From Kant to Husserl: Selected Essays 9780674065420

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FROM KANT TO HUSSERL

FROM KANT TO HUSSERL SELECTED ESSAYS

Charles Parsons

HARVARD

UNIVERSITY

Cambridge, Massachusetts London, England 2012

PRESS

Copyright © 2012 by the President and Fellows of Harvard College All rights reserved Printed in the United States of America Library of Congress Cataloging-in-Publication Data Parsons, Charles, 1933– From Kant to Husserl : selected essays / Charles Parsons. p. cm. Includes bibliographical references (p. ) and index. ISBN 978-0-674-04853-9 (alk. paper) 1. Philosophy, German—18th century. 2. Philosophy, German—19th century. 3. Philosophy, German—20th century. 4. Philosophy, Modern. 5. Kant, Immanuel, 1724–1804. 6. Frege, Gottlob, 1848–1925. I. Title. B2741.P37 2012 193—dc23 2011030877

For Jotham and Sylvia

CONTENTS

Preface

ix

Part I: Kant

Note to Part I

3

1 The Transcendental Aesthetic

5

2 Arithmetic and the Categories

42

3 Remarks on Pure Natural Science

69

4 Two Studies in the Reception of Kant’s Philosophy of Arithmetic

80

Postscript to Part I

100

Part II: Frege and Phenomenology

5 Some Remarks on Frege’s Conception of Extension

117

Postscript to Essay 5

131

6 Frege’s Correspondence

138

Postscript to Essay 6

158

7 Brentano on Judgment and Truth

161

8 Husserl and the Linguistic Turn

190

Bibliography

217

Copyright Acknowledgments 231 Index 233

PREFACE

The present volume is the first of two volumes collecting most of my essays on other philosophers, excluding those already reprinted some years ago in Mathematics in Philosophy. The rough division of the volumes is between essays on pre-twentieth-century and on later authors. Thus the projected second volume will be titled Philosophy of Mathematics in the Twentieth Century. The essays in the present volume are on the whole less focused on philosophy of mathematics than those in the projected second volume. Frege and Brentano are reasonably thought of as nineteenth-century figures, although they were intellectually active into the twentieth century, and some of their late work is discussed in these essays. Husserl is certainly a twentieth-century figure, and the work discussed in Essay 8 was all at least published in that century and mostly written then. But it seemed more appropriate to group that essay with those on Frege and Brentano, the more so since it is not at all about philosophy of mathematics. Of the Kantian essays, Essays 1, 2, and 4 grew out of my earlier work on Kant’s philosophy of arithmetic. However, Essay 1 is a general commentary on the Transcendental Aesthetic and does not attempt to present an interpretation of Kant’s philosophy of mathematics. It is well known that much of the latter is to be found in other, rather scattered, writings of Kant. Essay 1 is the only place where I have attempted to say something substantial about the distinction between appearances and things in themselves, but it limits its focus to the Aesthetic and so does not purport to be a full treatment of that theme even in the Critique of Pure Reason. It is an issue that I have always found difficult, and I am sympathetic to the view expressed by Allen Wood that it is not possible to resolve the main disputes on the basis of ix

PREFACE

the texts.1 Essay 2 takes as its point of departure an obvious question: Since, according to a tradition that Kant did not question, mathematics is the science of quantity, arithmetical notions should have a definite relation to the categories of quantity. The main task of the essay is to explore what that relation is. Important relevant information is gleaned from Kant’s lectures on metaphysics and related texts. Essay 4 concerns some texts in which Kant’s philosophy of arithmetic is discussed in the early years after Kant’s own publication, first the writings of his disciple Johann Schultz (1739–1805), written while Kant was still active, and then an early essay by Bernard Bolzano (1781–1848), published in 1810 and thus not long after Kant’s death. My hope was that studying some early reactions to Kant’s philosophy of mathematics would shed some light on disputed questions about its interpretation. That hope was realized to at most a limited extent, but the texts studied are of interest in their own right. Essay 3 stands apart from the other Kantian essays because it was prepared as comments on a paper on Kant’s philosophy of science by Philip Kitcher. It concerns what Kant meant by “pure natural science” and thus the relation between the first Critique and the Metaphysical Foundations of Natural Science. Up to that time scholarship on the latter text was largely German, but shortly after its publication Michael Friedman’s powerful studies of Kant’s philosophy of physics began to appear, and that has stimulated other work on Kant’s philosophy of science. I do not attempt to comment on that work here, but I hope that some points in my small essay are still found of interest. It will be clear that the two essays on Frege, Essays 5 and 6, are not focused on large issues concerning his logic and philosophy. They are distinctly less ambitious than my earlier essay “Frege’s Theory of Number” (Essay 6 of Mathematics in Philosophy). The distinctive character of Frege’s conception of extension, compared to the concept of set as it developed from Cantor on, was certainly worth pointing out, and since the first publication of Essay 5, more has been done on the subject by others. Others have written about Frege’s reaction to the discovery of Russell’s paradox. My essay attempts to focus specifically on the concept of extension in this context. The essay was, in addition, preliminary to some of my writing on the concept of set, in particular “What Is the Iterative Conception of Set?” (Essay 10 of Mathematics in Phi1

See his Kant, pp. 63–76. x

PREFACE

losophy). The discussion of the Frege–Russell correspondence in Essay 6 serves to amplify Essay 5, as does the brief discussion of the draft letter of 1918 to Karl Zsigmondy. Since Essay 6 originated as an extended review of Frege’s collected correspondence, it unavoidably takes up rather different themes. The reader may be surprised by the fact that I wrote Essay 7 on Brentano, whose view of formal logic was somewhat conservative and who did not contribute significantly to the philosophy of mathematics. In fact, it originated with my teaching of Husserl in the 1990s. I followed Dagfinn Føllesdal in prefacing exposition of Husserl with a brief treatment of Brentano, emphasizing the famous remarks about “intentional inexistence” in Psychology from an Empirical Standpoint. However, I was led to study other writings of Brentano, particularly the compilation Wahrheit und Evidenz, and decided to expand the part of the course devoted to Brentano and treat him more as a figure in his own right.2 I found his views on judgment and truth of particular interest and thus was ready to accept an invitation from Dale Jacquette to contribute to the Cambridge Companion to Brentano. In contrast to that in Brentano, my interest in Husserl is of long standing, originating in my graduate student days, through discussion groups on texts by Merleau-Ponty and Husserl. Although I have never felt close to any phenomenological school, phenomenology has exercised a more general influence on my approach to philosophy. But although there are remarks about Husserl in earlier writings of mine, Essay 8 is the only article-length discussion of themes in Husserl that I have written. It was prompted by an invitation to appear as a critic in an Author Meets Critics session on Michael Dummett’s Origins of Analytical Philosophy. It thus seemed suitable as a contribution to the volume on the history of analytical philosophy edited by Juliet Floyd and Sanford Shieh, intended to honor our common teacher Burton Dreben but published only after his death. Dreben was a major pioneer of the now flourishing study of the history of analytical philosophy. The essay is revealing about the nature of my own engagement with that enterprise. My writing on Frege was relatively early and did not lead to the sustained scholarly engagement of Dummett, Tyler Burge, Thomas Ricketts, Richard Heck, and others. I have also not been attracted to scholarly work on Russell, 2

The interest of Wahrheit und Evidenz had been urged on me some time before by Per Martin-Löf. xi

PREFACE

Wittgenstein, and the Vienna Circle. On the other hand, I have taken an interest in figures on the periphery of this history, who were not analytical philosophers as that is understood by historians but were close enough to it either to exercise an influence or to be objects of (sometimes polemical) attention. Husserl plays this role in Essay 8 of this volume, and Brouwer and Hilbert also played such a role, although that is not emphasized in what I have written about them. One might say the same about Gödel, who has been the subject of my most sustained scholarly endeavor concerning twentieth-century philosophy. In Essay 8 I also argue that Husserl is significant as an object of comparison with analytical philosophers during the development of that tendency. I believe that there are others who could be fruitfully studied from that point of view. Some interesting such work has been done by others, for example Michael Friedman in his book A Parting of the Ways and other writings. In writing about these figures, including Kant, I do not claim to be a historian of philosophy. Although I have taken an interest in a number of historical figures, and I have tried not to be unhistorical in my approach to them, I have not attempted to produce a full portrait of the thought of any of the figures I have written about or of the general development of sets of ideas that interest me. Thus the essays in this volume and its projected successor are essays and not monographs or fragments of monographs. Without very consciously addressing the question, I have thought that a larger-scale study of any of these figures would be too great a distraction from systematic work in logic and philosophy of mathematics. It can’t escape the reader’s notice that these essays reflect a bias in my attention toward figures who wrote in German. This may have begun with my interest in Kant, but it also reflects a wider interest in German culture and history first stimulated by my father. A fuller study of the history of the foundations of mathematics in the nineteenth and twentieth centuries would have to take in a number of British figures, Russell first of all but also nineteenth-century figures. And one would have to take in Poincaré and other French figures. And at least one American from before our own time, Charles Sanders Peirce, would have to be included in the story. But as a writer of essays, I make no apology for the fact that my choice of subjects rests to some extent on personal attitudes. I have supplied the essays on Kant and Frege with postscripts, in the case of Kant somewhat lengthy. The study of Kant’s philosophy of mathematics was transformed by a new generation of scholars, first Michael xii

PREFACE

Friedman and then a number of younger writers. I thought that I could not reprint writings of my own without addressing issues raised by this important work. The case of Frege was less pressing, but I thought it desirable to comment on some writing about Frege and at least one documentary discovery, that of his letters to Ludwig Wittgenstein. One general development that finds some reflection in the Frege postscripts should be mentioned here. That is the great growth of our knowledge of Frege’s biography. This can be seen as a product of the demise of the German Democratic Republic as it was before 1989. Nearly all of Frege’s life was spent in that territory. The East German scholar Lothar Kreiser was able to do quite a bit of relevant archival research, but the articles he published were little read. After German reunification, however, the Frege scholar Gottfried Gabriel became professor in Jena, and research on Frege and his milieu expanded greatly; in particular, Kreiser’s biography, Frege, containing a great deal of information about Frege and his environment, appeared in 2001.3 The essays are reprinted unrevised. However, some additions have been made to footnotes. The additions are signified by square brackets. However, Postscripts written for this volume attempt to come to grips with some of the work on the subjects of these essays that has been done since their original publication. The writings reprinted here reflect somewhat similar but not quite the same debts as do my other writings. Essays 2, 3, 5, and 6 were written when I was at Columbia University, and Essays 1, 4, 7, and 8 after my move to Harvard. I owe much to both institutions and to my colleagues there, as acknowledgments in particular essays will show. Burton Dreben and Hao Wang stimulated and encouraged my early interest in the history of the foundations of mathematics. W. V. Quine, though not a historical scholar, set an example by his knowledge of languages and his wide reading in earlier work in logic. I have had several stimulating interlocutors about Kant, particularly Robert Paul Wolff, Hubert Dreyfus, the late Samuel Todes, Stephen Barker, and Jaakko Hintikka in earlier years, and later Dieter Henrich, Paul Guyer, Carl Posy, John Carriero, Tyler Burge, Daniel Warren, Béatrice Longuenesse, and Daniel 3

Of other informative publications on this subject, I mention Gabriel and Kienzler. Another figure whose biography and intellectual background have been illuminated by work made possible by the political change is Rudolf Carnap, who studied in Jena and finished his doctorate there. See especially Awodey and Klein. xiii

PREFACE

Sutherland. On Frege, I owe much to Michael Dummett, Tyler Burge, Thomas Ricketts, Warren Goldfarb, George Boolos, and Richard Heck, although in some cases more to their writings. What I have written reflects only part of what I learned from these individuals and still less of what they had to teach me. On Husserl and phenomenology, I evidently owe much to Dagfinn Føllesdal, and earlier I learned much from Dreyfus and Todes, and later from Kai Hauser and Mark van Atten. Students have been a source of instruction and stimulation in these areas as in others. On Kant, I should mention Alan Shamoon, Pierre Keller, and Ofra Rechter at Columbia, and Emily Carson, Katalin Makkai, Arata Hamawaki, Thomas Teufel, and Andrew Roche at Harvard, as well as Frode Kjosavik (Oslo) and Katherine Dunlop (UCLA). On Frege, I should mention Michael Resnik (Harvard, 1963), who in particular introduced me to Frege’s Nachlaß. On Husserl, worthy of mention are Richard Tieszen, Gail Soffer, and Nathaniel Heiner at Columbia and Abraham Stone at Harvard, as well as Mark van Atten (Utrecht). I wish to thank Denis Buehler for his valuable editorial and other assistance with this volume. I also thank John Donohue of Westchester Book Services and the copy editor, Ellen Lohman, for their careful work and attention to detail, even when I did not always agree with their views. Thanks also to Wendy Salkin for preparing the index. This volume is dedicated to my children, Jotham and Sylvia Parsons. Both are scholars of more distant regions of the past than I have ventured into, and they have had to face an environment less hospitable to their sort of scholarship than I have had to deal with in my own career.

xiv

PART I

KANT

NOTE TO PART I

In these essays the Critique of Pure Reason is cited in the usual A/B manner. Other writings of Kant are cited by volume and page of the Academy edition, Gesammelte Schriften, which are given in the translations I have used and in many other translations, including those of the Cambridge edition. In Essays 1, 2, and 3 the Critique is quoted in Kemp Smith’s translation, sometimes modified. In Essay 4 and the Postscript the Guyer and Wood translation is used for quotations. I use the following short titles and other translations: (Inaugural) Dissertation: De mundi sensibilis atque intelligibilis forma et principiis (2:385–419). Translated by G. B. Kerferd in Kant, Selected Pre-Critical Writings, ed. Kerferd and Walford. Metaphysical Foundations (of Natural Science): Metaphysische Anfangsgründe der Naturwissenschaft (4:467–565). Translated by James Ellington. Prolegomena (4:255–382). Translated by Lewis White Beck (revising earlier translations). “Regions in Space”: “Von dem ersten Grunde des Unterschiedes der Gegenden im Raume” (2:377–383). Translated by D.  E. Walford in Kerferd and Walford, op. cit. Theology lectures: Religionslehre Pölitz (28:989–1126). Translated by Allen W. Wood and Gertrude M. Clark as Lectures on Philosophical Theology. Ithaca, N.Y.: Cornell University Press, 1978.

Translations other than those cited here are my own.

3

1 THE TRANSCENDENTAL AESTHETIC

Among the pillars of Kant’s philosophy, and of his transcendental idealism in particular, is the view of space and time as a priori intuitions and as forms of outer and inner intuition respectively. The first part of the systematic exposition of the Critique of Pure Reason is the Transcendental Aesthetic, whose task is to set forth this conception. It is then presupposed in the rest of the systematic work of the Critique in the Transcendental Logic.

I The claim of the Aesthetic is that space and time are a priori intuitions. Knowledge is called a priori if it is “independent of experience and even of all impressions of the senses” (B2). Kant is not very precise about what this “independence” consists in. In the case of a priori judgments, it seems clear that being a priori implies that no particular facts verified by experience and observation are to be appealed to in their justification. Kant holds that necessity and universality are criteria of apriority in a judgment, and clearly this depends on the claim that appeal to facts of experience could not justify a judgment made as necessary and universal.1 Because Kant is quite consistent about what propositions he regards as a priori and about how he characterizes the notion, the absence of a more precise explanation has not led to its being regarded in commentary on Kant as one of his more problematic notions, even though a reader of today would be prepared at least to entertain the

1

The relevant kind of universality is “strict universality, that is . . . that no exception is allowed as possible” (B3); thus it itself involves necessity. 5

KANT

idea that the notion of a priori knowledge is either hopelessly unclear or vacuous. It is part of Kant’s philosophy that not only judgments but also concepts and intuitions can be a priori. In this case the appeal to justification does not obviously apply. It is harder to separate what their being a priori consists in from an explanation that Kant offers, that they are contributions of our minds to knowledge, “prior” to experience because they are brought to experience by the mind. However, I believe a little more can be said. For a representation to be a priori it must not contain any reference to the content of particular experiences or to objects whose existence is known only by experience. A priori concepts and intuitions are in a way necessary and universal in their application (so that their content is spelled out in a priori judgments). In fact, Kant apparently holds that if a concept is a priori, its objective reality can be established only by a priori means; that seems to be Kant’s reason for denying that change and physical motion are a priori concepts.2 Although this consideration leads into considerable difficulties, they do not affect the apriority of the concepts of space and time or of mathematics. The concept of intuition requires more discussion. Kant begins the Aesthetic as follows: In whatever manner and by whatever means a mode of knowledge may relate to objects, intuition is that through which it is in immediate relation to them. (A19/B33) Later he writes of intuition that it “relates immediately to the object and is singular,” in contrast with a concept which “refers to it mediately by means of a feature which several things may have in common” (A320/ B377). To this should be compared the definition of intuition and concept in his lectures on Logic: All modes of knowledge, that is, all representations related to an object with consciousness, are either intuitions or concepts. The intuition is a singular representation (repraesentatio singularis), the concept a general (repraesentatio per notas communes) or reflected representation (repraesentatio discursiva).3

2 3

For change, see B3, but Kant is not entirely consistent; compare A82/B108. Logik, ed. Jäsche, §1, 9:91. 6

THE TRANSCENDENTAL AESTHETIC

An intuition, then, is a singular representation; that is, it relates to a single object. In this it is the analogue of a singular term. A concept is general.4 The objects to which it relates are evidently those that fall under it. That it is a repraesentatio per notas communes is just what the Critique says in saying that it refers to an object by means of a feature (Merkmal, mark) which several things may have in common. In both characterizations in the Critique, an intuition is also said to relate to its object “immediately.” Kant gives little explanation of this “immediacy condition,” and its meaning has been a matter of controversy. It means at least that it does not refer to an object by means of marks. It seems that a representation might be singular but single out its object by means of concepts; it would be expressed in language by a definite description. One would expect such a representation not to be an intuition. And in fact, in a letter to J. S. Beck of July 3, 1792, Kant speaks of “the black man” as a concept (11:347). Apparently he does not, however, have a category of singular non-immediate representations (i.e., singular concepts). He says that the division of concepts into universal, particular, and singular is mistaken. “Not the concepts themselves, but only their use, can be divided in that way.”5 Kant does not say much about the singular use of concepts, but their use in the subject of singular judgments is evidently envisaged. The most explicit explanation is in a set of student notes of his lectures on logic, where after talking of the use of the concept house in universal and particular judgments, he says: Or I use the concept only for a single thing, for example: this house is cleaned in such and such a way. It is not concepts but judgments that we divide into universal, particular, and singular.6 Thus it is not clear that there are singular representations that fail to satisfy the immediacy condition. 4

“It is a mere tautology to speak of general or common concepts” (Logik §1, note 2, 9:91). 5 Logik §1, note 2, 9:91. Alan Shamoon argues persuasively that this view is directed against Meier and thereby against Leibniz. See “Kant’s Logic,” ch. 5. Appreciation of this remark of Kant, and of Kant’s conception of singular judgments, derives mainly from Thompson, “Singular Terms and Intuitions.” 6 Wiener Logik (1795), 24:909. Shamoon, in commenting on this passage, remarks that a judgment is singular, and its subject concept has singular use, if it has in the subject a demonstrative or the definite article. (See “Kant’s Logic,” p. 85.) 7

KANT

Assuming that there are none, it does not follow that, as Jaakko Hintikka maintained in his earlier writings, the immediacy condition is just a “corollary” of the singularity condition,7 since the fact that the only “intrinsically” singular representations are intuitions would not follow from the singularity and immediacy conditions without the further substantive thesis that it is only the “use” of concepts that can be singular. Moreover, we have so far said little about what the immediacy condition means. Evidently concepts are expressed in language by general terms. It would be tempting to suppose that, correlatively, intuitions are expressed by singular terms. This view faces the difficulty that Kant’s conception of the logical form of judgment does not give any place to singular terms. In Kant’s conception of formal logic, the constituents of a judgment are concepts, and concepts are general. We are inclined to think of the most basic form of proposition as being ‘a is F’ or ‘Fa’, where ‘a’ names an individual object, to which the predicate ‘F’ is applied. How is such a proposition to be expressed if it must be composed from general concepts? Evidently the name must itself involve a singular use of a concept. Kant does offer examples involving names as cases of singular judgments,8 but also judgments of the form ‘This F is G’.9 Kant’s acceptance of the traditional view that in the theory of inference singular judgments do not have to be distinguished from universal ones (A71/ B96) implies that the subject concept in a singular judgment can also occur in an equivalent universal judgment.10 Relation to an object not by means of concepts, that is to say not by attributing properties to it, naturally suggests to us the modern idea of direct reference. That that was what Kant intended has been proposed by Robert Howell.11 It appears from the above that Kant’s view must be that judgments cannot have any directly referential constituents, and 7

“Kantian Intuitions,” p. 342. In his principal discussion of the matter, “On Kant’s Notion of Intuition,” Hintikka does not say explicitly how he understands the immediacy condition or its role, but indicates that he thinks the singularity condition gives a sufficient definition. But cf. note 11 of “Kant’s Transcendental Method and His Theory of Mathematics.” 8 ‘Caius is mortal’ in Logik §21, note 1 (cf. A322/B378), also in Logik Pölitz (1789, 24:578); ‘Adam was fallible’, in Refl. 3080 (16:647). 9 In addition to the passage from the Wiener Logik cited above, ‘This world is the best’ in Refl. 3173 (16:695). 10 Kant gives the example ‘God is without error; everything which is God is without error’ in Refl. 3080 (16:647). 11 “Intuition, Synthesis, and Individuation,” p. 210. 8

THE TRANSCENDENTAL AESTHETIC

indeed it has been persuasively argued that Kant has to hold something like a description theory of names.12 This is, however, not a decisive objection, since intuitions are not properly speaking constituents of judgments. This conclusion still leaves some troubling questions, particularly concerning demonstratives. If we render the form of a singular judgment as ‘The F is G’, then the question arises how we are to understand statements of the form ‘This F is G’ or even those of the form ‘This is G’. The latter form might plausibly (at least from a Kantian point of view) be assimilated to the former, on the ground that with ‘this’ is implicitly associated a concept, in order to identify an object for ‘this’ to refer to. But now how are we to understand the demonstrative force of ‘this’ in ‘This F is G’? It only shifts the problem to paraphrase such a statement as ‘The F here is G’. Although there is no doubt something conceptual in the content of ‘this’ or ‘here’ (perhaps involving a relation to the observer), in many actual contexts it will be understood and interpreted with the help of perception. It is hard to escape the conclusion, which seems to be the view of Howell,13 that in such a context intuition is essential not just to the verification of such a judgment and to establishing the nonvacuity of the concepts in it, but also to understanding its content. But it would accord with Kant’s general view that the manifold of intuition cannot acquire the unity which is already suggested by  the idea of intuition as singular representation without synthesis according to concepts, that one should not be able to single out any portion of a judgment that represents in a wholly nonconceptual way. In the Aesthetic, the logical meaning of the immediacy condition that we have been exploring is not suggested. Following the passage cited above Kant says that intuition is that to which all thought as a means is directed. But intuition takes place only in so far as the object is given to us. This again is only possible, to man at least, in so far as the mind is affected in a certain way. (A19/B33) The capacity for receiving representations through being affected by objects is what Kant calls sensibility; that for us intuitions arise only through sensibility is thus something Kant was prepared to state at the outset. It 12

Thompson, “Singular Terms and Intuitions,” p. 335; Shamoon, “Kant’s Logic,” pp. 110–111. 13 “Intuition, Synthesis, and Individuation,” p. 232. 9

KANT

appears to be a premise of the argument of the Aesthetic; if not Kant does not clearly indicate there any argument of which it is the conclusion.14 An earlier proposal of my own, that immediacy for Kant is direct, phenomenological presence to the mind, as in perception,15 fits well both with the opening of the Aesthetic and the structure of the Metaphysical Exposition of the concept of space (see below). One has to be careful because this “presence” has to be understood in such a way as not to imply that intuition as such must be sensible, since that would rule out Kant’s conception of intellectual intuition,16 and of course that human intuition is sensible was never thought by Kant to follow immediately from the meaning of ‘intuition’. That this is what the immediacy condition means can probably not be established by direct textual evi14

A remark at B146 is translated by Kemp Smith as “Now, as the Aesthetic has shown, the only intuition possible to us is sensible.” The German reads simply, “Nun ist alle uns mögliche Anschauung sinnlich (Aesthetik).” The remark does not make clear that Kant is doing more than simply refer to the Aesthetic as the place where that thesis was stated and explained. If it is the conclusion of argument rather than an assumption of Kant, then the argument is not explicitly pointed to in the Aesthetic. The most plausible theory about what such an argument might be would give it a form similar to that of the second edition Transcendental Exposition of the Concept of Space: Geometry is (in some sense to be explicated) intuitive knowledge; this is possible only if the intuition involved is sensible; therefore human intuition is sensible. As an argument for the existence of a priori sensible intuition this might possibly be discerned in the text of the Aesthetic. But something further would be needed to get to the conclusion that all human intuition is sensible. Although I have not systematically studied the use of the terms Anschauung and intuitus in Kant’s earlier writings, it seems clear that they emerge as central technical terms in the 1768–1770 period, when Kant makes the sharp distinction between sensibility and understanding and makes the decisive break with the Leibnizian views of space and sense-perception. Especially noteworthy is the fact that Kant’s early formulation of his views on mathematical proof in the “Untersuchung über die Deutlichkeit der Grundsätze der natürlichen Theologie und der Moral” (2:272–301), although it already makes the connection between mathematics and sensibility, does not use the term Anschauung in the principal formulation of its theses. It occurs only a few times in the entire essay. I would conjecture, then, that in Kant’s development the use of Anschauung as a technical term and the thesis that human intuition is sensible emerged more or less simultaneously and that he did not articulate theories in terms of the notion of intuition in abstraction from, or before formulating, the latter thesis. 15 “Kant’s Philosophy of Arithmetic,” p. 112. 16 Cf. B72 and elsewhere. A fuller explanation of the divine understanding as intellectual intuition is given in the theology lectures (28:1051). 10

THE TRANSCENDENTAL AESTHETIC

dence.17 What is in any case of more decisive importance is the question what role immediacy in this sense might play in the parts of Kant’s philosophy where intuition plays a role, particularly his philosophy of mathematics. The intent of Hintikka, apparently shared by some other writers on pure intuition whose views are not otherwise close to Hintikka’s,18 is to deny that pure intuition as it operates in Kant’s philosophy of mathematics is immediate in this sense at all, whether by definition or not. Whether this is true is a question to keep in mind as we proceed.

