Realistic Rationalism (Representation and Mind)

Acknowledgments

I acknowledge with gratitude the help of many friends and colleagues at various stages in the writing of this book. Special thanks are due to Glenn Branch, who has been especially helpful in the ªnal stages of the undertaking. Special thanks are also due to Arnold Koslow and David Pitt, who were in at the beginning and provided support and useful comments all along the way. All three of them have taught me a lesson about generosity. In addition, thanks are owed to Jody Azzouni, Mark Balaguer, Ned Block, Paul Boghossian, Arthur Collins, Hartry Field, Eric Hetherington, Keith Hosacks, Dan Isaacson, Gerrit Jan Kemperdyk, Juliette Kennedy, Noa Latham, Paolo Mancosu, Richard Mendelsohn, Elliott Mendelson, Sid Morgenbesser, Tom Nagel, Yuji Nishiyama, Robert Nozick, Paul Postal, David Rosenthal, Mark Sainsbury, Robert Tragesser, Palle Yourgrau, Hao Wang, and Linda Wetzel. My thanks also go to the members of my seminars at the Graduate Center in the springs of 1993, 1994, and 1996, and to the members of my seminar at King’s College, University of London, the audiences at my Oxford University lectures, and the audience at my Cambridge University Moral Sciences Lecture (1994 Michaelmas term).

Introduction

In the course of this century, a large segment of Anglo-American philosophy was persuaded to abandon the traditional conception of philosophy on which it is an a priori inquiry into the most general facts about reality. This conception was replaced with one or another of two naturalist conceptions of philosophy: philosophy as therapy designed to cure the linguistic illness of which philosophy itself is the cause, and philosophy as an a posteriori discipline within natural science. Expressing the former naturalist conception, Wittgenstein (1961 [1922], sec. 6.53) wrote in the Tractatus: The correct method in philosophy would really be the following: to say nothing except what can be said, i.e., propositions of natural science—i.e., something that has nothing to do with philosophy—and then, whenever someone else wanted to say something metaphysical, to demonstrate to him that he had failed to give a meaning to certain signs in his propositions. This conception remained with him throughout his life. In the Philosophical Investigations, Wittgenstein (1953, sec. 109) articulates his therapeutic conception of philosophy in the famous passage: [Philosophical problems] are, of course, not empirical problems; they are solved, rather, by looking into the workings of our language, and that in such a way as to make us recognize those workings: in despite of an urge to misunderstand them. The problems are solved, not by giving new information, but by arranging what we have always known. Philosophy is a battle against the bewitchment of our intelligence by means of language. In an equally famous passage, Quine (1969a, 83) expresses the latter— philosophy as natural science—conception of philosophy: Our very epistemological enterprise, therefore, and the psychology wherein it is a component chapter, and the whole of natural science wherein psychology is a component book—all this is our

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own construction or projection from stimulations like those we were meting out to our epistemological subject. There is thus reciprocal containment, though containment in different senses: epistemology in natural science and natural science in epistemology. Broadly, there are three doctrines called “naturalism” in contemporary philosophy. The ªrst, which may be called “ontological naturalism,” claims that the universe consists exclusively of natural objects, that is, spatiotemporal objects belonging to the causal order in nature. This is the contemporary naturalism closest to the doctrines that have traditionally been referred to as “naturalism.” The second doctrine, which may be called “epistemological naturalism,” claims that knowledge is knowledge of natural objects. The third doctrine, which may be called “methodological naturalism,” claims that the only way we can obtain knowledge of the universe is through prescientiªc and scientiªc investigations of natural objects. Ontological naturalists are epistemological and methodological naturalists, but epistemological and methodological naturalists may or may not be ontological naturalists. Since he thinks that our theories in natural science commit us to abstract objects because they involve ineliminable quantiªcation over them, Quine, who is a methodological naturalist, is neither an ontological nor an epistemological naturalist.1 The naturalist hegemony is well established today in the form of programs to naturalize philosophy and philosophize naturalistically, in agendas to deºate one or another philosophical concept, in revivals of the late Wittgenstein’s therapeutic positivism, in resuscitations of American pragmatism, and in construals of philosophy as an exclusively second-order discipline concerned with linguistic and/or conceptual analysis. These positions, which are at some points overlapping, at some points independent, and at some points even conºicting, are tied together by the privileged status they accord to natural objects and by their ªrm epistemological opposition to anything smacking of an autonomous metaphysics claiming to provide a priori knowledge about reality. This hegemony ºourishes despite a number of prominent philosophers, such as Blanshard, Chisholm, Ewing, Langford, Thomas Nagel, Pap, and P. F. Strawson, whose philosophizing is a continuation of just 1. In the philosophy of mind, the term “naturalism” is often used just to mean antiCartesianism. Here it is better to use a term like “materialism.” In any case, my argument in this book is not directed against philosophers who take themselves to be “naturalists” in this sense, unless, of course, they are also naturalists in any of the senses of “naturalism” in the text.

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such old-style metaphysics. The problem is not that such philosophers have failed to produce anything of recognized philosophical signiªcance, but rather that their work lacks a substantive metaphilosophical dimension, and that it either ignores important philosophical developments that have taken place within the prevailing naturalistic outlook or accommodates itself too much to that outlook. As a consequence, contemporary representatives of traditional philosophy have not articulated and defended a metaphilosophy with which to oppose the well-articulated and well-defended positions based on Wittgenstein’s and Quine’s naturalism.2 One consequence of this was that the stereotype of traditional philosophy as a series of interminable and inconclusive squabbles was allowed to go unchallenged. This stereotype has motivated critics of metaphysics from Kant to Carnap, who see such squabbling to be characteristic of metaphysical philosophizing and to arise from the metaphysician’s view that philosophy in and of itself is a legitimate source of a priori knowledge about reality. The critics are impatient with philosophical business as usual and, in the case of many of them, particularly the positivists and Quine, they are attracted by the ideal of a more amicable future in which an intellectual consensus of the sort that exists in science becomes the way of philosophy.3 The late Wittgenstein and his followers, of course, do not share this ideal. The critics agree in denying that philosophy can be a legitimate source of a priori knowledge about reality, but they disagree about why. There are two main diagnoses of what is wrong with thinking that philosophy can be a legitimate source of a priori knowledge about reality. According to the diagnosis popular among Wittgensteinians, logical empiricists, and ordinary language philosophers, the mistake is 2. Even the best of such philosophizing is disappointing from this perspective. Strawson’s (1985) championing of the rationalist tradition does nothing to articulate the metaphilosophy inherent in that tradition, and, as I (1990b, 344, n. 1) have argued elsewhere, the book’s discussion of intuition and even of naturalism muddies the waters. Nagel’s (1986) book, The View from Nowhere, which, I believe, makes signiªcant contributions to several metaphysical topics, is content simply to endorse a rationalist perspective against Quinean empiricism. Further, Nagel’s (1986, 105–9) overestimation of the force of Wittgensteinian philosophy of language, particularly the rule-following argument, leads to the absence of a discussion of the essential role that mathematical realism plays in the formulation of rationalism and in the development of a metaphilosophy based on realism and rationalism. 3. Of course, the positivist claim to eschew metaphysics did not go unquestioned. The positivists were accused, rightly I believe, of doing metaphysics, as it were, under the table. In particular, their criterion of cognitive signiªcance was criticized as either an a priori metaphysical principle or a self-defeating empirical one. But their explicit doctrine was that it was a convention. (See chapter 6, section 3.)

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thinking of philosophy as a ªrst-order discipline like the sciences, which, like them, has some aspect of reality as its subject matter. This is wrong because the sciences cover reality exhaustively. Once physics, chemistry, psychology, social science, history, and so on have staked their claims to their subject matters, there is no part of reality left for philosophy. With no domain of facts to settle issues between conºicting metaphysical claims about reality, it is no wonder that metaphysicians, laboring under this misconception, should be involved in endless controversies: the controversies have no objective resolution because they are not about reality. Mary Warnock (1995) has aptly described this diagnosis in her recent reºections on the vicissitudes of ethics in this century. Speaking of philosophy at Oxford during and after World War II, she (1995, 22) writes: The new philosophy was contrasted with a supposed golden age, when philosophers were metaphysicians, and did not bother about the concepts, or words, actually embedded in language. . . . philosophers were allowed to pontiªcate about [causation, mind, and so forth] . . . and use what concepts they chose to invent. In contrast, “the new philosophy,” she (1995, 21–22) writes, . . . was a ‘second-order’ subject. What we meant could be put in the following way: botanists, let us say, talk about plants and their genetic composition; and historians write about events and people of the past. Philosophy, however, has no subject-matter of its own. There are no philosophical objects to be examined. Philosophy, unlike botany or history, does not apply concepts to things; it is one step higher up the ladder of abstraction. . . . Philosophy considers the concepts that other subjects employ, and seeks clariªcation, or analysis, of them. It is out of this description of philosophy (which I still think is a good one), that there arose . . . the idea that philosophy is linguistic. The “new philosophers” took the phenomena about which traditional metaphysicians speculated to be natural phenomena belonging to the province of natural sciences like physics and psychology. They concluded that genuine knowledge about causation, mind, and so forth is empirical knowledge in natural science. In order for there to be something beyond natural phenomena, there would have to be nonnatural objects and a priori knowledge. But the strong strain of naturalism and empiricism is the background of these “new philosophers” assured them that there are no non-natural phenomena and there is no

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a priori knowledge. Hence, having no ªrst-order subject matter, either philosophy can deliver no genuine knowledge at all or it is a secondorder subject which delivers second-order knowledge in the form of linguistic and/or conceptual analyses of ªrst-order knowledge. There was, however, disagreement among these naturalists about how philosophy should be thought of in relation to the natural realm. The main issue is whether or not philosophy is kicked upstairs to become some sort of second-order discipline, with the task of clarifying the linguistic and conceptual matters in ªrst-order disciplines. While many of the philosophers about whom Warnock is talking saw philosophy as such a second-order discipline, the late Wittgenstein did not. He (1953, sec. 121) wrote: One might think: if philosophy speaks of the use of the word “philosophy” there must be a second-order philosophy. But it is not so: it is, rather, like the case of orthography, which deals with the word “orthography” among others without then being second-order. For him (1953, sec. 124), philosophy makes no contribution to human knowledge—even second-order knowledge: “It leaves everything as it is.” Philosophy is a kind of linguistic therapy. Wittgenstein (1953, secs. 122 and 123) thinks it should just help us to “command a clear view of the use of words” principally by helping us to “see connexions” in the uses of words through “ªnding and inventing intermediate cases”. Quine’s was the other diagnosis of what is wrong with the traditional metaphysical view that philosophy can be a legitimate source of a priori knowledge about reality. He agrees with other naturalists that the sciences cover reality exhaustively. Once physics, chemistry, psychology, social science, history, and so forth have staked claims to their subject matters, there is nothing left. Hence, there can’t be an autonomous metaphysical philosophy. But for Quine this doesn’t mean that philosophy cannot legitimately address questions about reality. It only means that it must do so within natural science. The trouble with traditional metaphysics is that it took itself to be an autonomous discipline with the right to speculate about reality independently of the experiential and methodological constraints internal to natural science. Without such constraints, the traditional philosopher’s conclusions were often unscientiªc speculations about scientiªc matters. On both diagnoses, the cure is to replace the traditional conception of the relation of philosophy to reality with a naturalistic one on which the sciences are the only ªrst-order disciplines. Beyond this, each form of twentieth-century naturalism has its own idea of how to understand the relation between philosophy and science. Except for the late

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Wittgenstein, naturalists see philosophy as at least a second-order discipline that is part of the scientiªc enterprise. But there is disagreement about the relation of philosophy to ªrst-order scientiªc disciplines, and hence about the role philosophy plays in scientiªc investigation. The disagreement turns on which aspect of what Quine calls “reciprocal containment” it is that a naturalistic metaphilosophy stresses—either “epistemology in natural science” or “natural science in epistemology.” Every scientistic naturalist accepts the view that philosophy is to some extent concerned with clarifying the linguistic and conceptual practices of ªrst-order disciplines. But those who stress natural science in epistemology tend to think that it is entirely legitimate for philosophers—with suitable basic training in science—to get down there in the trenches with the scientists, not only to provide more scientiªc troops but also to further their own quest for philosophical enlightenment. On the other hand, those who stress epistemology in natural science tend to think of philosophers who do not stick to metascientiªc analysis as having gone native. Each form of twentieth-century naturalism thought that a proper dose of its medicine would cure philosophy of the disease of metaphysics. The interminable and inconclusive squabbling of the past would disappear and philosophy would enjoy a future in which honest philosophical endeavor is rewarded with steady philosophical progress. But, as must by now be evident, we are not living in such a philosophical Canaan. Even a cursory look at the controversies in contemporary philosophy of language and logic, the philosophy of mathematics, the philosophy of mind, and so on shows that, although the cures have been tried, there is not the slightest sign of the disease going away. Indeed, it looks more as though the disease has spread to the hospital staff. Philosophical squabbles are going on as before with no serious prospect of abating, and, if anything, the range of controversy has only increased with the addition of the internal disagreements among Wittgensteinians and among Quineans and with the disagreements between Wittgensteinians and Quineans. Hence, some reassessment of the “revolution in philosophy” is surely in order. The present book is one reassessment. It is a radical reassessment. Its broad aim is to provide the metaphilosophy and the arguments to show that abandoning the traditional conception of philosophy in favor of one or another form of naturalism was a fundamental mistake. Not that traditional versions of the metaphysical conception of philosophy did not deserve criticism, but the critics threw out the baby with the bathwater. Part of my case for this claim was presented in my (1990b) earlier book, The Metaphysics of Meaning. That

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book took the necessary ªrst step of showing that Wittgensteinian and Quinean arguments do not justify abandoning the traditional metaphysical conception of philosophy. The present book takes the next step of formulating and justifying a new version of traditional realist and rationalist philosophy. This version is the position to which the title of this book refers. This enterprise of trying to revive the traditional conception of philosophy did not originate in nostalgia for the past. I spent my philosophically formative years during the ºowering of logical empiricism, Quineanism, and ordinary language philosophy; I was a naturalist and empiricist of the scientistic sort. Nevertheless, like nearly everyone who goes into philosophy, I was initially drawn in by the pull of philosophy’s uniquely puzzling questions: What is knowledge? What is the relation between mind and body? Is there free will? Are ethical values universal? and, particularly, its central question, What is philosophy? My disillusionment with naturalism, as I will explain below, came as the result, on the one hand, of ªnding that naturalist and empiricist philosophies do not provide satisfying answers to the questions that ªrst lure us into philosophy and, on the other, of coming to think that answering some of those questions requires a nonnaturalist position combining realism in ontology with rationalism in epistemology. Such a position differs from the naturalist positions in twentiethcentury philosophy in various ways. One way in which ªrst-order and second-order disciplines can differ is in terms of the questions they ask. The distinction in this case is that a ªrst-order discipline addresses questions about some domain of objects in the world and a secondorder discipline addresses questions about the linguistic forms or concepts employed in ªrst-order disciplines. Another way in which they can differ is in terms of their role in answering questions about the domain. The distinction in this case is that the ªrst-order discipline has a fact-ªnding and fact-systematizing role in the investigation of the scientiªc domain, and the second-order discipline does not. With respect to the former distinction between ªrst-order and second-order disciplines, our non-naturalist position says that philosophy is both ªrst-order and second-order. It thus rejects the naturalist positions that would restrict the questions it asks. Warnock’s “new philosophers” simply had too impoverished a conception of the range of questions a discipline can address. Mathematics addresses questions about a domain of numbers, sets, spaces, and so on, but, since metamathematics is part of mathematics, mathematics also addresses questions about the technical language within which mathematical accounts of those domains are given. My suspicion is that, not having separated

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the two distinctions between ªrst-order and second-order disciplines, those philosophers, already carried along by the linguistic turn, found it easy to assume that, since philosophy does not contribute to the empirical work of the natural sciences, its questions are, by default, restricted to the linguistic and/or conceptual structure of ªrst-order disciplines. With respect to the second distinction, I think that the naturalist’s diagnosis is a half-truth. The true part is that philosophy is not a ªrst-order discipline in the hands-on sense in which the sciences themselves are. There is surely some room for doubt about this in connection with ethics and aesthetics, where there is more plausibility in thinking of philosophy as a ªrst-order discipline than there is in the philosophy of science. But I believe that thinking of them in this way confuses the roles of moralist and moral philosopher and the roles of art critic and aesthetician. Ethics and aesthetics are better seen as second-order studies: respectively, studies of the work of moralists and art critics. Philosopher-moralists like Sartre and philosopher-art critics like Danto wear two hats. Furthermore, the universal scope of philosophy strongly suggests that it is a second-order discipline with general interests in the common epistemological and ontological problems of ªrst-order disciplines. Philosophy would hardly have this particular form of universal scope if it were literally a collection of ªrst-order disciplines with circumscribed domains of objects as their subject matters. The correct part of the naturalist’s diagnosis is that philosophy is a second-order discipline in the sense that it is not part of a scientiªc attempt to ascertain the facts about a domain and build a theory to explain them on the basis of deeper principles. On my non-naturalism, philosophy is a ªrst-order discipline only in asking questions about the world. The traditional philosophers who took philosophy to be an inquiry into general facts about reality did not, I think, want to say that philosophy is part of the scientiªc enterprise in a hands-on way. Rather, I think their view was that philosophy is part of the scientiªc enterprise in another way. It has the status of a second-order discipline in having no fact-ªnding or fact-systematizing role in scientiªc investigation, but that does not restrict its epistemic contribution to serving as conceptual referee in someone else’s ball game. The false part of the diagnosis is the assumption that not being a ªrst-order discipline in not having a fact-ªnding or fact-systematizing role in scientiªc investigation means that a discipline is not in a position to address substantive questions about reality. The possibility that the naturalist’s diagnosis overlooks is that some questions that arise in the course of a scientiªc investigation of reality are not questions that can be answered by broadening the investigation to attain a wider scientiªc

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knowledge of the facts or better scientiªc knowledge of their underlying principles. This is because the questions are not scientiªc questions. They are philosophical questions. They concern the nature and validity of the very methodology on which such knowledge rests. For example, physics is in no position to dispel skeptical doubts about how we know that the future will be sufªciently like the past to justify our conªdence in the scientiªc use of induction.4 Philosophical questions that arise in relation to mathematics are: Are numbers and sets objects? If so, what kind of objects are they? What does the mathematician’s knowledge of numbers and sets consist in? Does mathematical knowledge depend on natural facts? and Why does the mathematician’s knowledge of numbers and sets seem so much more certain than even the physicist’s knowledge of matter? Similar questions arise in connection with logic and linguistics. Such philosophical questions concern both the fundamental nature of the reality investigated in the ªrst-order discipline and the methods that can provide knowledge of it. Those questions receive no answer in ªrst-order mathematical, logical, and linguistic investigations, not simply because the focus of those investigations is on describing and explaining facts about the objects under study, but because those questions concern the status of the investigations and their methodological foundations.5 On our position, philosophy, conceived of as a second-order discipline with no role in the fact-ªnding and fact-systematizing of science, nonetheless answers certain questions about the objects in the domains of the sciences. How does it go about doing this? There is a long answer and a short answer to this question. The long answer, and it is only a partial answer at that, is this entire book, but particularly chapters 2, 4, 5, and 6. The short answer is that philosophy tries to answer such questions in the way that philosophers in the foundations of mathematics try to answer questions like what kind of things numbers are. Philosophical attempts to answer this question constitute the dialectic 4. Quine (1975, 68) claims that skeptical questions are scientiªc questions. I shall return to his claim below. I note here that his inability to say anything about how science might resolve such doubts argues in favor of the view in the text. See Stroud (1984, 209–54). 5. Although mathematicians, logicians, and linguists normally conªne themselves to answering questions about the structure of objects such as numbers, sets, propositions, and sentences, some, like Frege, Hilbert, Brouwer, Gödel, and Chomsky, have not, for one reason or another, been content to leave philosophical questions to the philosophers, but have stepped out of their role as scientists to address epistemological or ontological issues about their discipline in a way that contributes importantly to our philosophical understanding. Such scientists are rare, and we mark their special status by referring to them as both a scientist and a philosopher, as, for instance, in the title of the Schilpp volume Albert Einstein: Philosopher-Scientist.

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among realists, conceptualists, and nominalists that began in Greek philosophy and that continues, in a far more professionalized form, to the present. It is, moreover, hard to see how the foundations of mathematics, conceived of as a discipline the aim of which is to answer questions about the epistemology and ontology of mathematics, can be understood without taking it to be a second-order discipline (in the sense of the second ªrst-order/second-order distinction) that can provide knowledge of reality. Naturalism also was behind the positivist attack on traditional philosophy’s claim to provide a priori answers to substantive factual questions about reality. Note again the early Wittgenstein’s (1961 [1922], sec. 6.53) remark quoted above: The correct method in philosophy would really be the following: to say nothing except what can be said, i.e., propositions of natural science—i.e., something that has nothing to do with philosophy—and then, whenever someone else wanted to say something metaphysical, to demonstrate to him that he had failed to give a meaning to certain signs in his propositions. Schlick (1949, 285) thought that metaphysicians radically misjudge the nature of the philosophical questions they attempt to answer: . . . The error committed by the proponents of the factual a priori can be understood as arising from the fact that it was not clearly realized that such concepts as those of the colors have a formal structure just as do the numbers or spatial concepts, [which] determines their meaning without remainder. . . . Thus, [the sentences that are the show-pieces of the phenomenological philosophy] say nothing about existence, or about the nature of anything, but rather only exhibit the content of our concepts . . . they bring no knowledge, and cannot serve as the foundations of a special science. Such a science as the phenomenologists have promised us just does not exist. As I see it, the early positivists and more recent positivists such as Carnap and the late Wittgenstein overestimate the scope of linguistic meaning. Linguistic meaning is not rich enough to show either that all metaphysical sentences are meaningless or that all alleged synthetic a priori propositions are just analytic a priori propositions. The idea that linguistic meaning can be used for such purposes was Frege’s; in particular, it came from his expansion of the concept of analyticity, undertaken in order to provide a semantic basis for his logicist explanation of mathematical truth as analytic truth. To a philosopher like Schlick, Frege’s logical semantics together with Wittgenstein’s philo-

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sophical foundations in the Tractatus seemed capable of accounting for all of the phenomenologist’s examples of synthetic a priori knowledge without resorting to faculties such as intuition. I believe that the program to explain away such examples of synthetic a priori knowledge as analytic a priori knowledge fails just as logicism does. The only way in which analyticity might be made powerful enough for such a positivist program is to adopt something like Carnap’s (1956a, 222–32; 1956b) approach, but Quine (1953c, 32–57) shows that the approach doesn’t help. The approach provides no concept of analyticity, so there is no notion of the analytic a priori under which to bring the metaphysician’s synthetic a priori propositions. Arbitrarily putting the disputed propositions on a list with the uninterpreted term “analytic” at the top is hardly a refutation of metaphysics.6 Although our conception of philosophy conºicts with Quine’s (1974, 2) nonpositivistic, methodological naturalism, it shares Quine’s (1969a, 69) characterization of epistemology as “concerned with the foundations of the sciences.” What I reject are his claims that science is ªrst philosophy and that philosophy is a scientiªc concern with scientiªc knowledge. Philosophy, as I see it, is not continuous with science; it is not of a piece with science.7 Philosophy, or at least one large part of it, is subsequent to science; it begins where science leaves off. Quine’s case for naturalizing epistemology is based on what he (1969a, 75) refers to as the “[t]wo cardinal tenets of empiricism”: ªrst, “whatever evidence there is for science is sensory evidence,” and, second, “inculcation of meanings of words must rest ultimately on sensory evidence.” He (1969a, 75) refers to them, and hence naturalized epistemology, as “unassailable.” This, however, is something of an exaggeration. They have been assailed, to my mind quite successfully, from various philosophical standpoints. Stroud (1984, 209–54) has 6. This is not, as I see it, a shortcoming of the notion of analyticity, but rather an inevitable consequence of the Fregean notion of analyticity. As I will explain below, the original sin was to broaden the traditional Lockean and Kantian notion of analyticity in the way Frege did instead of revising it slightly to meet his criticisms. When it is revised, we retain a narrow analytic/synthetic distinction that vindicates the traditional metaphysical conception’s focus on the explanation of synthetic a priori knowledge. 7. My position further departs from Quine’s in rejecting his claim that formal science is continuous with natural science (and hence the extraordinary consequence of his naturalized Platonism that entities referred to in unapplied portions of mathematics do not exist). On a realist view of the formal sciences, they are about abstract objects, while on everyone’s view of the natural sciences, they are about natural objects. Hence, the epistemologies of the formal and natural sciences will differ in the way that traditional rationalists always claimed they do. We shall see, however, that this difference can be given a much sharper statement than it has received at the hands of traditional rationalists.

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criticized Quine’s naturalized epistemology from a Kantian standpoint for being unable, in principle, to explain the possibility of knowledge of the world. Kim (1993, 216–36) has criticized it from a traditional epistemologist’s standpoint as irrelevant to the philosopher’s attempt to understand knowledge. And I (1990b, 175–202, ch. 5) have criticized it from the standpoint of the philosophy of language as resting on arguments against the theory of meaning that simply do not work. The difference between science and philosophy, particularly philosophy of science, is a difference between their fundamental questions. The fundamental questions of philosophy—such as questions about our knowledge of universals and demonstrable truths, about the existence of the external world and other minds, about the relation of the future and the past to the present, and about the relation of the mind to the body—differ essentially from scientiªc questions. For one thing, some philosophical questions turn a Pyrrhonistic gaze toward science, calling into question the standards on which sciences evaluate conºicting scientiªc claims and asking whether those standards can be justiªed in the face of philosophical doubt. Quine (1975, 68) would dispute the claim that philosophical skepticism is outside science, but the issue is not whether a philosopher has a theory that allows him to argue that “sceptical doubts are scientiªc doubts,” but whether it is plausible to make the claim that skeptical questions are scientiªc questions. Given science as we know it, it seems far-fetched to claim, for example, that the Cartesian question of whether there is an external world is a scientiªc question. For another thing, philosophical questions lead to a kind of metaphysical vertigo: the deeper we go into them, the more all the possible answers are buffeted by new and often more difªcult objections, the more bewildered we become about whether anything works, and the more we come to suspect that perhaps there is something wrong with the questions themselves. We shall see below that even questions far removed from standard skepticism, such as those concerning the nature of mathematical truth, have engendered such a metaphysical vertigo in some of the best philosophers of mathematics. In an uncharacteristically essentialist moment, Wittgenstein (1953, sec. 133) once described philosophical questions as “questions which bring [philosophy] itself into question.” The crisis of faith that some philosophers have when they come to doubt whether there ultimately are answers to philosophical questions reºects the deep truth of Wittgenstein’s remark. Philosophical questions are, at the very bottom, the question, What is a philosophical question? That is hardly a scientiªc question. Thus, on the separation of philosophy from science, my view is more like the late Wittgenstein’s, but it is unlike his in virtually every other

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respect. This is particularly so with respect to the nature of philosophy. Wittgenstein lost faith that philosophical thinking, particularly as embodied in the logico-semantic work of Frege and Russell, could accomplish the positivist task he had set himself in the Tractatus. He came to think that their work was part of the problem and had to be just as vigorously opposed as “plain nonsense” as the more classical metaphysics he had opposed in the Tractatus. The full task of showing that metaphysical sentences are “plain nonsense” demanded a new conception of meaning and language. The shift to the new conception led to a comprehensive rejection of traditional philosophy. Everything goes: no theories, no explanation, no hidden essences, no analysis, no universals, no underlying meanings, and no discoveries in the traditional philosophical sense. “The real discovery,” Wittgenstein (1953, sec. 133) now wrote, “. . . is the one that gives philosophy peace, so that it is no longer tormented by questions which bring itself into question.” On my view, everything stays: theories, explanation, hidden essences, analysis, universals, meanings, and discoveries in the traditional philosophical sense. Furthermore, such discoveries are not limited to the sphere of language where philosophy and linguistics join forces to uncover hidden syntactic and semantic essences and exhibit their epistemological and ontological character. Philosophical discoveries include the extralinguistic sphere where philosophy joins forces with sciences to uncover other hidden essences, substantiate the existence of abstract objects, and exhibit their epistemological and ontological character. These reºections describe the ways in which the major naturalist and empiricist positions differ from the realistic rationalism I will set out here. How might someone holding one of the former positions come to think that it should be abandoned in favor of the latter position? I will try to answer this question on the basis of a sketch of my own route from the scientistic naturalism I once held to the position I now hold. The signiªcant issue for me concerning scientistic naturalism was one that divided the two principal forms of the position in the early sixties. The twin prophets of scientistic naturalism, Quine and Chomsky, both departed from Viennese and Oxford philosophy in taking the view that language is to be studied within the science of language. Philosophers should seek enlightenment about the nature of language in linguistics rather than in the elaboration of artiªcially constructed calculi or in the description of ordinary usage. Philosophers of language should be informed about the science of linguistics. But Quine and Chomsky disagreed over what kind of linguistics a scientiªcally informed philosopher ought to be informed about.

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Within linguistics, there were two quite different schools of thought about the nature of language. One was Bloomªeldian structuralism, which was the orthodoxy of the time. Quine, who favored this school, presented it to philosophers in his (1961b, 47–64) paper “The Problem of Meaning in Linguistics.” The other was the new school of generative grammar that Chomsky himself was developing in opposition to Bloomªeldian structuralism. Philosophers got their ªrst exposure to this approach in Chomsky’s (1957) Syntactic Structures. These two views presented very different pictures of the science of language, with very different implications for linguistic analysis in philosophy. On the Bloomªeldian picture, linguistics is a science of the distributional relations in speech. Linguistic orthodoxy conceived of sentences acoustically and meanings behavioristically. On the Chomskyan picture, linguistics is a science of the speaker’s knowledge of the language rather than a science of speech. Instead of thinking of grammatical investigation taxonomically, on the model of botany or stamp collecting, Chomsky conceived of it generatively, on the model of systems of logic. Chomsky’s Syntactic Structures seemed to me to present the more plausible conception of the scientiªc study of natural language. But, despite its novel proposals about syntax and phonology, the book was silent on semantics. This was disappointing to a philosopher interested in applying linguistics to philosophy, since semantics is the area of linguistics with the greatest potential for shedding light on philosophical issues. Apart from this disappointment, it seemed to me that the failure of Syntactic Structures to say something about semantics in generative grammar was also a problem for the Chomskyan theory of generative grammar. For one thing, since linguistics was traditionally concerned with meaningfulness and sameness of meaning, Chomsky’s theory appeared incomplete. For another thing, even the theory’s own goals, as explicitly set forth in Syntactic Structures, could not be achieved without a semantic theory being part of the theory of generative grammar. For example, the existence of nonambiguous expressions like “automated processing device” that have different syntactic structures, i.e., [[automated][processing device]] and [[automated processing][device]], shows that the phenomenon of ambiguity cannot be explicated (as Syntactic Structures claims) in terms of multiple nonequivalent syntactic derivations. A semantic theory is required to predict when two syntactically nonequivalent derivations are equivalent semantically. Thus, I was led (initially with Jerry Fodor and Paul Postal) to try to develop the semantic theory that was required for the theory of gen-

Introduction

xxv

erative grammar. However, the development of such a theory, which at the time I saw as a means of showing that Chomsky’s conception of scientistic naturalism was preferable to Quine’s, led eventually to my abandoning naturalism. Chomsky’s syntactic theory had taken linguistics from Bloomªeld’s nominalist view of the science, on which it is a branch of acoustic physics, to Chomsky’s own conceptualist view of linguistics, on which it is a branch of psychology. I came to think that the semantic theory originally developed to plug a gap in Chomsky’s theory of generative grammar is what leads to taking the next step: from linguistic conceptualism to linguistic realism. The problem that started the train of thought leading to this conclusion was the existence of two incompatible ways to interpret such a semantic theory. One came from the Chomskyan framework within which the semantic theory was developed and the other from the Fregean framework which provided the rationale for the theory’s posit of senses. Within the latter framework, senses are interpreted as abstract objects, but within the former framework, they are interpreted as concrete psychological objects. Hence, plugging the gap in the Chomskyan theory of generative grammar raises the question of whether the theory is to be interpreted realistically or psychologically. The overall theory of generative grammar cannot be a theory of concrete mental/neural reality and have a component semantic theory that is a theory of abstract objects. This situation has to be resolved in favor of a uniform realist interpretation or a uniform naturalist interpretation. Various considerations made a uniform realist interpretation seem preferable. Chomskyan linguistics distinguished between competence, the speaker’s knowledge of the language, and performance, the speaker’s exercise of that knowledge, but, in deªning competence as the speaker’s knowledge of the language, it also distinguished between the speaker’s knowledge and what the knowledge is knowledge of. Furthermore, although generative linguists frequently preached on the theme of linguistics as a natural science, as working linguists, they formulated principles about the structure of sentence types. Since what makes such principles true or false is the structure they are about, the structure of sentences rather than the structure of our knowledge of them, it seemed more in line with linguistic practice to take grammars to be theories of sentence types rather than theories of the speaker’s psychology. Another consideration in favor of adopting a realist interpretation of grammar was that a realist interpretation of senses makes it possible to explain how we can have a priori knowledge of the necessity of an analytic truth such as “Squares are rectangles.” Taking sentences to be

xxvi

Introduction

mental/neural entities cuts intensionalism off from such an explanation. The failure to explain analytic necessary truth would be a serious defect in the position. Similar considerations in connection with syntax and phonology reinforce the case for a uniform realist interpretation. In the cases of both syntax and phonology, a realist interpretation provides a better account of the generality of linguistic structures. Sentences, the structure of which makes grammatical principles true or false, are types, not utterance tokens or mental/neural tokens, and hence sentences are abstract objects. The phonology of a sentence is also an abstract object. If English phonological theory were just a theory of the human English speaker’s voicing apparatus, as Chomsky and Halle (1968) contend, then a human English speaker’s voicing apparatus would be the only instrument that could pronounce English sentences. But it is as implausible to claim that no other apparatus can produce the sound patterns of English sentences as it is to claim that no instrument other than the piano can produce the sound pattern of “Twinkle Twinkle Little Star.” If phonological theory is no more speciªcally about the human voicing apparatus than a tune is speciªcally for the piano, the structure represented in a phonological description, like the structure represented in a musical score, must be understood abstractly in order to be understood as pronounceable by any instrument that can produce tokens of the sound types the description represents. The initial break with naturalism resulting from these reºections was presented in my (1981) book Language and Other Abstract Objects. It argued that linguistics is a science of languages, collections of sentences—not minds—and its theories are thus about abstract objects in the same sense in which mathematical realists claim that mathematical theories are about abstract objects. There was another aspect of my semantic theory that contributed to my change of attitude toward naturalism. This was the theory’s nonFregean deªnition of sense. The virtually universally accepted deªnition of sense then, and no doubt still, is Frege’s (1952, 56–60) deªnition of sense as the determiner of reference. Carnap (1956b, 234), who was Frege’s vicar in our time, deªned it in much the same terms as Frege, as the “general conditions which an object must fulªll in order to be denoted by [a] word.” In stark contrast to this deªnition of sense in terms of a relation between language and the world, my (1972, 1–11) deªnition explained it in terms of factors purely internal to sentence structure. Sense is the determiner of sense properties and relations, like meaningfulness and synonymy, rather than the determiner of referential properties and relations, like denotation and truth.

Introduction

xxvii

The existence of such a non-Fregean deªnition of sense not only showed that a Fregean deªnition is not obligatory for intensionalists, but also raised the question of whether a Fregean deªnition is even desirable. In the late seventies, I began exploring the question of whether the difªculties from which the prevailing intensionalism was shown to suffer could mostly or entirely be laid at the door of Fregean intensionalism. I (1997) eventually came to think that the difªculties that Wittgenstein, Donnellan, Putnam, and Kripke had raised were difªculties for only an intensionalism based on Frege’s notion of sense. The difªculties raised by Wittgenstein’s “Moses” example, Donnellan’s “whale” example, Putnam’s “cat” and “water” examples, and Kripke’s “gold” example all derive from the Fregean requirement that intension determine extension. This requirement imposes too strong a constraint on the assignment of extensions to expressions. Wittgenstein (1953, sec. 84) had made the point in his criticism of Frege in his Philosophical Investigations: the application of a word is not everywhere bounded by rules . . . rules [which] never let a doubt creep in, but stop up all the cracks where it might. Putnam’s and similar examples show that it is impossible to stop up all the cracks, but non-Fregean intensionalists are not Fregean masons who are committed to the principle that the sense of a word gives its application. With a non-Fregean deªnition of sense, the difªculties attributed to intensionalism are shown to belong only to Fregean intensionalism. Consider Putnam’s (1975a) claim that “cat” does not have the sense of “feline animal” because “feline animal” does not determine the referent of “cat” in counterfactual situations where we are fooled into thinking robots are really cats. But on my deªnition of sense, according to which it does not determine reference, “cat” can mean “feline animal” and still refer to robots in Putnam’s counterfactual situation. Furthermore, if, as I (1994a) think, problems about identity and opacity in connection with proper names cannot be handled by direct reference semantics, there is a big advantage in having a non-Fregean intensionalism that is free of the difªculties plaguing Fregean intensionalism. Without our non-Fregean deªnition of sense, we are caught between two sets of very powerful but conºicting intuitions: on the one hand, the intuitions on which the counterfactual criticisms of descriptivism are based, and, on the other, the intuitions on the basis of which Frege introduced senses for names and other referring expressions. For, once we try to satisfy the former by eliminating senses, the

xxviii

Introduction

problems about identity and opacity resurface. Our intensionalism provides a way of doing justice to both sets of intuitions. It handles the counterfactual problems that the critics raise for Frege’s intensionalism because, without the requirement that sense determine reference, the senses of proper names can be purely metalinguistic, containing no substantive predicates to cause counterfactual trouble. The Fregean problems that the critics themselves face are handled simply by the fact that proper names have senses. Thus, on the basis of Semantic Theory and subsequent work on linguistic semantics, together with my (1986, 1990b, and 1992) related work in the philosophy of language, I came to reject Frege’s intensionalism as undesirable as well as unnecessary. The new intensionalism complemented the rejection of naturalism based on linguistic realism. In The Metaphysics of Meaning (1990b), I argued that Wittgenstein’s case for ontological and epistemological naturalism and Quine’s case for his uncompromising empiricism and methodological naturalism were based on weaknesses in the Fregean foundations for realism. Their criticisms were speciªcally tailored to Fregean intensionalism, exploiting problems that arise from its referential deªnition of sense, from its location of semantics within an attempt to construct a logically perfect language, and from its failure to extend realism about logic and mathematics to natural language. I argued that Wittgenstein’s and Quine’s criticisms, as a consequence, have no force against the radically different intensionalism that is available, given a non-Fregean deªnition of sense, the location of semantics within the study of natural languages in linguistics, and a realist philosophy of linguistics.8 The last step needed to arrive at the new version of traditional realist and rationalist philosophy presented in this book was prompted by the realization that, however strong this response to Wittgenstein and Quine, the overall strength of the case against empiricism and naturalism depends also on the success of realism in the philosophy of mathematics. Further, the arguments in both Language and Other Abstract Objects (1981) and The Metaphysics of Meaning (1990b), while putting considerable weight on realism, make almost no attempt to address the issue of realism in the philosophy of mathematics. Finally, as realism was anything but a popular success there, it was necessary to have a sequel to The Metaphysics of Meaning which would vindicate realism in the philosophy of mathematics. 8. As I (1990b, 135–62) showed, even Wittgenstein’s famous rule-following argument can be met on the basis of an account of the use of language that derives normativity from the abstract sentence and expression types of the language that are tokened in its use. See also chapter 4 for a discussion of Kripke’s (1982) rule-following argument.

Introduction

xxix

Nozick (1981, 1) rightly says that questions about the meaning of life, free will, and the nature of the self are the ones that ªrst move us to become philosophers, but he goes on to say, wrongly I believe, that the question of “whether sets or numbers exist can be fun for a time, [but] they do not make us tremble.” If the realist’s answer to the question of whether sets and numbers exist actually is the means of restoring the rationalist conception of knowledge, thereby reversing the naturalist and empiricist thrust of twentieth-century philosophy, that answer ought to shake us up quite a bit, even if we don’t tremble as we would on learning the meaning of life. The vindication of mathematical realism leads straightforwardly to the restoration of the traditional metaphysical conception of philosophy. The rationalist metaphilosophy required for this restoration will be there in the vindication. Thus, restoring the rationalist conception of philosophy depends on providing a convincing vindication of realism in the philosophy of mathematics. Realism has to succeed in the area of philosophy where the issue it addresses properly belongs and where philosophers have the expertise to decide it competently. Moreover, many, perhaps most, philosophers of mathematics today think realism has been refuted on the basis of certain well-known epistemological and semantic objections. Remarks like the following are not infrequent: “It may be true that realism has some advantages over other ways of understanding numbers and sets, but realism is a nonstarter, because, as everyone knows, there are overwhelming objections to saying that mathematics is about abstract objects.” In light of such sentiments in the philosophy of mathematics, and taking into consideration that philosophers outside of a highly technical ªeld defer to the specialists on questions in it, it is clear that an attempt to restore the traditional metaphysical conception of philosophy will not be convincing unless the objections to realism in the philosophy of mathematics are overcome. The bulk of this book is a systematic attempt to overcome them. There are basically three objections in the philosophy of mathematics to the realist claim that mathematics is about abstract objects. Each is based on a problem concerning an aspect of the claim. The most widely inºuential objection turns on the realist’s claim that numbers, sets, and other mathematical objects have no spatial or temporal location. If realism were right about the abstract nature of mathematical objects, we could not have causal access to them, and without that, the critics claim, we could not have mathematical knowledge. The locus classicus of this objection is Benacerraf’s (1983 [1973]) inºuential paper “Mathematical Truth.” A second objection turns on the realist’s claim that numbers, sets, and other mathematical objects are determinate objects.

xxx

Introduction

The criticism here is that reference to numbers (and other formal objects with a similar structure) is indeterminate. The locus classicus here, too, is a paper of Benacerraf’s (1983 [1965]), his “What Numbers Could Not Be.” The third objection turns on the traditional abstract/concrete distinction that the realist presupposes. The objection is that the distinction cannot be coherently drawn. There is no locus classicus in this case, and, in fact, the alleged counterexamples to the distinction do not, as far as I know, ªgure among objections to realism in the literature. Nonetheless, in a certain respect, these examples pose as serious a problem to realism as the ªrst two objections. The sentiment that the ªrst objection refutes realism is sometimes part of a general malaise about the philosophy of mathematics that stems from doubts that any philosophical position can provide an acceptable epistemology and an acceptable ontology for mathematics. Both Benacerraf (1983 [1973]) and Putnam (1994) have expressed the feeling that “nothing works” in the philosophy of mathematics. It should be noted, however, that such pessimism, if justiªed, bodes ill for philosophy as a whole. No part of philosophy is an island. We can’t be satisªed with an epistemology that works for the natural sciences but not for mathematics and the other formal sciences. There are general philosophical questions about which we can say nothing without a position on the ontology and epistemology of the formal sciences, questions like: Is there knowledge which is so certain that it is irrational to doubt it? Even for the skeptic to doubt it? Is there knowledge which is absolutely a priori? (I.e., can we know anything about the world without evidence from experience?) What kinds of things are there? and Can the normative force of value be ªt into a world of fact? These questions cut across the different areas of philosophy. If it were really the case that nothing works in the philosophy of mathematics, there would be little hope for all the other areas of philosophy where those general questions arise.9 More often, however, the sentiment that the ªrst objection refutes realism is not part of a general pessimism about the philosophy of mathematics. Most antirealists aren’t the least bit pessimistic about the prospects for their own philosophy of mathematics. They see the objections as showing that realism is the least plausible of the alternative positions. Their belief is based on the thought that, in having 9. No part of natural science is an island, either. Unless a nominalist program like Hartry Field’s (1980) can be carried through, knowledge of mathematical objects is a component in all scientiªc knowledge, so that no epistemology can work for the natural sciences without having a component that works for knowledge of numbers, sets, and so on. See Malament (1982) for technical objections to Field’s program, and section 2.2 of the next chapter for philosophical objections.

Introduction

xxxi

rejected a naturalistic view of the domain of mathematics, realists ªnd themselves up against the intractable epistemological problem of how the fact of mathematical knowledge can be explained. The consensus among such antirealists is that true realism of the kind Gödel advocated is to be dismissed out of hand. The force of these objections has been greatly exaggerated. The reason that antirealists believe that the epistemological objection is intractable is simply that that’s the way it looks when viewed through the lens of their own empiricism and naturalism. To be sure, all the objections are intractable within empiricism and naturalism. But whether they would be intractable outside them is another question. That question is begged when the argument for the intractability of any of the difªculties in question assumes, as I will try to show it does in the case of the Benacerraªan objections, that they must be overcome within an empiricist or naturalist framework. After all, realists ought to claim that no account of our knowledge of mathematics can be given in empiricist or naturalistic terms. The fact that few of them do make this claim attests to the inºuence of empiricism and naturalism in Anglo-American philosophy. The strength of empiricism derived in large part from its having an apparently unproblematical naturalist ontology which kept pace with the intellectual developments in this century to corroborate its epistemic vision. By contrast, rationalism had no such corroborating ontology. With an unproblematic realist ontology, rationalism could have provided a better explanation of the certainty of mathematical and logical truth than the empiricist’s conventionalist explanations, psychological explanations, and even, as I shall argue, Quine’s holistic explanation. For, as will be explained in chapter 3, section 5, realism can account for the certainty of mathematics and logic in terms of the necessity of mathematical and logical truths, and it can account for their necessity in terms of the unchangeable properties and relations of nonspatial and atemporal objects. Thus, with a corroborating realism, rationalists could have countered the inºuence of empiricism on the basis of the Leibniz-Kant criticism of empiricism that experience cannot teach us why mathematical and logical facts couldn’t be otherwise than they are. Since the absence of a respectable ontology was to a signiªcant extent responsible for rationalism’s troubles, a vindication of realism ought to go far towards resuscitating rationalism. But it works the other way around as well. Absence of a respectable theory of knowledge was to a signiªcant extent responsible for the epistemic vulnerability of realism. In his epistemological objection to realism, Benacerraf (1983 [1973]), 412–15) assumes that mathematical facts must ªgure casually

xxxii

Introduction

in the mathematician’s knowledge of them. He even says: “Some such view must be correct.” He would hardly have felt in a position to make such a strong claim without argument had there been a respectable rationalist theory of knowledge around. Accordingly, it is not unreasonable to expect that a resuscitated rationalist theory of knowledge can go a long way in defending realism against the charge that it cannot explain mathematical knowledge. In any case, a rationalist epistemology is realism’s only hope of explaining how we can have knowledge of objects with which we cannot causally interact. One of the major themes of this book is the inseparability of realism and rationalism. Realism without rationalism is unbelievable and rationalism without realism is unstable. We have seen how implausible realism can be made to seem when its critics are allowed to assume that an account of mathematical truth has to meet an epistemic requirement set in terms of an empiricist theory of knowledge. We will see in the next chapter how, without rationalism, realism easily slides over into a form of antirealism. The integration of realism and rationalism in a single position provides realism with epistemological credibility and rationalism with ontological stability. Here is the layout of the book. The core of its argument is contained in chapters 1, 2, 4, and 5. Their agenda is to show that the apparent force of the principal objections to realism rests on the implicit “divide and conquer” strategy which excludes rationalism from the defense of realism and realism from the defense of rationalism. I will argue that once they are integrated into a single position, there are strong replies to these objections. The replies to the epistemological and semantic objections are a matter of providing a comprehensive defense for Gödel’s formulation of realism. In the case of the epistemological objection, the defense must supply an appropriate rationalist theory of knowledge. This would block the much too fast dismissal of realism on the grounds that taking numbers to be abstract objects makes them unknowable. What is true is only that they are unknowable on the basis of an empiricist epistemology. In the case of the semantic objection, the defense must supply an appropriate intensionalist semantics. This would block arguments from the symmetry of intended and deviant interpretations within the mathematical sphere to the indeterminacy of reference to numbers. In the case of the ontological objection to realism, the defense is a matter of developing a new ontological theory within which the traditional abstract/concrete distinction can be coherently drawn. This new ontological theory turns out to have signiªcant bearings on many of the sciences as well as on a number of philosophical topics.

Introduction

xxxiii

The ªrst two chapters are a reply to the epistemological challenge to realism. Chapter 1 is concerned with preliminaries and chapter 2 contains the reply to Benacerraf (1983 [1973]) and the philosophers who have tried to turn his argument into a refutation of mathematical realism. It is a testament to the force of his statement of the epistemological criticism that it has also convinced a large number of uncommitted philosophers, many of whom recognize realism’s evident philosophical strengths and considerable prima facie plausibility in actual mathematics. For example, there is a body of substantial arguments for mathematical realism, such as Frege’s (1953, 1964) arguments in the Grundlagen and Grundgesetze and Benacerraf’s (1983 [1973]) own argument in “Mathematical Truth,” to the effect that, without mathematical realism, we can’t have the same (Tarskian) semantics for mathematical sentences that we have for other sentences and it is then unclear what we are to say about the semantics of mathematical sentences. Chapter 3 is a digression from the main line of argument, but it complements the reply to the epistemological challenge to realism in chapter 2 by posing an epistemological challenge to antirealists. It argues that antirealists face the epistemological challenge of explaining the special certainty of mathematical and logical knowledge, that Quine’s response fails, and, as a consequence, that antirealists stand no chance of meeting this challenge. If both the reply to the epistemological challenge to realism and this argument against antirealism work, we will have met the epistemological challenge to realism in a way that shows that it is the antirealist rather than the realist who faces an apparently insurmountable epistemic challenge. Hence, if the argument up to this point is correct, the proper attitude toward realism and antirealism ought to be the very opposite of what has been the received opinion in contemporary philosophy of mathematics. The doubts about the prospects for an adequate epistemology that have been widely directed toward realism are more appropriately directed toward antirealism. Chapter 4 replies to the semantic challenge to realism in Benacerraf’s (1983 [1965]) “What Numbers Could Not Be.” I take a novel approach to Benacerraf’s argument. I ªrst develop a strategy for blocking indeterminacy arguments generally and then show that Benacerraf’s argument for the indeterminacy of reference to numbers is a special case of such indeterminacy arguments. The strategy blocks not only Benacerraf’s symmetry claim about intended and deviant interpretations of arithmetic but also related symmetry arguments such as those in Quine’s argument for the indeterminacy of translation, Kripke’s rulefollowing argument, and Putnam’s argument for global referential indeterminacy.

xxxiv

Introduction

Chapter 5 replies to an ontological challenge to realism. This is a challenge to the coherence of realism based on examples that a number of recent philosophers, including some realists, have thought to undermine the traditional abstract/concrete distinction. Some are easily handled on the basis of considerations that have long been part of the realist position but the relevance of which has been overlooked in connection with the putative counterexamples. Others, particularly Frege’s (1953, 35) famous equator example, are more difªcult and far more interesting, requiring a signiªcant addition to ontological theory. I will argue not only that this addition shows that the distinction is not undermined, but also that it provides a new ontological theory with applications to philosophical and scientiªc questions. As we noted above, it would be bad news for philosophy as a whole if nothing works in the philosophy of mathematics. If the line of argument in chapters 1–5 is correct, the news that something works in the philosophy of mathematics ought to be good news for philosophy as a whole. Chapter 6 presents a rationalist metaphilosophy. It develops it out of the principles underlying the arguments of the previous chapters. The ªrst section of the chapter explains how the rationalist epistemology in chapter 2 for knowledge in the formal sciences can be extended to provide a rationalist epistemology for certain types of philosophical knowledge as well. Our aim is to construct a uniªed conception of what it is to explain synthetic a priori knowledge in the formal sciences and in their philosophical foundations. The second section of the chapter examines some of the philosophical implications of the metaphilosophy. Here I try to set out some new thoughts about the rationalism/empiricism controversy over the existence of synthetic a priori knowledge, Carnap’s positivist critique and Quine’s naturalist critique of metaphysical philosophy, the philosophical distinction between internal and external questions, and the place of skepticism in a world of knowledge.

Chapter 1 Philosophical Preliminaries

1.1 The Framework Poets often comment on the multifariousness of things. Hamlet’s rebuke of Horatio is familiar, and Louis MacNeice, in his poem “Snow”, tells us that “[t]he world is crazier and more of it than we think, incorrigibly plural.” Realists in the philosophy of mathematics make a speciªc claim about just how much crazier and more populous the world is than the familiar classiªcation “animal, vegetable, or mineral” suggests. Realists think there are things that (necessarily) have neither spatial nor temporal location: abstract objects, such as numbers, sets, propositions, proofs, sentences, and meanings. Being an object that (necessarily) has no spatial or temporal location is the core of the conception of an abstract object in realist thought from Plato to Gödel. It is also how an abstract object is understood in antirealist criticisms of realism. Thus, Goodman and Quine (1947) write: We do not believe in abstract objects. No one supposes that abstract entities—classes, relations, propositions, etc.—exist in space-time; but we mean more than this. We renounce them altogether. (Italics mine) Furthermore, taken as the essential deªnition of “abstract object,” this conception is the most compact one that ªts the usage of both realists and their critics. Subtracting nonspatiality or atemporality makes the deªnition inadequate, since the resulting deªnition no longer captures the notion of abstractness, while adding causal inertness, mindindependence, or some other property that abstract objects are generally taken to have, makes the deªnition redundant, since the original deªnition already implies those properties (as will be argued below). Moreover, when we add the deªnition of “concrete object” as something that can possibly have a spatial or temporal location, we concisely capture the intuitive distinction between the abstract and concrete.

2

Chapter 1

I will call the realist position I am defending in this book “general realism.” General realism is realism in general. It makes the indeªnite claim that there are abstract objects. General realism is a view about ontology. In addition to general realism, there are particular realisms, such as mathematical realism, logical realism, and linguistic realism. A particular realism makes a claim that the domain of a formal science contains one or another kind of abstract object, e.g., numbers, propositions, or sentences. A particular realism is a view in the foundations of a formal science concerning what type of objects knowledge in that science is about.1 What makes someone a realist is his or her acceptance of abstract objects; what makes someone a realist of a particular kind is his or her acceptance of abstract objects of that kind. Kinds here are kinds of structure that abstract objects have, e.g., mathematical, logical, or linguistic structure. Commonly, acceptance of abstract objects of a certain kind is the result of accepting theories about abstract objects of that kind, but being a realist of a certain kind does not depend on having much theoretical knowledge. Plato was a realist about numbers before there was a theory of arithmetic. The elucidation of the kinds of structure abstract objects have is the task of pure mathematics, pure logic, and pure linguistics. A pure science is, according to our realism, pure because it is about abstract objects pure and simple. Applied sciences are distinguished from pure ones by their concern with concrete objects, but what speciªc kind of objects those sciences are about is a more complex question, which we shall consider in chapter 5. If any particular realism is true, general realism is true, but, of course, the truth of general realism does not entail the truth of any particular realism. Moreover, philosophers all of whom would subscribe to general realism might hold different ontological opinions about different particular realisms. Someone might, for example, be a logical and mathematical realist without being a linguistic realist. Frege (1964, 13) took such a position. Some mathematicians seem to be arithmetic realists, but not geometric realists. It might even be possible for someone—Chomsky (1986, 33) seems to be an example—to be a linguistic conceptualist while being a mathematical realist, but there is room for doubt about this because objections to realism in one area seem to apply to realism in other areas. However, some particular realisms are clearly 1. For the time being, I will use “formal science” to denote mathematics, logic, and linguistics, without implying any doctrine about the nature of those sciences. At the end of chapter 3, I will suggest such a doctrine.

Philosophical Preliminaries

3

interdependent. For example, inscriptionalist nominalists like Goodman and Quine (1947) could not be linguistic realists. This way of formulating realist and antirealist positions has important consequences for the ontological controversies in the foundations of the various formal sciences. It suggests that the widespread practice of evaluating ontological positions exclusively within the foundations of the directly relevant formal science could be a mistake for one or another of the positions. It can be argued that isolating the controversy between mathematical realists and mathematical antirealists from ontological controversies in the foundations of the other formal sciences has kept realists from making as strong a case for their position as they could. An example is the following. Inscriptionalist nominalists in mathematics claim that we can do justice to mathematical practice without countenancing abstract objects by taking mathematics to be about mathematical expressions. Such nominalists must mean expression tokens, since expression types—in the standard Peircean sense—are abstract objects. But, since there are not enough actual expression tokens for mathematics to be about them, the nominalist’s program in mathematics requires, at the very least, some way of characterizing the class of possible tokens of mathematical expressions on the basis of a sample of actual tokens. Now, this nominalist program in mathematics is a special case of the Bloomªeldian (1936) nominalist program in linguistics. That program takes linguistic reality to be the acoustic phenomena of speech. Given how few sentences of a natural language are exempliªed in actual speech, Bloomªeldian linguists had to construct a categorical structure that characterizes the possible sentence tokens of the language on the basis of procedures for segmenting and classifying the items in a corpus of linguistic tokens. Chomsky (1975), however, showed that no such procedures exist, because there is no way to take the inductive step from distributional properties of actual tokens to grammatical categories. Since the program of the inscriptionalist nominalist in mathematics requires essentially the same bottom-up construction of essentially the same categorical structure, Chomsky’s argument applies equally to that program, putting the inscriptionalist nominalist in mathematics in the position of having to do something shown to be a lost cause in linguistics. If these considerations are right, mathematical realists have missed the opportunity to strengthen their case against one form of antirealism. In this book, I will assume general realism for the sake of argument. The assumption begs no questions, because the main line of argument here is not to establish realism but only to defend it against criticisms

4

Chapter 1

that we cannot have knowledge of abstract objects, that we cannot determinately refer to them, and that we cannot distinguish them from concrete objects. For example, the assumption that there are abstract objects bears no weight in the explanation of how we can have knowledge of abstract objects, since the antirealist who challenges the realist to explain how we can have such knowledge without contact assumes realism for the sake of argument—otherwise the antirealist would have no ontological position to challenge. In the same spirit, we can also assume mathematical, logical, and linguistic realism. The epistemological, semantic, and ontological challenges to particular realisms arise from their parallel claims that the objects in the domain of their various sciences are abstract, not from any aspect of their mathematical, logical, or linguistic structure. For example, since the question of how we know about objects of one kind with which we can have no contact does not differ from the question of how we can know about objects of other kinds with which we also can have no contact, an epistemology that answers the epistemological challenge for one kind of abstract object answers it for all kinds. This does not mean that the epistemologies for mathematical, logical, and linguistic knowledge, and even for varieties of such knowledge, such as arithmetic and geometric knowledge, will not differ from one another in various ways. But such differences will not affect the general question. It is not an aim of this book to provide a comprehensive argument for general realism or any particular realism. Explaining how knowledge of abstract objects is possible, how reference to them can be determinate, and how they differ from concrete objects does not establish that there are such objects. Nonetheless, at various points in the course of the book, I will try to strengthen the case for realism by supplying reasons to think that abstract objects exist and by exposing weaknesses of one or another form of antirealism. For example, in section 2.2, I will present two philosophical objections to Hartry Field’s (1980) antirealism, and in chapter 2, I will strengthen Benacerraf’s (1983 [1973]) argument that realism provides a better account of the semantics of mathematical sentences than antirealism. Even though it is not our aim to argue for realism, we should explain what form of argument the realist can give for the existence of abstract objects. To establish general realism, it sufªces to establish mathematical realism, logical realism, or linguistic realism. The argument for establishing one of them is an argument to show that the particular realism in question is preferable to its rival particular nominalisms and conceptualisms as an account of the objects of knowledge in the relevant formal science. Hence, to explain how a realist can argue

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systematically for the existence of abstract objects in the domain of a formal science, we have to look at what is involved in showing that realism provides the best account of the objects of knowledge in that science. A particular realism is an ontological position in the foundations of a particular formal science. We can distinguish the foundations of mathematics, the foundations of logic, and the foundations of linguistics from mathematics proper, logic proper, and linguistics proper. The former are branches of the philosophy of science, concerned with a philosophical understanding of the results in the particular sciences proper. Mathematicians, logicians, and linguists, like other scientists, typically conduct their professional business with little interest in the philosopher’s concern with the nature of numbers, sets, propositions, sentences, and so on. The attempt of philosophers of science to understand the nature of such entities takes the form of a dialectic among nominalists, conceptualists, and realists, where the issue is which position makes the best scientiªc and philosophical sense of the science proper. Mathematics, logic, and linguistics tell us that statements like (1)–(4) are true: (1) There is a perfect number less than seven. (2) There are propositions that imply everything. (3) There are English sentences with no phonologically realized subject. (4) There are inªnitely many numbers (propositions, sentences). In virtue of our accepting what mathematics, logic, and linguistics tell us, philosophers are committed to the existence of numbers, propositions, and sentences. Philosophers who accept the formal scientist’s claims about numbers, propositions, or sentences are required to acknowledge that there are such objects. But that acknowledgment does not require taking a stand on the issue of what kind of things numbers, propositions, and sentences are. Thus, it is mistaken to think that the ontology of mathematics is an issue that can be settled on the basis of a principle of ontological commitment. We may concede for the sake of argument that, as Quine (1961a, 13–14) says, “a theory is committed to those and only those entities to which the bound variables of the theory must be capable of referring in order for the afªrmations made in the theory to be true.” But this concession is not enough to establish that “[c]lassical mathematics . . . is up to its neck in commitments to an ontology of abstract objects.” The principle of ontological

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commitment may show that classical mathematics is up to its neck in numbers, but it can’t show that the numbers we are up to our necks in are abstract objects. This requires a philosophical argument to show that numbers are abstract rather than concrete objects. Realist arguments for the existence of abstract objects are arguments that numbers, sets, propositions, proofs, sentences, meanings, or similar objects are abstract objects. By the same token, antirealist arguments against the existence of abstract objects are arguments against those objects’ being abstract objects. Hence, as part of the philosophical dialectic in the foundations of the formal sciences, an argument for realism, as well as one for nominalism or conceptualism, is required to show that the ontology in question best accommodates the full range of facts in the formal science and best satisªes our philosophical intuitions. Thus, an argument that the objects of a formal science are abstract is successful just in case it shows that a realist ontology is best in these respects.2 All sides in this dialectic have a stake in seeing to it that the scientiªc conclusions from which their philosophical arguments proceed enjoy a large measure of independence from partisan philosophical intrusion. The more such intrusions that philosophers allow themselves, the more they open themselves to the charge that they are arguing from their own theory, and the less convincing their argument for their ontological position becomes. On the other hand, the more they base their ontological position on philosophically unadulterated scientiªc conclusions, the more their position can claim to have the backing of impartial science, and the more convincing it becomes. The recognition on all sides that they have a stake in keeping such intrusions to a minimum is the gyroscope that restores the balance of the dialectic whenever partisan ontological considerations intrude to deºect it from its proper course. 1.2 Two Forms of Antirealism In this section, I consider brieºy two forms of antirealism; in the next section, I want to consider three approaches to the foundations of the formal sciences that, at least as their advocates present them, are forms of realism. 2. The realist’s claim that abstract objects exist is sometimes criticized as inchoate because, as the criticism runs, abstract objects are characterized exclusively on the basis of the negative property of not having spatial or temporal location. This is simply false. Their characterization includes various positive properties: being an object, having a formal structure, having the properties and relations in that structure necessarily, existing necessarily (if they exist), being objective, and being knowable (if they are knowable) on the basis of reason alone.

