," in this volume pp. 135-158; Posy, "Kant's Mathematical Realism.," in this volume pp. 293-313 and "Mathematics as a Transcendental Science," in F~llesdal et. al. eds. Phenomenology and the Formal Sdences (University Presses of America. 1990); Thompson, "Singular Terms and Intuitions in Kant's Epistemology," this volume, pp. 81-107; Young, "Kant on the Construction of Arithmetical Concepts," Kant-Studien 73 (1982), pp. 17 -46, and "Construction, Schematism, and Imagination," in this volume, pp.159-175. 2 For more on this historical development, and Kant's place within it, see my "Algebra. Constructibility, and the Indetenninate," in Brittan, ed., Causality, Method, and Modality: essays in honor of jules Vuillemin (Kluwer, 1991). pp. 99-123. 3 In a crucial, albeit very difficult passage from the Methodology in the first Critique, Kant says: " ... in mathematical problems there is no question of ... existence at all, but only of the properties of the objects in themselves, [that is to say], solely in so far as these properties are connected with the concepts of objects" (A719JB747). 4 In "Singular Terms and Intuitions in Kant's Epistemology," Thompson argues persuasively that we cannot quantify over them. 5 See Posy, "Kant's Mathematical Realism," p. 307 6 See Hintikka, "Practical vs. Theoretical Reason - An Ambiguous Legacy," reprinted in Knowledge and the Known. 7 See the first Critique, A480!B508. 8 See Posy, "Brittanic and Kantian Objects," in den Ouden, ed., New Essays on Kant (Peter Lang, 1987), pp. 29-46. Posy argues that the contrast between the definability of mathematical concepts and the Wldefinabi1ity of empirical concepts, on which Kant insists (CPR, A728!B756ff.), is itself a corollary of the spontaneity/receptivity contrast. 9 See Thompson, "Singular Terms and Intuitions in Kant's Epistemology," pp. 86ff., whose accoWlt I follow closely. 10 Objects which cannot be given in concreto, under the forms of space and time, might be
ALGEBRA AND INTUITION
335
called ''transcendent'' objects. Examples are, of course, God and the souL Although a minimal conceptual detennination of these objects is possible, by way of certain analytic propositions (e.g., "God is omnipotent"), they cannot be further detennined by either "pure" or "empirical" intuitions. An object which cannot in principle be further determined is Dot properly an object of knowledge; the real possibility of its concept cannot be demonstrated. There is a kind of contin~ uum here: mathematical objects are completely determined, empirical objects are adequately determined, transcendent objects are inadequately determined, where "adequately" means "so as to meet the requirements of the Transcendental Deduction." 11 I realize that the consequent of this conditional needs a great deal more clarification and support, but its import should be apparent. 12 Notice that this premise is. independent of the requirements of the arguments of the Transcendental Deduction that an object of experience be adequately determined It is primarily this fact, I think, that allows mathematics to be discussed'first, in the Aesthetic, before the considerations central to the Analytic are advanced. 13 In the passage immediately preceding that quoted in footnote 3 at CPR, A719!B747, Kant says: "There is indeed a transcendental synthesis framed from concepts alone, a synthesis with which the philosopher is alone competent to deal; but it relates only to a thing in general, as defining the conditions Wlder which the perception of it can belong to possible experience." I take this to mean that a complete conceptual determination (one sense of "synthesis") of a "thing in general" is possible, but presumably every other object, things in particular, are conceptually under-determined. Of course, Kant intends 2. to illustrate and support the more general thesis, and not the other way around. 14 In Kant's Theory of Science I implied, mistakenly, that the conceptual Wlder-determinatiQn thesis entails the intuitivity thesis. Perhaps the converse holds, however, a fact that would rationalize the otherwise excessive attention paid to the explanations Kant offers in the case of such examples as " + 5 = 12" and "the sum of the interior angles of a triangle is equal to two right angles." Puzzling over these examples, without first having some grip on a general strategy, is futile and worse. 