II I now turn to the argument of the Aesthetic. The part of the argument called (in the second edition) the Metaphysical and Transcendental Expositions of the concepts of space and time (§§2–3 [through B41], 4–5) argues that space and then time are a priori intuitions. The further conclusions that they are forms of our sensible intuition, that they do not apply to things as they are in themselves and are thus in some way subjective, are drawn in the “conclusions” from these arguments (remainder of §3, §6) and in the following “elucidation” (§7) and “general observations” (§8, augmented in B). The framework is Kant’s conception of “sensibility,” the capacity of the mind to receive representations through the presence of objects. By means of outer sense, a property of our mind, we represent to ourselves objects as outside us, and all without exception in space. (A22/B37) 17

Two passages in the Dissertation are highly suggestive: For all our intuition is bound to a certain principle of form under which form alone can something can be discerned by the mind immediately or as singular, and not merely conceived discursively through general concepts. (§10, 2:396) That there are not given in space more than three dimensions, that between two points there is only one straight line, . . . etc.—these cannot be concluded from some universal notion of space, but can only be seen in space itself as in something concrete. (§15C, 2:402–403)

Both, it seems to me, support the claim that intuition is immediate in the sense at issue. The punctuation of the Latin in the first passage, however, suggests that singulare is being offered as explication of immediate, and thus rather goes against the claim that the connection between immediacy and ‘seeing’ obtains by definition. It is not, on the other hand, something for which Kant argues. 18 For example Pippin, Kant’s Theory of Form, ch. 3. 11

KANT

“Outside us” cannot have as its primary meaning just outside our bodies, since the body is in space and what is inside it is equally an object of outer sense.19 Kant alludes at the outset to what is in fact the background of all his thinking about space (and to a large extent time as well): the issue between what are now called absolutist and relationist conceptions of space and time, represented paradigmatically by Newton and Leibniz: What, then, are space and time? Are they real existences? Are they only determinations or relations of things, yet such as would belong to things even if they were not intuited? (A23/B37) Early in his career Kant’s view of space was relationist and basically Leibnizian. This was what one would expect from the domination of German philosophy in Kant’s early years by Christian Wolff’s version of Leibniz’s philosophy. Kant was, of course, influenced from the beginning by Newton and was never an orthodox Wolffian. In 1768, in “Regions in Space,” he changed his view of space in a more Newtonian direction;20 this was the first step in the formation of his final view, which is in essentials set forth in the Inaugural Dissertation of 1770. The Metaphysical Exposition of the Concept of Space gives four arguments, the first two evidently for the claim that space is a priori, the second two for the claim that it is an intuition. (i) The first argument claims that “space is not an empirical concept which has been derived from outer experiences” (A23/B38). The representation of space has to be presupposed in order to “refer” sensations to something outside me or to represent them as in characteristic spatial relations to one another. This argument might seem to prove too much, if its form is, “In order to represent something as X, the representation of X must be presupposed.” If that is generally true, and if it implies that X is a priori, the argument would show that all representations are a priori. Kant, however, seems rather to be claiming that the representation of space (as an individual, it will turn out from the third and fourth

19

Although I don’t know of specific comments by Kant on “proprioceptive” sensations, it follows that such objective content as they have would belong to outer sense. 20 This essay is generally represented as (temporarily) completely buying the Newtonian position. Reasons for caution on this point, in my opinion justified, are given in William Harper, “Kant on Incongruent Counterparts.” 12

THE TRANSCENDENTAL AESTHETIC

arguments) must be presupposed in order to represent particular spatial relations. The argument should be seen as aimed at relationism. Leibniz would be committed to holding that space consists of certain relations obtaining between things whose existence is prior both to that of space and to these relations. However, it seems open to the relationist to say that objects and their spatial relations are interdependent and mutually conditioning.21 The argument is stronger if it is viewed as calling attention to the fact that it is the spatial character of objects that enables us to represent them as distinct from ourselves and from each other. This is not the plain meaning of the text. That it may be Kant’s underlying intention, however, is suggested by a parallel passage in the Dissertation: For I may not conceive of something as placed outside me unless by representing it as in a place which is different from the place in which I myself am, nor may I conceive of things outside one another unless by locating them at different places in space. (§15A, 2:402) (ii) The second argument claims that space is prior to appearances, in effect to things in space: We can never represent to ourselves the absence of space, though we can quite well think it as empty of objects. (A24/B38–39) In what sense of “represent” can we not represent the absence of space? The existence of space is not necessary in the most stringent sense; in whatever sense we can think things in themselves, we can think a nonspatial world. On the other hand, Kant has to claim more than that we are incapable, as a “psychological” matter, of imagining or representing in some other way the absence of space.22 Kant’s conclusion will be that space is in some way part of the content of any intuition, and in that way any kind of representation that allows representing the absence of space will not be intuitive. Thus he 21

As was apparently urged against Kant by Eberhard’s associate J. G. E. Maass; see Allison, Kant’s Transcendental Idealism (1st ed.), p. 84, and The Kant-Eberhard Controversy, pp. 35–36. 22 This psychologistic reading has been advocated by some commentators, e.g., Kemp Smith, Commentary, p. 110. It is somewhat encouraged by the German: Wir können uns niemals eine Vorstellung davon machen, daß kein Raum sei. Although our inability to imagine the absence of space is not what Kant is ultimately after, it is of course an indication of it, and has some force as a plausibility argument. 13

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says that it is “the condition of the possibility of appearances” (A24/ B39). I doubt that one can single out at the outset, independent of the further theory Kant will develop, a notion of representation in which we can’t represent the absence of space. That space is a fundamental phenomenological given that in some way can’t be thought away is a very persuasive claim. But it would take a whole theory to explain what it really means, and Kant seems to have to appeal to more theory in order to explicate it himself. We can think its absence, but we can’t give content to that thought in the sense of “content” that matters: relation to intuition. But that way of putting the point presupposes not only the claim that outer intuition is spatial, but the claim that concepts require intuition in order not to be empty. Kant says we can think space without objects. This is in one way obviously true; for example it is what we do in doing geometry. It is not clear, however, that Kant means to appeal to geometry at this point, and if he does one could, at least from a modern point of view, object to his claim on the ground that in geometry we are dealing with a mathematical abstraction, not with physical space (or at least that it is then a substantive scientific, and in the end empirical, question whether our description of space fits physical reality). In any event, it is not clear that the thought of space without objects is not really just the thought of space with objects about which nothing is assumed. This understanding, which seems weaker than what Kant intended, is sufficient for Kant’s claim that space is a priori but possibly not for his case against relationism. (iii–iv) The third and fourth arguments of the Metaphysical Exposition are, as I have said, concerned to show that space is an intuition. Strictly, the claim is that this is true of the “original representation” of space (B40), since from Kant’s point of view there clearly must be such a thing as the concept of space, to be a constituent of judgments concerning space.23 23

In fact, he ought to distinguish between what he calls the “general concept of space” (A25), which would apply to portions of space, and the concept that applies uniquely to the “one and the same unique space” (A25/B39). The latter could, however, be a “singular use” of the former, although that would oblige us to view it as expressed by a demonstrative attached to the word “space” in its general meaning. Kant in the Dissertation speaks more freely of “the concept of space” and writes for example, The concept of space is therefore a pure intuition. For it is a singular concept, . . . (§15C, 2:402)

while in the Critique he writes, 14

THE TRANSCENDENTAL AESTHETIC

Part of Kant’s claim, what is emphasized in the third argument, is that the representation of space is singular. This has a clear and unproblematic meaning. That when it refers to the space in which we live and perceive objects, or to the space of classical physics, ‘space’ is singular is an obvious datum of what one might call grammar; moreover, its having reference in the former usage surely rests on the fact that there is a unique space of experience, and it is reasonable to suppose that the uniqueness of space in classical physics derives from this. It is abstractly conceivable, however, that we could have characterized space in some conceptual way from which uniqueness would follow (as might be the case with a conception of God in philosophical theology). Then we would have, not an intuition but a singular use of a concept. Kant clearly intends to rule out this possibility. Now this would be, if not exactly ruled out, rendered idle if Kant could claim that the representation of space is not only singular but also immediate in the sense of one of the interpretations mentioned above, of involving presence to the mind analogous to perception. Kant seems to be saying that when he begins the fourth argument with the statement “Space is represented as an infinite given magnitude” (B39; cf. A25). In any event Kant needs, and clearly intends to claim, a form of immediate knowledge of space; otherwise the question would arise whether what he has said about the character of the representation of space does not leave open the possibility that there is just no such thing. Kant also claims that the representation of a unitary space is prior to that of spaces, which he conceives as parts of space. (The modern mathematical notion of space, roughly a structure analogous to what is considered in geometry, is not under consideration.) Spaces in this sense can only be conceived as in “the one all-embracing space” (A25/ B39); unlike a concept, the representation of space contains “an infinite number of representations within itself” (B40). Consequently, the original representation of space is an a priori intuition, not a concept. (B40)

How far this represents an actual difference of view on Kant’s part and how much it is a matter of more careful formulation, I do not know. Even in the second edition of the Critique Kant titles the section we are discussing “Metaphysical Exposition of the Concept of Space.” (This contrast between the Dissertation and the Critique was noted by Kirk Dallas Wilson, “Kant on Intuition,” p. 250.) 15

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Whatever the precise sense of ‘immediate’ in which Kant’s thesis implies that the representation of space is immediate, there is a phenomenological fact to which he is appealing: places, and thereby objects in space, are given in one space, therefore with a “horizon” of surrounding space. The point is perhaps put most explicitly in the Dissertation: The concept of space is a singular representation comprehending all things within itself, not an abstract common notion containing them under itself. For what you speak of as several places are only parts of the same boundless space, related to one another by a fixed position, nor can you conceive to yourself a cubic foot unless it be bounded in all directions by the space that surrounds it. (§15B, 2:402) This way of putting the matter has the virtue of describing a sense in which space is given as infinite (better “boundless”) which does not commit Kant to any metrical infinity of space (that is, the lack of any upper bound on distances), although his allegiance to Euclidean geometry did lead him to affirm the metrical infinity of space. Kant says that space is given as “boundless”; he also wishes to say that, without the aid of the intuition of space, no concept would accomplish this: A general concept of space . . . cannot determine anything in regard to magnitude. If there were no limitlessness in the progression of intuition, no concept of relations could yield a principle of their infinitude. (A25) Kant does not, so far as I can see, argue in the Aesthetic that the infinity of space could not be yielded by “mere concepts” at all, still less that no infinity at all could be obtained in that way. His arguments seem at most to say that “a general concept of space” could not do this and are not in my view of much interest. It seems very likely that from Kant’s point of view there can be a conceptual representation whose content would in some way entail infinity (that of God would again be an example24). From a modern point of view, we can describe (say, by logical 24

In his theology lectures, however, Kant discusses the “mathematical infinity” of God and says that “the concept of the infinite comes from mathematics, and belongs only to it” (28:1017). To say that God is infinite in this sense is to compare his magnitude with some unit. Since the unit is not fixed, one does not derive an absolute notion of the greatness of God, even in some particular dimension (such as understanding). It is doubtful that from Kant’s point of view the statement that 16

THE TRANSCENDENTAL AESTHETIC

formulae) types of structure that can have only infinite instances; an axiomatization of geometry would be an example. Such a description would use logical resources unknown to Kant, and that he would have recognized the possibility of a purely conceptual description of mathematically infinite magnitude is doubtful.25 But even if he did, there would be the further question of constructing it, which would be the equivalent for Kant of showing its existence in the mathematical sense. Construction is, of course, construction in intuition. By the “progression of intuitions” in the above quotation from A25 Kant presumably means some succession of intuitions relating to parts of space each beyond or outside its predecessor; such a succession would “witness” the boundlessness of space. A similar appeal to intuition is needed also for the construction of numbers, so that arithmetic does not yield a representation of infinity whose non-empty character can be shown in a “purely conceptual” way. What is accomplished by the Metaphysical Exposition? Kant makes a number of claims about space of a phenomenological character that seem to me on the whole sound. That space is in some way prior to objects, in the sense that objects are experienced as in space, and in the sense that experience does not reveal objects, in some way not intrinsically spatial, that stand in relations from which the conception of space could be constructed, seems to me evident. The same holds for the claim that space as experienced is unique and boundless (in the sense explained above). Furthermore, it seems to me that these considerations do form a formidable obstacle that a relationist view such as Leibniz’s has to overcome. However, they are not a refutation of such a view, since God is infinite in this sense is free from reference to intuition. Kant also considers the notion of God as “metaphysically infinite”: In this concept we understand perfections in their highest degree, or better yet, without any degree. The omnitudo realitatis [All of reality] is what is called metaphysical infinity (28:1018, trans. p. 49).

Kant concludes that the term “All of reality” is more appropriate than “metaphysical infinity.” (A briefer remark with the same purport is in Kant’s letter to Johann Schultz of November 25, 1788, 10:557.) I would conclude that although a purely conceptual characterization of God does entail that God is infinite, in what Kant considered the proper sense this implication cannot be drawn out without intuition. 25 On this point see §II of Friedman, “Kant’s Theory of Geometry,” which contains an interesting discussion of these passages. Compared to my own discussion in the text, Friedman downplays the phenomenological aspect. 17

KANT

phenomenological claims of this kind would not suffice to show that in our objective description of the physical world, we would not in the end be able to carry out a reduction of reference to space to reference to relations of underlying objects such as Leibniz’s monads. It is another question how much of a case Kant has yet made for the stronger claims of his theory of space. Regarding the claim that space is a priori, part of the content of this is surely that propositions about space will be known a priori, and it is hard to see so far that anything very specific has been shown to have this character. But the propositions in question will be primarily those of geometry, and we have not yet examined the Transcendental Exposition or other evidence concerning Kant’s view of geometry. The kind of considerations brought forth in the Metaphysical Exposition also hardly rule out possible naturalistic explanations. It could be objected that our experience is spatial because we have evolved in a physical, spatio-temporal world. Such an explanation would of course presuppose space, but it would be empirical in that it made use of empirical theories such as evolution (or some alternative naturalistic account). It would view the inconceivability of the absence of space as a fact about human beings. In a way it could not have been otherwise: beings of which it is not true would not be human beings in the sense in which we use that phrase. But although we can’t conceive how it could turn out to be wrong, it is in some way abstractly possible that it should turn out to be wrong; some change in the world, which our present science is incapable of envisaging, could lead us to experience the world (and ourselves) as, say, in two spaces instead of one. Now we should probably understand the claims made in the Metaphysical Exposition as ruling out the kind of naturalistic story just sketched. When Kant says that the representation of space “must be presupposed” in one or another context, the necessity he has in mind is something stricter than the natural necessity that is the most stringent that one could expect to come out of the naturalistic story. This does not change the philosophical issue, since the naturalist would respond that in so far as they make this strong claim, the claims of the Metaphysical Exposition are dogmatic. I shall leave the issue at this point, because the notion of necessity will come up at some further points in our discussion of the Aesthetic, in particular in connection with geometry. 18

THE TRANSCENDENTAL AESTHETIC

Since I have said that the Metaphysical Exposition, although it poses a real difficulty for relationism, does not refute that view, we should not leave it without noting that it does not contain Kant’s whole case against the relationist position. Kant’s break with relationism came in “Regions in Space” in 1768. There he refers to an essay by Euler which argues for absolute space on the basis of dynamical arguments which go back to Newton.26 Kant says that Euler’s accomplishment is purely negative, in showing the difficulty the relationist position has in interpreting the general laws of motion, and that he does not overcome the difficulties of the absolutist position in the same domain (2:378). Kant then deploys his own argument, the famous argument from “incongruent counterparts.” Although this argument does not occur in the Critique, it is used for different purposes in other later writings of Kant, up to the Metaphysical Foundations of Natural Science of 1786.27 By incongruent counterparts Kant means bodies, in his examples three-dimensional, that fail to be congruent only because of an opposite orientation. (The same term could be applied to figures representing their shapes.) One can think of right and left hands, with some idealization, as such bodies. He considers them “completely like and

26

Euler, “Réflexions sur l’espace et le tems.” Kant’s own final position about absolute space is presented in the Metaphysical Foundations, according to which absolute space is a kind of Idea of Reason. The manner in which he discusses the question, both briefly in the 1768 essay and more fully in the Metaphysical Foundations, should dispel a somewhat misleading impression created by the exposition in the Aesthetic, from which a reader could easily conclude that in developing his theory of space and time, Kant was not concerned with the considerations about the foundations of mechanics that were central to the debate between Leibniz and Newton and have played a central role in debates about relationist and absolutist or substantivalist views down to the present day. (See Michael Friedman, “Metaphysical Foundations of Newtonian Science”; cf. §IV of Friedman, “Causal Laws.”) 27 In the §15C of the Dissertation, Kant appeals to incongruent counterparts in arguing that the representation of space is an intuition (2:403). In §13 of the Prolegomena (2:285–286) and more briefly in the Metaphysical Foundations (4:483– 484), it is offered further as a consideration in favor of the view that space is a form of sensibility not attaching to things in themselves. It has been maintained that Kant’s different uses of the argument are inconsistent (for example, Kemp Smith, Commentary, pp. 161–166). A thorough discussion of Kant’s use of the argument, which undertakes to rebut this accusation, is Buroker, Space and Incongruence, chs. 3–5. 19

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similar” (2:382), in particular in size and the manner of combination of their parts. Yet their surfaces cannot be made to coincide “twist and turn [it] how one will,” evidently by continuous rigid motion. Nonetheless, Kant considers the difference to be an internal one, and he says: Let it be imagined that the first created thing were a human hand, then it must necessarily be either a right hand or a left hand. In order to produce the one a different action of the creative cause is necessary from that, by means of which its counterpart could be produced. (2:382–383) Kant claims that the Leibnizian view could not recognize this difference, because it does not rest on a difference in the relations of the parts of the hands. He concludes that the properties of space are prior to the relations of bodies, in accordance with the conception of absolute space and contrary to relationism. Kant’s claim has been defended in our own time by noting that the existence of incongruent counterparts depends on global properties of the space.28 We can already see this by a simple example: In the Euclidean plane, congruent triangles or other figures can be asymmetrical; they can be made to coincide by a motion only if it goes outside the plane into the third dimension. Similarly, it is the three-dimensionality of space (which Kant emphasizes) that prevents incongruent counterparts from being made to coincide; this could be accomplished if they could “move” through a fourth dimension. Moreover, in some spaces topologically differing from Euclidean space, called non-orientable spaces (a Möbius strip would be a [two-dimensional] example), the phenomenon could not arise. Relationist replies to an argument based on these considerations are possible, but I shall not pursue the matter further here.29 28

See Nerlich, “Hands, Knees, and Absolute Space”; also Buroker, Space and Incongruence, ch. 3. 29 For two recent mathematically and physically informed treatments, see Earman, World Enough and Space-Time, ch. 7, and Harper, “Kant on Incongruent Counterparts.” Both concentrate on the argument of “Regions in Space” but also have something to say about the later versions. Harper is more sympathetic, especially to the claim of the Dissertation and later writings that intuition is needed to distinguish incongruent counterparts. Harper’s paper contains a number of references to further literature. Earman’s discussion places the argument in the context of 20

THE TRANSCENDENTAL AESTHETIC

III I now turn to the Transcendental Exposition. I understand by a transcendental exposition the explanation of a concept, as a principle from which the possibility of other a priori synthetic knowledge can be understood. (B40) The claim of the Transcendental Exposition is that taking space to be an a priori intuition is necessary for the possibility of a priori synthetic knowledge in geometry. It is therefore a premise of this argument that geometry is synthetic a priori. Kant clearly understood geometry as a science of space, the space of everyday experience and of physical science. Thus for us, it would be very doubtful that geometry on this understanding is a priori;30 indeed, the development of non-Euclidean geometry and its application in physics were, historically, the main reasons why Kant’s theory of geometry and space came to be rejected. With regard to geometry, as with mathematics in general, Kant, however, does not see a need to argue that it is a priori; it is supposed to follow from the obvious fact that mathematics is necessary (B14–15). In this, Kant was in accord with the mathematical practice of his own time. The absence of any alternative to Euclidean geometry, and the fact that mathematicians had not sought for sophisticated verifications of the axioms of geometry, cohered with the absence of an available way of interpreting geometry so as to give space for the kind of distinction between “pure” and “applied” the development of the absolutist-relationist controversy from Newton to the present day. 30 In fact, that the geometry of space is empirical was held a generation after Kant by the great mathematician C. F. Gauss. Kant’s view that it is only in transcendental philosophy that it is established that mathematics yields genuine knowledge of objects probably implies that although it is a synthetic a priori truth that physical space is Euclidean, this is not intuitively evident in the way geometrical truths are. (Cf. Friedman, “Kant’s Theory of Geometry,” p. 469 and n.20, also p. 482n.36 [of original].) But I do not see that there could be a Kantian argument for the conclusion that physical space is Euclidean that did not take as a premise that space as intuited, as described in the Aesthetic, is Euclidean. [It is all too easy to represent Kant’s view as being that philosophy tells us that space is Euclidean. Any Kantian philosophical argument for the Euclidean character of physical space would take as a premise that space as conceived and studied in geometry is Euclidean.] 21

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geometry that would imply that only the latter makes a commitment as to the character of physical space.31 It seems that there should not be any particular problem with Kant’s assertion that characteristic geometric truths are synthetic, so long as we understand geometry as the science of space. But we must now, as we have not before, take account of the analytic-synthetic distinction. Kant gives the following explanation: In all judgments in which the relation of a subject to the predicate is thought . . . , this relation is possible in two different ways. Either the predicate B belongs to the subject A, as something which is (covertly) contained in this concept A; or B lies outside the concept A, although it does indeed stand in connection with it. In the one case I entitle the judgment analytic, in the other synthetic. (A6–7/B10) When a concept is “contained” in another may not be very clear. As a first approximation, we can say that a proposition is analytic if it can be verified by analysis of concepts. Kant thinks of such analysis as the breaking up of concepts into “those constituent concepts that have all along been thought in it, although confusedly” (A7/B11); this would give rise to a narrower conception of what is analytic than has prevailed in later philosophy. Kant suggests as a criterion of synthetic judgment that in order to verify it, it is necessary to appeal to something outside or beyond the subject concept. This may be experience, if the concept has been so derived, as in Kant’s example ‘All bodies are heavy’ (B12, also A8), or if experience is otherwise referred to. In the case of mathematical judgments it is, on Kant’s view, pure intuition. 31 In the second edition of the Critique (B15) and even more in the Prolegomena Kant talks of “pure mathematics.” I know of only one use of this phrase in the first edition (A165/B206) (but mathesis pura occurs in the Dissertation; see note 44 below). Kant does not say explicitly with what non-pure mathematics he is contrasting it, but the A165/B206 passage suggests that the contrast is with applied mathematics, although he does not use that term there or, so far as I know, elsewhere in the Critique. Additional evidence that that is the contrast Kant intends is that he distinguishes pure from applied logic (A52–53/B77–78) and contrasts pure with applied mathematics in a note to his copy of the first edition of the Critique (Refl. XLIV, 23:28). (I owe the latter observation to Paul Guyer; cf. Guyer, Kant and the Claims of Knowledge, p.  189. I am also indebted here to Michael Friedman.)

22

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In arguing that mathematical judgments are synthetic, Kant emphasizes the case of arithmetic, where he seems (reasonably in the light of history) to have anticipated more resistance. The geometrical example that he gives, that the straight line between two points is the shortest (B16), might be more controversial than some alternatives, which either involve existence or had given rise to doubt. The parallel postulate of Euclidean geometry would meet both these conditions. It is hard to see how by analysis of the concept “point external to a given line” one could possibly arrive at the conclusion that a parallel to the line can be drawn through it, unless it is already built into the concept that the space involved is Euclidean. That latter way of looking at such a proposition, however, is alien to Kant. We can well grant Kant’s premise that geometrical propositions are synthetic; the hard questions about the analytic-synthetic distinction arise with arithmetic and with non-mathematical subject matters. But his view of geometry as synthetic a priori is tied to the mathematical practice of his own time. If we make the modern distinction between pure geometry, as the study of certain structures of which Euclidean space is the oldest example, but which include not only alternative metric structures but also affine and projective spaces, and applied geometry as roughly concerned with the question which of these structures correctly applies to physical space (or space-time), then it is no longer clear that pure geometry is synthetic; at least the question is bound up with more difficult questions about the analytic-synthetic distinction and about the status of other mathematical disciplines such as arithmetic, analysis, and algebra; and the view that applied geometry is a priori would be generally rejected. If we do grant Kant’s premises, however, then the conclusion that space is an a priori intuition is, if not compelled, at least a very natural one. That it is precisely intuition that is needed to go beyond our concepts in geometrical judgments might be found to require more argument, particularly since he does admit the possibility of synthetic a priori judgments from concepts.32 That empirical intuition will not do

32

The modern discussion of the analyticity or syntheticity of arithmetic might be taken to show that the fact that arithmetic is not analytic in Kant’s particular sense does not show that it depends on intuition. So long as one holds to the conception of geometry as the science of space, it is not clear how to apply this line of thought to geometry. 23

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is implied by the premise that geometry is a priori and therefore necessary. Kant does supply such an argument in his account of the construction of concepts in intuition, in the context of describing the difference between mathematical and philosophical method, to which we will now turn. This account has rightly been seen as filling a gap in the argument of the Aesthetic.33 It has been the focus of much of the discussion in the last generation about Kant’s philosophy of mathematics. To construct a concept, according to Kant, is “to exhibit a priori the intuition which corresponds to the concept” (A713/B741). An intuition that is the construction of a concept will be a single object, and yet “it must in its representation express universal validity for all possible intuitions which fall under the same concept” (ibid.). It is clear that Kant’s primary model is geometrical constructions, in particular Euclidean constructions.34 It is construction of concepts that makes it possible to prove anything non-trivial in geometry, as Kant illustrates by the problem of the sum of the angles of a triangle. The proof proceeds by a series of constructions: one begins by constructing a triangle ABC (see Figure 1), then prolonging one of the sides BC to D, yielding internal and external angles whose sum is two right angles, then drawing a parallel CE dividing the external angle, and then observing that one has three angles α ’, β ’, γ whose sum is two right angles and which are equal respectively the angles α , β , γ of the triangle.35 In this fashion, through a chain of inferences guided throughout by intuition, he [the geometer] arrives at a fully evident and universally valid solution of the problem. (A716–717/B744–745) Intuition seems to play several different roles in this description of a proof. The proof proceeds by operating on a constructed triangle, and the operations are further constructions. They are constructions in 33

For example by Hintikka. It does not follow that it is to be read as independent of the connection between intuition and perception or sensibility. The latter view is effectively criticized in Capozzi Cellucci, “J. Hintikka e il metodo della matematica in Kant.” 34 The importance of Euclid for Kant’s philosophy of mathematics was stressed by Hintikka; see in particular “Kant on the Mathematical Method.” Particular Euclidean constructions are stressed by Friedman, “Kant’s Theory of Geometry.” 35 This proof occurs in Euclid, Elements, Book I, Prop. 32. 24

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A E

B

D C

Figure 1.

intuition; space is, one might say, the field in which the constructions are carried out; it is by virtue of the nature of space that they can be carried out. Postulates providing for certain constructions are what, in Euclid’s geometry, play the role played by existence axioms in modern axiomatic theories such as the axiomatization of Euclidean geometry by Hilbert. But not all the evidences appealed to in Euclid’s geometry are of this particular form; in particular, objects given by the elementary Euclidean constructions have specific properties such as (to take the most problematic case) being parallel to a given line. On Kant’s conception, these evidences must also be intuitive. A third role of intuition (connected with the first) is that we would represent the reasoning involving constructive operations on a given triangle as reasoning with singular terms (to be sure depending on parameters). Kant clearly understood this reasoning as involving singular representations. Free variables, and terms containing them, have the property that Kant requires of an intuition constructing a concept, in that they are singular and yet also “express universal validity” in the role they play in arguing for general conclusions.36 A difficult question concerning Kant’s view is whether the role of intuition can be limited to our knowledge of the axioms (including the postulates providing constructions), so that, to put the matter in an idealized and perhaps anachronistic way, in the case of a particular proof such as the above-discussed one, the conditional whose antecedent is the conjunction of the axioms and whose consequent is the theorem 36

This analogy was first noted by Beth, “Über Lockes ‘allgemeines Dreieck’.” 25

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would be analytic. Such a view seems to be favored by Kant’s statement that “all mathematical inferences proceed in accordance with the principle of contradiction”: For though a synthetic proposition can indeed be discerned in accordance with the principle of contradiction, this can only be if another synthetic proposition is presupposed, and if it can be discerned as following from this other proposition. (B14) These remarks have generally been taken to imply that it is only because the axioms of geometry are synthetic that the theorems are.37 On the other hand, Kant describes the proof that the sum of the angles of a triangle is two right angles as consisting of “a chain of inferences guided throughout by intuition” (see above). Interpretations of Kant’s theory of construction of concepts by Beth, Hintikka, and Friedman have all taken that to mean that, according to Kant, mathematical proofs do not proceed in a purely analytical or logical way from axioms.38 It is clear (as has been given particular emphasis by Friedman) that had Kant believed that they do, the Aristotelian syllogistic logic available to him would not have provided for a logical analysis of the proofs. In fact, one anachronistic feature of the question whether the conditional of the conjunction of the axioms and the theorem is analytic is that our formulation of such a conditional would use polyadic logic and nesting of quantifiers, devices that did not appear in logic until the nineteenth century.

37

See for example Beck, “Can Kant’s Synthetic Judgments Be Made Analytic?” pp. 89–90. In his work Prüfung der kantischen Critik der reinen Vernunft, vol. 1, Kant’s pupil Johann Schultz, who was professor of mathematics at Königsberg and who clearly discussed philosophy of mathematics with Kant, seems to have understood Kant’s view in this way. His argument for the synthetic character of geometry is largely, and his argument for the synthetic character of arithmetic is almost entirely, based on the fact that these sciences require synthetic axioms and postulates. Regarding arithmetic, however, there are clear differences between Kant and Schultz (see my “Kant’s Philosophy of Arithmetic,” pp. 121–123). [See also Essay 4 in this volume.] 38 Beth, “Über Lockes ‘allgemeines Dreieck’ ”; Hintikka, “Kant on the Mathematical Method” and other writings; Friedman, “Kant’s Theory of Geometry.” Interestingly, Kurt Gödel expresses this view in an unpublished lecture draft from about 1961 (thus conceivably influenced by Beth but not by the others). [See now Gödel, Collected Works, 3:384. An equally likely source of influence is Russell, Principles of Mathematics, §434.] 26

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It is not literally true that Kant could not have formulated such a conditional; it is not that these logical forms could not be expressed in eighteenth-century German.39 But it would be more plausible to suppose that Kant thought of mathematical reasoning in terms of which he had at least the beginnings of an analysis. What we would call the logical structure of the basic algebraic language, in which one carries out calculations with equations whose terms are composed from variables and constants by means of function symbols, was well enough understood in Kant’s time. Such calculations are described by Kant as “symbolic construction.”40 And of course Kant would not describe the inference involved in calculation as logical. Friedman has illuminated a lot of what Kant says about geometry by the supposition that basic constructions in geometry work in geometric reasoning like basic operations in arithmetic and algebra. And in a language in which generality is expressed by free variables, and “existence” by function symbols, the conditional of the conjunction of the geometric axioms and a theorem could indeed not be formulated, so that the question whether it is analytic, or logically provable, could not arise. We do not have to decide this issue, because in any event Kant’s account of mathematical proof gives clear reasons for regarding geometrical knowledge as dependent on intuition. Nonetheless the Transcendental Exposition is probably not intended to stand entirely on its own independently of the Metaphysical Exposition. That the intuition appealed to in geometry is ultimately of space as an individual does not follow just from a “logical” analysis of mathematical proof 41 or even

39

Formulations of axioms and postulates for geometry that would lend themselves to expressing such a conditional are given by Schultz, Prüfung, 1:65–67. 40 A717/B745. It is not possible for me to go into this notion or how Kant understands the role of intuition in arithmetic and algebra. See Parsons, “Kant’s Philosophy of Arithmetic”; also Thompson, “Singular Terms and Intuitions,” §IV; J. Michael Young, “Kant on the Construction of Arithmetical Concepts”; Friedman, “Kant on Concepts and Intuitions in the Mathematical Sciences.” 41 An influential recent tradition of discussion of Kant’s theory of construction of concepts, represented by Beth, Hintikka, and Friedman, ignores the more “phenomenological” side of Kant’s discussion of these matters. Beth and Hintikka in fact reduce the role of pure intuition in mathematics to elements that would, in modern terms, be part of logic. Hintikka draws the conclusion, natural on such a view, that Kant’s view that all our intuitions are sensible is inadequately motivated. (See “Kant’s ‘New Method of Thought’ and His Theory of Mathematics,” pp. 131–132.) The same tendency is present in Friedman’s writings, but because geometry gives particular constructions, there is a clear place in his account for the intuition 27

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from the observation that what is constructed are spatial figures. Kant presumably meant here to rely on the third and fourth arguments of the Metaphysical Exposition. Before I turn to the further conclusions that Kant draws from his arguments, I should comment briefly on the Metaphysical and Transcendental Expositions of the Concept of Time. These discussions bring in no essentially new considerations. The arguments of the Metaphysical Exposition parallel those of the Metaphysical Exposition of Space rather closely. Since there is not obviously any mathematical discipline that relates to time as geometry relates to space, one may be surprised that a Transcendental Exposition occurs in the discussion of time at all. That time has the properties of a line (i.e., a one-dimensional Euclidean space) Kant evidently thinks synthetic a priori, and he appeals to properties of this kind (A31/B47).42 Kant also adds that “the concept of alteration, and with it the concept of motion, as alteration of place, is possible only through and in the representation of time” (B48). The concepts of motion and alteration are, for Kant, dependent on experience,43 which makes Kant’s statement here misleading, but he did allow synthetic a priori principles whose content is not entirely a priori (B3). Some writers on Kant have thought that Kant thought that arithmetic relates to time in something close to the way in which geometry relates to space. This view finds no support in the Transcendental Exposition or in corresponding places in the Dissertation.44 Though time

of space. (See his “Kant’s Theory of Geometry,” pp. 496–497.) He also gives an extended account of the role of time, even in geometry. For discussion of Friedman’s views, I am much indebted to Ofra Rechter. I regret that time and the format of this article have not permitted me to do them justice here. 42 That “different times are not simultaneous but successive” is perhaps a way of formulating the fact that instants of time are linearly ordered. 43 For motion see A41/B58, also Prolegomena §15 (4:295), for alteration B3. The problems surrounding these views are discussed (with references to other literature) in Essay 3 in this volume. 44 In fact, the latter text seems to give this role to “pure mechanics”: Hence pure mathematics deals with space in geometry, and time in pure mechanics. (§12, 2:397)

For a view of what Kant might have meant by this statement, see Friedman, “Kant on Concepts and Intuitions,” §5. 28

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and arithmetic do have an internal connection, it is difficult to describe and not really dealt with in the Aesthetic.45

IV I now want to turn to the conclusions Kant draws from his discussion of time and space in the Aesthetic. The one with which Kant begins is the most controversial, and in some ways the most difficult to understand: Space does not represent any property of things in themselves, nor does it represent them in their relations to one another. That is to say, space does not represent any determination that attaches to the objects themselves, and which remains when abstraction has been made of all the subjective conditions of intuition. (A26/B42) Kant’s distinction between appearances and things in themselves has been interpreted in very different ways, and accordingly the question what Kant’s fundamental arguments are for holding that “space does not represent any property of things in themselves” is controversial. A second conclusion Kant draws is that “space is nothing but the form of all appearances of outer sense,” or, as he frequently expresses it, the form of outer intuition or of outer sense. One might mean by “form of intuition” a very general condition, which might be called formal, satisfied by intuitions or objects of intuition. This is part of Kant’s understanding of the notion. One must distinguish between the general disposition by which intuitions represent their objects as spatial, and what space’s being a form of intuition entails about the objects of outer intuition, that they are represented as in space, and that they stand in spatial relations that obey the laws of geometry. The latter seems properly called the form of appearances of outer sense. Kant’s doctrine of pure intuition is that this form is itself known or given intuitively.