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1.2.1 The Kantian Compromise Kant sought a compromise between empiricism and rationalism because he thought neither can explain the full range of our knowledge, especially our mathematical knowledge. Rationalists went too far in allowing reason unrestricted speculative freedom in metaphysics, while empiricists went too far in trying to curb the excesses of reason in metaphysics. As Kant saw it, Hume’s view that all genuine knowledge falls into either the category Matters of Fact or the category Relations of Ideas throws out mathematics along with metaphysics. The arithmetic truth “Seven plus ªve equals twelve” falls outside the category Relations of Ideas because its predicate “is twelve” is not a component of the concept “seven plus ªve.” Such truths, as Locke had observed, are not triºing, that is, not analytic in Kant’s sense of literal concept containment, but express “real knowledge.” Yet they also fall outside the category Matters of Fact because they are necessary. Experience, as Kant argued, can teach us that a judgment is true, but not that it couldn’t be otherwise. Hence, mathematics as well as metaphysics falls between the two Humean stools. Accordingly, if Hume throws out metaphysics because of the epistemic status of its principles, he has to throw out mathematics too. Mathematical truth is as much a mystery for Humean empiricism as metaphysical truth. Humean empiricism explains why “Squares are rectangles” is true: the deªnition of “square” categorizes squares as a certain kind of rectangle. But it provides no more of a notion of why “Seven plus ªve equals twelve” is true than of why a metaphysical principle like “Every event has a cause” is. Kant was awakened from his dogmatic slumbers by the clatter of mathematics going out the window. Rationalism, as Kant saw it, was principally responsible for the speculative excesses of metaphysics. On the one hand, it draws the sharpest possible distinction between the world and the cognitive faculties of our minds and, on the other hand, it imposes no curb on the use of those faculties. Rationalism thus allows us to use our reason to try to obtain knowledge of objects to which our faculty of sensible intuition can bear no relation. In this, reason’s reach exceeds its grasp. This is what Kant saw as the source of the striking lack of progress in metaphysics, and he concluded that the window has to be shut both to keep mathematics in and also to keep metaphysical excesses out. Kant’s Copernican revolution nails the window shut. It makes the existence of objects in the world depend on our cognitive faculties. At the beginning of section 22 of the B-version of the Transcendental Deduction—entitled “The Category has no other Application in Knowledge than to Objects of Experience”—Kant ([1787] 1929) states his basic objection to the rationalist claim that we can have knowledge of objects to which our faculty of sensible intuition can bear no relation:

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To think and to know an object are thus by no means the same thing. Knowledge involves two factors: ªrst, the concept, through which an object in general is thought (the category); and secondly, the intuition, through which it is given. For if no intuition could be given corresponding to the concept, the concept would still indeed be a thought, so far as its form is concerned, but would be without any object, and no knowledge of anything would be possible by means of it. So far as I could know, there would be nothing, and could be nothing, to which my thought could be applied. Kantian philosophy came to be highly inºuential in the foundations of mathematics. Brouwer ([1913] 1983) developed his “neointuitionism”—what is generally called “mathematical intuitionism”— on the basis of, as he put it, (Kant’s) “old intuitionism.” The new intuitionism came from the old by “abandoning Kant’s apriority of space but adhering the more resolutely to the apriority of time.” The mind divides the stream of moments of time into “qualitatively different parts,” thereby “creat[ing] not only the numbers one and two but also all ªnite ordinal numbers.” Kantian philosophy is also easily recognized in Chomsky’s (1965, 1986) conceptualist view of linguistics. Although Chomsky’s linguistic conceptualism is more sophisticated than Brouwer’s mathematical conceptualism in its account of how the mind creates sentences and is more scientistic in its strongly biological ºavor, it is the linguistic counterpart of Brouwer’s Kantian conception of mathematics.3 But Kant’s Copernican revolution rests on philosophical doctrines too dubious to ground particular conceptualisms like Brouwer’s and Chomsky’s. Everyone is familiar with the standard problems with transcendental idealism, such as the fact that Euclidean geometry, alleged to be an a priori necessary truth, turned out to be an a posteriori contingent falsehood. Not only did the fate of Euclidean geometry deal a heavy blow to conªdence in the Kantian explanation of synthetic a priori knowledge generally, but the role of geometry in relativity theory provided a paradigm for treating other alleged a priori necessary truths as a posteriori contingent truths—highly theoretical but nonetheless empirical in nature. The proposal to replace classical logic with a more suitable quantum logic can be seen as based on that paradigm. 3. Here is another case where it is a good idea for realists to avoid the parochial practice of defending a particular realism in isolation from the controversies involving other particular realisms. For many of the arguments that I and Postal (1991) brought against Chomsky’s linguistic conceptualism can, with only trivial adaptation, be brought against Brouwer’s mathematical conceptualism.

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Of more direct concern in the present context are two other problems. One is that Kant’s transcendental idealism does not succeed in solving the problem about necessity that he raised in connection with empiricism. However Kant’s transcendental idealism is understood, it locates the grounds of mathematical facts within ourselves in at least the minimal sense that it entails that such facts could not have existed if we (or other intelligent beings) had not existed. But, as Frege (1964, 1–25) pointed out, locating the grounds of necessity within us does not explain the necessity of mathematical truth. It at best explains why we naturally take mathematical truths to be necessary.4 In treating necessity as an aspect of the psychology of contingent beings, Kant shifts the basis of the explanation from mathematics to our contingent psychology. The latter provides no grounds to explain why the truths of mathematics couldn’t be otherwise. The other problem is the veriªcationism in Kant’s position. It is clear from the last sentence in the preceding quote that he is saying that the possibility of objects—and hence the possibility of knowledge of them—depends on the possibility of verifying their existence through acquaintance in intuition or inner experience. But why is the fact that “no intuition could be given corresponding to [an] object” a reason to think that the concept couldn’t have an object? Frege (1953, 101) famously observed that we can know a great deal about mathematical objects to which our faculty of sensible intuition can bear no relation: Nought and one are objects which cannot be given to us in sensation. And even those who hold that the smaller numbers are intuitable, must at least concede that they cannot be given in 1000 intuition any of the numbers greater than 10001000 , about which nevertheless we have plenty of information. Kant himself does not provide an external reason to think that the possibility of objects of one or another kind depends on the possibility of our being able to verify their existence in intuition or inner experience. Attempting to motivate such veriªcationism from within transcendental idealism would beg the question, since the core 4. To see this, consider a contemporary Kantian position like Brouwer’s ([1913] 1983). On Brouwer’s intuitionist position, the justiªcation for formal beliefs depends on some sort of introspective contact with internal objects. But, as Brouwer ([1913] 1983, 69) readily admits, the objects of mathematical knowledge are created by the mind out of mental stuff. Since the created objects share the contingency of their creator and the mental stuff of which they are created, mathematical conceptualism takes numerical relations to be contingent relations, and truths about such relations to be contingent truths. Psychologistic accounts, to echo Frege, at best explain why we think of arithmetic truths as necessary, but not why they are necessary.

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veriªcationist idea that what exists depends on our cognitive makeup is a basic assumption of transcendental idealism. Moreover, there does not seem to be any motivation for veriªcationism outside transcendental idealism either. The irredeemable failure of the most notable attempt to motivate veriªcationism independently of transcendental idealism, namely the positivists’s linguistic attempt to equate the meaningfulness of a sentence with its veriªability and the meaning of a sentence with its veriªcation conditions, is well known.5 Veriªcationists have failed to motivate an epistemic constraint on what there is, and no account of our cognitive faculties, scientiªc or otherwise, supports the idea that the limits of those faculties should be the touchstone of existence. Despite the appeal of veriªcationism as a quick refutation of metaphysics, veriªcationists have yet to explain why objective reality should have to pass a knowability test framed in terms of human knowledge. There is little behind veriªcationism but epistemic chutzpah. 1.2.2 Fictionalist Nominalism In this subsection, I will present arguments against Field’s (1980, 1989) ªctionalist nominalism. Field’s argument against mathematical realism is an attempt to turn Quine’s and Putnam’s indispensability argument against their conclusion that we are committed to abstract objects because they are indispensable in doing natural science. Field argues that natural science can be done without numbers, and, on the basis of this, that simplicity requires us to limit our ontology to natural objects. From the standpoint of general realism, it is initially questionable for Field to base his case against realism on the dispensability of numbers for doing natural science. It is, of course, quite legitimate for Quine and 5. The equation is contravened by the simplest facts of meaning in natural language, e.g., the veriªability conditions of “John is ªve feet tall” and “If John were one foot taller, he would be six feet tall” are the same, but the sentences are not synonymous. Moreover, as a number of philosophers have observed, the equation condemns itself as meaningless because it is not veriªable. Some veriªcationists have tried to escape this consequence by saying that the equation is a convention, analytic, or meaningful in a noncognitive sense, as ethical injunctions are sometimes thought to be. Taking it as a convention pulls its teeth. Only if one is antecedently in agreement with the positivist is there any point in adopting that convention rather than one the metaphysician might propose. Taking the equation as analytic is not helpful either, since the only sense of analyticity on which it is analytic is one based on Carnapian meaning postulates, but again we can adopt anything we like as a meaning postulate. Finally, the positivist cannot get off the hook by taking the equation to have the force of an ethical injunction. A sentence expressing the equation still has to be meaningful in the cognitive sense. How can someone use the sentence to recommend that meaningfulness be taken as veriªability if the sentence has no cognitive content?

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Putnam to use an indispensability argument restricted to natural science to argue for realism, but Field’s use of dispensability to argue against realism begs the question. Unless Field assumes epistemological naturalism, the dispensability of reference to numbers in natural science does nothing to show that a commitment to abstract objects can be avoided, since they might be indispensable in formal science. Even if, as Field claims, we can do natural science without a commitment to abstract objects, this provides no reason to think that formal science can be done without such a commitment. Field does not consider the prospect of establishing realism on the basis of an argument for the indispensability of abstract objects in pure mathematics, logic, or linguistics. What he (1980, viii, 6) says is that Quine’s and Putnam’s indispensability argument is the only serious argument for realism that he knows, and that other arguments for realism are “unpersuasive.” But he doesn’t say what other arguments he has considered or why he ªnds them “unpersuasive.” There are serious arguments for mathematical, logical, and linguistic realism, from Frege’s arguments to current arguments in the foundations of linguistics. No doubt Field ªnds them “unpersuasive,” but, having provided no critical examination of them, the real work of refuting realism is yet to be done. It may be here that a tacit assumption of epistemological naturalism enters Field’s argument. But this assumption will again beg the question against the realist if there is no argument to support it. I can ªnd no direct argument for the assumption in Field’s work, but it is plausible to think that he would want to argue that there is simply no plausible alternative to epistemological naturalism. Field (1989, 59) thinks that realists have “to postulate some aphysical connection, some mysterious grasping,” and hence he presumably would want to say that this rules out non-naturalist epistemologies. If this is the argument for epistemological naturalism, the next chapter will show that it is not a good one. Field’s positive view is that the truths in mathematics proper are truths in a ªction. The problem with this view has been missed because philosophers have been too parochial, failing to consider issues in the foundations of mathematics in relation to issues in the foundations of the other formal sciences. From an unparochial standpoint, a test of the adequacy of a view taken in the foundations of one formal science is whether the view is adequate in the foundations of other formal sciences. Hence, we ask: Is the sort of view that Field has about the truths of mathematics adequate as a view about the truths of linguistics? What happens when we try to carry it over to the foundations of linguistics?

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The counterpart in the foundations of linguistics of Field’s claim that there are no numbers is the claim that there are no sentences. The counterpart to Field’s claim that mathematical truths like “Two plus two is four” are not about numbers is the claim that linguistic truths like “‘Visiting relatives can be annoying’ is ambiguous” are not about sentence types. Types are abstract objects. Further, the sentences that are intended to express Field’s ªctionalist nominalism can’t be treated as tokens of grammatical types, since if there are no sentence types, there are no tokens of sentence types either. This raises the problem of whether sense can be made of the discourse in which Field expresses his mathematical ªctionalism, say, Science without Numbers, Field’s positive view. To avoid the consequence that his view is self-defeating because it makes its own linguistic expression impossible, Field has to interpret his discourse as consisting of concrete objects, deposits of ink on paper. But so construing the discourse drives Field’s ªctionalist nominalism right back to the dubious inscriptionalist nominalism that he originally—and for good reasons—was at such pains to avoid. Recall Field’s (1980, 6) statement that his nominalism . . . is in sharp contrast to many other nominalistic doctrines, e.g., doctrines which reinterpret mathematical statements as statements about linguistic entities or about mental constructions. Field was quite right to try to avoid inscriptionalist and conceptualist attempts to understand the vastness of mathematical reality in terms of a paltry ªnite collection of deposits of ink, graphite, and chalk, or of mental events. My point is that, in the end, Field can’t avoid this. He cannot restrict his view to the foundations of mathematics, since the success of linguistic realism entails the success of realism in general. Since Field has to defend inscriptional nominalism in linguistics, not only is the plan for a clean surgical strike against realism that Field announces in Science Without Numbers unworkable, but the problems with inscriptionalist nominalism, which doomed the Goodman and Quine (1947) enterprise, undercut Field’s nominalist program. (See Katz 1996c for a discussion of these issues.) There is a further criticism concerning Field’s (1980, 1989) claim that reference to numbers is reference to ªctional entities and mathematical truth is truth in the ªction of mathematics. Field (1989, 2–3) says: “The sense in which ‘2 + 2 = 4’ is true is pretty much the same as the sense in which ‘Oliver Twist lived in London’ is true.” For Field, the former statement is true according to the well-known arithmetic story, while the latter statement is true according to the well-known Dickens story. Some such view is needed in Field’s position, since

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otherwise it does not distinguish truths like ‘2 + 2 = 4’ from falsehoods like ‘2 + 2 = 17.’ There is, however, an essential difference between mathematics and ªction that shows that truth in mathematics cannot be taken to be truth in ªction. It is that consistency is a necessary condition for truth in mathematics but not for truth in ªction. A ªctional character’s having incompatible properties (in the ªctional corpus) doesn’t rule out the existence of the ªctional character, but a mathematical object’s having incompatible properties does rule out the existence of the mathematical object. If I recall correctly, Dr. Watson is attributed incompatible properties. At one place in the Sherlock Holmes corpus, Watson’s Jezail bullet wound is in his shoulder, while at another place in the corpus, the wound is in his leg. However, the discovery of this inconsistency does not show that Dr. Watson does not have the ªctional existence we took him to have. The inconsistency doesn’t show that he has the status of Hamlet’s children. Nor does it show that the adventures of Sherlock Holmes are committed to implying everything, e.g., that Holmes is Inspector Lestrade or that Watson is Professor Moriarty. Contrast this with the discovery of an inconsistency in mathematics, which automatically establishes nonexistence. (This criticism does not depend on the example. It is pointless to quibble about examples, e.g., by arguing that the locations are mentioned in different Arthur Conan Doyle stories [so one of them must be false in the Sherlock Holmes corpus]. There are other actual cases in literature, but hypothetical cases will do just as well.)6 Further, it does not help to try to distinguish mathematical ªction from, as it were, ªctional ªction. It doesn’t help to say that it is part of our logical pretense about mathematics (but not about ªction) that nothing can have incompatible properties. If arithmetic and literature are both ªction, why should there be this logical pretense in the one case but not in the other? If mathematics is simply a story that mathematicians tell, in which part of the story is that consistency is a condition for existence, then, on the one hand, their story could have been like ªctional ªction in tolerating inconsistency, and, on the other hand, their story could change so that future mathematical ªction is like ªctional ªction in tolerating inconsistency. But it is clear that neither of these scenarios is possible. There can’t be inconsistency in mathematics—that’s a logical impossibility; there can be inconsistency in ªction— 6. Kaufmann (1961, 375–377) discusses the prima facie contradiction in Macbeth and two clearcut contradictions in Goethe’s Faust. Goethe is quoted as saying, “The more incommensurable and incomprehensible . . . a poetic production is, the better.”

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that’s both a logical possibility and an actuality. There is a basic difference between mathematics and literature: consistency is an absolute constraint in mathematics but not in ªction. The explanation is obviously that ªction is ªction and mathematics is fact. 1.3 Wrong Turns that Point in the Right Direction To prepare for the response to the epistemological challenge to realism in the next chapter, I want to examine a number of unsuccessful epistemological approaches to knowledge in the formal sciences that philosophers, including realists, have taken. My interest in these approaches is with what can be learned from them. I will argue that the nature of their failure suggests the direction that we should take in looking for a successful account of formal knowledge. We will look at three epistemological approaches: the classical Platonist’s, the contemporary Aristotelian’s, and the naturalized realist’s. I will argue that all of them pursue a reconstruction of formal knowledge as knowledge that is au fond a matter of acquaintance and that this underlying empiricism is what has been responsible for the failure of realists who take one of these approaches. Neither the notion of acquaintance with abstract objects nor the notion of acquaintance with concrete objects is a viable option for realists. The only way for realists to have any chance of meeting the epistemological challenge is for them to eschew an epistemology based on acquaintance. As will be seen in the next chapter, this means that experience cannot be allowed to enter in any way, shape, or form—that nothing outside reason can provide grounds for mathematical, logical, or linguistic knowledge. I want to show that, even though classical Platonism, contemporary Aristotelianism, and naturalized realism claim full realist credentials, and even though they share important features of traditional realism and rationalism, their epistemology is empiricist at the core, and hence their failure has no implications for a genuinely rationalist approach. If this can be shown, it will make it clear that their failure does not reºect adversely on the realist’s prospects for meeting the epistemological challenge. 1.3.1 Classical Platonism Plato’s doctrine of anamnesis was the ªrst epistemology proposed for realism. It is condemned by antirealists as myth, but this is too strong, since myths can be compelling ways of expressing ideas that would have been hard or impossible to express literally at the time. Plato’s myth of the metals in the Republic can be seen in retrospect as a ªgurative expression of the idea of genetic determinism. But Plato’s

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myths of the cave and of recollection, even demythologized in a way that provides us with a literally expressed epistemology, ought to be rejected, especially by realists, as articulating an incoherent epistemology for abstract objects. The strategy behind the doctrine is to extend the range of the perceivable to include abstract objects. But the strategy makes knowledge of abstract objects depend on acquaintance with them. The idea is that the source of the knowledge we obtain through recollection is perceptual contact with the objects known. What is bad about this idea is that it buys into the core empiricist notion that all our knowledge ultimately derives from acquaintance. That notion is not transplantable from a naturalistic ontology to a realist ontology. Since abstract objects are outside the nexus of causes and effects and thus perceptually inaccessible, they cannot be known through their causal effects on us. In buying into the empiricist idea that knowledge is based on acquaintance, classical Platonists render their overall position incoherent. It is as senseless to suppose that we can be acquainted with atemporal abstract objects prior to our entrance into the spatiotemporal world as it is to suppose that we can be acquainted with them afterwards. Acquaintance requires a point of contact, some temporal position that both we and the object occupy, but there cannot be such a point in the case of objects that have no temporal location whether during the soul’s existence in this world or prior to its incarnation.7 It has seemed to many—myself (1981, 200–202) included—that Gödel was a classical Platonist of some kind. In contemporary philosophy of mathematics, Gödel’s ([1947] 1983, 483–84) remarks about perceiving abstract objects are widely interpreted as expressing the view that we have perceptual acquaintance with abstract objects. But, despite the fact that this interpretation seems to ªt those remarks themselves, one cannot help having qualms about attributing so obvious an inconsistency to so subtle and powerful a thinker. Such qualms have induced no less an authority on Gödel than Hao Wang to suggest that those problematic references should be taken as a metaphor for some noncausal form of apprehension. As Yourgrau (1989, 399) reports: . . . we are able to “see” [mathematical objects] only because there “is” [an objective] mathematical world. How can we, however, apart from using our ªve senses, see anything that is not in our 7. Even supposing there are beings in another world that can make some sort of contact with abstract objects, they would have to be atemporal and then the same problem would arise when we try to imagine that we could be those beings or be continuous with them.

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minds? The “interaction” must be something different from that between us and the physical world. But, desirable as it is to try to ªnd a more charitable interpretation, this suggestion does not help. The metaphorical construal removes the incoherence by denying that the interaction is anything like causal interaction with natural objects. But that is all it does. As Benacerraf ([1973] 1983, 415–16) pointed out, in telling us only what contact is not, the construal does not tell us what Gödel’s special sort of contact is. It affords us no idea of what the metaphorical use of “perception,” “seeing,” or “interaction” might amount to. We can set up the Aristotelian scheme: grasping is to abstract objects as perceiving is to concrete ones. But once the notion of sensory contact is factored out, as it must be in order to obtain a charitable interpretation of Gödel’s references to perception, all that is left of the interaction analogy is the unhelpful claim that mathematical intuition and sense perception are similar in that both are ways of coming to know. It is clear that the strategy of trying to extend the range of perceivable objects to include abstract objects presents the classical Platonist with the dilemma of choosing between literalness and incoherence on the one hand or nonliteralness and inanity on the other. But Gödel ought not to be interpreted as a classical Platonist. True enough, his statement that we have “something like a perception . . . of the objects of set theory” seems to encourage such an interpretation, but, in the broader context, his use of “perception” does not seem to have been intended in this way. The passage that immediately follows the one containing this statement suggests that Gödel is anything but a classical Platonist. Gödel (1947 [1983], 484) says: It should be noted that mathematical intuition need not be conceived of as a faculty giving an immediate knowledge of the objects concerned. Rather it seems that . . . we form our ideas also of those objects on the basis of something else which is immediately given. Only this something else here is not, or not primarily, the sensations. . . . It by no means follows, however, that the data of this second kind, because they cannot be associated with actions of certain things on our sense organs, are something purely subjective, as Kant asserted. Rather, they, too, may represent an aspect of objective reality, but, as opposed to sensations, their presence in us may be due to another kind of relationship between ourselves and reality. Gödel is clearly saying that the relationship between ourselves and mathematical reality is not the kind of relationship that we have to

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17

physical reality in the case of sensations. Mathematical knowledge, he says explicitly, is not the causal effect of “actions of certain things on our sense organs.” Gödel is not much more informative about what he does mean when he talks about this relationship than Frege is when he talks about grasping—though Gödel is more explicit than Frege about what he does not mean. Gödel’s reluctance to say what this relationship is can be easily explained. Gödel was reluctant to try to characterize the relationship because he had no epistemology to put in place of classical Platonism. This explanation not only ªts what he says, but it ªts the fact that Gödel spent a great deal of time studying Husserl’s phenomenology in the hope of obtaining insight into the given in mathematics (Wang, personal communication). On this more charitable interpretation, Gödel should be credited with taking the signiªcant step of breaking with classical Platonism. Realists who recognize the dilemma in which classical Platonism puts them ought to acknowledge the importance of Gödel’s break with classical Platonism and its strategy of trying to extend the range of perceivable objects to include abstract objects. They should follow Gödel’s lead in looking for a different strategy. 1.3.2 Contemporary Aristotelianism Like classical Platonists, the philosophers I am calling “contemporary Aristotelians,” who sometimes call themselves “Platonists,” also pursue the strategy of seeking to extend the range of the perceivable to include the objects of mathematics, but, unlike classical Platonists, they locate the objects of mathematical knowledge in the natural world. Maddy (1980, 1989, 1990) is a prominent example of such a philosopher. Her aim is to avoid the problem of access to abstract objects, thought of as denizens of a Platonic realm, on the basis of the idea that mathematical knowledge is knowledge by acquaintance. According to her (1980, 179), if there are three eggs to be seen in a carton, then the set of the eggs is to be seen there too. On this approach, we have a posteriori perceptual knowledge of sets just as we have of eggs. No doubt Maddy’s position avoids the dilemma facing classical Platonism, since, on her position, acquaintance with mathematical objects is acquaintance with natural objects. But it looks as if her attempt to naturalize abstract objects is a classic case of out of the frying pan into the ªre. Since natural objects are not nonspatial, atemporal, or causally inert entities, her set located in that egg carton must be just another concrete object like the eggs. But if sets are sets and her position is Platonism, the set can’t, logically speaking, be a concrete object like the egg carton. So we have a new dilemma: either the terms in question

18

Chapter 1

have their standard meanings and her Platonism is as incoherent as classical Platonism, or the terms do not have their standard meanings and it is misleading of her to refer to her position as “Platonism.” Maddy has faced up to this dilemma. Confronted with the fact that, on the standard sense of the term “abstract,” her notion of a mathematical object is not that of an abstract mathematical object, she (1990, 59) responds: “So be it; I attach no importance to the term.” For her, then, mathematical objects are simply natural objects, as spatially and temporally located, as causally active, and presumably as contingent, as spotted owls. Nonetheless, she claims to be a Platonist, albeit one with an eccentric ontological terminology. No doubt she would dismiss the complaint that she is presenting a very un-Platonistic claim about mathematical objects under the term “Platonism” as a mere quibble about who gets to use the term “Platonism.” The term “Platonism” thus goes the way of the term “abstract.” Initial appearances to the contrary, it turns out that there is no contribution to meeting Benacerraf’s challenge to realism to be found in Maddy’s position. Maddy’s real position is Aristotelian. This becomes clear when the term “abstract” in her various claims about mathematical objects is replaced with the term “concrete”: mathematical objects are concrete objects within the natural universe of space and time, mathematical knowledge is natural knowledge, and mathematical epistemology is empiricist in perhaps some broadly Quinean sense. Such an Aristotelian view certainly escapes the epistemological questions about realism, but it faces more difªcult questions. How can numbers and pure sets be naturalized? How can it even be meaningful to ascribe physical location to a number or a set? How can there can be enough natural objects for all the numbers and sets? Where is the null set? Is it in more than one place? What explanation can be given for the special certainty, if not the necessity, of mathematical and logical truths? In the case of the last question, the obvious move is to go Quinean, deny necessity, and explain mathematical and logical certainty in terms of centrality. Such a move will enable Maddy to use Quine’s doctrine of science as ªrst philosophy to explain the commitment to numbers and sets. But how does her Aristotelian view that numbers and sets are concrete objects square with Quine’s Platonist view of numbers and sets as abstract objects? That Platonist view is supposed to follow from mathematics and the Quinean doctrine of science as ªrst philosophy. Furthermore, even though Quine’s empiricism offers a better account of the certainty of mathematical and logical truths than Mill’s empiricism, it is incoherent, as I shall argue in chapter 3. If both that argument and the argument in chapter 2 that realism can meet Benacerraf’s challenge succeed, Maddy has traded an epistemological challenge

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that can be met for one that can’t. (See also the criticism of Maddy in Chihara 1992.) 1.3.3 Naturalized Realism To escape the epistemological challenge to realism, some philosophers have tried to frame a position between the extremes of classical Platonism and contemporary Aristotelianism. Instead of trying to extend the range of the perceivable to include mathematical objects or trying to make such objects perceivable by making them natural, their strategy is to combine a realist ontology with an empiricist epistemology. Scientiªc knowledge of the abstract objects in a formal domain is to be explained on the basis of an empirical investigation of the mental/neural structures that constitute our knowledge of mathematical, logical, and linguistic reality. The proponents of such a “naturalized realism” hope to obtain the virtues of both an empiricist epistemology and a realist ontology without incurring the vices of either. Since psychological theories are about natural objects, naturalized realism will be free of the epistemological difªculties about causal contact that are supposed to plague realism. Since naturalized realism accepts the existence of abstract objects, it can hold that formal truths are about such objects, and hence avoid the difªculties of trying to account for mathematics, logic, and linguistics on an exclusively naturalist ontology. Such a position sounds too good to be true—and it is. Its problems arise from the very distinction on which naturalized realism is based. This is the separation of the (abstract) objects that formal knowledge is about from the (natural) objects with which, according to the position, our epistemic faculties interact in the acquisition of such knowledge. Given this separation, the abstract objects that the claims of formal scientists are about are not the objects that those scientists study to determine the truth or falsehood of those claims. Rather, they examine our inner mental states or their neurophysiological underpinnings. Empirical facts about those states or their underlying wetware are the basis for claims about numbers, propositions, and sentences in mathematics, logic, and linguistics. What rationale is there for denying the highly intuitive and widely accepted principle that the nature of the objects that constitute the subject matter of a discipline determines the nature of the discipline? The proposition that six is a perfect number asserts that the number six is equal to the sum of all its divisors except for itself, but, according to naturalized realism, it is not the number six that we focus on to determine if it is a perfect number. Physics and biology are parts of natural science because they study natural objects, and the issue of

20

Chapter 1

whether mathematics is also part of natural science is the issue of whether numbers are psychological objects, as Kant and Brouwer thought, or abstract objects, as Frege and Gödel thought. George (1989) offers a rationale for distinguishing the ontological nature of the objects with which a science is concerned from the ontological nature of the science. He agrees with the linguistic realist that linguistics is concerned with grammars in the abstract sense, but holds nonetheless that linguistics is a psychological science. George (1989, 106–7) claims that linguistic realists are “confused” because they slide from the view that linguistics is not about [internal mental states] to the view that linguistics is not psychological. [Katz] seems to assume that the nature of the objects one is investigating determines the nature of one’s investigation. George (1989, 98) rejects this assumption on the grounds that “Entities can be referred to in many different ways,” arguing that Just as an inquiry into the identity of Z’s favorite planet is not plausibly considered part of planetary astronomy, so an inquiry into the identity of Z’s grammar is not plausibly considered part of mathematics. . . . identiªcation of that grammar, an abstract object, is a fully empirical inquiry. Since the case “Z’s favorite number” is completely parallel to the case of “Z’s favorite grammar,” it follows, by parity of reasoning, that arithmetic is “a fully empirical inquiry.” Since so momentous a substantive conclusion could hardly be gotten with such paltry linguistic means, the reasoning must be fallacious. The fallacy results from an ambiguity in phrases like “inquiries about Z’s favorite planet.” Such phrases have both a referential sense, on which the inquirer can be an astronomer investigating a certain planet—which just happens to be Z’s favorite planet—and a nonreferential sense, on which the inquirer can be a psychologist investigating Z’s taste in planets (Nishiyama in conversation). The sentence “Linguistics is an inquiry into the grammar that a speaker knows” is ambiguous in the same way. On its referential sense, it expresses the claim that linguistics is an inquiry into an abstract object—which happens to be referred to here in a scientiªcally quaint way. On its nonreferential sense, it expresses the claim that linguistics is a psychological inquiry into the speakers’ epistemic states, namely, an inquiry to discover which grammar they know. Conºating these two senses, George (1989, 89) infers that “identifying a speaker’s grammar, an abstract object, is already part of the psychological enterprise.” When the two senses are kept apart, it is clear that George’s conclusion about

Philosophical Preliminaries

21

the nature of linguistics does not follow. On the referential sense, there is no identiªcation that is a matter of psychology; on the nonreferential sense, there is such an identiªcation, but the only conclusion that can be drawn is one about the nature of the speaker’s psychology. Finally, as suggested by the parallel of George’s argument for arithmetic, his view entails a collapse of the formal sciences into psychology, with the consequence that there is no room left for the study of abstract mathematical, logical, and linguistic objects themselves. The shift from the mathematical domain to the domain of psychology replaces discoveries about the structure of numbers, propositions, and sentences with discoveries about the cognitive states of human beings. The mathematical, logical, and linguistic investigations into the structure of numbers, propositions, and sentences has been lost. McGinn (1993) proposes an answer to the objection that naturalized realism provides no place for the study of the structure of the abstract objects in the domain of a formal science. His answer is that the mathematical and logical competences that are investigated in psychology mirror the structure of the abstract objects that mathematical and logical truths are about. Given this mirroring relation, those competences can serve as the source of knowledge of the abstract objects of which they are knowledge. Hence, mathematical and logical investigations into the structure of numbers and propositions have not been lost. They are alive and well in the naturalistic areas which study the human mind/brain. But what entitles McGinn to assume that our mathematical and logical competences mirror the abstract mathematical and logical reality of which they are knowledge? The assumption is not a necessary truth, because whether the mirroring relation holds depends on contingent spatiotemporal creatures. It is thus possible that our mathematical, logical, and linguistic competences do not mirror how things actually stand with numbers, propositions, and sentences. Given this possibility, unless there is an argument for the assumption, it begs the question against realists (e.g., Frege 1964, 12–15) who claim that there is a difference between how we take or represent abstract reality and how it actually is. But since such an argument would have to be based on an independent way of ªnding out how abstract reality actually is, if there were such a way, then naturalized realism would be otiose. The doubt that our mathematical, logical, or linguistic competence mirrors how things actually stand with numbers, propositions, and sentences is not a skeptical doubt about whether we have knowledge of them. We can have knowledge of them in the usual sense, but that knowledge can fall short of mirroring their structure. The possibility arises from the fact that the knowledge of relation is loose enough to

22

Chapter 1

allow a competence to diverge in signiªcant ways from the abstract structures of which it is knowledge. Consider the case of linguistic competence. It is empirically possible that the competence of speakers of English takes the form of a ªnite list of sentences (plus their structural descriptions) that is nonetheless so long that it contains every sentence that they could ever collectively produce. It is also empirically possible that our performance mechanisms are such that the grammatical judgments of English speakers and their use of language are the same as they now are. Given this, the evidence on which linguists presently base their view that grammatical rules of English contain recursive mechanisms that generate an inªnite set of sentences would be the same too. Hence, in our hypothetical situation, linguists will conclude, as they do now, that there are inªnitely many English sentences. But, since ex hypothesi the English speaker’s competence represents English as having only ªnitely many sentences, the competence of English speakers does not mirror the structure of English. Soames (1986) gives empirical reasons for thinking that the psychologist’s representation of grammatical competence in fact diverges signiªcantly from the linguist’s representations of the structure of the language. He points out that, because the competence system must be accessed in on-line speech production and comprehension, its representation of information has to take the form of heuristics suitable to on-line processing. The same failure of mirroring can arise in the case of mathematical or logical competence. For example, we obtain a direct counterpart of the linguistic case just considered by putting the propositions of logic in place of the sentences of English. Still other instances of such divergence are ones in which mathematical or logical competence consists of deviant rules that continue a mathematical or logical series normally until a point well beyond that which it is humanly or physically possible to compute the members of the series and then begins to generate abnormal values. Such possibilities make it necessary to substantiate the mirroring assumption. This requires independent examinations of our knowledge of a formal structure and of the formal structure of which it is knowledge. Only on the basis of both examinations can we make the comparison that enables us to say that our competence mirrors the abstract facts, and hence that a naturalistic investigation of the mind/brain provides reliable information about the abstract mathematical, logical, and linguistic structures themselves. But now it is easy to see that naturalized realism begs the question. Naturalized realism takes the roundabout route through naturalistic investigations of competence

Philosophical Preliminaries

23

because it supposes that realists cannot provide a direct route to knowledge of abstract reality. But if an indirect, naturalistic way of knowing about abstract reality is possible only if we also have a direct route to knowledge of that reality, the game is up. Since prior nonnaturalistic knowledge of abstract reality is required for naturalistic knowledge of abstract reality, naturalized realism leaves the original problem about knowledge of abstract objects exactly where it was. 1.3.4 Moral The moral for the realist is clear. The realist must avoid both an epistemology based on acquaintance with abstract reality and an epistemology based on acquaintance with a concrete (physical, psychological, or neurological) reality. Such epistemologies treat knowledge of facts, laws, and theories in the formal sciences as a posteriori knowledge. The realist requires an epistemology that treats knowledge of facts, laws, and theories in the formal sciences as purely a priori knowledge, that is, an epistemology that explains knowledge in the formal sciences on the basis of reason alone. Since linguistic conceptualists such as Chomsky (1965, 48–54, 205–07; 1986) and mathematical conceptualists such as Kant or Brouwer ([1913] 1983) sometimes also claim to be rationalists, it is worth stressing here that “reason alone” means reason alone. Consequently, our notion of a priori knowledge has to be understood in a stronger sense than the conceptualist’s. There are two respects in which our sense is stronger. First, our notion has to be understood as applying only to knowledge the justiªcation of which can be entirely independent of experience. Supporters of a formal principle do not have to turn to experience to provide grounds for its truth, nor, by the same token, can critics of the principle turn to experience to provide grounds for its falsehood. Second, our notion involves a more comprehensive construal of “experience” than the conceptualist’s. To the conceptualist, “experience” means “experience of things outside of the body,” whereas, to the realist, “experience” means “experience of things outside reason.” In the former case, independence is independence of information coming from perception of the external world. In the latter, independence is also independence of information coming from introspection of our internal nonratiocinative states.8 Realists must eschew an epistemology 8. I am not disputing the right of conceptualists to use the term “rationalist.” Their opposition to the empiricist claim that all our knowledge comes to us through the senses might be taken to entitle them to the term. But I want to distinguish what a conceptualist such as Chomsky calls “rationalism” from what we are calling “rationalism” here. The former is, I believe, more aptly called “nativism.” It doesn’t amount to anything more

24

Chapter 1

that says that knowledge comes from introspection of internal objects to avoid falling back into the claim that formal knowledge comes from acquaintance with concrete objects. For the realist, reason alone must be the source of formal knowledge. Hence, the realist must try to develop an epistemology along the lines of the traditional rationalism of philosophers like Descartes and Leibniz, that is, one on which mathematical, logical, and linguistic knowledge is knowledge of a priori truths knowable on the basis of reason alone. than nativism because the conceptualist is precluded from claiming that linguistic knowledge is a priori in the strong sense of “independent of experience” and from claiming that natural languages contain genuine necessary truths.

Chapter 2 The Epistemic Challenge to Realism

2.1 Introduction The epistemic challenge to realism in Benacerraf’s ([1973] 1983) paper “Mathematical Truth” is to explain how spatiotemporal creatures like ourselves can have knowledge of objects with no spatiotemporal location. Although it seems clear that Benacerraf thinks that realism is unable to explain mathematical knowledge, his aim in this paper is not to refute realism. His aim is to make both sides in the controversy between realism and antirealism face up to their problems: doing justice to mathematical knowledge in the case of the realist and doing justice to mathematical truth in the case of the antirealist. Benacerraf’s message is that no philosophy of mathematics, as it presently stands, is equal to the task of explaining both mathematical knowledge and mathematical truth. In contrast, antirealists such as Gottlieb (1980, 11) and Field (1980, 98) claim that the perceptual inaccessibility of abstract objects refutes realism because it exposes realist epistemology as a form of mysticism. Most philosophers, though they are not prepared to dismiss realism outright, nonetheless ªnd the circumstantial evidence of epistemological malfeasance sufªcient to judge realism guilty until proven innocent. It is especially easy for them to reach this conclusion since their naturalist outlook inclines them to be suspicious of non-natural objects. All in all, few philosophers think that realism has much of a chance of meeting Benacerraf’s epistemological challenge. But arguments, not numbers, count in philosophy. In this chapter, I will try to show that Benacerraf’s argument that mathematical knowledge is impossible if mathematical objects are beyond our causal reach, which underlies both the verdict that realism is guilty outright and the conclusion that it is guilty until proven innocent, is unsound. The argument rests on the false assumption that information from causal interaction with natural objects is a necessary feature of justiªcation in any form of knowledge. The assumption is rarely questioned in

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Chapter 2

contemporary philosophy because of the prevailing empiricist outlook and the absence of a rationalist epistemology that provides an alternative to experience as a basis for taking mathematical propositions to be true. But, since empiricism can be questioned and such a rationalist epistemology cannot be ruled out, the assumption is hardly invulnerable. We saw in the introduction that many realists also accept the assumption. Restricting themselves to an acquaintance epistemology, they have ended up with an incoherent epistemology, or a position that is not realism, or empty hands. Realists, however, are in fact the last philosophers whose ontology requires them to explain mathematical knowledge on the basis of perceptual acquaintance. Since it is only a naturalist ontology that forces a philosopher to adopt the empiricist principle that all knowledge rests in part on causal interaction with natural objects, realists ought to be the ªrst philosophers to reject the idea that evidence about natural objects is essential to knowledge about non-natural objects. Because realists who have bought into an acquaintance epistemology are the majority of inºuential realists, and the rare inºuential realists such as Gödel who reject it have typically been misrepresented as accepting it, realism has yet to receive a defense against the charge of epistemological malfeasance. Since realism rejects all forms of naturalism, ontological, epistemological, and methodological, its explanation of our knowledge of pure mathematics and other formal sciences should be based on the rationalist’s notion that the truths of pure mathematics and other formal sciences are truths of pure reason. 2.2 Truth or Knowledge The issue is what is the best overall philosophy of mathematics and the formal sciences. An adequate philosophy of mathematics, Benacerraf ([1973] 1983) points out, has to provide both a plausible semantics for number-theoretic propositions and a plausible epistemology for mathematical knowledge. He argues that none of the available philosophies of mathematics can do this. Each satisªes one of these requirements at the expense of the other. “Separately,” Benacerraf ([1973] 1983, 410) says, “[the requirements] are innocuous enough,” but “jointly they seem to rule out almost every account of mathematical truth that has been proposed.” I ªnd Benacerraf’s imposition of these requirements to be acceptable, and his assessment that no available philosophy of mathematics satisªes them to be entirely accurate. As indicated, I reject the view that our best bet for obtaining an adequate philosophy of mathematics is further study of the concept of

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truth in mathematics. This view sees the principal obstacle to be either that some of the accounts that we have of mathematical truth are not fully or properly formulated or else that we do not as yet have all the accounts of mathematical truth. In contrast, I think that our best bet is further study of the concept of knowledge in mathematics. Insofar as the problem is the product of both a semantic and an epistemic requirement, the obstacle to an adequate philosophy of mathematics could be epistemological as well as semantic. Benacerraf ([1973] 1983, 409) rightly requires that “an account of mathematical truth . . . must be consistent with the possibility of having mathematical knowledge,” but that requirement, although weak enough to be generally acceptable, is not, as it stands, strong enough to rule out an epistemic solution. To do that, the requirement would have to be bolstered with some restriction of its notion of possibility to what the causal theory of knowledge allows as possible knowledge. Such a restriction is in place within the epistemology that Benacerraf thinks is broadly right. Referring to the “core intuition” of epistemologies like Goldman’s (1967), Benacerraf ([1973] 1983, 413) claims that “some such view must be correct.” This claim rules out an epistemic solution. The question is then what reason there is for thinking that such a causal epistemology is generally acceptable. Benacerraf’s ([1973] 1983, 413–14) reason for wanting an ontology that allows causal connection seems to be exclusively epistemic. If numbers are abstract objects, “then the connection between the truth conditions for the statements of number theory and any relevant events connected with people who are supposed to have mathematical knowledge cannot be made out,” since our “four-dimensional space-time worm does not make the necessary (causal) contact with the grounds of the truth of [those statements].” This point, though undeniable, does not yet show that we cannot come to know abstract objects. It only shows that we cannot come to know them in the way we come to know concrete objects, that is, via a causal connection between ourselves and the objects of knowledge. To be sure, the condition that has to be satisªed to know number-theoretic statements cannot be one the satisfaction of which ensures that we are causally connected to the mathematical facts. But what grounds are there for thinking that the condition has to involve a causal relation to mathematical reality rather than some other epistemic relation? The answer is empiricism. It is the empiricist principle that all knowledge depends on experience that sanctions generalizing the causal condition appropriate to cases of empirical knowledge such as Benacerraf’s ([1973] 1983) case of Hermione’s knowledge that “the

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black object she is holding is a trufºe” to cases of mathematics and other types of formal knowledge. The price, however, is that now the acceptability of a causal condition on formal knowledge depends on the acceptability of empiricism. But, since empiricism hasn’t been established, there is no argument that everyone has to accept the strengthened requirement that the justiªcation condition for formal knowledge cannot be met without appealing to experience, and hence there are no uncontroversial grounds on which to invoke the strengthened requirement. Saying that causal contact with the objects of knowledge is a necessary condition for mathematical knowledge makes the requirement strong enough to rule out realism, but now the requirement is too strong to be generally acceptable. Benacerraf’s ([1973] 1983, 412–15) claim that the realist account of mathematical truth does not mesh with “our over-all account of knowledge” has no force against a realist account on which mathematical knowledge is purely a priori. Given what he means by “our over-all account of knowledge,” his claim comes down to the assertion that a realist account of mathematical truth does not mesh with empiricism. But since it has yet to be shown that empiricism is the best overall theory of knowledge, the claim cannot be used to argue against a rationalist epistemology without begging the question. It is thus an open possibility that the condition for such knowledge is one that, if satisªed, ensures correspondence to mathematical facts, not, as it were, courtesy of our senses, but purely a priori, as rationalists have always thought. Benacerraf recognizes this danger. In the unpublished manuscript from which much of the material in “Mathematical Truth” came, he (1968, 53) notes that the focus in that manuscript might also have been on the concept of knowledge. He indicates that the assumption that the source of the problem is our understanding of mathematical truth rather than our understanding of human knowledge reºects his personal conªdence in a causal theory of knowledge, but he explicitly recognizes that arguing from that assumption might be criticized as “stacking the deck.” Benacerraf (1968, 53) says that his . . . claim is that with the concepts of knowledge and truth, extricated as I have suggested, we do not seem to have adequate accounts of mathematical truth and mathematical knowledge. I am open to suggestion on how the analysis of either concept might be improved to remedy this defect. We will proceed on the assumption that the obstacle to an adequate philosophy of mathematics can be located in the empiricist concept of knowledge.

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2.3 Toward an Epistemological Solution In this section, I want to show that, in addition to there being an option of seeking an improved concept of mathematical knowledge, there are prima facie reasons for thinking that an improved concept of knowledge is a better prospect than an improved concept of truth. On the one hand, Benacerraf’s criticism of antirealism—the positions he ([1973] 1983, 406–7, 416) calls “combinatorial”—is a much stronger argument against antirealism than his formulation suggests, and, on the other hand, there are difªculties with the extension of the empiricist concept of knowledge to the formal sciences. (I will discuss Benacerraf’s criticism at some length because it will have further applications later in the book.) Benacerraf ([1973] 1983, 405–12) pointed out that realism has the advantage of allowing a uniform semantic treatment of mathematical and nonmathematical sentences, that is, one that treats the logical forms of mathematical sentences and corresponding nonmathematical sentences in a parallel way. On realism, (1) and (2) are both straightforward instances of (3), (1) There are at least three perfect numbers greater than seventeen. (2) There are at least three large cities older than New York. (3) There are at least three FG’s that bear R to a. but, on the antirealist approaches he calls “combinatorial,” (2) is not an instance of (3). Benacerraf ([1973] 1983, 410–12) bases his preference for a uniform semantics on the success of Tarskian semantics generally, the absence of an alternative semantics for combinatorial approaches, and the difªculty of coming up with an appropriate multiform semantics for such approaches.1 The difªculty of coming up with a multiform semantics might not be the worst of it for combinatorial approaches. The intuitive appeal of treating the semantic form of mathematical sentences like (1) and nonmathematical sentences like (2) in the same way suggests that we wouldn’t want a multiform semantics even if we could come up with one. Burgess (1983, 1990, 7) made a start toward such a criticism in observing that the choice between a uniform and a multiform semantics for natural language belongs to “the pertinent specialist professionals” 1. This argument against antirealism must also contain an argument against a noncombinatorial nominalist view like Field’s (1980, 1989) that takes reference to numbers to be reference to ªctional entities and mathematical truth to be truth in a certain type of ªction. See the criticisms in section 2 of the previous chapter for such an argument.

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in linguistics and suggesting that they would ªnd a multiform treatment of sentences like (1) and (2) unacceptable. I think that Burgess is certainly right that the choice has to be made on the basis of which semantics best ªts into the linguist’s account of the grammar of natural language. He is also right, as I shall now argue, that a multiform treatment of sentences like (1) and (2) is unacceptable in linguistics. If the intuition that (1) is as much an instance of (3) as (2) is correct, then what is wrong with treating (2) but not (1) as an instance of (3) is that sentences with essentially the same grammatical structure are treated as essentially different in grammatical structure. To be sure, there are cases in linguistics where it might seem that sentences with the same grammatical structure are treated as different in grammatical structure. A classic example of such a case in the history of generative linguistics is the postulation of different underlying structures for (4) and (5). (4) John is easy to please. (5) John is eager to please. However, the linguist’s treatment of such cases differs from the combinatorialist’s treatment of cases like (1) and (2). The difference is that cases like (4) and (5) are ones where the appearance of sameness of grammatical structure does not go below the surface: “John” is the direct object of the inªnitive in (4) and the subject of the inªnitive in (5). The linguist’s desire to account for those facts—in accord with the prevailing theory of grammatical relations (see Chomsky 1965)—motivated a transformational analysis of (4) and (5) on which surface similarity masks deep grammatical difference. Hence, the linguist’s treatment sacriªces no more than a surface grammatical similarity, and the sacriªce is for the higher good of preserving a deeper grammatical truth. Since the different grammatical treatment of sentences like (4) and (5) is grammatically driven, such sentences are the “exceptions” that prove the rule that grammatically similar sentences are to be similarly described. The transformational analysis of (4) and (5) is both appropriate to linguistics and properly implemented. The linguist’s aim is to obtain a more encompassing analysis of grammatical structure than is possible on a description that preserves surface similarity. The analysis is properly implemented because the assignment of different underlying syntactic structures is based on grammatical facts about the sentences. In contrast, the multiform analysis of (1) and (2) is neither appropriate to linguistics nor properly implemented. The combinatorialist’s aim is the linguistically irrelevant one of trying to avoid a commitment to realism

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31

in the foundations of mathematics. The implementation of the analysis is improper not only because there is no strong grammatical intuition reºecting a difference in their grammatical structure, but also because there is not even a hint of the kind of underlying grammatical differences we ªnd in connection with cases like (4) and (5). From a grammatical perspective, (1) and (2) are each as much an instance of (3) as the other. (1) is not an isolated case. There are inªnitely many sentences of English in which number terms occur in a referential position. Hence, if the concerns of a partisan viewpoint in the philosophy of mathematics are allowed to decide questions of grammatical structure, distinctions reºecting no grammatical differences will be made over a wide segment of the language. Since such distinctions are only philosophically motivated, a multiform semantics would compromise the autonomy of linguistics. Linguistic argumentation would degenerate into philosophical debate—Why should the linguist let the concerns of antirealism decide? Why not the concerns of realism? And so on. To preserve the autonomy of linguistics and the integrity of its argumentation, only linguistic considerations can be allowed to determine the description of sentences.2 Related to the foregoing semantic doubts about the extension of the empiricist concept of knowledge to the formal sciences are doubts about whether there is anything in the natural world to which knowledge about numbers, sets, propositions, and sentences can be causally connected. Not only is there, as it were, no trufºe, there doesn’t seem to be anything to which such knowledge can be causally related, not even in the remote way in which theoretical truths in physics can be causally related to events in cloud chambers or pictures from radio telescopes. There seem to be no natural objects to serve as the referents of the terms in (6)–(8) and no natural facts to which such truths may be taken to correspond. (6) Seventeen is a prime number. (7) No proposition is both true and false. 2. Someone might reply that it would be a good thing if the present disciplinary boundaries were to disappear and if questions within disciplines could be decided partly on the basis of arguments from other disciplines in a fully interdisciplinary way. It is not clear to me what would be so good about this, either for scientiªc disciplines or for philosophy. If decisions within disciplines became so radically interdisciplinary, it would wreak havoc with argumentation about the nature of phenomena in a scientiªc discipline, since its practitioners could then substitute philosophical arguments for scientiªc considerations. We would forfeit the constraint to save the phenomena. Nor would it be good for philosophy to be without an independent scientiªc characterization of knowledge by which to judge philosophical accounts of knowledge.

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(8) An anagram is an expression that is a transposition of the letters of another expression. Other doubts about whether the causal condition generalizes stem from the argument of rationalists such as Arnauld, Leibniz, and Kant that experience cannot provide knowledge of the fact that mathematical and logical truths couldn’t be otherwise. Quine and other naturalists have denied that there are necessary truths, but, in the fullness of time and modality, those arguments have not proven, as, for example, Marcus (1990) has argued, to have the force they were once thought to have.3 Given that there is nothing in the natural world that can explain the necessity of such truths, they are at least prima facie counterexamples to a causal condition on all formal knowledge. Benacerraf quite correctly claims that a realist account of mathematical truth does not mesh with an empiricist account of knowledge. From the perspective of considerations such as those above, this can be taken to mean only that we cannot know truths about mathematical objects in the same way we know truths about natural objects. Furthermore, from that perspective, the existence of mathematical knowledge shows that there must be a different way of knowing mathematical truths. 2.4 Mysticism and Mystery Some antirealists would dismiss the possibility of an epistemology for abstract objects. They take the view that any epistemology for any form of realism is mysticism. Gottlieb (1980, 11) says, “Abstract entities are mysterious and must be avoided at all costs.” Field (1989, 59) says that the realist “is going to have to postulate some aphysical connection, some mysterious grasping.” Chihara (1982, 215) says that the realist’s appeal to Gödelian intuition is “like appealing to experiences vaguely described as ‘mystical experiences’ to justify belief in the existence of God.” Dummett (1978, 202) says that Gödelian intuition “has the ring of philosophical superstition.” These antirealists think such things be3. Given that the argument for the causal condition in the case of mathematical knowledge is so weak, it seems that the prevalence of the naturalistic outlook is the only thing that instills conªdence in the prospects of empiricism as a general theory of knowledge. Naturalism supplies the premise required to generalize from a causal condition on justiªcation in the uncontentious case of empirical knowledge to one on justiªcation in all cases of knowledge. Having an ontology that says that all objects of knowledge are uniformly natural objects and also accepting a causal condition on knowledge of natural objects exerts overwhelming pressure to generalize the causal condition to knowledge of numbers, sets, propositions, sentences, meanings, and the like. Katz (1990b) argues that the principal arguments that twentieth-century philosophers have given for naturalism are deeply ºawed.

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cause they think that, in the absence of objects acting directly or indirectly on us, abstract objects might be any way whatsoever for all we would know. Hence, they think that the realist’s claim that we know how things are in the realm of abstract objects in spite of having no natural connection to it can only be based on the pretense that we have a supernatural connection. There are two things wrong with this antirealist view. One is the unwarranted assumption of an empiricist epistemology on which we have already commented and on which we will have more to say in the following sections. The other is a confusion between mysticism and mystery. No one would deny that there is a mystery about how we can have knowledge of abstract objects. But such extreme antirealists make far too much of that particular philosophical mystery. Philosophy is full of such mysteries. Every philosophical problem is one. Each reºects something we ªnd incomprehensible about the world or our knowledge of it. In the case at hand, we ªnd truth in mathematics and other formal sciences incomprehensible. But the obscurity of the rational mechanism is no grounds for dismissing the claim that pure reason does what, from the common-sense standpoint, it appears to do. Why should the realist’s project of explaining how spatiotemporal creatures like ourselves obtain their knowledge of causally inert objects deserve more disparagement than the naturalist’s project of explaining the equally mysterious process by which we obtain conscious experience from the physical effects of material objects or the empiricist’s project of explaining why it is rational to believe that the future will be like the past? To be sure, some realists have strayed off into mysticism, but it is just guilt by association to criticize all realists for the sins of some. A philosophical mystery is no grounds for crying “mysticism.” Mysticism involves the claim to have a means of attaining knowledge beyond our natural cognitive faculties. Those who cry “mysticism,” “superstition,” and the like perhaps need to be reminded of the fact that our sensory faculties do not exhaust those faculties. Sophisticated empiricists recognize an autonomous rational faculty as essential for knowledge. For example, Benacerraf’s ([1973] 1983, 413) statement that “knowledge of general laws and theories, and, through them, knowledge of the future and much of the past” is “based on inferences based on [perceptual knowledge of medium-sized objects]” seems to recognize that the inferential operations of reason can’t be reduced to the operations of our senses. Such empiricists do not think of reason as a device for cataloguing data based on similarity comparisons of sensory information, but as an inferential engine that, together with sense experience, is necessary for the justiªcation of our beliefs about the

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world. The bottom line is that the epistemology for mathematical knowledge to be set out in the next section, which is based on our natural cognitive faculty of reason, should dispel the charge of mysticism. Antirealists will no doubt be less than happy about this epistemology, but they will not be able to dismiss it as mysticism. 2.5 An Outline of a Rationalist Epistemology Our project is intended to provide an “improved” concept of knowledge, improved in the sense that it overcomes the principal fault of traditional realism: its lack of a plausible epistemology. As we observed in the last chapter, Gödel ([1947] 1983, 484) points us in the right direction. He makes it clear that the “mathematical intuition”—we may say, the rational faculty—on which mathematical knowledge rests “cannot be associated with actions of certain things upon our sense organs” or with “something purely subjective, as Kant asserted.” Gödel ([1947] 1983, 484) went on to say that its presence in us “may be due to another kind of relationship between us and reality.” These remarks focus the task for developing a realist epistemology that can meet the epistemological challenge. The task is to provide an account of this other kind of relationship that explains how we come to stand in that relationship to the realm of abstract objects and, with no window on that realm, come to know what things are like there. 2.5.1 Epistemic Conditions Like Benacerraf ([1973] 1983, 414), I will assume for the sake of argument that knowledge is justiªed true belief. Belief, truth, and justiªcation will not be understood in any special philosophical sense; rather, they will be taken in a sense as close to their familiar sense as possible. Someone has a belief about something when he or she takes a proposition about it to be true. A proposition is about something when one of its referring terms refers to it. A proposition is true when the things it is about satisfy its truth condition, that is, when the facts are as the proposition says they are. Our belief about something is justiªed when we have adequate grounds for taking the proposition about it to be true. These characterizations ªt the epistemic assumptions of Benacerraf’s statement of the epistemic challenge and are compatible with a wide range of views in the theory of knowledge. 2.5.2 The Belief Condition No aspect of the realist’s epistemology can entail causal contact with abstract objects. But the constraint that mathematical and other formal knowledge be based on reason alone, though it applies to the truth and

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justiªcation conditions, does not apply to the belief condition. In the case of the belief condition, the states that constitute the taking of a proposition to be true can derive features of their ideational content from experiential relations to natural objects, since abstract objects are not involved in such relations.4 The possibility that the ideational content of doxastic states depends in some respects on experience shows that nativism does not have to be a component of the rationalist position. Nativism is a theory about the acquisition of the concepts with which our cognitive faculties work. It can be understood as the claim that the concepts required to form beliefs about (inter alia) abstract objects are either themselves inherent constituents of our cognitive faculties or else derivable from concepts that are inherent constituents on the basis of principles that also belong to those faculties. (See Katz 1979; 1981, 192–220.) The most inºuential contemporary form of nativism is Chomsky’s (1965, 47–59). His nativism hypothesizes that the child has innate knowledge of the grammatical structure of natural language, and also innate principles for putting this knowledge to use in acquiring (tacit) knowledge of a natural language (i.e., competence). Given that the child’s innate knowledge represents the full range of possible competence systems for natural languages, the child’s task is to choose, on the basis of a sample of utterances of a language, the system in this range that will give it ºuency in the language. There are various speciªc hypotheses about the way in which the child chooses, such as parameter setting, hypothesis testing, and so on, but the nativist’s general claim is that the child’s innate grammatical structure is so rich that experience does no more than determine which competence system is the right one for the sample of utterances to which it is exposed. Now a philosopher might well think, as I do, that both nativism and rationalism are correct. One might think that the former is correct because some such theory as Chomsky’s accounts best for the facts concerning the child’s acquisition of a competence system, and one might think that the latter is correct because some such theory as the one I will describe best accounts for the epistemological and semantic facts about knowledge in mathematics and other formal sciences. But equations of nativism and rationalism, such as Chomsky’s (1965, 1966), conºate psychological questions with epistemological questions. Chomsky’s conºation doubtless stems from the prior conºation of psychology and epistemology that comes with conceptualism. 4. This discussion is a revision of the view about the belief condition in Katz (1995, 500–502). It was prompted by a query from Glenn Branch.