15 For present purposes it is Wlimportant whether he had anything like the modern concept of a mathematical proof in mind or how he understands his problematical distinction between demonstration and discursive proofs (CPR, A735!B763ff.). 16 They include, in more recent times: LoW. Beck, "Can Kant's Synthetic Judgme-nts Be Made Analytic?" Kant-Studien, 47 (1955), pp. 166-81; Gordon Brittan, Kant's Theory of Science (Princeton University Press, 1978), chapters 2 and 3; Ernst Cassirer, "Kant und die modeme Mathematik," Kant-Studien, 12 (1970), pp. 1-40; Gottfried Martin, Kant's Metaphysics and Theory of Science, trans. Lucas (Manchester University Press, 1955). 17 They include: E. W. Beth, "Uber Locke's 'Allgemeines Dreieck'," Kanr-Studien, 48 (195657), pp. 361-80; Bertrand Russell, The Principles of Mathematics (Cambridge University Press, 1904), especially #434; and the papers by Friedman, Hintikka, Parsons (who endorses this view for arithmetic), and YOWlg mentioned in the first footnote. 18 See the CPR, B268: "there is no contradiction in the concept of a figure which is enclosed within two straight lines, since the concepts of two straight lines and of their coming together contains no negation of a figure." .19 See Alfred Tarski's paper, "What is Elementary Geometry?" in Henkin, Suppes, and Tarski, ·eds., The Axiomatic Method (North-Holland, 1959). Depending on [he precise formulation of "elementary geometry" chosen, of course, different metamathematical results can be proved. 20 On this variation, the Godel results, far from showing that Kant is wrong about the
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decidability of arithmetic, show that Kant is right (as reconstructed) in denying that truth in mathematics can be identified with provability in a fonnal system; their truth must rest on extraconceptual or ''perceptual'' considerations. GOdeI himself seems to urge something like this possibility in "What is Cantor's Continuum Problem," reprinted in Benaceraff and Putnam, ed5., Philosophy o/Mathematics: Selected Readings (Prentice-Hall, 1964). 21 All of it in the two articles by Friedman cited above. Despite my many reservations about his positive account of the issues involved, he has devastated attributing this sort of conceptual under-deterrnmation thesis to Kant. 22 First Critique, Al 64JB204. 23 Both Friedman, "Kant on Concepts and Intuitions in the Mathematical Sciences," and Hintikka, "Kant's Theory of Mathematics Revisited," try in interesting ways to reconcile this passage with that quoted earlier at B 14. Perhaps it is enough to regard these passages as supporting the conceptual under-detetmination and intuitivity theses respectively (the passage at Bl4 is clearly directed against Leibniz and the claim that mathematical objects can be conceptually determined, whereas the passage at B744-745 contrasts the "intuitive" methods of the mathematician with the "conceptual' methods of the philosopher) and that more in the way of reconciliation is not needed. 24 Interestingly, all of the recent work on Kant's philosophy of mathematics mentioned in the first paragraph of this paper makes contact at one point or another with this line (parsons seems to think that the first line of interpretation is more plausible in connection with geometry and the second in the case of arithmetic). All tend to identify construction, at least in arithmetic. with calculation. and all tend to assimilate the idea of calculation to "symbolic construction." I here follow Friedman in "Kant's Theory of Geometry." Hintikka begins in the same place. the disti~ tion between monadic and polyadic quantification, but moves in a,rather different direction. 25 "To construct a concept means to exhibit a prior{ the intuition which corresponds to the concept," CPR, A713!B741. This characterization is notoriously problematic. It would seem to follow from it that concepts for which no intuition can be exhibited a priori are not constructible. But without any further qualification, the concepts (or, equivalently in the case of geometry, the curves) proscribed by this criterion would seem to be few in number. e.g., curves everywhere continuous, but nowhere differentiable. Everything turns, of course, on what it means to "exhibit a priori" an intuition corresponding to a concept. It turns out, on the present line of interpretation, that in the paradigm case this involves not so much drawing a line as calculating a number. 