Relevant texts are the argument for the syntheticity of ‘7 + 5 = 12’ (B15–16), the characterization of number as the “pure schema of magnitude” (A142–3/B182), and Kant’s letter to Schultz of November 25, 1788 (10:554–558). For two related but still differing interpretations of the connection, see Parsons, “Kant’s Philosophy of Arithmetic,” §§ VI and VII, and Friedman, “Kant on Concepts and Intuitions.” 45

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That outer intuition has a “form” in this sense does not by itself imply that space is subjective or transcendentally ideal. It seems that intuitions might have this “form” and the form be itself given intuitively without its following that the form represents a contribution of the subject to outer representation and knowledge of outer things.46 Kant, however, denies this. Space is “the subjective condition of sensibility, under which alone outer intuition is possible for us” (A26/B42). Kant’s arguments, both in the Aesthetic and in corresponding parts of the Prolegomena, are based on the idea that the fact that a priori intuition is possible can only be explained if the form of intuition derives from us, as we will see. There are two different things that are to be explained, one specific to the Aesthetic and one not: first, the fact that there is a priori intuition of space; second, the fact that there is synthetic a priori knowledge concerning space, in particular in geometry. Of course, the existence of such knowledge is one of Kant’s arguments for a priori intuition. But in arguing for the subjectivity of space Kant appeals specifically to a priori intuition rather than to synthetic a priori knowledge. Thus even in the Transcendental Exposition he writes: How, then, can there exist in the mind an outer intuition which precedes the objects themselves, and in which the concept of these objects can be determined a priori? Manifestly, not otherwise than in so far as the intuition has its seat in the subject only, as the formal character of the subject, in virtue of which, in being affected by objects, it obtains immediate representation, that is intuition, of them, and only so far, therefore, as it is merely the form of outer sense in general. (B41) Kant appeals to the same consideration in arguing that space and time are not conditions on things in themselves: For no determination, whether absolute or relative, can be intuited prior to the existence of the things to which they belong, and none, therefore, can be intuited a priori. (A26/B42)

46 Some later writers influenced by Kant seem to have taken the idea of a form of intuition in this way. This is not to say that the form represents things as they are in themselves in Kant’s or some other sense; rather it means merely that whether this is so is a further question.

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Were it [time] a determination or order inhering in things themselves, it could not precede the objects as their condition, and be known and intuited a priori by means of synthetic propositions. But this last is quite possible if time is nothing but the subjective condition under which all intuition can take place in us. (A33/B49) Kant thus argues on the same lines both to the conclusion that a priori intuitions do not apply to things in themselves and to the conclusion that space and time are forms of intuition. In the presentation of the argument in §§8–9 of the Prolegomena, Kant makes clearer that what is advanced is a consideration specific to intuition: Concepts, indeed are such that we can easily form some of them a priori, namely such as to contain nothing but the thought of an object in general; and we need not find ourselves in an immediate relation to an object. (4:282) Thus with regard to a priori intuition, there is a problem about its very possibility; with regard to a priori concepts, the problem only arises from the fact that to have “sense and meaning” they need to be applicable to intuition, and at this stage it is not evident that that intuition has to be a priori.47 Why should it be obvious that a priori intuition which “precedes the objects themselves” must “have its seat in the subject only”? It is tempting to see this in causal terms: there could not be any causal basis for the conformity of objects to our a priori intuitions unless this basis is already there with the intuition itself. We could imagine Kant arguing as Paul Benacerraf does in a somewhat related context:48 we can’t understand how our intuitions yield knowledge of objects unless there is an adequate causal explanation of how they conform to objects, and in

47

Kant could presumably argue that the subjectivity of space is needed to explain synthetic a priori knowledge in geometry by appealing to the “Copernican” hypothesis that “we can know a priori of things only what we ourselves put into them” (Bxviii). The more specific claim about intuition Kant evidently thought more directly evident. Thus Kant says of the Copernican hypothesis that in the Critique itself it will be proved, apodeictically not hypothetically, from the nature of our representations of space and time and from the elementary concepts of the understanding. (Bxxii n.) 48

Benacerraf, “Mathematical Truth.” 31

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the case of a priori intuitions, such an explanation is impossible unless the mind is causally responsible for this conformity. It would be rash to suppose that Kant never thought in this way, and many commentators, perhaps most eloquently P.  F. Strawson in his conception of the “metaphysics of transcendental idealism,”49 have read Kant as saying that the mind literally makes the world, along the way imposing spatial and temporal form on it. Two views about intuition that we have considered above, that an intuition has something like direct reference to an object and that an intuition involves phenomenological presence of an object, may be of some help here. There can’t be direct reference to an object that isn’t there; thus there may be puzzlement as to how an object can be intuited “prior” to its existence (whatever exactly “prior” means here). We have to ask exactly what the object of the intuition is. That to whose existence the a priori intuition is prior is presumably an empirical object. But then maybe the answer is that that object, strictly speaking, isn’t intuited prior to its existence (and perhaps that it can’t be), so that the proper object of the intuition is a form instantiated by it rather than the object itself. Then the claim becomes that the only way in which the form of a not-yet-present object can be intuited is if this form is contributed by the subject. It is not clear to me how the force of this claim is specific to intuition or how it is more directly evident than other applications of the Copernican hypothesis. The phenomenological-presence view seems to me to defeat the literal sense of the claim in Kant’s argument. Since imagination is immediate in the required sense, immediacy of a representation does not imply the existence of its object at all, so that it seems it can perfectly well be “prior” to it. Again, however, a general claim about a priori knowledge survives this observation: Kant can reply that if, in an imaginative thought experiment, I have intuition from which formal properties of objects can be learned, the only assurance that these properties will obtain for subsequent empirical intuitions of what was imagined is if the form is contributed by me. We have to examine more closely the meaning of the conclusion that things in themselves are not spatial or temporal; this might offer hope of greater insight into Kant’s argument. This leads us, however, into one of the worst thickets of Kant interpretation: the concept of 49

Strawson, Bounds of Sense, part four. 32

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thing in itself and the meaning of Kant’s transcendental idealism. Since, according to Kant, transcendental idealism finds support from arguments offered in the Analytic and Dialectic as well as the Aesthetic, we can in the present discussion deal with only one aspect of the issues. One might begin by distinguishing the claim that we do not know that things, as they are in themselves, are spatial (or that our knowledge of things as spatial is not knowledge of things as they are in themselves) from the claim that things as they are in themselves are not spatial. A long-running debate concerns the question whether Kant’s arguments might prove, or at least lend plausibility to, the first claim and yet not prove the second, although it is often suggested by Kant’s language. Kant, it has been claimed, leaves open the possibility, traditionally called the “neglected alternative,” that although we don’t know that things in themselves are spatial, or that they have the spatial properties and relations we attribute to them, nonetheless, without its being even possible for us to know it, they really are in space and have these properties and relations.50 Kant might reply to this objection by appealing to the arguments of the Antinomies, particularly the Mathematical Antinomies.51 That would, however, leave him apparently making a dogmatic claim in the Aesthetic, with no indication that an important part of its defense is deferred. A more interesting reply is that when the concept of thing in itself and Kant’s argument in the Aesthetic are properly understood, it will be clear that the “neglected alternative” is ruled out. One understanding of the contrast of appearances and things in themselves would be that our intuitions represent objects as having certain properties and relations, but in fact they don’t have them. Kant occasionally comes close to saying this: What we have meant to say is . . . that the things we intuit are not in themselves what we intuit them as being, nor their relations so constituted in themselves as they appear to us. (A42/B59) It is hard to see how, on this view, Kant avoids the implication that our “knowledge” of outer objects is false: the objects we perceive are 50

This claim has a long history in writing about Kant; see Allison, Kant’s Transcendental Idealism (1st ed.), pp.  110–114, and Kemp Smith, Commentary, pp. 113–114. 51 Cf. Ewing, Short Commentary, p. 50. 33

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perceived as spatial, but “in themselves,” as they really are, they are not spatial. One might call this general view of the relation of appearances and things in themselves the Distortion Picture. It arises naturally from viewing things in themselves as real things, of which Kant’s Erscheinungen are ways these things appear to us. It identifies how things are in themselves, in Kant’s particular sense, with how they really are.52 This view certainly rules out the “neglected alternative.” But it seems to do so by fiat. It is difficult to see how, on this interpretation, the thesis that things in themselves are not spatial is supported by argument.53 Indeed, if the idea that things in themselves are spatial merely means that their relations have the formal properties that our conception of space demands, the thesis that they are not is pretty clearly incompatible with the unknowability of things in themselves. Space has to be what is represented in the intuition of space, as it were as so represented. A plausible line of interpretation with this result, favored by several passages in the Aesthetic (e.g., that from B41 quoted above), might be called the Subjectivist view. This is what is expressed in Kant’s frequent statements that empirical objects are “mere representations.”54 A better way of putting it might be that for space and time and therefore for the objects in space and time, the distinction between object and representation collapses, or that an “empirical” version of the distinction can 52

Such an identification may be encouraged by §4 of the Dissertation, where Kant writes, Consequently it is clear that things which are thought sensitively are representations of things as they appear, but things which are intellectual are representations of things as they are. (2:392)

This remark is, however, the conclusion of an argument that Kant would have disclaimed in application to space and time in the Critique, appealing to the variability of the “modification” of sensibility in different subjects, as Paul Guyer points out (Kant and the Claims of Knowledge, p. 341). Also, the formulation itself seems to be criticized in the Critique (A258/B313); see Prauss, Kant und das Problem der Dinge an sich, p. 59n.13. Still, the passage encourages the idea that the Distortion Picture is the view with which Kant started when he first came to the view that space is a form of sensibility representing things as they appear. 53 Indeed, it may lead to actual inconsistency, as Robert Howell, who seems to adopt this view, argues in “A Problem for Kant.” 54 Such statements are, however, rare in passages added in the second edition, and the argument where this conception is most strongly relied on in its simple form, the “refutation of idealism” in the Fourth Paralogism, is omitted; in the new Refutation empirical objects are more clearly distinguished from representations. 34

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only be made in some way within the sphere of representations.55 According to this view, the neglected alternative is ruled out because there would be a kind of category mistake in holding that things in themselves, as opposed to representations, are spatial. Paul Guyer, in his discussion of the Aesthetic’s case for transcendental idealism, relies heavily on an interpretation of an argument from geometry in the General Observations to the Aesthetic. I see his interpretation as making this argument turn on just such a subjectivist view. Commenting on Kant’s first conclusion concerning space, Guyer says that Kant assumes that it is not possible to know independently of experience that an object genuinely has, on its own, a certain property. Therefore space and time, which are known a priori, cannot be genuine properties of objects and can be only features of our representations of them.56 Guyer objects to this assumption on the ground that one might conceivably know, because of constraints on our ability to perceive, that any object we perceive will have a certain property; our faculties would restrict us to perceiving objects that independently have the properties in question, so that it would not follow that the objects cannot “on their own” have them. According to Guyer, Kant nonetheless relies on this assumption because he conceives the necessity of the spatiality of objects and their conformity to the laws of geometry as absolute; he holds not merely (1) Necessarily, if we perceive an object x, then x is spatial and Euclidean; but rather (2) If we perceive an object x, then necessarily, x is spatial and Euclidean.57 This has to be a condition on the nature of the objects, not merely a restriction on what objects we can perceive. Hence, according to Guyer, this view commits Kant to the view that spatial form is imposed on objects by us. 55

As Kant suggests in the Second Analogy, A191/B236. Kant and the Claims of Knowledge, p. 362. 57 Ibid., p. 366. 56

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Guyer discerns an appeal to (2) in the second clause of the following remark: If there did not exist in you a power of a priori intuition, and if that subjective condition were not also at the same time, as regards its form, the universal a priori condition under which alone the object of this outer intuition is itself possible; if the object (the triangle) were something in itself, apart from any relation to you, the subject, how could you say that what necessarily exist in you as subjective conditions for the construction of a triangle must of necessity belong to the triangle itself? (A48/B65) Here the first “necessarily” can express the kind of necessity expressed in (1), but the second necessity does not have the form of being conditional on the subject’s construction, intuition, or perception. Guyer states that that the absolute necessity claimed in (2) “can be explained only by the supposition that we actually impose spatial form on objects.”58 It is, indeed, a reason for not resting with the “restriction” view that Guyer regards as the major alternative.59 Apart from its relevance to questions about the distinction between appearances and things in themselves, the point is relevant also to another controversial point: whether Kant’s argument for transcendental idealism in the Aesthetic makes essential appeal to geometrical knowledge, or whether it needs to rely only on the kind of considerations presented in the Metaphysical Exposition. Clearly the Metaphysical Exposition yields at best conditional necessities of the general form of (1); an argument from absolute necessity to transcendental idealism has to rely on geometry. In my view, Guyer’s exegesis of the argument from the General Observations is quite convincing, and this argument is clearer than what can be gleaned from the arguments that proceed more directly from a priori intuition (i.e., B41, A26/B42, and Prolegomena §§8–9, all commented on above).60 58

Ibid., p. 361. Regarding the power of a priori intuition as “the universal a priori condition under which alone the object of this outer intuition is itself possible” (emphasis mine) hardly squares with the restriction view. 60 Guyer seems to suppose that the argument he derives from the General Observations is the same argument as that of the above passages. That seems to me doubtful. He does, however, point to other passages in Kant’s writings where he is pretty clearly arguing from necessity. 59

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The claim (2), however, is more defensible than Guyer allows, at least with regard to geometry: The content of geometry has to do with points, lines, planes, and figures that are in some way forms of objects, and not with our perception. If we accept the usual conception of the necessity of mathematics, what will be necessary will be statements about these entities. There is nothing in the content of these statements to make their necessity conditional on our perceiving or intuiting them. Thus it seems to me likely that Kant was not sliding from conditional necessity to absolute necessity, but rather applying the idea that mathematics is necessary, which he would have shared with his opponents, to the case of the geometry of space. The objection to this is the now standard one, that we do not have reason to believe that the geometry of actual space obtains with such mathematical necessity. Even if we grant Kant this premise, however, it is questionable that he attains the “apodeictic proof” of his Copernican principle that he claims. Whether the essential is a priori intuition or “absolute” necessity, in either case the claim must be that non-application to things in themselves is the only possible explanation. The merit of the Subjectivist view is that it offers a view of appearances as objects that fits with that explanation. The Subjectivist view does not directly imply the Distortion view, but can lead to it naturally. The relation depends on how one thinks of the object of representations. If appearances are representations, it is natural to think of things in themselves as their objects. And Kant clearly sometimes does think of them that way, as for example in places where he says that the notion of appearance requires something which appears: We must yet be in a position to think [objects] as things in themselves; otherwise we should be landed in the absurd conclusion that there can be appearance without anything that appears. (Bxxvi) The same conclusion also, of course, follows from the concept of an appearance in general; namely, that something which is not in itself appearance must correspond to it. (A251) But if the object of our empirical representations is a thing in itself, and these representations represent their objects as spatial, then we have the Distortion view. But this conception of the object of representations 37

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is not the only one that Kant deploys even within the Subjectivist conception, as one can see from the discussions of the concept of object in the A deduction (esp. A104–105) and the Second Analogy (A191/ B236). I would like now to introduce a third possible meaning of the nonspatio-temporality of things in themselves, what I will call the Intensional view. According to this view, the conclusion from the argument of the Aesthetic is that the notions of space and time do not represent things as they are in themselves, where, however, “represent” creates here an intensional context, so that in particular it does not entitle us to single out things in themselves as a kind of thing, distinct from appearances. The manner in which we know things is not “as they are in themselves,” but rather “as they appear.” But talk of “appearances” and “things in themselves” as different objects is at best derivative from the difference of modes of representation. However, there is an inequality between the two, in that representation of an object as it appears is full-blooded, capable of being knowledge, while representation of an object as it is in itself is a mere abstraction from conditions, of intuition in particular, which make such knowledge possible. Assuming that it has been shown that knowledge of things as spatial is not knowledge of them as they are in themselves, on this view there cannot be a further question whether things as they are in themselves are spatial; either “things in themselves are not spatial” merely repeats what has already been shown, or it presupposes that there is a kind of thing called “things in themselves.” This is a philosophically attractive idea, and it is supported by many passages where Kant expresses the distinction as that of considering objects as appearances or as things in themselves, as in the following striking remark: But if our Critique is not in error in teaching that the object is to be taken in a twofold sense, namely as appearance and as thing in itself; if the deduction of the concepts of understanding is valid, and the principle of causality therefore applies only to things taken in the former sense, namely, insofar as they are objects of experience—these same objects, taken in the other sense, not being subject to the principle—then there is no contradiction in supposing that one and the same will is, in the appearance, that is, in its visible acts, necessarily subject to the law of nature, and 38

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so far not free, while yet, as belonging to a thing in itself, it is not subject to that law, and is therefore free. (Bxxvii–xxviii) Gerold Prauss has supported a version of this view by a careful textual analysis of Kant’s manner of speaking about things as they are in themselves.61 Prauss acknowledges, however, that Kant’s way of speaking is far from consistent and that his usage often lays him open to the interpretation of things in themselves as another system of objects in addition to appearances. In fact, Kant often says in virtually the same place things that seem to support the intensional view, and things that contradict it.62 I shall not go into the many questions the intensional view raises. In spite of the above passage from the preface to the second edition, it has often been claimed that this understanding of the distinction will not suffice for the purposes of Kant’s moral philosophy, and indeed Kant’s ethical writings contain passages that would be very difficult to square with it. Clearly, it is beyond the scope of this article to go into such matters. We do, however, have to consider whether the intensional view can offer a sensible interpretation of Kant’s arguments for his conclusions in the Aesthetic. The difficulty lies in the fact, noted above, that Kant in the statement of his conclusions understands the form of sensibility as contributed entirely by the subject, so that the spatiality of objects and their geometrical properties are due entirely to ourselves.63 This is sometimes expressed in the language of the Subjectivist view, as in the claim that a priori intuition “contains nothing but the form of sensibility” (Prolegomena §9, 4:282). That is to say, it is not just conditioned by my own subjectivity, so that it therefore represents them in a way that, in particular, would not be shared by another mind whose forms of intuition were different, but it is conditioned entirely by my own subjectivity. This is the essential element of the conclusion that Guyer draws from the argument from the necessity of geometry in the General

61

Kant und das Problem der Dinge an sich, ch. 1. As Manfred Baum remarks concerning B306–308 in “The B-Deduction and the Refutation of Idealism,” p. 90. The Phenomena and Noumena chapter seems to me on the whole to favor the intensional view, but not consistently, as Baum rightly observes. 63 It is this that gives rise to the temptation to think of the matter causally, which in turn leads naturally to the idea of “double affection,” which the intensional view avoids. 62

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Observations. It is very naturally interpreted by the Subjectivist view of objects. It is not clear, however, that either the conclusion that spatiality arises entirely from the subject or the Subjectivist view of empirical objects is incompatible with the intensional view, which should perhaps be seen primarily as an interpretation of the conception of thing in itself. A difficulty that has been raised for it is the following: According to it, we know certain objects in experience, and we can think these very objects as they are in themselves. But our very individuation of objects is conditioned by the forms of intuition and the categories. How can we possibly have any basis for even thinking of, for example, the chair on which I am sitting “as it is in itself,” when there is no basis for the assumption that reality as it is in itself is divided in such a way that any particular object corresponds to this chair? The only possible reply to this objection is the one suggested by Prauss: when one considers this chair as it is in itself, “this chair” refers to an empirical object, so that its consideration as an appearance is presupposed.64 So long as there is some distinction between empirical objects and representations, this way of understanding talk of things in themselves is available. The conclusion that the intensional view is most concerned to resist, that there is a world of things in themselves “behind” the objects we know in experience, is not forced by Kant’s subjectivist formulations, unless one takes the conditioning by our subjectivity in a causal way. It seems to me clear that Kant intended to avoid taking it in that way, but a discussion of the matter would be beyond the scope of a treatment of the Aesthetic. This is not to deny that Kant’s conclusion is more subjectivist than many who are sympathetic to Kant’s transcendental idealism will be comfortable with. The modern idea of the “relativity of knowledge,” that all our knowledge is unavoidably conditioned by our own cognitive faculties, or language, or “conceptual scheme,” so that we can’t know or even understand how the world would “look” from outside these (for example from a “God’s eye view”) no doubt owes important inspiration to Kant.65 In his conception of forms of intuition, Kant claimed 64

Kant und das Problem der Dinge an sich, pp. 39 ff. It is in turn reflected in Kant commentary, for example in Allison’s idea of “epistemic conditions,” which underlies his interpretation of Kant’s transcendental idealism. 65

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to identify aspects of the content of our knowledge that are conditioned entirely by our own subjectivity but are still knowledge of objects, reflected in the most objective physical science. That one should be able to identify such a “purely subjective” aspect of objective knowledge is surprising and even paradoxical. Even granted a priori knowledge of necessary truths about space, I have found Kant’s arguments in the Aesthetic for this conclusion less than apodeictic. But that premise does give them enough plausibility so that it is not surprising that more modern views that reject this particular radical turn of Kant’s transcendentalism also reject the premise. The Aesthetic is of course not the only place where Kant argues for transcendental idealism or says things bearing on its meaning. In particular, the Analytic probably contributed more to the development of the modern conception just alluded to. I should end by emphasizing once again the very limited scope of the present discussion of transcendental idealism.66

66

I wish to thank the editor for his comments on an earlier version, for his explanation of his own views, and for his patience. I am also indebted to the participants in a seminar on Kant at Harvard University in the fall of 1989. 41

2 ARITHMETIC AND THE CATEGORIES

On its conceptual side, mathematics as Kant understands it involves in an essential way the categories of quantity. This much should be obvious to readers of the Critique of Pure Reason. To trace this connection in more detail, however, has not been a main concern of interpreters of Kant’s philosophy of mathematics, at least recent ones. No doubt it has been thought that the connection is bound up with traditional logic and with a conception of mathematics more restrictive than what has come to prevail since the rise of set theory and abstract mathematics. The questions concerning Kant’s conception of intuition and of construction of concepts that have dominated the literature on Kant’s philosophy of mathematics are more directly connected with philosophical debates of recent times. Nonetheless, an investigation of the relation of arithmetic at least to the categories of quantity might promise to be instructive for several reasons. First of all, it should clarify how Kant understands the basic concepts of arithmetic, that of number in particular. Second, Kant’s conception of number and therefore of arithmetic is bound up with the schematism of the categories, since he describes number as the schema of quantity (A142/B182), and thus with problems in Kant’s philosophy that go beyond his philosophy of mathematics. Third, on just the point of the relation of number and schematism, Kant appears to have changed his view after the first edition of the Critique, as we shall see below. The purpose of the present essay is to explicate Kant’s understanding of arithmetical concepts and their relation to the categories of quantity. This will require some exposition of Kant’s conceptions of quantity, for which we have to rely on Reflections and on his lectures on Metaphysics. With this background we will address Kant’s view of 42

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number and compare what is said in the first edition of the Critique with some texts from 1788–1790. This comparison yields some puzzles of interpretation having to do with the place of number with respect to the pure and schematized categories. I will preface this whole discussion with some remarks about Kant’s view of mathematical objects in general. This story has no overwhelming moral. It does show that however different his picture of the basic concepts of mathematics was from our own, however confused it may have been when measured against what we can now do with the help of set theory and modern logic, Kant had more to say about the concept of number and related concepts than has been appreciated.