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2.5.3 The Truth Condition The correspondence of the mathematical proposition and the mathematical fact in virtue of which the proposition is true involves no contact between an abstract and a concrete object because, on our realism, such correspondence holds among abstract objects. On the one hand, mathematical facts are facts about abstract objects. On the other hand, both of the principal conceptions of propositions, the Fregean and the Russellian, enable us to construe number-theoretic and other formal propositions as abstract objects. On the Fregean conception, propositions are senses of sentences, and it is quite natural for the mathematical realist to say, in accord with linguistic realism, that senses of sentences are abstract objects. On the Russellian conception of propositions, a proposition is an ordered n-tuple consisting of the object(s) the proposition is about and the property or relation the proposition ascribes to it (them). Here too mathematical propositions can be taken as abstract objects because, for us, all of the components of such propositions—the mathematical objects in the n-tuple as well as the properties and relations—are abstract objects. Since both mathematical propositions and the facts they are about are abstract, mathematical truth is simply an abstract relation between abstract objects. 2.5.4 The Justiªcation Condition On our epistemology, what counts as adequate grounds for the truth of a proposition depends on the nature of the proposition, which, in turn, depends on the nature of the objects the proposition is about. So, propositions about natural objects are empirical and propositions about abstract objects are nonempirical. We can take something like Benacerraf’s ([1973] 1983, 413) concept of empirical knowledge as specifying what counts as adequate grounds for the truth of propositions about natural objects. We require a concept of a priori knowledge to specify what counts as adequate grounds for the truth of propositions about abstract objects. In Language and Other Abstract Objects, I (1981, 200–216) made a suggestion about how to understand a priori knowledge of abstract objects. It involved two thoughts. The ªrst was that the entire idea that our knowledge of abstract objects might be based on perceptual contact is misguided, since, even if we had contact with abstract objects, the information we could obtain from such contact wouldn’t help us in trying to justify our beliefs about them. The epistemological function of perceptual contact is to provide information about which possibilities are actualities. Perceptual contact thus has a point in the case of empirical propositions. Because natural objects can be otherwise than they actually are (non obstante their essential properties), contact is

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necessary in order to discover how they actually are. In some possible worlds, gorillas like bananas, while in others they don’t. Hence, an information channel to actual gorillas is needed in order for us to discover their taste in fruit. Not so with abstract objects. They couldn’t be otherwise than they are. They have all their intrinsic properties and relations necessarily.5 Purely abstract properties and relations of abstract objects cannot differ from one world to another. The way abstract objects actually are with respect to their intrinsic properties and relations is the way they must be. Hence there is no question of which mathematical possibilities are actualities. Unlike what is actually the case about gorillas, what is actually the case about numbers is what must be the case about them. In virtue of being a perfect number, six must be a perfect number; in virtue of being the only even prime, two must be the only even prime. Since the epistemic role of contact is to provide us with the information needed to select among the different ways something might be, and since perceptual contact cannot provide information about how something must be, contact has no point in relation to abstract objects. It cannot ground beliefs about them. In On the Plurality of Worlds, Lewis (1986b, 111–12) expresses a similar thought. He says that the necessity of a mathematical proposition exempts it from the requirement on empirical propositions to show that they counterfactually depend on the facts to which they correspond. According to Lewis (1989b, 111), counterfactual dependency is not required for mathematical propositions because “nothing can depend counterfactually on non-contingent matters. For instance, nothing can depend counterfactually on what mathematical objects there are. . . . Nothing sensible can be said about how our opinions would be different if there were no number seventeen.” (Italics mine) Field (1989, 237) rightly objects that we can sensibly say how things would be different if the axiom of choice were false. Furthermore, if what Lewis says were so, there would be no reductio disproofs of necessarily false statements in the formal sciences, since such proofs 5. My claim is that all intrinsic or formal properties and relations of abstract objects are necessary. Sometimes it is held that some properties and relations of abstract objects are contingent, e.g., the relation I bear to the number seventeen when I am thinking of it. On my view, this is not the case (see chapter 5, section 3). A somewhat trickier case: a word has the property of being coined at a particular time (but might have been coined at another time). Since I want to treat the words of a language as types in Peirce’s sense, and hence as abstract objects, coining a word is not creating a word of the language. As I (1981, chapter 5) argue elsewhere, what happens when a word is coined is that speakers of a language begin to use tokens of the word as tokens of that type under conditions that lead to a change in their competence. Clearly, a word type’s having (or not having) a representation in the competence of speakers is not one of its intrinsic or formal properties and relations.

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begin with the supposition that a necessarily false statement is true, which, on Lewis’s view, is “nothing sensible.” For example, in a reductio proof that the square root of two is irrational, we suppose counterfactually that (9) is true (9) There is a rational number equal to the square root of two. and then, reasoning from this supposition, we go on to spell out how things would be different, given the existence of such a rational number. We say such things as that there would be numbers that are both even and odd. The absurdity of a reductio is precisely what it is sensible to say about how things would be different if the necessarily false supposition were true. At one time, Lewis (1973, 24–26) held the opposite view. Presumably, the change took place because of the difªculty of reconciling the earlier view that something sensible can be said about how our opinions would be different if a necessarily false proposition were true with Lewis’s possible worlds conception of propositions.6 Insofar as we have no commitment to the possible worlds conception of propositions, we have no reason to stick to it at the prohibitive cost of not being able to make sense of reductio proofs.7 To express the otiosity of contact in the case of formal knowledge unproblematically, we require an alternative conception of propositions on which necessarily false sentences express propositions. One such conception is that propositions are senses of sentences. (See Katz 1972, 1977; Smith and Katz in preparation.) With this conception, we can 6. In possible worlds semantics, propositions are sets of possible worlds: the proposition P is the set of possible worlds in which P is true. Either contradictions are true in the null set of possible worlds or they express no proposition. But, if we say that they are true in the null set of possible worlds, it does not seem possible to maintain the distinction in Lewis’s earlier position between its being sensible to say “p! + 1 is prime” and “p! + 1 is composite” on the supposition (1) “there is a largest prime p!”, but not on “there are six regular solids” or “pigs have wings.” Since the former are taken to be sensible things to say, it would also have to be sensible to say “pigs have wings” on the supposition (ii) “pigs have wings and are wingless.” But, since ex hypothesi a contradiction is true in exactly the null set of possible worlds, all contradictions express the same proposition. Since (i) and (ii) express the same proposition, if it is sensible to say “p! + 1 is prime” and “p! + 1 is composite,” it must also be sensible to say “pigs have wings.” So, it seems that, to avoid having to concede that the latter is sensible, Lewis decided to say that neither is. (See Katz 1996a for a discussion of the shortcomings of possible worlds semantics.) 7. Even if reductio proofs are not necessary because a direct proof can be given for each theorem established by a reductio proof, the cost of not being able to make sense of such proofs is still prohibitive. The existence of direct proofs doesn’t change the fact that reductio proofs are proofs.

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specify the condition for supposability as the meaningfulness of the clausal complement of the verb “suppose.” A sentence expresses no supposition when and only when, like in (10), (10) Suppose that seventeen loves its mother. the clausal complement of “suppose” has no sense, and hence provides no object for the propositional attitude. When the complement has a sense, even one that is necessarily false, like the complement in (11), (11) Suppose that some propositions are both true and false. there is an object for the propositional attitude, namely the sense of the clause, and the whole sentence expresses a supposition. Thus, on our alternative conception of propositions, there is something to be supposed in the case of necessarily false sentences, and hence we can make sense of reductio proofs. Given this condition for supposability, reductio proofs in the formal sciences are tests of necessary truth based on an exploration of the logical consequences of supposing a necessary truth to be false. In such proofs, we ªrst suppose that what a necessary truth asserts is not the case. Then, by deriving an explicit inconsistency, we expose the fact that the proposition we supposed is a contradiction, from which fact we then infer that the denial of the proposition is true. Hence, something sensible can be said about how things are on a necessarily false supposition, namely, that things are every way they can supposably be.8 In expanded form, this is the ªrst of the two thoughts about knowledge of abstract objects in Language and Other Abstract Objects (Katz 1981). The second thought was that it is no loss for the epistemic function of contact not to carry over to the realm of abstract objects because our reason is an appropriate instrument for determining how things must be in that realm. It is a truism of mathematical practice that reasoning can show that mathematical objects could not be otherwise than as mathematics presents them. When mathematicians proved the proposition that the square root of two is irrational, they showed, on the basis of reason alone, that its truth conditions are satisªed no matter what one supposed about the numbers. But, if the light of reason enables the mind’s eye to see that the square root of two is 8. Moreover, since, on that notion, necessarily false sentences with different senses can express different propositions, separate reductio proofs are to be kept separate. We have to keep them separate to be able to say that things are the same on the supposition that there is no number seventeen as they are on the different supposition that there is no number two or that there is a largest prime.

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irrational, of what relevance can it be that the body’s eye cannot see numbers? Reductio proofs provide the most straightforward way of showing that a mathematical truth is a necessary truth. This is because such proofs (explicitly) begin with the supposition that the proposition is false and go on to exclude every possibility of the supposition’s being true. They show that there is only one possibility of how the mathematical objects in question might be because other putative “possibilities” are impossibilities. The following argument (A) that two is the only even prime is an example. (A) We see that two is an even prime. Supposing that another number is even and prime, that number must be either less than two or greater than two. If the number is less than two, it has to be one, but then it is not even. But, if the number is greater than two, then, since it is even, it is divisible by two, and hence not prime. Since the law of trichotomy cannot fail here, there is no even prime other than the number two. A proof provides us with adequate grounds for knowledge of a proposition about abstract objects by showing that it is impossible for the objects to be other than as the proposition says they are. But the question arises of how a proof establishes the necessity of its conclusion when, typically, the conclusion is simply a statement that the mathematical objects in question have a certain property rather than a statement expressing a modal predication about how they have the property. For example, the conclusion of (A) simply states that every number other than two lacks the property of being even and prime. It is not itself a statement that they necessarily lack the property. So, how can (A) establish that its conclusion is necessary? The answer is that a proof of a proposition P that is not itself a modal statement establishes the modal statement “Necessarily, P” in virtue of the fact that it is a proof of P. The essence of proof consists in reasoning so close-textured, so tight, that it excludes every possibility of the conclusion’s being false. Mathematicians sometimes say that there are no “holes” or “gaps” in a proof. This is not just a matter of logical structure. The tightness of a proof derives not only from the absence of any counterexample to any of the steps from premises to conclusion, but also from the absence of any counterexample to any of the premises. The argument (A) is a proof that two is the only even prime not only because none of its inferential steps can be faulted, but also because its premises cannot be faulted either. Hence, there is no possibility of two not being an even prime. Thus, (A) shows that two is

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necessarily the only even prime, not in virtue of a modal conclusion, but in virtue of reasoning so tight that every possibility of another even prime is ruled out.9 2.6 The Order of Knowledge There are two respects in which the foregoing account of knowledge falls short of meeting Benacerraf’s epistemic challenge to realism. First, it is only an explication of some of the rational methods for acquiring formal knowledge, and hence we have yet to establish that none of the remaining methods presupposes contact with abstract objects. Second, the account does not explain how a full rationalist epistemology would meet the epistemic challenge. The present section completes our sketch of the rationalist epistemology. The next section explains how the challenge is met. 2.6.1 The aspects of reason remaining to be discussed are those that are responsible for the steps from knowledge of simple mathematical facts to knowledge of mathematical laws and theories. Since causal contact plays no role in explaining knowledge of the simple mathematical facts that underlie knowledge of mathematical laws and theories, dependency of these aspects of reason on contact with abstract objects seems unlikely on the face of it. Nevertheless, to be sure there is no depend9. Tightness is different from the properties of informativeness and depth. A completely tight proof may be less informative about what is going on mathematically—may give less insight into the mathematical structure—than a proof with a hole. Further, tightness and informativeness are both different from depth, what the proof tells us about the bigger mathematical picture. Truth table proofs of tautologies and solutions to chess problems are completely tight and completely informative, but they are not mathematically deep. Thus, a proof with a hole in it can be mathematically signiªcant if it is informative or deep, but, of course, for the purpose of having mathematical beliefs on which we can rely absolutely, an uninformative or shallow proof is as good as an informative or deep one. It is widely held that knowledge is reliable in the sense of being theoretically and practically dependable. In empirical knowledge, reliability is typically explained on the grounds that the knowledge rests on evidence from either direct or indirect causal contact with the natural objects it is about. Since the process of arriving at empirical beliefs monitors those objects, empirical investigation provides grounds for conªdence in the evidence, and hence the beliefs it supports. In the case of mathematical knowledge, we can explain reliability in terms of tightness of proof. The tightness of mathematical proofs underwrites our conªdence that their conclusions represent the numbers as they are. Every possibility of the numbers being otherwise has been excluded because there are no gaps.

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ency, we need to see how a rationalist epistemology handles the ascent from knowledge of basic mathematical facts to knowledge of mathematical laws and theories. The ascent from basic facts to knowledge of laws and theories is a feature of both a priori and a posteriori knowledge. Recognizing that there are theoretical cases of empirical knowledge in addition to basic cases like Hermione’s knowledge that what she is holding is a trufºe, Benacerraf ([1973] 1983, 413) qualiªes his causal condition as “an account of our knowledge about medium-sized objects, in the present” and goes on to say: Other cases of knowledge can be explained as being based on inference based on cases such as these. . . . This is meant to include our knowledge of general laws and theories, and, through them, our knowledge of the future and much of the past. This is, in effect, to introduce an order of knowledge into empiricist epistemology: there is basic knowledge of “medium-sized objects, in the present” and transcendent knowledge of “general laws and theories, and, through them, . . . knowledge of the future and much of the past.” Since there are general laws and theories in the formal sciences, a rationalist epistemology will have to describe a similar order of knowledge in the formal sciences. Just as Benacerraf’s empiricist epistemology posits basic, observational knowledge of properties of medium-sized objects, our rationalist epistemology correspondingly posits basic ratiocinative knowledge of evident properties of abstract objects, e.g., the knowledge that four is composite. Just as Benacerraf’s empiricist epistemology posits transcendent knowledge of empirical laws and theories, a rationalist epistemology correspondingly posits transcendent knowledge of formal laws and theories. A sharp observational/theoretical distinction in the natural sciences has proven notoriously difªcult to draw, so much so that many philosophers of science have given up trying to draw it. A sharp basic/transcendent distinction in the formal sciences, though, as far as I know, little investigated, is unlikely to prove more tractable. But, in the present investigation, it is no more necessary to try to draw such a distinction for the formal sciences than it was for Benacerraf to draw one for the natural sciences. We are no more concerned with describing science than he was. Nonetheless, we can sketch a rough distinction. In the case of the natural sciences, for the most basic of basic knowledge, we have the case of seeing the color of the litmus paper with our own eyes. As we move away from this extreme to less basic knowledge, we have cases of observation that depend more and more on such artiªcial devices

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as electron microscopes, radio telescopes, and the like that boost the power of our natural faculties. Since our conªdence in such devices depends on laws and theories, theoretical considerations play a role in connecting what we directly perceive with what we count as observation. As we move from basic knowledge to transcendent knowledge, we have a shift in focus to establishing general laws and theories. Observational methods still play a role, but systematic considerations increasingly dominate. The distinction in the formal sciences is similar. At the extreme of basic knowledge, we have the case of seeing—though not with our eyes—that four is composite. In other cases of basic knowledge, such as our knowledge that two is the only even prime, our insight is extended on the basis of reasoning and the use of computational devices that boost the power of our natural ratiocinative faculties. Since our conªdence in such devices depends on laws and theories, theoretical considerations play a role. As we move from basic knowledge to transcendent knowledge, we have a shift in focus to establishing general laws and theories.10 Insight still plays a role, but systematic considerations increasingly dominate. In the formal sciences, it is common to refer to seeing that something is the case as “intuition” and to take such immediate apprehension as a source of basic mathematical knowledge.11 Mention of intuition raises two immediate concerns. One is the cry of “mysticism” on the part of some radical antirealists. In this connection, the comments earlier in this chapter on the difference between mysticism and mystery apply. In addition to those comments, I will quickly discuss Wittgenstein’s criticisms of intuition. Wittgenstein (1953, sec. 213) dismissed intuition as a source of knowledge, referring to it as “an unnecessary shufºe.” He had two reasons for thus dismissing it. First, he thought that nothing is gained by invoking intuition because the signs that make up the speech of the “inner voice” of intuition would require interpretation just as much as the signs that make up the speech of the public language. In The Metaphysics of Meaning, I (1990b, 159–61) argued that 10. There is a story about a famous mathematics professor (different versions of the story mention different professors) who introduced a lemma as intuitively obvious in the course of proving a theorem. When a student didn’t see that the lemma is intuitively obvious, the professor retired to examine the lemma, and, after an hour, returned to announce, “I was right; it is intuitively obvious.” 11. The emphasis on the common notion of intuition should not obscure cases of basic mathematical knowledge that do not depend on rational operations encompassed within a single grasp of structure (though they might be thought of as concatenated intuitions). Examples are the theoretically unmediated inferences underlying our knowledge that two is the only even prime and our knowledge that a cube has twelve edges.

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Wittgenstein’s inner voice characterization of intuition is no more than a caricature of intuition in mathematics, logic, and linguistics, and I (1990b, 135–62) showed that within a realist framework there is no problem about a regress of interpretations. This solution is essentially a version of the solution to Kripke’s puzzle about rule following that I shall propose in chapter 4. Wittgenstein’s other reason is based on the observation that intuition is not always reliable: “if it can guide me right, . . . it can also guide me wrong.” Wittgenstein concludes that it is a mistake to appeal to intuition to pin down the interpretation of an initial segment of a series so that we can go on in the right way. This criticism, insofar as it goes beyond the previous one, is simply a misrepresentation of the commitment that comes with accepting intuition as a source of knowledge of basic mathematical and logical truths. There is no strong commitment to the infallibility of intuition. To be sure, intuition does not always do its job properly. Intuitive “seeing” is sometimes not reliable, but in this respect it is no different from visual seeing. Since we don’t dismiss sight as a legitimate source of basic knowledge because there are times when our eyes deceive us, it is hard to see why we should dismiss insight as a legitimate source of basic knowledge because there are times when it deceives us. Like judgments based on sight, those based on insight can be checked against similar judgments and against theories based on them. Once intuition is integrated into a systematic methodology that enables us to correct unclear and deceptive cases on the basis of a broad range of clear cases and principles derived from them, Wittgenstein’s worry that intuition sometimes gives the wrong guidance disappears. The other concern is to avoid a possible misunderstanding of the notion of intuition. The notion of intuition that is relevant to our rationalist epistemology is that of an immediate, i.e., noninferential, purely rational apprehension of the structure of an abstract object, that is, an apprehension that involves absolutely no connection to anything concrete. The misunderstanding that I want to avoid is a confusion of this notion of intuition with a Kantian notion. The danger is real, since the most serious attention that the notion of intuition has received in recent literature in the philosophy of mathematics has been Parsons’s (1980) development of a Kantian concept of intuition that involves a connection to something concrete in sense perception. I want to make it clear that Parsons’s concept is not what I mean, because that connection prevents it from playing any role in a strictly rationalist epistemology for pure mathematics or any other pure formal science. This is not to say that Parsons’s concept isn’t relevant to our approach to the foundations of the formal sciences. It may be useful in connection

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with what I call composite objects in chapter 5, particularly those that involve the type/token relation, such as drawn geometric ªgures and linguistic utterances (and inscriptions). I believe that the discussion in that chapter, particularly the distinctions drawn between composite objects and abstract objects and between the application of pure arithmetic and the application of pure geometry, is essential to seeing Parsons’s work in the right light. It is also crucial to the notion of intuition in our sense that intuitions are apprehensions of structure that can reveal the limits of possibility with respect to the abstract objects having the structure. Intuitions are of structure, and the structure we apprehend shows us that objects with that structure cannot be certain ways. Consider some examples. The intuition of the number four as a composite of two and two shows the impossibility of four’s being a prime number. The intuition of the logical structure of an instance of modus ponens shows the impossibility of the truth of the premises without the truth of the conclusion. The intuition of the grammatical structure of “I saw the uncle of John and Mary” shows the impossibility of the sentence’s having just one sense. As Descartes pointed out, when intuition is clear and distinct, as it is in such cases, we “could have no occasion to doubt it.” What is present to our minds in a clear and distinct intuition of abstract objects is the fact that their structure puts the supposition of their being otherwise than as we grasp them to be beyond the limits of possibility. The rationale for claiming that it is intuition that is the source of basic mathematical and other formal knowledge is something like the precept that Holmes recommends to Watson in The Sign of Four. Holmes says, “How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, is the truth?” There are cases in which we can eliminate everything but intuition as a possible explanation of how it is known that a premise or a step in a proof has no counterexample. In cases like the compositeness of four, the pigeon-hole principle, the indiscernibility of identicals, and the ambiguity of “I saw the uncle of John and Mary” or the well-formedness of “The cat is on the mat,” there is no explanation other than intuition for the fact that ordinary, unsophisticated people, without expert help, immediately grasp the truth. Consider the pigeon-hole principle. Even mathematically naive people immediately see that, if m things are put into n pigeon-holes, then, when m is greater than n, some hole must contain more than one thing. We can eliminate prior acquaintance with the proof of the pigeon-hole principle, instantaneous discovery of the proof, lucky guesses, and so on as “impossibilities.” The only remaining explanation for the immediate knowledge of the principle is intuition.

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Rationalists have sometimes extended this Holmesian defense of intuition, arguing that our knowledge of logic itself rests au fond on the exercise of intuition. Ewing (1947, 26) points out that an inferential step . . . must either be seen immediately or require further argument. If it is seen immediately, it is a case of intuition; if it has to be established by a further argument, this means that another term, D, must be interpolated between A and B such that A entails D and D entails B . . . , but then the same question arises about A entailing D, so that sooner or later we must come to something which we see intuitively to be true, as the process of interpolation cannot go on ad inªnitum. If the critics of intuition want to challenge the claim that it is a source of basic formal knowledge, they owe us an alternative explanation of our knowledge in cases of the kind that rationalists explain in terms of intuition. Intuition, like empirical observation, has both a horizontal and a vertical limit, which make a transcendent mode of knowledge necessary: on the one hand, there are too many fundamental objects in the domain, e.g., natural numbers, and, on the other, there are principles that express relations among the objects of intuition that are too general to be apprehended in those objects, e.g., mathematical induction. Transcendent formal knowledge, like transcendent empirical knowledge, is based on inferences based on basic knowledge. Such inferences generalize basic knowledge and bring basic and transcendent knowledge into an integrated, coherent, total system under the guidance of an ideal of theoretical systematization. The ideal is best thought of as part of the general concept of knowledge rather than as a special feature of the concepts of empirical and formal knowledge. Thus, many aspects of theoretical knowledge in the natural and formal sciences—for example, the simplicity of theoretical systems—are imposed, as it were, from above. In the formal sciences as well as in the natural sciences, theoretical considerations can be brought to bear in the process of systematization to determine the status of intuitively unclear cases and to correct intuitive or theoretical judgments that have been mistakenly accepted. As suggested already, systematization compensates for the fallibility of the processes of acquiring basic and transcendent knowledge by bringing considerations from one part of the overall system to bear on issues in another. Thus, the ideal of systematization gives a holistic character to justiªcation in the formal and empirical sciences, but this is an epistemic holism that has nothing to do with a semantic holism such

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as Quine’s (1961c, 43). Our holism does not claim that “the whole of science” is the smallest independently meaningful linguistic unit. Rather, assuming a prior correlation of sentences and senses in natural languages, our holism concerns the ways in which propositions in a particular system in a formal science obtain their support from one another and from the basic knowledge on which the theory rests. It is clear, according to our rationalist epistemology, that formal knowledge is not an exclusively bottom-up affair in which all basic knowledge is established prior to and independently of transcendental knowledge and systematization. Exactly how much foundationalism there is can be left open here, just as Benacerraf left the parallel question open in his account of empiricist epistemology. The extent to which transcendent knowledge must be anchored in basic knowledge can be treated as a more general question about reason’s overall ideal of systematization, and, as such, the question is independent of the issues here. Thus, our methodological holism is compatible with various forms and degrees of foundationalism. 2.6.2 The holistic character of systems of formal knowledge allows for the possibility that not everything counted as knowledge at the level of transcendent knowledge can be shown to be necessary. There is, as far as I can see, no way to show that the justiªcation of formal knowledge is uniformly a matter of excluding every possibility of falsehood. This does not mean that only principles for which we can exclude every possibility of falsehood count as knowledge in the formal sciences. Some principles at the transcendent level might count as knowledge, even though we cannot show that there is no possibility of their falsehood, because we can show that there is no possibility of achieving the best systematization of the science without them. I will call such principles “apodictic.” Possible examples might be Church’s thesis and the (number-theoretic) principle of mathematical induction. Although in the former case, we seem to be unable to prove that recursiveness is effective computability, we might argue that application of the ideal of systematization to transcendent logicomathematical knowledge shows that the thesis is essential to its best systematization. The introduction of the category of Apodictic Principle does not commit us to formal knowledge’s being demonstrably necessary or demonstrably apodictic. As realists, we are committed to there being a fact of the matter in the case of the apodictic truths of formal science, as we are committed to there being one in the case of the necessary truths of formal science, but we are not committed to

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always being able to know the facts.12 Realists have no epistemic chutzpah. To give our conception of transcendent formal knowledge systematicity, I will assume that the category Apodictic Principle encompasses all transcendent formal knowledge that cannot be directly shown to be necessary. As far as I can see, there is no way to establish that formal knowledge can either be directly shown to be necessary on the basis of intuition, some intuitionlike form of reason, or proof, or else be shown to be essential to the best systematization of the body of knowledge in question. But then again, a very similar assumption is made in connection with transcendent natural knowledge. Both seem to derive from the higher-order notion of systematization. Given this assumption, the question of whether formal knowledge is uniformly a priori is the question of whether knowledge of apodictic principles is a priori. The answer I want to give is that, in spite of the fact that apodictic principles cannot be directly shown to be necessary, knowledge of them is a priori because it is established on the basis of reason alone, on the basis of necessary truths themselves established by reason alone. Since knowledge of basic formal facts is a priori, since the step from that knowledge to transcendent knowledge of formal laws and theories is also a priori, and since, as a consequence, ªlling gaps and correcting errors is a priori too, formal knowledge always has the a priori warrant of pure reason. Systems of a priori formal knowledge to which we add apodictic principles thus remain a priori. Hence, our rationalism about basic formal knowledge can be extended to transcendent laws and theories. 2.6.3 Even though justiªcation in the formal sciences is a priori, propositions in those sciences are revisable in principle. We ºatly reject Quine’s (1961c, 42–46) equation of apriority with unrevisability (as well as his equation of apriority with analyticity). Of course, the revisability of a priori propositions in the formal sciences is something quite different from the revisability of a posteriori propositions in the natural sciences. Their revisability is revisability in the light of further pure ratiocination, not revisability in the light of further empirical discoveries. This sharp separation of rational and empirical revisability is a consequence of the fundamental difference between formal and empirical knowledge im12. It might be that an argument showing that a principle is essential to achieving the ideal of systematization in the formal sciences is a kind of transcendental argument for the principle. If so, there has to be an explanation of the nature of such arguments, particularly, of how, given our realism, they differ from Kantian transcendental arguments.

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plicit in the preceding sections (and to be stated explicitly in section 2.8). This difference explains why it isn’t possible to argue against the a priori nature of knowledge of mathematics just on the grounds of the revisability of mathematical beliefs, as, for example, Kitcher (1983) tries to do. In criticizing Kitcher, Hale (1987, 148) observes that “if revisability is to conºict with apriority, it must be revisability for empirical reasons.” There is, however, a well-known sort of argument that purports to show that mathematics is revisable for empirical reasons because continuing to maintain an alleged a priori mathematical truth would be irrational if a better overall empirical theory can be obtained once the proposition is given up. Perhaps the best-known example of this sort of argument is Putnam’s (1975b, xv–xvi) claim that abandoning Euclidean geometry in physics is a counterexample to its apriority: “[s]omething literally inconceivable had turned out to be true.” He (1975b, xv–xvi) writes: I was driven to the conclusion that there was such a thing as the overthrow of a proposition that was once a priori (or that once had the status of what we call “a priori” truth). If it could be rational to give up claims as self-evident as the geometrical proposition just mentioned, then, it seemed to me that there was no basis for maintaining that there are any absolutely a priori truths, any truths that a rational man is forbidden to even doubt. Putnam’s case does not support his conclusion. It is wrong to say that the proposition that was overthrown “was once a priori” or “once had the status of what we call “a priori” truth.” To be sure, people once believed that it is a priori that Euclidean geometry is a true theory of physical space, and they could not conceive of its not being a true theory of physical space, but Euclidean geometry was never determined a priori to be a true theory of physical space. From the fact that it is believed that p is a priori, it does not follow that p is a priori. In fact, the grounds for accepting Euclidean geometry as a true theory of physical space were straightforwardly a posteriori. The abandonment of Euclidean geometry in physics was a revision of an empirical theory. What everyone believed—and what Einstein showed to be false—was a theory in natural science that claimed that the geometric structure of physical space is Euclidean. Putnam’s case shows at most that there are no absolutely indubitable propositions in natural science. Since the case concerns an empirical application of Euclidean geometry, the case is one in which an a posteriori applied geometry was falsiªed on empirical grounds, not one in which an a priori pure geometry was falsiªed on such grounds. Thus, the Einsteinian revolution provides no

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basis for the claim that pure mathematics is in some broad (perhaps Quinean) sense empirical.13 Putnam-style arguments to show that mathematics is empirically revisable presuppose Quine’s holistic empiricism. Presupposing that scientiªc knowledge (whether formal or natural) forms a single integrated empirical theory ensures that the disputed a priori truth is part of the all-encompassing empirical theory. Without that presupposition the critic of a priori knowledge can no longer argue that holding on to a mathematical proposition would be irrational if a better overall empirical theory can be obtained by giving it up. If, instead of constituting one all-encompassing system of empirical beliefs, science were to be divided into two separate domains of belief, one containing formal theories about abstract objects and the other containing empirical theories about concrete objects, the disputed a priori mathematical truth in Putnam-style arguments would belong to the former and the empirically refuted proposition would belong to the latter. The disputed a priori mathematical truth would thus be a different proposition from any a posteriori proposition that could be given up to improve the current empirical theory. The Quine-Putnam thesis that mathematics is legitimized in virtue of the indispensability of numbers for natural science is a form of what we called methodological naturalism in the introduction. This view, that the only way to have knowledge is through natural science, also presupposes a Quinean empiricism. Without that thesis, there is no reason to think that the legitimacy of mathematical theories about numbers and the like depends solely on their role in natural science. The view thus ignores arguments for their legitimacy based on their indispensability in pure mathematics. If they are indispensable for doing pure mathematics, they are legitimate, and empirical science doesn’t have to enter the picture. As I argued in the last chapter in connection with Field’s nominalism, the grounds for acknowledging mathematical objects are not restricted 13. On a realist view, a pure geometry, Euclidean or otherwise, is a theory of a class of abstract spatial structures. In a complete theory, its principles express the possibilities of ªgures and relations among them within a space. Anything that conºicts with the principles is an impossibility in the space. Grammars, as I (1981) have argued, can be conceived in a similar way, as theories of a class of abstract sentential structures the principles of which express the possibilities of linguistic forms and grammatical relations within a language. In making a place for the notions of necessity and possibility in connection with pure geometries and pure grammars, we can bring geometric and grammatical knowledge under the scope of our rationalist epistemology. In chapter 5, I present an account of the distinction between pure and applied geometries, pure and applied grammars, and so on in terms of the different kinds of objects they study.

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to their role in natural science. We could establish their existence even if there were no empirical science. We could do enough mathematics and philosophy of mathematics prior to the development of empirical science to know about the plenitude of numbers and hence to have an argument that numbers cannot be identiªed with natural objects. It is even conceivable that we could do enough mathematics as disembodied Cartesian beings or in dreams to have a basis for positing numbers, sets, and other mathematical objects. Since the Quine-Putnam argument for the legitimacy of mathematics rests on Quinean empiricism, it rests on not just a contested empiricism, but, as we shall see in the next chapter, on an incoherent empiricism. 2.7 Have Any Questions Been Begged? This completes our sketch of a rationalist epistemology to explain why contact with logical and mathematical objects is not necessary for logical and mathematical knowledge. It does not, however, complete our response to Benacerraf’s epistemic challenge. To do that, we need to deal with two general doubts that might be raised about whether the epistemology that we have sketched is an adequate response to the challenge. 2.7.1 The ªrst doubt is that our rationalist epistemology might beg the question, since the reasoning it sees as justifying principles in mathematics and logic sometimes rests on the very principles that the reasoning is supposed to justify.14 Since the acceptability of such reasoning depends on the acceptability of the principles, we would already have had to accept the principles to accept the grounds that the reasoning provides. This doubt raises a more general doubt, one that goes well beyond the question of our success here. Logicians too are in the situation of attempting to justify inferential principles on the basis of inferences the validity of which depends explicitly on the principles they purport to 14. Circularity would be the wrong way to describe the alleged difªculty. An argument is circular when the very same proposition appearing as the conclusion appears as a premise. On a realist view of mathematical and logical principles, this could never happen. Realism sharply separates the objects of formal knowledge themselves, i.e., numbers, sets, logical principles, proofs, and the like, from our inner epistemic states and processes that provide us with knowledge of them. The former are abstract, objective, and autonomously existing, while the latter are concrete, subjective, and mind-dependent. Thus, the reasoning that provides us with grounds for taking mathematical and logical principles to be true cannot contain those abstract principles.

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justify. It is no coincidence that their situation is the same as ours, since our account of reasoning in the formal sciences was developed as an explication of rational practices in formal science. Hence, if it is concluded that we beg the question in attempting to defend realism, then logicians beg the question in attempting to justify theories of logical structure on the basis of inferences that depend on those principles. Hence, logicians cannot legitimately claim to justify logical principles and to have knowledge of them. Hence, there can be no science of logic. Since, furthermore, all other branches of science rely on principles of logic, not much is left. These consequences are a reductio of both charges of question-begging. The criticism that logicians have failed to provide a science of logic has to be rejected, and, consequently, the criticism that we have begged the question in providing the desired “improved” concept of knowledge has to be rejected. Moreover, it is not hard to see that what is wrong with both criticisms is that the doubt on which they are based is the doubt of the philosophical skeptic. Nothing less than the philosophical skeptic’s challenge to the prevailing standards for knowledge in the formal sciences would be strong enough to undermine formal knowledge across the board. To avoid begging the philosophical skeptic’s question, we would have to provide a basis independent of those standards that explains why it is reasonable to use them rather than others that might be used instead. Empiricists and rationalists are in the same boat with respect to philosophical skepticism. At the very least, empiricists also accept logical principles as the basis for scientiªc inferences to general empirical laws and theories. Hence, if there were grounds for suspicion that a rationalist epistemology begs the question, there would be the same grounds for suspicion that an empiricist epistemology begs the question.15 As a consequence, the charge is not one that empiricists can use against rationalists. In talking about acceptance of the prevailing standards for knowledge in the formal sciences, I am not suggesting that we know precisely what those standards are, that they are beyond criticism, or that their 15. Empiricists cannot argue that there is a disanalogy between their commitment to logical principles and the rationalist’s because the role of such principles in empiricism is only to transfer knowledge while their role in rationalism is to create knowledge. This may or may not be so, but, even if it is, the transfers are essential to empiricist epistemology and they cannot be made without logical principles. Consequently, the loss of those principles to skepticism would be just as serious a loss for empiricists as it would be for rationalists. (Thanks to Glenn Branch [personal communication] for raising this issue.)

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formulations at present are adequate. There is, no doubt, much work to be done to provide a full explication of our prevailing standards. I am assuming only that our rough idea of them is clear enough to distinguish questions that call scientiªc knowledge into question from their own standpoint from ones that call them into question from the standpoint of philosophical skepticism. I will return to this distinction in chapter 6. There is an independent skeptical challenge to the empiricist’s account of basic empirical knowledge. Exploiting Goodman’s (1955, 59– 83) counterinductive strategy, the Humean skeptic will ask Hermione how she can claim to know that what she is holding is a trufºe when, for all the evidence she might have from causal contact, she could just as well be holding a truffrock as a trufºe (a truffrock is a trufºe until some time in the future and a rock thereafter). Since every extrapolation consistent with all the evidence—including even extrapolations about medium-sized objects, since they involve the posit of permanence— have the same empirical ground for their truth, there is no reason to prefer the inductive extrapolation to a trufºe over a counterinductive extrapolation to the truffrock. It is question-begging for Hermione to reply that the former extrapolation was obtained on the basis of the inductive rule. The skeptic can present the fact that the latter extrapolation was obtained on the basis of a counterinductive rule, say, one that presupposes that the world is a gruesome place that will show its true colors in another year, and press Hermione to explain why extrapolation on the basis of her rule is more rational. She will fail, of course, to satisfy the skeptic that her inductive policy is more rational. If empiricists have their Humean skeptic, rationalists have their Cartesian skeptic. As the Cartesian skeptic sees it, rationalists are in a parallel and equally hopeless epistemological situation with respect to logical and mathematical knowledge. Descartes (1970, 150–51, 236–37) explains that it is possible for logical principles to have been false because “the power of God cannot have any limits.” How can we rationally claim to know that noncontradiction is a necessary truth on the grounds that reason shows that it could not be false when God could “make it not be true . . . that contradictories could not be [true] together”? To the Cartesian skeptic who complains that we have no right to claim to know that laws of logic are necessary truths, we cannot answer that God could not make something conºict with the principle of noncontradiction because a contradiction cannot be true. To answer that a contradiction cannot be true begs the question. For it is precisely the necessity of a contradiction’s falsehood that the Cartesian skeptic

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is challenging us to justify when he introduces the idea of God’s absolutely unrestricted omnipotence. Since both empiricists and rationalists perforce beg the philosophical skeptic’s question and since the former can hardly grant themselves immunity to skepticism while using it against the latter, the empiricist is in no position to use the fact that our reply to the epistemic challenge begs the philosophical skeptic’s question against us. There is a further consideration, too. As we saw earlier, Benacerraf ([1973] 1983, 413) has to appeal to inference based on the results of observation in cases of “medium-sized objects, in the present” to account for “our knowledge of general laws and theories, and, through them, our knowledge of the future and much of the past.” This appeal makes the empiricist account of natural knowledge dependent on principles of logic, and hence opens it up to the same skeptical challenge that besets our rationalist account of formal knowledge. Since empiricists are also vulnerable to the Cartesian skeptic, it would do them no good to grant themselves immunity to the Humean skeptic. The fact that a rationalist epistemology begs the philosophical skeptic’s question is neutral with respect to the claim that instances of the reasoning that a rationalist epistemology describes provide completely adequate grounds for mathematical and other formal knowledge.16 It is clear why this is so: the notion of acceptability relevant to satisfying the philosophical skeptic is different from the notion of acceptability relevant to providing grounds for conclusions in the formal sciences. The former is acceptability as a response to a philosophical skeptic’s challenge to our prevailing standards and the latter is acceptability on the basis of the prevailing standards. Once we distinguish these two notions, it no longer appears that failing to satisfy the philosophical 16. The further question of whether or not we have to concede ultimately that philosophical skepticism is right is irrelevant to the present issue. Perhaps showing that the realist can provide an epistemology that handles formal knowledge as well as the antirealist’s causal theory handles natural knowledge would in some sense be pointless if philosophical skepticism did turn out to be right. But, as we have no reason to think it will, we have no reason to think that my reply to Benacerraf is pointless. It is also worth noting that a refutation of Humean skepticism does not confer an advantage on empiricism, since what we now accept as mathematical knowledge is certainly not going to be taken as mere opinion unless something like Cartesian skepticism is established at the same time. Not only is it hard to see how philosophical skepticism might be refuted in one case and carry the day in the other, but, even if something like Cartesian skepticism were established, that would confer no real advantage on empiricism in the present controversy. There would be no rationalist epistemology, but that would only be because there would be no formal knowledge for it to explain. It would not have been shown that realism, as opposed to empiricism, cannot meet the epistemic challenge.

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skeptic prevents us from maintaining that our rationalist epistemology answers the epistemic challenge to realism. Accordingly, the only legitimate question that arises about Hermione’s claim to know that what she is holding is a trufºe is whether she has an adequate basis, relative to prevailing mycological standards, for claiming to have such knowledge. If Hermione’s teacher asks her how she knows that she is holding a trufºe, and she produces evidence that it is a fungus, grew underground, has a warty surface, is blackish in color, exudes an earthy aroma, and so on, she has justiªed her claim to know that she is holding a trufºe. It would be absurd for her teacher to ºunk her because this evidence does not exclude the skeptical possibility that what she is holding is something gruesome. Similarly, the only question that arises about Hermione’s claim to know that two is the only even prime is whether she has an adequate basis, relative to prevailing mathematical standards, for claiming to have this knowledge. Suppose Hermione’s mathematics teacher asks her how she knows that two is the only even prime, and she produces a proof like (A). “There,” she says, “that’s how I know that there couldn’t be another even prime!” It would be equally absurd for her mathematics teacher to ºunk her because the reasoning does not exclude the skeptical possibility that God violated the law of trichotomy in order to create other even primes. Hermione’s demonstration that she fully understands the proof shows that she has an adequate basis for claiming to know that two is the only even prime. Having met the prevailing standards, she has to be credited with the knowledge. The same goes for Hermione’s claim to know logical principles on the basis of reasoning that depends on the principles that the reasoning is intended to justify. If, for instance, her reasoning depends on the logical principle that contradictories cannot be true together, then, of course, the reasoning would not be acceptable were it not the case that contradictories cannot be true together. But it would be absurd for Hermione’s logic teacher to ºunk her because she hasn’t excluded Descartes’s supposition about God’s omnipotence. Once philosophical skepticism is out of the picture, the only remaining question is whether a body of evidence or a piece of reasoning measures up to the prevailing standards. Neither our rationalist epistemology nor Benacerraf’s empiricist epistemology has any worry on this score. Each sanctions reasoning that will meet the prevailing epistemic standards in its respective area of science. This is no surprise. Each was developed as an explication of those standards. No doubt both epistemologies, as they stand, fall short of being fully adequate explications, but their shortcomings are irrelevant here, since both can be corrected without affecting the present issues.

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2.7.2 The second doubt is about whether we have begged the question about the ability of reason to determine necessary truth. Since such determinations are commonly based on conceivability intuitions, the doubt arises from traditional philosophical criticisms of the claim that such intuitions can establish facts about necessity and possibility. Arnauld (1984, 140) argued against Descartes’s inference from the fact that he can conceive of himself as pure mentality to the conclusion that he is essentially pure mentality, claiming that conclusions about what is possible and impossible cannot be drawn from premises about what is and is not conceivable. Mill (1874, vol. 2, ch. 5, sec. 6) drew attention to the gap between a psychological premise concerning the conceivability of something and a metaphysical conclusion about “the possibility of the thing in itself.” Mill’s distinction went right to the heart of the matter. The premise concerns what is, as Mill put it, “in truth very much an affair of accident, and depends on the past history and habits of our own mind,” but the conclusion expresses a logical fact which can be neither contingent nor relative. Descartes confesses in Meditation I that he found it inconceivable that he could sanely doubt that his hands and body are as they appear to him until it occurred to him that he had been deceived by appearances in dreams. If conceivability is relative to the circumstances (e.g., dependent on what things we do and do not think of), how can it provide grounds for beliefs about metaphysical possibility and impossibility? Why then should the fact that we ªnd the falsehood of a proposition inconceivable mean that not believing it is irrational? Why should it mean anything more than that we are laboring under a psychological restriction that prevents us from forming a conception of what the proposition asserts is the case? Yablo (1993) has argued that such general doubts about conceivability as a basis for knowledge of possibility and impossibility are of a piece with general philosophical doubts about our other cognitive faculties. Given that intuitions based on conceivability play an essential role in logic, general doubts about conceivability ought to be treated with the same caution with which we customarily treat general philosophical doubts about our other cognitive faculties. Yablo tries to show that the doubts of Arnauld, Mill, and others are based on confusion about the concept of conceivability. He concludes that none of those skeptical arguments works, and that recent successes of “substantive modal metaphysics” ought to encourage us to direct our energies into developing a better theoretical understanding of modal inquiry. As I understand Yablo, his line of argument is similar in spirit to the line taken above in connection with philosophical doubts about the

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effectiveness of reason in answering logical and mathematical questions. Accordingly, doubts about conceivability like Arnauld’s and Mill’s can be taken either of two ways. They can be taken as skeptical doubts intended to philosophically question conceivability as a standard for modal knowledge. Or they can be taken as a nonskeptical call to explicate the right kind of conceivability—the kind that actually provides the grounds for our knowledge about what is possible and impossible. Once we see that doubts about conceivability like Arnauld’s and Mill’s can be taken in both these ways, we can defuse them with the same strategy used above to absolve a rationalist epistemology of the charge of begging the question. We concede that we cannot meet the philosophical skeptic’s challenge to conceivability, but argue that this concession is of no more signiªcance than our conceding that we cannot meet the Cartesian or the Humean skeptic’s challenge. From the standpoint of the present defense of reason’s ability to determine necessary truth, all that needs to be shown is that intuitions based on the proper sort of conceivability measure up to the prevailing standard for modal knowledge. Defending conceivability is not the only way of defending reason’s ability to determine necessary truth. We could base the defense on reason’s power to recognize inconsistency. This seems to be a different defense. Before it occurred to Descartes that he had been deceived by appearance in dreams, he found it inconceivable that he could be sane and think he had neither hands nor a body, but he presumably did not ªnd his sanity incompatible with his thinking that he had neither hands nor body. Descartes’s subsequent thought that he might be dreaming does not show that it wasn’t inconceivable to him that he could be sane but think he has no hands and no body, but it does show that it wouldn’t be inconsistent for him to believe that he is sane and that he had no hands and no body. Of course, it might be said that the appropriate notion of inconceivability is an idealization expressing what is conceivable for us in principle, but it is arguable whether even ideal conceivability coincides with possibility (in the discussion of naturalized realism in the preceding chapter, I explained why what an ideal speaker knows does not necessarily coincide with the objective facts about the grammar of the language). If we base our epistemology on reason’s power to recognize a proposition as contradictory rather than its power to conceive of something as existing, we can exploit the evident logical fact that what is contradictory is impossible. The contradictoriness of the characterization “barber who shaves all and only those who do not shave themselves” entails the impossibility of such a barber. There is no corresponding evident fact to exploit in the case of what even an ideal conceiver would

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ªnd inconceivable. To be sure, we can think something is contradictory when it is not, but such failures to recognize contradictory structures can be corrected a priori, and therefore do not matter. What matters is that when we are right that something is contradictory, it follows that it is impossible. In contrast, when we are right that something is inconceivable, there may still be a question whether or not it is impossible. Yablo (1993, 36–40) himself offers no ªnal assurance that there is a notion of inconceivability strong enough to provide satisfactory metaphysical grounds for thinking that inconceivability implies impossibility. But wouldn’t the Cartesian skeptic challenge the step from the inconsistency of the characterization “barber who shaves all and only those who do not shave themselves” to the impossibility of such a barber? God, as Descartes argued, could “make it not be true . . . that contradictories could not be [true] together.” But doubts about the inference to the conclusion that such a barber could not exist are those of the philosophical skeptic. Here we again return to our earlier strategy for defusing skeptical doubts about the use of reason to answer logical and mathematical questions. Such doubts, as we have seen, are not relevant, because the question is only whether the inference is in accord with the prevailing logical standards. We do not have to answer the skeptic’s challenge to those standards. In the present context, we might have to worry about the possibility of something literally inconceivable turning out to exist, but we don’t have to worry about the possibility of something literally contradictory turning out to exist. 2.8 A New Distinction between Natural and Formal Knowledge As traditional realism draws the distinction between the formal and the natural sciences, it is entirely a matter of ontology. The formal sciences are about one kind of object and the natural sciences are about another kind. This is, of course, right as far as it goes, but it doesn’t go far enough. As history shows, it hasn’t prevented realists from thinking that realist epistemology requires some form of causal contact with the objects of mathematical knowledge. What must accompany the traditional realist’s ontological distinction between the formal and the natural sciences is a corresponding epistemological distinction. In addition to saying that the natures of the objects studied in the formal and the natural sciences are different, we need to explain precisely how knowledge in the formal sciences differs from knowledge in the natural sciences. Possibly gorillas like bananas, but possibly they don’t. Natural science asks which possibility is actual. Supposably there is one even prime, but supposably there is more than one even prime.

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Formal science asks which supposition is necessary (or essential to the systematization of the necessary). Painting the picture in broad brush strokes, we can say that investigation in the natural sciences seeks to prune down the possible to the actual, while investigation in the formal sciences seeks to prune down the supposable to the necessary.17 Given that the aim of investigation in the natural sciences is to determine what possibilities are actualities, while the aim of investigation in the formal sciences is to determine what supposabilities are necessities, natural science and formal science require appropriately different epistemic method and produce correspondingly different kinds of knowledge. Pruning down the supposable to the necessary requires reason, the whole of reason, and nothing but reason. Pruning down the possible to the actual requires perceptual contact as well. Since pruning down the supposable to the necessary requires only reason, formal knowledge is a priori knowledge. Since pruning down the possible to the actual requires interaction with natural objects as well as reason, natural knowledge is a posteriori knowledge. Philosophers who jump to the conclusion that mathematical and other formal knowledge must be based on causal contact do not realize that the task of the formal sciences is to prune down the supposable to the necessary. If they did, they would see that perceptual contact can play no role in the justiªcation of formal knowledge. But, thinking that the task of the formal sciences is, like the task of the natural sciences, to prune down the possible to the actual, they quite reasonably conclude that perception must play a role in the justiªcation of formal knowledge. Hence, they are driven to think either that mathematical knowledge cannot be knowledge of abstract objects, as many naturalists think, or that there is perceptual contact with abstract objects, as many realists think. It is thus not surprising that many naturalists believe that realists must end up as mystics, and that many realists do indeed end up that way. 2.9 A Tenable Dualism Quine’s methodological naturalism and his doctrine that science is ªrst philosophy led him to countenance abstract objects in the domain of mathematics. As Quine (1961c, 46) sees it, “Ontological questions . . . are on a par with questions of natural science.” He thus allows abstract 17. One qualiªcation that might be required is that the supposable may in certain special cases be pruned down to the apodictic rather than the necessary. Whether such a qualiªcation is required for the possible depends on linguistic and metaphysical issues that are beyond the scope of this book.

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objects into the domain of science, but only, as it were, through the experiential back door and then only those with the proper credentials of applicability in natural science. Our position agrees with Quine’s abstract/concrete dualism, but disagrees with the naturalist/empiricist basis on which he posits the existence of abstract objects, and disagrees even with the notion that they are posits. On our position, it is formal science, not natural science, that provides the basis for claims about the existence of abstract objects, and it is the foundations of the formal sciences, not the sciences themselves, where it is shown that talk about numbers, sets, and other objects discussed by the formal sciences is best understood as talk about abstract objects. We have heard again and again in the course of this century that dualisms are untenable. Doesn’t the dualism of the abstract and the concrete face the same threat of incoherence as Cartesian dualism? Quine (1961c, 44–45) gets his dualism off the hook by taking an instrumentalist view of science on which it is “a tool, ultimately, for predicting future experience in the light of past experience” and on which “[e]pistemologically [abstract mathematical objects] are myths on the same footing with physical objects and gods, neither better nor worse except for differences in the degree to which they expedite our dealings with sense experience.” Rejecting both the monism of ontological and epistemological naturalism and Quine’s empiricist way of getting his dualism off the hook, we have to show that abstract/concrete dualism does not face the threat of incoherence that Cartesian dualism does. The reason that it doesn’t is simply that not all dualisms are the same. The dualism of the abstract and the concrete differs from the dualism of the mental and the physical in one critical respect. Both of them posit domains of causally incommensurable objects, to be sure, but the former pays no price for its ontological bifurcation because, unlike Cartesianism, realism is under no pressure to explain how ontologically incommensurable objects can causally interact with one another. Descartes’s substance dualism is under such pressure because our experience is replete with clear signs of what appear to be causal interactions between the mental and the physical. But, in the case of realism, there are no signs of causal interaction between the abstract and the concrete. Indeed, at this point in the book, we see that nothing could be a sign of such interaction. Without a counterpart of psychophysical correlations, the realist is under no pressure to explain away the appearance of causal connection between correlated but causally incommensurable objects. So, there is no pressure pushing the realist into an incoherence of the sort that physicalists believe the pressure from psychophysical correlations push Cartesians into.

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Even in the absence of a counterpart to psychophysical correlations, many antirealists have thought that the dualism of the abstract and the concrete creates an epistemological pressure to explain causal connections between abstract and concrete objects. It should now be clear that they think this only because they assume that the causal theory of knowledge must apply to formal knowledge as well as natural knowledge. And it should also now be clear that this assumption is false because there is a rationalist theory of knowledge in mathematics and the other formal sciences.

Chapter 3 The Epistemic Challenge to Antirealism

3.1 The Challenge to Antirealism The challenge to realism was to explain how mathematical knowledge is possible if mathematical objects are abstract. The challenge to antirealism is to explain how mathematical knowledge is possible if mathematical objects are concrete. The challenge to antirealism derives from the traditional rationalist argument that knowledge of what contingently is cannot ground knowledge of what must be, but is a broader argument that applies to Quineans, who reject the notion that there is anything that must be. The challenge is to explain the special certainty of mathematical and other formal truths on the basis of a naturalist concrete ontology and an empiricist epistemology. This challenge is, in a sense, complementary to the epistemological challenge to the realist to account for mathematical knowledge. In both cases, ontology raises a problem for epistemology. Realism’s rationalism seems the right sort of epistemology to explain a priori knowledge, but it is prima facie unclear how such an epistemology is supposed to work in the formal sciences when non-naturalism says that what we know in those sciences are facts about abstract objects. Antirealism’s empiricism seems the right sort of epistemology to explain a posteriori knowledge, but it is prima facie unclear how such an epistemology is supposed to work in the formal sciences when naturalism says that there is nothing to know except facts that are au fond about concrete objects. How can the special certainty of theorems about numbers, sets, and the like rest on a posteriori facts about contingent objects? That special certainty is an objective feature of those theorems. It is different from the subjective certainty we have about our beliefs in them. The former is something like an exclusion of all possibility of doubt based on the prevailing standards in the discipline. White (1972, 4–5), for example, says that propositions are certain when the question of their truth is settled in virtue of the exclusion of all reasonable

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possibility of falsehood.1 Subjective certainty is the mental state which we describe with such adjectives as “conªdent,” “assured,” or “certain.” The mathematical proposition that the circle cannot be squared is objectively certain, and it would remain so even if the general level of mathematical competence were to drop to the point at which everyone became a would-be circle squarer. The objective certainty of a proposition consists in the fact that different people who recognize its certainty are recognizing the same logical relations, whereas the subjective certainty of a belief consists in the fact that different people who are equally certain of the same proposition are in similar mental states but not in the same token mental state. Subjective certainty, as Rollins (1967, 67) observes, may be justiªed or not—and hence such certainty may be justiªed in one person’s case but not in another’s—while objective certainty obtains or doesn’t. As with many signiªcant notions in philosophy, there is no fully satisfactory explication of the notion of objective certainty. Questions of many kinds arise about the above description. How is possibility to be understood? We apparently do not want to understand it as logical possibility because we want to allow some contingent statements to be certain. Again, this description of objective certainty takes the exclusion of all reasonable possibility of a proposition’s falsehood to be a consequence of its relation to other propositions, but what relations are meant? Logical relations? Rational ones? A priori ones? Metaphysical ones? Inductive ones? Further, there is no demonstration that formal truths have a special certainty that common empirical truths lack. There are, however, clear signs that the certainty of the former is special. One is the relative imperviousness of knowledge in the formal sciences to doubt based on counterfactual hypotheses, in contrast to the relative susceptibility of empirical knowledge. Contrary hypotheses in the case of empirical knowledge are far-fetched or off the wall, as is, for example, the hypothesis that the moon is made of swiss cheese, but contrary hypotheses in the case of formal knowledge are impossible, as is, for example, the hypothesis that four is not composite. Another sign of the difference can be seen in adverbial modiªcations of “certain.” The certainty of truths in the formal sciences is customarily described with adverbs such as “completely” and “absolutely,” whereas such adverbs are typically eschewed in describing the certainty of even the most secure empirical truths. Instead, their certainty 1. This may seem to come dangerously close to explicating necessity rather than certainty, but in the last section, I will argue that this appearance simply reºects the close connection between necessity and certainty in the area of formal truth.

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is described using adverbs such as “theoretically” and “empirically.” In the former case, we express an unconditional exclusion of doubt, while, in the latter, we qualify the exclusion of doubt, relativizing it to a theory or a body of empirical evidence. Fortunately, it is not necessary for our purposes to have either an explication of special certainty or a demonstration that formal truths have special certainty. On the one hand, the notion is clear enough, and, on the other hand, it is generally acknowledged that formal truths do have a special certainty. Even empiricists like Mill and Quine who deny the existence of necessary truth acknowledge that mathematical and logical truths have a special certainty, as is clear from the fact that they recognize the obligation to provide an empiricist explanation of it. The acknowledgment is also seen in Quine’s dissatisfaction with Mill’s explanation of the certainty of mathematics and logic in terms of the more extensive empirical conªrmation of mathematical truths. Quine (1966, 100–101) writes: [Mill’s] doctrine may well have been felt to do less than justice to the palpable surface differences between the deductive sciences of logic and mathematics, on the one hand, and the empirical sciences ordinarily so-called on the other. Worse, the doctrine derogated from the certainty of logic and mathematics; but Mill may not have been one to be excessively disturbed by such a consequence. Perhaps classical mathematics did lie closer to experience then than now. . . . It is, of course, open to naturalists ºatly to deny that mathematical and logical truths have such a special certainty, but such stonewalling is implausible in the face of such wide recognition of “the palpable surface differences between the deductive sciences of logic and mathematics, on the one hand, and the empirical sciences ordinarily so-called on the other.” Historically, empiricists who have kept faith with the doctrine that all truths are just a posteriori truths about contingent objects have had a hard time trying to explain how mathematical and logical truths can have the special certainty they are generally acknowledged to have. Mill’s answer was an acute embarrassment to empiricists. Arithmetic truths are laws about natural objects, differing from what we commonly take as such laws only in being of a higher order. The arithmetic truth “1 = 1” is about natural objects weighing one pound, being a foot tall, and so on. Since they are a posteriori truths, they have to be falsiªable. Mill’s (1874, vol. 2, ch. 6, sec. 3) explanation is that “1 = 1” could be false because one natural object weighing a pound need not have exactly the same weight as another weighing a pound. The

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arithmetic equation “3 = 2 + 1” is about natural objects because it expresses the physical possibility of grouping three ungrouped natural objects into two groups—to which Frege (1953, 9–17) replied, “What a mercy, then, that not everything in the world is nailed down.” Mill explained the special certainty of mathematical truths as a matter of their receiving far more extensive empirical conªrmation than other a posteriori truths. We know the truth of a mathematical equation like “3 = 2 + 1” either on the basis of direct induction from a sample of three natural objects or on the basis of inference from more general inductive truths such as “The sums of equals are equals.” Frege’s (1953, 9–17) criticism of this account is generally regarded as about as close to a total refutation as philosophical criticisms get. Moreover, Mill’s explanation of the certainty of mathematical truths transparently fails to establish a genuine difference between their certainty and that of a posteriori truths. On Mill’s view of “1 = 1”, it can at best have a probability of 1, but even that does not capture the certainty of that equation. If anything is required to drive the point home, note that an objective probability of the occurrence of one event with respect to another can be 1 even though there are indeªnitely many instances in which the one event occurs without the other, whereas the certainty of an arithmetic equation is lost with one counterinstance. Further, it is hard to see why esoteric mathematical truths should be better conªrmed than commonplace empirical ones. Unable to accept Mill’s inductivism, empiricists ºirted for a time with a conventionalist account of logical and mathematical truth, but, as Quine (1936) showed, it, too, fails as an explanation. It was only with Quine’s empiricism that the empiricist’s situation was turned around. It made full-blooded empiricism respectable. Quine’s (1961c, 46) empiricism treats posits of mathematical objects and hypotheses about their relations on a par with posits of physical objects and hypotheses about theirs, simply as a means to “expedite our dealings with sense experiences.” Quine’s holism solves both the problem of accounting for the certainty of logical and mathematical truths and the problem of accounting for them as truths about the natural world. In a nutshell, Quine’s (1961c) explanation of the certainty of mathematical and logical truths is that “total science” is a single system in which those truths, though revisable in principle, are nonetheless so far removed from the system’s experiential boundary and so intertwined with everything else in the system, that revision of them is not really a live option in normal science. This doctrine saved the uncompromising empiricist’s account of mathematics as a posteriori knowledge. Since Quine, empiricists have stopped worrying about the certainty of mathematical and logical truths.

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In this chapter, I will argue that Quine’s epistemology is, in fact, no more successful than inductivism and conventionalism. Since the failure of Quine’s epistemology is tantamount to the failure of contemporary uncompromising empiricism, if my argument goes through, contemporary empiricism will be shown to have no explanation of the special certainty of mathematical and logical truths and no explanation of how they can be about natural objects. Without those explanations, it will have no account of mathematical knowledge. 3.2 Antirealist Responses Since the antirealist claim is that everything that exists is a concrete object in the sense of something that can have spatial or temporal location, antirealists may be monists or dualists with respect to the mind/body problem. Antirealists can subscribe to the existence of both Cartesian egos and material objects like sticks and stones, and hence be mind/body dualists. Or they can be mind/body monists, subscribing to idealism, materialism, or neither, i.e., they might straddle the issue because, taking their cue from Russell (1927), they are neutral on the basic nature of reality. Our argument in this book is directed at antirealists of every kind, but we are most concerned with materialist antirealists because the most vigorous contemporary forms of antirealism were inspired by naturalism in Danto’s (1967, 448–50) sense of a monism that says that everything that exists is a material or physical part of a single causally interrelated spatiotemporal realm. American naturalism in the early part of this century was an ontological and epistemological naturalism. It was an across the board “revolt against dualism,” opposed as much to the dualism of the abstract and the concrete as to the dualism of the mind and the body or the dualism of the natural and the supernatural. Such global naturalists had a rationale for their comprehensive stand against dualism. They saw dualism per se as the common cause of a number of philosophical problems. In bifurcating the world, dualism creates the problem of reconciling ontologically incommensurable entities. Cartesian dualism creates the problem of explaining causal interaction between the mind and the body, supernatural dualism creates the problem of explaining causal interaction between the spiritual and the natural, and Platonic dualism creates the problem of explaining causal interaction between the abstract and the concrete. Since all dualisms pose essentially the same problem, there can be a single solution to them all: reject dualism per se. In spite of the attraction of eliminating the source of all of these philosophical problems in one stroke, some philosophers adopt a

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naturalist stance on one issue or another but not on all. For example, Fodor (1990, 132 n. 6) is opposed to mind/body dualism, but not to abstract/concrete dualism. Those who forgo the simple and effective solution of rejecting dualism per se do so because they think there are overriding considerations, usually tied in with their needs in the case of the dualism that most matters to them, to allow certain non-natural objects full ontological citizenship. Accordingly, a more descriptive terminology for contemporary philosophers calling themselves “naturalists” would not categorize them as naturalists tout court, but rather as naturalists on the abstract/concrete issue or naturalists on the mind/body issue or naturalists on the natural/supernatural issue. Thus, on the abstract/concrete issue, global naturalists and local naturalists see the source of the problem of reconciling the incommensurable entities somewhat differently, but they have the same view about its cause: dualism. Although methodological naturalists can maintain an abstract/concrete dualism, as Quine does, they take the scientist’s and the epistemologist’s object of study to be “a natural phenomena, viz., a physical human subject” (Quine 1970, 83), and they take an uncompromisingly empiricist stand on the a posteriori status of all knowledge. Hence, unlike our dualism, the dualism that methodological naturalism tolerates is kept within the bounds of ontology. It is not allowed to spill over into epistemology. The two most inºuential forms of naturalism in contemporary philosophy are Quine’s and Wittgenstein’s. Whereas the former is dualist, empiricist, and scientistic, the latter is monist, critical, and ascientistic. These two forms of naturalism have given rise to two distinct naturalistic explanations of the special certainty of mathematics and logic. I will discuss Quine’s paradigm in this and the next section, and Wittgenstein’s in section 3.4. Uncompromising empiricism explains all our knowledge as based, to some extent or other, on information about the natural world obtained through our senses. Such information can undergo certain ratiocinative processing once inside us, but our reasoning, in and of itself, never provides the sole grounds for a piece of knowledge.2 Since only natural objects can affect our senses, all scientiªc knowledge, the formal and the natural alike, must ultimately, if very indirectly, depend on 2. Pure naturalism requires an epistemology that sees all scientiªc knowledge as about the natural world. The general requirement goes the other way too. An empiricist epistemology requires a naturalist ontology. A rationalist epistemology does not mesh with a naturalist ontology in which there are only objects with contingent properties that can be known a posteriori, and an empiricist epistemology does not mesh with a realist ontology in which there are abstract objects that have necessary properties and can be

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sensory information about such objects. But if so, how can logical and mathematical knowledge be so much more certain than natural knowledge when it rests on the same sensory information as the latter and when that information itself would seem to be of a distinctly lower order of certainty than the knowledge that rests on it? The palpable inadequacy of Mill’s view led logical empiricists to deny that mathematical and logical knowledge is a posteriori. Carnap (1963, 64) wrote: Since empiricism had always asserted that all knowledge is based on experience, this assertion had to include mathematics. On the other hand, we believed that with respect to this problem the rationalists had been right in rejecting the old empiricist view that the truth of “2 + 2 = 4” is contingent upon the observation of facts, a view that would lead to the unacceptable consequence that an arithmetical statement might possibly be refuted by new experiences. Our solution . . . consisted in asserting empiricism only for factual truth. By contrast, the truths of logic and mathematics are not in need of conªrmation by observations. . . . Carnap (1963, 64) distinguished “logical truths,” which “do not state anything about the world of facts [but] hold for any possible combination of facts,” from “factual truths,” which make empirical statements and do not hold for any possible combination of empirical facts. This reconstruction of Hume’s distinction between relations of ideas and matters of fact gave rise to a new empiricism. Thanks to Frege’s logical semantics, particularly his conception of analyticity, arithmetic truths like “7 + 5 = 12” are no longer synthetic a priori, but analytic a priori truths in Carnap’s extended sense of “analytic.” Hence, logical empiricists are not subject to Kant’s criticism of Hume for throwing out mathematics along with metaphysics. This left logical empiricists with the question of what kind of truth Carnapian analytic truth is if it is neither factual nor observationally conªrmed. There were basically two answers. One was that it is truth by convention. Hempel’s ([1945] 1983) “The Nature of Mathematical Truth” is a classic formulation of this answer. The other answer abandons the idea that stipulation can deliver truth in favor of a noncognitivism about logical and mathematical truth. As Carnap (1963, 64) puts it, theories “for the construction of a metalanguage for the analysis of the language of science” are meant “not as assertions, but rather as known, if at all, only a priori. This general requirement is part of the pressure to construct philosophical systems. In its fullest form, the controversy between rationalism and empiricism is a controversy between entire systems of philosophy.