26 Euclid's actual procedure, given by Friedman. is somewhat more complicated. but the essential point is the same: the "existence" of an infinite number of points is guaranteed by the iterability of particular constructive procedures. 27 For one thing, Kant's text abounds with arguments which are expressly analytic and patently polyadic, e.g., at CPR, A766/B794. 28 The most detailed criticism of Friedman's use of the concept of infinity, and of his reading of the passage at CPR, B40 on which a case for that use is based, was made by Manley Thompson in comments at the American Philosophical Association Central Division Meeting in the spring of 1987. Hintikka·s position has been subjected to extensive criticism, perhaps in the most damaging way by Parsons, "Mathematical Intuition," Proceedings of the Aristotelian Society, N.S. 80 (1979-80), pp. 142-168,
"constructive" although it is not without many difficulties. See. for example, Parsons, "Infinity and Kant's Conception of the 'Possibility of Experience'," reprinted in Mathematics in Philosophy. 30 See the first Critique, A52/B76, A151/B19Off., and the opening paragraphs of his lectures on logic. 31 See R.K. Meyer and K. Lambert, "Universally Free Logic and Standard Quantification Theory," Journal a/Symbolic Logic, 33 (l968), pp. 8-26. 32 See Friedman, "Kant's Theory of Geometry," p. 202: "the case of arithmetic is primary." 33 See Thompson, "Singular Terms and Intuitions in Kant's Epistemology." p. 98: "A numeral is thus a symbolic construction of a number, and in arithmetic by further symbolic constructions from numerals (by calculation) we discover further numerical properties of numbers, just as in . geometry by further ostensive constructions from figures we discover further geometrical properties of figures." 34 See Parsons, "Kant's Philosophy of Arithmetic," p. 67: "The considerations about the role of symbolic operations in mathematics apply equally to logic and therefore undetmine Kant's apparent wish to distinguish them on this basis.," and Thompson, "Singular Terms and Intuitions in Kant's Epistemology," p. 106, n. 23: "Since general logic so conceived [as first-order quantification theory + identity] contains symbolic constructions and demonstrations, it would seems at least in this sense to be something Kant would have to regard as a branch of mathematics." 35 Following Friedman, "Kant on Concepts and Intuitions in the Mathematical Sciences," who in turn draws on Parsons. Thompson. and Young. 36 See Parsons, "Arithmetic and the Categories." p. 149: "finite iteration is an abstract coun~r part of the notion of successive repetition. But to describe it was quite beyond the logical and mathematical resources of Kant and his contemporaries; the task was first accomplished in the 1880's by Frege and Dedekind." 37 Following Young, "Construction, Schematism, and Imagination," although Young is careful to point out, as noted below, that for Kant the constructions on which arithmetical reasoning rests are in fact ostensive. 38 Ibid.• p. 124. 39 Friedman, "Kant on Concepts and Intuitions in the Malhematical Sciences," argues for including both arithmetic and algebra under the heading of symbolic construction. In so doing he seems to me to miss the point of Kant's distinctions. Young, "Kant on the Constructions of Arithmetical Concepts," thinks that it would be a proper extension of Kant's view to regard calculations as ~ymbolic constructions, for reasons already advanced. but he notes correctly that in the basic case arithmetical constructions are ostensive. Indeed, he finds it more plausible to regard our knowledge of arithmetical truths as resting on ostensive constructions than geomenical truths. for a collection of n things, in his view, can represent the number n in a way that allows us to use the collection to gain knowledge about the number, whereas the parallel claim about the geometrical constructions has little support. 40 We have to distinguish between the process of calculation and its product. In arithmetic, only the latter is ostended. Only because of the underlying isomorphism with the ostensive construction is the correctness of the symbolic construction with numerals assured. In this sense the numerals "represent" and do not simply "symbolize" as do algebraic variables. 4,' As in a letter to Johann Schultz of November 25, 1788. The letter is included in Amuif Zweig, trans. and ed. Kant: Philosophical Correspondence, 1759-1799 (University of Chicago Press. 1967).