I From our modern point of view, a noteworthy feature of Kant’s philosophy of mathematics is the absence of an articulated account of mathematical objects. Kant does talk in a highly general way about objects, in particular in saying that the categories spell out “the concept of an object in general.” But even the pure categories, once they are distinguished from the forms of judgment, envisage concrete objects, since they include substance, causality, and community. Kant’s fullblooded notion of object is that of an object of experience, that is, a spatio-temporal object.1 Thus Kant rarely expresses a philosophical commitment to specifically mathematical objects, although passages that we would read as involving reference to such objects abound in his writings. Exceptions are the statement that ‘7 + 5 = 12’ in a singular proposition (A164/ B205) and the statement that “we can give it [the concept of a triangle] an object wholly a priori, that is, construct it” (A223/B271). In another passage Kant’s language is even stronger: As regards the formal element, we can determine our concepts in a priori intuition, inasmuch as we create for ourselves, in space and time, through a homogeneous synthesis, the objects themselves— these objects being viewed simply as quanta. (A723/B751)

1

On this point see §2 of my “Objects and Logic” [or of Mathematical Thought and Its Objects]. 43

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In the second of these three places, Kant partly takes away what he has given in saying that the triangle is “only the form of an object,” thus apparently shifting from a use of “object” that would comprehend mathematical objects to one that does not.2 Even when he is most explicit about mathematical objects, Kant does not attribute existence to them. In fact he seems to reject such an attribution in saying that “in mathematical problems there is no question of . . . existence at all” (A719/B747).3 The pure category of existence is schematized as existence at a definite time (A145/B184); it implies actual existence (Wirklichkeit). To know the actual existence of something requires connection with an actual perception by means of the analogies of experience (A225/B272). For this reason it seems clear that mathematical existence is not a form of actuality. There are indications rather that Kant thought of it under the category of possibility. This is said quite explicitly by Kant’s disciple Johann Schultz, in criticizing Eberhard for interpreting Kant’s concept of the objective reality of a concept as meaning the actual existence of objects falling under it instead of their possibility: But unfortunately the example from pure mathematics does not fit, for in mathematics possibility and actuality are one, and the geometer says there are (es gibt) conic sections, as soon as he has shown their possibility a priori, without inquiring as to the actual drawing or making of them from material.4 What plays the role of mathematical existence in Kant’s usage is constructibility. It is tempting to regard this as possible existence: the construction of a concept shows the possible existence of an object whose form is given by the construction. Given Kant’s understanding of possibility, however, construction in pure intuition is not sufficient 2

Kant’s notion of an object of experience as explicated by the schematized categories does give place to one type of “object” that is at least not a spatio-temporal thing, namely the accidents or states of substances. This would license an analogous shift in the use of “object.” Probably Kant thought of the “forms” of objects as quanta as similarly provided for by the categories of quantity. 3 See Thompson, “Singular Terms and Intuitions in Kant’s Epistemology,” pp. 338– 339; also Parsons, “Kant’s Philosophy of Arithmetic,” Postscript, p. 148. 4 Review of vol. 2 of Eberhard’s Philosophisches Magazin (1790), in Ak. 20:386n. This review was written in close collaboration with Kant and is partly based on  manuscripts by Kant; however, the passage quoted does not occur in those manuscripts. 44

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to show such possible existence without the aid of certain philosophical considerations. To be possible is to agree “with the formal conditions of experience, that is, with the conditions of intuition and of concepts” (A218/B266; emphasis mine). The latter conditions are of course the categories. In his discussion, Kant is quite explicit about the relevance to the mathematical realm of this conception of possibility. When he says, as we noted above, that a constructed triangle is “only the form of an object,” he goes on to say that to “determine” the possibility of an object of which it is the form, it must be the case that such a figure is “thought under no conditions save those upon which all objects of experience rest” (A224/B271). These are not only space, as a condition of outer appearance, but that “the formative synthesis through which we construct a triangle in imagination is precisely the same as that which we exercise in the apprehension of an appearance, in making for ourselves an empirical concept of it.” These are just the considerations advanced in the Axioms of Intuition. But the consequence seems to be that knowledge of the objective reality of mathematical concepts, that is, the possible existence of instances of them, is philosophical rather than purely mathematical knowledge.5 This state of affairs poses a dilemma for Kant’s philosophy with regard to the status of mathematical knowledge. Kant’s conception of mathematical knowledge as resting on demonstrative proof in which the essentially mathematical element is construction in pure intuition makes it of a quite different character from philosophical; of course that contrast is the main theme of the Discipline of Pure Reason in its Dogmatic Employment. It seems quite clear that Kant thinks of such knowledge as independent of philosophy. But mathematical demonstration seems not to yield knowledge of objects in the genuine sense, unless it is supplemented by some philosophical reflection. A much-cited remark in the second edition Transcendental Deduction illustrates the difficulty: Through the determination of pure intuition we can acquire a priori knowledge of objects, as in mathematics, but only in regard to their form, as appearances; whether there can be things which must be intuited in this form, is still left undecided. Mathematical concepts are not, there, by themselves knowledge, except 5

Cf. Thompson, op. cit., p. 339. 45

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on the supposition that there are things which allow of being presented to us only in accordance with the form of that pure sensible intuition. (B147) One possible resolution would be to admit an ambiguity in the phrase “knowledge of objects”; mathematical knowledge, unaided by philosophy, is knowledge of objects in a weaker sense, in which the objects known are forms such that so far as mathematics is concerned it is “left undecided” whether they are the forms of real objects. That suggestion seems to commit Kant to mathematical objects. But other versions of this resolution might make Kant something like what is nowadays called a modalist; that is, constructibility of a concept would entail the possible existence of a (physical) object of the form involved, but the notion of possibility would have to be attenuated when compared with that explicated in the Postulates; it would be a version of what recent writers have called mathematical possibility.6 The resolution most immediately suggested by this passage, however, would still leave mathematical knowledge as knowledge of objects in the full-blooded sense. But although mathematical demonstration would yield knowledge of such objects (since the supposition Kant mentions is true), it would not establish that the concepts involved are objectively real. This suggestion still leaves open the interpretation of quantifiers in mathematics, and thus seems to require either one of the two other solutions mentioned above, or the more extreme view of Thompson that a Kantian canonical language for mathematics would not contain quantifiers at all and would express generality only by free variables.7

6 For example Putnam, “Mathematics without Foundations,” esp. pp. 49, 58–59; also my Mathematics in Philosophy, esp. pp. 21–22, 183–186. Though it is stricter than the notion of logical possibility that does occur in Kant, such a notion of possibility would still have a formal character. 7 Thompson, op. cit., pp.  340–341, but Thompson expresses this position with some diffidence. Thompson suggests “there is constructible” as a reading for the particular (“existential”) quantifier in mathematics, but says that since one must “see (intuit) the constructibility, . . . [t]here is no need for a special symbol by which one represents discursively (asserts that there is) the constructibility one must intuit” (340). This seems to be a non sequitur. Nonetheless, Thompson’s thoughtful discussion raises some difficult issues concerning Kant’s distinction between demonstrations and discursive proofs, which I have not attempted to deal with here.

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Something like the second of these three solutions may be read into the above-cited remark by Schultz, but more direct evidence that Kant faced the issue of the “ontology” of mathematics is lacking. It is instructive to ask, however, whether Kant could have adopted the first solution and accepted mathematical objects, as he indeed seems to do in some passages cited above. He would have to acknowledge a use of quantifiers wider than over ‘objects’ in his full sense of objects of experience, but his conception of the “logical use of the understanding” seems to make this acknowledgment already. If, with Schultz, he were to read the particular quantifier as “es gibt,” this would not connote Dasein or Wirklichkeit, but that would not commit Kant to a real theory of “nonexistent objects” of the sort that is attributed to Meinong or inspired by him.8 A possibly more serious question that would arise for a Kantian conception of mathematical objects, and of mathematics as knowledge of such objects, comes from his view that knowledge of objects requires intuition. When Kant speaks in this vein, he does regard construction of concepts in pure intuition as yielding such objects; in that sense, there would be intuition of them. But strictly speaking, this probably applies only to what Kant calls ostensive construction, which is characteristic of geometry, as contrasted with symbolic construction, characteristic of algebra (A717/B745). It is the former that is said to be “of the objects themselves.” This leaves somewhat unclear in what sense it would be open to Kant to say that construction gives the objects of arithmetic and algebra. J. Michael Young seems to me reasonable in describing the construction involved in the intuitive verification of ‘7 + 5 / 12’ in the second edition Introduction as ostensive.9 But although Kant does speak of seeing the number 12 come into being (B16), what is constructed is clearly a set or configuration of twelve objects. In passing in this passage, and more explicitly in the Schematism (A140/B179), he refers to such a configuration as an image of the number (see below). We shall see below that Kant’s remarks about number frequently show a conflation of the notions of a particular number n and of a set of n objects. This may have prevented him from facing the 8 Cf. the comparison of Kant with Frege in “Objects and Logic,” pp. 494–495, [or Mathematical Thought and Its Objects, pp. 5–7]. A Meinongian view would be of course quite foreign to Kant. 9 “Kant on the Construction of Arithmetical Concepts,” pp. 30–31.

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question whether numbers, strictly speaking, can be constructed in intuition. It is noteworthy that both in the Introduction and in the Axioms of Intuition Kant focuses on singular propositions about numbers, so that the question how to interpret generalizations about them is not raised. It is at the latter point that we ourselves are inclined to see the problem of “ontological commitment to numbers” as arising. Young suggests that Kant might regard statements about numbers as statements about finite sets, but he considers only a singular example.10 In one way or another, Kant must regard some objects of arithmetic and algebra as at a conceptual remove from the intuitions that found statements about them. This, rather than his conception of existence, seems to me to be the most principled difficulty in the way of Kant’s adopting the “mathematical-objects picture.” In some cases, such as rational numbers, it seems that Kant would fall back on the notion of symbolic construction. Positive and negative rational numbers are talked of in the context of a calculus, in which there are definite rules for manipulating expressions of the form ± m/n, where m and n range over natural numbers. By adding symbols for roots, we can similarly accommodate algebraic real numbers. Kant did, however, make a distinction of status between rational and irrational numbers. When, in a letter of September 1790, August Wilhelm Rehberg asked why the understanding cannot “think √2 in numbers” (11:206), Kant does not challenge the formulation; for him “number” meant primarily “whole 10

Ibid., p. 37. Of course we can now develop first-order arithmetic in a theory of finite sets; the essential ideas for this development were first discovered by Ernst Zermelo in 1908. Such a development has the uncomfortable feature that it singles out more or less arbitrarily a certain sequence of sets as “the” natural numbers. A more neutral procedure had been devised earlier by Richard Dedekind in his Was sind und was sollen die Zahlen? Dedekind reads statements about “the” natural numbers as general statements about any “simply infinite system,” that is, structure satisfying the Dedekind-Peano axioms. But to develop number theory in this way requires either a second-order theory of finite sets or an axiom of infinity. [Shortly after this last remark was written, W. W. Tait convinced me that this interpretation of Dedekind is incorrect. See my Mathematical Thought and Its Objects, pp. 46–48.] There is, however, a third possibility which fits Kant’s way of talking a little better. This would be to replace talk of numbers by talk of sets modulo cardinal equivalence. Instead of operations on numbers we would have operations on sets: disjoint union for sum, and Cartesian product for product. Identity of numbers would be replaced by cardinal equivalence of sets. The prior question, how appropriate it is to talk of sets in the Kantian context, is discussed below. 48

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number.” It is geometric construction that shows that the concept of √2 is not empty, but such a root is “not a number, but only the rule of approximating of it.”11 But Kant’s remark that such a quantity “can never be thought in numbers” suggests that the representation of the rational number m/n does allow us to think it completely in numbers. However this may be, the geometric construction once again yields not √2 “itself” but rather a representative of √2, in the form of a pair of lines whose length has ratio √2. But then the question arises what, if anything, it means to speak of √2 “itself”; this question would lead us into questions about mathematical objects that Kant did not consider.

II Kant’s conception of the categories of quantity combines two kinds of notions: “quantity” as understood in logic in his time, and conceptions of whole and part. The connection between these two kinds of ideas is not very clearly made. The first is reflected in the Table of Judgments, in which judgments are classified with respect to quantity as universal, particular, or singular (A70/B95). In the universal and particular cases, quantity is what we would express by the quantifiers ‘all’ and ‘some’. Kant’s conception of a singular judgment is less clear. It would be most natural to us to count as a singular judgment one of the form ‘a is B’, where a is a singular term, and indeed Kant gives such examples.12 But in the language of concepts, that would suggest that a singular judgment is one in which a concept of a different type (or perhaps even not a concept at all, but an intuition) is the subject. Kant repudiates this suggestion in saying that it is not concepts but their use that can be singular.13 Kant gives his most explicit explanation when, after talking of the use of the concept house in universal and particular judgments, he remarks: Or I use the concept only for a single thing, for example: This house is cleaned in such and such a way. It is not concepts but judgments that divide into universal, particular, and singular.14 11

Letter to Rehberg, September 1790, 11:210. Logik §21, Note 1, 9:102; Metaphysik Volckmann, 28:396. 13 Logik §1, Note 2, 9:91; Wiener Logik, 24:909. See Thompson, “Singular Terms and Intuitions,” pp. 316–318. 14 Wiener Logik, 24:909. 12

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This would suggest, as Alan Shamoon remarks in commenting on this passage, that a judgment is singular, and its subject concept has singular use, if it has in the subject a demonstrative or the definite article.15 Thus singular judgments, like universal and particular ones, would have an expression of quantity, in effect a quantifier. But Kant does not offer a theory of proper names. The above passage indicates a clear enough distinction of a formallogical kind between singular and other judgments. By comparison the justification in the Critique of Pure Reason for including singular judgments in the table is unclear and appeals to epistemological considerations (A71/B96). At all events singular judgments are not at center stage in Kant’s logic. Where the singular/general distinction is fundamental in Kant is not in formal logic but in the distinction between intuitions and concepts. I will not venture to explicate the category of unity as Kant understands it. In the Table of Categories, we already find notions of whole and part. The categories of quantity are unity, plurality, and totality (A80/ B106). In Kant’s explanations of these categories, in the Critique and elsewhere, the whole/part notions dominate over those of logical quantity. In view of the fact that number arises from these categories according to Kant, it is disappointing that their connection with logical quantity is not more clear, although Kant is explicit enough about the connection of totality with universality, as we shall see.16 Modern analyses of number have connected it closely with quantification, but this is not a matter about which Kant achieves much clarity. It requires some explanation to see how the unity, plurality, and totality of the Table are related to the notions of quantity explored in the Axioms of Intuition, which officially presents the principle governing the schematized categories of quantity. Indeed, how Kant understands 15

“Kant’s Logic,” p. 85. It has been disputed whether in the correspondence between the forms of judgment and the categories, Kant intended unity to correspond with singular judgment and totality with universal, as one would expect, or vice versa, as the order in the two tables of the Critique suggests. In my view Michael Frede and Lorenz Krüger have made a convincing case for the former correspondence; see their “Über die Zuordnung der Quantitäten des Urteils und der Kategorien der Grösse bei Kant.” [This issue has continued to be controversial. For an opposing view see Thompson, “Unity, Plurality, and Totality.”] 16

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the categories of quantity as pure categories is not entirely clear. Although Kant’s explanations are often obscure and sometimes inconsistent with one another, the issues involved in both these matters concern a subject of modern discussion, namely the relation of the set/ element relation to the whole/part relation. To learn more about how Kant understood notions of quantity, whole and part, we will turn to Kant’s Reflections attached to the sections of Baumgarten’s Metaphysica dealing with whole and part17 and to the notes from Kant’s lectures on Metaphysics, which generally contain a section corresponding to the same place in Baumgarten.18 In the Axioms, Kant tells us that an extensive quantity is one “in which the representation of the parts makes possible that of the whole (and therefore necessarily precedes it)” (A162/B203). “All appearances are intuited as aggregates (multiplicities) of previously given parts” (A163/B204). The term translated “multiplicity” is Menge, later used by Cantor and now the standard German term for set. How it should be translated in Kant is a problem; “plurality,” “collection,” and “multitude” are also possibilities; my choice of “multiplicity” is somewhat arbitrary.19 It is suggested by the fact that in one place Kant equates

17

§§155–164, “Totale et partiale,” reprinted in Ak. 17:58–61. Some, but not all, of Kant’s analysis follows Baumgarten. The close connection between ideas of quantity and of whole and part is shared with Baumgarten; indeed it can be traced back to Aristotle’s Categories. The role of a concept in conceptions about quantity (see below) is not in Baumgarten. The Reflections we cite are dated by Adickes between 1780 and the beginning of the 1790s; they are in vol. 18 of Ak. and are cited merely by number. Earlier Reflections are briefer and, on the whole, less independent of Baumgarten. (But see note 26 below.) 18 The relevant sections, all in vol. 28 of Ak., are Metaphysik Volckmann (c. 1784/ 1785), pp. 422–428, esp. 422–424; Metaphysik von Schön (c. 1789/1790), pp. 504– 506; Metaphysik L2 (WS 1790(1), pp. 560–562; Metaphysik Dohna (1792/1793), pp.  636–637; Metaphysik K2 (early 1790s), pp.  714–715. The passage from Metaphysik L2 agrees verbatim with the corresponding section of Pöhlitz, Immanuel Kants Vorlesungen über die Metaphysik, pp. 31–32 of the 1924 reprint. These materials are all cited merely by page number. With reference to notes from Kant’s lectures, a statement such as “Kant says . . .” should be regarded with caution; in my usage it should be regarded as an abbreviation for “Kant is reported to say . . .” 19 I am agreeing with Kemp Smith’s translation of Menge at A140/B179, but here he translates it as ‘complex’. For my own purpose, a uniform translation is desirable. 51

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Menge with the Latin multitudo.20 In the Axioms, where what is primarily at issue is the schematized categories of quantity, Kant is talking of the relation of extended objects to their spatial parts. What Kant calls an aggregate or multiplicity is therefore closer to a mereological sum. This spatial model is evidently not conceived by Kant to be the only form taken by the schematized categories of quantity; indeed he generally, though not always, regards time as more fundamental than space. We shall turn to this question. Let us now turn to the pure categories of quantity. Kant says that totality “is nothing but plurality considered as unity” (B111). This should remind you of Cantor’s explanation of the notion of set.21 This is reinforced by the following remarks from lectures on metaphysics: Vieles insofern es Eins ist, ist die Allheit. Id, in quo est omnitudo plurium, est totum.22 Kant does not distinguish very clearly between the whole/part and the set/element relation. I will show, however, that there is some basis, even though not clearly articulated, for Kant to make such a distinction. Something like the latter relation is needed to make sense of the relation of the categories to the concept of number. Kant’s most elementary notion concerning whole and part is that of a compositum, which seems to be simply an object in which parts can be distinguished. He is concerned to distinguish compositum from quantum, in which the parts must be homogeneous,23 but also from totum.24 The latter distinction is not too clear. Two distinguishable ideas are that a totum is not part of something further, or at least not 20

Metaphysik Volckmann, p. 422. In his German translation of the Inaugural Dissertation, Klaus Reich translates multitudo in §1 as Menge; see Kant, De mundi, pp. 4–5. 21 Especially Cantor’s characterization of a set as “jedes Viele, welches sich als Eines denken lässt,” Gesammelte Abhandlungen, p.  204. I am not pressing any claim of an anticipation of Cantor by Kant; rather, it seems to me that Cantor’s explanations are based on older ways of thought and that ideas about whole and part are not entirely absent from his own conception of what a set is. Kant, however, associates Menge with the category of plurality (B111). This passage was pointed out to me by Pierre Keller. 22 Metaphysik L2, p. 560. 23 B203, also Refl. 5836, 5842. Cf. B201n. 24 Allheit, the third category of quantity in Kant’s table, is rendered in Latin as totum in the above-quoted passage, but as totalitas in Refl. 5838. A distinction 52

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represented as such,25 and that the concept of a totum involves unity of the plurality of parts.26 The former idea seems to dominate in the Reflections on these matters, the latter in the explanation of the category of totality in the Critique (B111; but cf. B114). It is the former idea that has an obvious connection with the universal form of judgment. Nonetheless it seems to me that it is the latter idea that is the more interesting one, and the more relevant to the concept of number. It is the idea that is found later in Cantor. If a whole of parts is thought of as one object, to which, however, a definite conception belongs of what the parts are, then a set of parts is at least determined. Where the objects are spatio-temporal, what distinguishes a sum from a set is precisely that the latter has definite elements. One Kantian manner in which what we would call the elements might be given is by a concept. In fact we find Kant saying: A thing can be seen as a compositum (in a series) but without totality (of aggregate). Therefore the concept of the compositum is not yet that of a totum. To be a quantum requires homogeneity, to be a compositum not. The totum is always considered as a quantum according to a certain concept. Totality belongs to the concept of a compositum as homogeneous, that is as quantum. (Refl. 5843) What this passage suggests is that the “homogeneity” of parts that will make a compositum a quantum is their falling under a common concept; then that concept imparts unity to the plurality of parts, so that they constitute a totum. On this reading, the totum is determined by the set of parts falling under the concept in question. But now we would distinguish a whole that has a certain set of parts from the set of parts itself; indeed, the concept defining the set might naturally allow the whole as an “improper” part, so that it will be an element of the set. But the notion of a whole also suggests a different role for a concept, namely a sortal concept that the whole object falls under. Then there will be derivative concepts applying to the parts, marking them as parts of this whole, or of a whole of this kind. It is not easy to see between the two might be made along the lines of that between quantum and quantitas (A163/B 204), but Kant does not do so very explicitly. 25 “A compositum, insofar as it is not a part, is a totum” (Refl. 5834). 26 Refl. 5833, 5840. In the former matter is said to be compositum, body totum. 53

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how the parts can be “homogeneous” except by falling under some such derivative concept, which returns us to the first reading of the above passage.27 We have been considering remarks of Kant that are on an abstract level and could plausibly be taken to be explicating pure categories. In the case of spatio-temporal objects, however, Kant evidently thinks that spatio-temporal extension itself constitutes the basic form of division into homogeneous parts. To that extent, the parts of an object are homogeneous simply by virtue of being parts of that object, and clearly there is a deeper homogeneity of the spaces occupied by the parts; because the representation of a spatial whole is a result of synthesis, the synthesis is of the sort Kant calls “mathematical” (B201n.). Both of these forms of homogeneity will be bound up with a third, there being some concept that offers unification of the second of the two types mentioned above.28 Kant evidently intended his definitions concerning quantity to cover both discrete and continuous quantity, and the distinction seems still to be defined in abstraction from space and time: A quantum by whose magnitude the multiplicity of its parts is undetermined, is called a continuum; it consists of as many parts as I wish to give it; it does not consist of individual parts. On the other hand, every quantum through whose magnitude I wish to represent the multiplicity of its parts is discrete.29 A quantum through whose concept the multiplicity of its parts is determined, is discrete; one through whose concept of quan-

27

In one text, Refl. 4822 (1775/1779, 17:738), Kant complicates the matter further by saying that in a quantity (Grösse) the whole must be homogeneous with the parts. Here he seems to be thinking of “quantities” in the sense in which it is filled out by a mass term; his example is a quantity of money, and he seems to reject the idea of a quantity of ducats. On this conception, a quantity differs both from a mereological sum and from a set. 28 One would not expect Kant to conceive recognition of the same object at different times on the model according to which an enduring object is a whole that has as parts temporal “stages.” Nonetheless, the mathematical representation of time as a line, on which Kant lays great stress, means that persistence through time will have some formal features of extension in space. 29 Metaphysik L2, p. 561. 54

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tity the multiplicity of the parts is in itself undetermined, is a continuum.30 Note that Kant says that in the case of a continuum the multiplicity (Menge) of parts is undetermined. Kant certainly held the pre-Cantorian view that ‘number’ means finite number.31 If a quantum were a continuum if its concept did not determine the number of parts, that would then make every quantum with infinitely many parts a continuum. That does not follow from what Kant says. What he evidently means is that the concept of a continuum does not determine what the parts are. Although these definitions are abstract, space and time are of course the paradigms of continua. Kant considers the parts of space and time to be spaces and times, rather than points. It follows that they do not have simple parts; presumably, since they can be divided in arbitrary ways, neither has a definite set of parts.32 In his theory of matter, Kant in effect holds that objects in space are similarly continuous. Of course the application of arithmetic, and even the development of the mathematics of continuity, requires that some quantities be identified as discrete. Evidently Kant accommodates this by making what are the parts of a quantum depend on how it is conceived, as for example in the above quotation from Reflection 5844. If, as Reflection 5847 has it (see note 32), all real quanta are indefinitely divisible, it must be that the concept that “determines” the parts of a discrete 30

Refl. 5844. In both these passages, Menge could quite appropriately have been translated ‘set’. 31 In one place, however, Kant intimates a distinction between infinity in the sense of nonfiniteness, and unsurpassably large quantity: The former [the concept of the infinite] does not determine at all, how large something is; however, the concept of maximum does determine quantity. The concept of the infinite shows that my quantum is larger than my power of measuring. Therefore “God is the infinite being” does not say as much as “God is the greatest being.” (Metaphysik K2, p. 715) 32

At one point, however, Kant seems to view this as characteristic of quanta in general: Every quantum is a compositum whose parts are homogeneous with it. Consequently it is a continuum and does not consist of simple parts. (Refl. 5847)

Here he goes beyond his usual characterization of a quantum in assuming that the parts are homogeneous not only with each other but with the whole, but the situation is not the special one envisaged in Refl. 4822 (see note 27 above). He is here apparently thinking of spatio-temporal quanta. 55

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quantity does not stop further division; that is, further division is possible, although the resulting parts would not any longer fall under the concept. That must be the situation if we are to make sense in these terms of attributions of cardinal number. Kant sometimes regarded this concept as not intrinsic to the quantum, so that a quantum that is continuous if one considers its possible divisions into parts can be considered as discrete: Quantum discretum is that whose parts are considered as units; that whose parts are considered as multiplicities is called a continuum. We can also consider a continuum as discrete; for example, I can consider the minute as unit of the hour, but also as set which itself contains units, namely 60 seconds.33 If, stretching Kant’s explicit formulations, we allow nonconnected “objects” to count as wholes, we can accommodate the assignment of cardinalities in the physical realm: the number of people in this room would attach to their mereological sum, conceived as having individual people as parts (as opposed to some other conceivable division). Elsewhere Kant describes a “discrete quantity per se” as one “in which the number of parts is determined arbitrarily by us.”34 The text goes on: Number is therefore called quantum discretum. Through number we represent every quantum as discrete. The situation evidently results from combining the dependence on a concept, of a division into parts that gives a definite number and the taking of this concept as not intrinsic to the quantum. In fact Kant goes further in treating number as dependent on our representation. But some backtracking will be necessary before we can go into this. 33

Metaphysik Volckmann, p. 423. The issue is complicated by a distinction made in this text between a quantity that is in itself discrete (an sich discretum) and a continuous one that is represented as discrete. It appears that only the latter case will occur in the realm of appearance, but the example of a bushel of corn as a quantum that is discrete because of having parts whose parts are heterogeneous may be intended to illustrate the notion of in itself discrete quantity. 34 Metaphysik L2, p. 561. Per se is reminiscent of an sich in the corresponding passage of Metaphysik Volckmann (see note 33), but the characterization of per se is almost opposite. One or the other hearer may have misunderstood Kant. 56

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Up to now we have concentrated on Kant’s purely abstract discussion of part, whole, and quantity; to all appearances these notions belong to the pure categories. Some considerations concerning space and time have, however, crept in. When Kant begins to talk of number, the amount that can be said on the pure categorical level seems to be very limited. Already in the Inaugural Dissertation (§1), Kant finds an abstract intellectual conception of the composition of a whole of parts to be possible, but to “follow up” such a conception and represent it in the concrete involves temporal conditions: Thus it is one thing, given the parts, to conceive for oneself the composition of the whole, by means of an abstract notion of the intellect; and it is another thing to follow up this general notion, as one might do with some problem of reason, through the sensitive faculty of knowledge, that is to represent the same notion to oneself in the concrete by a distinct intuition. The former is done by means of the concept of composition in general, insofar as a number of things are contained under it (in mutual relations to each other), and so by means of ideas of the intellect which are universal. The second case rests upon temporal conditions, insofar as it is possible by the successive addition of part to part to arrive genetically, that is by synthesis, at the concept of a composite, and in this case falls under the laws of intuition. (2:387) The same duality arises again when, in §12 of the Dissertation, Kant refers to the concept of number: In addition to these concepts there is a certain concept which in itself indeed is intellectual, but whose actuation in the concrete (actuatio in concreto) requires the assisting notions of time and space (by successively adding a number of things and setting them simultaneously beside one another). This is the concept of number, which is the concept treated in arithmetic.35 I shall not try to sort out what, at this stage, belongs to the abstract concept and what to its “actuation in the concrete.” From Kant’s later critical standpoint, any construction that would yield models of math35

2:397. Reich translates actuatio in concreto as Darstellung im Einzelnen (Kant, de mundi, p. 35). 57

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ematical notions such as that of number will involve the forms of intuition; this seems to be true even of the most basic notion of a compositum. In the Critique of Pure Reason, the status of the pure categorial notions is obscured by Kant’s characterizing number as the schema of quantity (A142/B182) and by the fact that most of Kant’s explanation of notions of quantity occurs in the Axioms, where he is principally concerned with the schematized categories. Later texts return to a position close to that of the Dissertation, as we shall see. The problem that Kant faces is how much beyond some basic definitions he can develop without construction, which on his own account will involve intuition. With respect to number, a further factor is that he tends not to distinguish a multiplicity’s having a certain number from our knowledge of that fact; indeed from the point of view of transcendental idealism the two should be essentially connected. He tends even to characterize number in epistemic terms: To know a multiplicity distinctly by adding of unit to unit is to count. A number is a multiplicity known distinctly by counting.36 Very often, when Kant talks of the relation of number and arithmetic to time, time seems to play the role of a subjective condition of apprehension. Needless to say, this does not strengthen Kant’s case for the view that arithmetic is synthetic and dependent on intuition. On this matter, I have already written elsewhere.37 The above citation illustrates another phenomenon that is frequent in Kant’s remarks about number. That is that he tends not to distinguish, for a given number n, between a “multiplicity” with cardinal number n and the number n itself.38 This conflation illustrates the lack, discussed above, of an articulated theory of mathematical objects in Kant, and with respect to the idea of ostensive construction of numbers may have contributed to it. Note also that by ‘number’ Kant evidently means primarily cardinal or ordinal number, at all events whole 36

Metaphysik Volckmann, p.  423 (in Latin); cf. Metaphysik von Schön, p.  506; also Metaphysik Dohna, pp. 636–637. 37 “Kant’s Philosophy of Arithmetic,” esp. pp.  128–142. But on the treatment there of Kant’s conception of mathematical objects, see the Postscript, pp. 147– 149, which §I of the present essay amplifies. [On this subject see further the Postscript.] 38 Cf. B16; Metaphysik L2, p. 561 (cited above). 58

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number as opposed to what we would call rational, real, or complex number.