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proposals.” Driven by his perennial positivism about the nature of philosophical disagreement, Carnap claimed that issues about logical or mathematical truth that arise outside linguistic systems with explicitly formulated rules for theoremhood are meaningless. Moreover, since there is no cognitive basis for external debate about the correctness of such systems, philosophers can do no more than propose linguistic systems. Anything (everything?) is okay as long as it is explicitly formulated. I shall discuss this position in chapter 6. Quine ([1936] 1983) famously refuted the ªrst answer, showing that stipulation cannot even explain the truth of logical laws (much less their certainty). Logical truths, being inªnite in number, must be captured as instances of general principles, but, as logic is required for this enterprise, conventionalism offers no explanation of logical truth. (See also Benacerraf’s ([1973] 1983, 419–20) for an important addendum to Quine: convention does not guarantee truth; it gives us no distinction between “those cases in which it provides for it [and] . . . those in which it does not”.) Quine (1961c) also mounted a powerful attack on the second answer. Carnap’s noncognitivism has two components: a positivistic motivation for the claim that linguistic systems are essentially unconstrained proposals and a conception of analyticity as a basis for the understanding of logic and mathematics. Since positivism was widely seen as a failed program, the issue came down to whether Carnap’s apparatus of meaning postulates can explain analyticity. Quine (1961c, 32– 37) argued that such a recursive speciªcation of the “analytic” sentences of a language tells us which of its sentences are to be assigned this label, but it does not tell us what property is attributed to the sentences with this label. As Quine (1961c, 33) put it, the label “might better be untendentiously written as ‘K’ so as not to seem to throw light on the interesting word ‘analytic’.” There is nothing to distinguish the recursively speciªed class of sentences with the label “analytic” from any other class of sentences, except for the fact that someone chose to label them such. Carnap (1963, 918) replied that analyticity is no worse off than other notions of formal logic, but this reply is grist for Quine’s mill. Construed along Carnapian lines, “S is a logical truth in L,” “S implies S′ in L,” and so on are, of course, no worse off than analyticity, but, as Quine sees it, they are no better off either. Quine (1961c, 36) wrote: Appeal to hypothetical languages of an artiªcially simple kind could conceivably be useful in clarifying analyticity, if the mental or behavioral or cultural factors relevant to analyticity—whatever

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they may be—were somehow sketched into the simpliªed model. But a model which takes analyticity merely as an irreducible character is unlikely to throw light on the problem of explicating analyticity. This criticism of Carnap’s analytic/synthetic distinction was the ªrst battle in Quine’s campaign to replace logical empiricism with an uncompromising empiricism in the Millian spirit. Quine’s (1961c, 37–42) criticism of the veriªcationist account of synonymy was the second. The story is familiar: the relation of statements to experiences, on which the notions of conªrmation and disconªrmation depend, presupposes the conªrmability and disconªrmability of statements in isolation, and this, in turn, presupposes “a cleavage between the analytic and the synthetic.” When, as a consequence, “the unit of empirical signiªcance” is taken to be “the whole of science” rather than the term or the statement, we get a new uncompromising empiricism free of the defects of Mill’s empiricism. This is Quine’s (1961c, 42–46) famous conception of knowledge as a single fabric of empirically revisable beliefs with maximally revisable beliefs at the observational periphery and minimally revisable ones at the center. Quine’s epistemology does a far better job of explaining the special certainty of logical and mathematical knowledge than Mill’s. More central beliefs are more certain, not just because they have been more frequently conªrmed in experience, but because of their larger role in knitting together the entire fabric. Central beliefs, being involved in more interconnections, receive more empirical support from the total predictive success of the system, but there is another, perhaps more signiªcant, source of their certainty. Quine (1961c, 44) points out that “our natural tendency [is] to disturb the total system as little as possible.” This orders beliefs in the system so that the more centrally located statements are sacriªced only after other, less drastic, measures have failed to accommodate “recalcitrant experience.” Hence, maintaining the equilibrium of the system means sacriªcing mathematical and logical beliefs only as a last resort. Further, mathematical and logical statements enjoy a special certainty because the fate of the total system is bound up with the central mathematical and logical beliefs. Abandoning them removes the strands that hold the web of belief together, so that the system becomes a tissue of lies. This is worse than the opening up of an embarrassing hole here or there as the result of abandoning less central beliefs, because the unit of signiªcance is the whole system. Loss of the main connecting strands maximally disrupts the entire network of

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interconnections in the system, making the whole of our experience literally incomprehensible.3 3.3 The Revisability Paradox Full appreciation of the power of Quine’s empiricist account of logic and mathematics must compel admiration from even the staunchest rationalist. It is quite surprising that an account based on so extreme a form of empiricism can steer contemporary empiricism past the Scylla of Millian inductivism and the Charybdis of logical empiricism and come so close to capturing the special certainty of mathematics and logic. Nonetheless, Quine’s holistic conception of knowledge does not in the ªnal analysis enable contemporary empiricists to provide a satisfactory account of the special certainty of logical and mathematical truth, because the conception is inconsistent. Quine’s epistemology is an account of the way we adapt our system of beliefs to changing experience. It is an epistemology of reevaluation: A conºict with experience at the periphery occasions readjustments in the interior of the ªeld. Truth values have to be redistributed over some of the statements. Reevaluation of some statements entails reevaluation of others, because of their logical interconnections . . . . Having reevaluated one statement we must [sic] reevaluate some others, which may be logically connected with the ªrst or may be the statements of logical connection themselves. (1961c, 42) There are three principles that are constitutive of this epistemology. One tells us when we “must” reevaluate, one tells us where we can reevaluate, and one tells us how we should reevaluate. The when-principle is, in effect, the principle of noncontradiction. It mandates revision when there is inconsistency. The system as a whole must be restored to consistency within itself and with its observational periphery. The where-principle in Quine’s principle of universal revisability. It says that no statement of the system is “immune to revision.” As Quine (1961c, 43) puts it: 3. This is why it is child’s play for Quine to meet Grice and Strawson’s (1956) objection that the distinction between “not believing something and not understanding something” can be used to draw the analytic/synthetic distinction. Quine can agree with them that “it would be absurd to maintain that [the former distinction] does not exist,” but he can say that the latter distinction is nothing more than the distinction between the very central and the not very central.

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Revision even of the logical law of excluded middle has been proposed as a means of simplifying quantum mechanics; and what difference is there in principle between such a shift and the shift whereby Kepler superseded Ptolemy, or Einstein Newton, or Darwin Aristole? In permitting no exceptions to the rule that “the tribunal of sense experience” is the supreme court in which all knowledge is judged, this principle is the heart of Quine’s uncompromising empiricism. The how-principle is the principle of simplicity. It provides guidance about which statements are the best to revise in conºicts with experience. (See Quine (1960, 20–21) on conservatism.) On Quine’s epistemology, noncontradiction, universal revisability, and simplicity are different from other principles in our system of beliefs in this respect: they are constitutive of the epistemology of the system. The epistemology is a belief-revision epistemology and those principles comprise the basic mechanism of belief revision. They thus serve as essential premises in every argument for reevaluating a belief. Every such argument has to assume the principle of noncontradiction as a rationale for departing from an assignment of truth to logically conºicting statements. The principle is required to initiate the process of revising presently accepted statements, otherwise we have to tolerate a radically laissez-faire epistemology on which anything goes. Further, every argument has to assume that the class of revisable statements is the class of all statements of the system, otherwise the epistemology will no longer be the uncompromising empiricism it was intended to be. Finally, every argument has to assume simplicity or something like it to narrow down the class of potentially revisable statements. Here is the paradox of revisability. Since the constitutive principles are premises of every argument for belief revision, it is impossible for an argument for belief revision to revise any of them because revising any one of them saws off the limb on which the argument rests. Any argument for changing the truth value of one of the constitutive principles must have a conclusion that contradicts a premise of the argument, and hence must be an unsound argument for revising the constitutive principle. Consider a special case of the paradox. Given universal revisability, the principle of noncontradiction is revisable in principle. If it is revisable in principle, there is a possible belief-revision argument for its reevaluation. But, as we have seen, since the principle of noncontradiction is a constitutive principle, it must appear as a premise of the argument. But if it is right to revise a belief in the system, that belief was wrong all along, and if it was wrong all along, it cannot be a part

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of a sound argument. The argument for revising the belief would be unsound and provide no grounds for the revision. Hence, there can be no sound argument for revising the principle of noncontradiction, and it is not open to revision. Nonetheless, since all beliefs are revisable, the principle of noncontradiction must be revisable, and hence it is both revisable and not revisable. The paradox was set out on the basis of Quine’s own notion of reevaluation, namely, redistribution of truth values. But reevaluation might mean just dropping a principle from the web of belief. This option construes revision to consist in a statement’s changing from being marked true to disappearing from the system, rather than construing revision to consist in a statement’s changing from being marked true to being marked false. This option does not avoid the basic paradox. All that happens with this change is that the argument for revising the principle of noncontradiction goes from being unsound to being invalid. For, instead of the argument having a false premise, it now lacks a premise essential to drawing the conclusion that the principle of noncontradiction ought to be revised (i.e., dropped from the web of belief). The argument for revision is now invalid. Since all beliefs are revisable, noncontradiction must be, but since, again, noncontradiction is not revisable, it is, again, both revisable and not revisable. Looked at from the right angle, universal revisability already ºashes the signal Paradox! Paradox! Paradox! Unrestricted universality sanctions the dangerous move of self-application, which is a familiar feature of paradox. From the application of the belief-revision epistemology to itself, it follows that a revisable principle is unrevisable. Hence, just as the barber paradox proves that there is no actual barber who shaves all and only those who do not shave themselves, so the revisability paradox proves that there is no actual epistemology that says that everything including itself is revisable. It might make sense to contemplate replacing any and every plank in Neurath’s boat, but it makes no sense to contemplate replacing the basic principle of ship construction that says that there have to be planks between us and the water. The revisability paradox shows that no form of uncompromising empiricism, at least none that we are presently aware of, can meet the epistemic challenge to antirealism. The paradox also undercuts the Quinean explanation of how truths of mathematics and truths of logic can be taken to be about natural objects in the Quinean (1961c, 44) sense of being part of “a device for working a manageable structure into the ºux of experience.” Thus, both of the prima facie advantages of Quine’s uncompromising empiricism over Mill’s empiricism are lost.

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3.4 Wittgenstein’s Naturalism Wittgensteinian naturalism provides an alternative way of taking an implacable stand against the rationalist position on a priori knowledge, the only way left. The price is, of course, abandoning empiricism as a philosophical position, since, for Wittgenstein, empiricism is another philosophical theory, as much a “house of cards” as any other. Indeed, the alternative entails the radical reconceptualization of philosophy of Wittgenstein’s (1953) late philosophy, which rejects philosophical theories and the metaphysical issues that deªne them. Wittgenstein sought to eliminate what he took to be the linguistic confusions that underlie metaphysical issues. For Wittgenstein, traditional philosophy is not bad philosophizing as it was for Kant, nor bad science as it is for Quine, but simply nonsense resulting from metaphysical pictures that get us tangled up in the rules of language and prevent us from having a clear view of our linguistic practices. Good philosophy is linguistic therapy. It tries to give us a clear view of those practices by providing reminders of what our actual use of language is, accurate descriptions of particular linguistic techniques, and so on. Wittgenstein’s account of certainty seems to be the only option remaining for ontological and epistemological naturalists to account for the special certainty of logical and mathematical knowledge. Wittgenstein (1956, pt. 2, sec. 39) says: “To accept a proposition as unshakably certain—I want to say—means to use it as a grammatical rule: this removes uncertainty from it.” For Wittgenstein, characterizations of mathematical results as absolutely necessary are only a “somewhat hysterical way of putting things” (1956, pt. 5, sec. 46). The “must” that mathematicians and philosophers typically use to express mathematical or logical compulsion is no more than the expression of an attitude towards the technique of calculation . . . . The emphasis of the must corresponds only to the inexorableness of this attitude both to the technique of calculating and to a host of related techniques. The mathematical Must is only another expression of the fact that mathematics forms concepts. (1956, pt. 5, sec. 46) Thus, Wittgenstein (1969, sec. 47) says, “Forget transcendent certainty, which is connected with your concept of spirit.” Wittgenstein (1969, sec. 39) thinks that absence of doubt in the case of mathematical calculation is a feature of our linguistic techniques and practices: “This is how calculation is done, in such circumstances a calculation is treated as absolutely reliable, as certainly correct.”

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Furthermore, the possibility of doubt is itself built into those techniques and practices, but in such a way as to make some expressions of doubt intelligible and others not. As Wittgenstein (1969, sec. 154) puts it: “There are cases such that, if someone gives signs of doubt where we do not doubt, we cannot conªdently understand his signs as signs of doubt.” While Quine’s account of the certainty of mathematical and logical truth is unacceptable logically, Wittgenstein’s is unacceptable linguistically. He claims that an expression of philosophical doubt in connection with a truth like “3 = 2 + 1” or with a properly carried out, fully conclusive mathematical demonstration is literally unintelligible. Wittgenstein (1953, sec. 119) would say that the words of a philosopher who expresses a skepticism about the certainty of a piece of logic or mathematics are “plain nonsense”, “bumps that the understanding has got by running its head up against the limits of language.” (See also Wittgenstein (1969, secs. 446 and 370).) To be sure, we would be confused if someone were to express doubt about a simple arithmetic truth. If Hermione says, quite sincerely, “‘3 = 2 + 1’ might be false,” we would be at a loss to know what’s up with her. But it is just because the English sentence she used, “‘3 = 2 + 1’ might be false,” is meaningful that she is unintelligible to us. We are at a loss to know what she was up to in saying what she said. We know she knows the language and we also know she knows more than enough mathematics to know better than to doubt the addition. It is, therefore, because the sentence “‘3 = 2 + 1’ might be false” is meaningful in English, because the context seems to leave us no choice but to understand her utterance as a token of this sentence type and hence to have the meaning of the type, and because we know enough mathematics ourselves, that we can’t ªgure out what her motive might be for asserting it. Thus, Wittgenstein’s account runs counter to clear intuitions about what is and is not meaningful. Consider this scenario. We learn that Hermione has been professor of mathematics at the College de France for many years. Has she gone nuts? No, she acts perfectly sane and passes every psychological test for normality with ºying colors. What’s the answer? Finally, we are told: she’s read Descartes on what it means for God to be all-powerful, and that has shaken her faith in the certainty of arithmetic. Here, too, it is clear that it would be quite wrong to say that Hermione’s assertion is “plain nonsense.” Here Hermione’s utterance is even intelligible to her audience. If Wittgenstein were right, the passages expressing skepticism in Descartes’s Meditations, Hume’s Treatise, Goodman’s Fact, Fiction, and Forecast, Kripke’s (1982) Wittgenstein on Rules and Private Language, and

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even Wittgenstein’s (e.g., 1953, sec. 185) Philosophical Investigations would be literally unintelligible, plain nonsense, literally signs without sense. But, whatever complaints there have been about such works, none has been about the unintelligibility of the language in them expressing the skeptical arguments of those philosophers. When we want genuine examples of meaningless language, we have to turn to literary works like Lewis Carroll’s Jabberwocky or to coinages from linguists (e.g., Chomsky’s “Colorless green ideas sleep furiously”). There is real plain nonsense. The contrast between such cases and expositions of skepticism in and out of philosophical texts show that Wittgenstein was simply wrong about the linguistic distinctions between meaningfulness and meaninglessness, and hence wrong about certainty. As we saw in chapter 2 in connection with Burgess’s (1990, 7) point about mathematical sentences, the question of what to say about the meaningfulness and meaninglessness of sentences of natural language belongs to “the pertinent specialist professionals” in linguistics. Linguistic intuition seems quite clearly to say that the philosophical texts in question, whatever else their defects, are not meaningless. In philosophy, we are puzzled about how to answer the skeptical questions in those texts, but in linguistics, which abstracts away from philosophical content, there is not even a hint of meaninglessness about the sentences which express the questions. (In chapter 6, I shall present a similar argument against Carnap’s positivism.)4 3.5 The Necessity and Certainty of Formal Truths If the arguments in the previous sections are correct, antirealism faces an epistemological challenge that it has no resources to meet. I now want to show that realism, which of course faces the same challenge, has the resources to meet it. If realism can account for the special certainty of mathematical and logical truths, then, combining such an 4. Wittgenstein’s account of certainty rests on arguments in the sections of the Philosophical Investigations leading up to and including the paradox about rule following. Those arguments purport to show that there are no objective facts about meaning in the traditional sense of the term. But, as I (1990b) argued, the assumption of those arguments is that the basic facts for deciding questions about meaning in natural language are facts about use. This assumption is not defended, and is open to question because there is another conception of the basic facts about meaning in natural language. On this conception, the basic facts are facts about the sense properties and relations of expression and sentence types (e.g., that the word “bank” is ambiguous, the expressions “sister” and “female sibling” are synonymous, the words “blind” and “sighted” are antonymous, and the expression “free gift” is redundant). Facts about use are derivative. They derive from the speakers’ semantic knowledge of the senses of expression and sentence types,

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account with the results of the last chapter, we have a strong case for the epistemological superiority of realism over antirealism. Realism explains the special certainty of formal truths in terms of their necessity, and their necessity in terms of the abstractness of the objects they are about.5 Necessary formal truths are necessary because they describe unchangeable properties and relations of unchangeable objects. Given that abstract objects can have neither spatial nor temporal location, no property of their intrinsic structure and no relation they bear to one another solely in virtue of their intrinsic structure is subject to change.6 If no intrinsic aspect of an abstract object is subject to change, then no aspect of an abstract object can be otherwise than it is, their pragmatic knowledge about the use of tokens, and their knowledge of features of the context of use. Hence, the basic facts for deciding questions about meaning do not have to be, as Wittgenstein assumed, facts about use. I (1990b, 21–133) argued further that such a conception of the basic facts about meaning leads straightforwardly to a theory of meaning for sentence types that is not subject to Wittgenstein’s criticisms, and that the facts about use that Wittgenstein deploys against theories of meaning can be derived from facts about sentence types and the circumstances of their tokens. For example, the facts about family resemblance that Wittgenstein uses against the classical intensionalist claim that meanings are universals are shown to be derivable from the meaning of general terms and features of the circumstances of their application. To take another example, Wittgenstein’s rule-following paradox is shown not to arise because, on this theory of meaning, we can specify the semantic fact in virtue of which a speaker’s application of a word is correct or incorrect. (In the next chapter, I present a similar treatment of Kripke’s related rule-following argument.) 5. If Frege were right about the analyticity of mathematical and logical truths, another explanation of their necessity would be at hand. But the failure of Frege’s logicism blocks such an explanation. I would also argue that it is blocked by the failure of his attempt to provide a sufªciently comprehensive referential semantics. Frege’s semantics purports to bring the entire range of deªnitional truth under his broadened notion of analyticity. This enabled Frege to explicate analyticity in a way conducive not only to arguing for logicism, but also to arguing for the analyticity of logic. The basis of his broadened notion of analyticity was his deªnition of sense as “mode of referential determination,” which enabled him to understand containment in terms of principles of the theory of reference. I (1992) argued that Frege’s deªnition of sense makes reference the central concept in the theory of sense, thereby ineluctably blurring the boundaries between sense and reference and thereby bringing purely linguistic truths within the referentially speciªed class of analytic truths. To draw the proper boundaries, sense has to be deªned without reference to reference. It has to be deªned, not as the determiner of reference, but as the determiner of sense properties and relations, such as having a sense, having more than one sense, having the same sense, having opposite senses, having a redundant sense, and so forth. (See Katz 1992.) Once this is done, the boundaries of linguistic truth are drawn sufªciently narrowly so as properly to circumscribe the area of the necessity that is a matter of language. (For a systematic development of these ideas, see Katz [in preparation].) 6. Only extrinsic aspects, such as being thought about by someone or being instantiated in this or that concrete object, are subject to change. See the further discussion of the point in chapter 5.

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and connections that depend on only such aspects are necessary. Since the predication expressed by a formal truth holds in virtue of the fact that the property or relation that it attributes to the abstract objects it is about is one of their intrinsic (structural) properties and relations, there is no possibility that the objects do not have the property or relation. Hence, such a formal truth is a necessary truth.7 This account is uniform. Necessity is the same thing in connection with mathematical, logical, and linguistic truths. To be sure, linguistic, logical, and mathematical objects differ in ways that reºect differences among linguistic, logical, and mathematical structure. The necessary connection between being the number two and being the only even prime is an aspect of the abstract arithmetic structure of numbers. The necessary connection between being a consequence of true premises and being true is an aspect of the abstract logical structure of propositions. And the necessary connection between a sentence’s being analytic and its being nonsynonymous with every synthetic sentence is an aspect of the abstract semantic structure of senses. But, even though necessary connections can be aspects of different structures in different kinds of abstract objects, such differences do not affect the nature of the necessity. The unchangeableness of the connections within and among the structures of abstract objects turns on their common abstractness. In note 1 of chapter 1, I indicated that I would be using the term “formal truth” simply as an abbreviation for “mathematical, logical, or linguistic truth.” I will continue to use it this way outside the context of realistic rationalism, but when speaking about matters inside this position, “formal truth” will be used with the more descriptive meaning “proposition that is true of abstract objects in virtue of their form.” 3.6 Eliminativism and Supervenience The above account of the necessity and certainty of formal knowledge makes no effort to explain them in nonmodal terms. I have not tried to eliminate the modal idiom because I think there isn’t a prayer of doing it. Furthermore, I don’t think that modal properties and relations stand in need of an account in terms of nonmodal properties and relations to avoid circularity, as eliminativist naturalists about modality 7. A natural question about the above explanation is whether realism says that abstract objects have necessary existence. If some possible worlds do not contain some abstract objects, then the realist can’t say that formal truths about abstract objects are necessary truths, since formal truths about such abstract objects are not true in all possible worlds. The realism I have developed in this book takes abstract objects to have necessary existence. Chapter 5 explains why realists should take this position on abstract objects.

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suppose. Rather, their circularity criticisms reºect a misconception on their part of the nature of explanation in the area. The situation here is parallel to Quine’s (1961c, 27–32) eliminativist criticism of the theory of meaning. As I (1990b, 175–202) have argued, the requirement of a noncircular explanation to certify synonymy or any other concept in the theory of meaning is just a reductionist ªction. Quine promoted this requirement in the philosophy of language because he (1961a, 11) himself had so strong a prejudice against meanings as entities that he dismissed them without so much as a hearing. Quine (1961b, 48) urges us to: . . . resolve to treat “alike in meaning” in the spirit of a single word “synonymous,” thus not being tempted to seek meanings as intermediary entities. . . . [and to] treat the context “having meaning” in the spirit of a single word, “signiªcant,” and continue to turn our backs on the suppositious entities called meanings. The only reason Quine doesn’t advocate the same strategy with sentences—treating “is a well-formed sentence” in the spirit of a single word “grammatical” and turning our backs on sentences—is that he does not have it in for them. Somehow one kind of abstract object, a sentence type, is kosher, but another kind, a meaning, isn’t (see Quine 1987, 216–19). This intolerance led Quine (1961b) to embrace the similarly intolerant Bloomªeldian linguistics. Quine’s (1961c, 27–32) argument against synonymy in “Two Dogmas of Empiricism” is an application of Bloomªeldian operationalism. Quine himself makes this clear when he (1961b, 56–57) adopts Bloomªeldian operationalist substitution criteria as the proper approach to the problem of synonymy, and points out that his argument against synonymy in “Two Dogmas of Empiricism” is an application of this operationalism to semantics. Quine’s (1961b, 57) claim that the attempt to make objective sense of synonymy results in “something like a vicious circle” is an artifact of the operationalist demand that the attempt be carried out without the use of intensional concepts, and his overall argument against intensional concepts is a non sequitur because it leaves the possibility of theory construction as a way of explaining them open. The requirement of a noncircular explanation for modal properties and relations is similarly just a reductionist ªction, motivated by a prejudice against unreduced modal notions. It is certainly not the case that reduction is required for semantic or modal concepts to do their work. I (1972, 1987) have tried to show that explanations of sense properties and relations in terms of the structure of senses are semantically illuminating. Explanations of modal properties in terms of the

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structure of abstract objects are metaphysically illuminating. In connection with the explanation of the necessity of mathematical and logical truth, illumination comes from their necessity being grounded ontologically, rather than simply being left as a consequence of the deªnition of necessary truth as truth in all possible worlds. Our realist account explains why mathematical and logical truths are true in all possible worlds on the basis of the ontological status of the objects that such truths are about and the nature of the mathematical or logical structure of those objects. These considerations bear on Kim’s (1993) attempt to develop a naturalistically based normative epistemology. Kim eschews a reduction of the normative to the natural, but he requires that the normative supervene on the natural in a way that makes it possible to provide criteria for justiªed belief in terms of subvenient naturalistic phenomena. This requirement has teeth because Kim (1993, 235–36) makes the signiªcance of an evaluative or normative concept depend on its being governed by criteria expressible naturalistically. Kim (1993, 216–19) adopts the quasi-eliminativist requirement that the criteria of justiªed belief must be formulated on the basis of descriptive or naturalistic terms alone, without the use of any evaluative or normative ones, whether epistemic or of another kind. (Italics in original) I applaud Kim’s rejection of reductionism, but I can’t go along with this requirement. He (1993, 218) says that the difªculty with the use of epistemic terms is just that they “are themselves essentially normative.” I ªnd this explanation unhelpful because Kim’s requirement is already written to preclude “the use of any evaluative or normative [terms],” presumably on grounds of circularity. But there is no argument to show that there is vicious circularity, and hence to preclude an explanation of justiªcation in normative terms. Why isn’t it legitimate to use normative concepts to explain the concept of justiªcation, just as it is legitimate to use the concept of sense to explain the concepts of analyticity and synonymy, to use logical and mathematical concepts to explain other logical and mathematical concepts, or to use the ontological concept of an abstract object to explain a modal concept? How would Kim handle justiªcation in mathematics and the other formal sciences? Global supervenience obtains trivially: since mathematical and logical truths hold in all possible worlds, worlds that coincide with respect to truths involving natural properties and relations coincide with respect to truths involving mathematical and logical properties and relations. But the supervenience relation required for

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Kim’s naturalistically based epistemology must provide criteria for particular justiªed mathematical and logical beliefs in terms of particular subvenient naturalistic phenomena. This requires a local supervenience relation between particular mathematical and logical truths and particular natural truths, but this supervenience relation, I submit, cannot be generally supposed to obtain in the case of theorems about numbers, sets, spaces, and even sentences. There is nothing in the natural world that can serve as an appropriate supervenience base for the explanatory task in the case of proofs of theorems about such entities. Moreover, if realism is right, there is no alternative to normative concepts’ serving as the basis for other normative concepts. If, as I argued in the last chapter, we can have knowledge of abstract objects a priori and the formal sciences are essentially different from the natural sciences, and if, as I will argue in the next chapter, abstract objects in the domain of a formal science are the norms governing correctness in the science, then there is no need for a requirement of supervenience on natural aspects of reality. Normative judgments in the formal sciences (or, as I shall argue in the ªnal chapter, in their philosophical foundations) are true just in case they reºect the structure of the relevant abstract objects. For example, since the norms for sentencehood in English are the sentence types of English, the claim that an utterance or inscription is well-formed in English is true just in case it is a token of a sentence type of English. Kim (1993, 236) claims that rejecting the supervenience of the normative on the natural “would sever the essential connection between value and fact on which . . . the whole point of our valuational activities depend”. This claim has no force within a realist framework where valuational activities derive their normativity from abstract aspects of reality.8 3.7 Conclusion Naturalists do not have a satisfactory account of the certainty of formal truths, and hence they fail to meet the epistemological challenge to antirealism. I do not pretend to have ruled out all possibility of antirealists’ coming up with another plausible naturalistic explanation of the special certainty of logical and mathematical truths to replace Quine’s (or Wittgenstein’s). But, whatever hopes for the future natural8. Though one cannot but have some sympathy for Moore’s (1942, 588) position, it has never been clear to me what natural fact the good might supervene on. Moore does not satisfactorily explain moral uses of “good” as in “good person” and “good human being.” Kim (1993, 235) does not come to his aid.

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ists may entertain, at this point, they have no explanation of the special certainty of such truths and no prospects for obtaining one. If this conclusion is taken together with the rationalist epistemology set out in the preceding chapter, the true picture of the realist/antirealist controversy emerges. In contrast to the picture that antirealists have been promoting over the years, the true picture is that it is antirealism, rather than realism, that faces an epistemological challenge it has not met and, as far as one can see, lacks the resources to meet.

Chapter 4 The Semantic Challenge to Realism

4.1 Introduction The semantic challenge to realism comes from another of Benacerraf’s papers, “What Numbers Could Not Be.”1 In that paper, he ([1965] 1983) argues that there is no principled way of deciding which of the settheoretic models of Peano arithmetic is the numbers, that there is no principled way of deciding what system of objects is the numbers, and hence that we cannot make sense of the idea that numbers are determinate objects. These conclusions are intended to motivate his ([1965] 1983, 291) structuralist view that Arithmetic is . . . the science that elaborates the abstract structure that all progressions have in common merely in virtue of being progressions. It is not a science concerned with particular objects—the numbers. Concluding with a remark that is reminiscent of Quine’s (1960, 76) claim that there are no “language neutral meanings,” Benacerraf ([1965] 1983, 294) says, “if the truth be known, there are no such things as numbers.”2 1. “What Numbers Could Not Be” does not represent Benacerraf’s (1996) present thinking on the issues in question, but I shall focus on it because it has become the classic statement of skepticism about the determinacy of reference to the numbers, and, as such, it long ago took on a philosophical life of its own. Moreover, from a realist perspective, Benacerraf hasn’t changed his basic view all that much. In this connection, I will consider some of the things he (1996) now says about the issues. 2. In Katz (1996b) I expressed unease about taking Benacerraf’s statement at face value because of his ([1965] 1983, 294) ªnal remark, “which is not to say that there are not at least two prime numbers between 15 and 20.” Benacerraf (1996, 51 n. 10) says that this remark was meant to indicate that his statement “should not be taken as contradicting established mathematical results.” Recognizing that this cannot be an absolute requirement, he gives a higher priority to hermeneutics than to revisionism. Then he (1996, 51–52) says that “the reduction [in Benacerraf ([1965] 1983)] does not answer the fundamental question regarding what makes the true [mathematical statements] true,” adding “On the assessment of WNCNB, they wouldn’t be true.” This explanation removes the

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There are two responses to Benacerraf’s argument that should be dismissed at the outset. One is that, although it is a challenge to the mathematical realist’s claim that numbers are determinate objects, it is not, strictly speaking, a challenge to general realism, which could be true even if mathematical realism were false. Nothing should be made of this. On the one hand, mathematics is so much the central case in the philosophical controversy over the existence of abstract objects that indeterminacy of reference to numbers will translate immediately into doubts about the tenability of general realism. On the other hand, Benacerraf’s argument, as we shall see, is close enough to indeterminacy arguments against logical realism and linguistic realism that, were it to work in the case of mathematical realism, we could expect that something quite similar would work against them. The other response is that Benacerraf’s structuralism seems to fall prey to his own empiricist scruples about knowledge of abstract objects. Since the structure that “all progressions have in common merely in virtue of being progressions” is, on his own admission, an “abstract structure,” there must also be a problem about how spatiotemporal creatures like ourselves can have knowledge of that structure. Here, too, I think we should make nothing of the point. The problem that Benacerraf is raising about the determinacy of our reference to numbers is independent of the question of whether his structuralism can be reconciled with his position on mathematical knowledge in “Mathematical Truth.” Benacerraf’s troubles would be cold comfort if we were forced to concede that reference to numbers is indeterminate. Contrary to general wisdom, I think Benacerraf-style arguments for the indeterminacy of reference to numbers and other mathematical objects are irredeemably ºawed. I think that the ºaw has escaped notice for two related reasons. One is that philosophers have not recognized that those arguments are special cases of indeterminacy arguments generally; as a consequence, indeterminacy arguments in the philosophy of mathematics have not been viewed from an abstract enough perspective to make the ºaw in them apparent. The other reason is that unease, but replaces it with confusion about Benacerraf’s priorities. One would have thought that such an assessment contradicts established mathematical results and hence represents at least a signiªcant downgrading of hermeneutics. The deeper issue between us is whether hermeneutics can be separated from ontological and epistemological questions in the foundations of the formal sciences. The view in chapter 1 is that hermeneutics is a feature of controversies like that among realism, conceptualism, and nominalism, and that between empiricism and rationalism. Hermeneutics serves as a criterion for assessing such positions. Roughly, the best philosophical position, other things being equal, is the one that preserves the broadest range of established mathematical results and philosophical intuitions.

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the ºaw in indeterminacy arguments in the philosophy of language, the paradigm cases of indeterminacy arguments, has itself not been widely recognized. I will try to show that, once both these conditions are corrected, the ºaw in Benacerraf-style indeterminacy arguments in the philosophy of mathematics becomes evident, and also that linking indeterminacy arguments in the philosophy of mathematics and the philosophy of language provides a deeper understanding of the nature of such arguments and of the forms of skepticism based on them. My approach will be as follows. First, I will present analyses of the two principal indeterminacy arguments in the philosophy of language that reveal their ºaw. On the basis of those analyses, I will develop a conception of the structure of indeterminacy arguments in general. Next, I will explain how knowledge of the ºaw in question can be used to develop a general strategy for resisting indeterminacy arguments. Finally, I will show that Benacerraf-style arguments fall under our general conception of indeterminacy arguments, and then show how our strategy can be applied to block such arguments. 4.2 Indeterminacy Arguments in the Philosophy of Language Indeterminacy arguments are skeptical arguments. They claim that we lack the means to distinguish among the things we have to distinguish among in order to legitimize our belief that our talk about certain objects is talk about determinate objects. One characteristic of such skeptical arguments is that they are based on an allegedly unbreakable symmetry between the intended interpretation of such talk and certain deviant interpretations. The skeptic challenges us to break the symmetry. Indeterminacy arguments differ from one another with respect to the kind of knowledge in question, e.g., about meanings, numbers, and so on, and also in the considerations they offer for the allegedly unbreakable symmetry. Nonetheless, they form a distinct class of skeptical arguments exhibiting a common pattern of philosophical reasoning. The class of indeterminacy arguments includes several celebrated arguments about meaning and reference in the philosophy of language. The most celebrated of them is, of course, Quine’s (1960) argument for the indeterminacy of meaning and translation (and the derivative argument he [1969b] gives for the inscrutability of reference). Quine’s indeterminacy argument has served as a model for philosophers in constructing other indeterminacy arguments. I will treat it as the paradigm of the class of indeterminacy arguments. Wittgenstein’s (1953) well-known argument about rule following is sometimes taken as an indeterminacy argument, but, strictly speaking,

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it isn’t one. Wittgenstein is not a philosophical skeptic putting forth an argument designed to show that we don’t know what we are talking about. Given his metaphilosophical views, Wittgenstein would be the last philosopher to be in the business of creating new philosophical problems or trying to prove that no sense can be made of our ordinary talk. As Wittgenstein (1953, sec. 133) wrote, “. . . the clarity that we are aiming at is indeed complete clarity. But this simply means that the philosophical problems should completely disappear.” Wittgenstein’s rule-following argument is rather an attack on certain philosophical theories that he thinks confuse us about the use of language, in particular those according to which private mental states have the normative force to ªx meaning. Wittgenstein’s (1953, sec. 201) famous paradox is the culmination of an argument—beginning, in effect, at sec. 139—designed to refute such philosophical theories and to locate the normative force to ªx meaning in the rules of our public linguistic practice. For these reasons, Wittgenstein’s rule-following argument will not be one of my examples of indeterminacy arguments in the philosophy of language, though I will consider it at the very end of the chapter in connection with Putnam’s Skolemite skepticism. In contrast, Kripke’s (1982) rule-following argument is put forth as a general skeptical argument. His paradox is intended as a philosophical problem for everyone. Moreover, since Kripke’s argument is a highly inºuential indeterminacy argument and also since I wish to call attention to a feature of it in which it contrasts with Quine’s indeterminacy argument, I will discuss Kripke’s argument too. 4.2.1 Quine’s Argument Quine’s (1960, 27) indeterminacy thesis comes from reºection on radical translation. Quine (1960, 78–79) says that the discontinuity of radical translation tries our meanings: really sets them over against their verbal embodiments, or, more typically, ªnds nothing there. He (1960, 53–54) explains: We could equate a native expression [“gavagai”] with any of the disparate English terms “rabbit,” “rabbit stage,” “undetached rabbit part,” etc., and still, by compensatorily juggling the translation of numerical identity and associated particles, preserve conformity to stimulus meanings of occasion sentences. The consequence is indeterminacy of translation. Given two analytical hypotheses about the translation of “gavagai,” such as “rabbit” and “rabbit stage,” Quine (1960, 72) claims:

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Both could doubtless be accommodated by compensatory variations in analytical hypotheses concerning other locutions, so as to conform equally to all independently discoverable translations of whole sentences and indeed all speech dispositions of all speakers concerned. And yet countless native sentences admitting of no independent check . . . may be expected to receive radically unlike and incompatible English renderings under the two systems. Recently, Quine (1990) has again stressed that his indeterminacy doubts “come from reºecting on radical translation.” This point is also stressed by prominent Quine scholars, e.g., Gibson (1994). The problem with Quine’s argument is not that there is some difªculty in getting indeterminacy from radical translation. The problem is that it is far too easy. It is as easy getting the cards one has stacked the deck to get. In the game of radical translation, the deck containing the semantic evidence from which the linguist can draw in trying to choose among the available translations contains only extensional cards. Since the intensional cards have been left out entirely, it is hardly surprising that Quine’s ªeld linguist can ªnd no semantic evidence to distinguish between extensionally equivalent translations like those pairing “gavagai” with “rabbit,” “rabbit stage,” and “undetached rabbit part.” Quine (1960, 28–29) presents radical translation as “translation of a language of a hitherto untouched people,” that is, as a situation that is nothing more than actual translation purged of “hints” from previous translations and shared culture. Quine (1960, 28, n. 2) even suggests that such situations are sometimes encountered by real linguists who have developed some “techniques” for working in them. Thus, radical translation is presented to readers of Word and Object as if it were an unobjectionable idealization of actual translation, one that merely abstracts away from biasing complications like “historical or cultural connections” between languages. We are encouraged to think that the deck is a new one, containing no cards with nicked edges or bend marks that might serve to mark them. But, in addition to the idealization’s being set up to exclude genuinely biasing factors, it is set up to exclude all evidence about sense properties and relations of expressions. Such evidence seems prima facie precisely what is required to break the symmetry between referentially equivalent translations. It is, after all, a fact of some sort that speakers of a natural language make judgments about the meaningfulness, ambiguity, synonymy, and other sense properties and relations of expressions in their language. Mightn’t a bilingual speaker tell us that “gavagai” is synonymous with “rabbit” but not with “rabbit stage” and

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“undetached rabbit part”? Since the question at issue is whether the intension of a sentence can in principle be objectively determined for actual translation, in excluding the most promising evidence for objectively determining intensions from the radical translation situation, Quine has set things up so that translation cannot help but come out indeterminate. I am not suggesting that there was any deliberate deck stacking. What I think is that Quine constructed the radical translation situation to reºect the situation that Carnap (1956b) uses in describing his method for determining intensions in “Meaning and Synonymy in Natural Languages.” (Quine almost always had Carnap in mind when thinking about semantics.) Carnap (1956b, 236–40) presents a method for choosing between translations that he thought established the superiority of intensionalism over extensionalism. Carnap imagines a situation in which a ªeld linguist asks his German-speaking informant Karl questions designed to provide evidence that will choose between “horse” and “horse or unicorn” as translation of the German term “Pferd.” Carnap thinks that the linguist can only obtain the necessary evidence by switching from queries about Karl’s application of “Pferd” to actual animals to queries about his application of the term to possible ones. Carnap thinks that Karl’s responses to queries in German like whether he would apply “Pferd” to something that looks like a horse but has a horn in the middle of its forehead will settle the issue of translation. Now, Carnap (1956b, 239) thinks of the intension of a predicate extensionally, as the objects (in possible worlds) to which the predicate applies, and, consequently, the situation of the linguist that Carnap assumes in his method involves no evidence about sense properties and relations. As a reconstruction of this Carnapian situation, Quine’s radical translation situation assumes that the linguist’s method involves no such evidence. Quine’s insight was to see that Carnap’s method does not work when the extensional evidence cannot discriminate between the competing translations if their application is the same in all possible cases. Because “rabbit,” “rabbit stage,” and “undetached rabbit part” are such competing translations, they refute Carnap’s method. Quine’s error was to present what is essentially Carnap’s translation situation as the general situation of translation. As a general argument against the determinacy of translation, Quine’s argument fails because it rules out evidence about sense properties and relations by ªat. It begs the question against the intensionalist who does not understand intension extensionally as Carnap does and accordingly takes a different view of intensional evidence. (See Katz 1990a, 1994b.)

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If actual translation is not seen through the lens of radical translation, then at least initially it is an open question, on the one hand, whether there are “linguistically neutral meanings,” and, on the other hand, whether linguists can obtain evidence about them that, together with standard scientiªc methodology, will determine a best choice among competing translations. Since there is no a priori reason not to accord the judgments speakers make about the meaningfulness, ambiguity, synonymy, and other sense properties and relations of expressions the same beneªt of the doubt that is accorded to prima facie evidence in other infant sciences, the only non-question-begging way to decide the issue of whether the intensions of sentences can be objectively determined is to try. Collect evidence about sense properties and relations, try to construct a theory that systematizes them, and see what comes of it. The hypothesis of determinate meanings must end up like the phlogiston hypothesis, but it also might end up like the gene hypothesis. Here’s how a non-question-begging attempt might go. Let the ªeld linguist investigate not only the referential properties and relations of expressions of the language under study but also their sense properties and relations. Let the ªeld linguist collect the judgments informants would make about the sense properties of “gavagai” and the relations its sense bears to the senses of other expressions. Such expressions are not restricted to the vocabulary of the informant’s language, but, with bilingual informants, include expressions like “gavagai” in the informant’s language and expressions like “rabbit,” “rabbit stage,” and “undetached rabbit part” in the language into which the ªeld linguist intends to translate the former expressions. Now, our method, as opposed to Carnap’s, is for the ªeld linguists to elicit a wide range of judgments about the sense properties and relations of the expression to be translated and its possible translations and use these judgments as evidence on which to decide which of the translations is best. Translation is sameness of sense (synonymy) between expressions of different languages, and hence the best translation is the one with the same sense properties and relations. To elicit judgments, the linguist can ask an informant questions about the sense of “gavagai,” starting with the direct question “With which, if any, of the English expressions ‘rabbit,’ ‘rabbit stage,’ or ‘undetached rabbit part’ is the term ‘gavagai’ synonymous?” A choice of one of these expressions is certainly evidence for the translation of “gavagai” as that expression. If, for some reason, the informant is unable to choose, there are other informants and less direct questions. For example, the linguist can ask “Is ‘gavagai’ closer in sense to ‘infancy,’ ‘adolescence,’ and ‘adulthood,’ or to ‘infant,’ ‘adolescent,’ and ‘adult’?” or “Does the

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sense of ‘gavagai’ bear the same part-whole relation to the sense of some word in the informant’s language that the sense of ‘ªnger’ bears to the sense of ‘hand’ in English?” If some informants can make some such judgments, there is evidence relevant to distinguishing among translations like “rabbit,” “rabbit stage,” and “undetached rabbit part.” Once such evidence is acknowledged, we have no reason to doubt that, in combination with methodological principles such as simplicity, such evidence can in principle enable us to choose a best translation. As noted, our method involves the use of a bilingual informant. How things could be otherwise when the method is supposed to deal with translation is hard to imagine. Quine is on record as objecting to bilinguals. He (1960, 74) thinks that, once we introduce bilingual informants with a common linguistic competence, we are already assuming determinate translation: Quine raises the possibility that different bilinguals have different linguistic competences, and so correlate different expressions with “gavagai”. This is, of course, a real possibility. Quine’s objection would go through if we assumed that there actually are bilingual informants with a common linguistic competence, since that would mean that translation is determinate. But our method only assumes the possibility of such bilingual informants. Since that does not imply that translation is determinate, Quine’s objection does not go through. The possibility of a homogeneous community of bilingual informants is on a par with the possibility of a homogeneous community of monolingual speakers, which is something customarily assumed when linguists investigate the grammar of a single language. Thus, to entertain the possibility of such a community of bilingual informants is only to suppose it in the spirit in which an investigation in any science supposes that there are laws to be found. Such suppositions get investigations going and are tested in the course of them. The supposition of laws of translation, like the supposition of any laws, will be conªrmed or disconªrmed depending on whether the investigation succeeds or fails in providing us with suitable conªrmed statements of them. Since the issue is whether there are intensions of sentences and whether they can be objectively determined, no question is begged in assuming natural languages have intensional structure for the sake of determining whether or not the assumption is true. Contrast this hypothetical acceptance of an intensionalist framework, which begs no question against the extensionalist, with Quine’s insistence that radical translation is the right model for actual translation, which, because it requires categorical acceptance of an extensionalist framework, begs the question against the intensionalist.

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If reºection on radical translation were all there is to Quine’s argument against intensionalism, we could dismiss the argument as ruling out intensionalism arbitrarily. But Quine had another argument in his (1961c) famous paper “Two Dogmas of Empiricism.” If this attack on the analytic/synthetic distinction had been successful, it would have ruled out intensionalism in an entirely principled way. As will be recalled, Quine argued that we cannot make objective sense of synonymy and analyticity in any of the ways we have for explaining legitimate logico-linguistic concepts. He then argued, quite convincingly, that lexical deªnition, explication, abbreviation, substitution procedures, and meaning postulates are inadequate to make objective sense of the analytic/synthetic distinction. This he took to be enough (1961c, 37) to conclude: “That there is such a distinction to be drawn at all is an unempirical dogma of empiricists, a metaphysical article of faith.” As I (1988; 1990b, 175–202) explained elsewhere, the argument overlooks what is, in fact, the most promising type of explanation for notions like synonymy and analyticity. This type, which is characteristic of mathematics and logic—and which subsequently became the standard type of explanation in generative linguistics—is to deªne a family of concepts within an axiomatic theory of the structure of a system of objects.3 I will not repeat the details of this criticism here. The essential point is that, in taking substitution procedures to be the proper approach to explaining concepts in linguistics, Quine was buying into the pre-Chomskyan taxonomic conception of the science. Quine makes it quite clear not only that he (1961b, 56–57) is assuming that the explanation of concepts in linguistics, from phonology to semantics, is based on “substitution criteria, or conditions of interchangeability” but also that this assumption underlies his argument that attempts to explain synonymy via substitution procedures “involv[e] something like a vicious circle.” Since this means that “Two Dogmas of Empiricism” contains no criticism of an approach that deªnes intensions within an axiomatic theory, Quine’s exclusion of an intensionalist approach to translation is arbitrary. One further point. Katz (1990b, 197–98) and Gemes (1991) have noted a paradoxical aspect to the symmetry between competing translations in the radical translation situation. “Rabbit,” “rabbit stage,” and “undetached rabbit part” are supposed to be semantically equivalent translations of “gavagai”—as Quine (1960, 27) says, they are equally 3. As I (1990b, 184–93) argued elsewhere, and apparently as Quine (1990, 199; see also Clark 1993) now agrees, the question of whether there can be a theory of meaning is entirely a matter for linguistic research to decide.

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“compatible with the totality of speech dispositions.” And yet they are also supposed to be “radically unlike and incompatible.” How can they be equivalent and also incompatible? Since the question of indeterminacy turns on whether there is a general relation of synonymy—in the terminology of “Two Dogmas of Empiricism,” a relation of sameness of meaning that holds for variables S and L—determinacy is a condition for synonymy within a language as much as it is a condition for translation between languages. Thus, the earlier question arises about a single language: how can there be competing hypotheses about the synonymy of expressions of the language that are “compatible with the totality of speech dispositions” of its speakers and yet the question of the synonymy of the expressions be indeterminate because they are “radically unlike and incompatible”? The incompatibility seems to establish the existence of dispositions to judge the expressions as different in meaning, as indeed the incompatibility of “rabbit,” “rabbit stage,” and “undetached rabbit part” does, but then the hypotheses cannot be “compatible with the totality of speech dispositions.” The paradox seems to arise because senses are smuggled into the radical translation situation to keep the linguist’s choice from degenerating into a choice among synonyms. As we shall see below, this paradoxical element is, in one form or another, found in indeterminacy arguments generally and will be important for our response to Benacerraf’s indeterminacy argument. 4.2.2 Kripke’s Argument Kripke (1982) asks what fact about my ªnite mind grounds the claim that I mean 125 in literal uses of “sixty-eight plus ªfty-seven.” The fact must somehow justify my thinking that the referent of that expression is the third member of the triple in the inªnite set of triples of numbers that is the plus function. It must not equally justify my thinking that the referent is the third member of the corresponding triple in an inªnite set of triples such as Kripke’s (1982, 8) quus function, i.e., the triple . As Kripke puts it (1982, 54), “the idea in my mind is a ªnite object” so why “can it not be interpreted as the quus function, rather than the plus function?” The ªnitude of our ideas makes successful reference depend on our justifying a projection from ªnitely many known addition triples to the inªnite plus function. But such a projection situation faces us with the Humean problem that a ªnite number of observed cases is compatible with any prediction about unobserved cases. Since an idea in my mind can only represent a ªnite set of triples, and since the ªnite set it represents is a proper subset of the inªnite quus function as well as a

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proper subset of the inªnite plus function, the claim that “sixty-eight plus ªfty-seven” means 125 can be no better supported than a claim that it means 5. Kripke’s doubt about whether the available evidence determines a unique referent for our uses of “sixty-eight plus ªfty-seven” is parallel to Quine’s doubt about whether the available evidence determines a unique translation for the native’s term “gavagai.” Furthermore, their reason for skepticism is the same. Quine and Kripke both think that the evidence is inadequate to justify the intended interpretation because it supports the intended and deviant interpretations equally well. Last, but by no means least, the evidence supports them equally well because it is conªned to extensional facts about expressions, and, in Kripke’s case as in Quine’s, the restriction to extensional evidence is a product of the way in which the semantic situation has been constructed. Just as intensional evidence is unavailable in choosing a translation for “gavagai” because Quine constructed his radical translation situation to exclude facts about sense properties and relations, so intensional evidence is unavailable for ªxing the reference of “sixtyeight plus ªfty-seven” because Kripke constructed his rule-following situation to exclude facts about them. In the former situation, there is no appeal to information about the senses of “rabbit,” “rabbit stage,” and so on, and, in the latter, there is no appeal to information about the senses of “plus,” “quus,” and so on. If, like Quine’s, Kripke’s puzzle arises because senses have been painted out of the semantic picture, the solution to Kripke’s puzzle, like the solution to Quine’s, ought to emerge once we paint them in again. Moreover, unlike Quine’s overall argument, which contains the criticism that we cannot make objective sense of senses, Kripke’s overall argument does not contain any criticism of senses. Consequently, the resolution of Kripke’s puzzle ought to consist in introducing the senses of “plus,” “quus,” and so on, and describing their role in the reference of mathematical expressions. Senses are the linguistic objects that speakers have knowledge of in virtue of their semantic competence. On linguistic realism, a sense is an abstract object, but it is nonetheless a ªnite object because it is compositionally constructed from the ªnite senses of the ªnitely many lexical items in a sentence. Kripke is certainly right that our minds, and hence our ideas, must be ªnite things, but when senses are in the picture, we not only have the ªnite idea of the extension of the numerical expression “sixty-eight plus ªfty-seven,” that is, the number one hundred twenty-ªve, we also have the ªnite idea of the intension of the expression, that is, the sense of “sixty-eight plus ªfty-seven.” More generally, as competent speakers of English, we know the sense of

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numerical expressions and the sense of “m plus n,” which we may suppose is given by (P). (P) If n is 0, the sum is m, and if n is greater than 0, the sum is the number reached from m by the process of taking its successor, then the successor of the successor, and so on n times. Knowledge of this sense, knowledge of the senses of the numerical terms which can replace “m” and “n,” and knowledge of compositionality constitute knowledge of the compositional sense of expressions like “sixty-eight plus ªfty-seven.” This knowledge too is ªnite, just as are the senses themselves, since both our knowledge of senses and the senses of which it is knowledge are the product of ªnitely many combinations of ªnitely many elementary units—mental entities in the former case, abstract entities in the latter. Recall that Kripke’s puzzle is a Hume-style problem of projecting from a ªnite number of observed cases to an inªnite number of unobserved cases. With the recognition of senses and our knowledge of them, the puzzle disappears because we are no longer faced with a problem of projecting from the ªnite to the inªnite. On the one hand, both our knowledge of the sense of “plus” and the sense itself are ªnite objects, and, on the other hand, neither of them serves as a basis for projection. We do not refer to the third member of the triple in using “sixty-eight plus ªfty-seven” because we cleverly employ what we recall from our acquaintance with addition triples to rule out projections to the quus function. Rather, we refer to 125 because, in virtue of our grammatical knowledge as English speakers, we grasp the ªnite sense of “sixty-eight plus ªfty-seven,” because we use that expression with the intention to refer to the referent of its sense, and because, in virtue of falling under the sense of “sixty-eight plus ªftyseven,” the number 125 is the referent of the expression.4 Hence, the fact about my ªnite mind that grounds my meaning 125 in literal uses of “sixty-eight plus ªfty-seven” is the ªnite fact that my communicative intention reºects my grammatical knowledge of the compositional sense formed from the sense given by (P) and the senses of “sixty-eight” and “ªfty-seven.”5 Kripke (1982, 54) seems to have a reply prepared in advance: 4. On my new intensionalism, it is not generally in the nature of senses to do this, but it doesn’t matter here whether 125 is the referent of “sixty-eight plus ªfty-seven” because, as Fregeans would say, it is the nature of the sense of that expression to determine 125 as its referent or because, as I would say, the relevant contextual factors are in place and 125 falls under the sense of that expression. 5. I anticipate the question how we can know that the expressions “and so on” and “iterate” in (P) have a sense with the concept “sameness” rather than a sense with the concept “schmameness” so that “plus” expresses the notion of doing the same up to the

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But ultimately the sceptical problem cannot be evaded, and it arises precisely in the question how the existence in my mind of any mental entity or idea can constitute “grasping” any particular sense rather than another. The idea in my mind is a ªnite object: can it not be interpreted as determining a quus function, rather than a plus function? This reply is not to our solution, but to some other (psychologistic) one. The pronoun “it” in Kripke’s question refers to an idea in my mind, and hence in asking the question, Kripke is only pointing out that this idea can be interpreted under a projection to the quus function. Once senses are in the picture, it is clear that trying to apply Kripke’s argument in this passage to our view would involve an illegitimate shift from talking about (grasping) a sense to talking about “[t]he idea in my mind.” If the pronoun in Kripke’s question were to keep its reference to an objective sense, the answer to Kripke’s question would be that there is no interpreting of senses: senses simply determine one function to the exclusion of others.6 Let us review the main aspects of our solution to Kripke’s puzzle about rule following. There are only two points at which it makes any sense to claim that the puzzle arises. One is the relation between my ideas and the sense of “plus” that they enable me to grasp. The other is the relation between the sense of “plus” and its referent, the plus function. To repeat our explanation of why the problem cannot arise at the former point, Kripke’s problem is a Hume-style problem of what justiªes projecting from an observed ªnite number of addition triples to the inªnite plus function, and hence, when what is grasped is the sense of “plus,” what is grasped is ªnite too, so that there is no longer a problem of projecting from ªnite sample to inªnite population.7 n-kth case and doing something else thereafter (e.g., taking the square of the successor). This question is a repeat of the Quinean question discussed above. We thus explain how we can know this in the same way that we explained how we can know that the sense of “rabbit” involves the concept “object” rather than the concept “temporal slice of an object” or the concept “undetached part of an object.” We can verify that we are grasping the senses we think we are grasping in the case of the word “plus” by comparing its sense properties and relations with those of the expressions “and schmo on” and “schmiterate” in the same manner in which we compared the sense properties and relations of “rabbit” with those of “rabbit stage” and “undetached rabbit part” to determine their nonsynonymy. See my (1990b, 166–67) discussion of such veriªcation in connection with one of Kripke’s examples. 6. It is thus mistaken to think that the problem must resurface as a problem about grasping senses; see also Boghossian (1994) and my (1994b and 1997, 10–11). 7. It is, moreover, hard to see how there is any other relevant problem about how our ªnite ideas enable us to grasp ªnite senses. No problem can come from doubts about the existence of ideas or senses, since neither behaviorism nor skepticism about intensions

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The other point at which it might make sense to claim that Kripke’s problem arises is in the relation between the sense of “plus” and the plus function that is its referent. Here there is a relation between something ªnite and something inªnite, but even so, Kripke’s problem cannot arise because there is no projection once senses enter the picture. The relation of the sense of “sixty-eight plus ªfty-seven” to its referent the number 125 is simply that the plus function exempliªes the sense of “plus”. As Kripke (1982, 11) puts his problem, it is one the solution of which must be “an account of what fact it is (about my mental state) that constitutes my meaning plus.” That problem cannot arise, since determination of the inªnite plus function is out of our hands—or, perhaps we should say, outside our heads. Determination of the inªnite plus function is an abstract semantic relation between two abstract objects. Kripke (1982, 54) himself says that there is no problem in connection with the inªnite sequence falling under a ªnite concept: The “sense” in turn determines the addition function as the referent of the “+” sign. There is no special problem . . . as to the relation between the sense and the referent it determines. Kripke (1982, 54) explicitly allows that the plus function, not the quus function, falls under the sense of “plus” in saying that “[i]t simply is in the nature of a sense to determine a referent.” Thus, Kripke can hardly object to the claim that we succeed in determinately referring to 125 in our literal uses of “sixty-eight plus ªfty-seven” because 125, being the sum of sixty-eight and ªfty-seven, is the referent that the sense of “sixty-eight plus ªfty-seven” determines. Finally, like Quine’s radical translation situation, Kripke’s rulefollowing situation contains a paradoxical element. In Quine’s radical translation situation, the expressions “rabbit,” “rabbit stage,” and “undetached rabbit part” were supposed to be equivalent but “incompatible English renderings” of “gavagai.” In Kripke’s rule-following situation, the reference of “sixty-eight plus ªfty-seven” is supposed to be indeterminate between the semantically indistinguishable inªnite enters the picture. Also, it is an uncontroversial fact that ºuent speakers grasp senses; for example, they grasp senses when they recognize an ambiguity. Further, they can be presumed to grasp senses on the basis of some mental representation of the grammatical structures that relate sentences to senses. There are, to be sure, psychological problems about syntactic and semantic competence and philosophical problems about the mental and the physical, but neither is relevant here. Furthermore, as indicated in chapter 2, no relevant epistemological problem could arise without the assumption that an idea is an idea of something in virtue of their causal connection.

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triples of numbers that are the plus function and those that are the quus function, but, as the terms “plus” and “quus” have to be nonsynonymous expressions that respectively denote those functions, paradoxically, the functions are semantically distinguishable and the reference of “sixty-eight plus ªfty-seven” determinate. 4.3 The General Form of Indeterminacy Arguments The common pattern of reasoning that we have found in Quine’s and Kripke’s indeterminacy arguments is the general form of such arguments. They begin with an explicit or implicit account of our informal knowledge of some domain, typically embodied in the skeptic’s representation of the situation we face in relation to theories of the domain. The skeptic assumes the account to be a complete representation of that knowledge at least insofar as issues of indeterminacy are concerned, and, on the basis of this assumption, the skeptic argues for an unbreakable symmetry between the intended interpretation and certain deviant interpretations of the theory which precludes our ruling out the deviant interpretations. At this point, indeterminacy arguments go their separate ways. Those that are put forth as skeptical arguments, such as Quine’s, proceed from there being no way to break the symmetry to the conclusion that no interpretation (e.g., translation) is correct because there is no fact of the matter. Those that are not put forth as skeptical arguments, such as Kripke’s, stop with the symmetry puzzle and simply ask what is to be done. Going for the robust skeptical conclusion, however, involves a dubious inferential step. As I (1988; 1990b, 183) argued against Quine, symmetry at best delivers the unknowability of language neutral meanings, so unknowable meanings, like Kant’s noumena, could still exist. If realism is right—and it hasn’t been ruled out—it is absurd to think that existence of objects in a realm that is independent of us depends in any way on our cognitive powers. The skeptic must do something to shore up the ªnal step in the argument, but the available props themselves are in need of bracing.8 It is even more critical for the skeptic to support the assumption that the representation of our informal knowledge on which the symmetry puzzle rests is complete. As we noted, Quine sought to do this by trying 8. Benacerraf (1996, 26) recognizes this weakness in the argument, and he (1996, 52–53 n. 15) argues that Wright’s (1985, 122) attempt to shore up the argument at this point does not work. I think the issue goes deeper because it involves the veriªcationist theory of meaning. That, as I see it, is what Wright’s concept of perfect understanding rests on, but I can’t pursue the question here.