336
29 There is something misleading, in any case. about attributing the view that mathematics requires models containing at least a denumerable number of elements to Kant, for it i.s clearly his view that all numbers are finite. Numbers are determinate quantities and "a determinate yet infinite quantity is self-contradictory" (CPR, A521/B555). Kant's view of infinity is thoroughly
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42 It is possible that Kant identified arithmetic proper with the four basic combinatorial opera_ tions and distinguished it from general arithmetic or a1gebra in part on the basis of the fact that the latter includes the extraction of roots as well. As Michael Young has pointed out to me. the expectation of closure under the four basic operations would lead to the introduction of negative numbers, but not to the introduction of irrational numbers. Moreover, there is a sense in which negative numbers can be "constrUcted," as the difference between pairs of positive (and hence ostendibie) numbers, while L"Tational numbers cannot be "constructed" in the same way as ratios of pairs of positive numbers. This consideration would support Friedman's emphasis on the rationa1{llTIltional number distinction as the key to the difference between arithmetic and algebra. but, first, it is not in my view fundamental (the operation on variables is) and, second, I have not been able to determine whether Kant thinks extraction of roots does not belong to arithmetic proper. What could be a principled reason for its exclusion, particularly if one took arithmetical constructions as symbolic rather than ostensive? 43 "Kant on Concepts and Intuitions in the Mathematical Sciences." 44 Kant had a copy of Descartes' Geometry in his library and was presumably familiar with Descartes' views on algebra and analytic (as contrasted with "synthetic") geometry. An adequate description of the algebraic background of Kant's thought would also have to include Leibniz, whose work on algebra is extensive, and Euler, whose Anleitung zur Algebra (published in 1770) is widely regarded as the best algebra text of the century. Newton's views on algebra will be discussed shortly. 45 See the letter to K.L Reinhold of May 19, 1789: "The mathematician cannot make the least claim in regard to any object whatsoever without exhibiting it in intuition (or, if we are dealing merely with quantities without qualities. as in algebra, exhibiting the quantitative relationships thought under the chosen symbols) ... "In The Kant-Eberhard Controversy, trans. Allison (Johns Hopkins Press, 1973), p. 167. The intuition to which the arithmetician appeals is not, at least in the basic case, "symbolic." although it is. of course, "representative." 46 It is simply a fact, according to Kant, that these constructions show the "reality" of whole and rational numbers and their various rule-governed combinations. Perhaps it should be made explicit that it is only insofar as magnitudes also have qualities that they can be given an ostensive construction, only insofar as they have qualities that their "reality" be shown. Mere quantities can only be given a symbolic construction. 47 Note in the passage at CPR, A7161B744, that a symbolic construction "exhibits in intuition ... all the various operations through which the magnitudes are produced and modified" (my italics), whereas an ostensive construction exhibits an objecT corresponding to a concept. Note also as against those who emphasize the time-taking, hence "intuitive" character of arithmetical calculation that Kant here talks about representing the operations involved in algebraic manipulation and not carrying them out. 48 The mathematical tradition until- well into the 18th century (and in many cases beyond) rejected negative roots of quadratic equations as "false" and "unreal" simply because they could not be represented, although they could, of course, be symbolized. 49 In this tradition, algebra is identified with "analysis." Thus Fran~ois Viera (1540-1603), the "father" of modem algebra, entitled his work Introduction to the Analytical Art. 50 The Kant-Eberhard Controversy, p. 110. In an important gloss on this passage in his letter to Reinhold of May 19, 1789, Kant reinforces the point: if Eberhard understood the example, he would see "that the definition which Apollonious gives, e.g., of a parabola, is itself the exhibition of a concept in intuition, namely the intersection of a cone under certain conditions, and in establishing the objective reality of the concept, that the definition here, as always in geometry, is at