III I shall now turn to the discussion of number in the Schematism and to the texts of 1788–1790 that seem to be inconsistent with it. Kant appears in the Schematism to reject the idea expressed in the Dissertation and implicit, though not consistently held to, in the Metaphysics lectures, of describing the concept of number in terms of the pure categories. In the Schematism, Kant uses a numerical example in the course of explaining the notion of schema and distinguishing it from that of image (Bild). If I put five points one after another, he tells us, • • • • •

this is an image of the number five (A140/B179). Its relation to its object will seem to us quite different from that in the other cases he mentions, such as the concepts of triangle and dog (A141/B180). At all events he continues: But if, on the other hand, I think only a number in general, whether it be five or a hundred, this thought is rather a representation of a method whereby a multiplicity, for instance a thousand, may be represented in an image in conformity with a certain concept, than the image itself. (A140/B179) It is not entirely clear whether he is here describing the thought of number in general, that is, the entertaining of the general notion of natural number, or giving a general description of the thought of a particular number (so that it is the description, rather than the thought described, that is general over the natural numbers). The former reading seems to me slightly more likely. However, even the thought of a particular number will have to be distinguished from an image of it; moreover, the thought of a number as large as 1,000 will in practice have to involve general operations on numbers. However, even for a number like 5, for which there is no difficulty in obtaining the sort of thing Kant calls an image, we do not have “a method of representing a multiplicity in an image in conformity with a 59

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certain concept,” unless the multiplicity itself is determined by a concept, in the example at hand something like dot on the page. This is just Frege’s point that a number attaches to a concept.39 We have already seen Kant wrestling with this issue and attempting to fit it into a conception of “multiplicities” based on whole/part ideas. It is curious that when Kant comes to enumerate the schemata of the individual categories, it is only for the categories of quantity that he describes an “image,” and what he says does not exactly fit what he has said previously: The pure image of all magnitudes (quantorum) before outer sense is space; that of all objects of the senses in general is time. But the pure schema of magnitude (quantitatis), as a concept of the understanding, is number, a representation which comprises the successive addition of homogeneous units. Number is therefore simply the unity of the synthesis of the manifold of a homogeneous intuition in general, a unity due to my generating time itself in the apprehension of the intuition. (A142–143/B182) No doubt what is meant by calling space and time “pure images” of quanta is that their structure relevant to the application of the categories of quantity can be represented by spatial or temporal structure. In particular, the image of a number in the sense of the previous passage will be spatio-temporal. Indeed, Kant’s emphasis on successive addition in descriptions of the concept of number makes it possible that here he conceives the image to be essentially temporal: the points are an image of the number five by being put one after the other (hintereinander); thus, they constitute an image of a number by virtue of being generated in succession.40 Kant at this time seems to have rejected the distinction of the Dissertation between the “intellectual concept” of number and its “actuation in the concrete.” The abstract conception of whole, part, and quantity 39

It is unlikely that it is this concept, rather than the concept of number in general or of a single number such as 5 or 1000, that Kant has in mind when he speaks of representing in an image “in conformity with a certain concept.” For it seems clear from the last sentence of the paragraph that it is the latter concept whose schema is being described; hence it must be the concept of totality or perhaps of number. 40 In A140/B179, Kemp Smith translates wenn ich fünf Punkte hintereinander setze as “if five points be set alongside one another,” thus losing the implication of successive “setting.” 60

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is little in evidence in the Critique, in particular not where number is discussed. Nonetheless, the identification of number as a schema would have its difficulties, for it attributes a temporal content to the notion of number itself. Kant may have been prepared to accept this consequence, for more than one possible reason: any construction that would give rise to the series of numbers would generate them successively, each one by addition of one more from the previous ones. In particular, coming to know the number of a multiplicity by counting involves the generation of a sequence (of acts or tokens) isomorphic to the numbers up to a given one. On transcendental idealist grounds, Kant might have resisted the distinction between a multiplicity’s having a certain number and the condition being fulfilled for our knowing this in a canonical way. But the strongest reason would probably have been his conviction of the necessity of construction for arithmetic.41 41

In talking of “symbolic construction” in algebra, Kant does say that algebra “abstracts completely from the properties of the object that is to be thought in terms of such a concept of magnitude” (A717/B745). How far does this “abstraction” extend? Does it make algebra applicable to objects in general, independently of the forms of intuition? If the role of intuition is only that the signs of a formal calculus are objects of intuition, and the conformity of steps to rules is intuitively checkable, then perhaps there is no reason to attribute to the operations any spatio-temporal content or to the limit the applicability of algebra to spatio-temporal objects. No such limitation is suggested in Kant’s first formulation of these ideas, in Untersuchung über die Deutlichkeit der Grundsätze der natürlichen Theologie und der Moral (1764, esp. 2:291–292). With regard to applicability, in the 1788 letter to Schultz discussed below, Kant says quite unequivocally that mathematics is applicable only to sensible things (10:557). With respect to the content of pure algebra the matter is less clear; see below. There is some support for the thesis of Alan Shamoon (“Kant’s Logic,” p. 221n.) that for that domain Kant still held in the Critical period the formalist view expressed in 1764. Shamoon’s dissertation contains an interesting discussion of symbolic construction and its relation to ideas of Lambert. Concerning the Deutlichkeit, my own remark (“Kant’s Philosophy of Arithmetic,” p.  138) that it exhibits a connection in Kant’s mind between sensibility and the intuitive character of mathematics before he developed the theory of space and time of the Aesthetic was aimed at Jaakko Hintikka’s thesis that his own essentially logical analysis of the role of intuition in mathematical proof describes a “preliminary” or “earlier” stage of Kant’s philosophy of mathematics, at which no connection between intuition and sensibility is made. (See his paper “Kant’s Transcendental Method and His Theory of Mathematics.”) In this connection the reader’s attention should be called to Mirella Capozzi Cellucci, “J. Hintikka e il metodo della matematica in Kant.” My remark is elaborated on 61

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There is also a conceptual gap which, whether or not Kant was conscious of it, makes his definitions of discrete quantity fall short of capturing the notion of finite quantity, which he would need for his own conception of number. A discrete quantity, as Kant defines it, will have a definite number of parts, but there is no necessity that this number should be finite; in fact, on this level Kant does not offer much of a conceptual basis for comparing magnitudes and for formulating answers to questions about the magnitude of particular quanta.42 Kant’s appeal to “successive repetition” was possibly an attempt to capture the notion of finiteness. Consider: The concept of magnitude in general can never be explained except by saying that it is that determination of a thing whereby we are enabled to say how many times a unit is posited in it. But this how-many-times is based on successive repetition, and therefore on time and the synthesis of the homogeneous in time. (A242/B300) We might compare the situation with that obtaining once we have the set-theoretic notion of cardinality. In his definition of discrete quantity and identification of number with it, Kant leaves open the possibility of infinite number, even though other remarks of his reject it. But he does not take the key step taken by Cantor, giving a general definition of when two sets have the same cardinal number, and what he says about greater and less is somewhat crude.43 But even when all this has been done, two further steps need to be taken for a set-theoretic theory of cardinal number: the notion of cardinal has to be related to that of ordinal; from Cantor on it has been accepted that an informative answer to the question of the cardinality of a set will place it in the sequence of pp. 241–243, but the paper contains a number of further criticisms of Hintikka’s conception of a “preliminary” Kantian theory. She is perhaps the only one of Hintikka’s critics to engage him on his own grounds, with respect to his use of a Euclidean conception of mathematical proof; see especially §7 on ekthesis and logic in Kant. 42 Since Kant’s discussion of quantity comprehends the continuous as well as the discrete, knowledge of magnitude involves more than just determination of cardinalities (i.e., counting); it will involve measurement. I have not gone into such issues at all here. Some commentators have read the Axioms of Intuition as concerned with the possibility of measurement of physical phenomena. See for example Gordon Brittan, Kant’s Theory of Science. 43 E.g., Metaphysik Volckmann, p. 424; Metaphysik Dohna, p. 637. 62

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ordinals.44 Second, finiteness has to be characterized. The finite ordinals and ordinals in general are often explained in terms of different notions of iteration; finite iteration is an abstract counterpart of the notion of successive repetition. But to describe it in abstract terms was quite beyond the logical and mathematical resources of Kant and his contemporaries; the task was first accomplished in the 1880s by Frege and Dedekind. Whatever considerations may have motivated Kant’s position of 1781, in some later texts he returns to a view close to that of the Dissertation, and holds that at least some essentials of the concept of number are intellectual and presumably derive from the pure categories. This may have been made possible for him by his reworking of the Transcendental Deduction for the second edition of the Critique, with its distinction between a more abstract level of the argument, presented in §§15–20, which considers the synthesis of a given manifold of intuition in general, without making any assumptions about our particular forms of intuition, and the application of these abstract considerations to our forms of intuition, in the argument of §§24–26. In particular, Kant distinguishes in this context between intellectual and figurative synthesis (B151). The former is that “which is thought in the mere category in respect of the manifold of an intuition in general.” How this new formulation works out for the categories of quantity and the notion of number is not very explicit in the second edition of the Critique. It is reasonable to conjecture, however, that Kant saw the notion of intellectual synthesis as a framework into which to fit the abstract conceptions of quantity developed in his lectures. Note that he characterizes the concept of a quantum as “the consciousness of the manifold [and] homeogeneous in intuition in general” (B203).45

44

Hence the centrality to the theory of cardinals of the axiom of choice, which implies that every cardinality can be located somewhere in the sequence of ordinals, and of the continuum problem, which is the question where in the sequence of ordinals the cardinality of the continuum lies. 45 Kemp Smith translates “consciousness of the synthetic unity of the manifold . . . ,” following Vaihinger, who emended “Bewusstsein des mannigfaltigen Gleichartigen” to “Bewusstsein der synthetischen Einheit des mannigfaltigen Gleichartigen.” (See Kant, Kritik der reinen Vernunft, ed. Schmidt, p.  217 [or ed. Timmermann, p. 261]. As an interpretation, this seems to me reasonable enough. [I no longer think so. Daniel Sutherland has argued at length and convincingly against this proposed emendation. See “The Role of Magnitude,” pp. 418–426.] 63

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So far, Kant has not put the concept of number into his framework. But that is just what he seems to do in his letter to Johann Schultz of November 25, 1788. There he says that arithmetic had for its object “merely quantity (Quantität), i.e. a concept of a thing in general by determination of magnitude” (10:555), and he goes on to say: Time, as you quite rightly remark, has no influence on the properties of numbers (as pure determinations of magnitude), . . . and the science of number, in spite of the succession, which every construction of magnitude requires, is a pure intellectual synthesis which we represent to ourselves in our thoughts. (10:557) Kant might seem to be responding to the point, later much emphasized by Frege, that the concept of number applies to objects in general, independently of such conditions as those Kant associates with sensibility. But although, according to Kant, we may have such an intellectual concept of number, it is applicable only to sensible things (sensibilia). This much would, however, be to be expected if what is at issue is application to yield knowledge of objects in the full sense. But what Kant says by way of argument for it may just as well include pure mathematics: Insofar, as quantities are to be determined in accordance with it [the science of number], they must be given to us in such a way that we can take up their intuition successively, and so this taking up must be subjected to the condition of time, so that we can still subject no object to our estimation of quantity by numbers except that of our possible sensible intuition. (Ibid.) Kant thus leaves doubt about how much of a “science of number” there can be without intuition and time; it is not entirely clear that the difference between his position here and that of 1781 is more than terminological. Kant’s response to Rehberg seems, however, to be more emphatic. Rehberg challenges the formulations of the Schematism. He admits that the application of arithmetical truths to sensible appearances would be subject to the condition of time, but he claims that to see the “truth of the arithmetical propositions themselves” no intuiting of the form of sensibility is necessary since no intuiting of time is required, in order to carry out arithmetical and algebraic proofs, which are rather immediately evident 64

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from the concepts of numbers, and only require sensible signs, from which the concepts are recognized during and after the operation of the understanding. (11:205–206) He expresses puzzlement as to why “the understanding, in the generation of numbers, which are a pure act of its spontaneity, is bound by the synthetic propositions of arithmetic and algebra.” In particular, the form of our sensibility does not prevent us from “thinking in numbers” in the way in which the nature of space prevents us from thinking “straight lines that would be equal to certain curved ones.” The problem Rehberg raises is, in effect, that of the difference between geometry as a theory of space, and arithmetic, whose relation to space and time must on any account be more indirect, a perennial problem for the interpretation of Kant’s philosophy of arithmetic.46 In reply, Kant seems to concede the existence of a “mere concept of the understanding of a number” and that the understanding “makes for itself the concept of arbitrarily” (11:209, 208). No synthesis in time is required for the mere concept of the square root of a positive quantity; even the impossibility of a square root of a negative quantity can be known “from mere concepts of quantity.” As soon as, however, instead of a,47 the number of which it is the sign is given, in order not merely to designate its root, as in algebra, but to find it, as in arithmetic, the condition of all generation of numbers, namely time, is unavoidably presupposed. (11:209) This remark expresses a constant view of Kant, that time is involved necessarily in mathematical construction, at least ostensive construction. This holds for geometry as well as arithmetic, as is indicated by remarks to the effect that thinking of a line involves “drawing it in thought” (B154). However, one might find in it the startling view that algebra, and therefore presumably symbolic construction, is independent of conditions of time, at least as regards its objects. Could we go on to say that the 46

Cf. “Kant’s Philosophy of Arithmetic.” With respect to the remarks there (pp. 120ff) about Leibniz’s proof in the Nouveaux Essais of ‘2 + 2 = 4’, it can now be observed that similar proofs of ‘8 + 4 = 12’ and ‘3 × 8 = 24’ are to be found in Herder’s notes from Kant’s lectures on Mathematics, 29:57–58. The lectures would have been in 1762/3, but the notes may be inauthentic or contain later additions by Herder; see 29:658. [For more on these proofs, see the Postscript to Part I.] 47 That is, instead of the symbol ‘a’. 65

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“science of number” which in 1788 was said to be a “pure intellectual synthesis” is in fact just algebra, where one crosses the line from algebra to arithmetic and constrains one’s objects by the forms of intuition, as soon as one undertakes to calculate actual values of algebraic expressions for particular given arguments? If so, Kant missed an opportunity to say so in the letter to Schultz, where in fact there is no word of symbolic construction; instead he says that this pure intellectual synthesis is one which “we represent to ourselves in thoughts.” Further doubt on such an interpretation is cast by one of Kant’s preliminary sketches for his reply to Rehberg. There he says that although the objects of arithmetic and algebra are “with respect to their possibility not under conditions of time,” such conditions do govern the construction of the concept of quantity in their [the objects’] representation through the synthesis of imagination, namely composition, without which no object of mathematics can be given.48 So far, the force of Kant’s remark could be limited to ostensive construction. But he goes on to characterize algebra as the art of bringing under a rule the generation of an unknown quantity through numeration (Zählen), independently of every actual number, only through the given relations of the quantities. This quantity to be generated is always a rule of numeration.49 Since they differ in emphasis from the actual letter, these remarks do not necessarily represent Kant’s considered position. But it is hard to imagine his having written them if he had consistently in this period thought of algebra as containing only purely intellectual concepts. Kant evidently found suggestive the fact that a geometric construction was needed to give an adequate intuitive representation of an irrational quantity; it fit in neatly with the view of the Refutation of Idealism that space and time are interconnected in such a way that consciousness of things in space is necessary for me to locate myself in objective time. Both Reflection 13 and the last paragraph of Kant’s letter argue that 48

Refl. 13 (1790), 14:54. Ibid., emphasis mine. By Zählen Kant evidently means something more general than counting; “√2” is called a Zeichen des Zählens, because the concept it expresses contains a rule for approximating it by rational numbers. 49

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space and time are interconnected in mathematical construction. With respect to the concept of number, Kant in one text argues that both space and time are necessary to the determinate representation of a number: We cannot represent to ourselves any number other than by successive enumeration in time and then taking together this multiplicity in the unity of a number. This latter, however, cannot occur otherwise than by my putting them next to one another, for they must be thought as given at the same time, that is, as taken together into a single representation; otherwise this representation forms no quantity (number).50 That by “next to one another” he means next to one another in space is clear from the context. The conclusion to be drawn from examining these texts, in my opinion, is that Kant did not reach a stable position on the place of the concept of number in relation to the categories and the forms of intuition. One could find connections between this difficulty and other problems in Kant’s philosophy, for example that concerning the status of the “intellectual” representation “I think” (B423n.). As regards arithmetic, one might take Kant’s problem to be solved by a modern distinction between, on the one hand, characterizing the natural numbers as an abstract structure and developing “arithmetic” as the theory of what must be true in such a structure and, on the other hand, actually constructing an instance of the structure (or some initial part of it). The former would belong to the realm of “mere concepts,” and neither time nor anything else Kant would regard as involving intuition would be part of its content. Time would enter as a condition of construction, for example, such that models for the numbers can be constructed in it if any can be constructed at all.51 In its general lines, this seems to me a defensible position about the relation of the intuitive and the abstract with respect to arithmetic.52 But there is no neat division of labor, as is 50

Refl. 6314 (1790). This is one of a group of texts in which Kant returns to the ideas of the Refutation of Idealism. For a discussion of them in that connection, see Guyer, “Kant’s Intentions in the Refutation of Idealism.” 51 Cf. “Kant’s Philosophy of Arithmetic,” p. 140. 52 Cf. my “Mathematical Intuition”; also “Intuition and the Concept of Number.” [These themes are discussed at much greater length in Mathematical Thought and Its Objects, especially chapters 3, 5, and 6. In the remark in the text, I must have 67

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shown by the role of calculation in developing the consequences of an abstract characterization of the structure of numbers.53 More generally, the duality of abstract conceptualization and intuition in mathematical thought is exhibited in the philosophical differences about the foundations of mathematics, with logicism and settheoretic realism emphasizing the former, and the different forms of constructivism the latter. The fact that these differences persist should remind us that the task of striking the right balance in describing this duality has not become an easy one even after the great advances in the foundations of mathematics since Kant’s time.54

taken it as obvious that a purely conceptual development of the theory of the numbers as an abstract structure requires a more powerful logic than Kant had at his disposal. Cf. the closing remarks of Essay 4 of this volume.] 53 Cf. “Kant’s Philosophy of Arithmetic,” pp. 138–139, and Young’s discussion of calculation in “Kant on the Construction of Arithmetical Concepts,” esp. §II. 54 Earlier versions of this essay were presented at the Robert Leet Patterson conference on Kant’s philosophy of mathematics at Duke University in March 1983 and at colloquia at Columbia University and at the Graduate Center of the City University of New York in May 1983 and March 1984 respectively. I am indebted for their comments to all three audiences. The questioning of Jerrold J. Katz and others at the Graduate Center concerning the relation of ideas of whole and part to space and time influenced the final version considerably. I wish to thank Dieter Henrich for helpful conversation, and I owe a special debt to Carl Posy, without whom the essay would not have been written. 68

3 REMARKS ON PURE NATURAL SCIENCE

In attempting to crack the hardest nut in Kant’s philosophy of science, his conception of an a priori or “pure” part of science, Philip Kitcher shows both courage and an appreciation of what is central to Kant’s philosophy.1 The issues that in the Critique of Pure Reason are the subject of the Transcendental Analytic are discussed in the Prolegomena under the heading “How is pure natural science possible?” Some of the most difficult issues faced by interpreters of Kant could thus be represented as concerning how Kant answers that question. But what does the question itself mean? What part of natural science is pure? ‘Pure’ is clearly closely related to ‘a priori’, but are they the same, and if not, how do they differ? What principles in or about science are a priori? Writers on Kant’s philosophy of physics do not agree on such questions. The issues are not resolved by turning to the Metaphysical Foundations of Natural Science, a work that raises as many questions of interpretation as it answers. The distinction Kant makes in the Introduction to the Critique of Pure Reason (B3) between a priori and pure knowledge is obviously relevant to the Metaphysical Foundations, but it turns out to be not nearly so straightforward as it seems. The idea seems to be that a proposition is pure only if there is nothing empirical in its content, so that a paradigm example of an impure proposition would be an analytic one involving empirical concepts, such as “Gold is yellow” (at least for the first of the two men referred to at A728/B756). Then one can see how synthetic a priori propositions that are not pure can arise: a logical model for the notion would characterize them as propositions that can be proved by arguments whose premises are either a priori propositions 1

Kitcher, “Kant’s Philosophy of Science.” 69

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involving only pure concepts, or logical truths, or analytic propositions, in which evidently only the third can involve empirical concepts essentially. ‘Pure natural science’ must on any account involve essentially concepts that are in some sense empirical. About matter, the subject of the Metaphysical Foundations, Kant could hardly be more explicit. In explaining in the Preface what he means by calling his investigation metaphysical, Kant contrasts the “transcendental part of the metaphysics of nature” with metaphysics of nature in a more special sense. His remarks about the latter suggest the model of nonpure a priori knowledge I have just sketched: it “occupies itself with the special nature of this or that kind of things, of which an empirical concept is given in such a way that, besides what lies in this concept, no other empirical principle is needed for cognizing the things” (4:470). In the next sentence he mentions the “empirical concept of matter.” Whatever Kant means by ‘pure natural science’, in the case of physical science (which seems to be the only genuine case), it will be a development of the special metaphysics of physical nature that is the official subject of the Metaphysical Foundations and is based on the empirical concept of matter. The sense in which the concept of matter is empirical is controversial, as we will see. Even if we do not disturb the apparent straightforwardness of Kant’s account, a terminological inconsistency emerges, in that ‘pure natural science’ either itself contains or depends on propositions that in the sense of B3 are not pure.2 One might be tempted to suppose that the distinction Kant makes between “transcendental” and “special” metaphysics of nature turns on the absence in the former and presence in the latter of empirical concepts. This would seem to contradict B3, where the example proposition, “Every alteration has a cause,” is said to be not pure because “alteration is a concept that can only be 2

It is well known that this inconsistency surfaces in the Introduction itself: Kant’s example of an impure a priori proposition is “Every alteration has a cause.” But that very statement is referred to soon after as a “pure a priori proposition” (B4–5). In replying to a critic who pointed out the inconsistency, Kant says that “pure” is ambiguous, and that in B3 it meant “with no admixture of anything empirical” and in B5 “dependent on nothing empirical” (“Über den Gebrauch teleologischer Prinzipien in der Philosophie,” 8:183–184, my translation). I do not find the second characterization at all clear. At all events Kant’s emphasis in B4–5 is on necessity and strict universality. 70

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drawn from experience.” This would suggest that containing empirical concepts essentially is a feature that the propositions of the Metaphysical Foundations share with some of those of the first Critique, presumably at least the Dynamical Principles. Nevertheless, this admission is certainly not made clearly in the Critique. One of the merits of Philip Kitcher’s essay is that he examines how concepts that are in some way empirical can figure in a priori knowledge. He does not take the empirical/a priori contrast for granted, as one may be tempted to do with Kant even if one does not in one’s own philosophy. With respect to the propositions of the Metaphysical Foundations, at least those closest to Newtonian laws, Kitcher sees Kant as taking them to be a priori only in an attenuated sense; they “admit of something like an a priori proof.” It is worth noting that the question whether and in what sense Kant holds basic Newtonian principles to be a priori has received rather divergent answers in recent commentary. Kitcher’s reading of Kant is in fact definitely more aprioristic than that of the two recent writers in English who have discussed the Metaphysical Foundations most extensively, Gerd Buchdahl and Gordon Brittan (see Kitcher’s note 21). The general tendency of German writers seems to be the opposite; they tend to take Kant at his word and assume that the statement from the Preface quoted above applies at least to the formal content of the Metaphysical Foundations, for example, the Propositions (Lehrsätze), which in the Mechanics include the conservation of matter, the law of inertia, and the equality of action and reaction.3 Kant’s explicit statements about what he is doing certainly favor the German view. Kitcher offers a systematic reason, which, however, seems to apply to any principles that contain empirical concepts essentially. A proposition, even if true, does not express knowledge of objects unless the 3

See also B17–18. The German writers I have in mind are Peter Plaass, Kants Theorie der Naturwissenschaft; Lothar Schäfer, Kants Metaphysik der Natur; and Hansgeorg Hoppe, Kants Theorie der Physik. (I am indebted to Ralf Meerbote for calling the latter two works to my attention and correcting my all-too-uncritical reliance on Plaass.) German writers have stressed the connection between the Metaphysical Foundations and the Opus Postumum; in addition to Hoppe I might mention the very interesting article by Burkhard Tuschling, “Kants ‘Metaphysische Anfangsgründe der Naturwissenschaft’ und das Opus postumum.” Tuschling maintains that early in the work reported in the Opus Postumum Kant abandoned some of the central theses of the Metaphysical Foundations. In preparing this essay, I have not attempted the enormous task of delving into such matters. 71

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“objective reality” of the concepts constituting it has been established. In the case of empirical concepts, this can only be by experience. It follows that any proposition containing empirical concepts essentially will have an empirical presupposition for its expressing knowledge of objects. It is chiefly by emphasizing this rather simple point that Kitcher differs from the more aprioristic commentators. Kitcher’s notion of “conceptual legitimacy,” however, departs consciously from Kant’s explicit notion of objective reality as possible exemplification in experience, in order to account for the role of idealization in science. In the case at hand, we must consider what is meant by saying that the concept of matter is empirical. Kant’s basic conception of matter is of “the movable in space”; because the representation of space is certainly a priori, anything empirical in its content would have to come from the concept of motion.4 What would be empirical in the content of this notion is not any more clear, as it seems to involve merely the notion of an object’s changing its location in space, and thus the categories, space, and time. This consideration lends support to the hypothesis of Peter Plaass that in its content the notion of matter is in fact a priori; experience is needed only to establish its objective reality.5 A similar hypothesis would deal with the empirical character of the concept of change or alteration (Veränderung), which Kant mentions among the “predicables” or “pure, but derived concepts of understanding” in the Critique (A82/B108).6 Plaass’s view seems to me much the clearest view that has been offered of how the empirical enters into the formal content of the Metaphysical Foundations. Plaass apparently holds that the role of experience in establishing the objective reality of the concept of matter is almost trivial: “for, that a concept has objective reality, can be completely proved by a single example.”7 Kitcher can criticize this view effectively even without appealing to his extension of the notion of objective reality to that of empirical legitimization. To establish any example of motion, we would have to make the distinction between real and apparent motion. Even granted the relativity of this distinction to a frame of reference, it seems we would need to set up such a frame, thus applying a theory to the 4

Prolegomena §15, 4:295. Plaass, Kants Theorie der Naturwissenschaft, chs. 4 and 5. 6 Ibid., p. 84. 7 Ibid., p. 89. 5

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world. Of course, descriptions of motion have implications about acceleration and therefore about the distribution of forces. Thus, even if the role of the empirical is minimized, as it is on Plaass’s hypothesis, some significant attenuation of the a priori character of fundamental physics seems unavoidable, along the lines Kitcher suggests. At this point we might mention Kitcher’s suggestion that the “empirically legitimized concepts” that enter into what he calls “quasi a priori” knowledge might be concepts we have prior to the construction of theories, and thus—unless one supposes them to be innate—concepts of a commonsense character. I confess this seems un-Kantian in spirit and at odds with Kant’s explanations of the concepts of matter and motion, which tend rather to connect them with technical notions of his philosophy. The picture Kitcher suggests is probably an improvement on Kant’s, in that one begins theory construction with rough and ready concepts, which are modified as theory construction proceeds. Kitcher’s notion of the “quasi a priori” has another difficulty, of a kind faced by many interpretations of the relation of the Critique and the Metaphysical Foundations. For it is not easy to see how the attenuation of apriority that Kitcher discerns in the latter work is completely escaped by the Dynamical Principles of the Critique. As we have seen, Kant holds that the objective reality of the concept of alteration which occurs in his principle of causality can only be established by experience. Again, it may seem that the empirical element is trivial, in that virtually any experience will reveal change. But what Kant specifically means is alteration of the state of a substance; he is actually operating with a distinction like that between a “real” and a “mere Cambridge” change.8 But then the identification of objective changes is a theoryladen matter; in particular, uniform motion in a straight line is not a change of the state of the moving body and therefore does not require a cause, while acceleration is a change of state (A207n/B252n). The consideration involved is quite general: in order to identify objective change, we must “categorize” what is given so that the states of the objects that are said to change are singled out. At this point one might object that on the basis of a single experience we can be sure that something alters; what is then more “theoryladen” is the identification of the object that changes, its location in one substratum rather than another. I am not sure how to spell this out 8

Ibid., p. 97. 73

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in Kantian terms without making the objective reality of the concept of alteration a priori, because the only presupposition of its objective reality would be that experience really is possible. If we consider the same question in the more specific case of motion, we encounter new puzzles. In a single experience we can certainly discern motion, even if latitude is left as to what is said to move and what is said to be at rest. Kant’s statement that “the fundamental determination of a something that is to be an object of the external senses must be motion, for thereby only can these senses be affected” (4:476) seems to imply that any experience will contain motion, but Kant’s view of the status of this proposition is unclear. Plaass attempts an a priori proof of the statement just quoted, which he calls a “metaphysical deduction” of the concept of motion.9 If this proof captures Kant’s intention, Kant took it to be a priori true that any outer experience would contain motion, thus placing motion on the same plane as alteration, except for the qualification “outer” (which is discussed below). Plaass’s argument seems to me fallacious. One can perhaps accept his assertion that an object of the outer senses must contain “an objective connection” of spatial and temporal determinations and that this connection is made by the concept of motion; however, he offers no argument that this role must be played by the concept of motion rather than some other. Moreover, the question remains whether Kant intended this statement with the generality that Plaass gives it or even as an a priori truth; one could object with Ralph Walker that the statement only says “what must be so for us, because of the way our senseorgans are constituted.”10 I do not know whether Plaass or Walker is right concerning Kant’s meaning. The dispute reveals an unclarity in Kant’s statement on the relation of the Critique and the Metaphysical Foundations, which is in my opinion bound up with problems of the interpretation of the Critique itself. At the beginning of the Foundations Kant distinguishes the

9

Ibid., pp. 98–99. Walker, “Status of Kant’s Theory of Matter,” p. 593. Hoppe regards the statement we are considering here as “not at all a critical result, but rather a residue of tradition, not overcome by transcendental philosophy” (Kants Theorie der Physik, p. 64, my translation). 10

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“transcendental part of the metaphysics of nature,” evidently what is contained in the Analytic of Principles, from metaphysics that “occupies itself with the special nature of this or that kind of things” (4:470), which he then identifies as metaphysics of “corporeal or thinking” nature. The fact that the categories are schematized only in terms of time is supposed to give the Analytic of Principles an abstract generality that cuts across distinctions in the sensible world, such as that between the physical and the mental. But the matter is in fact not so neat, as Kant admits in the second edition of the Critique, when he says that outer intuitions are needed to establish the objective reality of the categories (B291). He goes on to say: “In order to exhibit alteration as the intuition corresponding to the concept of causality, we must take as our example motion, that is, alteration in space. . . . The intuition required is the intuition of the movement of a point in space” (B291–292). The last remark complicates the issue, because, as Kant makes clear at B155n, this “intuition” is not of motion in the physical sense. Although it does indeed give intuitive content to the concept of alteration, it falls short of establishing its objective reality. The dispute between Plaass and Walker would arise concerning the meaning of “we must” in the passage just cited. Kant’s appeal to a purely geometrical notion of movement seems to give some support to Plaass. At the same time it also seems to be a confusion; clearly only the real possibility of physical motion would establish in this way the objective reality of the concept of alteration. Walker is, in my view, quite convincing in arguing that outer experience as such does not require physical motion. Before leaving the subject of the sense in which the content of the Metaphysical Foundations is a priori, we might comment on the notion of a priori knowledge Kitcher uses in his reconstruction. The fact that the interpretation was originally devised for the purpose of incorporating a notion of a priori knowledge into naturalistic epistemology makes one suspicious about its application to Kant.11 In fact, Kitcher seems to understand his notion of a priori procedure in causal terms: “An a priori procedure for a proposition is a type of process such that . . . if it were followed, would generate knowledge of the proposition” 11