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to show that beliefs involving putative symmetry-breaking sense properties and relations do not qualify as legitimate knowledge of the semantic domain, and hence that those properties and relations can be excluded from a full account of our informal semantic knowledge. Kripke makes no such argument. This, I believe, is because he is not playing skeptic, but only devil’s advocate. Genuine skeptics, however, cannot neglect this aspect of their argument, since, as we have seen in the case of Quine’s and Kripke’s arguments, failure to exclude symmetry-breaking properties opens up the possibility of a determinate speciªcation of the objects in the domain based on those very properties and relations. An argument to illegitimatize the symmetry-breaking properties and relations is essential for an effective indeterminacy argument for skepticism about our knowledge of the domain. The properties and relations that make the interpretations that enter into the symmetry puzzle incompatible are, for this very reason, potential symmetry-breaking properties. Indeed, it must be clear even on the basis of the skeptic’s own account of the puzzle that the intended and deviant interpretations are incompatible, since otherwise nothing stops us from saying that the objects involved in the deviant interpretation are just the objects involved in the intended interpretation under a deviant description. If skepticism is not to be lost for lack of a response to the claim that the objects in the allegedly deviant model are just the meanings or the numbers under a deviant description, then it must be clear in what respects the deviant interpretations conºict with the intended one. But if this is clear, the skeptic’s indeterminacy argument is threatened on the grounds that those differences show that our knowledge of the objects in the domain is rich enough to distinguish the objects assigned to expressions on a deviant interpretation from the objects assigned on the intended interpretation. The symmetry is unstable because the interpretations have to be both equivalent and incompatible—equivalent in order to be symmetrical and incompatible in order to be rivals. Recognition of the incompatibility among the various interpretations in indeterminacy arguments reºects the fact that what is being challenged is something we in fact know or know how to do—such as distinguishing synonymous and nonsynonymous expressions or referring determinately to numbers.9 9. The ship of Theseus is not a symmetry puzzle in an indeterminacy argument, because it is not a skeptical challenge to something we know or know how to do. Rather, it reºects a gap in our knowledge of the conditions for something’s identity over time. There is no issue about how to distinguish a ship we know to be Theseus’s from a ship we know not to be. Nor is there an issue about how to distinguish the two ships; they are distinguishable historically, spatially, and materially. The issue is which ship is Theseus’s.

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For example, in the case of Quine’s skeptical challenge, the incompatibility shows that we know that the expressions “rabbit,” “rabbit stage,” and “undetached rabbit part” are nonsynonymous. The reason that there is a symmetry issue at all is that, at least tacitly, we have the knowledge required to distinguish the intended interpretation from deviant ones. Here we can go on the offensive against the skeptic. If our knowledge were as meager as the skeptic wishes us to think, the interpretations in symmetry puzzles would not be incompatible. Thus, on the one hand, the skeptic has to underrate our knowledge to make us think that we are impotent to make such distinctions, and, on the other hand, the skeptic has to rate our knowledge properly in order to construct a symmetry between competing interpretations. Since skeptics can’t have it both ways, they face a dilemma: either the account of our knowledge on which their symmetry claim rests is complete and the interpretations are not rivals, or the interpretations are rivals and the account is not complete. In the former case, there is symmetry, but only because the interpretations are equivalent, and hence not rivals. In the latter case, there is nonequivalence and rivalry, but only because the interpretations are nonsymmetrical. In neither case is there a real symmetry puzzle.10 10. Simply in virtue of his description of the “gavagai” puzzle in Word and Object, we know that English contains the sense properties and relations necessary to distinguish among the putatively symmetrical translations “rabbit,” “rabbit stage,” and “undetached rabbit part.” If these expressions did not have different senses, they could not be competing hypotheses. It would be absurd for Quine (1960, 27) to claim that manuals for translating . . . can be set up in divergent ways, all compatible with the totality of speech dispositions, yet incompatible with one another. (Italics mine) Moreover, our knowledge of the nonsynonymy of those expressions is knowledge of the sense differences in their compositional and decompositional structure. For example, the different meanings of “stage” and “part” that combine with the meaning of “rabbit” to form meanings for the whole expressions “rabbit stage” and “undetached rabbit part” exhibit such a sense difference. The meaning of “stage” involves the concept “temporal slice of some enduring object, event, or process,” whereas the meaning of “part” involves the concept “one of the divisions of a whole.” Once we recognize such sense differences, the puzzle about translation disappears, since that recognition reºects speech dispositions that conªrm one of the divergent translations and disconªrm the others. In the case of Kripke’s puzzle about rule following, our knowledge of the sense of “plus”—something like that given by (P)—and Kripke’s (1982, 7–54) deªnitions of “quus” and “quaddition” (expressed in terms of the sense of “plus”) makes the puzzle comprehensible, but, as explained above, that semantic knowledge also enables us to resolve the puzzle because it explains why there is no step of projecting numerical reference. Similar points apply to Kripke’s (1982, 19–20) version of the argument involving the terms “table” and “tabair.” See my (1990b, 163–74) discussion of this and the other versions of his argument.

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The dilemma shows the need for the skeptic to have an argument that illegitimatizes the symmetry-breaking properties and relations. To escape the dilemma, the skeptic must establish that our recognition of the incompatibility of the intended and the deviant interpretations does not show that we have knowledge of differences between them that breaks the symmetry. The skeptic has to argue, as Quine argued in “Two Dogmas of Empiricism,” that the properties and relations that are involved in the incompatibility are not, in the ªnal analysis, legitimate (e.g., they cannot be made objective sense of). Skeptics have to show something like what Quine tried to show when he argued that our belief in synonymy and analyticity is due to our subscribing to the myth of the museum, that our recognition of incompatibility is not based on knowledge, but on false belief. If skeptics do not illegitimatize the features on which our recognition of incompatibility depends, either because the argument intended to do so doesn’t work or because no argument is given, the symmetry is broken and the indeterminacy argument fails. A successful illegitimatizing argument is the only way skeptics can avoid having their account of our informal knowledge discredited for not doing that knowledge full justice. 4.4 The Strategy for Resisting Indeterminacy Given these considerations, there is a straightforward strategy for resisting indeterminacy arguments. First identify the paradoxical aspect of the symmetry puzzle. Then focus on the properties and relations of the intended interpretation that make it and the deviant interpretation(s) rivals, that is, those features of the intended interpretation but not the deviant one(s) that prevent us from treating the latter as just the former under a deviant description. On this basis, we take those features of the objects in the intended model to be ones those objects have, and hence we take them to be the features the skeptic has omitted in representing our informal knowledge of the domain. The next step, which may or may not be necessary depending on the nature of the skeptic’s argument, is to counter the skeptic’s claim that the features are illegitimate. Since the claim cannot be backed up with an appeal to indeterminacy without begging the question, the skeptic must come up with an independent way of excluding symmetry-breaking properties, one that explains why we are mistaken to think that they carry the weight we have put on them. This requires the skeptic to invoke some philosophical perspective that enables the skeptic to impose a general condition on legitimate properties and relations. In Quine’s case, it was the naturalist/empiricist perspective that made the taxonomic theory

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of grammar with its operationalist methodology of “substitution criteria” or “conditions of interchangeability” seem to be the proper test of concepts in linguistics. Using the Quinean example as a guide to analyzing a skeptic’s attempt to come up with an independent way of excluding the symmetry-breaking properties, we expose the philosophical perspective the skeptic has invoked to judge what are and are not legitimate properties and relations. The ªnal step is to observe that the skeptic’s claim of illegitimacy is of illegitimacy from that philosophical perspective but not from our philosophical perspective. This frees us to employ the properties and relations in question to rule out the skeptic’s deviant interpretation(s). This strategy can be illustrated with an indeterminacy argument based on Frege’s (1953, 68) famous Caesar question. In criticizing “Leibnizian deªnitions” of the individual numbers, Frege puts the Caesar problem in these terms: . . . we can never—to take a crude example—decide by means of our deªnitions whether any concept has the number JULIUS CAESAR belonging to it, or whether that same familiar conqueror of Gaul is a number or is not. Let us suppose that a skeptic challenges us to distinguish between an interpretation of Peano arithmetic on which “seventeen” refers to the number 17 and an interpretation on which it refers to Julius Caesar. Now the Caesar question is easily answered. We can reject identity statements like “17 = Caesar” on the grounds that Caesar has the property of being a concrete object, while the number 17 has the property of being an abstract object, or that Caesar was a sentient creature, a human being, a Roman, and so on, while the number 17 is none of these things.11 11. We don’t have to go to all the trouble of developing structuralism to solve problems like the Caesar problem. Of course, once we develop it, we can say, as Shapiro (1996) does, that since 17 is an element in a structure and Caesar is not, their identiªcation is a category mistake. But, from our perspective, developing structuralism for this purpose is overkill. Further, from our perspective, a structuralist solution to such symmetry problems perpetuates the myth that we have no choice but to cave in to Benacerraf. Moreover, we cave in without really getting out of the woods. Frege (1953, 68) pointed out that a solution would provide us with “authority to pick out [numbers] as selfsubsistent objects that can be recognized as the same again.” Aren’t the integer seventeen, the rational number seventeen, the real number seventeen, and the natural number seventeen “the same again”? Despite Wittgensteinian philosophers of mathematics who would deny it, it is pretty plausible to say that they are. But even if one thinks that they are not all the same, the structuralist ought at least to say that they have a structural property in common. This, however, as Kastin (1996) observes, is “precisely what is ruled out by Shapiro’s polystructuralism.”

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Benacerraf rejects identity statements like “17 = Caesar” on the stronger grounds that they are “senseless or ‘unsemantical’” because the terms in them do not belong to a common superordinate category. He ([1965] 1983, 286–87) writes: If an expression of the form “x = y” is to have a sense, it can only be in contexts where it is clear that both x and y are of some kind or category C, and that it is the conditions which individuate things as the same C which are operative and determine its truth value. There is no such C in the case of 17 and Caesar. The basic ontological category of the former is Abstract and the latter Concrete, and these categories are disjoint. Hence, Benacerraf recognizes that we can rule out all interpretations that identify numbers with Julius Caesar, Marcus Brutus, Lassie, the moon, or other concrete objects. None of those things could be a number because each of them has or has had spatial or temporal location, whereas numbers can have neither spatial not temporal location. It perhaps goes without saying that putative identities in indeterminacy arguments do not have to be shown “senseless” or “unsemantical” for us to have grounds for rejecting them. It is enough if they are shown false. In the question of whether numbers are sets, unlike the Caesar question, all of the objects in the intended and deviant interpretations belong to the category Abstract. Nonetheless, we can reject the identiªcation of numbers with sets on the grounds that sets but not numbers have members, have subsets, contain the null set, and so on, while numbers but not sets can be prime or composite, even or odd, perfect or imperfect, and so on. There are, of course, many mathematical systems of abstract objects with the same structure as the numbers. Thus, we have questions like whether the natural numbers are distinguishable from the even numbers. Again, the identiªcation of the natural numbers with a deviant system of numbers is refutable on the basis of the features of the system that make it another collection of objects. In this example, a symmetry claim identifying the even numbers with the natural numbers is refuted by the fact that none of the former but every other one of the latter is odd, the fact that the latter contains an odd prime but the former doesn’t, and so on. The parallel question in a similar but nonmathematical case is whether numbers are English sentences, supposing that the sentences of English form a recursive sequence. The sentences of English are types, so they too belong to the category Abstract. In this example too, the putative identiªcation of the numbers with the deviant model is

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neither “senseless” nor “unsemantical.” Still, we can reject the identiªcation of numbers with sentences as false, since sentences but not numbers are interrogatives and declaratives, have subjects and direct objects, have morphemes as constituents, and so on, while numbers but not sentences are odd or even, prime or composite, and so on. Cases like the one in which the deviant interpretation is the system of even numbers are trivial, since the application of our strategy does not require going beyond the properties in Peano arithmetic. In cases like those in which the deviant interpretation consists of Roman emperors, sets, and sentences, the application of our strategy requires going beyond the properties and relations that appear in the mathematical theory. This might seem to be something we ought not to do if we share the assumptions behind Frege’s criticism of “Leibnizian deªnitions” as improper. On the basis of Frege’s semantical view that sense is the determiner of reference, a proper deªnition of a mathematical term determines its referent, and hence a proper deªnition of number decides inter alia that Julius Caesar is not a number. The assumption that deªnitions of number terms do what a Fregean deªnition is supposed to do pushes the philosopher of mathematics into seeing the relation between the Dedekind/Peano deªnition of arithmetic and the numbers in terms of the Fregean relation between sense and reference. The question whether numbers are objects thus comes to depend on the question whether Peano arithmetic uniquely determines them as its model. Since Peano arithmetic does not do what a Fregean deªnition is supposed to do, but nonetheless expresses the mathematician’s concept of arithmetic, structuralism follows as the most plausible explanation of number theory in the absence of determinate numbers. This is how, as I see it, Frege’s semantics misleads philosophers of mathematics. If they didn’t accept Frege’s notion of deªnition, they wouldn’t think that the properties that enter into the mathematical deªnition of arithmetic bear the entire weight of determining the numbers, or, to put it the other way around, they wouldn’t think, as Benacerraf ([1965] 1983, 291) does, that “the properties of numbers which do not stem from [their arithmetic structure] are of no consequence whatsoever.” Given that we are not committed to Frege’s semantics, our strategy can presuppose that Peano arithmetic, since it is silent on whether numbers are abstract, have members, have grammatical properties and so on, is an incomplete account of our informal knowledge of the numbers that is involved in determinate reference to them. If there is an argument, independent of Fregean semantics, that such nonnumber-theoretic properties are illegitimate, as we shall see there is in Benacerraf’s indeterminacy argument, we have to analyze it to ªnd out

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what philosophical perspective it invokes. But here that is not necessary. We can simply argue that, in providing features that establish the incompatibility of those interpretations, the skeptic has provided us with the features required to break the symmetry.12 4.5 Benacerraf’s Argument At one point in his argument, Benacerraf ([1965] 1983) assumes that, if we want to say that one set-theoretic account of the numbers is correct, we must be able to justify saying it. One might balk at this assumption in the same spirit in which, as I suggested in section 4.3 above, one might balk at Quine’s indeterminacy argument: granting everything he says about radical translation, language neutral meanings could exist unknowably. So why couldn’t numbers exist unknowably? Realists, after all, hold that numbers are abstract objects that exist independently of us and our epistemic capacities, so, of all philosophers, realists are in the best position to insist that numbers can be unknowable thingsin-themselves. In reºecting on his argument, Benacerraf (1996, 26) acknowledges that “the need to refer to [the assumption] is a genuine weakness of the argument.” Nonetheless, I don’t want to make anything out of this weakness. I agree with Benacerraf’s ([1965] 1983, 284) earlier claim that “the position that this is an unknowable truth is hardly tenable.” Dependency on the principle that there cannot be a correct account without some way to show that it is correct is a weakness of his indeterminacy argument, to be sure, but, by the same token, it would be a weakness of realism if it had to say that there is a correct account but concede that it has no way to show that it is. It would be theoretically lame for realists to take this position, since it would raise questions about realism just as Kant’s things-in-themselves raised questions about his transcendental idealism. Benacerraf’s argument moves from a preliminary stage in which he concludes that numbers could not be sets to a ªnal stage in which he concludes that numbers could not be any other objects either. If the 12. Accepting our strategy does not require one to deny that there are cases of theories of which two or more members of a set of competing models are equally good interpretations, where it is indeterminate which of the models is the “right” interpretation of the theory. But it requires one to say that, in such cases, the “indeterminacy” is a consequence of the incompleteness of the theory. When an argument for the indeterminacy of the choice between one and another model for a theory turns on an incompleteness in the speciªcation of the theory, we have an argument for completing the speciªcation rather than an argument for skepticism about the choice. The completion of a theory resolves the choice of an interpretation for the theory in favor of one of the models.

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ªnal stage of his argument were just a generalization of the preliminary stage, we could apply our strategy to Benacerraf’s overall argument and conclude that it is self-defeating (because, e.g., descriptions of Ernie’s and Johnny’s set-theoretic constructions enable us to distinguish their interpretations of Peano arithmetic from the intended interpretation, the numbers themselves). This would be the end of it. Benacerraf ([1965] 1983, 290) would have been right to claim that “‘objects’ do not do the job of numbers singly,” but wrong to take that as grounds for claiming that “numbers could not be objects at all.” Singly, they and the objects of a deviant interpretation that structurally masquerade as numbers exhibit conºicting properties that our strategy can use to distinguish one from the other. But the ªnal stage of the argument is more than just a generalization of the preliminary stage. In the ªnal stage, Benacerraf attempts to illegitimatize properties that could be used to rule out deviant interpretations. Benacerraf ([1965] 1983, 291) is aware that there are such properties: “. . . it would be only [the properties of numbers that do not stem from the relations they bear to one another in virtue of their forming a progression] that would single out a number as this object or that.” And he takes speciªc steps to illegitimatize them. In this respect, Benacerraf’s argument is like Quine’s rather than Kripke’s. Thus, the central question concerning Benacerraf’s argument is whether the argument provides a compelling reason for thinking that properties of numbers that do not arise from the relations they bear to one another in virtue of their forming a progression are illegitimate for us to use in distinguishing the numbers from other systems of objects that also form a progression. Benacerraf ([1965] 1983, 291) claims that “[t]he search for which independently identiªable particular objects the numbers really are (sets? Julius Caesars?) is a misguided one.” It is “misguided” because, as he puts the driving principle of his ([1965] 1983, 290) argument, If one theory can be modeled in another . . . then further questions about whether the individuals of one theory are really those of the second just do not arise. It is in explaining why such questions do not arise that Benacerraf makes his Quine-like attempt to illegitimatize properties that rule out deviant interpretations. Benacerraf ([1965] 1983, 290) says that such questions do not arise because “the mathematician’s interest stops at the level of structure,” and hence it is “mistaken” for philosophers “not [to be] satisªed with [the mathematician’s] limited view of things” and “to want to know more.”

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If Benacerraf were right, then the only properties available to us for discriminating between the individual of one theory and the individuals of another theory in which the ªrst is modeled would be the structural properties represented in the theories (or in mathematics generally?). Depending on how widely the notion of a structural property is taken, we might still be able to resist Ernie’s and Johnny’s identiªcations of numbers with sets, since properties of sets like having members and properties of numbers like being prime are mathematical properties.13 But the Caesar question and others would be reopened, since abstractness, the various properties it entails (e.g., causal inertness), grammatical properties, and so on are not mathematical properties. Not only would questions like the Caesar question be reopened, but our general strategy for resisting indeterminacy arguments would be invalidated. If the only properties we can use for our strategy are those in which the mathematician takes an interest, the discovery of a paradoxical element would no longer guarantee that our strategy applies. We could not generalize from the way in which indeterminacy was resisted in the cases where Peano arithmetic is modeled in set theory to all cases where Peano arithmetic is modeled in another theory. Such a generalization assumes that any properties of the deviant interpretation, including nonmathematical ones, can be used in applying our strategy. The real weakness of Benacerraf’s argument is his claim that it is “mistaken” for philosophers “not [to be] satisªed with [the mathematician’s] limited view of things.” Benacerraf ([1965] 1983, 291) thinks that philosophers who “want to know more” are making a mistake because there is nothing more of consequence to know about the numbers than the properties of them that “stem from the relations they bear to one another in virtue of being arranged in a progression.” Other properties of the numbers “are of no consequence whatsoever,” Benacerraf ([1965] 1983, 291) argues, because . . . in giving the properties . . . of numbers you merely characterize an abstract structure—and the distinction lies in the fact that the “elements” of the structure have no properties other than those relating them to other “elements” of the same structure. This begs the question. In the quotation that heads Benacerraf’s ([1965] 1983, 272) paper, R. M. Martin says that the philosopher who is “more 13. A property is structural relative to the domain of objects under consideration. Hence, not having members is not a structural property of numbers, but it is a structural property of the null set.

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sensitive [than the mathematician] to matters of ontology” thinks there is more to be said about the properties of numbers than arithmetic structuralism allows. Benacerraf ([1965] 1983, 290) believes that Martin is “mistaken,” but his reply can’t just be a reassertion of the claim that there is no more to be said about the properties of numbers than arithmetic structuralism allows. Benacerraf ([1965] 1983, 290) is surely right to say that a concern with the answer to philosophical questions about numbers “miss[es] the point of what arithmetic, at least, is all about,” but that’s hardly decisive in a philosophical controversy. He is surely wrong to think that a concern with the answer to philosophical questions about numbers misses the point of what the controversy concerning the indeterminacy of numbers is all about. That controversy is at least as much philosophical as it is mathematical. It is hard to see why philosophers who “want to know more” are not simply philosophers wanting to do their own philosophical thing. Wetzel’s (1989b, 282) careful analysis of Benacerraf’s argument ªngers premise (A) as the real culprit, (A) Structural properties (i.e., the properties of numbers that stem from the relations they bear to one another in virtue of being arranged in a progression) do not sufªce to individuate [numbers]. but, although I agree with her that (A) is mistaken, I think that the real culprit is (B). (B) The essential properties of numbers are all structural. Numbers, she (1989b, 282) observes, are not the sorts of things that have members, but, essential as that negative property is to numberhood, it isn’t a structural property (Zermelo “numbers” and von Neumann “numbers” both have members). It is essential to seventeen that it has no spatiotemporal location, that it is causally inert, that it is not mind-dependent, and so on, but these are not structural properties. (B) is mistaken because, although some essential properties of numbers— perhaps even those that matter most for most purposes—derive from the relations they bear to one another in virtue of being arranged in a progression, others, at least as the realist sees it, derive from the relations that numbers bear to one another, to sets, to propositions, to sentences, and so on in virtue of all of them being abstract objects.14 14. The point here does not depend on assuming realism. In providing nonmathematical properties of numbers such as necessary absence of spatiotemporal location, causal

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The good name of mathematics does not suffer because our knowledge of the numbers goes beyond mathematics. Arithmetic is not intended to be more than a theory of the mathematical structure of the numbers. Mathematics in the broadest sense has nothing to say about some essential properties of numbers because it has no metaphysical aspirations. This is not, of course, to criticize mathematics, but only to impugn Benacerraf’s philosophical assumption that number-theoretic properties exhaust those that are relevant to distinguishing numbers from other things. Hence, the fact that a property is one “in which the mathematician professes no interest” does not prevent it from playing a role in distinguishing the intended interpretation of arithmetic from a deviant interpretation. Benacerraf’s attempt to illegitimatize nonmathematical properties is the equivalent of Quine’s attempt to illegitimatize nonreferential properties such as synonymy and analyticity. Like Quine, Benacerraf has to exclude the objectionable properties in order to establish his indeterminacy thesis, but, also like Quine, his grounds for exclusion are inadequate. The seriousness of this inadequacy can be appreciated from the perspective of a general conception of indeterminacy arguments. The cost to Benacerraf of failing to illegitimatize nonmathematical properties is the failure of his argument for the indeterminacy of reference to numbers, just as the cost to Quine of failing to illegitimatize intensional properties was the failure of his argument for the indeterminacy of translation. For Benacerraf’s ([1965] 1983, 291) claim that “The number words do not have single referents” follows only if number theory encompasses the full range of properties that can be used to exclude unintended models of arithmetic. Since it doesn’t, his indeterminacy argument cannot take number theory as a complete explication of our knowledge of the numbers, and the alleged symmetry on which the argument rests can be rejected on the same grounds on which we rejected the alleged symmetries on which Quine’s and Kripke’s arguments rest. They underestimate our informal knowledge of the domain. inertness, and so on, mathematical realism makes it easy to refute identity statements like “17 = Julius Caesar.” But on mathematical conceptualism and mathematical nominalism too, we can make short work of those identities. The non-mathematical properties that function as counterexamples change, reºecting the respective psychological and inscriptionalist or ªctionalist conception of numbers. Conceptualists can refute “17 = Julius Caesar” by observing that Caesar, unlike 17, crossed the Rubicon. Inscriptionalist nominalists can refute it by observing that 17, unlike Caesar, is a construction out of orthographic symbols. Fictionalist nominalists can refute it by observing that 17, unlike Caesar, never existed in reality.

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4.6 The Metaphysics of Number-Theoretic Skepticism Reºection on the selectiveness of formalization and theory construction ought in itself to have made us suspicious about the claim that only arithmetic properties are legitimate to use in distinguishing numbers from other things. It is in the nature of formalization and theory construction to select those properties that have a distinctive role in the structure chosen for study. Moreover, selectiveness is essential in the formal sciences because numbers and the other objects they study are not the private property of any one discipline. Numbers belong to the domain of mathematics in virtue of the mathematician’s concern with their arithmetic structure, but they also belong to the domain of philosophy in virtue of the philosopher’s concern with the most general facts about reality. The domains ought to be thought of not as disjoint but as overlapping. Numbers are, as it were, communal property. What is proprietary is only the special interest a discipline takes in them. The mathematician’s special interest in numbers is with their arithmetic structure; the philosopher’s is with their ontology and epistemology. From the standpoint of the inherent selectiveness of formalization and theory construction, the assumption of Benacerraf’s argument that we know nothing about the numbers except what is in number theory seems truly bizarre. Or, rather, it seems so today. Back in the sixties when Benacerraf wrote “What Numbers Could Not Be,” the assumption seemed anything but bizarre. At that time, philosophy was in the throes of a naturalistic revolt against traditional metaphysics. The idea of philosophy as an a priori discipline with an independent subject matter consisting of the most general facts about reality was widely seen as a dangerous piece of speculative metaphysics. To some of those in the vanguard of the revolution, the idea was nonsense; to practically all, it was responsible for most of the ills of traditional philosophy. The new movements of the day—logical empiricism, Oxford philosophy, late Wittgensteinianism, and Quinean naturalism—all took the naturalistic view that philosophy has no subject matter of its own. Only the sciences have subject matters of their own. Philosophy, properly conceived, is a second-order discipline, concerned with the semantic clariªcation of ordinary and/or scientiªc language. It is no wonder that, writing at such a time, Benacerraf regarded it as a mistake for philosophers to inquire into philosophical facts about numbers that fall outside the mathematician’s sphere of interest. There are none. The metaphysical philosopher who sets out to discover such facts is, in Wittgenstein’s (1953, sec. 94) words, “in pursuit of chimeras.” It is no

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wonder, therefore, that the assumption that mathematicians tell us whatever there is to know about numbers did not seem bizarre in those halcyon days when it seemed to almost everyone that the metaphysical storms of the past had ªnally blown themselves out.15 4.7 Putnam and Wittgenstein Putnam (1983, 423) says that, as well as showing that no formal system can capture our intuitive notion of a set, Skolemite considerations show that no “formalization of total science” (or even of our total belief system) can capture our intuitive notion of the world. Putnam (1983, 424) sees this “paradox” as a serious problem for any philosopher or philosophically minded logician who wishes to view set theory as the description of a determinate independently existing reality. From the perspective of this chapter, this is a “serious problem” only for those who think that axiomatic set theory exhausts the information on the basis of which we determinately refer to sets, or even that it exhausts the information on the basis of which it is taken to be a description of the domain of sets. Putnam (1983, 424) is clearly aware that mathematical realists can challenge the reasoning underlying the “paradox” on the grounds that “it is natural to think that something else—our ‘understanding’” rather than axiomatic set theory captures the intuitive notion of a set. Since this, in its inchoate form, is one of the thoughts underlying our strategy, we need to look at why Putnam rejects the thought that our understanding is what captures the intuitive notion of a set. 15. In reºecting on “What Numbers Could Not Be,” Benacerraf (1996, 28) considers three ways that realists might try to escape the argument in that paper. One of them, “stick[ing] to [their] realist guns and refusing [. . .] to yield on the matter of the possibility of establishing the right answer as the right answer,” is the approach that we have taken in this chapter. Benacerraf (1996, 28–29), however, ªnds this approach the least “appealing” of the three possible escape routes because Unless a strong argument can be mounted for its realist under-pinnings that is itself grounded in a persuasive analysis of mathematical practice, . . . [this escape route] is “simply too theoretical, too metaphysical.” Since Benacerraf’s attack on the determinacy of reference to numbers itself relies on a naturalist metaphysics to support the claim that philosophy is a second-order discipline with no aspect of reality as its subject matter, his claim that the realist’s escape route is “simply too metaphysical” is something of the pot calling the kettle black. Furthermore, given the epistemological explanation in chapter 2, the kettle is not so black anymore.

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Putnam (1983, 424) says: But what can our “understanding” come to, at least for a naturalistically minded philosopher, which is more than the way we use our language? The Skolem argument can be extended . . . to show that the total use of the language . . . does not “ªx” a unique “intended interpretation.” So the naturalistically minded philosopher is in the soup. The Platonistically minded philosopher, according to Putnam (1983, 424), has an immediate response to the Skolem argument, namely, . . . he will take this as evidence that the mind has mysterious faculties of “grasping concepts” (or “perceiving mathematical objects”) which the naturalistically minded philosopher will never succeed in giving an account of. . . . the extreme positions— Platonism and veriªcationism—seem to receive comfort from the Löwenheim-Skolem Paradox; it is only the “moderate” position (which tries to avoid mysterious “perceptions” of “mathematical objects” while retaining a classical notion of truth) which is in deep trouble. Putnam is wrong to think realists take “comfort” from the indeterminacy arguments, which purport to show that the intended interpretation cannot be uniquely ªxed. I imagine that Putnam is thinking that realists might claim, as Benacerraf ([1965] 1983) notes, that they can hold that the intended interpretation is correct even though they are unable to justify it as the correct interpretation. But, as I argued in section 5, such a claim is not only theoretically lame but it raises doubts about realism parallel to the doubts raised by Kant’s claim that there are unknowable things-in-themselves. Hence, realists, no less than naturalists, need to explain why Putnam’s extension of the Skolem argument “to show that the total use of the language . . . does not ‘ªx’ a unique ‘intended interpretation’” doesn’t prevent them from holding that we can refer determinately to mathematical objects. If, as Putnam (1983, 423) claims, “a formalization of total science . . . could not rule out unintended interpretations,” the realist may not be in the “deep trouble” that “the naturalistically minded philosopher” is in, but the realist’s trouble is deep enough. But, in fact, realists are not in any trouble at all, providing they subscribe to the right semantics. Benacerraf (1985, 110) has put his ªnger on what is the nub of Putnam’s skepticism: Of course, . . . no explanation [of how an unintended interpretation can be ruled out] can be satisfactory. . . . The reason resides

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in our logocentric predicament: Any explanation must consist of additional words. Words which themselves are going to be said to need interpretation. [Putnam’s] strategy has a wondrous simplicity and directness. He will construe any account we offer as an uninterpreted extension of our already deinterpreted theory— by explaining we merely produce a new theory which, if consistent, will be as subject to a plethora of (true) interpretations as was the old. (“Any interpretation is itself susceptible to further interpretation.”) Wittgenstein’s rule-following argument is the engine that drives Putnam’s Skolemite argument. This might seem at ªrst to be bad news for the realist, but, on closer examination, it turns out that the realist, unlike Wittgenstein’s conceptualist interlocutor in the Philosophical Investigations, is in a good position to avoid Wittgenstein’s paradox. Wittgenstein’s (1953, sec. 141) criticisms of Frege’s (and Russell’s) approaches to semantics were supposed to show that his own usebased approach to meaning and his interlocutor’s mentalistic approach are the only ones left as serious contenders for our semantic allegiance. If those criticisms had succeeded, Wittgenstein’s use-based approach would have emerged as the clear winner because his rule-following argument refutes the idea that “what comes before the mind” provides the normative basis for ªxing meaning in the use of language. But if those criticisms do not succeed in eliminating all other approaches to meaning, a refutation of the mentalistic approach does not settle the issue of what provides normative force. In The Metaphysics of Meaning, I (1990b, 21–133) argue that Wittgenstein’s criticisms do not work against an approach to semantics that is realist rather than mentalistic and non-Fregean rather than Fregean. Wittgenstein’s criticisms are too closely tailored to Fregean semantics and the Fregean conception of an ideal—logically perfect—language. I argue that the criticisms do not work against a theory in which senses are non-Fregean and reside in natural languages, understood as a system of sentence types systematically correlated with abstract senses. I then go on to argue that abstract senses supply the normative element missing in mentalistic approaches, and non-Fregean intensionalism supplies a conception of how grammatical norms are applied in the use of language that avoids Wittgenstein’s criticisms of the interlocutor’s Fregean conception of the application of language. On the interlocutor’s conception of meaning, what comes before the mind when we understand a word, being an uninterpreted symbol, requires a “method of projection” to interpret it, but the addition of such an interpretation is only the addition of another uninterpreted

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symbol. Thus, as Wittgenstein (1953, sec. 198) says, “[it] hangs in the air along with what it interprets, and cannot give it any support. Interpretations by themselves do not determine meaning.” This is the “logocentric predicament.” Something more than symbols, words, and other specimens of orthography is required to determine meaning. Our linguistic realism provides that something more: a semantics of abstract senses. Abstract senses have the normative semantic force that mental meanings lack because, rather than uninterpreted symbols, they are the semantic content of the sentence types that conªgurations of symbols express in the use of language. On my (1990b, 86–111, 142–55) account of the use of language, it is a top-down affair in which a speaker’s knowledge of the sentence-sense correlations of the language results from his or her ability to use conªgurations of spoken or written symbols as tokens of sentential types. The literal use of language consists in the speaker’s realizing an intention to use a token of a word type with the sense that the type has in the language. Since the abstract sense of a word type is not a symbol but a semantic universal, there is no appeal to interpretations that are only further symbols, and hence there is no “logocentric predicament.” Furthermore, since the sense of a word type is the norm for the literal use of its tokens, the sense has the force to determine their proper application. Since sense determines “what we call ‘obeying the rule’ and ‘going against it’ in actual cases,” this account of the use of language avoids Wittgenstein’s (1953, sec. 201) paradox about rule following. Once senses of the right sort enter the picture, the predicament disappears. The logocentric predicament is a senseless predicament. In form and resolution, Putnam’s argument is close to Kripke’s, which shouldn’t be surprising in light of their common debt to Wittgenstein. But in one respect, Putnam’s argument is closer to Quine’s and Benacerraf’s arguments: it contains, not a full-ºedged argument for illegitimacy, as their arguments do, but a reason for thinking about abstract objects—abstract senses, too, presumably—are not philosophically legitimate. The reason, as the above Putnam quotations show, is that naturalism precludes such “mysterious” objects. In characterizing realism as resorting to “mysterious ‘perceptions’,” Putnam lines up with Gottlieb, Field, Chihara, and Dummett in unwarrantedly disparaging realism. Thus, the criticism of them in chapter 2, section 4 applies to Putnam: he confuses mystery with mysticism. Further, as shown in chapter 1, section 3.1, and in chapter 2, talk of “faculties of ‘grasping concepts’ (or ‘perceiving mathematical objects’)” does not necessarily imply a connection to abstract objects that puts such faculties beyond a naturalistic psychology of our cognitive proc-

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esses. There is, as far as I can see, no reason whatsoever to think that scientiªc psychology cannot in principle explain the rational faculties on which our account of mathematical knowledge is based, though, to be sure, current scientiªc psychology is far from telling us much about them. While it is true that some realists are guilty of obfuscation in talking about grasping facts about abstract objects in terms of causal connections, others—Gödel in particular—are entirely innocent. Hence, Putnam’s implicit illegitimatizing argument fails too. 4.8 Conclusion If the argument in the present chapter is right, then the true signiªcance of the fact that there are indeªnitely many models that qualify structurally as well as the numbers themselves as interpretations of arithmetic is not, as Benacerraf ([1965] 1983) claimed, that we cannot refer determinately to the numbers, or, as Putnam (1983) claimed, that we cannot refer determinately to “the real world,” but rather that determinate reference to the numbers and to the real world depends as much upon our language and philosophy as it does upon our science.

Chapter 5 The Ontological Challenge to Realism

5.1 Introduction The ontological challenge to realism is posed by certain alleged counterexamples to the traditional abstract/concrete distinction. Games such as chess and checkers, linguistic entities such as words, sentences, and natural languages, putatively natural objects such as the equator or points in space, and impure sets (i.e., sets some of the members of which are not abstract objects) seem to be abstract, but they also seem to have spatial or temporal location. According to the traditional distinction, if something is abstract, it cannot have either spatial or temporal location. Consider the example of words and sentences. In contrast to their utterances and inscriptions, words and sentences are types, and hence, in accord with C. S. Peirce’s (1958, 423) classic statement of the type/ token distinction, they are abstract objects. If sentences are abstract objects, then the collections of them that form natural languages are abstract objects too. Nonetheless, words, sentences, and natural languages are customarily taken to have temporal properties. People say that English has a history: it didn’t exist at the time of the ancient Greeks, it exists now, and it won’t exist after the destruction of our solar system. But if some abstract objects have temporal location, the traditional abstract/concrete distinction cannot be saved. In contrast to the epistemic and semantic challenges, such purported counterexamples to the traditional abstract/concrete distinction have not, to my knowledge, been put forth as a challenge to realism. Moreover, it might seem that such putative counterexamples are not a special problem for realists, since realists as well as antirealists bring up such examples as difªculties for the traditional abstract/concrete distinction, and, as we saw in chapter 2, antirealists make use of the distinction in their epistemological criticisms of realisms. However, the realist cannot claim that everyone is in the same boat. These examples pose a special problem for realism because they threaten only the abstract side of the distinction. Realism cannot be formulated without the concept of an

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abstract object, and hence the alleged counterexamples threaten to leave realism with no coherent formulation. There is no similar threat to conceptualism and nominalism. Since they claim that there are only objects with spatiotemporal locations, they can be formulated without the concept of an abstract object. To be sure, conceptualists and nominalists still have the problem of accounting for such things as types, but that does not make those positions unformulatable. Generally speaking, philosophers have accepted the alleged counterexamples as genuine and tried to revise the notion of abstractness to allow abstract objects to have some sort of spatial or temporal location. I think that this approach is both unsatisfactory and unnecessary. It leads to highly unintuitive concepts of the abstract, since there is no getting around the basic intuition that asking where or when questions about abstract objects such as universals, numbers, sets, and types involves some kind of category mistake. Further, the realist position as we understand it here seems unformulatable on any of the contemplated revisions of the notion of abstractness. Thus, before we say the alleged counterexamples are real counterexamples, we ought to be sure that there is no other way to handle them. I think that such a more prudent approach will show that nothing compels us to take the apparent counterexamples at face value. I will pursue this approach in the present chapter. I will conclude that none of the counterexamples is real. Nonetheless, the attitude I take toward the examples is not dismissive. Quite the contrary. I think the alleged counterexamples have much to teach us, though not what those who have taken them at face value and sought a new abstract/concrete distinction have supposed. I will argue that the traditional distinction is basically sound, though, as do other traditional distinctions in philosophy, it needs signiªcant reconstruction. 5.2 Ontology This section sketches the conception of ontology underlying our reconstruction of the traditional abstract/concrete distinction. The sketch is intended to make it easier to understand the thinking about ontological issues in this chapter and also to see how this chapter together with the previous chapters and the ªnal chapter form an integrated rationalist/realist position. Except for one minor argument occurring at the very end of this chapter, nothing important in the defense of the traditional abstract/concrete distinction depends on acceptance of our conception of ontology. On our conception, pure ontology is a foundational discipline of foundational disciplines. The various sciences take one or another

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aspect of the world as their subject matter. Their investigations tell us of the existence and structure of such things as numbers, propositions, sentences, supernovae, atoms, and genes. The foundations of mathematics, of logic, of physics, and of the other sciences take those sciences as their subject matter and investigate the nature of the sciences and of the objects they study. Pure ontology concerns the foundations of the sciences. It studies the philosophical concepts that are used to express the nature of the sciences and the objects they investigate.1 This conception is not a foundationalism. Since Wittgenstein (1956, 171), the standard complaint against a foundationalist view of mathematics is that mathematics does not stand in need of philosophical foundations. The results of mathematics are grounded in mathematical practice and require no other grounding. I have no wish to quarrel with this view. As I see it, philosophy is not necessary to provide the underpinnings for mathematical practice, but only to provide understanding of it. The foundations of the sciences are intended to answer philosophical questions (e.g., what kind of objects are numbers and sets?) that arise about the sciences. The aim of the philosophical foundations of a science is to shed light on ontological and epistemological aspects of the objects the science studies, and not, as indicated at the end of chapter 4, on aspects of those objects with which the science itself is concerned. (Mathematics may require foundations to deal with the discovery of a paradox, but foundations in this sense are mathematical foundations, though they may be accompanied by considerable philosophical commentary.) Gödel’s mathematical realism, Brouwer’s mathematical conceptualism, and Hilbert’s mathematical nominalism (and their counterparts in the foundations of logic and linguistics) present different conceptions of the nature of the reality studied in the formal sciences. The foundations of mathematics, logic, and other formal sciences are concerned inter alia with whether the reality studied in those sciences is abstract or concrete. As the foundational study of such foundational disciplines, pure ontology is concerned inter alia with the nature of the properties of abstractness and concreteness, with understanding what it is to be abstract or concrete. This chapter is concerned with such understanding. The focus is on the deªnitions of the categories Abstract and Concrete, but, as those categories are systematically related to other ontological categories, their deªnitions have to mesh with the deªnitions of the related 1. The ontological argumentation that goes on in the foundations of a science about the issue of whether the objects of the science are abstract or concrete is not pure ontology as the term is used here.

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categories. Thus, what is required are deªnitions that form a system of ontological categories that speciªes the categories belonging to the system in an intuitively satisfying way and accounts for their relations to one another in virtue of which they form a division of what there is at the most general level of conceptual structure. To arrive at such a system of deªnitions, we pursue the customary philosophical practice of seeking deªnitions that provide the best insight into concepts and their role in philosophical investigations. The concern is, therefore, not exclusively with what the terms “abstract” and “concrete” mean in our language, though such semantic considerations can be expected to play a role in determining a system of ontological deªnitions. The emphasis is on discovering concepts of abstractness and concreteness that provide the most philosophically illuminating account of the ontological positions in the foundations of the sciences. 5.3 The Traditional Abstract/Concrete Distinction We now have to be more explicit about what is intended in the traditional distinction, because aspects of it typically left implicit or not sufªciently emphasized play an important role in blocking the alleged counterexamples. One implicit aspect of the traditional distinction that needs ºeshing out is that the notions of location in space and location in time are intended to be taken as broadly as possible. They are intended to refer to any spatial or temporal location, not necessarily a ªxed, precise, or well-circumscribed one. Spatial or temporal location, however loose, imprecise, or ill-circumscribed, sufªces for concreteness. Given this broad construal, having (lacking) spatial and temporal location comes to the same thing as having (lacking) spatial and temporal properties. This raises the familiar question of whether someone’s having the property of thinking about the number seventeen at time t means that the number seventeen has the property of being thought about at that time. Some philosophers try to avoid this problem by distinguishing between properties with an intentional component and those without one. I think that the best way to deal with the problem is to argue that the temporal property is not applied to an abstract object in such cases but to something else. It does not follow from the fact that the number seventeen stands to someone in the relation of being thought about by them at t that seventeen has a temporal property. What is the case is that two facts obtain: someone has a thought at t and the thought is about the number seventeen. Here the

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temporal designation expresses the time at which the person has the thought, but it has no application beyond the ªrst fact. It is not constitutive of the relation of the thought to its object. That relation is an atemporal correspondence relation like truth. The situation is the same for other temporal indicators, as in “John frequently (rarely, intermittently, always, never) thinks about the number seventeen” and also for spatial indicators, as in “John thinks about the number seventeen in the shower (on the subway, at the seashore)”. The temporal or spatial designation in such sentences expresses a parameter of the thinking, telling us how often, when, or where it takes place, but says nothing about the object of the thought. Abstract objects have a variety of formal or intrinsic properties and relations, that is, properties and relations that are aspects of their mathematical, logical, and linguistic structure. Mathematical, logical, and linguistic objects subclassify (e.g., into arithmetic, set-theoretic, and geometric objects) in virtue of the kind of mathematical, logical, and linguistic structure they have. Numbers are mathematical abstract objects because they have an arithmetic structure, which is a kind of mathematical structure, and sentences are linguistic abstract objects because they have a syntactic structure, which is a kind of linguistic structure. Harking back to chapter 3, section 5, mathematical, logical, and linguistic objects have their mathematical, logical, and linguistic properties and relations necessarily because they are intrinsic properties and relations of abstract objects. It is implicit in the traditional abstract/concrete distinction that the abstract has no taint of the concrete and the concrete no taint of the abstract. An abstract object can have nothing about it that is spatially or temporally locatable, and a concrete object can have nothing about it that is neither spatially nor temporally locatable. One indication that abstract objects are understood in this way is the (misguided) practice of assigning them to a world separate from the sensible world of concrete objects. A further indication of this understanding is that their untainted abstractness is what makes abstract objects anathema to empiricists and ontological naturalists. It is precisely this pure abstractness that conºicts with their view that knowledge depends on causal connection. I shall call this feature of abstract and concrete objects homogeneity. I shall say that an object is homogeneous when, roughly speaking, its constituents each have the same ontological nature as the object itself. Abstract objects are, as it were, abstract all the way down, and concrete objects are concrete all the way down. The constituents of an object are objects out of which the object is constituted, so that every constituent

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of a constituent of an object is a constituent of the object.2 The constituents of a structured object are the objects between which the relations in its structure hold (e.g., the constituents of a sentence are all the phonemes, morphemes, words, phrases, and other linguistic constituents which make up the sentence). We can thus say that an object O is homogeneous with respect to the concept C if and only if, for any atomic object, it is homogeneous with respect to C just in case the object falls under C, and, for any structured object, it is homogeneous with respect to C just in case it and all its constituents fall under C. If an object is not homogeneous with respect to C, it is heterogeneous with respect to C. Numbers are homogeneous with respect to the concept “abstract” for the realist and homogeneous with respect to the concept “mathematical” for everyone. Also, for Descartes, animals and our bodies are homogeneous with respect to the concept “matter” and the ego is homogeneous with respect to the concept “mind.” Some concrete objects can have a discontinuous existence. A car can be taken down to its minimal automotive parts and reassembled. If natural events are taken as concrete, the same is true of them. A football game does not go on during half time, but it resumes after half time, and it is constituted by the play in both halves. Since the question of homogeneity does not arise when there is only a collection of automotive parts and at half time, that is, when there is nothing to locate, a natural way to understand “homogeneous with respect to C” is to understand it to mean that the things it applies to are homogeneous with respect to C during every phase of their existence. One can, of course, say that the car is homogeneous with respect to the concept “physical” even when it is disassembled, because all of its constituents are physical, but this does not provide a general condition, since one cannot say a similar thing in the case of events. None of the constituent events of a football game (e.g., screen plays, instances of interference with a receiver, or penalties for unnecessary roughness) occur during half time. Finally, a feature of the traditional distinction that also needs ºeshing out is the status of the modal in the statement that abstract objects cannot have either spatial or temporal location while concrete objects must have one or the other kind of location. I think that it is clear that the intention is for the modals here to be taken to express metaphysical necessity. The intention can be expressed by saying that an abstract 2. The reader may assume for the sake of convenience that division into constituents stops with atomic objects, which have no constituents, but there is no reason for us to impose this as a condition and to do so would exclude non-well-founded set theory (i.e., set theory dealing with inªnitely descending membership sequences).

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object has no spatial or temporal location in any possible world where it exists and that a concrete object has either spatial or temporal location in every possible world where it exists.3 But what the intention is concerning the source of the necessity is less clear. Presumably, the source of the metaphysical necessity is either logical or semantic. In the former case, an attribution of a temporal or spatial property to an abstract object is meaningful but logically false, while in the latter, the attribution is not (fully) meaningful (i.e., sentences expressing such an attribution express no proposition). There are considerations favoring a semantic source. Intuitively, we want to distinguish sentences like (1) from sentences like (2) and (3). (1) The number seventeen spent a week visiting its mother in Brooklyn. (2) The number seventeen is prime and is not prime. (3) The number seventeen is even. The latter are (logically or mathematically) false. They express necessarily false propositions; hence they express propositions; and hence they are meaningful. But (1) commits a kind of category mistake. It is not meaningful, and hence does not express a proposition, and hence is not (logically or mathematically) false. Again, (4a) is doubly contradictory. (4a) The number seventeen is even and is not prime. (4b) The number seventeen is even and spent a week in Brooklyn. In this respect, it is repetitive. (4b) isn’t. It also commits some kind of category mistake. Unlike (4a), (2), and (3), where the predicate applies to the subject but is incompatible with it, (4b) and (1), because of their application of the temporal and spatial predicates to their abstract subjects, are deviant. This difference, if sustainable, supports a preference for a semantic source of the metaphysical necessity. However, nothing in our account of the abstract/concrete distinction makes use of more than the fact of metaphysical necessity, and hence we can leave the question of what explains this fact open. 3. Possible worlds are used in this chapter as in chapter 2 because of their convenience as an expository device, but neither the reconstruction of the abstract/concrete distinction in this chapter nor the epistemology in chapter 2 depends on a possible worlds explication of necessity and possibility. See Kripke (1980, 15–20) for an explanation of the use of possible worlds terminology.

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We can reformulate the traditional abstract/concrete distinction in terms of the deªnitions in (D): (D) An object is abstract just in case it lacks both spatial and temporal location and is homogeneous in this respect. An object is concrete just in case it has spatial or temporal location and is homogeneous in this respect. The existence of an abstract object is existence without spatiotemporal location, while the existence of a concrete object is existence at a spatial or temporal location. 5.4 Preliminary Reºections on the Adequacy of (D) Traditional realists characterize abstractness and concreteness on the basis of properties in addition to having or lacking spatiotemporal location. This raises the question of whether (D) is too narrow in failing to include or entail essential properties of abstractness and concreteness. In this section, I will argue that the essential properties in question can be derived from (D) together with some generally accepted assumptions, and also that certain properties wrongly taken to be essential are not similarly derivable. Given this, (D) can be taken to be the most economical way of explicitly stating the traditional abstract/ concrete distinction. 5.4.1 Plato characterized the Forms as atemporal, incorporeal, nonsensible, and transcendent, and correspondingly characterized concrete objects as temporal, corporeal, sensible, and immanent. It is not clear whether all of these properties were intended to be deªnitional or whether only some were deªnitional and others were intended to be just necessary. In any case, the fact that, except for the atemporality of abstract objects and the temporality of concrete objects, Plato’s characterization of abstractness and concreteness goes beyond that in (D) obligates us to argue that each property in Plato’s characterization not mentioned in (D) can be shown to follow from (D) (with acceptable assumptions), or else shown not to be a necessary aspect of the distinction in the sense of (D). The incorporeality of abstract objects follows immediately from the fact that a body is essentially something spatially extended, something located in the space marked by the limits of the material that constitutes it. The corporeality of concrete objects would follow too if all concrete objects were bodies (or dependent aspects of them, e.g., shadows and rainbows). But concrete objects do not have to be bodies. Cartesian egos

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and thoughts are concrete because they have temporal position, but they are not bodies, because they have no spatial position. Thus, it would be a mistake to regard corporeality as an inherent feature of concrete objects, even if one is a materialist on the mind/body problem. An ontology that says that all concrete objects are corporeal would preclude concrete objects that are mental in the Cartesian sense, and hence make the classical mind/body problem unstatable. In deªning concrete objects as those with either temporal location or spatial location or both, (D) avoids an ontology with such unacceptable consequences. The property of being nonsensible is surely an essential property of abstract objects. To do justice to the fact that spatiotemporal creatures like ourselves cannot causally interact with objects that are not in space-time, it is best to express the property as the modal property of being necessarily inaccessible to sensory reception. The counterpart property of being sensible is, correspondingly, the modal property of being possibly accessible to sensory reception. Let us make the (uncontentious) assumptions that sensory reception begins with the impingement of stimuli on the surface of a sensory organ and that impingement must occur at some time and place. Given that sensory reception presupposes spatiotemporal position, abstract objects are necessarily inaccessible to sensory reception while concrete objects are possibly accessible to sensory reception. These properties follow from (D). The properties of being sensible and being nonsensible are special cases of the more general properties of causal activeness and causal inertness. Abstract objects have customarily been taken to be necessarily causally inert and concrete objects to be possibly causally active. Assuming that the concept of causation involves a schema that temporally locates cause and effect with respect to each other, there are necessarily (even in the case of instantaneous causation) temporal properties and relations of the cause and its effect, and hence the customary view about the causal status of the abstract and the concrete follows from (D).4 Early in the development of the theory of Forms, Plato seems to have held a “two world” view. In addition to the world we inhabit together with other spatiotemporally located and causally interrelated things, there is an independent world of nonspatial, atemporal, and causally inert Forms. This view explains the objectivity and autonomy of the concepts examined in dialectic by taking the abstract to be transcendent (dwelling outside our world) and the concrete to be immanent 4. Backwards causation does not preclude the causal inertness of abstract objects and the causal interactiveness of concrete objects from being consequences of (D).

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(dwelling in our world). Plato’s account of how we know the Forms would seem to commit him to the “two world” view. However, realists in our sense should not subscribe to it. The transcendence of abstract objects is not something to which (D) forces realists to subscribe, although with plausible assumptions about the natural world the immanence of concrete objects may be a consequence of (D). Plato’s epistemology may commit his realism to a “two world” view, but clearly ours doesn’t commit our realism to one. Our epistemology allows the realist to take a “one world” view on which the totality of all things includes those that fall under the ontological category Abstract as well as those that fall under the ontological category Concrete. Moreover, to subscribe to the transcendence of abstract objects would hand our world over to the naturalist without a ªght, making realism harder to defend because, in order to argue for the existence of abstract objects, the realist would then have to argue ªrst for the existence of another world. Furthermore, the antirealists can quite reasonably deny the possibility of a world other than ours. They can argue that the world is everything there is viewed as a whole, and since ex hypothesi it contains no abstract objects, there are no such objects. (In section 5.7, we shall see that realism needs to reject the “two world” view in order to handle the second class of putative counterexamples to the abstract/concrete distinction.) Hence, we reject transcendence as a property of abstract objects. 5.4.2 It is instructive to compare (D) with Dummett’s (1973, 471–511) explication of “concrete” and “abstract.” On his explication, concrete objects are those objects to which we can refer on the basis of an ostensive act, such as pointing, together with a verbal cue to indicate the target of the gesture. In contrast, abstract objects are those that cannot be so referred to, but must be picked out using an expression containing terms standing for abstract objects, e.g., “the number of bagels” and “the shape of that bagel.” Noonan (1976) and Hale (1987) bring up counterexamples to Dummett’s explication. But I want to look at what I believe is a deeper conceptual difªculty, namely, that Dummett’s explication does not stand on its own but presupposes the traditional abstract/concrete distinction or something like it, and hence does not provide an alternative to our explication (D). Intuitively, Dummett’s criterion for concreteness puts the relation the wrong way around: objects are not concrete because they can be referred to ostensively: they can be referred to ostensively because they

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are concrete. Ostension is directing the attention of an audience to an object via placing a ªnger on it, indicating a line from a pointer to it, or in some other way putting it on salient display. For such directing of attention to make sense, there must be a spatial position (or spatial positions) that is occupied (or successively occupied) by the target of the ostension. Because the Empire State Building is there at Fifth Avenue and Thirty-fourth Street, it can be the target of an ostension that directs attention to that location. Correspondingly, abstract objects cannot be referred to ostensively because there is nowhere they are at. Hence, unless it is already assumed that concrete objects can occupy spatial position and abstract objects cannot, we would be unable to claim that the former but not the latter can be referred to ostensively. Hence, instead of replacing the traditional notion of the concrete, Dummett’s ostensive criterion presupposes it. There is another sense in which Dummett’s account depends on something like the traditional abstract/concrete distinction. In making reference to terms in the language that stand for abstract objects, the account presupposes a prior semantics for those terms according to which their senses say enough about what it is to be an abstract object for them to have an extension consisting of abstract objects. Dummett’s account of the abstract appears unobjectionable as long as we don’t ask about the concepts that express the condition something has to satisfy for it to be the referent of a term like “seventeen.” Once we see that it is the concept of an object having neither spatial nor temporal location that does the semantic work here, we see that Dummett’s criterion does not provide an alternative to the traditional abstract/concrete distinction because it depends on a concept of abstractness of the sort that the traditional distinction provides. 5.5 Modes of Existence The condition for the possible existence of concrete and abstract objects is that their full description be consistent in itself and consistent with the relevant theoretical truths about other objects of the same kind.5 What about their actual existence? Since, as far as possible, we want our account of the existence conditions for objects to ºow from the 5. In connection with concrete objects, the notion of a kind is customarily taken to be a matter of categorization in the natural sciences. In connection with abstract objects, the notion of a kind is similarly a matter of categorization in the formal sciences. Extensionally, a kind, e.g., number, ªgure, or sentence, encompasses all the abstract objects the existence of which involves the existence of each of the others within a common system, e.g., the number system, Euclidean geometry, and English grammar.

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deªning features of the category to which they belong, the existence conditions for concrete objects ought to be (EC). (EC) An object O exists in a possible world W (at time t) just in case O has a spatial or temporal location in W (at time t) (EC) does not, however, tell us what the mode of existence is for concrete objects. It is plausible to say that concrete objects exist contingently, i.e., for any concrete object, there is a possible world in which that object exists and a possible world in which it does not exist. Glenn Branch (personal communication) points out a theological complication. God, being eternal and the creator of the world, has temporal properties, and hence, according to (D), is concrete; yet if, as is presumed, God’s existence is necessary existence, He is not a contingent being, and hence not concrete. So there is room for doubt about whether concreteness entails contingency on the grounds that God is concrete but not contingent. The complication might be gotten around by arguing that the conception of God as a concrete object is paradoxical. It is essential to God that He created the universe, but He couldn’t have done that without creating space and time. But if He created them, He is their cause and has a location prior to their creation. But it doesn’t seem possible for Him to occupy a temporal position prior to creation: time would have existed already and God could not have created it. Moreover, (D) requires that a concrete object have spatial or temporal location during its entire existence, but this means that either God is not a concrete object or He couldn’t have created the universe. Of course, on other conceptions of God, his eternality is timelessness; that is, God is outside time. Anselm (1962, 71) says, “But [Thou], although nothing exists without thee, nevertheless dost not exist in space or time, but all things exist in thee.” This works against the objection to the thesis that concreteness entails contingency, since, in this case, God is, by deªnition, not a concrete object. There is an independent reason for thinking that God cannot be concrete. There are possible worlds without concrete objects. The fact that Plato’s world of Forms contains no concrete objects does not show it to be impossible. Since the objection we are considering assumes that God exists necessarily, God exists in Plato’s world of Forms, too. Since nothing in that world is concrete, God cannot be concrete. God is perhaps best left to the theologians. Whatever one believes about supernatural concrete objects, natural objects are contingent, and hence there is no way to tell a priori whether the actual world is one of the possible worlds in which there are particular natural objects such

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as gorillas with a taste for bananas. To tell whether such things exist in the actual world, a posteriori investigation is necessary. What about the mode of existence of abstract objects? Given that concrete objects exist contingently, symmetry considerations suggest that abstract objects exist necessarily. Concrete objects have their natural properties and relations contingently and exist contingently; abstract objects have their formal properties and relations necessarily and exist necessarily. The deªnitions of “concrete object” and “abstract object” reinforce this suggestion. (D) says that concrete objects are the kind of objects that have spatial or temporal location, while (EC) says that their existence consists in their occupying a spatial or temporal location. Since the deªnition of “abstract object” says that abstract objects are the kind of objects that have no spatial or temporal location, occupancy of a spatial or temporal location can have nothing to do with the existence of abstract objects. Since, furthermore, this negative clause, except for the homogeneity requirement, is the entire deªnition of “abstract object” and abstract objects have all their formal properties necessarily, there is no non-formal condition they have to satisfy to exist. Neither of the two possible sources for such a condition, the deªnition of “abstract object” and differences in the formal properties and relations of such objects across possible worlds, provide content for it. Hence, there is nothing for the existence of abstract objects to be contingent on. This, however, doesn’t seem to be enough for realists to claim that abstract objects exist necessarily. Possible worlds differ with respect to what is actually in them, but not with respect to what is possibly in them. Possible worlds do not differ with respect to the possibility of containing a golden mountain, the number seventeen, or a talking dog, nor, alternatively, with respect to the impossibility of a largest integer, a round square, or a colorless color. Thus, for any consistent conception of objects of a certain kind, objects of that kind are a possibility in all worlds. For something to exist in a world is for there to be something in the world that instantiates the appropriate concept. Since the possibility of something in a world is simply the existence of a consistent concept of it, the existence of possibilities is the same as the existence of concepts capable of instantiation. For example, the existence of the possibility of a golden mountain in a world is the existence of a concept of a golden mountain for something in the world to fall under.6 6. Since all objects that are not necessarily nonexistent are possibilities in every world, the concepts of those objects exist in every world. Hence, the senses of “golden

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Now, consider statements formed by substituting a term “t” with a sense s (which expresses a possibility) into the schema “Possibly necessarily x exists.” If s is inherently inconsistent (e.g., s is the sense of “the round square”), then “Possibly necessarily s exists” is false. If “t” is a concrete term such as “ships,” “shoes,” “sealing wax,” “cabbages,” or “kings,” then the statement “Possibly necessarily s exists” is also false, since the statement claims that it is possible that something contingent exists necessarily. In contrast, if “t” is a term the sense s* of which picks out abstract objects in the sense of (D), there is no corresponding inconsistency. Since it is possible that the objects s* picks out exist in every possible world, the proposition “Possibly necessarily s* exists” is true. Unlike round squares or concrete objects such as ships, shoes, sealing wax, and so on, it is possible that abstract objects such as the number seventeen exist in all possible worlds. Given that realists accept a modal logic at least as strong as S5, they can conclude “Necessarily s* exists.” (Note that, since realists claim that seventeen exists in the actual world, they also claim that it is not possible that seventeen necessarily does not exist, and, consequently, there is no parallel argument from “Possibly necessarily seventeen does not exist” to “Necessarily seventeen does not exist.”)7 In his deªnitive study of Frege’s philosophy of mathematics, Dummett (1991, 307–308) makes a similar point about the necessary existence of certain abstract objects. Dummett (1991, 307) compares the existence of a system of mathematical objects to the existence of God (presumably a being who is all-powerful, perfectly good, and the creator of the world), saying “. . . one may believe in it or disbelieve mountain,” “the number seventeen,” “talking dog,” and every other consistent concept exist in every possible world. Since those concepts are ex hypothesi abstract objects, abstract objects of one kind have necessary existence. Moreover, since consistent senses belong to the semantic kind of abstract objects, and since, as we have assumed, the existence of an abstract object of some kind presupposes the existence of the other objects of that kind, we can infer that the sense of “largest integer,” “round square,” and other inconsistent expressions also exist in every possible world. 7. This argument for the necessary existence of abstract objects would be rejected by philosophers like Armstrong (1989, 80–81) who claim that the fact that it is wrong to argue from possibility to existence in the case of concrete objects shows that it is also wrong so to argue in the case of abstract objects. Armstrong bases this claim on the supposition that there is nothing special about abstract objects that would sanction the inference in the latter case when it is not sanctioned in the former. But this misses the distinction that the realist can draw between knowledge of the existence of concrete space-time objects, which requires empirical evidence, and knowledge of the existence of abstract objects, which does not. We can’t establish a priori the existence of gorillas with a passion for bananas or the nonexistence of a king of France, but we can establish a priori the existence of Cantorian transcendental numbers and the nonexistence of a largest integer.