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the same time the construction of the concept." (Ibid., p. 168). He adds that the actual drawing of the parabola, given the parameter, follows as a practical corollary and has nothing to do with the theoretical point at issue. 51 The case of Newton is typically complex and illustrates the lack of clarity concerning the relationship between geometry and algebra, synthetic and analytic methods, and the foundation for each, which in fact extends well into the 19th century. On the one hand, he says in his Universal Arithmetic (1707) with apparent approval that "the Modems advancing yet much further [than the plane, solid, and linear loci of the Greeks] have received into Geometry all lines that can be expressed by Equations," and we know that in his own work he made rather free-wheeling use of analytic methods. Yet on the other hand, in a letter to David Gregory (1661-1708), Newton remarks that "Algebra is the analysis of bunglers in arithmetic," and he continued to take the Greek mathematical tradition as his standard of rigor. Perhaps for reasons of the "elegance" which Kant mentions in the Eberhard controversy (see below), Newton composed the Principia according to the synthetic-geometrical method, apparently no more than assuming that the synthetic proofs he offered were adequate to reach the results that he had in fact arrived at analytically. See Morris Kline, Mathematics: The Loss of Certainty (Oxford University Press, 1980), chapterV. 52 The Kant-Eberhard Controversy, p. 111. 53 According to Kline, Mathematics: The Loss of Certainty, p. 125. Newton tried to ground algebra by arguing that "the letters in algebraic expressions stand for numbers and no one can doubt the certainty of arithmetic." What this means, apparently, is that algebraic results are simply generalized versions of arithmetical truths and in some sense reducible to them. It is noteworthy that Kant does not avail himself of this strategy, nor could he, on my interpretation, sin~ arithmetic proper affords ostensive constructions on the basis of which its truths are immediately verifiable. 54 See the Critique of Pure Reason, A7341B762. 55 Notably in the Axioms of Intuition, A1631B204. 56 More than anyone else, Gottfried Martin in Arithmetic and Combinatories: Kant and His Contemporaries, trans. Wubnig (Southern Illinois University Press, 1985) has stressed the role played by algebra in Kant's philosophy of mathematics and the ways in which it is "creative" and hence, in another sense of the word than we have used so far, "synthetic." Martin goes astray in insisting that Kant was the first to formulate certain "axioms" of arithmetic (such as the rule of commutation), but he was right to draw attention to the status of the rules used in combinatorial operations. 57 In which case Kant's "decidability" thesis needs to be reconstrued. - 58 Descartes held that algebraic "objects" were admissible just in so far as they could be given a geometrical (spatial) construction. Newton held that algebraic "objects" were admissible just in so far as they could be given an arithmetical (numerical) construction. In distinguishing between arithmetic and geometry, on the one hand, and algebra, on the other, Kant would seem to reject both of these moves. But when the possibility of such ostensive constructions (even at second hand) is rejected, intuition as so far understood comes to playa reduced role. 59 See William S. and Martha Kneale, The Development of Logic (Oxford University Press, 1962). p. 308. 60 The Kant-Eberhard Controversy, p. 110. . 61 I have learned a great deal from all ofthe authors listed in the first paragraph. Michael Young , has made very helpful comments on an earlier draft of section 4 of the paper. James Allard raised the right sort of questions throughout, though I wasn't able to answer all of them.
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JAAKKO HINTIKKA
KANT'S TRANSCENDENTAL METHOD AND HIS THEORY OF MATHEMATICS
I. THE AIM OF THIS PAPER
·This paper has a dual aim. On the one hand, it is a part of a larger attempt to understand the nature of Kant's ideas of transcendental method and transcendental knowledge and their implications, for instance, the question as to what the objects of transcendental knowledge are. On the other hand, I am outlining once again what I take to be the true argumentative structure of Kant's doctrines of the mathematical method, space, time, and the forms of inoer and outer sense. The link between the two is that on my interpretation Kant's theory of mathematics offers an excellent example of the applications of his transcendental method. Moreover, after having recently defended my construal of Kant's views on mathematical reasoning and their foundation on historical and textnal gronnds, it may be in order· to try to vindicate it in another way, to wit, by relating it to the overall natnre of Kant's philosophy, including his idea of transcendental knowledge. I suspect that this may be a better way of convincing my colleagnes than nitty-gritty analyses of Kantian texts. At the same time, this approach offers me a chance of indicating some of the consequences of my results concerning Kant's theory of mathematics for the rest of his philosophy. It tnms out that the observations we can make in pursuing this line of thought have also interesting consequences for our contemporary thought in the philosophy of logic and mathematics. 2. KANT'S CONCEPT OF THE TRANSCENDENTAL
,.
A starting-point is offered by Kant's crucial concept of the transcendental. How does Kant use the term? Kant introduced the term "transcendental" in a slightly different way in the first and in the second editions of the Critique of
Pure Reason: A version:
I'call transcendental all knowledge which is concerned, not so much with objects. as with
B version: I call transcendental all knowledge which is concerned. not so much with objects, as
341 Carl J. Posy (ed.), Kant's Philosophy of Mathematics, 341-359. IB tOQ?
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