See Kitcher, “A Priori Knowledge,” p. 4. 75

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(italics added).12 Kant himself lays himself open to such a causal interpretation of the a priori in characterizing a priori knowledge as knowledge that is independent of experience (see, for example, B3). Pressing a causal interpretation would wreak havoc with transcendental philosophy as Kant understands it. I believe that this aspect of Kitcher’s understanding of the a priori does little work in his preceding essay. What matters is the identification of certain key conceptions and modes of argumentation as a priori. In finding the argument of the Metaphysical Foundations mainly unsuccessful, Kitcher is in agreement with many earlier commentators.13 Although I do not intend to challenge this conclusion, I believe a more positive account of the relation of the Analytic of Principles and the main parts of the Foundations is possible. I also take issue with portions of Kitcher’s diagnosis of the weaknesses of Kant’s argument. In any sustained attempt at Kantian reconstruction, there is a risk that one of the main actors in the drama of Kant’s philosophy will be left out. In Kitcher’s reconstruction I miss the categories. Kitcher chooses the Dynamics for detailed discussion. Architectonically, the categories that should be at work there are those of quality. These are murky notions even in the Critique; it is not too surprising that Kitcher does not find the connection.14 If we turn to the Mechanics, however, we find a clear enough connection of the propositions with principles 12

Cf. Kitcher, “How Kant Almost Wrote ‘Two Dogmas of Empiricism,’ ” p. 218. Kitcher is more explicitly psychologistic and causal in “A Priori Knowledge,” where, however, he is not primarily concerned to interpret Kant. He does refer to the explication there offered as having “Kantian psychologistic” underpinnings. Historically, Kant has been appealed to both for and against psychologism; my own inclination, in contrast to Kitcher’s, is toward an antipsychologistic interpretation. In view of the attention Kitcher pays in “How Kant Almost Wrote ‘Two Dogmas’ ” to Kant’s equation of the necessary and the a priori and the difficulties that gives rise to, I might hazard the conjecture that it is just to escape a psychologistic causal interpretation of the a priori that Kant gives so much emphasis to this equation. Consider the following passage, which closely anticipates the definition of a priori truth given by Frege at the beginning of the Grundlagen: “If we have a proposition which in being thought is thought as necessary, it is an a priori judgment; and if, besides, it is not derived from any proposition except one which also has the validity of a necessary judgment, it is an absolutely a priori judgment” (B3). 13 And, if Tuschling is right (see note 3), with Kant himself. 14 A more positive account of this connection is given by Schäfer, Kants Metaphysik der Natur. 76

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for the categories of relation. On the other hand, we find a more fundamental source of weakness in the arguments than a mere architectonic prohibition of the use of mathematics. The conservation of matter (Proposition 2 of the Mechanics) is obviously an application of the First Analogy, the law of inertia (Proposition 3) of the Second Analogy, and the equality of action and reaction in the communication of motion (Proposition 4) of the Third Analogy. The force of “application” in this context is problematic. In each case, Kant’s argument rests on a particular interpretation of a categorial concept. The key step in Kant’s proof of the conservation of matter is this passage: “Hence the quantity of the matter according to its substance is nothing but the multitude of the substances of which it consists. Therefore the quantity of matter cannot be increased or diminished except by the arising or perishing of new substance of matter” (4:542). Kant has already identified quantity of matter with the number (Menge) of its movable parts (4:537), and undertaken to motivate this interpretation by appeal to the notion of substance. He emphatically rejects (4:539–540) the notion that matter should have a “degree of moving force with given velocity” (that is, momentum) which can be taken as an intensive quantity. This idea in turn seems to rest on the identification of matter as substance in space: But the fact that the moving force which matter possesses in its proper motion alone manifests its quantity of substance rests on the concept of substance as the ultimate subject (which is not a further predicate of another subject) in space; for this reason this subject can have no other quantity than that of the multitude of its homogeneous parts, being external to one another. (4:541) We may see Kant as dealing with the following sort of problem: How are we to make sense of the notion of substance in space—that is, to make judgments involving this category in application to our actual outer intuitions? The schematization of the category in terms of time does only part of the work. Even if one takes as inevitable the identification of substance in space (Descartes’s extended substance) with matter, it is another step to think of an extended portion of matter as consisting of parts that are themselves substances. Kant may have had in mind arguing that they must be substances because they are subjects 77

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of motion; that is, once one has identified extended substance as the movable in space, it will follow that the subject of motion must be a substance. But the best result this consideration can accomplish is to force the question back to one concerning the idea that motion must be the fundamental determination of something that affects the outer senses (see above). Indeed, there seems to be a factor in the interpretation of the category of substance in the context of space that is not deduced from the pure category and the nature of space itself. Where time instead of space is involved, this is exactly what happens in the schematism of the categories; Kant’s argument requires something like a second schematization of the category in terms of space. This point is perhaps clearer when we turn to the connection between the Second Analogy and the law of inertia. In Kant’s proof (4:543) he simply assumes that motion (in effect, uniform motion in a straight line) is a state and that therefore only acceleration is an alteration in the sense of a change of state (as he explicitly states in the Critique, A207n/B252n). Without some such assumption there is no way to advance from the principle of causality to Kant’s conclusion. Without an assumption of this general form, we are unable to apply the category of causality to matter and motion. Commentators often represent Kant as concerned in the Metaphysical Foundations with the “mathematizability” of phenomena, in other words, concerned with showing that a mathematical theory of the physical world can be constructed and elaborating a philosophical account of how this is possible. In so doing, Kant interprets the categories of substance and causality in quantitative and spatial terms. ‘Pure natural science’ might develop what Kitcher calls a “projected order of nature” in the form of a mathematical model of a world in space and time conforming to the Kantian categories. On any interpretation, Kant’s conception of a scheme of this kind leaves much to experience. But Kant did not show convincingly that even his basic interpretations of the categories were not optional. Here is a brief sketch of a picture of ‘a priori science’ somewhat different from Kitcher’s. One might single out certain concepts because they involve only space, time, very general categories, and fundamental and abstract notions concerning our cognitive faculties. Obviously, a theory sketched in terms of such concepts has highly general application if it even approximates the truth. Indeed, a problem with such a theory might be finding a “handle” for empirical verification and falsification. 78

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If the theory has a high degree of intrinsic plausibility, it may resemble logic and mathematics from an epistemological point of view. If a theory so developed turns out to be false, it may well require some revision in our notions of the relation of our cognitive faculties to the world. In fact, the revision of classical physics early in this century exhibited this character.

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The present essay takes its point of departure from a thought I have had at various times in thinking about interpretations of Kant’s philosophy of mathematics in the literature, in particular that offered by Jaakko Hintikka. That was that if the interpretation is correct, shouldn’t one expect that to show in the way that Kant’s views were understood by others in the early period after the publication of the first Critique? That reflection suggests a research program that might be of some interest, to investigate how Kant’s philosophy of mathematics was read in, say, the first generation from 1781. I have not undertaken such a project. However, I will make some comments about two examples of this kind. In doing so I haven’t always kept my eye on Kant, because the figures involved are of interest in their own right. The first is Johann Schultz (1739–1805), the disciple of Kant who was professor of mathematics in Königsberg. The second is Bernard Bolzano (1781–1848), who in an early essay of 1810 offered a highly critical discussion of Kant’s theory of construction of concepts in intuition. In one way, I think the result of this little experiment is negative, in that it does little toward settling disputed questions about the interpretation of Kant. On the other hand, I think it brings out some problems of Kant’s views that could be seen either at the time he wrote or not long after. We might recall some of the disagreements in the literature on Kant’s philosophy of mathematics. One might see these as arising from challenges to a traditional and natural view, that what is synthetic in mathematical truths is entirely reflected in axioms from which they are derived. In opposition to this tradition, E. W. Beth and Jaakko Hintikka offered proposals according to which the most essential role of intuition is in certain mathematical inferences, which can now be captured by first-order quantificational logic. Hintikka offered a controversial 80

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interpretation of the concept of intuition itself. It is characterized by Kant as a singular representation in immediate relation to its object (e.g., A320/B376–377). The meaning and significance of the immediacy criterion were debated, with the main issue being whether its significance is epistemic and whether it implies some analogy with perception.1 Michael Friedman’s work is in the tradition of Beth and Hintikka, in that he regards intuition (at least in mathematics) as playing mainly a logical role and its role as making possible mathematical inferences that the logical resources available to Kant could not analyze and constructions not only witness what we would formulate as existence statements but even give meaning to mathematical statements.2

I Let me turn now to Schultz. Schultz is explicit about some mathematical matters about which Kant is not. This has made him of value to interpreters of Kant, but it has led to disagreement about the extent to which what he says reflects Kant’s views or work or is original with him. The view I defended many years ago is that there is no convincing reason to believe that the mathematical material that Schultz brings to bear in defending Kant, where it is not found in Kant’s writings, is not original with him.3 On the whole I still uphold this view; see the appendix to this essay. But in any case my present strategy is to treat Schultz as a figure in his own right and ask how he understood Kant. Although his Prüfung der kantischen Kritik der reinen Vernunft 4 is not 1

For my own presentation of different views on this issue, see the postscript to my “Kant’s Philosophy of Arithmetic,” pp.  142–147. However, my most considered view is presented in §I of “The Transcendental Aesthetic” (Essay 1 of this volume). [See also the Postscript to Part I.] 2 More recent work on Kant’s philosophy of mathematics has in many ways moved beyond these issues. However, it is about them that I will interrogate Schultz and the early Bolzano. 3 “Kant’s Philosophy of Arithmetic,” pp. 121–123. I was criticizing Gottfried Martin’s dissertation, subsequently published in expanded form as Arithmetik und Kombinatorik bei Kant. I can’t forbear to comment that chapter 6 of that book [absent from the dissertation] seems to me to show distinct influence from my essay, although Martin does not cite it. (I had sent him a copy before publication.) It was added by the translator to the bibliography of the translation. 4 Much of part II is devoted to replies to articles in Eberhard’s Magazin; the systematic discussion of the Aesthetic that the reader might expect is not presented. 81

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specifically a work on the philosophy of mathematics, that subject occupies a prominent place in it, no doubt in part because the author was a mathematician, and in part because it deals almost entirely with the Introduction and the Aesthetic.5 A natural question to put to Schultz is how he understood the term ‘Anschauung’, what was his conception of intuition. So far as I could determine, there isn’t an explicit discussion of the meaning of this term in the Prüfung. That leaves not as clear as one would wish where he stands on the singularity and immediacy of intuition. Kant’s discussion of mathematical proof brings out the importance of the singularity of intuitions, and the third argument of the Metaphysical Exposition of the Concept of Space is generally read as arguing that the original representation of space is singular, although in the characterization of intuition at the beginning of the Aesthetic only the immediacy criterion is mentioned. The fourth argument seems to be to the effect that the representation is immediate, but as we have noted the force of this in Kant’s philosophy of mathematics has been controversial. What can be found in Schultz bearing on these questions is disappointing. The most informative passage is probably the following: If, however, the representation of space is . . . not a product of any concept, but an immediate representation, that, as e.g. the representation of color, precedes the concept and must first offer to the understanding the material for the formation of the concept, then it [the representation of space] is undeniably a sensible representation, or, as Kant very suitably calls it, an intuitive representation, [an] intuition. (Prüfung, I, 58–59) This passage does not emphasize at all the singularity of intuition and indeed would by itself be compatible with an understanding of intuition as not essentially singular. Such a reading of Schultz might be encouraged by the fact that he often argues for the necessity of intuition in geometry by observing that some terms in geometry must be primitive. He is critical of Euclid’s notorious “definitions” of basic notions Translations from this work are my own, although passages from part I devoted to arithmetic are translated in the translation of Martin. 5 I was struck by the fact that the phrase “Philosophie der Mathematik” occurs in the preface to part II (p. v). But it already occurs in the Critique, A730/B758. 82

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like point and remarks that leading mathematical works of his time do not make any use of them. However, in the passage in which he says this, he does say of the representation that the geometer has of points, lines, surfaces, and solids that “he has created them from no general concept, but he rather presupposes them as something immediately known to him” (Prüfung, I, 55). Also, he argues that concept formations in geometry presuppose the representation of space, with the latter pretty clearly understood as singular. But “immediate” for him seems to have the meaning of something like “not derived” or “given.” He doesn’t bring up the contrast between sensible intuition and intellectual intuition. In Kant’s own writing, one can certainly distinguish a logical from an epistemic use of “immediate,” where the former occurs in the characterization of intuition at A320/B377, where a concept is said to relate to an object “mediately, by means of a mark that several things can have in common,” and the latter is at work, for example, when Kant describes certain propositions as immediately certain. I haven’t located a passage in the Prüfung where the logical use is clearly in play. But that is in the main due to his not articulating the distinction. There is one passage bearing on the matter in Schultz’s earlier Erläuterungen über des Herrn Professor Kant Critik der reinen Vernunft of 1784. In talking of the contrast of intuitions and concepts at the beginning of his exposition of the Aesthetic, Schultz says that concepts are “representations that are referred to the object only mediately, by the aid of other representations” (pp. 19–20 of the 2nd ed.). This last phrase might have been suggested by A320. But it is not really very explicit and is less rather than more informative than Kant’s own characterization in that place. As regards Schultz’s view, however, this earlier passage should dispose of the idea that he did not regard intuitions as essentially singular.6 As was first brought out by Martin, Schultz offers axioms and postulates for arithmetic and uses them in his argument for the claim that arithmetical judgments are synthetic. Interesting as this is, it was unsatisfying to me in my earlier work because it left Schultz with little to say about the evident difference from Kant’s point of view between arithmetic and geometry. Schultz could not have simply missed Kant’s claim 6

It seems possible that Schultz at one time attended Kant’s lectures on logic. But I do not know of definite evidence on the matter. 83

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that arithmetic has no axioms, since it is repeated in Kant’s well-known letter to him of 1788.7 So with regard to the axioms, we have a clear disagreement. There is at least a difference also about postulates, since Kant in speaking of postulates of arithmetic does not seem to have in mind general principles like those stated by Schultz. In “Kant’s Philosophy of Arithmetic,” I wrote: Kant does not seem to have had an alternative view [to that of Schultz] of the status of such propositions as the commutative and associative laws of addition. He can hardly have denied their truth, and it seems that if they are indemonstrable, they must be axioms; if they are demonstrable, they must have a proof of which Kant gives no indication. (1969, p. 123) Some recent writers, beginning with Michael Friedman, have suggested what view Kant might have held about the status of such principles as associativity and commutativity.8 If something along the lines they propose is correct, then there is a disagreement between Kant and Schultz, and for reasons I will explain shortly Schultz seems to me to have on the whole the better case. It is possible that Schultz did not understand Kant’s view well enough to see this disagreement clearly. But very likely Schultz was not inclined to advertise disagreements with Kant; when he expressed some criticism of the Transcendental Deduction in an anonymous review in 1785, the episode seems to have caused severe strain between them.9 The interpretation proposed by Friedman seems to amount to the claim that for Kant these laws are not propositions at all, so that the question of their truth should not arise. They are “procedural” or “operational” rules. The magnitudes that arithmetic and algebra are ap7 However, in Schultz’s exposition of the Axioms of Intuition in Erläuterungen, this claim is not mentioned. In the Prüfung, it is possible that Schultz has the passage of the Axioms of Intuition in mind when he writes, “It seems initially as if arithmetic knew of no axioms” (I, 218). He then proceeds to discuss principles that Kant considers analytic. 8 Friedman, Kant and the Exact Sciences, pp. 112–114; see also Longuenesse, Kant and the Capacity to Judge, p.  282. Shabel, “Kant on the ‘Symbolic Construction’,” seems to express such a view with reference to algebra but is silent about arithmetic. 9 See Beiser, Fate of Reason, pp. 206–207 and 360n.57, and Kuehn, Kant, p. 321. The review is the one to which Kant responds in the well-known footnote in the preface to the Metaphysical Foundations of Natural Science (4:474n.).

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plied to come from elsewhere, in the first instance from geometry but not only from geometry. Arithmetic and algebra are quite “independent of the specific nature of the objects whose magnitudes are to be calculated” (113). They merely “provide operations . . . and concepts . . . for manipulating any magnitudes there may be” (ibid.). This general character must already be possessed by the singular propositions (such as ‘7 + 5 = 12’) on which Kant focuses attention, so that it is not itself sufficient to make what for us would be truth-value-bearing propositions not such for Kant. Evidently the idea is that the associative, commutative, and related laws function as rules of inference. Given that genuine propositions must occur as premises and conclusions of these inferences, the question of their soundness can hardly be evaded, at least once attention is called to them as Schultz did. In her discussion of algebra, Lisa Shabel seems to attribute to Kant the view that in the case of the application of algebraic methods to a geometric problem, it will in the end always be possible to cash in the result of the algebraic manipulations by a geometric construction. That would allow algebraic rules to have a nonpropositional character, but then their soundness would be a problem for particular domains of application. It would be solved for the case of applications to Euclidean geometry by the well-known constructions of arithmetic operations. Beyond this geometric setting, how generally was this problem solved in the eighteenth century? Schultz distinguishes “general” from “special” mathematics; instances of the latter are concerned with a specific kind of quantum, as is geometry. In contrast, general mathematics abstracts completely from the different qualities of quanta, so it deals only with quanta as such and their quantity, and it only examines all the possible ways of combining the homogeneous, by which the magnitude of a quantum in general is generated and can be determined.10 Schultz then describes addition and subtraction as the two main ways of “generating quantity by combining the homogeneous.” Multiplication as iterated addition he seems to regard as derivative, although essential to giving a number as answering the question how many times (Prüfung, I, 214–215). 10

Prüfung, I, 212, Wubnig’s translation. 85

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Schultz’s conception of general mathematics is developed at length in the first part of his Anfangsgründe der reinen Mathesis (1790),11 published between the two parts of the Prüfung. The subject begins with the general concept of quantity and the most general combinations of quantities. He writes that things are called different12 insofar as there is something in the one that is not in the other, and the same13 insofar as they are not different (§1). Things are called homogeneous, insofar as one looks to that in them that is the same, inhomogeneous [or] heterogeneous insofar as one looks to that in them which is different. (§3)14 The determination, how many times something homogeneous with it must be combined with itself in order to generate [the thing] is called a quantity [Quantität]. (§4)15 A thing in which quantity [Quantität] takes place is called a quantity [Quantum]. (§5)16 Mathematics is (conventionally) defined as the science of quantity. Here Schultz uses the term Größe, indicating clearly that both Quantität and Quantum are meant to be included. The general theory of quantities (mathesis universalis) investigates the generation of quantities in general (§8). The most general types of such generation (Erzeugung) are addition and subtraction. However, about this Schultz writes: But through this the quantum is not yet determined as a quantum, that is with respect to its quantity (Quantität), but the latter requires the determination, how many times just the same homogeneous is combined with itself in order to generate the quantum (§4). The determination of the how many times is possible only 11

Because this work is almost unknown, I have included a fair amount of quotation from it. 12 Verschieden (diversa). 13 Einerlei (eadem). 14 Dinge heißen gleichartig, homogen, so fern man auf das sieht, was in ihnen einerley ist; ungleichartig, heterogen, so fern man auf das sieht, was in ihnen verschieden ist. 15 Die Bestimmung, wie vielmal zur Erzeugung eines Dinges ein ihm gleichartiges mit sich selbst verknüpft werden muß, heißt eine Größe oder Quantität. 16 Ein Ding, in welchem Quantität statt findet, heißt eine Größe, ein Quantum. All these quotations are from p. 2. 86

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through a number. Therefore all further generations of quantities except general addition and subtraction rest on numbers. Since, however, every number is again a quantum that is generated from numbers, the general theory of quantities, except for general addition and subtraction, consists merely in the science of numbers or arithmetic.17 Schultz assumes something that Kant does not state and conflicts with the view that arithmetic has no axioms. That is that a science that deals generally with quantity, applying, as Friedman says, to whatever quanta there may happen to be, will have general principles statable as propositions. But one of the principles (his first postulate) is that quanta can be added: To transform several given homogeneous quanta through taking them together successively into a quantum, that is into a whole.18 Since it gives a closure property, this seems to put a constraint on what quanta there are.19 The same would be said of the second postulate:

17

Allein hiedurch wird das Quantum noch nicht als Quantum, d.i. in Ansehung seiner Quantität bestimmt, sondern diese erfordert die Bestimmung, wie vielmal eben dasselbe Gleichartige mit sich selbst verknüpft werden muß, um das Quantum zu erzeugen (§4). Die Bestimmung des Wievielmal aber ist nur durch eine Zahl möglich. Also beruhen, ausser der allgemeinen Addition und Subtraction, alle übrigen Größenerzeugung auf Zahlen. Da aber jede Zahl wieder ein Quantum ist, daß aus Zahlen erzeugt wird, so besteht die allgemeine Größenlehre, ausser der allgemeinen Addition und Subtraction, bloß in der Zahlwissenschaft oder Arithmetik (§9, p. 3). 18 Mehrere gegebene gleichartige Quanta durch ihr successives Zusammennehmen in ein Quantum, d.i. in ein ganzes zu verwandeln (Anfangsgründe, p. 32 §7). A different formulation occurs in Prüfung, I, 221. 19 Kant, in his draft of comments on Kästner’s essays in Eberhard’s Philosophisches Magazin, makes a comment that relates to Schultz’s first postulate. He says of the statement that a line can always be extended, That does not mean what is said of number in arithmetic, that one could increase it, always and without end, by the appending of other units or numbers (for the appended numbers and quantities that are thereby expressed are possible by themselves, without its being the case that they may belong to a whole with the previous ones). (20:420)

Schultz takes this comment into his review almost without change, although it may appear to conflict with his first postulate. Kant’s main point, however, is the contrast with geometry: there is no presupposition of something like space within which a line can be extended. The claim seems to be that the “appended numbers” 87

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To increase and to decrease every given quantum in thought without end.20 To increase, and to decrease, any given quantum as much as one wants, i.e. to infinity.21 For adding small numbers, such as 7 and 5, I have to imagine the units out of which the number 5 is composed, according to the series individually; then I have to add one after the other onto the number 7 and so generate the number 12 by means of successive combining. (Prüfung, I, 223)22 The postulate seems to allow the mathematician to treat 7 + 5 as defined, but the procedure described (essentially that of Kant, B15–16) reduces the defined character of ‘+5’ to that of ‘+1’. So it appears that only that special case of the postulate is used. But so long as one talks generally of quantities as Schultz does, and does not single out and deal separately with the natural numbers, the rational numbers, and the real numbers, one can’t derive the generally defined character of addition in this way. Schultz remarks later (Prüfung, I, 232) that a “laborious synthetic procedure” is needed to see that 7 + 5 = 12. An interesting feature of Schultz’s procedure is that in the Anfangsgründe he undertakes to treat multiplication as derived. So it is not an accident that he does not state in either work any axioms concerning multiplication. His definition seems to presuppose that the second argument is a whole number: Multliplying a quantum a by any number n means finding a quantum p that is generated from the quantum a in just the manner in which the number n is generated from the number 1. (p. 61)23

are possible independently of belonging to any whole such as space with those to which they are appended. 20 Jedes gegebene Quantum in Gedanken ohne Ende zu vermehren und zu vermindern (Anfangsgründe, p. 40). It seems reasonable to regard Schultz’s postulates as prior to his axioms of arithmetic, but in the Prüfung the axioms are stated first. However, the postulates do come first in the Anfangsgründe. 21 Prüfung, I, 221. Schultz held this even of infinite quantities (ibid., I, 224). 22 Schultz’s argument that his axioms of commutativity and associativity are needed to derive ‘7 + 5 = 12’ occurs on pp. 219–220, just after the statement of the axioms. 23 Ein Quantum a durch irgend eine Zahl n multiplizieren, heißt ein Quantum p finden, das aus dem Quanto a auf eben die Art erzeugt wird, als die Zahl n aus der Zahl 1. 88

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Schultz has in a rudimentary way the idea of multiplication as iterated addition. He offers a proof of the distributive law for multiplication by a number (p.  63) by a step-by-step procedure that, to become a proper proof by our lights, would have to proceed by induction, and a similar proof of the commutativity of multiplication of numbers (p. 64).24 Friedman states that Kant’s view is that in arithmetic and algebra “there are no general constructions” analogous to the basic Euclidean constructions (1992, p. 109n.24). This would serve to account for a difference between his remarks in his letter to Schultz and the Prüfung: when he talks of postulates, he clearly has in mind numerical formulae. Something like Schultz’s postulates seem to be needed in actual mathematical practice, as it was before the modern axiomatic treatments of the number systems. And even in those, there is a functional equivalent in treating certain functions at the outset as defined or in making explicit existence assumptions. And perhaps the postulates do amount to general constructions. Kant, as interpreted by Friedman, still has a point: addition, for example, does not always function in mathematics as a construction that serves as a building block for other constructions, although I think it possible that Schultz thought of it that way in relation to multiplication, without getting far in thinking through the problems involved. But his own remarks about ‘7 + 5 = 12’ illustrate the fact that sometimes the result of an addition is the result of a potentially complex procedure, which can be mirrored by a proof. On this latter point there may be a clear disagreement with Kant, since in the letter he says that ‘3 + 4 = 7’ is a postulate because it requires “neither an instruction for resolution nor a proof” (10:556). Schultz does not say explicitly that a proof is necessary, but he does

What Schultz means by “number” would be a subject for further discussion. The evidence known to me is compatible with the suggestion made by W. W. Tait in the discussion in Chicago that he would not have distinguished whole numbers from finite sets. 24 In Parsons, “Kant’s Philosophy of Arithmetic,” footnote 9, it is remarked that the distributive law would be needed to derive formulae involving multiplication such as ‘2 × 3 = 6’, and that Schultz does not remark on this. Schultz very probably thought his understanding of multiplication allowed him to prove the instances of distributivity that are needed, and indeed such special cases are not affected by his lack of a clear conception of proof by induction. 89

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seem to say that one is possible, and he uses the example as a reason for assuming associativity and commutativity as axioms.25 So I don’t think that Schultz rejects the idea of proving such propositions, and he clearly did not go along with Kant’s regarding them as postulates. Schultz clearly saw something that Kant did not acknowledge, that proofs in arithmetic, and therefore in higher mathematics built on it, require general principles. Even if the “procedural rule” interpretation gives Kant a stronger position than it seems to me it does, one quickly comes to the proof of general theorems, as Kant hardly denies. Though mathematical induction had been identified as a distinctive method of proof a long time before, the whole problem posed by rules of inference in mathematics really only came to consciousness some time later. On the whole, the Kantian way of thinking was not favorable to this consciousness-raising. Kant may have seen clearly that the existing logic was not adequate to mathematical inference. There is in modern formulations a trade-off between axioms and rules of inference, so that with at least some principles (most familiarly induction) there is a choice as to whether to formulate them as axioms or as rules. Arithmetic is a clear case where one cannot just rely on constructions (which we could formulate as existence axioms) and parametric reasoning that could be rendered by propositional logic with operations on variables and function symbols. Schultz identified associativity and commutativity as principles that had to be used. Beyond saying (apparently under Kant’s prodding) that they are synthetic, he does not offer a

25

Longuenesse gives a reason why Kant would have rejected the Leibnizian proof, apparently even as improved by Schultz. I have had some difficulty understanding her argument. The key statement is probably Addition does not owe its laws of associativity and commutativity to its temporal condition, but to the rules proper to the act of generating a homogeneous multiplicity. Thus the proof of Mathematik Herder [Ak. 29, 1:57—CP] was both useless and deceptive, for its validity was derived from the very operation whose validity it was supposed to ground. (op. cit., p. 282)

I don’t have an argument to the effect that Kant did not think of the matter in the way Longuenesse claims. But why should one not try to state the rules she refers to precisely and derive some from others? Then one can see if the circularity suggested by the second quoted remark actually obtains. Longuenesse might reply that this procedure is incompatible with denying that the rules in question express “properties of an object” rather than “pertaining to the very act of generating quantity.” But Kant did apparently think that such acts could be represented symbolically and enter into reasoning in algebra. 90

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philosophical account of them, and we have mentioned difficulties for some proposals of a Kantian view of them. Although Schultz was not as explicit about induction as some other mathematicians of his time and earlier, he implicitly appeals to it in his treatment of multiplication. This is a case where granting more to concepts than Kant’s philosophy of mathematics provides for is something interpreters might agree about.