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in it, but one cannot intelligibly say that it exists but might not have, or does not exist but might have done.” However, he bases his point on Frege’s context principle, saying that It must . . . be from the possibility of our knowing [the] existence [of a system of mathematical objects] a priori that the necessity of its existence derives; and this entails that the coherence of the conception of the system is sufªcient, in light of the context principle, to justify the assertion of its existence. Our argument shows that we do not have to rely on Frege’s context principle. This is fortunate, since that principle is inadequate. In the Grundlagen, Frege (1953, 71) states the principle: “It is enough if the sentence as a whole has a sense; it is through this that its parts obtain their content.” This principle conºicts with compositionality. Roughly speaking, compositionality is the principle that the sense of a sentence is a function of the senses of its syntactic constituents and their syntactic relations.8 How can the sense of a sentence be built up from the senses of the words in it if those words only “obtain their content” top-down from the sense of the sentence? Words have to be independently meaningful in order to supply the senses that are the “building stones” of thoughts. Given the intuitive persuasiveness of compositionality (e.g., the contrast of idiomatic and non-idiomatic senses, as of “kick the bucket,” demonstrates the existence of compositional structure), it is the context principle that has to go (see Katz (in preparation)). Another aspect of Dummett’s position on abstract objects is also in stark contrast to ours. Although he (1991, 308) thinks that mathematical objects exist of necessity, he does not think that all abstract objects do: “By no means all abstract objects exist of necessity: the Equator does not, for one.” I do not think that this is a coherent position, since nothing in Dummett’s argument for the necessary existence of abstract objects (or mine) depends on anything more than the abstractness of an abstract object. The kind of an abstract object it is does not enter into the argument. The equator of the earth does not, to be sure, exist necessarily, since its existence depends on the existence of the earth, 8. Since compositionality is a principle about abstract senses and abstract syntactic forms, it should not be understood in psychological terms or be justiªed on the basis of psychological considerations like the speaker’s ability to understand novel sentences. The justiªcation of compositionality comes from the fact that it expresses the dependency of sense properties and relations of syntactically complex expressions on the sense properties and relations of their constituents and syntactic structure. Hence, criticisms of compositionality that understand it or its justiªcation psychologically are irrelevant here.

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and hence it has a temporal history, i.e., it came into existence with the formation of the earth and will cease to exist when the earth ceases to exist. Furthermore, the equator is located in our solar system and many people cross it each year. Hence, the conclusion that Dummett should have drawn is that the equator is not an abstract object. Why would Dummett take it to be an abstract object? I think that there are two possible reasons. First, Dummett might have mistakenly taken Frege’s explanation in the Grundlagen of why number is not subjective to have shown that the equator is an abstract object. Frege (1953, 35) says: I distinguish what I call objective from what is handleable or spatial or actual. The axis of the earth is objective, so is the centre of the solar system, but I should not call them actual in the way the earth itself is. We often speak of the equator as an imaginary line; but it would be wrong to call it an imaginary line in the dyslogistic sense; it is not a creature of thought, the product of a psychological process, but is only recognized or apprehended by thought. If to be recognized were to be created, then we should be able to say nothing positive about the equator for any period earlier than the date of its alleged creation. But Frege does not even use the equator as an illustration of an abstract object, only as an illustration of an objective one. The equator is not like a child’s imaginary friend. It belongs in the category of things outside of us—things that do not depend for their existence on us and to which we cannot have the unique epistemic relation we have to our own ideas. But that category is not exhausted by things the are “actual in the way the earth is” and things that are abstract in the way numbers are. Frege doesn’t present the equator as an abstract object. Second, Dummett might take the equator to be an abstract object because it has mathematical properties; for example, it has all the mathematical properties of a circle. This, of course, is why the equator is one of the alleged counterexamples to the traditional abstract/concrete distinction. I will explain below why having such mathematical properties does not entail that the equator is an abstract object, and what kind of object it is. 5.6 The First Class of Putative Counterexamples The putative counterexamples to the traditional abstract/concrete distinction divide into two classes. The ªrst includes examples like games such as chess and checkers, natural languages, sentence types, signiªcant utterances, speeches, poems, and novels. Hale, for example,

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claims that games and natural languages, although abstract objects, nonetheless have temporal location. He (1987, 49) writes: . . . it seems an undeniable fact that games, unlike shapes or directions, are invented or devised, and so come into existence at a particular time; similarly, for languages. And it is hard to see any reason for denying them abstract status that is not simply ad hoc. In this section, I will argue that the examples in the ªrst class fail as counterexamples because they are abstract objects pure and simple. Such examples are mistakenly thought to have temporal properties because the realist position is not understood broadly enough. The second class includes the earth’s equator and impure sets such as the set of my children. In the sections to follow, I will argue that the examples in the second class fail to be counterexamples because they are not abstract objects. They are rightly thought to have temporal and spatial properties. Such examples are mistakenly taken to be abstract objects because ontology is not conceived broadly enough. Peirce’s (1958, 423) token/type distinction provides a reason for thinking that linguistic objects such as words and sentences are abstract objects: There will ordinarily be about twenty “the”’s on a page, and of course they count as twenty words. In another sense of the word “word,” however, there is but one “the” in the English language; . . . it is impossible that this word should lie visibly on a page or be heard in any voice. Quine (1987, 216–17) reiterates and clariªes Peirce’s explanation of linguistic types and tokens: ES IST DER GEIST DER SICH DEN KÖRPER BAUT: such is the nine word inscription on a Harvard museum. The count is nine because we count der both times; we are counting concrete physical objects, nine in a row. When on the other hand statistics are compiled regarding students’ vocabularies, a ªrm line is drawn at repetitions; no cheating. Such are two contrasting senses in which we use the word word. A word in the second sense is not a physical object, not a dribble of ink or an incision in granite, but an abstract object. In the second sense of the word word it is not two words der that turn up in the inscription, but one word der that gets inscribed twice. Words in the ªrst sense have come to be called tokens; words in the second sense are called types.

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Since the type/token distinction is as uncontroversial as any philosophical distinction gets, the realist can safely take the words and sentences of a language, as opposed to their utterances and inscriptions, and also languages in the sense of collections of sentences, to be abstract objects. Moreover, philosophers on both sides of the issue about uninstantiated universals accept the distinction. Armstrong (1989, 1–20) uses it to set up the problem of universals; I (1990b, 275–80; 1996c) use it to argue for linguistic realism. Games such as chess, as we shall see, can also be taken to be abstract objects. But if linguistic types and games really are abstract objects, they couldn’t come into existence at some time, have a history, or be found in one location or another. They couldn’t be counterexamples to (D)— they could only appear to be. Those who take them to be actual counterexamples must therefore be overlooking something. What they are overlooking, I submit, is how a comprehensive realist position interprets what it means to say that sentences, languages, and games have temporal and spatial properties. The mathematical realist holds that mathematical knowledge is the discovery of mathematical facts rather than the creation of them. Cantor discovered the transªnite numbers every bit as much as Madame Curie discovered radium. Both Cantor and Curie came to know about something that already existed. Thus, if chess is a trivial sort of mathematical system, as G. H. Hardy (1940) among others has claimed, chess could not have come into existence at any point in time or have its locus in any region of space. Since on their view mathematical systems are discoveries, realists will say that talk about chess’s having originated at some time and in some place should be interpreted to mean that the discovery of chess occurred at that time and place. Failure to attend to the realist’s distinction between discovery and creation results in confusing the (coherent) notion of the history of the discovery of an abstract object with the (incoherent) notion of the history of the object. In the case of the Eiffel Tower, the Mona Lisa, Paris, or any other genuinely created object, there is a history of both the creation and the created. But, in the case of abstract objects such as chess defenses or proofs of the inªnity of prime numbers, there is only a history of the discovery. The discovered object is an independently existing entity of which it is true to say at any time prior to the discovery that it exists. There is no such thing as a history of pi. Books like Petr Beckman’s The History of Pi and Eli Maor’s e: The Story of a Number chronicle discoveries, not inventions. The fact that Ludolph van Ceulen computed the approximation 3.14159+ in the sixteenth century is a piece of history concerning how we have come to know what we know about pi. Hence, the historical events of the sort put forth as

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counterexamples to the abstract/concrete distinction do not show that numbers, games, and other mathematical abstract objects have temporal or spatial properties. Linguistic realism handles the case of sentences and languages in essentially the same way. Katz (1981) and Katz and Postal (1991) have put forth a linguistic realism on which natural languages and their sentences are abstract objects. On this position, the study of a natural language is the study of the structure of its sentence types, in contrast with Chomsky’s (1965, 1986) linguistic conceptualism, on which the study of a natural language is the study of the linguistic knowledge of its speakers (their linguistic competence). On linguistic conceptualism, linguistics is a branch of psychology or biology, while, on linguistic realism, it is a branch of mathematics, as independent of psychology and biology as number theory. This independence is based on the distinction between the speaker’s (tacit) knowledge of a language and the language that the knowledge is knowledge of. This distinction is the logical next step in the process Chomsky began when he (1965, 3–18) ªrst distinguished between knowledge of a language (competence) and exercise of such knowledge (performance). Using this distinction, Chomsky argued contra Bloomªeld’s linguistic nominalism that linguistics is the study of linguistic competence rather than performance. Using our further distinction between knowledge of a language and the language it is knowledge of, linguistic realists argue that linguistics is the study of natural languages rather than (tacit) knowledge of them, as in Chomsky’s linguistic conceptualism. To claim that the objects of study in linguistics are natural languages is no more to deny that there is a study of the speaker’s knowledge of the language than Chomsky’s claim that the objects of study in linguistics are competences is to deny that there is a study of performance. The claim only says that such a psychological study is distinct, at least conceptually, from a grammatical study of natural languages. Insofar as the two studies are distinct, there needs to be some reason for equating them. There is no more of an a priori reason for identifying the subject matters of these studies than there is for identifying the mathematically ºuent calculator’s knowledge of the numbers with the number system itself, or the logically ºuent reasoner’s knowledge of logical implication with logical implication itself. There is no a posteriori reason either, since empirical ªndings can tell us about the speaker’s psychology but not whether language can be reduced to an aspect of that psychology. We can use the realist’s further distinction to make essentially the same response to the claim that natural languages and their sentences

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have a history as was made to the claim that games such as chess have a history. We can say that the claim about linguistic abstract objects fails to distinguish knowledge of a language and the language it is knowledge of. Talk about a language coming into existence, changing, or disappearing is to be understood as talk about changes in the linguistic knowledge of the members of a speech community (see Katz 1981, 7–9). The picture is this. As the result of largely unknown social and biological factors, aspects of the (tacit) grammatical knowledge of the members of a speech community change over time. The members of the community have one competence prior to the change and another competence—or a signiªcantly altered form of the original one—afterwards. Such changes may take place relatively quickly or slowly; they may be large or small. Some kinds of changes, such as the disappearance of a syntactic construction, usually occur rarely, while others, such as the addition of a word, are occurring all the time. These matters are the province of diachronic linguistics. On a realist view, diachronic linguistics is a study of the development of the competence of a community of speakers from knowledge of one language (or dialect) to knowledge of another. Diachronic linguistics is the sociological counterpart of developmental psycholinguists (which studies the infant’s transition from its preverbal stage to mastery of the language). On this picture, what happens in “language change” is that the members of a speech community acquire knowledge of a different natural language (or dialect) from the one that they had knowledge of prior to the change. The original competence, C, is knowledge of a language (or dialect) L, namely, the one with the properties that C represents its object as having. The resulting competence, C′, is knowledge of the language (or dialect) L′—namely, the one with the properties that C′ represents its object as having. Hence, what happens in language change is that the members of a community end up with the new competence C′. But, in virtue of having C′, they stand in the knowledge of relation to an abstract object L′ different from the abstract object L to which they stood in the knowledge of relation prior to the change. Being nothing more than a change in the epistemic states of speakers, linguistic change alters only the extrinsic knowledge of relation between the speakers in the community and the abstract objects in the realm of natural languages. With, as it were, no effort on the part of a natural language, it can at one time be the language that the speakers of a community know and at another time not be. Just as there is no intrinsic change in the number seventeen when someone stops thinking of it and starts thinking of the number eighteen, but only a new extrinsic relation between the person and the realm of numbers, so

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there is no intrinsic change in a natural language when the speakers of a community cease having knowledge of it and begin having knowledge of another language, but only a new extrinsic relation between the community and the abstract languages. What we call the history of a language is really the history of a language community, beginning when its members ªrst acquire knowledge of the language and ending when they no longer have it. We do not, of course, speak of discovering a language, but this does not show that the process is not one in which the language community acquires knowledge of something already existing. The oddity of “discover” in this context is due, I think, instead to the fact that the change that takes place when knowledge of a language emerges is very slow, a collective affair on the part of a whole people, and entirely tacit in comparison to the change that takes place when knowledge of a game such as chess or a number such as pi emerges. It is a process in which something previously (tacitly) unknown becomes (tacitly) known. When the process is quick, the participants are few, and, most importantly, the knowledge is explicit—as in the discovery of transªnite numbers or non-Euclidean geometry—the use of “discover” is not at all odd despite the fact that what is discovered is something abstract. We now have a full response to the allegation that games such as chess, languages, and sentences are counterexamples to the traditional conception of the abstract/concrete distinction. The mathematical realist uses the distinction between discovery and creation to handle the case of games. The mathematical realist can say that talk about the history of chess is ªgurative, referring to the history of the knowledge of chess on the part of a community. Extending that distinction to language, the linguistic realist can use it for the case of sentences and languages. The linguistic realist can say that talk about the history of a language is similarly ªgurative, referring to the history of the knowledge of the language on the part of members of a community. Hale (1987, 49) says that a position like ours . . . squares rather badly with much of our thought and talk about natural languages, requiring us, for example, to regard what we are pleased to call the history of the English language as really a record of a succession of replacements of one language by another. This is a straight non sequitur. What we call the history of English has nothing to do with whether we think of languages and sentences as abstract objects or not. It has to do with the nature of the principle of individuation for natural languages, not with the ontological status of the entities that are individuated under the principle. The principle

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need not be framed, and, on linguistic grounds, ought not be framed, as individuating sets of sentences, as Hale imagines, but as individuating sets of sets of sentences (which is how the linguistic realist would think of the relation of idiolects to dialects and dialects to languages). In this case, the “record” is a chronicle of a succession of sets of sentences all of which belong to the set that is English.9 If Hale thinks that what is unattractive is that a description of a sequence of abstract objects cannot be a history of a language because it omits the changes that take place in its speakers and the causal mechanisms of those changes, then the reply has already been given. On linguistic realism, what is referred to as “the history of a language” or “language change” is a historically given sequence of different competences, and hence a description of such a sequence would represent changes in the linguistic competence of the speakers and the causal mechanism for the changes. The view that languages and sentences are abstract objects in the sense of (D) is thus not committed to the unattractive consequence that the history of a language is a description of a sequence of abstract objects. Such a sequence is only a description of the languages of which the stages of the history are knowledge. 5.7 The Second Set of Putative Counterexamples The second class of alleged counterexamples to (D) is quasimathematical objects, objects such as the equator and impure sets. The equator is a mathematical circle with properties that only an abstract object can have, e.g., having a center equidistant from every point on its circumference. Hence, the equator is an abstract object. But, since the equator did not exist before the earth came into existence and will not exist once it disappears, the equator has a history. Further, during that history, numerous people have crossed it. Hence, the equator is not an abstract object. On the basis of these two conclusions, the critic claims that (D) leads to a contradiction. A similar contradiction is supposed to arise in the case of impure sets. An impure set, such as the set of my children and me, is a set, so it is an abstract object. But, since sets do not exist if their members do not exist, the set of my children and me will not exist when one of us dies. Since the set of my children and me has a history, it is not an abstract object. Hence, as before, the critic draws the conclusion that (D) leads to a contradiction. 9. On the question of whether the sentences of a natural language form a set, see Langendoen and Postal (1984) and Katz (1996c).

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At the outset, it is clear that these putative counterexamples cannot be handled in the way putative counterexamples like games, languages, and sentences were handled in the last section. We cannot argue that the temporal and spatial properties ascribed to the examples in the second class do not really belong to those objects but rather to something else, like our knowledge of them. There is no similar distinction on which to base a defense of (D). Unlike games, languages, and sentences, objects like the equator and impure sets have the temporal and spatial properties they are taken to have. Therefore, if the examples on the second class are to be shown not to be counterexamples to (D), a different response is necessary. Since those examples are conceded not to be abstract objects, one might thank that the proper response is to adopt the reverse of the response taken in the case of the ªrst class of putative counterexamples. This strategy would be to take the equator and impure sets to be concrete objects and try to explain away the mathematical properties as an appearance resulting from a confusion of those properties with certain of their natural properties. We would have to treat the equator as a natural object like the annual rings in a tree or an artiªcial object like the painted lines used to divide the lanes on highways. But, since annual rings are sections of wood having only a roundish form and vague boundaries, and dividing lines are layers of paint having only a somewhat linear form and vague boundaries, neither is deªnite enough for the mathematical properties in question to apply to them. Moreover, no part of the surface of the earth can be identiªed with the equator, because the equator would persist intact even if that part of the surface of the earth were to disappear. Furthermore, a natural band of the surface of the earth or a painted line on the surface of the earth could count as the equator only if it circled the earth at exactly the right place, but that place is the equator, so such a material band or line can’t be the equator. The strategy doesn’t work because the equator is in some sense a mathematical circle. With impure sets, the strategy of taking them to be concrete objects breaks down quickly. Impure sets cannot be assimilated to the more concrete reality of their members. As Boolos (1984, 448) observes, “It is haywire to think that when you have some Cheerios, you are eating a set,” since you are just “eating THE CHEERIOS.” Boolos rightly argues that the reference to the Cheerios in the bowl is not singular reference to a set but plural reference to the Cheerios themselves. There is, of course, the impure set of the Cheerios in the bowl. We refer to that set when we say such things as, “The set of the Cheerios in the bowl has more than seventeen subsets.” But since the set can’t be identiªed with the concrete reality of its members, we can’t, as the

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strategy requires, naturalize the set referred to under the description “the set of the Cheerios in the bowl” by taking it to be the objects referred under the description “the Cheerios in the bowl.” We are thus driven to the conclusion that objects like the equator and impure sets belong neither to the category Abstract nor to the category Concrete. This conclusion might seem tantamount to an admission that the examples in the second class are counterexamples to (D), but, in fact, it is no such thing. Logically speaking, this conclusion could not establish them as counterexamples because (D) does not say that the division of objects into abstract and concrete is exhaustive. In order for the conclusion that the examples in the second class belong to neither the category Abstract nor the category Concrete to refute (D), we would have to have said, and we have not, that ontology requires that everything belong to one or the other of those categories. To assess the situation, consider (D1)–(D3). (D1) If something is an object, then it is homogeneous. (D2) If something is homogeneous, then it is either an abstract object or a concrete object. (D3) If something is an object, then it is an abstract object or a concrete object. The examples in the second class are counterexamples to (D) only if we accept (D3). (D3) does not follow from (D). Only (D2) does. Hence, to obtain (D3), it is necessary that (D) belong to an ontological theory that contains (D1). But nothing in the traditional abstract/concrete distinction, in what I have said thus far, or in the nature of ontology warrants (D1). Therefore, the equator and impure sets can be taken to be counterexamples to (D1) rather than to (D). The argument that they are counterexamples to (D) thus falsely assumes that the basic ontological division under the category Object is dichotomous. But nothing requires us to say that the categories Abstract and Concrete are the only basic divisions among objects in the ontological system. The categories Music and Drama do not preclude the category Opera. Since there are no grounds for (D1), and hence for claiming that homogeneous objects are the only kinds of objects there are, there are no grounds for claiming that abstract objects and concrete objects are the only kinds of objects. The mistake in connection with the ªrst class of putative counterexamples was a failure to construe realism broadly enough to include linguistic realism. The mistake in connection with the second class is a failure to construe ontology broadly enough to include a category for objects that are a mixture like opera.

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To construe it broadly enough, we replace (D1) with (D4). (D4) If something is an object, it is either homogeneous or heterogeneous. This makes a place for a category of objects that are neither abstract nor concrete. I propose a category for objects, which I will call “composite objects,” that are heterogeneous in containing both abstract and concrete objects as constituents. A composite object, like other complex objects, is a whole formed from objects in virtue of a relation (or pattern of relations) among them. The relation is “creative,” as we shall say, because, when the relation holds among some number of appropriate objects, there is a new object over and above them (with them as its components). When a creative relation holds of abstract and concrete objects, there is a composite object with them as its components. Not all relations are creative. The relations number of, identity, between, greater than, and inside of are not creative, since, when they hold, there are just the relation, its relata, and the fact that the former holds of the latter. New York’s being between Boston and Washington creates no new object with New York, Boston, and Washington as its components. There is just a relational geographical fact. Again, there is a composite object, Susan’s rattle, the components of which are the container with some loose hard objects inside and a handle, but there is no composite object composed of Spot’s stomach and the dog biscuits inside. The distinction between creative and uncreative relations, like any metaphysical distinction, gets tricky when we try to spell it out. Without trying to do that, I want to make a general point about creative relations that should be kept in mind in the discussion of composite objects below. In the case of artifacts, the intention of the producer or the intention of the user can be an aspect of the creative relation. Sometimes, as in the case of Susan’s rattle, the balls are put there to be part of the mechanism which makes the rattling sound, and sometimes, as in the case of a primitive rattle made from a gourd in which its dried seeds are the noisemakers, the required intention comes in via use. The intention must, however, be of the right sort. The dog biscuits did not get into Spot’s stomach unintentionally, but they were put there for reasons that had to do with their taste. They are not part of the mechanism that performs the function of a stomach. Deªning “component” as in (D5), we can now set out the basic (D5) A component of a composite object is one of the objects among which its creative relation holds. (A component of a component of an object is not a component of the object.)

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deªnitions of the theory of composite objects: (P1.1) Composite objects are simple, complex, or compound. (P1.2) A composite object is simple just in case its only components are one or more abstract objects and one or more concrete objects (with a creative relation holding among them). (P1.3) A composite object is complex just in case its only components are one or more abstract or concrete objects and one or more composite objects (with a creative relation holding among them). (P1.4) A composite object is compound just in case its only components are two or more composite objects (with a creative relation holding among them). The unit set the only member of which is me is a composite object. Its components are an abstract object, viz., the null set, and a concrete object, viz., me. Its creative relation is the relation of containment in the sense of “having within,” that is, the relation of a container to the things in it, whatever their other relations to it may be. Analogously, Spot’s body contains his bones, muscles, and organs, but also those dog biscuits that he bolted down plus miscellaneous objects eaten over the years. So, although the subset relation and the membership relation are different, proper subsets and members are both contained in the sets of which they are subsets and members. Thus, the Metropolitan Museum of Art’s collection contains both individual works of art, such as Botticelli’s Annunciation, the sculpture Diskos Thrower, the ªgurine of a hippopotamus known as “William,” and so on, and collections like the Robert Lehman Collection, the collection of Greek bronzes, the collection of Egyptian art, and so on. It is because there is a containment relation that covers inclusion and membership that special pains are needed in teaching set theory to ensure that the student registers the difference. Hence, the components of the composite object in the case of impure sets generally will be their proper subsets and members, and the components of the composite object in the case of the set the only member of which is me are the null set and me. Further, that impure set is a simple composite object, while the set the members of which are my two children and me is a complex composite object. Besides the null set, the set of my children and me has other proper subsets as further components. One is the set of just my two children; another is the set of just my oldest child and me. Since such subsets are composite objects too, the set of my children and me is a complex composite object. We shall see below that the equator is a simple composite object. Examples of compound composite objects will be discussed later.

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Given that the notion of a component is not the notion of a part in classical mereology, our conception of composite objects conºicts with Lewis’s (1986a, 38–39) conception of composition, which denies that there is a “general notion of composition, of which the mereological form is . . . only a special case” and claims that “mereology already describes composition in full generality.” His (1986a, 37) grounds are that . . . when a unit set is made of its sole member, one thing is made out of one thing; whereas composition is the combining of many things into one. And he (1986a, 37–38) ampliªes: There is no sense in which my parents are part of me, and no sense in which two numbers are parts of their greatest common factor; and I doubt that there is any sense in which Bruce is part of his unit set. This is the position Moore (1962, 14) took in Notebook I: With the ordinary meaning of “class” it is impossible that any class should have only one member or none. To say so & so is a class, is to say that it is the extension of some concept, which applies to at least two different things. Thus, there can be no class which is the class of the eldest sons of Edward VII. Moore continues: It is obvious, however, that if we are to talk of the extension of a concept which applies to only one thing, the thing in question has not got to the concept in question the same relation which the extension of a concept which applies to several things has to it, & which we assert the extension to have when we say it is the extension of the concept. For if we say men are the extension of the concept “human being,” we imply that the extension is something to which the concept does not apply; whereas the Duke of Clarence was the eldest son of King Edward, & is not therefore the extension of that concept. This argument begs the question of whether there is a unit set of the eldest sons of King Edward VII. If there is no such unit set, then the relations to the concepts are different, as Moore thinks, but if there is, the relations are the same. Just as the concept “human being” applies to men and not to the set of which they are members, so the concept “eldest son of King Edward VII” applies to the Duke of Clarence and not to the set of which he is the sole member.

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Lewis’s grounds for identifying composition with mereological composition are none too plausible either. It makes perfectly good sense to say that someone is part of his unit set. Consider the case of the arrogant dean who appoints himself as the sole member of the standing committee on promotions and tenure. It is in the nature of committees to be bodies delegated to consider, investigate, take action about, and report on matters within their purview, but it is not in their nature to have some speciªc number of members. The colloquium committee of the Ph.D. Program in philosophy at the Graduate Center had one member during the academic year 1996–97, David Rosenthal. The cases of the arrogant dean and our colloquium committee member are examples of someone’s being part of his or her unit set. Isaacson (see Lewis 1991, 30) explained how collections can contain just one thing on the basis of the example of an art collection that has only one painting because the museum ran out of funds. Lewis (1991, 30–31) criticized Isaacson’s example: “. . . this thought is worse than useless. For all those allusions to human activity in the forming of classes are a bum steer.” This is not a good criticism. To be sure, Isaacson’s example, like all analogies, has features that do not carry over to all the cases in question, but that doesn’t prevent it from being illuminating with respect to the feature meant to be analogous. Isaacson’s example of the impoverished art museum is not in itself a “bum steer” because a teacher can use the example to prepare students for their encounter with “inªnite and miscellaneous” sets by putting the example into context. The teacher can point out that pure sets are abstract objects while impure sets are not, and hence it is only the latter that can be formed with our help. Lewis (1991, 31) is right that the sets the students will learn about in their study of set theory are too vast for us to have had a hand in their creation, but, once the students understand that those sets are abstract objects, they will see that there is no question of our having a hand in creating them. It is not a matter of resting set theory on theology, but of resting it on mathematical realism. Lewis’s (1986a, 37; 1991, 30) claim that “composition is the combining of many things into one,” if not mere stipulation, must be an appeal to ordinary usage. But it is perfectly good ordinary language to reply to a question about the composition of the new tenure committee by saying, “It has one member, the Dean himself.” But even if Lewis’s claim about the term “composition” were backed up by ordinary usage, his philosophical claim would not be established. So, let’s suppose for the sake of argument that Lewis is right that a student who is introduced to set theory on the basis of Cantor’s slogan that a set is a “many, which can be thought of as one” has “just cause for student protest”

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when he or she comes across singletons (or the null set too, for that matter). But the failing of which the set theory teacher is guilty is simply pedagogical conservatism: the teacher too rigidly followed Cantor in his or her choice of expression. Had the teacher instead said that a set is a one that is the collection of some number of things, the student would have had no grounds for protest, since then he or she would have received sound guidance concerning what singletons and their members have to do with each other. The one element of the singleton is some number of things (and so, too, are the zero elements of the null set). The advocate of nonmereological composition does not need the explanation that Lewis puts into the teacher’s mouth to explain unit sets. The part of our ontological system that classiªes objects has a tree structure of the following sort. The category Object is among maximally generally categories (together with Property, and so on). An object is a unity, a one not a many, and, when constituted of many objects, it is a unity in virtue of a unifying relation (or pattern of relations).10 Directly subordinate to the category Object is the category Homogeneous Object. Directly subordinate to the category Homogeneous Object are the categories Abstract Object and Concrete Object. Under Abstract Object will be such categories as Mathematical Object, Logical Object, and Linguistic Object, and under Concrete Object will be such categories as Physical Object and Mental Object. However, Homogeneous Object is not the only category directly subordinate to the category Object. There is also the category Heterogeneous Object. Directly subordinate to it is the category Composite Object.11 Directly subordinate to it are the categories Simple Composite Object, Complex Composite Object, and Compound Composite Object. We can now complete the deªnition of compositeness with the principles (P2)–(P7): (P2) Composite objects are identical when they have the same components and the same creative relation holds of them in the same way. (P3) Abstract, Concrete, and Composite are disjoint categories. (P4) Abstract, Concrete, and Composite are exhaustive categories. 10. I make no attempt to explain the notion of an object. That perhaps dubious enterprise is fortunately not necessary for our purposes. 11. The categories Heterogeneous Object and Composite Object are intensionally distinct. As seems clearly to be the case, they are not also extensionally distinct, but I will not try to show it here.

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(P5) The condition for the existence of a composite object is the sum of the conditions for the existence of their components plus the condition that the relevant creative relation obtain among them. (P5′) Composite objects have the same mode of existence as their concrete constituents. (P6) Composite objects sort into kinds (e.g., mathematical, logical, linguistic) and subkinds depending on the kinds and subkinds of their abstract object component(s) in the case of simple and complex composite objects, and depending on the kinds and subkinds of their composite objects in the case of compound composite objects. (P7) The condition for a composite object to be objective is that all of its constituents are objective. (P2) seems to be the obvious identity condition for composite objects. A relation holds of one sequence of objects in the same way as it holds of another sequence of objects just in case corresponding objects saturate the same places of the relation. (P3) is a consequence of the disjointness of the categories Abstract and Concrete, the fact that both of them fall under the category Homogeneous, and the fact that Composite falls under the disjoint category Heterogeneous. (P4) is a consequence of the fact that every object is either homogeneous or heterogeneous. (P5′) follows from (P5) and the fact that a composite object must have at least one concrete object component. Composite objects are contingent, since they contain concrete objects as components and concrete objects are contingent. Hence, despite the fact that the abstract object components of the composite object exist necessarily, the composite object exists in some but not all of the possible worlds in which its abstract components exist. Thus, the impure set of my children and me, the impure set of my children, me, and Superman, and the impure set of Superman and Lois Lane are contingent objects. Since it is a necessary condition for the existence of a composite object in a possible world that its concrete members have spatial or temporal location in that world, neither the impure set of my children, me, and Superman nor the impure set of Superman and Lois Lane exists in the actual world.12 12. There is, of course, the further question about the existence of impure sets such as that of my children, me, and Socrates, that of Plato and Aristotle, and that of the offspring of my children (which ex hypothesi do not but will exist). Should we treat the existence

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(P6) is straightforward. The impure set of my children and me, having a pure set as its abstract component, is a mathematical, and in particular a set-theoretic, composite object. The equator, having a geometric type as its abstract component, is a mathematical, and in particular a geometric, composite object. Our well-formed utterances are linguistic composite objects, and performances of symphonies, sonatas, and the like are musical composite objects. The example of operas suggests that, in addition to composite objects of these pure kinds, there are also composite objects of mixed kinds. (P7) uses the notion of objectivity in the Fregean sense. Something is objective if it is outside us, has its existence independent of ours, and does not stand to anyone in the relation of possession that a conscious experience stands to the person having it. Not all composite objects are objective. The equator is, but a circular afterimage is not. The theory of composite objects provides a reason internal to realism for rejecting a “two world” view: creative relations must hold of objects in the same world. If they did not, we could not explain why composite objects are in one of the worlds but not the other (their heterogeneity prevents them from being in both). Given that objects have to be in the same world for a creative relation to hold among them, a “two world” view would prevent us from handling the second class of putative counterexamples as composite objects. Assuming the customary conception of the universe (or the world) as everything there is taken as a whole, (D) together with (D4) and extensional equivalence of the heterogeneous and the composite (see fn. 11) entail that the universe is a composite object. The universe is not a concrete object because it has abstract components (and hence isn’t homogeneously concrete), and the universe cannot be an abstract object either, because it has concrete components (and hence isn’t homogeneously abstract). Since the universe is an object and contains abstract and concrete objects, it is heterogeneous, and hence a composite object. It can be characterized as the maximal composite object: the composite object that is not a component of any other composite object. (There is no unit set the member of which is the universe.) We can now argue, contrary to many versions of the modal ontological argument for the existence of God, that “Possibly necessarily God exists” is false. Since every object is either homogeneous or heterogeneous, God is either homogeneous or heterogeneous. God cannot be heterogeneous, since then God’s existence would depend on the of such impure sets in the same way we treat the existence of their components, or should we say they exist even though some of their components no longer exist or do not exist as yet? This is not an easy question to answer, but since it will arise for any account of impure sets, it does not require an answer here.

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existence of a concrete object, and hence God’s existence would be contingent. If God were homogeneous, God would be either abstract or concrete. God cannot be abstract because, in God’s bringing the world into existence, an abstract object would have to be, per impossible, causally active.13 Hence, God must be concrete, but in this case God does not have necessary existence, since, as we saw in section 5.5, there is a possible world that contains only abstract objects. Since God cannot be homogeneous or heterogeneous, God is an ontological impossibility, and hence “Possibly necessarily God exists” is (necessarily) false. The equator of the earth is a composite object. Its abstract component is a mathematical circle, its concrete component is a part of the surface of the earth, and its creative relation is the type/token relation. The equator is a different composite object from the impure set of the circle that is the abstract component of the equator and the part of the earth’s surface that tokens it. Both have the circle and the same part of the earth’s surface as components, but the creative relation in the latter case is the containment relation. The equator does not have the null set as a component; the equator is a simple composite object, while the impure set is a complex composite object. Finally, the equator is objective because its abstract and concrete components are objective. The type/token relation is usually thought of as a linguistic relation (perhaps because Peirce introduced it on the basis of the linguistic example of a word). But there are other cases. There are geometric cases of the type/token relation, including objects like the equator and the drawn ªgures used in geometric demonstrations, and there are musical cases. Tokens of types are composite objects where the abstract component is the type and the concrete component is some physical or psychological particular. The chalk or graphite triangles, squares, and circles that we write on blackboards and on paper are tokens, as are the utterances of sentences we produce in speech. The structure of a token is the structure of the abstract object that is the type of the composite object. The point is illustrated in the case of ambiguous ªgures like the Necker 13. It might be suggested, following Spinoza, that God’s creation is logical rather than causal, that is, the result of divine understanding rather than divine will. Latham (in conversation) glosses this idea as the idea that God’s understanding a universe to be the best is his creating it. But the notion of logical novelty does not have the right kind of content to provide the required informative analogy. In logical creativity, a conclusion that is unexpected on the basis of inspecting each premise separately emerges when deductions are made from the premises together. Here, however, our reasoning stays at the level of thought, in contrast to divine creation which goes from thought to object (God’s understanding to our universe). Since the notion of non-causal creation is empty, it remains unclear how divine thinking could be creation without being a form of causation.

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Figure 5.1

Figure 5.2

cube (ªgure 5.1). In such ªgures, the concrete object, which is a particular pattern of deposits of ink on the surface of a piece of paper, remains the same two-dimensional pattern, even though, at one moment, it is a component of a token of a projection of a cube with one front and back orientation, and, at the next, it is a component of a token of a projection of a cube with the opposite front and back orientation. Perhaps an even more striking example comes from cases in which the tokening is a near miss, as, for example, impossible ªgure of a tubular trident (ªgure 5.2). Here the concrete object does not make it as a two-dimensional projection of either the three-dimensional tritubular form or the three-dimensional bifurcate form. The ªgure as a whole does not have the structure of three tubular prongs or the structure of a two-tined fork, not even in the oscillating manner in which the Necker cube as a whole ªrst has the structure of a cube with one orientation and then the structure of another cube with the opposite orientation. In the Necker cube case, our attention alternates between two tokens, but the case of the tubular trident is different. Contrary to what Gregory (1966, 235) says, it is not that “the brain cannot make up its mind.”

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“The trouble,” Gregory thinks, “comes from the ambiguity of depth— the eye is not given the essential information to locate the parts in depth.” This assimilation of the impossible ªgure case to a case of ambiguity like the Necker cube is mistaken. In the latter, there are types to recognize the tokening of, while, in the tubular trident case, there are none. It is not that our brain fails us. No more decisive or more cognitively powerful brain could succeed where ours fail. There is no failure because there is no success. The two-dimensional drawing of the tubular trident is as good a concrete object as the two-dimensional drawing of the Necker cube, but, in the former case, there is no three-dimensional geometric type, whereas in the latter case, there are three-dimensional geometric types. There is no tokening in the case of the tubular trident for the same reason there is none in the case of the round square: there is no type. As we noted in section 5.5, Frege (1953, 35) opposed the common view that the equator is an imaginary line. An imaginary line is a line in imagination, not something objective on the surface of the earth. The equator is an objective boundary on the surface of the earth between the Northern and Southern Hemispheres—New York City is on one side of it and Rio de Janeiro on the other. It cannot, as indicated above, be identiªed with a particular physical feature of the surface of the earth such as a natural contour, since the equator cannot be altered or destroyed even by large-scale dirt removal operations. The concrete component of the equator cannot be destroyed by local physical changes. It would take a global physical change like the incineration of the earth. Despite erosion, earthquakes, and other changes in the topography of the earth, the concrete component of the equator is the same object that we refer to and that different travelers cross from one occasion to the next. The sameness of the equator over time requires the sameness of the concrete object component. One way to account for the sameness of the concrete object component is to take the aspect of the earth that tokens the geometric type circle to be the space-time worm of temporal contour slices that, at successive times, occupy the position on the surface of the earth determined by a plane passing through its center perpendicular to its axis of rotation. There are other ways, but no need to catalogue them. The equator has been seen as a counterexample to the traditional abstract/concrete distinction because predicates with disjoint ranges of application apply to it. For example, both predicates like “is a circle” and “has equal radii” and predicates like “has existed a long time” and “is crossed frequently” apply to the equator. The application of predicates of both kinds would be a problem if there were no alternative to (D1), but, once it is replaced with (D4) and the equator is taken to be

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a composite object, predicates of both kinds can apply to the equator. When we say that the equator is a circle or has existed a long time or has been crossed frequently, we are making a relational predication. We are ascribing a property to a composite object relative to its abstract or concrete components. In the case of the former predications, what we are saying absolutely is that the abstract object component of the equator is a circle, and in the case of the latter predictions, what we are saying absolutely is that the concrete object component of the equator has existed for a long time or has been crossed frequently. There is nothing ad hoc in explaining the predication of having equal radii relative to one component of the equator and the predication of being crossed frequently relative to the other component. Such relative predication is found in a wide variety of cases. We often say that things have this or that property when it is only one of their components that has the properties in an absolute sense. For example, we say that pokers and branding irons are red-hot when only one component of them is red-hot. Similarly, we ªnd clear cases of such relative predications in connection with the tokening of linguistic types. We say that sentence (5) is analytic, but since (5) is ambiguous, (5) A bank is a ªnancial institution. it cannot be analytic absolutely. It can only be analytic relative to one of its senses; that is, only one of its senses is analytic absolutely. Similarly, (6) (6) The detective dusted the table. (7) The detective removed dust from the table. analytically entails (7), but again, since (6) is ambiguous, (6) cannot analytically entail (7) absolutely. It can only analytically entail it relative to one of its senses; that is, one of the senses of (6) analytically entails the sense of (7) absolutely. It makes no sense to say that these semantic predications of analyticity and entailment are absolute in the way that the syntactic predication that (5) and (6) are well-formed is absolute. We cannot say that (5) is analytic absolutely, because (5) also has a nonanalytic sense on which it falsely asserts that raised ground bordering a body of water is a ªnancial institution; we cannot say that (6) entails (7) absolutely, because (6) has a sense (that of “putting dust on the table,” e.g., to look for ªngerprints) on which (6) contradicts (7). Predications such as that the equator is everywhere equidistant from its center, has existed a long time, and has been crossed frequently are relative predications, like the predications that (5) is analytic and that (6) analytically entails (7). In the same way that sentence (5) has the property of analyticity in virtue of having an analytic sense, the equator

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has the property of having all the points lying on it being equidistant from its center in virtue of having an abstract object component, and has the property of duration relative to its concrete object component. This account is generally applicable to all ascriptions of spatial or temporal predicates to composite objects. Thus, for example, saying that the impure set of my children and me did not exist at the time of the Big Bang, exists now, and will not exist in another ªfty years is making predications in the same relative way in which we ascribe temporal or spatial predicates to the equator.14 5.8 On Creative Relations In the case of creative relations, there is a new entity over and above the relation and its relata, while in the case of uncreative relations, there is none. To say that a composite object exists when a creative relation holds of abstract and concrete objects is to say that there is an object with properties they do not have that exists in virtue of the relation holding of them. For example, the impure set of my children and me has the property of being a nonempty set and the property of having both abstract and concrete components. These properties, which apply to the composite object absolutely, do not apply to the relation, the null set, or my children and me. An utterance or inscription of a sentence is a token of a linguistic type. The token, the composite object, is distinct from the type/token relation that creates it, its abstract component, and even its concrete component. The concrete component is a certain deposit of some substance like ink or a certain acoustic entity. For example, the string of inscriptions or sounds represented in (8) is not in itself a token of any string of sign types or letter types or word types.15 (8) Ah ’key ess ’oon ah ’may sah. (8) might be a sound sequence produced by a malfunctioning voice synthesizer, orthography produced by a monkey at a typewriter, or articulations of a parrot. Even assuming that (8) is a string in a universal phonetic system, it might be a token of a Spanish sentence that means “a table goes here” or a token of a Yiddish sentence that means “a cow 14. To say that properties like being everywhere equidistant from its center and having existed for a long time apply absolutely only to the abstract and concrete components of a composite object is not to say that there are no properties that apply absolutely to composite objects. “Is an object,” “is heterogeneous,” “is structured,” and “has components,” apply absolutely because there is no abstractness or concreteness condition for their application. 15. I got this example from Wetzel (1989a, 195), who says she got it from Lon Berk, who says he got it from George Boolos, who says he got it from Warren Goldfarb.

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eats without a knife.” It is possible to imagine a Spanish/Yiddish bilingual answering both a Spanish speaker who asks about furnishing the room and a Yiddish speaker who asks about bovine eating habits with one and the same utterance of (8). Since the acoustic or orthographic entity that (8) represents is not in itself a token of anything, it makes no sense to ask questions like, Is it a difªcult sentence to pronounce? and Does the sentence have a generic sense? When the acoustic or orthographic entity that (8) represents is a concrete object in a composite object with an appropriate grammatical type related to it by the type/token relation, we can ask whether it is a difªcult sentence to pronounce and whether it has a generic sense.16 Imagine that a can on a jiggling scaffold has leaked red paint onto a board in the form of letters that look exactly as if someone had written “Danger!” The paint deposit is a concrete object; it is not a token of a linguistic type. Putnam (1981, 1–2) imagines a similar case, in which an ant traces a line in sand that looks like a caricature of Winston Churchill. Putnam observes—though not in these terms—that the concrete object, the furrow in the sand, is not a Winston Churchill caricature, because there was no intention involved in its causal antecedents. To be a token of the English expression type “Danger!” an intention is similarly necessary. The intention can enter in various ways, however. For example, a worker who has been told to put a warning sign in the ground in front of a hole might well save himself or herself the trouble of painting a sign by nailing the board with the red paint deposit to a stick and putting it in front of the hole. As a consequence of the intentional act of using the board with the paint deposit as a danger sign, the deposit tokens the English expression “Danger!” Note that the token of the English expression “Danger!” is neither the type/token relation nor the abstract linguistic type nor the concrete paint deposit—which remains the same concrete object it was prior to becoming a sign. The token is that concrete object standing in the type/token relation to the abstract linguistic type in virtue of the intentional act of putting up the board as a warning sign. This is another instance of the point made earlier that, in the case of artifacts, the intention of the producer or the intention of the user can be an aspect of the creative relation. The intention of the causal antecedents of this rather atypical inscription is the intension that it serve as a literal token of the linguistic type “Danger!”17 16. Given the stratiªcation of linguistic types from phonetic types to syntactic types, linguistic tokens will typically be complex and compound composite objects rather than simple composite objects. I will ignore this technicality in the present discussion. 17. See Katz (1990b, 135–62) for an explanation of the notion of a literal use of language in terms of the speaker’s intention for the sense of his or her utterance to be a subordinate of the sense of the type of which the utterance is a token.

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The best of intentions can do nothing in the case of uncreative relations. The objects related to an integer under the uncreative relation number of simply instantiate it. Unlike the paint leaked from the can on the jiggling scaffold, the seventeen cherries spilled from a cherry picker’s basket are seventeen cherries, pieces of fruit that collectively numbered by the number seventeen. There is no object over and above the number seventeen and the seventeen cherries that is constituted by the number and the cherries. The difference between creative and uncreative relations is that the content of the former involves the notion of some further object of which the objects satisfying the relation are components, whereas the content of the latter does not. The content of the type/token relation involves the notion of a further object, a token, that exists in virtue of abstract and concrete objects standing in that relation. Similarly, the content of the containment relation involves the notion of a further object, an impure set, that exists in virtue of abstract and concrete objects standing in that relation. Wollheim (1968, 64–79) contrasts universals with types. He (1968, 75–76) observes that the relationship in which types stand to their tokens is more “intimate” or “intrinsic” than the relationship in which universals stand to their instances: [The] universal is present in all its instances. Redness is in all red things. With types we ªnd the relationship between the generic entity and its elements at its most intimate: for not merely is the type present in all its tokens like the universal in all its instances, but for much of the time we think and talk as though it were a kind of token, though a peculiarly important or pre-eminent one. In many ways, we treat the Red Flag as though it were a red ºag (cf. “We’ll keep the Red Flag ºying high”). Wollheim (1968, 66) further observes that this difference in intimacy between the generic entity and its elements is reºected in what he calls “transmitted properties,” i.e., properties that the generic entity and its elements share and that one of them has because the other one does. He (1968, 67) writes: Now there would seem to be two differences in respect to transmitted properties which distinguish universals from types. First, there is likely to be a far larger range of transmitted properties in the case of types than with universals. The second difference is this: that in the case of universals no property that an instance of a certain universal has necessarily, i.e., in virtue of being an instance of that universal, can be transmitted to the universal. In the case of types, on the other hand, all and only those properties

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that a token of a certain type has necessarily, i.e., that it has in virtue of being a token of that type, will be transmitted to the type. Examples would be: Redness . . . may be exhilarating, and if it is, it is so for the same reason that its instances are, i.e., the property is transmitted. But redness cannot be red or coloured, which its instances are necessarily. On the other hand, the Union Jack is coloured and rectangular, properties which all its tokens have necessarily: but even if all its tokens happen to be made of linen, this would not mean that the Union Jack itself was made of linen.18 In connection with abstract objects, Wollheim’s term “transmission” has a misleading connotation of causal process, but we can understand what he wants to say in terms of copossession. Another difference between instantiation and tokening is this. Not only is there copossession of properties in the case of tokening but not in the case of instantiation (because the universal U cannot itself instantiate U), but copossession in the case of tokening does not obtain in virtue of the concrete object component’s having the property that the type and token copossess. When an abstract type is tokened, the composite object, the token, copossesses the properties and relations in the structure of the abstract object component, the type, but the concrete object component does not enter into the copossession relation. The type has them absolutely and the token has them relatively, but the concrete object does not possess them at all. For example, the concrete object (8) does not per se have the property of containing a prepositional phrase. In contrast, in instantiation cases, such as being seventeen, it is the concrete objects per se that have the numerical property.19 18. Wetzel (1989a, 191) criticizes Wollheim’s second point on the grounds that we cannot say that the word “cat” and its tokens share the property of being composed of three letters, because the former has the property of being composed of three letter types while the latter has the property of being composed of three letter tokens. Taking Wetzel’s case to be a compound composite object, her question is about components of the composite object that consists of the type/token relation, an orthographic letter type, and physical marks. Once the case is put this way, it is easy to see that, in light of the discussion in the text, there is no substantive issue. 19. This conception of the type/token relation contrasts with a conception of it as modeling, that is, on which a type is thought of as transmitting properties to its tokens in a manner analogous to the manner in which a sculptor fashions the curve of the nose of a sculpture from a model. One problem with the model metaphor is that it forces us to think of the properties transmitted to tokens as produced in them in the modeling process. But, to take one example, underlying syntactic or semantic features of linguistic tokens cannot be thought of this way since they are not part of or in utterances or inscriptions. See Katz (1990b, 276–80).

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5.9 Seeing As This section provides an account of seeing as in the theory of composite objects as an alternative to Wittgenstein’s (1953, sec. 74) account of seeing us. Wittgenstein (1953, 193–208) uses this phenomenon to argue for his view of meaning as reºecting mastery of a technique. The discussion of seeing as is thus part of his (1953, secs. 65 and following) general case against theories of meaning according to which the application of a sign is a matter of recognizing something as falling under a general concept. The arguments concerning meaning that precede section 74—the criticisms of Fregean concepts in sections 65–71—are intended to undercut explanations of semantic phenomena such as ambiguity that appeal to a general concept of sense or meaning. Wittgenstein’s discussion of seeing as is continuous with his rulefollowing criticism of theories of meaning. That criticism attempts to show that intensionalists are driven from one interpretation to another ad inªnitum, so that application of the sign can never take place. Wittgenstein (1953, sec. 201) says: What this shews is that there is a way of grasping a rule which is not an interpretation, but which is exhibited in what we call “obeying a rule” and “going against it” in actual cases. His (1953, sec. 202) conclusion is that “‘obeying a rule’ is a practice.” The point in connection with perception is the same. If our account of visual perception is to avoid a similar regress, there must, Wittgenstein (1953, 195) believes, be a form of perception in which recognition does not depend on interpretation, of seeing something as something else. Furthermore, he (1953, 208) believes that a satisfactory account of the phenomenon of seeing as is only possible if we take the use of signs to depend only on the recognition of similarities between cases. He thinks that an account that invokes a shift in our inner mental picture to explain the shift in aspect that occurs when we look at a duck/rabbit drawing merely relocates the problem. The mental picture has to be an exact copy of the physical picture, since the shift in aspect changes nothing in the physical picture, yet if it is an exact copy, it exhibits a shift that stands in need of interpretation. In The Metaphysics of Meaning, I (1990b, 135–62) argued that our non-Fregean intensionalism avoids Wittgenstein’s rule-following argument. While Wittgenstein is right to think that his paradox dispatches “mental picture” semantics, he is wrong to think that it dispatches every intensionalism on which the application of signs depends on general concepts. A semantics in which senses are abstract objects of the right kind can avoid Wittgenstein’s rule-following paradox in a

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manner similar to the manner in which such a semantics was shown in chapter 4 to avoid Kripke’s rule-following paradox. The approach to Wittgenstein’s use of seeing as is similar. He (1953, sec. 74) denies that “if you see this leaf as a sample of ‘leaf shape in general,’ you see it differently from someone who regards it as, say, a sample of this particular shape.” He believes that it is a mistake to think that there are different shape types, both of which have the leaf as one of their tokens. His concern here as elsewhere in the Philosophical Investigations is to construe such differences simply as differences in use. This case of seeing a leaf as exemplifying different shapes is— importantly for him—on all fours with the case of hearing a sentence as meaning different things. There he thinks that we make the parallel mistake when we think that a sentence’s ambiguity consists its having two or more senses, since thinking this leads us to say that there are two or more things that are senses of the sentence. Wittgenstein wants to say that the ambiguity of a sentence consists in nothing more than its being used in two or more different ways. The theory of composite objects provides an account of seeing as that, unlike the mentalistic account, does not relocate the problem of explaining the shift in aspect from one thing to another. On this theory, seeing the deposits of ink that form the duck-rabbit picture ªrst as a duck or a rabbit and then as a rabbit or a duck is ªrst perceiving a token of the one picture type and then perceiving a token of the other picture type. We perceive a token of the type duck ªgure or a token of the type rabbit ªgure and then somehow our perception shifts to a token of the other type. Thus, on our theory, seeing the drawing as a duck picture is perceiving it to be the concrete component of a composite object the abstract component of which is a duck picture type, and seeing it as a rabbit picture is perceiving it to be the concrete component of a composite object the abstract component of which is a rabbit picture type. More generally, we can see X as a Y when we recognize X to be the concrete component and Y to be the abstract component of the same composite object. In such recognition, the abstract animal proªles are the meanings of the concrete object in the same way that, in the semantic recognition of (6), the abstract senses are the meanings of the concrete object. Seeing the contour of the concrete object differently when the shift occurs is seeing a token of a different type. The shift between the two tokens pivots on the part of the drawing that has the form of a double protrusion in the way that the shift between semantically different tokens of the sentence type (6) pivots on the occurrence of the word “dust.” Just as this occurrence is understood as a verb with the meaning of “remove dust” and also as a verb with the meaning of

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“spread dust,” so the protrusion is perceived as a picture of ears and also as a picture of a bill. The mixing of visual and cognitive content in the relation of a sign to meaning in rebus representations makes them even more complex than linguistic representations. Since the issue with which we are concerned is the same in both, the mixing of visual and cognitive content is a confounding element which is best factored out by switching to linguistic cases like the ambiguous sentences (9) and (10). (9) The uncle of the boy and the girl arrived late. (10) Visiting relatives can be annoying. The shift in semantic aspect in such cases is explained as different composite objects with the same concrete object component. Now, in contrast to an ambiguous rebus, we have a way of talking about the types that not only is uniform but also allows us to bring a rich body of linguistic theory to bear on the problem of characterizing the tokens between which the shift takes place. The types are complex grammatical structures in which different deep syntactic structures (with different senses) have the same surface structure.20 When the token (9) is the composite object the abstract object component of which is the sentence type in which “the boy” is the object of the prepositional phrase, (9) means one thing, and when the token (9) is the composite object the abstract object component of which is the sentence type in which “the boy and the girl” is the object of the phrase, (9) means another. Similarly, (10) means different things depending on whether (10) is a token in which “visiting” modiªes the subject “relatives” in the sentence type or whether (10) is a token in which “relatives” is the direct object of “visiting” in the sentence type. Contextual disambiguation is a matter of indicating which of the two tokens is meant, say, by replacing the verb phrase in (9) with “kissed his nephew and niece” or adding “if they overstay their welcome” to (10). The linguistic cases make it even clearer that ambiguity and other such properties of tokens do not apply to their concrete objects. The condition for the application of the predicate “is ambiguous” is the existence of multiple grammatical structures, but no multiple grammatical structures are phonetically or orthographically there in the 20. The disagreement with Wittgenstein in connection with the deep structure of sentences is perhaps the most fundamental disagreement between contemporary linguistics and his late philosophy. For a discussion of the issue from a realist standpoint, see Katz (1990b, 21–162).

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sound waves or the deposits of ink, graphite, or other substances that constitute the concrete components of tokens (9) and (10). The acoustic or orthographic contour of the concrete objects is too impoverished for such rich grammatical structure. The ambiguity of (9) is (in part) a function of the double bracketing that marks different scope relations in the noun phrase, and the ambiguity in (10) is (in part) a function of the underlying indeªnite subject of the verb “visit,” but nothing in the deposits of ink that form the concrete objects (9) and (10) corresponds to such a double bracketing structure or to such a subject constituent.21 Wittgenstein (1953, 196) observes that “The expression of a change of aspect is the expression of a new perception and at the same time of the perception’s being unchanged.” Our account of seeing as easily accommodates this observation. The change of aspect that takes the form of “a new perception” is the shift from one composite object to the other, which is a change from one type to another, e.g., from the type duck picture to the type rabbit picture, or from the meaning of “one’s visiting one’s relatives can be annoying” to the meaning of “one’s relatives visiting one can be annoying.” The “perception’s being unchanged” consists in the fact that the concrete object component of the preshift composite object and the postshift composite object is the same—thus the physical stimulus remains the same preshift and postshift. Wittgenstein (1953, 196) asks, “. . . what was [my visual impression] like before and what is it like now?” He says that, if I represent my visual impression of an ambiguous inscription before the shift of aspect and after by “a good representation of it,” “no change [in the representation] is shewn.” Of course. With “an exact copy,” no change would be shown. An exact copy of a concrete object is simply another one sufªciently like the original to count as another token of the same ambiguous type. This is a problem for explanations of seeing as in terms of what comes before the mind because the only structure mental images have, at least on Wittgenstein’s construal, is phenomenologically presented surface structure. The problem does not arise for our approach on which abstract objects have a deep structure. The surface syntactic structures of sentence types like (9) and (10) represent the unchanging inscriptions of their ambiguous sentence types. Their deep syntactic structures represent their changing meanings. Hence, if I 21. This is the familiar point from the Chomskyan criticism of American Structuralists (which the famous example of the different subject-inªnitive relations in “John is easy to please” and “John is eager to please” illustrates). Chomskyans also noted that the acoustic form of an utterance does not even contain all of the phonemic features of the utterance. See Studdert-Kennedy (1974).

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represent my visual impression of an ambiguous sentence, no change is shown, but if I provide a “good representation” of the sentence that the visual impression tokens, a change from one term of the ambiguity to the other is shown. Even if mental pictures were taken to have an underlying, unconscious component, they would not have the normative force to ªx meaning in the way Wittgenstein requires. Wittgenstein (1953, 196) cautions us that the perception “is not the drawing, but neither is it anything of the same category which I carry within myself.” He adds, “The concept of the ‘inner picture’ is misleading.” This is because, as he (1953, secs. 139–41) argued earlier, “what comes before our mind” is impotent to ªx meaning. I (1990b, 135–62) agree with Wittgenstein that an inner picture does not have the normative force to ªx the correct use of language. Something that comes before the mind can’t distinguish action in accord with a rule from action that conºicts with it. However, abstract linguistic objects can. They are normative. As I (1990b, 135–62, ch. 4) argued, the sentence types of a language (including their senses) are the linguistic norms for the literal use of language. Hence, the abstract components of the composite objects (9) and (10) are the criteria in virtue of which it is correct to say that those sentences are ambiguous in English. With reference to the literal use of language, speech acts that are in accord with the norms of English are uses that conform to the structure of English sentence types. For example, utterances like “its being obvious that pigs can’t ºy” and “that pigs can’t ºy is obvious” conform to the syntax of English, while utterances like “that pigs can’t ºy seems” do not. Syntactic conformity to English is tokening a well-formed expression or sentence type of English. Semantic conformity is syntactic conformity in which the sense of the token is the sense of the type (or, more precisely, a subordinate of it). 5.10 Demonstration with Drawn Geometric Figures There is a puzzle about how geometrical reasoning that refers to ªgures drawn on a physical surface can yield general conclusions about a class of abstract geometric forms. Of course, the use of such ªgures is only a crutch we rely on in cases where reasoning about the abstract mathematical objects themselves is psychologically difªcult, but nonetheless we often reach conclusions on the basis of reasoning from such ªgures that count as mathematical knowledge. So there is a question of how we can establish a theorem of geometry when we reason with reference to concrete objects like deposits of chalk on a blackboard which are no more than approximations—and usually quite bad ones at that—of the geometric ªgures that the conclusions are about.

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No Millian answer will satisfy us. Piling up more diagrams, even a great many more, wouldn’t establish a universal conclusion about a class of geometric ªgures. Inductive inference from a very large sample of cases couldn’t establish that the square of the hypotenuse of a Euclidean right triangle is necessarily equal to the sum of the squares of the other sides. Hence, if the drawn ªgures to which geometric demonstrations refer were concrete objects and nothing more, it would be a mystery how those demonstrations could provide knowledge of the Pythagorean theorem. Even though such demonstrations may not be the best way to obtain mathematical knowledge, it seems too strong to deny that they ever provide us with mathematical knowledge. The theory of composite objects explains how we sometimes get genuine mathematical knowledge from such demonstrations. In a nutshell, the explanation is that the points, lines, triangles, circles, and other geometric ªgures to which we refer in the demonstrations are tokens of geometric types, and hence composite objects. Their concrete components are the ªgures drawn in the dirt, on paper, or on blackboards; their abstract components are the geometric forms about which we draw conclusions; and their creative relation is the type/token relation. This also explains, on the one hand, why the drawn ªgures are ultimately unnecessary, and, on the other hand, why they are a basis on which we can reach valid geometric conclusions when the ªgures are indispensable from a practical standpoint. The ªgures are ultimately unnecessary because we can in principle reason about the pure geometric types directly; we can reach mathematical conclusions from the ªgures because our reasoning in reaching them is about the abstract object components of the composite ªgures. That our geometric reasoning in connection with drawn ªgures is about composite objects can be seen from the fact that the objects to which the reasoning refers exhibit the now familiar pattern of prima facie contradictory predications that are characteristic of composite objects. We say that a circle drawn on a piece of paper is a circle, that is, a curve all of the points of which are equidistant from its center, but we also say that I drew it with my Mongol 482 pencil. But if the objects of my geometric reasoning were just the graphite on the paper, I could say that I drew it with my Mongol 482 pencil, but not that it has radii of equal length. If the object were just the abstract circle, I could say that it has radii of equal length, but not that I drew it with my Mongol 482 pencil. Since it makes sense to say both things, it is a composite object.22 22. Thus, drawn geometric ªgures are similar to the equator in involving a geometric type, but dissimilar to it in having an artifact rather than a natural object as their concrete

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Supposing that such geometric reasoning is about composite objects, it is easy to see how, even though the reasoning refers to a single case, it can provide us with knowledge of a necessary truth about a class of geometric forms. On this supposition, the conclusion about the class of geometric forms is not inferred from the particular construction—from the concrete component of the composite object. That construction only helps us to have clear and distinct ideas about the structure of the abstract component. Rather, the conclusion is inferred from premises about the abstract component—about the geometric type. Since the geometric type in a demonstration of the Pythagorean theorem is Euclidean right angled triangle, the conclusion holds of all such triangles, and it holds of them necessarily, since, being an abstract object, the type has its geometric properties and relations necessarily. Finally, the theory of composite objects explains why geometrical reasoning that refers to a particular ªgure drawn on one physical surface at one time and place can hold also for other ªgures on other surfaces at other times and places. It also explains why different mathematicians can establish the same conclusion on the basis of reasoning that refers to temporally, spatially, and artistically very different ªgures. The explanation is that the ªgures are all tokens of the same type. Each is a composite object with the same abstract object component, and the conclusion drawn is a truth about the geometric structure of that abstract object. 5.11 Linguistic Composite Objects Linguistic utterances and inscriptions are composite objects, since they too exhibit the characteristic pattern of prima facie contradictory predications. Jones’s utterance was loud, occurred at dawn, and was heard all over the neighborhood, but it was also grammatical, an interrogative, and ambiguous. The former predications apply to the concrete object component of the composite object, while the latter apply to the abstract object component. Utterances and inscriptions are tokens of grammatical types. Like tokens of geometric types, tokens of grammatical types are studied on the basis of drawn diagrams similar to drawn geometric ªgures. In syntax, for example, they are what are called “phrase markers,” tree diagrams in which the nodes are labeled with symbols components. Accordingly, we use geometrical properties of the abstract object to make inferences about the concrete object in the case of a composite object like the equator, whereas we use the concrete object to make inferences about the geometric properties of the abstract object in the case of a composite object like a drawn ªgure.

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standing for syntactic categories and the bracketing structure assigns segments of their terminal strings to syntactic categories. Grammatical investigation no more depends on such diagrams than geometrical investigations depends on drawn ªgures, but they are often as useful to linguists as drawn geometric ªgures are to mathematicians. Given this similarity, the epistemological reºections concerning geometric tokens and types in the last section carry over to grammatical tokens and types. The same puzzle about reasoning in connection with geometric diagrams with which we began the previous section arises about reasoning in connection with grammatical diagrams too. On being presented with (9) or (10) for the ªrst time, a native English speaker knows that the type it tokens is ambiguous. As in the geometric case, such knowledge could not be grounded on perception or introspection of the concrete object (9) or (10) per se. As in the geometric case, we explain such knowledge as grounded on an intuition about a composite object. Thus, conclusions based on reasoning from a grammatical diagram can be grounded on a single utterance or inscription because they are about the grammatical type that is the abstract component of a composite object. Furthermore, we can explain how different linguists are able to establish the same conclusion about a matter of grammar on the basis of reasoning that refers to temporally, spatially, and artistically distinct diagrams. They are all composite objects with the same abstract object component, and the conclusions of the various linguists are about the grammatical structure of the common abstract object.23 5.12 On the Difference Between Pure and Applied Science The senses of a sentence make no contribution to the conditions for something to be one of its tokens because senses themselves are not types or parts of types. Only the syntactic and phonological (or orthographic) structure of a sentence type contributes to the conditions for its tokens. Note that geometric and musical types have tokens even though such types, unlike sentence types, have no senses. Also, despite the fact that a token of a sentence type in a nonliteral use of language has a meaning different from the meaning of its type, it is every bit as 23. This solution is not open to linguistic nominalists or linguistic conceptualists, since, according to them, a linguist’s basic knowledge of a sentence comes from acquaintance with a single (or at least very few) concrete object(s). This raises the questions of how they can account for the fact that the linguist draws conclusions about the grammatical properties and relations of the sentence (which the concrete object[s] token), and how they can account for the fact that different linguists draw conclusions about a common object of investigation.

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much a token of the type as tokens in literal uses.24 Even in literal uses of language, tokens customarily exhibit one sense, while the sentences of which they are tokens very frequently have more than one. Elsewhere I (1990b, 88–90, 137–55) have argued for a semantic monism on which there is just one kind of sense for grammar and pragmatics. The senses correlated with grammatical types in a language are the very same objects as the senses correlated with utterances and inscriptions in communication. There are two means relations. In the language, the means relation holds between a sentence (or some other syntactic type) and a meaning (one or more senses). This means relation is part of the structure of the language. It is the one involved when we say that the English sentence (10) means both—has both senses—that relatives’ coming to visit one is annoying and that going to visit relatives is annoying. Let us call this the “semantic” relation. In language use, the means relation holds between an utterance or inscription and a sense (either one of the senses of its type or some other, perhaps more explicit, sense). This means relation is part of the communication situation (along with, for example, Gricean [1989, 22– 57; see also Sperber and Wilson 1986] conversational implicatures). It is the one involved when we specify what a token of sentence (10) means in the context of the utterance. Let us call this the “pragmatic” relation. The pragmatic relation can be used to express the difference between a token with literal meaning and one with a nonliteral meaning. In the case of a token with literal meaning, the concrete object in the composite object means at least one of the senses in its abstract object component, while, in the case of a token with a nonliteral meaning, the concrete object means some sense (or senses) not in its abstract object component. The pragmatic relation, like the numbering relation, is not creative. Just as the states of the Union number ªfty, so a token of sentence (10) uttered on the occasion of an irksome visit from relatives means that the visit of relatives can be annoying. There is no new (complex composite) object with that token and that meaning as components; there is only an uncreative relation between the token and a sense (or senses). Thus, on the basis of the theory of composite objects, we can distinguish the domains of grammar and pragmatics on the grounds that the 24. Reference aside, literal uses are ones in which the sense of an utterance or inscription is the same as one of the senses of the sentence type of which it is a token, and nonliteral uses are ones in which the sense of an utterance or inscription is different from any of the senses of the sentence type of which it is a token. A complete account of literalness and nonliteralness would require extensive discussion, but we can use this approximation for present purposes. For a fuller account see Katz (1990b, 144–55).

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objects of study in the former are exclusively abstract objects while the objects of study in latter are composite objects. The relation of sentences to their senses is an internal relation (within the structure of a sentence), while the relation of utterances and inscriptions to their senses is an external one. In line with this distinction, we can distinguish between two notions of proposition (see Katz in preparation). On one notion of proposition, familiar from the philosophy of language, propositions are senses of sentence types. This is Frege’s notion of thought minus his understanding of senses. On this notion, propositions belong to the category Abstract Object, and the pragmatic relation between an utterance or inscription to the proposition it expresses is a relation between a composite object and an abstract object. On the other notion of proposition, familiar from philosophical logic, propositions are the bearers of truth values. There are various ways to construe the notion of propositions as bearers of truth values. On one currently popular construal, they are Russellian propositions. The proposition expressed by (11) (11) Bill Clinton is a politician. consists of the ordered pair of Bill Clinton and the property of being a politician. (Russellian propositions are also sequences of n objects paired with an n-placed relation.) Such propositions, on the present approach, are for the most part composite objects. For example, the proposition expressed by literal uses of (11) are composite objects consisting of a concrete object, i.e., Bill Clinton, an abstract object, i.e., the property of being a politician, and the creative relation of attribution. Now, given this notion of proposition, the pragmatic relation is, in part, a relation between composite objects. Accordingly, grammar concerns the relation of one set of abstract objects, sentences, to another set of abstract objects, senses, while pragmatics concerns the relation of composite objects, utterances or inscriptions, both to abstract objects, senses, and to composite objects, Russellian propositions. This distinction between grammar and pragmatics is a distinction between a pure science, grammar, and an applied science, pragmatics. Generalizing this distinction over the formal sciences, we can draw the distinction between pure and applied science on the grounds that the pure formal sciences exclusively concern the structure of abstract objects, while the corresponding applied formal sciences concern composite objects in which those abstract objects are components. The distinction between pure geometry and applied geometry and the distinction between pure set theory and applied set theory are, therefore, distinctions drawn in terms of a domain of abstract objects and a

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domain of composite objects. The pure formal sciences are all alike in that each is about a domain of abstract objects—geometric forms, sets, or numbers—but their applied counterparts are not. Given this generalization, the application of arithmetic to count the membership of collections differs from the application of pure set theory to collect concrete objects or the application of pure geometry to represent the shapes of concrete objects. The difference is that, while the applications of the latter involve an applied geometry and an applied set theory, the application of the former does not involve an applied arithmetic. This is because the number of relation is not creative. There are no objects over and above the abstract objects in the domain of mathematics and the concrete objects in the world that could serve as the subject matter of an applied arithmetic. 5.13 Compound Composite Objects Compound composite objects are composite objects all of the components of which are composite objects. Linguistic compound composite objects are compound composite objects all of the composite objects of which are linguistic composite objects, i.e., composite objects with linguistic abstract objects as components. Examples are literary speech, such as Lincoln’s address at Gettysburg, Ransom’s original copy of “Bells for John Whitesides’ Daughter,” and Tolstoy’s draft of War and Peace, and more mundane discourses such as commencement addresses, after-dinner speeches, lectures to children on their table manners, and locker room pep talks.25 It is widely recognized that expressions like “Lincoln’s Gettysburg Address,” “Gödel’s incompleteness proofs,” and “Beethoven’s Waldstein” are ambiguous between a token sense and a type sense. (12) Lincoln’s Gettysburg Address was the topic of the article. (13) Lincoln’s Gettysburg Address is a literary masterpiece. (14) Lincoln’s Gettysburg Address was audible to its entire audience. (12) has the ambiguity. (13) and (14) disambiguate it. (13) is true and (14) false. The objects that they are true and false of are different objects, which customarily are taken, respectively, to be a discourse type and a discourse token of that type. The theory of composite objects represents 25. Construing a discourse as a simple composite object, that is, as a sequence of abstract objects and a sequence of concrete objects, misconstrues its components. The components of discourses are utterances and inscriptions, not concrete acoustic and orthographic objects. Those at Gettysburg who fell asleep halfway through Lincoln’s address heard some of its utterances and missed others.

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these facts straightforwardly. It takes the expression “Lincoln’s Gettysburg address” in (13) to refer to an abstract object consisting of a sequence of abstract objects (sentence types). And it takes the expression “Lincoln’s Gettysburg Address” in (14) to refer to a compound composite object consisting of a sequence of tokens of those sentence types. Hence, the theory explains the ambiguity of “Lincoln’s Gettysburg Address” as an ambiguity between a sense of the expression referring to a compound composite object (i.e., the speech Lincoln gave at Gettysburg) and a sense of it referring to an abstract object (i.e., the common semantic content of the address that Lincoln delivered at Gettysburg and all the other discourse tokens of the discourse type Lincoln’s Gettysburg Address). The creative relation for a compound composite object is the succession relation. It relates the composite object components of a compound composite object to one another on the basis of the position of their abstract object components in the progression of abstract objects in the discourse type. In virtue of this creative relation, there is an emergent object, e.g., a conversation, lecture, address, poem, novel, and so on, which is a token of a literary type. Two such tokens, compound composite objects, are identical just in case they have the same components standing in the same creative relation. The existence condition for a compound composite object is the sum of the existence conditions for its component composite objects and the relation of succession holding among them. Thus, compound composite objects, like their composite object components, are contingent objects. Compound composite objects have properties relative to their composite objects, which in turn have them relative to their component abstract or concrete objects. Thus, a compound composite object like Gödel’s original inscription of his incompleteness proof can be said to be quite legibly written and to have taken Gödel more than a second to write down, but it can also be said to be valid and to prove that no formal theory answers all mathematical questions. Gödel’s proof has the former properties relative to the concrete objects in the sequence of composite objects and the latter properties relative to the abstract objects in the sequence of composite objects. We are taking a discourse type to be a progression of sentence types. We are taking a discourse token to be a token of a discourse type. Someone’s discourse is a token of a discourse type D just in case the sentence or expression type in the ªrst composite object in his or her discourse is the ªrst member of D, the sentence or expression type in the second composite object is the second member of D, and so on. We can think of a literary or musical form (or subform), such as the address, the poem, the sonnet, the novel, the sonata, and so forth, as a set of constraints on the structure of the composite objects in a

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compound composite object. These constraints do not express clauses in the deªnition of the notion of compound composite object. Rather, they express an independent classiªcation of such linguistic or musical objects. We are now in a position to respond to the criticism that realism fails to do justice to the fact that the contributions of authors and composers are creations rather than discoveries. In responding to this criticism, let us begin by employing the above distinction between discourse types and discourse tokens to remove the ambiguity of expressions like “Lincoln’s Gettysburg Address,” “Beethoven’s Waldstein,” and “Gödel’s incompleteness proofs” (and their consequent equivocal reference). On the one hand, reference to discourse types is reference to abstract objects, and, accordingly, talk about the work of an author or composer in such cases is talk about an abstract object. Here, an author’s or composer’s contribution to the subject can only be the making of a discovery. On the other hand, reference to a discourse token is reference to a compound composite object. Being contingent, such objects—manuscripts, performances, scores, and the like—do not exist prior to being brought into existence through the efforts of the authors or composers. Here, their contribution is a creation. Creation, in the strict sense, is ªrst-tokening—producing the ªrst concrete objects to fulªll the existence conditions for the compound composite literary or musical object (i.e., the concrete objects that are the constituents of the composite objects in the compound composite object).26 Given this, this criticism of realism is put to rest. It is true that people sometimes speak of creation when referring to types, but such speech cannot be taken in a strictly literal way to mean that a type, something without spatial or temporal location, has been causally brought into existence. Rather, the speaker wants to say something about the writer’s or composer’s originality in connection with the type, but is vague about the distinction between type and token. Once we are clear about the distinction, we have a perfectly good way of expressing what the speaker wants to say about the writer’s or composer’s originality in connection with a type. Originality here is ªrst-tokening of the type. Realists thus have a straightforward answer to how they can recognize an author’s creative achievement or his or her special relation to the artistic work understood as the abstract object. The artistic work in the intended sense is a creation as well as a discovery. Since the artist’s creative achievement, special relation to the work, and originality lie 26. I say “in the strict sense” because of cases where two or more independent ªrsttokenings occur at about the same time. There are, of course, other complications, but they are only matters of adjusting temporal and cultural parameters.

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in his or her ªrst-tokening, the debt we owe to the artist is for making it possible for us to appreciate the abstract type as well as exceptional tokens of the type. Hence, on realism, discussions of creation and creativity in the arts can proceed as usual within the theory of composite objects. Everything one would want to say about Lincoln’s rhetorical achievement or Beethoven’s musical achievement and about their special relations to their works can be said in the theory.27 The theory of composite objects suggests an interesting distinction within the arts between literary and musical composition, on the one hand, and the static visual arts, e.g., drawing, painting, sculpture, and photography, on the other. In literary and musical composition, the artwork or aesthetic object is the abstract object, while in the static visual arts, the artwork is the composite object. The drawing, painting, or sculpture is the aesthetic object. Goya’s The Third of May, 1808, the physical thing that hangs in the Prado, is a masterpiece of Western art. We treasure it both aesthetically and historically. Destroy it and we destroy the artwork, while destroying Lincoln’s ªnal text does not destroy an artwork, but only a valuable historical document. A particular composite object such as Lincoln’s ªnal text of the Gettysburg Address is treasured as a relic of a signiªcant phase of our history and as a memento of a revered historical ªgure, but not per se aesthetically. We treasure the abstract Gettysburg Address aesthetically, as a masterpiece of the rhetorical form. (The dynamic or performing arts, e.g., theater, dance, and concerts, are the mixed case. In them, there are two artworks, both the abstract object and the composite objects.) The fact that the physical painting The Third of May, 1808 is the art object while the physical text of Lincoln’s Gettysburg Address is not the art object explains why a fake of the former is an art fraud while 27. The above considerations have application to the controversy between Kivy (1983, 1987, and 1988), who takes the realist view that musical works are abstract objects, and Levinson (1990, chs. 1 and 10), who takes the conceptualist view that they are creations of human agency. Extending Kivy’s position with the theory of composite objects enables it to handle Levinson’s concerns without embracing what he (1990, 216 n. 4) calls “qualiªed Platonism,” a position on which objects that necessarily exist atemporally and independently of the causal powers of concrete objects are nonetheless brought into existence through human action. For example, Levinson (1990, 218) is legitimately concerned that we be able to say that Columbus’s discovery of America wasn’t logically his [the sense of the relation of unique possession taken to obtain in the case of artists and their works] in virtue of his discovering it. But Ives’s symphonic essay The Fourth of July is irrevocably and exclusively his. This concern is met without embracing qualiªed Platonism because Ives’s action is a ªrst-tokening, and, as we have seen, the theory of composite objects explains both such strong possession and uniqueness.

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a fake of the latter is not. Since the art object is the Goya painting hanging in the Prado, fakes can be passed off as the real The Third of May, 1808. In contrast, since the art object is the discourse type of which Lincoln’s ªnal text is a token, there can be no fake. Lincoln’s rhetorical masterpiece, being an abstract object, cannot itself be reproduced or copied, just tokened. Someone’s forgery of Lincoln’s ªnal text would not be a reproduction or copy of the aesthetic object, only another token of the type Lincoln’s Gettysburg Address. Selling it as the original for an exorbitant price is commercial fraud, not art fraud. To be sure, there are art frauds in literature or composition in the special sense of works falsely represented as a newly found Shakespeare sonnet or Schubert sonata. This is the generic, not the indeªnite particular, sense of the expression “an art fraud.” Such cases are fraudulent in that they are not instances of a Shakespeare or Schubert tokening, that is, of a tokening by the artist represented as tokening them.28 5.14 The Plenitude of Composite Objects The plenitude of composite objects will not sit well with some philosophers. Armstrong (1989, 19–20) tells us that, in ontology as elsewhere, “entities are not to be postulated without necessity.” Philosophers with this view of the scope of Occam’s razor will doubtless see the plethora of composite objects as a frightening violation of that venerable principle. They will see the ontological commitments of the theory of composite objects as too high a price to pay even for providing us with a way of preserving our intuitive concepts of abstractness and concreteness. The theory of composite objects cannot be defended on the grounds that the traditional abstract/concrete distinction, (D), or something like it has to be preserved at all, or these, costs. Realists may be compelled to save it, but they would be hard pressed to demonstrate to nonrealists that its virtues outweigh an enormous proliferation of entities. There are two much better lines of defense. One accepts the claim that ontological theories are subject to Occam’s razor and the other rejects it. On the ªrst line, we can argue that the work done by the theory of composite objects compensates for the entities to which it commits us. More importantly, we can deny that the theory of compos28. Despite the multiplicity of impressions, lithographs, woodblock prints, etchings, and tombstone rubbings, each is an art object, as can be seen from the fact that the artistic value of different impressions is different depending to a large extent on how well the impression comes out.

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ite objects commits us to an excessive plenitude of objects. Given that the domain of abstract mathematical objects is already so unsurpassably vast, it is simply false to say that the theory of composite objects increases this vastnesses. The second line of defense challenges the use of Occam’s razor in ontology on the grounds that its extension from its habitual abode in natural science to philosophy is illegitimate. Let us ªrst look at its application in the natural sciences to see whether the conditions of its legitimate application obtain outside those sciences, in formal science and in philosophy. In the natural sciences, a preference for hypotheses that explain the available evidence with the fewest postulated entities is intrinsic to empirical conªrmation. If a hypotheses H explains the same evidence as a hypothesis G, but does so by postulating more entities than G, then, other things being equal, the evidence has to bear greater weight in the case of H than in the case of G, and hence the amount of support it gives H is (proportionately) less than that it gives G. Thus, the rationale for preferring G is that the evidence supports it more strongly than H. Moreover, insofar as G explains the evidence as well as H does, without the extra entities and the more complex causal system that H postulates, we have no evidence for the extra entities and causal relations postulated in H, and hence no reason to think they exist. Occam’s razor ensures that the evidence backing up empirical claims is sufªcient to provide proper support. Hence, for Occam’s razor to apply in the formal sciences, evidence would have to play essentially the same role it does in the natural sciences. But, on our account in chapter 2 of how formal sciences differ from natural sciences, formal sciences are not evidence-driven. Natural sciences require a causal link to the objects in their domain because choosing among the various possibilities of how natural objects might be requires empirical evidence about how they actually are. Natural sciences are evidence-driven because their fundamental task is to prune the possible down to the actual. Formal sciences do not require a causal link to the objects in their domain because abstract objects couldn’t be other than they are. The fact that mathematical truths are necessary truths makes evidence otiose in mathematics, since evidence, being obtained in the actual world, cannot tell us how things are in all possible worlds. It is proof (together with intuition when, like proof, it excludes all the other possibilities) that constitutes adequate grounds for mathematical knowledge. Formal sciences are proof-driven because only proof prunes down the supposable to the necessary. Hence, on our rationalist epistemology, empirical evidence plays no role in the formal sciences. Mathematical and other formal knowledge is a priori knowledge.

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Kitcher (1983) claims that mathematical and logical knowledge cannot be a priori because empirical information that a proof has been carefully examined and found ºawless or that eminent mathematicians or logicians vouch for it constitutes empirical evidence for the theorem. Responding to Kitcher’s examples, Field (1996) argues that they are only grounds for weakening the thesis that empirical evidence cannot undercut an a priori belief to the thesis, which is usually taken to deªne the a priori, that a priori truths can be known independently of experience. I am dubious that Kitcher’s examples require even this concession. Checking a proof and hearing testimony from experts who claim to have checked it will certainly raise our conªdence that its conclusion is true, but the fact that they do so does not make either checking reasoning or testimony evidence for the theorem. They are no part of the grounds on which the truth of the theorem rests, though they may, of course, bolster our conªdence that it rests on suitable grounds. In fact, checking reasoning and getting expert testimony about such reasoning are not even evidence in empirical science, but only ways of assuring ourselves about the reliability of the methods used in obtaining the evidence. They do not make the truth of an empirical hypothesis clear to the understanding, but only bolster our conªdence that we can rely on the experiments and reasoning from which the evidence that does make it clear comes.29 Evidence from observation counts as a basis for a generalization in the natural sciences because, unless it is spurious, it reveals a causal process underlying the observed events that explains why the unobserved events fall under the generalization. Since there are no causal processes in the domain of pure mathematics, evidence does not count in mathematics. Proof counts. Veriªcations of Goldbach’s conjecture, even very many of them, do not provide evidence that establishes its truth, or even makes it probable in an objective sense. In spite of the fact that many true equations were computed in the nearly two hundred years from when Goldbach ªrst made the conjecture to when Schnirelmann demonstrated that every number is the sum of not more than three hundred thousand primes, the mathematical community took no credit for progress toward establishing Goldbach’s conjecture as a mathematical result. Starting with 4 = 2 + 2 and piling up true equations, one after another, about even numbers greater than two that 29. Witness testimony is evidence in a court of law, but, in the ªrst place, it is not about the reasoning in the courtroom but about the events the reasoning is ultimately about, and, in the second place, when true, it ªlls in a link in some causal chain concerning those events.

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are the sum of two primes, without doubt, may make mathematicians sit up and take notice, since interesting regularities among integers are infrequent enough to make piling up a large number of true instances noteworthy. Moreover, it may well increase the strength of mathematicians’ conªdence. But the conªdence of mathematicians is not mathematical knowledge.30 Moreover, mathematicians do not seem much bothered about the number of entities posited in a new theory. Their concern is rather with the rigorous development of principles and the consistency of systems. As the examples of set theory and number theory show, when these concerns are met, there is no remaining problem about the vastness of the domain of mathematical objects. This is, of course, not to say that mathematicians do not have a preference for simplicity. But simplicity in mathematical theories has to do with generality, symmetry, and similar features rather than with ontological economy.31 Some of the most highly prized mathematical theories, e.g., Cantorian set theory, are the most ontologically extravagant. When simplicity considerations enter into the choice of a formal theory, they are not a matter of ontological economy but a matter of systematic elegance. The impetus for the Occam’s razor criticism of the theory of composite objects comes from the assumption, explicit in Armstrong’s (1989, 19–20) discussion, that Occam’s razor is a universal principle to which every discipline must submit. But, since Occam’s razor is linked to evidential justiªcation and since the formal sciences are not evidence-driven but proof-driven, Occam’s razor has no role in the formal sciences. Hence, the Occam’s razor criticism of the theory of composite objects disappears. Furthermore, the similarity between ontology and mathematics is a reason for thinking that the criticism does not apply to ontology. Ontology is part of philosophy. Philosophy, to be sure, is not mathematics, but it is closer to mathematics than to the natural sciences. 30. Thus, Casti (1996, 105) is fundamentally mistaken in claiming that . . . nondeductive modes of reasoning—induction, for instance, in which we jump to a general conclusion on the basis of a ªnite number of speciªc observations— can take us beyond the realm of logical undecidability. So if we restrict our mathematical formalisms to systems using ªnite sets of numbers or nondeductive logic, or both, every mathematical question should be answerable. From our standpoint, on the former restriction, our deductive answers will not be to the right mathematical questions, and, on the latter restriction, our inductions concerning the right mathematical questions will not really provide knowledge of answers to them. 31. Of course, constructivist mathematicians are an apparent exception, but their concern with economy in mathematics is at bottom philosophical.

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Philosophical truths are not always necessary truths, and philosophical claims are not typically determined on the basis of proof, but their truth is no more determined on the basis of empirical evidence than is the truth of mathematical claims.32 Ontological claims are settled a priori on the basis of argument and counterexample. Philosophical argument, though a poor relation in the family of demonstrative methods, is nonetheless, like proof, a method of establishing truth by the purely rational means of eliminating possibilities. Since philosophical argument does not gain its force from the weight of evidence any more than does mathematical argument and since Occam’s razor applies only in regard to evidence, Occam’s razor is as inapplicable to ontology as it is to mathematics. Hence, the theory of composite objects cannot sin against Occam’s razor. The same conclusion can be reached directly from the fact that pure ontology is the foundations of the foundations of the sciences. The foundations of mathematics and other formal sciences do not concern themselves with postulating entities but with understanding the nature of the entities postulated in mathematics proper and other formal sciences. The issue is not whether there are numbers, which has already been settled within mathematics. The issue in applied ontology is what numbers are. Are they expressions, ªctional objects, mental constructions, abstract objects, points in an abstract structure, or perhaps something else? Hence, the theory of composite objects, which is a theory in the foundations of the foundations of the formal sciences, belongs to a level beyond that at which such questions of existence are addressed. Questions such as whether there is an equator or whether there are impure sets, utterances, and inscriptions, like the questions of whether there are circles, numbers, and so on, have already been answered at the levels of the sciences proper and the foundations of the sciences. Therefore, the theory of composite objects, which is a theory in a discipline at the level of the foundations of the foundations of the sciences, which simply explicates concepts, can bear no responsibility for postulating entities without necessity. One simplicity consideration is applicable to the theory of composite objects. Theories are criticized for introducing more categories than are 32. Philosophers sometimes talk about proving this or that philosophical conclusion, but it may well be that proof is not possible in philosophy, or that it is possible in principle, but, owing to the complexity of philosophical proof and the difªculty of attaining sufªcient rigor in philosophy, it is impossible in practice. Perhaps linguistics is between mathematics and logic, at the one extreme, and philosophy, at the other. At present, linguistics relies more on quasi-formal argument than on proof, but such argument is more like proof than philosophical argument, and linguistic reasoning is becoming more and more like mathematical reasoning as it becomes more sophisticated. These are important questions to which I hope to return sometime.

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necessary to do their explanatory or explicative work. Here, at least, is a relevant way of criticizing the theory of composite objects. But the category Composite Object is so small an addition to ontological theory and its explanatory payoff is so large that the theory of composite objects cannot be faulted on this score. Since the theory of composite objects brings order to the otherwise chaotic situation created by the second class of alleged counterexamples, and since it also sheds light on many topics in the philosophy of mathematics, logic, linguistics, aesthetics, and metaphilosophy, the introduction of the category Composite Object must be counted as a highly economical addition.33 33. Compared with other ways of dealing with the second class of alleged counterexamples (see, for instance, Hale 1987, 45–77), the addition of one extra category is a paragon of parsimony.

Chapter 6 Toward a Realistic Rationalism

6.1 The Path Back to Rationalism 6.1.1 Frege and the Linguistic Turn The inºuence of the linguistic turn, like that of other signiªcant intellectual revolutions, continues even though the linguistic turn is now history. However, the nature of its inºuence has not been fully understood, because the linguistic turn is generally seen in terms of the linguistic doctrines expressed in the slogans and manifestoes of its leading ªgures, and those linguistic doctrines are no longer taken seriously. But the inºuence that the linguistic turn exerts on philosophy is a matter of the continuing ascendancy of the positivist, empiricist, and naturalist views behind those doctrines rather than a matter of the temporary ascendancy of the doctrines themselves. The linguistic turn is mistakenly thought of as a nonpartisan attempt to reframe philosophical questions in manageable linguistic terms, and, through linguistic analysis (of one form or another), to eliminate philosophical questions that make no sense (e.g., Heidegger’s “Does the Nothing exist only because the Not, i.e., the Negation, exists?”). On the surface, it was often such a nonpartisan attempt, but, at a deeper level, it was another of the partisan attempts, which occur in the history of philosophy from time to time, to cleanse philosophy of metaphysical positions taken to be the cause of all its ills. The linguistic turn steered philosophy away from the traditional rationalist enterprise of explaining synthetic a priori knowledge and toward new forms of positivism, naturalism, and empiricism. The term “linguistic” had little to do with the deeper ends of the “revolution in philosophy.” It expressed the means chosen for effecting change and reºected the perception that dictated the choice of means, the perception that the linguistic (especially, the semantic) foundation of the doctrine of synthetic a priori knowledge is the soft underbelly of traditional rationalism. It is ironic that it was Frege, the greatest rationalist and realist metaphysician of nineteenth- and very early twentieth-century philosophy, who provided the ªrst phase of the linguistic turn with the

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linguistic and logical weapons for its assault on rationalist and realist metaphysics. In the ªrst phase, the early Wittgenstein, Schlick, Carnap, and their Viennese followers took both their linguistic cue and their logico-semantic tools from Frege (and Russell), using them to criticize traditional philosophy—much of which Frege (and Russell) subscribed to—as cognitively meaningless, and, on the model of Frege’s Begriffsschrift, to develop a conception of an ideal language and the novel thesis that, as Ayer (1946, 57) expressed it, “philosophy is a department of logic.” In the second phase of the linguistic turn, logical empiricism itself came under attack in large part for its Fregeanism. The late Wittgenstein and Quine directed their criticisms against the Fregean semantics underlying the forms of empiricism and positivism developed in the ªrst phase. Both saw Fregean semantics as the basis for the metaphysician’s claims about synthetic a priori knowledge, and both used those criticisms as a springboard for launching even more extreme forms of positivism and empiricism (see Kenny 1973, 113–14; Quine 1961c). The late Wittgenstein continued to argue for a naturalistic and positivistic view of philosophy on which sentences about nature are the only meaningful sentences and metaphysical sentences transcend the limits of language. Quine, eschewing positivistic naturalism, argued for a scientistic naturalism concerned with showing that much of traditional metaphysics is scientiªcally anachronistic speculation about the natural world. Quine’s (1969a, 82) naturalism replaces the positivist idea of exposing metaphysical statements as nonsense with the idea that philosophy goes on within natural science, either as a form of metascientiªc analysis of the conceptual and linguistic side of the scientiªc enterprise or “as a chapter of psychology and hence of natural science.” In this section, we will examine both the role played by Fregean assumptions in the development of the philosophical positions in the ªrst phase of the linguistic turn and the philosophical signiªcance of the Fregean focus of Wittgenstein’s and Quine’s criticisms of those positions in the second phase. I will try to show that the preoccupation with Frege has made the linguistic underbelly of rationalism appear far softer than it is by linking rationalism to Fregean intensionalism, thus obscuring other, less vulnerable intensionalist ideas. As we shall see, an intensionalism that breaks radically with Fregean intensionalism can provide concepts of meaning and analyticity that are not open to criticisms directed against the semantics assumed in the ªrst phase of the linguistic turn. These conceptions of meaning and analyticity are, moreover, too thin to be used for any positivistic purpose like those in the ªrst phase. Thus, such an intensionalism has the potential to restore

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explanation of synthetic a priori knowledge to its rightful place in philosophy. 6.1.2 The Vicissitudes of the Synthetic A Priori Passmore (1957, 157) writes: . . . in arguing that it is language that leads us astray or, again, in setting up the ideal of a perfect language which would not betray us because in it every expression would have a ªxed and deªnite sense, Frege, more than any other nineteenth century philosopher, anticipates the preoccupations of twentieth century positivism and its diverse progeny. To be sure, the metaphilosophical ideas with which Passmore credits Frege, together with the technical logical and linguistic machinery he invented and the important philosophical arguments he developed, provided the early Wittgenstein, Schlick, and other logical empiricists with the philosophical, logical, and semantic tools for their work. But Passmore’s epitome of Frege as farsightedly anticipating twentiethcentury philosophy puts the wrong slant on Frege’s relation to the revolution, missing his ambiguous role as the intellectual benefactor of the positivist, naturalist, and empiricist views that came to replace the traditional rationalist/realist view of philosophy as an autonomous a priori study of the most general facts about reality. It was Frege, the rationalist and realist, who made it all possible. Frege’s logico-mathematical innovations and his criticisms of Kant’s doctrines of analyticity and mathematical truth were the entering wedge for the early logical positivists’ attack on metaphysics, especially Kantian, neo-Kantian, Husserlian, and other forms of then current metaphysics. Before Frege, Kant’s criticism of Hume had stood as the bulwark against empiricistically and naturalistically inspired positivism. The vagueness of Hume’s notion of relations of ideas allowed Kant to explicate that notion as analyticity in his concept-containment sense. This linked mathematics with metaphysics as synthetic a priori knowledge. Since the former as well as the latter is outside both Hume’s categories of relations of ideas and matters of empirical fact, they share a common fate. If, as Hume advocates, metaphysics is “consigned to the ºames,” then mathematics is too. This enabled Kant to argue that, since mathematics cannot be abandoned, metaphysics cannot be abandoned either. The task of philosophy must thus be the metaphysical one of explaining how synthetic a priori knowledge in mathematics, science, and philosophy is possible. Kant’s argument assumes that there is no other explication of the notion of relations of ideas that is preferable to his own notion of

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analyticity. If there were another, one broad enough to bring mathematics under analyticity, it could cut the link between mathematics and metaphysics, thereby stripping metaphysics of the protection it receives from our unwillingness to abandon mathematics. This is precisely what Frege’s explication of analyticity as truth based on laws of logic and deªnitions purports to do. Coupled with Frege’s criticism of the Kantian notion of analyticity, Frege’s logical explication of analyticity creates an entirely new situation for naturalists and empiricists generally. Rejecting Kant’s account of mathematics as synthetic a priori knowledge in favor of Frege’s account of it as analytic a priori knowledge, the logical empiricists could reject Kant’s reductio of Hume’s positivism, his explanation of mathematical knowledge, and his rehabilitation of metaphysics. Dummett (1991, 111–24) also casts Frege—speciªcally, the Frege of the Grundlagen—as the initiator of the linguistic turn. Dummett (1991, 111) claims that Frege’s signiªcant philosophical move in initiating the linguistic turn was invoking the context principle to transform Kant’s question about our knowledge of numbers into the linguistic question about “how the senses of sentences containing numbers are to be ªxed.” Dummett exaggerates the importance to the linguistic turn of this relatively obscure use of the context principle in the philosophy of mathematics. It was of signiªcantly less importance than Frege’s general metaphilosophical ideas about the imperfections of language, his notion of an ideal language as a means for overcoming them, his criticism of the Kantian concept of analyticity as “unfruitful,” and his replacement of it with his own logical concept of analyticity. The logical empiricists seized on Frege’s concept of analyticity as showing that the Kantian concept of analyticity, due to its extreme narrowness, created the illusion of synthetic a priori knowledge which neo-Kantians, assorted rationalists, and Husserlian phenomenologists exploited to justify introducing special faculties such as intuition. Dubious appeals to intuition might have to be swallowed if Kantian analyticity were the only explication of the notion of relations of ideas, but, after Frege, there was no longer a necessity to swallow them. The early Wittgenstein and the logical positivists saw metaphysical explanations of alleged synthetic a priori knowledge as both misguided and unnecessary—misguided because they falsely claim to provide a priori insights into the nature of things, and unnecessary because Frege had provided the means for an alternative, analytic, explanation of a priori knowledge. Frege’s broadening of the category of the analytic opened up the prospect of accounting for all allegedly synthetic a priori knowledge as

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analytic a priori knowledge and thereby resuscitating Hume’s empiricism. Ayer (1946, 85), who popularized the new empiricism, wrote: Our knowledge that no observation can ever confute the proposition “7 + 5 = 12” depends simply on the fact that the symbolic expression “7 + 5” is synonymous with “12,” just as our knowledge that every oculist is an eye-doctor depends on the fact that the symbol “eye-doctor” is synonymous with “oculist.” And the same holds good of every a priori truth. To the early positivists, the view that mathematical truths are analytic in the sense of holding in all possible cases meant that those truths do not have factual content. Carnap (1963, 47) commented on this: . . . What was important in this conception from our point of view was the fact that it became possible for the ªrst time to combine the basic tenets of empiricism with a satisfactory explanation of the nature of logic and mathematics. Previously, philosophers had only seen two alternative positions: either a non-empiricist conception, according to which knowledge in mathematics is based on pure intuition or pure reason, or the view held, e.g., by John Stuart Mill, that the theorems of logic and mathematics are just as much of an empirical nature as knowledge about observed events, a view which, although it preserved empiricism, was certainly unsatisfactory. Mathematics no longer falls between the two Humean stools of relations of ideas and matters of empirical fact, and hence the empiricist is no longer in the embarrassing position of consigning mathematics to the ºames along with metaphysics. Wittgenstein and Quine, at ªrst sympathetic to the program of accounting for allegedly synthetic a priori knowledge as analytic a priori knowledge along the lines of Fregean analyticity, came, for different reasons, to see the program not as the solution for the ills of philosophy but as part of the problem. Both saw Frege’s theory of meaning as the principal culprit—in Quine’s case, as presented in Carnap’s semantics. Their criticisms were largely directed at it. For Wittgenstein, disenchantment resulted, as Allaire (1966) and Kenny (1973, 103–19) have pointed out, from his losing faith in Frege’s theory of meaning largely due to the difªculty about color incompatibility that Wittgenstein ([1922] 1961, secs. 6.375 and 6.3751) described in the Tractatus. Subscribing both to Frege’s conception of logical form and his view that logical necessity is the only necessity there is, Wittgenstein could not explain how sentences like “This spot is (entirely) red and (entirely) green,”

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which conjoin atomic (and therefore logically compatible [[1922] 1961, sec. 4.211]) propositions, could be necessary falsehoods. Wittgenstein (1953, e.g., sec. 92) came to think that this problem revealed a basic inadequacy of Frege’s account of meaning in natural language. Since there is no basis for logical necessity in the atomic form of such sentences, the idea that the logical powers of sentences derive from their logical forms, which are hidden beneath their surface syntactic forms, is wrong. Abandoning Frege’s semantics together with much of his own early philosophy, Wittgenstein developed a new account of meaning on which the logical powers of sentences do not have the force of necessity because they derive from the use of words. Nonetheless, he maintained his basic positivism on which all of traditional philosophy is to be rejected as nonsense. In the new account of meaning, however, the nonsense derives from the misuse of words. Wittgenstein thus continued to press his attack on metaphysics, though now on the basis of other, as he came to see them, more adequate linguistic means. Quine saw Frege’s theory of meaning as the principal obstacle to an uncompromising empiricism. Quine’s (1961c, 23) characterization of the notion of analyticity that he attacks in “Two Dogmas of Empiricism”—what a logical truth turns into when we put synonyms for synonyms—is Frege’s notion of a truth that rests on logical truths and deªnitions. Quine’s focus was on Carnap, who used his formalization of Fregean analyticity as a basis for developing a compromising empiricism that rejects Mill’s view that “the theorems of logic and mathematics are just as much of an empirical nature as knowledge about observed events” as “unsatisfactory”. Quine, however, saw Mill’s empiricism as essential to an uncompromising empiricism and Fregean analyticity as the last refuge for the rationalist doctrine that logical and mathematical knowledge is a priori knowledge. In his attempt to restore what he saw as the only true empiricism, Quine targeted the Fregean theory of meaning as updated in Carnap’s semantics, in particular on the notion of synonymy on which Frege’s extension of the class of logical truths to the class of analytic truths depends. Carnap (1956a, 222–29) had updated Frege’s semantics with the introduction of meaning postulates. This promised to provide Fregeans with a way to get around Wittgenstein’s problem about ascribing modal properties to sentences like “This spot is (entirely) red and (entirely) green”. On the meaning postulate approach, the logical powers of sentences arising from their extralogical vocabulary are handled in the same way as the logical powers of sentences arising from their logical vocabulary. Given a set of logical postulates that state the contribution of the logical vocabulary to the extensional structure

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of L, the analytic sentences of L can be characterized as sentences of L that follow just from the logical and meaning postulates of L. Since meaning postulates can be formulated so that mathematical truths come out as analytic sentences of the language, the formal side of the logical empiricist’s resuscitation of Humean empiricism is complete. In “Two Dogmas of Empiricism,” Quine (1961c) tried to show that there is nothing in the areas of deªnition, linguistics, and logic—the areas to which, respectively, the notions of meaning, synonymy, and analyticity most naturally belong—that can be used to make objective sense of those notions. The story is familiar. Semantic deªnitions rest on prior synonymy. Abbreviation is irrelevant. An objective notion of linguistic synonymy requires a substitution test that distinguishes between synonymous and nonsynonymous pairs of expressions, but all such tests are circular. Meaning postulates are neither general enough nor explanatory enough to explicate analyticity. Quine’s solution to Wittgenstein’s problem about the logical powers of elementary sentences was simply to deny that such sentences have logical powers. Since the extralogical vocabulary of a language contributes nothing to the logical powers of sentences, there is no need for intensions or the apparatus to represent them. This nihilistic solution ªts perfectly with Quine’s epistemological outlook. From the perspective of his Millian instincts, Quine saw the introduction of intensions as compromising both the extensionality of logic and the purity of empiricism. Carnap’s (1963, 64) logical empiricism, as we saw in chapter 3, explicitly recognized that “the rationalists had been right in rejecting the old empiricist view that the truth of ‘2 + 2 = 4’ is contingent upon the observation of facts.” On Carnap’s Humean empiricism, only nonformal knowledge depends on experience, while, on Quine’s Millian empiricism, all knowledge depends on experience with the natural world. One side of the Quinean coin was to eliminate the logical empiricist’s restriction of empiricism to nonformal knowledge; the other side was to provide an epistemology for knowledge generally on which the allegedly a priori truths of logic and mathematics are at bottom empirical. The sticking point for Carnap (1963, 64) was Millian empiricism’s “unacceptable consequence that an arithmetical statement might possibly be refuted tomorrow by new experiences.” Quine (1961c, 44) takes care of this with his account of logical and mathematical statements as central components of our total fabric of scientiªc beliefs. Even though logical and mathematical statements are revisable in principle in the light of new experiences, their centrality in the total fabric of our scientiªc beliefs makes their revision something that can occur only as a last resort.

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Despite their clear differences, the Wittgensteinian and Quinean positions that displaced logical empiricism have much in common. Both adopt a form of naturalism, both are anti-intensionalist in the extreme, both reject absolute necessity, both deny the existence of synthetic a priori knowledge, and, as a consequence, both are implacably opposed to the traditional metaphysical conception of philosophy. As Quine (1960, 76–77) himself notes, their common target was theories of meaning based on the concept of linguistically neutral meaning. Once those theories are removed and replaced with a conception of meaning based on holism à la Wittgenstein or holism à la Quine, the broad notion of analyticity, which served as the logical empiricists’ basis for a priori knowledge, disappears, and with it the traditional rationalist enterprise. Thus, the only issue left unresolved by the linguistic turn, the issue that today divides Anglo-American philosophy, is whether it is Wittgenstein’s ontological naturalism with its positivistic and therapeutic emphasis or Quine’s epistemic naturalism with its scientistic and pragmatic emphasis that goes in its place. I said at the outset that the path back to rationalism is through a critical examination of Frege’s role in the arguments of the philosophers who brought about the two phases of the linguistic turn. As we have seen, Schlick, Ayer, Carnap, and other logical empiricists used Frege’s broadened conception of analyticity as a basis for their unmetaphysical conception of the a priori. And, as we also saw, Wittgenstein and Quine accordingly took Frege’s conception of analyticity and his theory of meaning as their target. Now I concede that Wittgenstein’s and Quine’s arguments were entirely successful in refuting that conception of analyticity and that theory of meaning. What I do not concede is that, in successfully refuting them, Wittgenstein’s and Quine’s arguments do what they have to do to refute all of the theories of meaning on which the a priori and rationalism can be based. Here we return to the problem concerning the apparent logical powers of extralogical words. If Carnap’s, Wittgenstein’s, and Quine’s solutions exhaust the approaches to this problem, then the arguments of Wittgenstein and Quine have done all they have to do to refute them, but if their solutions do not exhaust all of the approaches, the entire second phase of the linguistic turn is in question. An alternative approach has to be intensionalist, so it has to retain the idea that meanings are senses of linguistic expressions, analyticity is a property of senses, and synonymy is a relation among them. Further, since Wittgenstein’s problem arises in connection with logically elementary sentences, the sense structure of sentences cannot always be—and in the case of logically elementary sentences such as “Bachelors are single” isn’t—reºected in their logical syntax. Finally, the Frege

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principle, which Wittgenstein ([1922] 1961, secs. 6.37 and 6.375) embraces in the Tractatus, that logic is the only source of necessity, has to be dropped. In opposition to Frege, we have to say that linguistic meaning is an independent source of necessity. We have to say that logically elementary sentences can have richly structured senses. This move is the key to solving Wittgenstein’s problem concerning the logical powers of logically elementary sentences (see Katz [to appear]). The appearance that the powers are logical is misleading. The powers are not logical but semantic. Semantically rich nonlogical senses make it possible to overcome the logically elementary character of sentences such as “Bachelors are single.” To make this approach work, we require a non-Fregean deªnition of sense, one that characterizes the semantically rich senses of syntactically simple expressions nonlogically. I provided such a deªnition in Semantic Theory and subsequent publications. I (1972, 1–10) deªned sense as the aspect of the grammatical structure of expressions and sentences that determines their meaningfulness, synonymy, antonymy, ambiguity, redundancy, analyticity, and other sense properties and relations. Frege’s arguments that senses are necessary to explain the informativeness of identity statements, substitution into oblique contexts, and meaningfulness in the absence of reference motivate the introduction of a concept of sense, but they do not motivate the introduction of just Frege’s concept. Our concept of sense does just as well for these purposes. And it does better against attacks on intensionalism ªrst by Wittgenstein and Quine and later by Donnellan, Kripke, and Putnam. As I (1986, 1990b, 1994a, 1997) have argued, the success of these attacks reveals, not a weakness of intensionalism, but a weakness of Fregean intensionalism. What makes my deªnition of sense radically different from Frege’s deªnition is that it uses no concepts from the theory of reference. In deªning sense as the determiner of referential properties and relations such as denotation, truth, and logical equivalence, Frege deªnes the concept of sense in terms of the vocabulary of the theory of reference, thereby reducing the theory of sense to the theory of reference. On our deªnition, the theory of sense is fully independent of the theory of reference. It has nothing to do with the relation of language to the world. It is an autonomous theory about one aspect of the internal grammatical structure of sentences. With this new deªnition of sense, the Fregean grip on the notion of analyticity can be broken. Frege (1953, 4) had deªned an analytic proposition as a consequence of logical laws plus deªnitions without any assumptions from a special science. Frege’s (1953, 99–104) rationale for this deªnition was that it is more “fruitful” than the LockeanKantian deªnition. To be sure, fruitfulness is a virtue if analyticity is

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to serve as the basis for explaining logical truths as analytic truths in a logicist program to reduce mathematics to logic. Independently of such an ulterior motive, fruitfulness might well be a vice. If we are looking for a deªnition to explain analyticity in natural language, a Lockean-Kantian deªnition, just because of its unfruitfulness, might be preferable. I (1972, 171–200, 1986, 1988, 1990b, 1994a, 1995, 1997, and in preparation) have argued that a reconstruction of the Lockean-Kantian deªnition of analyticity ªts the facts about sense structure in natural language better than Frege’s and further that, since Wittgenstein’s and Quine’s criticisms were very largely tailored to refute Fregean intensionalism, their criticisms do not refute intensionalism per se, and that the differences between our intensionalism and Frege’s enable the former to meet their criticisms. The principle consequence of this argument, for our present concerns, is that the problem of explaining synthetic a priori knowledge for nearly the same broad range of propositions that Locke and Kant considered to be synthetic a priori is reintroduced. On our narrow concept of analyticity, a sentence is analytic in case it has a referring term with a sense that contains the sense of the entire sentence. Containment means literal, “beams in the house,” containment, not the ªgurative, “plant in the seed,” containment of Fregean analyticity. Analyticity is not a species of logical truth where, for example, every proposition contains the disjunction of itself with every other proposition. (See Katz [1986, 62–3, in preparation].) Since the breadth of a notion of analyticity and the breadth of the range of propositions that are synthetic a priori on that notion are inversely related—the wider the notion of analyticity, the narrower the range of synthetic a priori propositions—Frege’s wide notion leaves little room for synthetic a priori propositions and Carnap’s far wider notion left (as he of course intended) no room for them at all. In contrast, the narrowness of our concept of analyticity entails a wide range of such propositions. Essentially the same class of a priori mathematical and other formal truths are synthetic propositions on our analytic/synthetic distinction as on Kant’s. Thus, as I (1990b, 297) argued at the end of The Metaphysics of Meaning, the reinstatement of the traditional analytic/synthetic distinction brings “twentieth century philosophy full circle round to the situation just prior to the Logical Positivist, Wittgensteinian, and Quinean attacks on metaphysics.” After nearly a century devoted to looking for ways to avoid Kant’s problem of explaining how synthetic a priori knowledge is possible, we have come face to face with it again. The problem reemerges in even more aggravated form because Kantian explanation is no longer an option. Attempts to ground synthetic a priori knowledge on our own nature are inadequate for various

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reasons, the most salient of which being that logical and mathematical truths tell us not merely that something is so but that it must be so. Human nature, however transcendentalized, cannot explain their necessity. The argument in The Metaphysics of Meaning thus seemed to me to leave us with but one approach to explaining synthetic a priori knowledge in the formal sciences. Such knowledge has to be grounded in the nature of the logical and mathematical facts. Further, since the epistemology of formal knowledge piggybacks on the ontology of the formal sciences, a rationalist epistemology must be based on features of logical and mathematical facts that explain why they couldn’t be otherwise. Finally, since the only conception of logical and mathematical facts that explains why they couldn’t be otherwise is realism, realism had to be available for the construction of a rationalist epistemology for logic and mathematics. The situation in the philosophy of mathematics was the other way around. Since causal contact with abstract objects is impossible, a rationalist epistemology had to be available for the construction of a realist position to answer Benacerraf’s charge that realism cannot explain mathematical knowledge. The upshot was clear: not only does a realist ontology have to be available for the construction of a rationalist epistemology, but a rationalist epistemology has to be available for the construction of a realist ontology. These reciprocal demands dictated an approach to the explanation of synthetic a priori knowledge on which rationalist epistemology and realist ontology are combined into a single, uniªed metaphysical position. This position is partly developed in the earlier chapters of this book. In chapters 2 and 3, I showed how it solves the epistemic puzzle for realism induced by the causal inaccessibility of mathematical objects by explaining the a priori character of mathematical knowledge on the basis of a realist account of the necessity of mathematical truths. This explanation of how we can have knowledge of abstract objects without causal access and how it can be synthetic a priori knowledge without resorting to Kantian idealism—on which it becomes knowledge of our own sensibility and understanding—is only a partial solution to the general problem of synthetic a priori knowledge. The remaining question is whether this explanation can be generalized to other kinds of synthetic a priori knowledge. How much of the overall problem of explaining synthetic a priori knowledge can be handled within the conception of a rationalist epistemology that has been sketched in the previous chapters? The rationalist epistemology developed thus far is incomplete along two dimensions. Horizontally, it needs to be spelled out more fully for the case of synthetic a priori knowledge in other formal sciences and in

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the natural sciences. The extension to logic seems straightforward, while the extension to linguistics, at least in its present state, is not. The extension to the natural sciences is trivial if the only synthetic a priori knowledge in them is the principles from the formal sciences they employ, but not otherwise. Vertically, our sketch of a rationalist epistemology has to be extended to synthetic a priori knowledge in philosophy. We cannot deal with all of these questions here, but, as the issue of philosophical knowledge has been at the center of the arguments in the linguistic turn which ushered in the present naturalist/empiricist hegemony, I will address the question of synthetic a priori philosophical knowledge.1 6.2 From Philosophy of Mathematics to Philosophy Two related factors have undermined the twentieth-century philosopher’s belief in the existence of a priori philosophical knowledge. One is the dissatisfaction that derives from the positivist’s invidious comparison of progress in metaphysics with progress in natural science. Carnap (1963, 44–45) once put the comparison in these terms: Even in the pre-Vienna period, most of the controversies in traditional metaphysics appeared to me sterile and useless. When I compared this kind of argumentation with investigations and discussions in empirical science or in the logical analysis of language, I was often struck by the vagueness of the concepts used and by the inconclusive nature of the arguments. . . . I came to hold the view that many theses of traditional metaphysics are not only useless, but even devoid of cognitive content. This comparison continued to be inºuential even after philosophers stopped taking the view that traditional metaphysics is senseless seriously. But as a consideration against synthetic a priori philosophical knowledge, it is a straightforward petitio, because it assumes, contrary to traditional metaphysics, that metaphysics and science are enough alike to be put on the same scale. The positivist cannot simply assume that 1. Strictly speaking, we do not require an account of natural knowledge on which it is uniformly a posteriori, since most philosophers who believe that there is synthetic a priori knowledge in natural science think that it consists in principles of systematization such as simplicity and principles of inference, which are, as it were, borrowed principles. Moreover, the Quinean holistic picture of knowledge, restricted to natural knowledge, would seem to support the claim that there are synthetic a priori truths native to natural science. This might be challenged, say on the basis of Koslow (1992), but this raises issues beyond the scope of this study.

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scientiªc progress is an appropriate yardstick for judging progress in metaphysics, and, seeing that the latter does not measure up to the former, conclude that something is wrong with metaphysics. In The Metaphysics of Meaning, I (1990b, 313–17) took some initial steps in arguing further that they are not enough alike for such a comparison. This chapter will continue the argument. Even the brief account of synthetic a priori philosophical knowledge that I will present here will explain why philosophy is not sufªciently sciencelike for it to be judged by scientiªc progress. The other factor is the apparent failure of metaphysics in the face of philosophical skepticism. This is a far more serious threat to our conªdence in the existence of a priori philosophical knowledge. Unlike the interpretation that the positivist puts on the facts about progress in metaphysics and science, the apparent failure of metaphysics to handle skepticism does not have to be argued for. Most philosophers would probably say that the failure is more real than apparent. The curious thing is that the threat of skepticism has been an impetus in converting many philosophers to naturalized epistemology. Stroud (1984, 209–54) shows that Quine’s naturalized epistemology offers no advantage of all over traditional metaphysics in dealing with skepticism. Goodman’s (1955) naturalized inductive epistemology, which is an attempt to escape Humean skepticism about induction, is equally unsuccessful. As I (1962, 48–49) once argued, the Humean problem resurfaces as a problem of what right we have to think that the inductive practices to be explicated are better than other conceivable (counterinductive) practices, that is, what right we have to think they are reliable in the long run. Goodman (1955, 64–66) conªdently assumes that Hume dissolved the problem, but this is not the case. Hume didn’t show that there is no rational justiªcation of induction, but only that there is none on the empiricist assumption that conclusions about unobserved events are either based on inferences from matters of fact in the past or on relations of ideas. Given the dependency of Hume’s argument on the empiricist theory of knowledge, Goodman’s naturalization is no more protection against the skeptic than Quine’s. In the remainder of this section, I will try to show that an extension of the rationalist epistemology set out in the previous chapters to philosophical knowledge provides us with a way in which skepticism can be isolated from a large class of issues about the nature of our scientiªc and philosophical knowledge. Nagel (1986, 67–71) rightly says that the problem of skepticism is an unavoidable consequence of the realist’s objective conception of the world, but I think that we can make living with the problem considerably easier. In the next section, I will try to show that Carnap’s critique of synthetic a priori

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philosophical knowledge fails because the semantics on which it is based has the same problems as the multiform semantics of combinatorial approaches to mathematical truth (as discussed in chapter 2, section 3). The starting point for our rationalist account of synthetic a priori knowledge in the formal sciences is the supposition that such knowledge is knowledge of abstract objects. Since abstract objects are unchangeable in their intrinsic structures and the relations they have to one another in virtue of such structures, the way abstract objects actually are is the way they must be, and, accordingly, basic knowledge and much transcendent knowledge in the formal sciences can be known a priori because the way they actually are can be determined by pure reason. The initial problem in extending this account to philosophical knowledge is that philosophy is not for the most part a study of abstract objects. This means that our account of basic knowledge in the formal sciences cannot be extended to philosophical knowledge wholesale, but only to certain limited cases of philosophical knowledge where intuition informs us about philosophically relevant properties of abstract objects. Philosophy can be about anything, but its main interest focuses on epistemological and metaphysical questions. What prevents the account in chapter 2 from applying to philosophical knowledge wholesale is that philosophy studies both the foundations of the sciences and the grounds for its own conclusions about them. But even though in the ªnal analysis we may not be able to extend our account of synthetic a priori knowledge in the formal sciences to philosophical knowledge generally, we can use the account to explain synthetic a priori philosophical knowledge in the foundations of the sciences. What is required for this is a speciªcation of the relation between philosophy in its role as a second-order discipline and the ªrst-order sciences that it studies. This relation will explain how philosophy can address the aspects of reality studied in the ªrst-order disciplines. Thus, we ask: (Q1) What is the relation of philosophy to the sciences on which philosophical synthetic a priori knowledge rests? (Q2) What is the nature of philosophical knowledge? Our aim is to restore a conception of philosophy as an independent discipline providing knowledge of the most general features of reality and the nature of knowledge in the sciences. Adopting this aim does not commit us to the view that philosophy is logically prior to science or that it has primary jurisdiction over knowledge in the sciences. There is a sense in which the sciences are ªrst, though it is not Quine’s sense of science as ªrst philosophy. Philosophy is a second-order discipline

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with respect to the sciences, studying their ontological commitments, conceptual structure, and epistemic character. There is also a sense in which philosophy is ªrst. It concerns issues arising from the ontological, conceptual, and epistemological principles on which scientiªc investigation depends. Since the sciences rest on those principles, trying to settle such issues within the sciences themselves would be a bootstrap operation. As will be recalled from chapter 3, this is where Quine’s neoempiricist epistemology comes to grief: an epistemology that makes scientiªc investigation settle all issues including those concerning its own constitutive principles is incoherent.2 An answer to (Q1) can be given in terms of the conception of ontology as a foundational discipline of foundational disciplines presented in chapter 5. We distinguished three levels of investigation: the sciences, the deeper level of their philosophical foundations, and the still deeper level of general philosophy. At the ªrst level, mathematics, logic, physics, and the like investigate numbers, sets, propositions, atoms, genes, and other objects in the world. At the next level, there are the philosophy of mathematics, the philosophy of logic, the philosophy of physics, and the philosophies of the other sciences, which concern the foundations of their respective sciences at the ªrst level. At the deepest level, there are the philosophical disciplines of pure ontology and epistemology. The disciplines at each level are concerned with questions that arise but are not answered in the prior investigations. Since the sciences are continuous with common sense, they try to answer questions that arise in the course of common-sense reºection on the world but cannot be answered by common sense. These are questions like: How many 2. Contrary to the early positivists and others—then and now—who have fallen under Wittgenstein’s spell, we maintain that philosophy seeks knowledge, that is, philosophical statements the truth of which is adequately grounded. Schlick (1959, 57) claims that “the error of ‘metaphysics’” was to suppose that the foundations of [science] was made up of “philosophical” statements (the statements of the theory of knowledge), and crowned by a dome of philosophical statements (called metaphysics). Schlick’s (1959, 57) argument for this is: For if, say, I give the meaning of my words through . . . other words, one must ask further for the meaning of these words, and so on. This process cannot proceed endlessly. It always comes to an end in actual pointings. This argument is too strong. If it shows that the metaphysician is caught in an inªnite regress and must forswear making statements, it will also show that the mathematician too is caught in an inªnite regress and must also forswear making statements. No doubt, the response would be that mathematicians are not engaged in a semantic enterprise, but, of course, neither is the metaphysician.