II My second example is an early writing by Bernard Bolzano, Beiträge zu einer begründeteren Darstellung der Mathematik, published in 1810, only six years after Kant’s death.26 This essay contains an appendix on Kant’s conception of construction of concepts in intuition, to which attention was drawn not long ago by a French writer, the late Jacques Laz, whose Bolzano critique de Kant comments on it extensively. Bolzano has often been mentioned as a pioneer in a way of thinking about logic and mathematics that in the long run undermined many aspects of a Kantian view. What is of interest to us, however, is his understanding of Kant at a time that was still historically close to that of Kant. Early in the main text of the Beiträge (I §6, p. 9), Bolzano expresses the view that there is an internal contradiction in the concept of pure or a priori intuition. The argument must be contained in the early sections of the appendix. In §1 Bolzano writes that Kant posed the question: What is the ground that determines our understanding to attach to a subject a predicate that is not contained in the concept of the subject? And he believed he had found that this ground could be nothing other than an intuition, which we connect with the concept of the subject, and which at the same time contains the predicate. What he says about Kant’s concept of intuition is brief; he describes it as representation of an individual. In §4, speaking for himself, he describes intuition as the representation occupying the place of X in judgments of the form “I perceive X,” where clearly there is no room for a priori intuition. Evidently the object of a perception is a representation; 26

Translations of quotations from this work are my own. 91

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it does not have to be sensible.27 But it does seem to be a representation as a particular event, so that “I perceive X” is unavoidably empirical. An implication of this formulation is that an intuition can be a constituent of a judgment, contrary to Kant’s stated view.28 It’s not very clear how Bolzano thinks intuition is meant to be related to perception on Kant’s conception. About a priori intuition he writes in §2: If we finally ask what an a priori intuition should be, I think that here no other answer is possible than: an intuition that is connected with the consciousness that it must be so and not otherwise. Only thus can intuition give rise to the necessity of the judgment based on it. On balance I am inclined to think Bolzano understands Kantian intuition generally on a perceptual model. Why else should he think that for intuition to be the basis of a judgment of necessity, the intuition itself should contain “consciousness” of necessity? Even with the help of Laz’s commentary, I am not able to see clearly what Bolzano’s argument against a priori intuition is. He complains that Kant has not given a clear definition even of the a priori–empirical distinction, and rightly observes that necessity is properly a property of judgments. Since an intuition is not a judgment, it cannot be necessary. But Bolzano’s own account surely doesn’t imply that an intuition does not have content that would have to be spelled out in propositional form. Bolzano turns more directly to the role of pure intuition in mathematics beginning in §7. He attributes to Kant the following reasoning: If I connect the general concept, e.g. of a point, or of a direction or distance, with an intuition, i.e. represent to myself a single point, a single direction or distance, then I find of these individual objects, that this or that predicate applies to them, and feel at the same time, that this is equally the case for all objects that fall under these concepts. 27

See the main text, II §15, p. 76. Bolzano in this text holds a theory of perception according to which the existence of an outer object has to be inferred from my representations, just the theory that Kant opposes in the Refutation of Idealism. 28 Laz appears to attribute this view to Bolzano’s interpretation of Kant; see op. cit., p. 74. 92

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How, asks Bolzano, can we come to this “feeling”? Is it through what is single and individual, or through what is general? Obviously through the latter, that is, through the concept and not through the intuition. A Kantian reply might have been to refuse this dichotomy, or at least its being applied in the way Bolzano applies it. “Construction of concepts in intuition” as Kant conceives it has to introduce representations that have the form of singular representations but are nevertheless in a certain way general, in that they represent the concepts that are thus constructed. It’s not easy to imagine how Bolzano might have reacted to the logical interpretation of Kantian intuition introduced by Beth and Hintikka and exploited by Friedman. But if he had had that in view in 1810, it’s hard to believe he would have reacted as he did to the idea of a priori intuition. Another way of putting the matter29 is that Bolzano’s reading does not make any room for a transcendental synthesis of imagination, which would be a priori but also unify the manifold with some sort of aim toward conformity to concepts. The synthesis of imagination is described as an “action of the understanding on sensibility” (B152). The result is that intuition as experienced has a content that is amenable to conceptualization, and insofar as the synthesis is a priori, by a priori concepts. In the footnote to B160 Kant writes that the unity of the manifold of space and time “precedes any concept” although it makes concepts of space and time possible. Bolzano may well have found remarks of this kind puzzling and thought that no sense could be given to them that would be consistent with the understanding of an intuition as a representation of an individual. In I §6 of the main text Bolzano mentions another disagreement with Kant; he denies “that the concept of number must necessarily be constructed in time and that accordingly the intuition of time belongs essentially to arithmetic” (p. 9). His discussion of this issue in §8 of the Appendix has a clear relation to Schultz’s defense of Kant’s philosophy of arithmetic. It is reasonable to conjecture that Bolzano knew Schultz’s Prüfung.30 Bolzano discusses Kant’s example, ‘7 + 5 = 12’. Simplifying the case to ‘7 + 2 = 9’, he sketches a proof on the Leibnizian model. 29

Suggested by some comments of Laz, op. cit., p. 75. The Anfangsgründe der reinen Mathesis is cited in the main text (I §5, p. 9). But the detail of the discussion of ‘7 + 5 = 12’ is not in that work. Bolzano certainly knew the Prüfung later; it is discussed in Wissenschaftslehre §79 and §305. 30

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But he makes clear that the associative law of addition is presupposed in the proof,31 which he glosses thus: that one in the case of an arithmetical sum attends only to the collection of the terms, not to the order (a concept certainly wider than sequence in time). This proposition excludes the concept of time rather than presupposing it. Bolzano is not concerned with the question whether the associative law or ‘7 + 5 = 12’ is synthetic, but rather with whether it depends on intuition. Kant, in the Introduction to the second edition of the Critique, is naturally read as deriving the former from the latter. Bolzano is an opponent of a priori intuition but not of the synthetic a priori, so that for him it is at least a possibility that arithmetical judgments  should be synthetic a priori judgments of a purely conceptual character.32 It is not easy to say where the disagreement with Kant (or for that matter Schultz) lies here, although undoubtedly there is one. Bolzano could be saying no more than that in the content of a statement like ‘7 + 5 = 12’ there is no reference to time, something with which Kant apparently agrees. But he evidently thinks it possible to reason mathematically with more general concepts such as that of order, without representing them by succession in time. That something like that is his quarrel with Kant and Schultz is indicated by the general remarks about mathematics in the main text, where he characterizes mathematics as the science dealing with “the general laws (forms) that things must conform to in their being (Dasein) (I §8, p. 11). But he glosses the latter by saying that mathematics does not give proofs of existence by concerns only conditions of the possibility of things. This is where he draws a contrast between mathematics and metaphysics. He is (and 31

Bolzano’s simplification means that he does not reach the point at which Schultz had to appeal to commutativity, and therefore we do not see whether Bolzano knew how to avoid that assumption. 32 Later, in Wissenschaftslehre §305, Bolzano does argue that ‘7 + 5 = 12’ is analytic. He relies on an explanation of a sum as “a totality . . . in the case of which no order of the parts is considered and parts of parts are regarded as parts of the whole.” He says explicitly that associativity is analytic; evidently he would have said the same about commutativity. His argument could be criticized on grounds like those on which, according to Laywine, “Kant and Lambert,” Lambert criticized Wolff: associativity and commutativity are in effect packed into the definition of addition. 94

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remained) critical of Kant’s claim that the methods of mathematics and philosophy are essentially different.33 The difficulty I had in understanding Bolzano’s quarrel with Kant over this issue arose from the fact that Bolzano’s remark doesn’t clearly say more than that the concept of the associative law doesn’t directly involve time, and this seems to be something Kant agrees with. One might infer, though, that Bolzano thought that if we have to represent the succession of numbers by succession in time, that is just a subjective condition of our consciousness of the relations of numbers, and would not detract from the purely conceptual character of arithmetic even if it were shown that time is an intuition, which would anyway be hardly compatible with Bolzano’s own conception of intuition as expressed in §4. Some of Kant’s own statements encourage the idea that time is only a subjective condition, for example that of the Schematism that number is the unity of the synthesis of the manifold of a homogeneous intuition in general “in that I generate time itself in the apprehension of the intuition” (A143/B182). Why, Bolzano might well ask, is a condition of the apprehension of the intuition part of the characterization of the relation of number to the category of quantity? One might ask this question even if one accepts the transcendental point of view that pervades Kant’s whole discussion of magnitude and quantity. Bolzano was even in this early work out of sympathy with that point of view. A way in which we might try to understand Bolzano’s claims in both of these arguments is that he is insisting on a rigorous distinction between a representation and what it is a representation of. Although almost none of that apparatus is present in the Beiträge, the logical platonism of Bolzano’s later period made it possible for him to make such distinctions across the board. Does the notion of a priori intuition involve compromising that distinction? If an intuition is a representation of an individual, one might ask, how can it still carry with it the fact that it reflects the construction of certain concepts? What an intuition does contain, according to Bolzano, is the consciousness that it must be so and not otherwise. What he rejects is something more specific than the very idea that an intuition might convey information about its object. Presumably if it conveys the information that its object 33

Although this matter is hardly at issue in the present essay, see Laywine, “Kant and Lambert,” for Lambert’s disagreement with Kant’s view, and Beiser, “Mathematical Method,” for an interesting history of Kant’s thesis in post-Kantian idealism. 95

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a is F, it will also convey the information that it is G, if G is a concept that is contained in F. But according to Kant, conditions of the construction of concepts lead to conclusions that are not contained in them. Although Bolzano does not really articulate an objection on these lines in the Beiträge, he may have thought that that feature of pure intuition was incompatible with a clear distinction between a representation and what it represents. Bolzano’s mathematical work might suggest a way in which, if Kant introduced a priori intuition in order to compensate for the expressive limitations of monadic logic, this was not fruitful for the further development of mathematics and its foundations. It is doubtful that the free variable languages that have been suggested to represent Kant’s conception of mathematics were adequate to the mathematics of Kant’s own day. Mathematicians beginning with Cauchy and Bolzano did not wait for logicians to develop a polyadic logic in order to exploit the capacity of ordinary language to express such notions.34 Rather, they set up definitions in those terms and reasoned with them as best they could. Bolzano already offered a splendid example in his Rein analytischer Beweis of 1817. The development of polyadic logic followed the development of a mathematics in which the reasoning with quantifiers was more complex; it did not precede it. Bolzano’s faith that “mere concepts” were adequate to the task of proving fundamental propositions of analysis and placing them in their proper order was in the end vindicated. The relevance of the limitation of monadic logic to the philosophy of mathematics in Kant’s time has been a matter of controversy, and our discussion of Schultz and the early Bolzano will hardly bring that controversy to an end. Friedman’s thesis that insight into this was the primary reason why Kant insisted that mathematics required intuition does not, it seems to me, get as much indirect support from Schultz’s writings as he might hope for. One controversy about Kant’s philosophy of mathematics was whether intuition plays a necessary role in mathematical inference and not merely at the stage of axioms and postulates. I regard that controversy as largely settled in Friedman’s favor,

34

This point is well made, with earlier examples than those I mention, in Rusnock, “Was Kant’s Philosophy of Mathematics Right for Its Time?,” pp. 433–435. Regarding the main argument of Rusnock’s paper directed against Friedman, it should be said that it concerns Friedman’s assessment of Kant’s philosophy of mathematics given his interpretation, not the interpretation itself. 96

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but Schultz says so little about inference in mathematics in his writings that they hardly strengthen the case. The text of Bolzano that we have discussed does not directly address this issue, but his rejection of a priori intuition was of a piece with the procedure noted above in his mathematical work, to go ahead with the kind of reasoning whose analysis in the end required polyadic logic, possibly trusting that in the end logic would catch up.

Appendix The investigation made here of Johann Schultz’s work and views offers an occasion to reconsider a question originally raised by Martin, what the revision might have been that Schultz made in part I of the Prüfung after receiving Kant’s letter of November 25, 1788 commenting on his draft and then discussing it with him. It is clear that the draft maintained that such arithmetic statements as ‘7 + 5 = 12’ are analytic and thus that Kant succeeded in convincing Schultz on this point. Martin makes the further claim that the mathematical material relevant to this issue, the axioms and postulates stated in the published Prüfung, were not in the draft and were either contributed by Kant or worked out in discussion between Kant and Schultz.35 As noted above, I questioned this claim in “Kant’s Philosophy of Arithmetic,” pp. 121–123. My view was and is that Schultz could well have argued that the axioms are analytic, as Leibniz did in the case of commutativity.36 It also seems a priori unlikely that Kant would have proposed axioms that would contradict his own thesis (reaffirmed in the letter) that arithmetic has no axioms. Concerning the postulates, matters are somewhat more complicated. The idea that arithmetic might have postulates of the sort that Schultz states was not original with either Kant or Schultz, since similar principles are regarded as such in Lambert’s Anlage zur Architectonic (1771, §76).37 It could also have been more difficult for Schultz to admit postulates, in formulation somewhat modeled on Euclid’s, 35

Martin, Arithmetik und Kombinatorik, p. 65. Martin makes the further claim that in the latter case the axioms should be credited to Kant “since he would doubtless have had the leadership in these discussions.” That this would be so about a mathematical matter is surely far from evident, particularly since in proposing his axioms Schultz contradicts Kant’s claim that arithmetic has no axioms. 36 See “Kant’s Philosophy of Arithmetic,” p. 123n.13. 37 See Laywine, “Kant and Lambert,” to which I am indebted on this point. 97

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and still argue that they are analytic. Therefore the conjecture that they were already in the manuscript on which Kant was commenting on is less likely. Kant certainly knew Lambert’s book, and one possibility is that he pointed out its relevance to Schultz. But it is also possible that Schultz was directly influenced by Lambert, who introduces his postulates without invoking an analytic-synthetic distinction.38 Evidently we do not have firm evidence concerning either the axioms or the postulates.39 The view that both were added at the last minute does not square well with Schultz’s remark at the beginning of the preface to the Anfangsgründe that the book is “the work of a laborious reflection of many years.” Why should one revisit this question, when it apparently cannot be resolved definitively? One reason would be Martin’s broader thesis, that books by disciples of Kant presented arithmetic axiomatically, and that this had an influence on subsequent developments leading in the end to the late nineteenth-century axiomatization of arithmetic. This thesis and the work beyond Schultz’s that he cites would be worth further examination. Schultz’s disagreement with Kant about whether arithmetic has axioms is a reason independent of the above discussion for giving Schultz a more autonomous role in this development than Martin credits him with. Martin himself cites another indication of this: In 1791 Kant’s pupil J. S. Beck defended as one of the theses for his habilitation, “It can be doubted whether arithmetic has axioms.”40 Even if Beck’s intention was to defend Kant’s position, the formulation leads Martin to conclude that this was a matter of dispute in the Kantian school. Martin makes another interesting observation about Schultz, which is apart from the main concerns of this essay but which connects him with Bolzano. He says that Schultz was quite clear on the point “that arithmetic, in particular of irrational numbers, and infinitesimal calcu38

Martin points out that postulates of arithmetic also occur in the earlier Neues Organon (1764); see the quotations in Arithmetik und Kombinatorik, p. 52, from Alethiologie §26 and §74. 39 Béatrice Longuenesse seems confused on this matter, surprising in so careful a scholar. She writes (Kant and the Capacity to Judge, p. 280) that Martin had seen the manuscript on which Kant comments in his letter, a claim for which I can find no warrant in Martin’s text. Although she cites my criticism of Martin, she adopts without comment a claim that I questioned, that the mathematical material in the Prüfung was not present in the earlier draft. 40 Martin, Arithmetik und Kombinatorik, p. 65. 98

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lus should be cut loose from all geometric accessories” (1972, p. 111).41 He is relying on the fact that Schultz puts these subjects in general mathematics and explicitly says that its proofs should be conducted independently of geometry (Anfangsgründe §21, pp. 10–11). This aspiration may give Schultz some historical importance. It may be a reason why Bolzano in citing this work says of Schultz that he “deserves much credit for the foundation of pure mathematics.”42

41

Martin attempts to trace this attitude of Schultz back to Kant as well. He does not mention the letter to A.  W. Rehberg of September 1790, which is at least a problematic text for this view. 42 I §5, Russ’s translation. A rough version of this essay was presented to the conference on Kant’s Philosophy of Mathematics and Science at the University of Illinois, Chicago, on April 28, 2001. I am greatly indebted to Daniel Sutherland and Michael Friedman for their organization of this stimulating event and to them, Lisa Shabel, W. W. Tait, and others for their comments. I don’t claim to have done justice to the points raised. Shabel in particular convinced me of the relevance of Schultz’s mathematical works, although I have been able to consult only the Anfangsgründe (1790), which I consider the most relevant to my theme. I am also much indebted to the editors for suggestions. 99

POSTSCRIPT TO PART I

“Arithmetic and the Categories” (Essay 2) was written just as a period of impressive growth in the study of Kant’s philosophy of mathematics was beginning. This beginning is marked by the early writings on the subject of Michael Friedman.1 He has continued to contribute up to the present day. One of his major contributions was to integrate the study of Kant on mathematics with that of his philosophy of physical science. His writings also stimulated work by a younger generation of scholars.2 Some reaction to this body of work is called for in the present reprinting. I will concentrate, however, on points in it that bear on what is said in these essays. However, I will not be able totally to avoid going back to “Kant’s Philosophy of Arithmetic.” Two older issues are addressed in these essays: Kant’s conception of intuition and the place, or lack of it, of a notion of mathematical object in Kant’s scheme. I will discuss intuition at some length and make some briefer remarks later about mathematical objects. Kant in well-known passages characterizes an intuition as a singular representation, and as a representation in immediate relation to its object. In “Kant’s Philosophy of Arithmetic,” I viewed these as distinct and essentially independent criteria. The latter claim was opposed to that of Jaakko Hintikka that the immediacy criterion is simply a corollary of the singularity criterion. There has to be some interdependence 1

Friedman, “Kant’s Theory of Geometry” and “Kant on Concepts and Intuitions.” I should mention Emily Carson, Katherine Dunlop, Ofra Rechter, Lisa Shabel, Daniel Sutherland, and Daniel Warren, although Warren’s work primarily concerns physical science. Since Essay 3, my only publication on Kant’s philosophy of science, is a short occasional piece, the present Postscript addresses issues only from the other three essays reprinted here. 2

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between them, since otherwise it would be possible to point to representations that, according to Kant, satisfy one criterion but not the other. What proved to be more controversial was the claim that immediate relation to objects “means that the object of an intuition is in some way directly present to the mind, as in perception.”3 The question of what the immediacy of intuitions consists in is revisited in the Postscript to that essay, in the discussion of a view of Robert Howell.4 Although I wrote there that the controversy had convinced me that my interpretation of the immediacy criterion was not so evident as I had thought, I did not make as clear as it should be where I then stood. The discussion of intuition in Essay 1 of this volume expresses a position that I still largely hold. It is admitted there that in the definition as expressed at A320/B377 “immediate” means “not mediate,” that is, not by means of marks that several objects can have in common. That may be the most basic meaning of the immediacy criterion. However, neither I nor the others who had written on the subject up to that time admitted the possibility of marks that are not possibly common to several objects. That Kant admitted such marks was documented later by Houston Smit.5 But even with this correction, if that is all that Kant means by “immediate,” it is hard to understand some aspects of Kant’s basic logical expositions, for example the remarks about the necessary connection of concepts and judgment in the section on the logical use of the understanding (A67–69/B92–94). I believe it is such considerations that lead Béatrice Longuenesse to write: “Kant’s characterization of intuition as ‘immediate’ representation essentially means, I think, that intuition does not require the mediation of another representation in order to relate to an object.”6 Although this may be suggested by the characterization of immediacy at A320/B377, it does not seem to me to be directly implied by it.7 3

Ibid., p. 112. Graciela De Pierris (Review of Guyer, p. 655) quotes a remark to the same effect from “The Transcendental Aesthetic” (p.  10) without noting that it is there described as “an earlier proposal” of mine. Friedman quotes the same remark (“Geometry, Construction, and Intuition,” p. 169) but notes in a footnote that it “refers back” to “Kant’s Philosophy of Arithmetic.” The changes in my views reflected in the later essay are not especially important for the comments they wish to make. 4 Ibid., pp. 144–145. 5 “Kant on Marks and the Immediacy of Intuition.” 6 Kant and the Capacity to Judge, p. 220n.15. 7 Other writers have expressed still different ideas of what immediacy amounts to. For example, Lorne Falkenstein seems to understand “immediate” as something 101

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It will be argued that these considerations remain in the logical sphere and do not yet imply that there is anything in Kant’s conception of intuition that would indicate that, in particular, the role of intuition in mathematics is not adequately explained by considerations that by our own lights belong to logic. I do not have any substantial additions to make to the considerations offered in previous writings in favor of the view that phenomenological presence is a significant part of what the immediacy of intuition amounts to. It would be hard to deny this in the case of empirical intuition, so that any real dispute concerns pure intuition. Here I refer the reader in particular to the discussion in Essay 1 of the third and fourth arguments in the Metaphysical Exposition of the Concept of Space. More important in the present context than these questions about the concept of intuition itself are questions about its role in mathematics. Essay 1 is somewhat brief in its treatment of issues about Kant’s philosophy of mathematics, but what is said there in connection with the Transcendental Exposition of the Concept of Space implies that more than one role for intuition emerges from Kant’s remarks about mathematical proof, particularly in the Discipline of Pure Reason in Dogmatic Use. Additional roles have been proposed by more recent writers (see below). There has been a long-running disagreement about the question whether, according to Kant, intuition must be appealed to in mathematical inferences or only in setting up the initial premises, in particular axioms. The former view was proposed a century ago by Bertrand Russell and has been developed by E. W. Beth, Jaakko Hintikka, and Michael Friedman. The latter view has been defended in our own time by Lewis White Beck and Gordon Brittan. The clearest texts supporting the former view directly concern geometry, as for example the discussion of geometric proofs in the Discipline, especially A713/ B741. However, my own writing on Kant’s philosophy of mathematics has not focused on geometry, although it naturally plays the role of an object of comparison with arithmetic. In “Kant’s Philosophy of Arithmetic” I did not question the Beck-Brittan view with respect to geometry, because it seemed to me to give an adequate account of why geometry should be synthetic and dependent on intuition; the problem like “prior to any processing of information by the subject”; see Kant’s Intuitionism, p. 60. He regards the immediacy criterion as the most basic meaning of “intuition” in Kant’s usage. 102

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why those things should be true of arithmetic was the starting point of the essay. The brief discussion of geometry in §III of Essay 1 in this volume does not take a position in this controversy, but what is said is compatible with the Russell-Friedman view, and in Essay 4 I say that the controversy is largely settled in its favor. The acceptance of the view that intuition plays an essential role in mathematical inference still leaves the question of its role in grounding the initial steps in mathematical proofs, in geometry construction postulates, axioms, and definitions. Furthermore, although influential advocates of the Russell-Friedman view have also held a purely logical interpretation of Kant’s concept of intuition, that interpretation is not essential to the claim that intuition plays an essential role in inferences. Friedman’s later writings on this subject deal almost entirely with geometry. He has come to agree that intuition in mathematics does have a phenomenological dimension,8 and since he has in no way given up the view that the role of intuition extends to inferences, it follows that he agrees that this issue is to some degree independent of the one concerning the nature of intuition. It should be added that there was never a dispute as to whether intuition has a logical dimension, which the singularity criterion ensures. It is somewhat difficult to locate what disagreement there may be between the views expressed in Essay 1 and those in Friedman’s later writings. Friedman undoubtedly gives a larger role to considerations from geometry, but there would be no disagreement with the claim that, in the Aesthetic, it is not only in the Transcendental Expositions that Kant relies on geometry. Broadly speaking, my treatment of the Metaphysical Exposition of the Concept of Space has phenomenological considerations standing more on their own feet than they do in Friedman’s account. But although geometry is not as much present as it is in Friedman’s reading, it is not completely absent either. I will consider only one case, the claim in the fourth argument that “Space is represented as an infinite given magnitude” (B40) and that in the third argument that the representation of a single space is prior to that of spaces. Appealing to a passage in the Dissertation, I wrote, “There is a phenomenological fact to which he is appealing: places, and thereby 8

“Synthetic History Reconsidered,” pp. 586, 592; see also his “Geometry, Construction, and Intuition.” 103

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objects in space, are given in a one space, therefore with a ‘horizon’ of surrounding space” (p. 19). That is the sense in which I take Kant to be entitled to describe space as “boundless,” but I explicitly say that it does not yield the metrical infinity of space and I do not suggest that one could obtain that independently of geometry. In my systematic writings, I describe phenomenological considerations like those just mentioned as “a step toward infinity” and do not see them as in any way getting one all the way.9 Friedman evidently also maintains that there is an appeal to geometry in Kant’s remark in the fourth argument that no concept, as such, can be thought as if it contained an infinite set of different possible representations within itself. Nevertheless, space is so thought (for all the parts of space, even to infinity, are simultaneous). (B40) In this he may be right, and it would be a weak defense to say that I did not assert the contrary, since I did not discuss this specific passage. What I do say about the parallel passage at A25 (Essay 1, pp. 16–17) is not very clear and does not make clearly a distinction that I do rely on in systematic writings. The idea that a further horizon is always there might be called weak boundlessness; it always invites a further step in such operations as extending a line segment. But it is another claim to say that such steps can be iterated indefinitely. Thus one exhibits the lack of a bound on distances in Euclidean geometry by indefinite iteration of the operation of laying out, on a given line, a segment equal to one chosen at the outset.10 How explicit a conception Kant had of indefinite iteration is not clear to me, but he does appear to have been explicit about some particular cases such as the one just mentioned, which model the idea of successive addition. I think that is more fundamental than the metrical considerations that might single out Euclidean space in particular.11 However, I think Friedman is right that at 9

See especially Mathematical Thought and Its Objects, §29. It appears that this is still meaningful in hyperbolic geometry, but in elliptic geometry there is a bound on distances. 11 Sutherland argues, in correspondence, that one can give phenomenological sense to some rudimentary metrical considerations and that they may have played a role in Kant’s thinking. I have no reason to dispute this. However, I don’t think such considerations can yield infinity without the iteration emphasized in the text. 10

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points where an argument requires appeal to it, Kant is at least implicitly relying on mathematics. The matter is complicated by the fact, brought out by Emily Carson, that Kant maintains that geometry must presuppose a given space that is infinite, within which the spaces generated by geometrical construction proceed.12 So it appears to be Kant’s view that the infinity of space is prior to geometry. I think the texts Carson cites show that the infinity of space does have a certain priority. Evidently the geometer must have this representation, if it is presupposed in geometric practice. Kant’s language indicates that he thinks that the geometer will be conscious of it.13 One thing Friedman was concerned to deny is that we have a full insight into the infinity of space independent of geometry. It is not clear that this follows from these texts, since Kant’s argument for the claim that such a space must be presupposed in the practice of geometry itself relies on descriptions of geometric procedure. Consider the following passage quoted by Carson: To say, however, that a straight line can be continued indefinitely means that the space in which I describe the line is greater than any line which I might describe in it. Thus the geometer grounds the possibility of his task of increasing a space (of which there are many) to infinity on the original representation of a single, infinite, subjectively given space.14 One might take the statement that the geometer “grounds” the possibility of indefinite continuation of a line on the original representation of an infinite space as meaning that the geometer has to appeal to something about that representation in justifying his own claims. But in general Kant takes geometry to be able to proceed without buttressing from philosophy. It seems more likely that Kant means that the ground of the possibility of continually increasing a space is the “single, infinite, subjectively given space,” but that this is revealed by philosophical reflection 12

“Kant on Intuition in Geometry,” esp. pp. 496–499. E.g., 20:419, from Kant’s partial draft of a review by Johann Schultz of volume 2 of Eberhard’s Philosophisches Magazin; cf. note 19 of Essay 4. 14 20:420. I have modified Carson’s translation. It is worth noting that if “increasing a space to infinity” has a metrical meaning then it would refer to something like indefinitely iterating the operation of laying a new segment of a line equal to the previous one. 13

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without being something that the geometer appeals to in his own arguments.15 To return to the fourth argument in the Metaphysical Exposition: Its opening sentence does not directly appeal to geometry, and I am less certain than Friedman is that he was implicitly making such an appeal in that place. However, when one spells the case out in the light of later writings, some appeal to geometry seems unavoidable. In all his writings on the subject, Friedman seems to want to bypass the role of intuition as a source of insight into the truth of mathematical or other propositions. In the case of geometry, such insight would operate in the practice of geometry; it would not offer justification prior to and independent of geometry, of the sort that Friedman is most concerned to deny. The idea of intuitive insight into truths such as mathematical axioms, and likewise into the correctness of inferences in mathematics, is very traditional. It seems to me that that is what Kant has in view in statements such as that geometrical knowledge is “immediately evident” (A87/B120) or that axioms are synthetic a priori principles that are “immediately certain” (A732/B760). The role of intuition in the representation of magnitudes has been explored in depth by Daniel Sutherland.16 Kant characterizes the concept of magnitude as “the consciousness of the homogeneous manifold in intuition in general, so far as through it the representation of an object first becomes possible” (B203).17 Note that it is said to be the consciousness of the homogeneous manifold in intuition; magnitude is tied to intuition, and intuition is essential to the representation of magnitudes. In a sense this is the most general role of intuition in mathematics, since Kant agrees with the common view of the time that mathematics deals with quantities, although he says that this is a consequence of the fact that only quantities can be constructed (A714/B742). A point that Sutherland has stressed is that this fact about magnitudes 15

One might add, as Katherine Dunlop suggests, that the metaphysician undertakes to show how we can have the representation of an infinite space, a task that is foreign to the geometer. 16 See first of all “The Role of Magnitude in Kant’s Critical Philosophy.” 17 On the translation of this passage, see Sutherland, “Role of Magnitude,” p. 418n.12. Kant makes clear that this is the definition of magnitude in the sense of quantum, not quantitas. Sutherland also notes that Kant speaks of intuition in general, so that at the level of explaining the notion of magnitude he is not assuming our particular forms of intuition. 106

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means that intuition can make distinctions that cannot be made by “mere concepts.”18 Up to now we have made no comments about matters specific to arithmetic, although that is the subject of Essays 2 and 4. In the more than twenty-five years since the publication of the earlier of these essays, the subject has been transformed, largely by closer attention to the background of Kant’s thought in the mathematics of his own and earlier times and by the analysis referred to above of his conception of magnitude. Those considerations would have some impact on what is said in my essay, but for the most part it would be more in the details (for example concerning the concept of homogeneity) than in the general line of my discussion. Some matters deserve comment, however. First, some writers beginning with Friedman regard mathematical objects as having no place in Kant’s scheme, while the general tenor of §I of Essay 2 is to explore various views that might be compatible with what Kant says while finding Kant not very articulate or definite on the matter. Kant makes clear that geometry concerns quanta and certainly talks of them as objects. They may be defective objects, as suggested by the remark that although we can give the concept of a triangle an object a priori, it is “only the form of an object” (A223/B271). A suggestion I have considered is that such objects come under the categories of quantity but not under those of quality and relation (and probably not under those of modality, at least as explicated in the Postulates), but I do not know of direct textual support for it.19 The more difficult questions concern arithmetic and algebra. Friedman maintained that they do not have distinctive objects. He proposed that the theory of magnitudes, which includes algebra and arithmetic, gets its objects from outside the theory, so that any general rules involved will govern “operations . . . for manipulating any magnitudes there may be.”20 Lisa Shabel’s account of symbolic construction in algebra and its background in eighteenth-century mathematics builds on this idea. It 18