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numbers are there? What does it mean to say there are inªnitely many of them? Why does dye turn liquids a uniform color? Why does water, unlike other liquids, expand on freezing? Why does adhesive tape stick? What is the difference between the pronouns in “It is crawling on the wall” and in “It is raining outside”? Why are “ºammable” and “inºammable” synonyms rather than antonyms? Similarly, at the level of the foundations of the sciences, philosophies of mathematics, physics, linguistics, and so on try to answer questions that arise in these sciences but perforce go unanswered in them. Typically, these questions concern the ontological nature of the objects in their domain (more generally, the reality that the sciences studies) and the epistemological character of knowledge in the science. Mathematics tells us about the vastness of the realm of numbers, but does not tell us what kinds of things numbers are. To answer that question and to explain how we acquire the knowledge we have of them, philosophers of mathematics formulate philosophical theories, such as Gödel’s realism and Brouwer’s conceptualism, and construct arguments to show that their theory is the most satisfying account of the mathematical facts and the relevant issues in the foundations of mathematics. Let us call the knowledge we obtain in the foundations of the sciences “foundational knowledge.” Foundational knowledge conforms to the naturalist’s claim that philosophers do not study scientiªc reality in the same hands-on way that scientists do. Even assuming that the naturalists were also right that scientiªc theories collectively cover reality completely, philosophical investigations can still inform us about reality, e.g., about whether numbers are abstract or concrete objects or not objects at all. Accordingly, it is not, as some naturalists have wanted to say, that second-order philosophical investigation can contribute nothing to our knowledge of the world, but rather that its contribution to our knowledge of the world is indirect, in contrast to the contributions of science. Since such investigation takes place below the level of scientiªc investigation, it concerns objects as they are represented in scientiªc theories, and consists largely in interpreting scientiªc theories. Even though philosophical contributions to our knowledge of the world are indirect, they address questions about the nature of reality because the questions are about objects the existence of which has already been vouchsafed by the science in question. Finally, pure ontology and epistemology address questions that arise at the level of the foundations of the foundations of the sciences. The questions they address arise in the controversies between rival philosophical theories in the foundations of the sciences, but are not answered in those controversies. While those philosophical theories are concerned with the nature of the reality studied and the knowledge

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acquired in the sciences, pure ontology concerns questions about the nature of the categories that those theories employ for representing the most general aspects of reality. As illustrated in chapter 5, ontology explains categories such as Abstract and Concrete in a way that accounts for their relations to one another and to other general categories. Pure epistemology concerns questions about the grounds of the principles on which scientiªc and foundational knowledge rest. It is especially concerned with the validity of those principles in the light of skeptical challenges to them. I will call the knowledge in pure ontology and epistemology “metaphysical knowledge.” We have distinguished between scientiªc knowledge and philosophical knowledge and between foundational knowledge and metaphysical knowledge. There is a third distinction that is important for our conception of philosophical knowledge. This is a distinction between internal knowledge and external knowledge. In application to our conception of philosophical knowledge, internal knowledge is knowledge within a framework where justiªcation only requires meeting internal standards of evaluation, and external knowledge is knowledge about a framework where the internal standards cannot play the role in justiªcation that they do within the framework. This terminology is adopted from philosophers like Kant and Carnap, but it marks a different distinction. In particular, unlike theirs, my terms “internal” and “external” are not also connected with a particular philosophical view about what a framework is or with philosophical doctrines like the positivist doctrine that external questions are meaningless. Justiªcation in the sciences only requires meeting the prevailing methodological standards. It would be nutty to insist that quantum physics isn’t justiªed until the physicists can explain how they know the future will be like the past or that Darwinian biology isn’t justiªed until biologists can explain how they know the world didn’t come into existence yesterday complete with apparent causal traces. This, I presume, goes without saying. But it is worth saying that justiªcation in the foundations of the sciences too only requires meeting the prevailing standards. This is a point that emerged in our response to the epistemic challenge to realism in chapter 2. There we saw that the success of a philosophical position in the foundations of mathematics does not depend on its success against skepticism. Since the prevailing standards of reasoning are not in question in the controversy in the foundations of mathematical knowledge, empiricism does not have to satisfy the Humean skeptic and rationalism does not have to satisfy the Cartesian skeptic. Similarly, philosophical skepticism is not an issue in the case of explications in the area of metaphysical knowledge. For example, whether the account of the abstract/concrete distinction presented in

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chapter 5 is the best explication of the distinction depends on how well it handles various alleged counterexamples, on systematic considerations such as simplicity, and on extrasystematic considerations such as the light it sheds on foundational and scientiªc issues. The question is internal to pure ontology. The explication does not have to refute philosophical skepticism concerning the categories Abstract and Concrete or any others that enter into the explication. Similarly, explications such as the explication of rational choice or conªrmation are internal matters within pure epistemology. The situation changes radically when we cross the line into external metaphysical knowledge. Here a large portion of our epistemic framework, even the entire framework in some cases, is called into question on the basis of a skeptical challenge to the epistemic standards that prevail within the framework. The skeptic adopts what is putatively a vantage point outside the framework and challenges us to show that we know what we take ourselves to know in spite of the skeptical possibility that the standards on which the knowledge rests led us astray. Skepticism unites philosophical foes. Although rationalists and empiricists, realists and nominalists, Cartesian dualists and materialist monists, and so on disagree about what internal knowledge we have, they perforce agree that we have external knowledge. Hence, they must now make common cause to defend the scientist’s claim to knowledge in the sciences and their own claims to knowledge in the foundations of those sciences. With respect to a particular skeptical issue, external metaphysical knowledge can thus be of three sorts: (A) Knowledge that there is internal knowledge. (B) Knowledge that there is no internal knowledge. (C) Knowledge that we cannot know whether there is internal knowledge or not. In case (A), we prevail over the skeptic. We have external knowledge that vindicates our belief in internal knowledge in the sciences and philosophy. In case (B), the skeptic prevails over us. We have external knowledge that vindicates the skeptic’s position that we do not have internal knowledge. In case (C), it’s a standoff. We have external knowledge that the issue of skepticism is undecidable. And, of course, there is the “none of the above” possibility. In this case, we fail to obtain external metaphysical knowledge. It is important to recognize that case (C) is a form of external metaphysical knowledge. Recognizing it prevents us from impatiently dismissing skepticism. Some philosophers dismiss skepticism out of

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hand because they despair of providing a justiªcation that will stand up to the skeptic. Even supposing that refuting the skeptic is hopeless, recognition of (C) shows that a reason is needed to claim that it is hopeless. Sometimes philosophers dismiss the issue of skepticism as pointless. Their sentiment here is that a reason to reject skepticism would only be a reason for believing what we cannot help believing anyway. The trouble with this sentiment is that, regardless of what our psychology might encourage us to believe, it is still worth knowing the truth about our beliefs if it can be known, even if it is the hard truth that what we believe is provably without rational justiªcation. Although there is some plausibility in thinking that we can’t help believing that we have internal knowledge, there is no plausibility, as things stand, in thinking that we cannot know the truth about that belief. In this connection, one is reminded of Prichard’s (1949) well-known argument for dismissing the issue of skepticism. He dismisses it because it depends on the Cartesian theory of knowledge and he takes its formulation of the epistemic task to be mistaken. He (1949, 15) argues that the theory of knowledge in this sense cannot exist because our customary knowledge “neither can be, nor needs to be, improved or vindicated by the further knowledge that it is knowledge.” A skeptical doubt about something we know is not a “genuine doubt but rests on a confusion.” We see that there can be no “genuine doubt” in a case where we know a proposition P once we notice that . . . what we are really doubting is not whether this consciousness [the knowledge that P] is really knowledge, but whether consciousness of another kind, viz., a belief [that P], was true. [A genuine doubt could only arise if] we have lost hold of, i.e., no longer remember, the real nature of our consciousness of yesterday. In this case, Prichard thinks that our only recourse is to regain that consciousness by repeating the cognitive steps by which we justiªed the belief a day ago. The problem with Prichard’s objection is that what he says is true only when the knowledge in question is internal. Since the prevailing standards do not come into question with common-sense knowledge, knowledge in the sciences, or foundational knowledge, the only issue in these areas is whether the standards were properly applied in the individual case or whether something we have taken to be the standard either isn’t the correct standard or isn’t correctly formulated (e.g., issues like those Hempel [1965, 3–79] raises about conªrmation or issues like the intuitionist challenge to double negation). To be sure, the only

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genuine doubt that could arise in a case of internal knowledge results from having “lost hold of” the fact that the appropriate standards have been properly applied, and the only recourse for dispelling the doubt is to repeat the steps taken in their correct application. But when the knowledge in question is external, what Prichard says isn’t true. His reºections do not even apply to the Cartesian claim that there is real skeptical doubt about propositions we ordinarily take ourselves to know and that the job of the theory of knowledge is to defend them against that doubt. Prichard’s criticism of the Cartesian theory of knowledge runs afoul of the internal/external distinction, since, on that distinction, skeptical doubts arise in an area where the standards are open to challenge. We are now in a position to extend our account of synthetic a priori knowledge in the formal sciences to synthetic a priori knowledge in the area of philosophical knowledge. On the basis of the previous considerations, we divide (Q2) into (Q2a) and (Q2b). (Q2a) What is the nature of internal philosophical knowledge? (Q2b) What is the nature of external philosophical knowledge? No question is begged, since we are making no antiskeptical claim at this point. As yet, the answer to (Q2b) is left open and hence no assumption is made about whether there is internal knowledge for our account of synthetic a priori philosophical knowledge to be an account of. We will turn to the issue of external philosophical knowledge below. I will consider two cases of synthetic a priori knowledge in connection with (Q2a): philosophical knowledge in the foundations of the formal sciences and internal metaphysical knowledge. Extending the explanation for synthetic a priori truths in the formal sciences to synthetic a priori truth in their foundations is straightforward.3 The extension begins with the response we gave to the epistemic challenge to realism in chapter 2. That response is an account of logical and mathematical knowledge that is based entirely on intuition and reason. Knowledge in the foundations of the formal sciences, like knowledge in the formal sciences, divides into a range of basic cases where intui3. It is not at all a trivial matter to convert our explanation of how synthetic truth in logic and mathematics can be known a priori into an explanation of how synthetic truth in the formal sciences generally can be known a priori. Even if the only other case we have to consider is linguistics, it will take a far better developed version of linguistic realistic rationalism than I have produced here to explain how our account of synthetic a priori truth in logic and mathematics can be adapted to the case of synthetic a priori truth in linguistics. The various things I have said about the case of linguistics in this and earlier works must sufªce for the present.

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tion and reason can exclude every possibility of falsehood and another range of cases where they cannot. In the former cases, no empirical information is required to establish these synthetic truths and hence the account is an explanation of how they can be known a priori. In the latter cases, foundational truths can be treated in the way apodictic knowledge in the formal sciences was treated in chapter 2. Some mathematical statements, e.g., Church’s thesis, although they have no proof, still can be shown to be a priori mathematical knowledge on the basis of an a priori justiªcation that shows them to be essential to the best systematization of a body of a priori mathematical knowledge. Although we are unable to prove that recursiveness is effective computability, accepting their equivalence can presumably be shown a priori to be essential for the best systematization of the overall body of mathematical knowledge. The same kind of a priori argument is possible for the apodictic status of propositions in the foundations of the formal sciences. To be sure, apodictic propositions in logic and mathematics belong to formal sciences that consist very generally of propositions we can prove, while propositions in the foundations of the formal sciences belong to a branch of knowledge that, for the most part, consists of propositions we make no pretense to be able to prove. But the body of knowledge with respect to which we determine what is essential to the best systematization can be taken broadly to include the knowledge in the formal sciences together with the knowledge in their foundations. When we have a body of knowledge that includes both the formal sciences and their foundations, we can argue that synthetic propositions in the latter that cannot be proved can nonetheless be known a priori because they can be shown on the basis of a priori argument to be essential to the best systematization of that comprehensive body of knowledge. As we noted, a complete account of the objects in the domain of the formal sciences depends on knowledge of their foundations. Without an ontological account of the objects in the domain of a formal science and an epistemology for our knowledge of them, the complete understanding for which we strive cannot be obtained. Benacerraf’s ([1973] 1983) combined semantic and epistemic requirement on accounts of mathematical truth is an expression of this need for a more complete understanding than the formal sciences themselves provide. This is not to deny that mathematicians, logicians, and linguists can get their technical work done without having a complete picture of of the nature of mathematical, logical, and linguistic objects. But as Quine (1961b, 47) once said

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. . . [the linguist’s ignorance concerning the ontology of semantics] is a theoretically unsatisfactory situation, as the theoretically minded among the linguists are painfully aware. This goes as well for ignorance about numbers, sets, propositions, sentences, and other objects of study in the formal sciences. Philosophical claims like Putnam’s (1975c) that mathematics needs no foundations are misleading. While it is true that mathematicians typically do their technical work without enlightenment about the philosophical foundations of mathematics, it is not true that the fuller understanding we seek about mathematical reality can be achieved without philosophy. As we argued at the end of chapter 4, the issue of whether there are such things as numbers depends as much on philosophy as on mathematics. Our approach to explaining knowledge in the formal sciences and their foundations contrasts with the approach Chisholm (1977, 34–61) calls “linguisticism.” That approach claims that a priori truths are true in virtue of rules of language. This doctrine is a holdover from the attempt during the linguistic turn to explain the truths of formal science as analytic a priori truths. The doctrine underlies Dummett’s (1978, 214) view that the notions of reference and truth in mathematical practice do not depend on an objective realm of objects, but on conceptual connections within the language of mathematical practice. Hale (1987, 124) also appears to endorse linguisticism when he claims that “the obvious and natural account of our knowledge of truths about abstract objects” for Platonists who abandon a contact epistemology is that the truth of a statement about abstract objects “is a matter ‘internal to language’ in the sense that truth is owed to conceptual liaisons.” Although our conception of a priori formal knowledge takes the faculty of reason to be the means of recognizing analytic necessary connection, it also takes it to be the means of recognizing synthetic necessary connection. We agree with linguisticism that there is an area in which recognizing necessity is entirely a matter of linguistic understanding, but claim that that area is much narrower than the one we get from Frege’s and Carnap’s notion of analyticity. On our approach, the area in which recognition of necessary connection is entirely a matter of linguistic understanding is just the class of metalinguistic sentences such as “‘John loves his mother’ is meaningful,” “‘Holmes dusted the table’ is ambiguous,” “‘Sister’ and ‘female sibling’ are synonymous,” “‘Open’ and ‘closed’ are antonymous,” “‘Unmarried bachelor’ is redundant,” “‘Squares are rectangles’ is analytic,” “‘Squares are not rectangles’ is contradictory,” and so on. If such sentences are the only cases for which linguistic considerations sufªce

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for knowledge of necessary truth, then the operation of reason is only in part linguistic, and the least signiªcant part at that (see Katz 1997, in preparation). Cases where recognizing necessary connections is more than a matter of understanding the meaning of sentences show that linguisticism’s claim that recognition of necessary relations is entirely linguistic is too strong. For example, we can fully understand the meaning of the sentence “Every even number greater than two is the sum of two primes” but still not know whether it is true or false. In fact, the reasoning from which we could obtain knowledge of the truth or falsehood of Goldbach’s conjecture can only get started after we grasp the meaning of the sentence. This objection cannot be deºected by moving to a stronger notion of understanding a sentence than the ordinary one of understanding its compositional meaning. Any notion strong enough to deliver the truth value of Goldbach’s conjecture would be far too strong to count as a notion of linguistic understanding. The other case to consider in answering (Q2a) is that of internal metaphysical knowledge. In ontology, internal knowledge will consist of a priori explications of concepts that occur in foundational discussions (e.g., the explication of “abstract” and “concrete” presented in chapter 5), and hence what goes for foundational knowledge goes for such internal metaphysical knowledge. In epistemology, internal knowledge will consist of explications and propositions stating that particular formal or foundational truths are a priori. Since intuition and pure reason are all that is required to show that an a priori truth is a priori, the propositions expressing the knowledge that truths in the formal sciences and their foundations are knowable a priori are themselves knowable a priori. We turn now to (Q2b). Earlier we described three possible outcomes to the search for external philosophical knowledge. Since we are only trying to formulate a rationalist/realist position and defend it against naturalist/empiricist positions, (B) can be ignored. If (B) obtains, we all go under together—those on Neurath’s boat as well as those sailing under realist and rationalist colors. Hence, assuming either (A) or (C) obtains, the question is what can be said about external metaphysical knowledge. Our explanation of synthetic a priori internal philosophical knowledge cannot be extended to external metaphysical knowledge. On the one hand, external philosophical knowledge is never knowledge of necessary truth. Philosophical skepticism shows that it is possible for propositions such as that the future will be like the past, that there is an external world that causes our internal percepts, and that there are other minds to be false. Such external propositions are not only

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supposably false, like logical and mathematical truths; they are possibly false as well. Hence, explanation of external philosophical knowledge cannot be based on reasoning that rules out the possibility of its falsehood. On the other hand, not being internal knowledge, external philosophical knowledge is not part of the system of knowledge embracing the formal sciences, their foundations, and the internal metaphysical foundations of their foundations. To be sure, propositions such as that the future will be like the past, that there is an external world that causes our internal percepts, and that there are other minds are assumed within the overall body of scientiªc knowledge, but that doesn’t transmute them into external knowledge. Further, external metaphysical propositions cannot be justiªed as apodictic within the overall body of internal knowledge, since the argument would have to be that they are essential to the best systematization. But such an argument begs the skeptic’s question. Skeptics have their own, quite different, ideas about the best systematization. One thing that can be said about external metaphysical knowledge, assuming that it exists, is that it is synthetic a priori knowledge. It is a small irony that the very skepticism that calls our internal knowledge into question shows that external metaphysical knowledge cannot be a posteriori. Experience is useless to defeat either Cartesian skepticism or Humean skepticism. The demon hypothesis can explain our experience, whatever the experience happens to be, just as our causal story can explain our experience, whatever it happens to be. Counterinductive hypotheses are compatible with the strongest evidence of regularity, just as our inductive hypothesis is compatible with it. In showing that experience cannot distinguish between the hypothesis that the actual world is a world in which the skeptic’s philosophical fantasy is fact and the hypothesis that it is not, skepticism establishes the irrelevance of experience to metaphysical investigation. Hence, external metaphysical knowledge, if it exists, is a priori knowledge. G. E. Moore’s (1959) celebrated a posteriori “proof” of our knowledge of the existence of external objects challenges this conclusion. Moore thought that the best way to prove the existence of external objects is to exhibit them empirically for all to see. So, Moore shows us one hand and then another, claiming thereby to have proved that two external objects exist, and hence that external objects exist. The demonstration purports to show that we cannot doubt the existence of external objects because the fact of their existence is already part of our common-sense knowledge of the world. All that is required for us to realize that we have knowledge of their existence is to have our attention called to it

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by an appropriate empirical exhibition. We have to show that this argument does not answer the Cartesian skeptic. From the viewpoint of prescientiªc thought about the world, Moore was certainly right to think that his argument is as strong a proof as might be given for anything (outside mathematics and logic, presumably). Further, even from the viewpoint of the sciences, Moore is right. The Cartesian question of the existence of external objects is not like the question of why objects fall down, why dyes turn liquids a uniform color, or why water expands on freezing. Such questions arise in prescientiªc reºection and get answered in scientiªc investigation. The question of the existence of external objects does not arise in our common-sense reºections about the world, and it also does not arise in the practice of science or the philosophy of science. This brings out the full scope of Moore’s argument. The argument not only applies to the area of common sense, it applies equally well, and for the same reason, both to science and to the foundations of science. Being built up from common-sense knowledge, science may revise our common-sense beliefs about the form or constitution of external objects. It can show that matter is discontinuous, that the earth is round, or that space is non-Euclidean. But science cannot, on pain of internal incoherence, revise the common-sense commitment to the existence of an external world. Moreover, given that Moore’s argument is valid for the sciences and given the relation between the sciences and their foundations, his conclusion is valid for their foundations too. But Moore’s argument has essentially the same problem as Prichard’s. Moore (1959, 147) rightly says: . . . we all of us do constantly take proofs of this sort as absolutely conclusive proofs of certain conclusions—as ªnally settling certain questions, as to which we were previously in doubt. But the fact that Moore’s proof is typical of the proofs we give in ordinary life and science—he cites proving that there are three misprints on a page as an illustration of what he means—should alert us to its vulnerability as a refutation of Cartesian skepticism. Moore’s proof is “absolutely conclusive” internally, but it is inconclusive externally. Stroud (1984, 110) criticizes Moore’s proof as follows: . . . the philosophical sceptic’s denial of our knowledge [is] the outcome of an investigation into the basis of all the knowledge or certainty we think we have about the world around us. That is why I think we feel it is not a “sufªcient refutation” of that scepticism simply [for Moore] to bring forward “a particular case

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. . . in which we do know of the existence of some material object.” The philosopher’s assessment of all of our knowledge of the world around us is meant to apply to every particular case in which we do think we know of some material object, so no case that could be brought forward would escape scrutiny. Since the skeptic questions our entire system of internal knowledge including the prevailing standards that ground that knowledge, neither the beliefs comprising the system nor the standards are of use against such an external challenge, and, as a consequence, Moore’s reminder that the existence of objects outside of us is already part of our common-sense system of beliefs is of no help against the skeptic’s challenge to the system as a whole. Moore’s premise that we know that here is one hand and there is another is certainly true. The Cartesian skeptic does not dispute that we have such knowledge about the existence of the two hands, but rather that it is knowledge of the existence of objects outside of us. The skeptic’s challenge is a challenge to the system as a whole because it provides a uniform alternative interpretation of our beliefs about hands and other objects of experience on which they are inner phenomenal objects rather than outer physical objects. The Cartesian skeptic no more denies that hands exist in the “vulgar sense” than Berkeley denies that stones exist in the “vulgar sense.” Moore’s holding up two hands no more refutes Descartes than Dr. Johnson’s kicking a stone refutes Berkeley. Hence, Moore’s argument is not an objection to the claim that external metaphysical knowledge (if there is any) is a priori. An epistemology for internal knowledge need not answer the question of skepticism to be acceptable, but it cannot be acceptable if it would in principle prevent us from learning the ultimate outcome of our struggle with philosophical skepticism. If, on the supposition that there is external metaphysical knowledge, it is a priori knowledge, then Quine’s epistemology does just that. Quine’s blend of empiricism and epistemic naturalism commits him to the continuity of empirical knowledge from the sciences to their foundations and from their foundations to metaphysical speculation. Empirical science is our ªrst and only philosophy. Furthermore, as Quine (1974, 2) says in The Roots of Reference, the skeptical challenge is “a challenge to natural science that arises within natural science.” And, in “The Nature of Natural Knowledge,” Quine (1975, 68) says: I am not accusing the sceptic of begging the question; he is quite within his rights in assuming science in order to refute science; this, if carried out, would be a straightforward argument by

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reductio ad absurdum. I am only making the point that sceptical doubts are scientiªc doubts. But if this is so and if scientiªc knowledge is a posteriori, as it is on Quine’s epistemology, then his epistemology makes it impossible for us to determine whether (A), (B), or (C) is the case. Since the point is the same for each of these cases, let us consider (A). If we construct an argument that refutes, say, the Cartesian skeptic, it constitutes metaphysical knowledge of why we are justiªed in claiming to know that there is an external world. As we have seen, such an argument must be a priori. Cartesian and Humean skepticism show, as we have seen, that, if there is an argument that can meet Cartesian or Humean skepticism, it cannot be one based on experience. Since experience is irrelevant, and since ex hypothesi there is an argument that refutes the Cartesian skeptic, then it is an a priori argument. But, since Quine’s epistemology allows only a posteriori arguments, it would rule out our knowing that (A) is the case. A priori arguments against skepticism seem promising only up to a point. Perhaps the most promising of them are the ones that attempt to exploit the rational presupposition of the skeptic’s argument. Skeptics cannot just express their doubts about our epistemic claims. They must provide an argument that we have to refute to show that we really do know what we think we do. But, in providing such an argument, skeptics assume linguistic and inferential principles which we can try to turn against them. Moreover, even though skeptical arguments arise in the area of external metaphysical knowledge, they presuppose internal principles, principles belonging to linguistics and logic. Since skeptics cannot disavow those principles without undercutting their skeptical challenge, they have a commitment to internal knowledge that offers us at least an entering wedge. The question is which forms of skepticism are vulnerable to this move and how vulnerable are they. I suspect that the use of such internal knowledge is one way to refute the logical skepticism that Descartes based on his conception of God’s omnipotence: Descartes’s skeptical argument against the principle of noncontradiction is selfrefuting because it presupposes the principle. It is not clear whether there are other forms of skepticism that are vulnerable to this kind of response. The question is how far the skeptic’s commitment to internal knowledge goes. Skeptics have to express their skepticism in language and sustain it logically, but they do not have to assume substantive principles about nature. Since the Cartesian challenge to our knowledge of the external world, the Humean challenge to our knowledge of the laws of nature, and so on do not involve a potentially

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self-defeating commitment to the standards that they challenge, it doesn’t look like they are vulnerable to this kind of response.4 We have considered a number of kinds of response to skeptical challenges and will consider another kind in the next section. None seems promising. The upshot for our rationalist epistemology—and for rival epistemologies too—is the unsurprising conclusion that nothing much can be said at present to show that we have the external metaphysical knowledge we take ourselves to have. But, in the present context, this conclusion is of less consequence for philosophy than it is frequently taken to be. As long as option (B) is not established, the foundations of the sciences and metaphysical explication can continue to provide a skeptic-free environment for one set of philosophical concerns. Skepticism has often been a central concern to philosophers, but at least as often philosophers have had other central concerns. For Socrates, Plato, Aristotle, Locke, Leibniz, Frege, and Quine, skepticism was of relatively marginal interest. It is ironic that it is the difªculty of the issue of skepticism, in particular the difªculty of establishing (B), that leaves us free to pursue business as usual in the areas of scientiªc and internal philosophical knowledge.5 6.3 Carnap’s Criticism of the Synthetic A Priori Failure to capitalize on this relation between the difªculty of the issue of skepticism and the freedom we have to pursue philosophical investigation into questions of internal knowledge has led to a degradation of reason and the consequent creation of a situation where naturalism, empiricism, and positivism are seen as saving philosophy from itself. Hume’s ([1739] 1958, 183) views that our reasoning about causes and effects derives from custom rather than from “the cogitative part of our natures” and that “school metaphysics” is nothing but “sophistry and illusion” obtain much of their force from such a degradation of reason. Viennese positivism—which substitutes “literal nonsense” for “sophistry and illusion” as the sin for which books of “school metaphysics” 4. Descartes’s own response was that our belief in an external world that causes our inner experiences must be true, since, if it weren’t, then, per impossibile, God would have done evil. To block skepticism about the external world, we have to prove the existence of God. As if that were not enough of a problem, Nozick (1981, 201–2) points out that such a proof is not sufªcient. 5. Moreover, new developments in science can be expected to add new dimensions to old problems in the foundations of the sciences and perhaps to raise new ones. Besides current problems in the foundations of science and metaphysics, we can expect that new results in the foundations of the sciences will stimulate new topics for metaphysical explications. Also, progress in metaphysical explication can very well change the picture in the foundations of the sciences.

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would be burnt—is like Humean positivism in also gaining its force from an exaggeration of the scope of skepticism. The early Wittgenstein ([1921] 1961, sec. 6.51) wrote: Scepticism is not irrefutable, but obviously nonsensical, when it tries to raise doubts where no questions can be asked. For doubt can exist only where a question exists, and an answer only where something can be said. Given what we have said about the scope of skepticism at the end of the last section, skeptically inspired pessimism about rational metaphysics no longer makes the positivistic quick ªx seem irresistible. Nor does skepticism bear on issues in the area of internal knowledge, such as the rationalist/empiricist issue. Hume’s skeptical doubts about our knowledge of causes and effects in and of themselves provide no support for a positivist criticism of internal metaphysical theories or an empiricist view of scientiªc knowledge. Hume’s argument for his positivism assumes his empiricist view that “[a]ll the objects of human reason or inquiry” are either relations of ideas, which are intuitively or demonstrably certain, or matters of fact and existence, which are based on sensory observation or on records and memory of such observations. Given this assumption, his empiricist view can receive no support from his skeptical arguments. If rationalists are helpless against Hume’s point that induction is neither intuitively nor demonstrably certain, then empiricists are helpless against Hume’s point that arguments for induction based on experience are circular. If Hume’s skeptical doubts were to sound the death knell for rationalism, they would sound it for empiricism as well. Thus, those doubts play no decisive role in the controversy between empiricists and rationalists, which is about the nature of internal knowledge in the sciences. Similarly, skeptical doubts can play no role in the realism/antirealism controversies in the foundations of the formal sciences and natural sciences. In chapter 2, we argued for the irrelevance of skepticism to the realism/antirealism controversy in the foundations of the formal sciences. Skepticism is also irrelevant to the instrumentalist claim that observational terms designate objects in nature but theoretical terms only provide convenient computing machinery. Here, too, skepticism cuts both ways: instrumentalists cannot use skepticism against the realist’s view of the theoretical vocabulary of science without having it turned against their realist view of the observation vocabulary. Given that skepticism has no implications for scientiªc and philosophical issues in the area of internal knowledge, positivist claims have to be assessed on the basis of the internal standards appropriate to them without consideration of the role that positivistic semantic

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doctrines were intended to play in saving us from skepticism. The skeptical doubts behind Wittgenstein’s concern with solipsism in the Tractatus cannot provide any basis for his positivist claim that metaphysical sentences are nonsense. The soundness of that claim would ªrst have to be adjudicated within linguistics and the foundations of linguistics. Similarly, the soundness of Carnap’s (1952) claim that statements like (1)–(6) are nonsense has to be evaluated on the basis of the appropriate linguistic standards. (1) Mathematical (logical, linguistic) objects are abstract. (2) Mathematical (logical, linguistic) objects are concrete. (3) Mathematical (logical, linguistic) objects are ªctions. (4) There is an external world. (5) The future will be like the past. (6) There are other minds. Since the claim that those English sentences do not have a sense is a linguistic claim like any other claim about English sentences, we can compare sentences (1)–(6) with genuine cases of nonsense sentences, such as (7)–(9), to determine whether (1)–(6) are sufªciently similar to them to count as nonsense too. (7) Quadruplicity drinks procrastination. (8) Colorless green ideas sleep furiously. (9) The number seventeen loves the taste of home cooking. But speakers of English will judge (1)–(6) to be nothing like (7)–(9). Further, it is hard to see how a sentence like, say, (5) could have no sense while sentences like (10) and (11) have a sense. (10) The future of London will be like the past of London. (11) Switzerland’s future will be like Switzerland’s past. Hence, Carnap’s claim is dubious linguistics. This argument against assimilating sentences like (1)–(6) to cases of semantic deviance like (7)–(9) is parallel to the argument in chapter 2 against the combinatorialist’s attempt to give a multiform semantics for English to justify construing mathematical sentences like (12) as having a different logico-grammatical form from sentences like (13). (12) There are at least three large cities older than New York. (13) There are at least three perfect numbers greater than 17.

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Our argument here is another development of Burgess’s (1990, 7) point that the question of what semantics is appropriate for mathematical sentences belongs to “the pertinent specialist professionals”. The question of what semantics is appropriate for metaphysical sentences also belongs to “the pertinent specialist professionals”. Once the former question is put in the hands of the linguists, it is independent of the epistemological issues of skepticism, and once the latter question is put in their hands, it is independent of the ontological issue of realism. As a consequence, there is nothing to counterbalance the force of the robust linguistic intuition that sentences like (1)–(6) are not meaningless in the manner of sentences like (7)–(9). From the ªrst expression of his positivism in his 1928 book Pseudoproblems in Philosophy (1969) to its mature expression in later works like “Empiricism, Semantics, and Ontology” (1952), Carnap’s claim that sentences like (1)–(6) are meaningless was based on his own homespun concept of meaningfulness. Meaningful sentences divide exhaustively into factual sentences, which have an appropriate empirical relation to the natural world, and formal sentences, which are nonfactual but reºect features of the framework concerned with expressing and manipulating sentences. There can be no evaluation of this conception on the basis of the results of investigations in linguistics, since it is simply a proposal of a framework (or class of frameworks) in Carnap’s (1952, 30–33) own sense of that term. Carnap’s critics—Quine, of course, but also Oxford philosophers like Strawson (1963, 503–18)—pointed out that Carnap’s claims to address substantive philosophical issues are “empty” without a reason to identify meaningfulness and meaninglessness in his special sense with meaningfulness and meaninglessness in natural language. In “Empiricism, Semantics, and Ontology,” Carnap (1952) repackages this distinction between the meaningful and meaningless in the terminology of “internal questions” and “external questions.” The former are the old factual and formal sentences, and the latter are the old metaphysical sentences. External questions concern the acceptance of a linguistic or theoretical system as a whole, and hence they are meaningless because their answers are unveriªable on the basis of reasoning within a scientiªc system. Carnap introduced this new way of drawing the old distinction to explain why theoretical systems containing terms referring to abstract objects can be used in formal systems without committing the user to Platonist metaphysics. Carnap’s explanation was that there is no commitment because the question whether to adopt a term within a theoretical framework can be answered on the basis of arguments internal to the system. As he sees it, the question of Platonist metaphysics, in contrast, is an external question about whether to accept the theoretical

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system as a whole, and such questions make no sense because there is no explicit framework within which to answer them. External questions are thus noncognitive, and decisions about them are practical matters, to be made on the basis of the uses to which one wishes to put the system. Hence, Carnap can have his cake and eat it: internal questions about terms referring to abstract objects are meaningful and can be employed in formal systems such as those described in Meaning and Necessity, while external questions such as the one at issue in the debate about mathematical realism are meaningless. Given its intended scope, Carnap’s meaningful/meaningless distinction is itself external. In this case, however, it is not an assertion and its “acceptance cannot be judged as being either true or false.” It is “a pseudo-statement without cognitive content.” It is a proposal that, as Carnap (1952, 31) says, “can only be judged as being more or less expedient, fruitful, conducive to the aim for which the language is intended.”6 The point extends to Carnap’s conception of logic. His principle of tolerance, which he (1937, secs. 17, 52) has espoused from The Logical Syntax of Language on, removes every consideration, even consistency, as a constraint on explications of the notion of a valid consequence: “In logic, there are no morals. Everyone is at liberty to build up his own logic, i.e., his own form of language, as he wishes.” With this denial of the normative in semantics and logic, choosing a framework within which to state principles of logic is liberty hall. Even a perverse whim to be logically inconsistent must be indulged, since there is no possibility of drawing a line between the permitted and the forbidden within Carnap’s amoral logical world. Allowing a moral preference for consistency would lead quickly to the conclusion, disastrous for Carnap, that logic is a normative discipline. But if there is nothing normative in semantics or logic, acceptance of a language system in which metaphysical sentences are meaningless has no cognitive status and the language system has no normative force. But, in such a totally amoral world, there is no normative ground on which Carnap can stand when he moralizes against metaphysics. The principle of tolerance leaves everything up to the individual. But if everyone can make philosophical choices entirely as he or she sees ªt, it is hard to see how Carnap’s statements about metaphysics can amount to more than positivist propaganda. It hardly needs saying that 6. But if this is the status of his meaningful/meaningless distinction, it is not only that Carnap never intended it to fall within linguistics, but that it couldn’t. Within linguistics, a distinction between the meaningful and the meaningless is a claim about linguistic fact, not a proposal that we are free to accept or reject on the basis of our desires (whatever they may be). In linguistics, distinctions between classes of sentences can only be judged as correct or incorrect depending on whether or not they ªt the grammatical facts.

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propaganda provides no reason for denying that there is synthetic a priori knowledge. The closest Carnap ever comes to appreciating these problems is his (1956b, 233–35) concern about Quine’s objection that, without the investigation of natural languages, “the semantic intension concepts, even if formally correct, are arbitrary and without purpose.” But Carnap doesn’t really get it. He (1956b, 235) says, I do not think that a semantic concept, in order to be fruitful, must necessarily possess a prior pragmatical counterpart. It is theoretically possible to demonstrate its fruitfulness through its application in the further development of language systems. But this is a slow process. Carnap doesn’t seem to see that the slower process of developing language systems does nothing to reduce the arbitrariness of semantic concepts and Carnapian language systems because it derives from the arbitrariness of the whole process of developing such systems under the dubious guidance of the principle of tolerance. He doesn’t seem to see that Quine, Strawson, and his other critics were questioning the relevance of an entirely unconstrained process of developing language systems to resolving linguistic issues and dissolving philosophical controversies.7 6.4 Conclusion The linguistic turn can be characterized by the problem it set out to solve, the adversaries it intended to defeat, and the means it chose to solve its problem and defeat its adversaries. Its problem was synthetic a priori knowledge. Its adversaries were positions that based metaphysical theories of knowledge and reality on the existence of synthetic a priori knowledge. The means for solving its problem and defeating its adversaries was a critique of the linguistic, particularly the semantic, assumptions on which such positions grounded their claims for the existence of synthetic a priori knowledge in science and philosophy. The linguistic turn made no mistake about the problem it had to solve or the adversaries it had to defeat. Its failure was due to the inadequacy of the means it used. The revolutionaries underestimated 7. On linguistic realism, there is a more sweeping criticism of the veriªcationism underlying Carnapian and other forms of positivism. If sentences are abstract objects, then grammatical properties of sentences are intrinsic features of the structures of abstract objects, and hence sentences are meaningful or meaningless independently of our cognitive capacities and the conditions in the natural world that determine what is and what is not empirically veriªable.

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the resources of metaphysics and overestimated their own linguistic criticisms. Their metaphysical adversaries were not restricted to positions like Kant’s that were strong epistemologically but weak ontologically or positions like Frege’s that were strong ontologically but weak epistemologically. As I have argued in the course of this book, there is a metaphysical position that is strong both epistemologically and ontologically. As I have also argued here (but more fully in The Metaphysics of Meaning), this position is also strong semantically, avoiding both the problems that Frege exploited in criticizing Kant’s analytic/synthetic distinction and the problems that Wittgenstein and Quine exploited in criticizing Frege’s intensionalism. The different positions on the issue of synthetic a priori knowledge were driven by the same desire to provide an adequate understanding of philosophy in the light of the impact on traditional philosophy of developments in nineteenth- and twentieth-century science. Except for the late Wittgenstein, the different sides on the issue shared a respect for scientiªc achievements such as Cantor’s set theory, Frege’s logic, and Einstein’s physics, the consequent disillusionment with Kantian and neo-Kantian metaphysics, and an appreciation of the need to provide a new explanation of the relation between science and philosophy. The positions that originated with the early Wittgenstein and Schlick and culminated with the late Wittgenstein and Carnap offered positivistically based explanations on which systematic knowledge is scientiªc. Quine’s naturalized epistemology offered the explanation that philosophy is natural science. These positions present a uniªed front only in opposing conceptions of philosophy on which philosophy is an autonomous discipline concerned with explaining synthetic a priori knowledge. Starting from the same desire for an adequate understanding of philosophy in the light of the developments in nineteenth- and twentieth-century science but embodying precisely this beleaguered metaphysical conception of philosophy, realistic rationalism provides a quite different explanation of the relation between science and philosophy. Unlike the positivist explanation, it recognizes the legitimacy of philosophical as well as scientiªc knowledge, and, unlike Quine’s explanation, it maintains that philosophical knowledge is distinct from scientiªc knowledge. Not only does it not assimilate philosophy to science in order to legitimatize philosophical knowledge, it maintains, contra Quine’s epistemic naturalism, that scientiªc knowledge is heterogeneous. Realistic rationalism holds a dualistic view about the ontology of science, claiming that the formal sciences study abstract objects and the natural sciences study concrete and composite objects. The aim of the former is to prune the supposable down to the necessary,

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211

while the aim of the latter is to prune the possible down to the actual. Given this difference between knowledge in the formal and natural sciences, the former is synthetic a priori knowledge and the latter is synthetic a posteriori knowledge. Philosophy addresses ontological and epistemological questions in the foundations of the sciences and the foundations of the foundations of the sciences that the sciences themselves do not address. The relation between philosophy and the sciences has both a vertical dimension on which philosophy attempts to understand the nature of the sciences and a horizontal dimension on which it attempts to understand aspects of the same reality studied in the sciences. On both dimensions, internal philosophical knowledge is synthetic a priori knowledge. Such knowledge is not the product of successful encounters with the skeptic. It is the product of the continuing dialectic among nominalists, conceptualists, realists, positivists, empiricists, and rationalists.

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Index

Abstract/concrete distinction, 1, 120–124 consequences of, 124–126 Dummett’s, 126–127 objections to, 6n2, 60–61, 193–194 (see also Ontological challenge to realism) Plato on, 124–126 putative counterexamples to, abstract, 132–138 putative counterexamples to, composite, 138–152 (see also Composite objects) revisionary deªnitions of, 18, 118, 126–127 Abstract objects. See also Abstract/concrete distinction arguments for, 4–6, 29–32 (see also Indispensability arguments) characteristics of, 1, 37n5, 77–79, 120–124, 124–127, 129–132 deªned as nonspatiotemporal, 1, 120–124 revisionary deªnitions of, 18, 118, 126–127 as sources of normativity, 82, 114–115, 160 Acquaintance epistemology. See also Empiricism inadequate for realism, 23–24, 25–26, 36–39, 58–59 unnecessary for realism, 39–41, 58–59 used by forms of realism, 14–18, 23–24 Allaire, Edwin B., 181 Analyticity Carnap on, xx–xxi, 69–70, 181–182 Fregean notion of, xx–xxi, 69, 78n5, 179–182, 184, 185–186 Lockean-Kantian notion of, xxin6, 7, 179–180, 185–186

non-Fregean notion of, 93, 178–179, 185–186 Quine on, 70–71, 72n3, 80, 93, 181–184 Anselm, 128 Antirealism. See also Nominalism; Conceptualism epistemic challenge to, 63–65 semantic challenge to, 29–32 Apodicticity, 47–48 A priori knowledge analyticity in explanations of, xx–xxi, 69, 180–181, 198–199 linguisticism about, 198–199 in philosophy, xi–xv, xx, 177–179, 188–190, 196–200 rationalist characterization of, 23–24, 36–41 and revisability, 48–50, 172, 183 synthetic, xx, 177–179, 186–188 Aristotle, 204. See also Contemporary Aristotelianism Armstrong, D. M., 130n7, 134, 170, 173 Arnauld, Antoine, 32, 56–57 Ayer, A. J., 178, 181, 184 Benacerraf, Paul. See also Epistemological challenge to realism; Semantic challenge to realism, Benacerraf’s indeterminacy on the philosophy of mathematics, xxx, 26–27 and semantic challenge to antirealism, 29–32 as structuralist about arithmetic, 85–86, 105–112 Berk, Lon, 152n15 Berkeley, George, 202 Blanshard, Brand, xii

220

Index

Bloomªeld, Leonard as linguistic nominalist, xxiv–xv, 3, 80, 135 Boghossian, Paul A., 97n6 Boolos, George, 139, 152n15 Branch, Glenn, 35n4, 52n15, 128 Brouwer, L. E. J. intuitionism of, 8, 9n4 as mathematical conceptualist, 8, 9n4, 20, 23–24, 119, 192 Burgess, John P., 29–30, 77, 207 Caesar problem, 103–105, 108 Carnap, Rudolf on analyticity, xx–xxi, 69–70, 181–182 criticism of traditional philosophy, 188, 206–209 on intensionalism, 90–91 on internal and external questions, 207–209 Carroll, Lewis, 77 Casti, John L., 173n30 Causal theory of knowledge, 27–28, 31–32. See also Acquaintance epistemology Certainty in the formal sciences antirealist inability to explain, 77, 82–83 plausibility of, 64–65 realist explanation of, 77–79, 83 subjective versus objective, 63–64 Chihara, Charles S., 19, 32, 116 Chisholm, Roderick M., xii, 198 Chomsky, Noam, xxiii–xxv. See also Conceptualism criticism of linguistic nominalism, xxiv–xxv, 3, 135, 159n21 as linguistic conceptualist, xxv, 2, 8, 23–24, 35, 135 nativism and rationalism, 23–24, 35 Classical Platonism, 14–17 Composite objects and relative predication, 150–152 simple, complex, and compound, 142–143 theory of, 141–143, 145–147 Composition, 121–122, 141–145 Compositionality, 131 Compound composite objects. See also Composite objects and discovery versus creation, 168–170 linguistic, 166–167

theory of, 142–143, 166–168 Conceivability, 56–58 Conceptualism in linguistics, xxv, 2, 8, 23–24, 35, 135, 163n23 in mathematics, 8, 9n4, 20, 23–24, 109–110n14 Concrete objects, 1, 124–129 Constitution. See Composition Contact epistemology. See Acquaintance epistemology Contemporary Aristotelianism, 17–19 Creation versus discovery. See Discovery versus creation Creative relations, 141, 152–154 (D), 124. See also Abstract/concrete distinction; Objects, categories of (D1), (D2), (D3), (D4), and (D5), 140–141. See also Objects, categories of Danto, Arthur C., xviii, 67 Descartes, René on conceivability, 56–58 on skepticism about logic, 53–55, 58, 203 Discovery versus creation, 134–135, 137, 168–170 Donnellan, Keith, xxvii, 185 Dummett, Michael, 32, 116, 126–127, 130–132, 180, 198 Eliminativism, 79–81. See also Supervenience Empiricism. See also Acquaintance epistemology; Causal theory of knowledge and certainty, 65–67, 68–69, 71–72, 74 in epistemic challenge to realism, 25–26, 27–28 inadequate for realism, 23–24, 25–26, 27–28, 31–32, 36–39, 58–59 inseparability from naturalism, xxxi, 32n3, 68–69n2 Mill’s, 65–66, 69, 71, 74 Quine’s, 65–67, 68–69, 71–74, 182–183 reason in, 33–34, 52, 54, 68–69 unable to explain knowledge of necessity, xxxi, 7, 32, 63, 187 used in forms of realism, 14–18, 23–24 Epistemic challenge to antirealism, 63–65. See also Certainty antirealism’s inability to meet, 77, 82–83

Index conventionalist response, 66, 69–70 Mill’s response, 65–66, 69 noncognitivist response, 69–71 realism’s ability to meet, 77–79, 83 Quine’s response, 66–67, 68–69, 71–72 Wittgenstein’s response, 75–77, 77–78n4 Epistemic challenge to realism dependence on empiricism, 25–26, 27– 28 rationalist solution to, 34–61 passim Epistemology, pure, 192–193 Ewing, A. C., xii, 46 Field, Hartry, 37 epistemological views of, 10–11, 25, 32, 116, 172 as ªctionalist nominalist, 11–12 as mathematical nominalist, xxxn9, 10–14, 29n1 Fodor, Jerry A., xxiv, 68 Formal sciences, 2n1, 79 distinguished from natural sciences, 58–59, 171–173 and Occam’s razor, 171, 173 pure and applied, 2, 163–166 Foundational disciplines, 5–6, 118–120, 174, 191–192 of foundational disciplines, 118–120, 191–193 independence of science, 5–6, 29–31, 77, 85–86n2, 174, 191–193, 206–207 Frege, Gottlob and analyticity, xx–xxi, 69, 78n5, 179–182, 184, 185–186 on the Caesar problem, 103, 105 context principle of, 130–131, 180 intensionalism of, xxvi–xxviii, 78n5, 93, 105, 114–115, 178–179, 184–186 (see also Intensionalism, criticism of Fregean) and linguistic turn, xx–xxi, 177–178, 179–180, 181–182, 184–186 as mathematical realist, xxxiii, 2, 20, 177, 179 as rationalist, 177–179 on senses, xxvi–xxviii, 78n5, 105, 185–186

221

God and metaphysics, 128, 130–131, 147–148 and skepticism, 53, 55, 203, 204n4 Gödel, Kurt epistemological views of, 15–17, 26, 34, 116 as mathematical realist, xxxi, xxxii, 1, 20, 26, 119, 192 Goldfarb, Warren, 152n15 Goldman, Alvin, 27 Goodman, Nelson, 1, 3, 12, 53, 76, 189 Gottlieb, Dale, 25, 32, 116 Gregory, R. L., 149–150 Grice, H. P., 72n3, 164 Hale, Bob, 49, 126, 132–133, 137–138, 175n33, 198 Halle, Morris, xxvi Hardy, G. H., 134 Heidegger, Martin, 177 Hempel, Carl Gustav, 69, 195 Hilbert, David, xixn5, 119 Hume, David, 7, 69, 179–180, 181, 189, 204–205 Husserl, Edmund, 17, 179, 186 Indeterminacy arguments. See Semantic challenge to realism Indispensability arguments, 10–11, 50–51 Intensionalism. See also Senses Carnap on, 90–91 criticism of Fregean, xxvii–xxviii, 77–78n4, 78n5, 93, 105, 114–115, 156, 178, 181–183, 184, 185–186 Fregean, xxvii–xxviii, 78n5, 93, 105, 114–115, 178–179, 184–186 non-Fregean, xxvii–xxviii, 78n5, 93, 114–115, 156–157, 178–179, 184–186 Intuitions of conceivability and inconsistency, 56–58 Gödel on, 15–17, 34 and Kantian intuitions, 44–45 as source of basic knowledge, 43–46 Wittgenstein’s criticism of, 43–44 Isaacson, Daniel, 144 Johnson, Samuel, 202

Gemes, Ken, 93 Geometry, demonstration in, 160–162 George, Alexander, 20–21 Gibson, Roger, 89

Kant, Immanuel. See also Analyticity, Lockean-Kantian notion of on a priori knowledge, 8, 179–180, 186

222

Index

Kant, Immanuel (cont.) criticism of empiricism, xxxi, 7, 32 criticism of traditional philosophy, 7–8 on Hume, 7, 179–180 as mathematical conceptualist, 7–10, 16, 20, 23 transcendental idealism of, 8–10, 186–187 veriªcationism in, 9–10 Kastin, J., 103n11 Kaufmann, Walter, 13n6 Kenny, Anthony, 181 Kim, Jaegwon, xxii, 81–82 Kitcher, Philip, 49, 172 Kivy, Peter, 169n27 Koslow, Arnold, 188n1 Knowledge. See also A priori knowledge; Skepticism of apodictic principles, 47–48 basic/transcendent, 42–43 formal versus natural, 58–59, 171–173 foundational, 192 holistic character of, 46–47, 71–72 internal/external, 193–194 as justiªed true belief, 34 metaphysical, 192–193 of necessary facts, xxxi, 7, 32, 56–58, 63, 187 observational/theoretical distinction in, 42–43 philosophical, 188–204 Kripke, Saul, xxvii, 123n3, 185. See also Semantic challenge to realism, Kripke’s rule-following Langendoen, D. Terence, 138n9 Langford, C. H., xii Latham, Noah, 148n13 Leibniz, Gottfried Wilhelm, xxxi, 24, 32, 204 Levinson, Jerrold, 169n27 Lewis, David, 37–38, 143–145 Linguistics, xiii–xiv conceptualism in, xxv, 2, 8, 23–24, 35, 135, 163n23 demonstration in, 162–163 nominalism in, xxiii–xxv, 3, 12, 80, 135, 159n21, 163n23 realism in, xxv–xxvi, xxviii, 2–4, 136–138, 162–163 Linguistic turn, xiv–xv, xx–xxi, 177, 184, 209–210

and Frege, xx–xxi, 177–178, 179–180, 181–182, 184–186 and logical positivism, 177–178, 179–181 and Quine and Wittgenstein, 178, 181–184 Locke, John, 204. See also Analyticity, Lockean-Kantian notion of Logical positivism. See also Ayer; Carnap; Schlick on analyticity, xx–xxi, 69–70, 181–182 on a priori knowledge, xiii–xiv, xx–xxi, 69–70, 180–181 criticism of traditional philosophy, xiii–xiv, xx, 178, 179–181, 188, 191n2, 204–205 in ªrst phase of linguistic turn, 177–178, 179–181 Maddy, Penelope, 17–19 Malament, David B., xxxn9 Marcus, Ruth Barcan, 32 Martin, R. M., 108–109 Mathematics conceptualism in, 8, 9n4, 20, 23–24, 109–110n14 nominalism in, 3, 10–14, 109–110n14, 119 realism in, xxviii, xxxi–xxxiii, 1–4, 5–6, 20, 26, 77–79, 109–110n14 McGinn, Colin, 21 Mereology. See Composition Mill, John Stuart on certainty, 65–66, 69, 71 empiricism of, 65–66, 69, 71, 74 and skepticism about necessity, 56–57, 65 Moore, G. E., 82n8, 143–144, 200–202 Mysticism distinguished from mystery, 33, 116 whether rationalism involves, 25, 32–34, 43, 116 Nagel, Thomas, xii, xiiin2, 189 Nativism, 23–24, 35 Naturalism. See also Philosophy, naturalist conceptions of disillusionment with, xxiii–xxviii inseparability from empiricism, xxxi, 32n3, 68–69n2 ontological, epistemological, and methodological, xii relation to materialism, xiin1, 67

Index Naturalized realism, 19–23 Necessity and certainty, 77–79 knowledge of, xxxi, 7, 32, 56–58, 63, 187 skepticism about, 32, 56–57, 63, 65, 75, 182, 184 sources of, 181–182, 184–185 Nishiyama, Yuji, 20 Nominalism ªctionalist, 11–14, 29n1 linguistic, xxxiii–xxxv, 3, 12, 80, 135, 159n21 mathematical, 3, 10–14, 109–110n14, 119 Noonan, Harold, 126 Nozick, Robert, xxix, 204n4 Objects, categories of, 145 abstract, 1, 120–124 composite, 141–143, 145–147 concrete, 1, 124–129 homogeneous, 121–122, 140–141 heterogeneous, 141, 145n11 Occam’s razor, 170–175 Ontological challenge to realism involves counterexamples to abstract/concrete distinction, 117 not usually viewed as challenge, 117 whether problem only for realism, 117–118 putative counterexamples, abstract, 132–138 putative counterexamples, composite, 138–152 Ontology as foundational discipline of foundational disciplines, 118–120, 191–193 independence of science, 5–6, 192– 193 pure, 192–193 (P1.1), (P1.2), (P1.3), and (P1.4), 142. See also Composite objects, theory of (P2), (P3), (P4), (P5), (P5’), (P6), and (P7), 145–147. See also Composite objects, theory of Pap, Arthur, xii Parsons, Charles, 44–45 Passmore, John, 179 Peirce, Charles Saunders, 37n5, 117, 133, 148 Philosophy

223

a priori knowledge in, xi–xv, xx, 177–179, 188–190, 196–200 as conceptual analysis, xii–xv, 111 dialectic of, xix–xx, 5–6, 85–86n2, 118–120, 173–174, 193–194, 211 as distinct from science, 5–6, 29–31, 77, 85–86n2, 174, 191–193, 206–207 whether ªrst-order or second-order, xiv–x knowledge in, 188–204 naturalist conceptions of, xi–xii, xiv–xvi, 67–68, 111–112, 178–179, 184–185 progress in, xiii, xvi, 188–189 as a posteriori science, xi–xii, xv, xxi–xxii, 178, 202–203 (see also Quine) as therapy, xi, xv, xxii–xxiii, 75, 88, 182 (see also Wittgenstein) traditional conception of, xi–xiii, xvi–xvii, 178–179, 184–187, 188–189 (see also Realistic rationalism) uniquely puzzling questions of, xvii, xxii, xxix Plato, 1, 2, 14–15, 124–126, 128, 204. See also Classical Platonism; Forms Postal, Paul M., xxiv, 8n3, 135, 138n9 Prichard, H. A., 195–196, 201 Proofs, 37–41 Propositions, 34, 165 as Fregean senses of sentence types, 36, 38–39, 165 and necessarily false statements, 37–38 as Russellian bearers of truth values, 36, 165 Putnam, Hilary, xxvii, xxx, 116, 153, 185, 198. See also Semantic challenge to realism, Putnam’s Skolemite on indispensability arguments, 10–11, 50–51 on revisability of a priori knowledge, 49–50 Quine, W. V. on analyticity, 70–71, 72n3, 80, 93, 181–184 and a priori knowledge, xv, 48, 182–183 and certainty, 66–67, 68–69, 71–72 conception of philosophy as a posteriori science, xi–xii, xv, xxi–xxii, 178, 202– 203 criticism of Carnap, xxi, 70–71, 90, 181–183, 207, 209

224

Index

Quine, W. V. (cont.) criticism of Fregean intensionalism, xxviii, 93, 181–182, 185–186 criticism of traditional philosophy, xv, 182–184 and empiricism, 65–67, 68–69, 71, 74, 182–183 epistemic holism, 66, 71, 72 inconsistency of epistemology of, 71–74, 191 on indeterminacy (see Semantic challenge to realism, Quine’s indeterminacy) and indispensability arguments, 10–11, 50–51 as linguistic nominalist, xxiii–xxv, 3, 12, 80 and naturalism, xi–xii, xv–xvi, xxi–xxii, 59–60, 66, 68, 111, 178, 184 naturalized Platonism, 59–60 and necessity, 32, 63, 65, 184 on ontological ignorance, 197–198 on radical translation (see Radical translation) on revisability, 48, 183 and skepticism, xxii, 189, 202–204 Radical translation. See also Semantic challenge to realism, Quine’s indeterminacy and bilingual speakers, 89–90, 92 as excluding intensional evidence, 89–92 paradoxical symmetry in, 93–94 source of indeterminacy of translation, 88–89, 92 Rationalism and apodictic knowledge, 47–48 and a priori knowledge, 36–41, 190 basic versus transcendent knowledge in, 41–47 expansions of, 187–188 inseparability from realism, xvii, xxviii, xxxi–xxxii, 23–24, 26, 68–69n2, 187 and nativism, 23–24, 35 and revisability, 48–51 and skepticism in general, 51–55 and skepticism about knowledge of necessity, 56–58 Realism, 1, 2 arguments for, 4–6, 29–32 (see also Indispensability arguments)

and discovery versus creation, 134–135, 137, 168–170 and empiricism, 14–18, 23–24 general versus particular, 2–4 inseparabilility from rationalism, xvii, xxviii, xxxi–xxxii, 23–24, 68–69n2, 187 linguistic, xxv–xxvi, xxviii, 2–4, 136–138, 162–163 mathematical, xxviii, xxxi–xxxiii, 1–4, 5–6, 20, 26, 77–79, 109–110n14 Realistic rationalism, xvii–xxi, xxii–xxiii, xxviii–xxix, xxxii, 210–211 Reductionism. See Eliminativism Relative predication, 150–152. See also Composite objects Rollins, C. D., 64 Rule-following. See Semantic challenge to realism, Kripke’s rule-following; Wittgenstein, rule-following argument Russell, Bertrand, xxiii, 67, 114, 178 Sartre, Jean-Paul, xviii Schlick, Moritz, xx–xxi, 178, 179, 184, 191n2, 210 Seeing as, 156–160 Semantic challenge to antirealism, 29–32 Semantic challenge to realism based on paradoxical symmetries, 99–103 general form of, 99–100 instability of, 100–102 skeptical versus nonskeptical, 87–88, 99–102 strategy for resisting, 102–106, 108 ultimate signiªcance of, 116 why ºaws in undetected, 86–87 and Wittgenstein’s rule-following argument, 87–88 Semantic challenge to realism, Benacerraf’s indeterminacy, 85 as challenge to general realism, 86 claims numbers are not objects, 106–107 claims numbers are not sets, 106–107 compatible with unknowable numbers, 106 and epistemic challenge to realism, 86 motivated by naturalism, 111–112 paradoxical symmetry in, 107, 110 presupposes structuralism about arithmetic, 107–110, 111 why ºaws in undetected, 86–87

Index Semantic challenge to realism, Kripke’s rule-following about justiªcation of projection to unknown, 94–95, 96, 97 as nonskeptical, 88, 99–100 paradoxical symmetry in, 98–99, 101n10 senses in solution of, 95–98, 101n10 Semantic challenge to realism, Putnam’s Skolemite Benacerraf on, 113–114 compatible with unknowable sets, 113 and conceptions of understanding, 112–113 presupposes structuralism about set theory, 112–114 realist response to, 114–115 relies on naturalism, 116 relies on Wittgenstein’s rule-following argument, 114–116 Semantic challenge to realism, Quine’s indeterminacy as based on criticism of analyticity, 93 as based on radical translation, 88–89, 92 (see also Radical translation) begs question against intensionalism, 89–92, 93 compatible with unknowable meanings, 88 paradoxical symmetry in, 93–94, 101 as skeptical, 99–100 Senses. See also Intensionalism and Fregean propositions, 36, 38–39, 165 Fregean theories of, xxvi–xxviii, 78n5, 105, 185–186 grasping of, 96–98 in Kripke’s rule-following semantic challenge, 95–98, 101n10 non-Fregean theories of, xxvi–xxviii, 78n5, 185–186 Shapiro, Stewart, 103n11 Ship of Theseus, 100n9 Skepticism Cartesian (about the external world), 189, 200–202, 203 Cartesian (about logic), 52, 53–55, 58, 193, 203 as challenge to rationalism, 51–52 as external challenge to knowledge, 194–195, 199–200 Humean, 53–55, 189, 193, 200, 203, 205

225

irrelevance to internal knowledge, 54–55, 193–194, 204–206 Moore on, 200–202 about necessity, 32, 56–57, 63, 75, 182, 184 not necessarily central to philosophy, 204 Prichard on, 195–196 Quine on, xxii, 189, 202–204 semantic challenges to realism as exhibiting, 87–88, 99–102 Wittgenstein on unintelligibility of, 76–77, 205–206 Smith, G. E., 38 Soames, Scott, 22 Socrates, 204 Spinoza, Benedict, 148n13 Strawson, P. F., xii, xiiin2, 72n3, 207, 209 Stroud, Barry, xixn4, xxi–xxii, 189, 201–202 Structuralism about arithmetic, 85–86, 103n11, 105–112 (see also Semantic challenge to realism, Benacerraf’s indeterminacy) about set theory, 112–114 (see also Semantic challenge to realism, Putnam’s Skolemite) Structural properties, 107–110 Studdert-Kennedy, Michael, 159n21 Supervenience, 81–82. See also Eliminativism Supposability, 38–39, 58–59 Transcendental arguments, 48n12 Transcendental idealism. See Kant, transcendental idealism Type/token distinction. See also Composite objects and abstract/concrete distinction, 117–118, 133–134 exempliªed in puzzling ªgures, 149–150 (see also Seeing as) in linguistic antirealism, 3, 12, 118 uncontroversiality of, 133–134 and universal/instance distinction, 154–155 Veriªcationism, xiiin3, 9–10, 99n8, 209n7 Wang, Hao, 15, 17 Warnock, Mary, xiv–xv, xvii

226

Index

Wetzel, Linda, 109, 152n15, 155n18 White, Alan R., 63–64 Wittgenstein, Ludwig and a priori knowledge, xx–xxi, 180–181, 184 on certainty, 75–77, 77–78n4 on color incompatibility, 181–182, 184 conception of philosophy as therapeutic, xi, xv, xxii–xxiii, 75, 182 criticism of Fregean intensionalism, xxvii, 77–78n4, 114–115, 156, 181–182, 184, 185–186 criticism of intuition, 43–44 criticism of traditional philosophy, xi, xv, xx, 75, 182, 184 on meaning, xx, 77–78n4, 88, 114–115, 156, 158n20, 160 and naturalism, xi, xx, 68, 75–77, 111, 178, 184 and necessity, 75, 181–182, 184–185 rule-following argument of, 77–78n4, 87–88, 114–116, 156–157 on seeing as, 156–160 and skepticism as unintelligible, 76–77, 205–206 Wollheim, Richard, 154–155 Wright, Crispin, 99n8 Yablo, Stephen, 56–58 Yourgrau, Palle, 15–16