See Sutherland, “Kant’s Philosophy of Mathematics,” also his “Kant on Arithmetic, Algebra, and the Theory of Proportions,” pp.  555–557, and, with respect to Frege’s criticism of the “units” view of number, his “Arithmetic from Kant to Frege.” 19 The idea is explored with respect to arithmetic in Rechter, “Syntheticity, Intuition, and Symbolic Construction,” pp. 185–192. 20 Kant and the Exact Sciences, pp. 113–114. 107

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implies that any objects for algebra would have to arise in applications, such as in geometry.21 The methods of algebra are on her view essentially tools for solving problems; in principle the problems can arise in any domain of magnitudes, but in practice the primary domains are those of arithmetic and geometry. In her paradigm examples, the final stage of the solution is a geometric construction. As noted in Essay 4, she is largely silent about arithmetic. A question that Shabel’s view raises is whether Kant is entitled to  say that algebra contains synthetic a priori judgments, or indeed whether there are any properly algebraic truths. Early in his letter to Schultz of 1788, Kant identifies “general arithmetic (algebra)” and “general theory of quantity” (allgemeine Größenlehre) and describes the former as an “amplifying science,” which is central to pure mathematics (10:555). Kant had held similar views since the pre-critical period.22 Sutherland, citing this and a great deal of other evidence, concludes that according to Kant algebra does have objects, namely magnitudes. That accords with the views just cited of Friedman and Shabel, since the objects that arise in application will be magnitudes. Since the magnitudes involved can be continuous, it is a little misleading on Kant’s part to refer to algebra as “general arithmetic.” However, that may be just the generality that Kant has in mind.23 It appears that the fact that magnitudes can be geometrically represented is enough to give algebra the foundation in intuitive construction that Kant’s general remarks about mathematics require. But Sutherland’s examination of a wide range of Kantian texts and their background, particularly in the Greek mathematical tradition, does not make out in detail how this is. The case concerning arithmetic is harder. Kant undoubtedly claims that there are synthetic a priori judgments in arithmetic, and he frequently talks of numbers. In Essay 2 I remark that Kant “tends not to distinguish, for a given number n, between a ‘multiplicity’ with cardinal number n and the number n itself” (p. 58 above). Sutherland and William Tait have pointed out that an ambiguity of this kind goes back

21

See her “Kant on the ‘Symbolic Construction’.” See Sutherland, “Kant on Arithmetic,” p. 549, and the passages cited there. 23 I owe this suggestion to Daniel Sutherland, who remarks that it can still be called arithmetic because it deals with operations such as addition and subtraction. 22

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to Greek conceptions of arithmetic.24 An attempt to make clear how Kant conceived finite multiplicities is made in §II of my essay. However, since such multiplicities would count as quanta, they are unlikely to be the objects of arithmetic in Kant’s considered view. In the letter to Schultz, Kant gives as a reason why arithmetic has no axioms that “it really does not have a quantum (i.e. an object of intuition as quantity) as object, but merely quantity (Quantität), i.e. a concept of a thing in general through determination of quantity” (10: 555). This leaves it unclear what Kant could be talking about when he talks of numbers. When, in the well-known argument for the syntheticity of ‘7 + 5 = 12’, Kant speaks of “see[ing] the number 12 arise” (B16), what is he referring to by “the number 12”? The reference to particular examples of multiplicities would suggest the interpretation we have just rejected. Sutherland argues that it is likely that Kant, along with many thinkers from ancient times well into the nineteenth century, thought of numbers as composed of “pure units,” which cannot be distinguished from one another qualitatively.25 The textual evidence for this is somewhat indirect, and it leaves unclear the status of units as objects. The points Kant mentions (B15, A140/B179) would share the essential property of pure units, but they are not alone in this; in the second place he describes the collection of points as “an image of the number five.” The indefinite article at least suggests that there are or could be others. Points can play the role of units, but the fact that there are alternatives would imply that we cannot say that it is a configuration of twelve points that ‘the number 12’ refers to. Concerning ‘7 + 5 = 12’, Kant writes: Although it is synthetic, however, it is still only a singular proposition. Insofar as it is only the synthesis of that which is homogeneous (of units) that is at issue here, the synthesis here can take place in only one way, even though the subsequent use of these numbers is general. (A164/B205) Although it is possible to read this as saying that the synthesis arrives at a single object 12, the contrast he immediately draws with the construction of a triangle counts against this, as many writers have observed. 24

Sutherland, “Kant on Arithmetic,” p. 535; Tait, “Frege versus Cantor and Dedekind,” §9. Both are probably indebted to Stein, “Eudoxus and Dedekind.” 25 Sutherland, “Kant on Arithmetic,” §5.2. 109

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The claim seems rather to be that there is a certain equivalence in the syntheses carried out using different “images” of the number five. It would be tempting to think of this equivalence as the existence (or coming to light) of a one-one correspondence, but as Sutherland has pointed out, nothing Kant says gives direct support to that reading. I don’t know just how Kant thought of the equivalence, but I conjecture that it was on the level of the act of the understanding, which would have a certain indifference to the particularities of the objects it is dealing with and even not to be directly about quanta but rather only about their quantitas.26 Probably the conclusion is that ‘12’ is not exactly a singular term as we would understand it, but Kant does not give evidence of having a theory of what else it might be. In Essay 4 of this volume, in discussing the disagreement between Kant and Schultz about whether arithmetic has axioms, I maintain that Schultz had the better case. But I have not done much to explain why Kant held the view he did. An ingenious explanation was developed by Friedman on the basis of the view noted above that the theory of magnitudes (which includes algebra and arithmetic) gets its objects from outside the theory. As noted above, a view of this kind seems to have the consequence that algebra does not consist of general propositions at all, and the same seems on Friedman’s view to be true of arithmetic. That leaves as a problem how arithmetic can even contain singular truths, as Kant emphatically asserts that it does. Any view leaves the puzzle as to how one can reason generally about quanta in order to arrive at a judgment like ‘7 + 5 = 12’. And then the question arises: If we can in these cases know that another unit can always be added, why would there not be a postulate that expresses this generally? In fact Schultz infers such a possibility from his second postulate.27 Kant seems in a couple of places to endorse an assumption like this, but without saying what its role is (or is not) in mathematical argument. 26

That would be in line with Longuenesse’s statement that the principles of arithmetic, “unlike the principles of geometry, are not dictated by the formal intuition that is its object, but are contained in the very act of constituting quantity or magnitude” (Kant and the Capacity to Judge, p. 281). However, in arguing that the general theory of quantity is science of numbers, Schultz writes that every number is itself a quantum (Anfangsgründe, p. 3, quoted above, p. 87). It is likely that he is identifying a number with a multiplicity of that number of elements. 27 See Prüfung, I, 223. 110

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In §III of Essay 2 I propose that Kant returns in later texts to a more “intellectualist” view of the concept of number than is expressed in the first edition of the Critique, especially in the characterization of number as the schema of the concept of magnitude, in the sense of quantitas (A142/B182). The position I found in texts of 1788–1790 would be closer to that of the Dissertation. This proposal has come under criticism. I will comment briefly on the criticism of it by Longuenesse. I wrote: Kant appears in the Schematism to reject the idea expressed in the Dissertation and implicit, though not consistently held to, in the Metaphysics lectures, of describing the concept of number in terms of the pure categories. (p. 59 above) Longuenesse comments: This is a strange thing to say if we recall that in the Schematism chapter, Kant writes that number is the schema of the category of quantity. Thus he does not abandon the definition of number “in terms of the pure categories,” unless “pure” is understood as meaning “having no relation to the sensible” (unschematized).28 In one way this remark misses my point, which was that it seemed to be a departure from what he had said previously to place the concept of number on the side of the schema rather than on the side of the category that has the schema. I certainly didn’t mean to say that either side ceases to play a role in the account of arithmetic in the Critical period, either in 1781 or later. If that misunderstanding is cleared up, I am not sure what disagreement remains. I conjectured that “Kant saw in the notion of intellectual synthesis a framework into which to fit the abstract conceptions of quantity developed in his lectures” (p. 63 above) and called attention to the definition of quantum given at B203, emphasized and analyzed in Sutherland’s work. I think I may have been confusing two senses in which categories might be pure: being thought without any relation to intuition and being thought in relation to intuition in general, in abstraction from the particular forms of intuition that we have. It is the latter that characterizes intellectual synthesis.

28

Kant and the Capacity to Judge, p. 256n.24. 111

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The texts I chiefly relied on were the letters to Schultz of 1788 and that to Rehberg of 1790. But the translation I gave of a key passage in the former letter can be improved. Kant wrote, Die Zahlwissenschaft ist, unerachtet der Succession, welche jede Konstruktion der Größe erfodert, eine reine intellektuelle Synthesis, die wir uns in Gedanken vorstellen. (10:557) It would have been better to render unerachtet der Succession as “not considering the succession” instead of “in spite of the succession” (see Essay 2).29 It thus appears that there is a certain abstraction involved in taking the science of numbers to be a “pure intellectual synthesis.” There still seems to be a difference with the position of 1781, but we may not be able to be sure what the difference is. One could read the letter to Schultz as saying that time is a subjective condition of carrying out mathematical construction (and thus arriving at mathematical knowledge) and also a constraint on the application of mathematics, but the content of arithmetic is quite independent of our particular forms of intuition. Some remarks in the letter to Rehberg that I called attention to would support this reading. In philosophy, for example in the first part of the B Deduction, Kant allows himself to reason about “intuition in general” in abstraction from our particular forms of intuition. It might solve problems for him, for example the puzzlement expressed above about arithmetical propositions, if he admitted such reasoning into mathematics itself. But whenever the opportunity to say that presents itself, he pretty clearly rejects it, and indeed it would not have fit well into his general philosophy to allow that mathematical reasoning could be about intuition in general, independently of our particular forms. In particular, would it be compatible with his conception of mathematical reasoning as involving construction of concepts in pure intuition? Although the matter arises only briefly in Essay 4, I will comment on one more issue, the Leibnizian proofs of arithmetical identities and what might have been Kant’s attitude toward them. As noted above,30 29

Zweig translates the phrase as “notwithstanding succession” (Correspondence, p. 285). I read that as closer to my earlier translation. 30 Essay 2, note 46. The proof is presented and discussed in Longuenesse, Kant and the Capacity to Judge, p. 279. Ofra Rechter observes (“The View from 1763,” pp. 34–35) that the proof is surrounded by remarks on numerical notation systems, in particular the contrast 112

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such a proof of ‘8 + 4 = 12’ occurs in Herder’s notes on Kant’s early lectures on Mathematics (29, 1:57). The proof as it stands would require associativity and commutativity, as Schultz observed later about a similar proof of ‘7 + 5 = 12’. Nowadays, we would prove such an identity using only the recursion condition for addition m + (n + 1) = (m + n) + 1, which Schultz probably would have regarded as a special case of associativity. Thus one would reason (1)

7 + 5 = 7 + (4 + 1) = (7 + 4) + 1

(2)

7 + 4 = 7 + (3 + 1) = (7 + 3) + 1

(3)

7 + 3 = 7 + (2 + 1) = (7 + 2) + 1

(4)

7 + 2 = 7 + (1 + 1) = (7 + 1) + 1 = 8 + 1 = 9

(5)

7 + 3 = 9 + 1 = 10 (by (3), (4))

(6)

7 + 4 = 10 + 1 = 11 (by (5), (2))

(7)

7 + 5 = 11 + 1 = 12 (by (6), (1)).

Thus this argument dispenses with commutativity. In effect, it reduces the evaluation of ‘7 + 5’ to that of ‘7 + 4’, and then to ‘7 + 3’, and so on. The proof in the Herder notes and that given by Schultz have in common a different procedure. We can render Schultz’s argument as follows: 7 + 5 = 7 + (4 + 1) = 7 + (1 + 4) = (7 + 1) + 4 = 8 + 4 8 + 4 = 8 + (3 + 1) = 8 + (1 + 3) = (8 + 1) + 3 = 9 + 3 9 + 3 = 9 + (2 + 1) = 9 + (1 + 2) = (9 + 1) + 2 = 10 + 2 10 + 2 = 10 + (1 + 1) = (10 + 1) + 1 = 11 + 1 = 12.31 This proof has the feature that it mirrors the description at B15–16 of how one arrives at ‘7 + 5 = 12’, namely that one starts with 7 and between base 2 and base 10, and the algorithms for addition, multiplication, and division. That at least gives a hint as to how he might have viewed the question of knowledge of arithmetic identities involving larger numbers. 31 Prüfung, I, 220. Schultz goes on to say, “That this is the only way by which we can arrive at insight into the correctness of the proposition that 7 + 5 = 12 is something that every arithmetician knows.” Did he have in mind writings on arithmetic by others? 113

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successively adds units to it.32 Although it is less explicit and does not note the use of associativity and commutativity, the proof in Herder’s notes has the same structure. Longuenesse notes the parallel between the early proof and the procedure of the Critique, but the difference she discerns leads her to make the remarks about Kant’s probable attitude to the proof that are discussed critically above (Essay 4, note 25). The question I raised there still stands: If there are rules “proper to the act of generating a homogeneous multiplicity,” why should one not state them as general rules and derive some from others? Kant may well have sensed that the time was not ripe for arithmetic to be treated axiomatically. But he seems to have avoided giving any account at all of general propositions in arithmetic.33

32

Cf. Longuenesse’s comments on the proof in Herder’s notes, op. cit., pp. 279–280. I am greatly indebted to Katherine Dunlop and Daniel Sutherland for comments on an earlier version of this Postscript. They are not responsible for failures on my part to take adequate account of their comments. 33

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PART II

FREGE AND PHENOMENOLOGY

5 SOME REMARKS ON FREGE’S CONCEPTION OF EXTENSION

In discussions of the elements of set theory, we find today two quite different suggestions as to what a set is. One appeals to intuitions associated with ordinary notions such as “collection” or “aggregate.” According to it, a set is “formed” or “constituted” from its elements. The axioms of set theory can then be motivated by ideas such as that sets can be formed from given elements in a quite arbitrary way, and that any set can be obtained by iterated application of such set formation, beginning either with nothing or with individuals that are not sets.1 According to the other, the paradigm of a set is the extension of a predicate. Terms denoting sets are nominalized predicates; and sets are distinguished (e.g., from attributes), by the fact that predicates true of the same objects have the same set as their extension. Generally, the axioms of set theory are viewed as assumptions as to what predicates have extensions.2 It would be hard to find an instance of a very pure account of the elements of set theory in terms of one of these suggestions to the exclusion of the other, so that perhaps neither offers by itself the basis of a complete account of the nature of sets. Mathematicians may also question whether the project of giving such an account is not metaphysical 1 For example Shoenfield, Mathematical Logic, pp. 238–240; Boolos, “The Iterative Conception of Set”; Wang, From Mathematics to Philosophy, ch. 6. [See now also Shoenfield, “Axioms of Set Theory.”] 2 Such a conception seems to underlie the widely held view that the “naive” or “intuitive” conception of set is expressed by the (inconsistent) universal comprehension schema. It seems to be expressed by W.  V. Quine in Set Theory and Its Logic, esp. pp. 1–2, and in The Roots of Reference. [For discussion see §VI of Essay 7 of Mathematics in Philosophy. However, the influence of this view, already in decline in 1976, has declined still further since then.]

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and therefore of no interest. This view, however, rests satisfied with a situation in which the concept of set is not so clear as it ideally could be, and in which the axioms accepted in set theory are not so evident as they could be. I am persuaded that the exploration of these conceptions is worthwhile. The first of the above suggestions claims ancestry in the writings of Cantor.3 The second occurs in a particularly pure and rigorous form in Frege. A close examination of Frege’s conception of extension is certain to be helpful in understanding the second suggestion and testing its adequacy. I do not propose to carry out such an examination in this paper, which is more limited in scope. I shall discuss some passages where Frege comments on something resembling the “Cantorian” explication. I shall then comment on the evolution of Frege’s views on extensions from his learning of Russell’s paradox until his death.

I There are a number of passages in Frege’s writings where he discusses a concept of set explained along the lines of the first suggestion. Curiously, he does not comment on Cantor’s explanations.4 But at the beginning of the Grundgesetze (pp.  1–3) he criticizes Dedekind; the same issue is discussed at greater length in an unfinished paper of 1906 on an essay by Schoenflies on the paradoxes.5 The ideas of both these comments can be traced back to criticisms in the Grundlagen of views according to which a number either is itself a “set, multitude, or plurality” (p.  38), or attaches to an “agglomeration of things” 3

For example Wang, From Mathematics to Philosophy, pp. 187–189. Except that in his 1892 review of Cantor’s Zur Lehre vom Transfiniten he says that Cantor is unclear as to what is to be understood by “set” and then quotes a passage in which he finds a hint of his own view (Review of Cantor, p. 164). This review contains Frege’s most interesting and judicious comments on Cantor, although he comments at length on Cantor’s theory of real numbers in Grundgesetze, vol. 2, §§68–85. [Because it was not directly relevant to my theme, I did not comment on the closing remarks of this review, where Frege clearly sees Cantor as an ally against the naturalistic empiricism influential in Germany at the time. Martin Davis does justice to this remarkable little exchange; see Review of Dawson, pp. 116–117.] 5 Nachgelassene Schriften, hereafter cited as NS, pp. 191–199, trans. pp.176–183. 4

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(Aggregat).6 Frege always understands an Aggregat as something composed of parts. Therefore it has two insufficiences as a bearer of number: first, it seems to be spatio-temporal and thus would leave unaccounted for the fact that non-spatio-temporal things can be numbered; second, what is composed of parts is not so composed in a unique way. Hence different possible decompositions of a whole into parts would give rise to different numbers. Thus an agglomeration as such does not have a definite number. In the Grundlagen, Frege does not have in view the mathematical concept of set, but rather a number of perhaps not very precise ordinary concepts. Some of these might now be “regimented” by means of the concept of set, others by a modern logic of the whole-part relation such as Lésniewski’s mereology or the Leonard-Goodman calculus of individuals. Frege tends always to interpret them in the latter sense. Frege interprets Dedekind as holding that his “systems” (i.e., sets) consist of their elements. Dedekind accepted the conclusion that a system with one element would be indistinguishable from the element itself.7 Because of extensionality (which Dedekind explicitly affirms) this can be true only for individuals and one-element sets: otherwise an object x and its unit set {x} must be distinguished because they do not have the same elements. Frege does not raise this difficulty but rather raises the point (parallel to one he made about the “agglomeration” theory of number) how there can be a null set: If the elements constitute the system, then where the elements are abolished the system goes with them.8 An empty concept has on the other hand no difficulty, and in view of the fundamental difference of concepts and objects, a concept under 6

Foundations, pp. 29–30, from a quotation from Mill, System of Logic, bk. III, ch. xxiv, §5. “Agglomeration” is apparently translated Aggregat in the translation Frege cites (by J. Schiel; see ibid., p. 9), but the term Aggregat is used by Frege with the same meaning in discussions without reference to Mill. I have used Mill’s “agglomeration” rather than “aggregate” throughout as an English version of it since the latter is often used as a synonym for “set” or “class.” 7 Was sind und was sollen die Zahlen?, par. 3. 8 Grundgesetze, 1:3. The German reads, “Wenn die Elemente das System bilden, so wird das System mit den Elementen zugleich aufgehoben.” 119

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which exactly one object falls cannot be confused with the object itself. That the extension of such a concept must be distinct from the object is not evident on Frege’s conception of extension. In fact, the conventional identification he makes between the two truth-values and certain extensions had the consequence that those are their own unit classes, and he suggests that such an identification be made for all objects that are “given independently of Wertverläufe.”9 This consideration would weaken the force of Frege’s argument against Dedekind, unless we interpret him to mean that Dedekind’s conception does not ever provide for a distinction between an object and its unit set, in which case it will fail in those cases where extensionality requires such a distinction. Frege does not go so far as to interpret Dedekind as taking sets to be agglomerations. In the discussion of Schoenflies he goes into a similar issue at greater length. There Frege says that the word “Menge” can be taken in two ways, which are most clearly expressed by the words “agglomeration” and “extension (Begriffsumfang).” But frequently these conceptions do not occur in their pure form, but mixed together and this makes for unclarity. The aggregative (aggregative) conception is the first to offer itself, but the requirements of mathematics pull towards the opposite side, and so confusions easily arise.10 Characteristic of an agglomeration is the presence of relations which make parts into a whole; the examples (except perhaps for “a corporation”) are all spatio-temporal. Moreover, the parts of a part are parts of a whole. This has of course the consequence that decomposition is not unique, which was in the Grundlagen a fatal obstacle to taking agglomerations as bearers of number. Frege finds the notion of agglomeration not precise enough to be a mathematical concept, a view which perhaps has been refuted by later developments. But these developments have also made even clearer that the notion is different from that of set. 9

Ibid., p. 18 n.1. NS, p. 196, trans. p. 181. The discussion of agglomerations and extensions on pp. 196–197 develops more explicitly a remark in Grundgesetze, 2:150. 10

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Frege devotes the last completed part of the paper to making clear the distinction of an extension from an agglomeration. From the plan (p. 191) he evidently intended to go on to discuss the notions of “Inbegriff . . . System, Reihe, Menge, Klasse.” It is natural to conjecture that he viewed any account of the last two that sought to distinguish either notion from that of extension as a “mixture” of the concepts of extension and agglomeration, and therefore as unclear, if not incoherent. It thus appears that Frege did not see any foundation to the idea, central to the sort of explanation of the concept of set according to our first suggestion that is used to block the well-known paradoxes, that the elements of a set must be “given” prior to the “formation” of the set. The only interpretation of this idea that Frege considered would be that a set consists of its elements, a view which he evidently took to be derived from the notion of agglomeration so that the model for it would have to be the manner in which a whole consists of parts. On the other hand, he does say that an extension “simply has its being (Bestand) in the concept.”11 It is clear that the model for this cannot be the part-whole relation. Could it give rise to a priority of the elements of an extension to the extension? Only, it seems, if there is a priority of the objects falling under a concept to a concept. It seems that Frege could get help from Russell. The above statement is one Russell could have subscribed to, taking concepts as propositional functions and extensions as classes. According to Russell, a propositional function presupposes its arguments, that is, the elements of its range of significance, not the arguments of which it is true. The arrangement of classes in a hierarchy of types is, in Russell’s account, a consequence of this principle.12 It may be that Russell has here tacitly introduced the concept of set that Frege rejects: is not the “range of significance” of propositional functions of lowest type a totality consisting of objects which is not explained by Russell’s own explanations of classes by way of propositional 11

NS, p. 199, trans. p. 183. Cf.: “On the other hand, what constitutes the being of the concept—or of its extension—are not the objects that fall under it but its marks (Merkmale), that is, the properties that an object must have in order to fall under the concept” (Grundgesetze, 2:150, my translation). This latter passage calls into question the view that Fregean concepts are not at all akin to intensional entities. 12 Principia Mathematica, 1:16, 54. 121

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functions? I shall not examine here whether this charge is true. There is another more direct conflict into which the proposed Russellian rescue would have placed Frege. Frege held that the range of significance of any concept-expression is absolutely all objects. Since the extension of a concept is an object, the Russellian principle would make the extension of a concept prior to the concept, contrary to the priority of concepts to extensions that Frege affirms more explicitly. The simplest way out for Frege would no doubt be to deny that extensions are really objects, that is, in effect to adopt a no-class theory. For Frege, this would be less complicated than it was for Russell: his logic was full (impredicative) second-order logic, which he seems never to have been tempted by the paradoxes to abandon. However, he would have had to give up either the identification of numbers with extensions, crucial to his logicism, or his thesis that numbers are objects. Dropping the identification of numbers with extensions is in fact the solution that Frege adopted at the end of his life, in the fragments of 1924–1925, but that went with rejecting extensions altogether (see below). A concept, according to Frege, is a function which has a value (the True or the False) for any object whatever as argument. Could Frege have dropped this view and approached the paradoxes on the basis of a Russellian idea that a function “presupposes its arguments,” but that its Wertverlauf need not be among those arguments? This is of course exactly the situation in set theory for a function defined on a set, where we can take the Wertverlauf to be the function as a set of ordered pairs. Such a step could hardly have failed to drive Frege in a direction deeply uncongenial to his previous thought. Consider a simple quantification ‘∀xFx’. If ‘Fx’ denotes a function that is not defined for certain arguments, then that it is true of these arguments is not implied by ‘∀xFx’. The latter cannot say of absolutely every object that it is F. Indeed, such absolute generality could be expressed only by a form of quantification not analyzed by Frege’s logical theory, perhaps by a “systematic ambiguity” of the quantifier parallel to such ambiguities as arise in Russell’s original theory of types. A similar ambiguity would have to attach to such predicates as ‘x = y’ that apparently apply to absolutely all objects.13 13

Cf. Essays 8 and 9 of Mathematics in Philosophy. In the object language of the theory of types, there need be no typical ambiguity, since each variable will have a 122

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Although Frege could thus in a way preserve the generality of the step from a concept to its extension, if the only “logical objects” available at the outset are the two truth-values, some principle of iteration is needed to obtain infinite classes. Frege’s reduction of arithmetic to logic would not be saved. Alternatively one might say that where the quantifier really is absolutely unrestricted, then ‘Fx’ does not denote a concept. It is not evident that this offers any advantage over saying that not all concepts have extensions; it seems to have the disadvantage that the latter still leaves second-order logic intact.

II I want now to make some remarks concerning the evolution of Frege’s views on the concept of extension after he learned of Russell’s paradox. The evidence known to me14 shows a gradually increasing skepticism, so that the rejection of extensions in 1924 does not come out of the blue. In the correspondence with Russell of 1902–1904 and the appendix to volume 2 of the Grundgesetze, he does not consider that extensions might be given up or so restricted that his analysis of number would have to be abandoned. He did consider the idea that Wertverläufe might be treated as second-class objects (uneigentliche Gegenstände).15 He apparently rejected this idea, before proposing to Russell the “way out” of the appendix. The subsequent fate of the Way Out in his thinking is obscure; it is not mentioned explicitly in the Nachgelassene Schrifien or in his publications after 1903. However, in the plan for the critique of Schoenflies Frege speaks of “concepts that agree in their extension, although this extension falls under one of the concepts but not the other.”16 This might be interpreted as presupposing the Way Out. However, this seems unlikely in the light of Frege’s analysis of the paradox in the appendix to volume 2 definite type index. But in Russell’s metalanguage it is essential to the general explanation he gives of the interpretation of the theory. This issue is independent of the difference between the simple and the ramified theory of types. 14 I have seen only part of Frege’s still unpublished correspondence. [But see Essay 6 in this volume.] 15 Letter to Russell, September 23, 1902; Grundgesetze, 2:254–255. 16 Begriffe, die im Umfange übereinstimmen, obwohl dieser Umfang unter den einen fällt, nicht aber unter den anderen. NS, p. 191, my translation. 123

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of the Grundgesetze. There Frege notes (p. 257) that it is the inference from the equality of value-ranges to the generality of an equality (i.e., from ‘ε’ f(ε) = ε’ g(ε)’ to ‘∀x[f(x) = g(x)]’) that yields the contradiction; the converse inference (which indeed is just an expression of extensionality) is innocent. He then generalizes the paradox argument to show, without any use of axiom V, that for any second-level function there are concepts which yield the same value as arguments of this function although not all objects falling under the one of these concepts also fall under the other. The above citation should be compared with the following: If it is permissible generally for any first-level concept that we speak of its extension, then the case arises of concepts’ having the same extension although not all objects falling under one also fall under the other.17 Since the counterexample is precisely the (common) extension of the two concepts, it seems that in the plan of 1906 Frege is just repeating the point made on pp. 257–261 of the appendix before he introduces the Way Out. In the former he concludes, “Mengenlehre erschüttert.” Does this mean that he thought already in 1906 that set theory is beyond repair? There is no other evidence of this; he may have meant no more than what he said of the paradox in the appendix: However, this simply does away with extensions of concepts in the received sense of the term.18 Nonetheless, the following statement, “Meine Begriffschrift in der Hauptsache unabhängig davon,” suggests a point of view that is expressed quite explicitly in one of his notes to Jourdain’s account of his work.19 This point of view seems to guide much of Frege’s writing from that time until 1919. Frege writes: 17

Grundgesetze, 2:260. Ibid. 19 Philip E. B. Jourdain, “The Development of the Theories of Mathematical Logic and the Principles of Mathematics: Gottlob Frege.” The notes are reprinted in Kleine Schriften. Apart from its containing these valuable notes, Jourdain’s article deserves recognition as the most accurate account of Frege’s work by another which had appeared up to that time. Of course Jourdain owed to Russell his appreciation of Frege’s importance. The notes are presumably translations by Jourdain of German originals, but the originals appear to be lost (see KS, p. 334). Note added in proof. In a letter dated March 22, 1976, I. Grattan-Guinness informs me that he has found that the German originals of Frege’s notes to Jour18

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And now we know that, when classes are introduced, a difficulty (Russell’s contradiction) arises. In my fashion of regarding concepts as functions, we can treat the principal parts of Logic without speaking of classes, as I have done in my Begriffsschrift, and that difficulty does not come into consideration.20 Frege repeats what he had indicated even before the paradoxes, that the laws of classes are not so evident as the “principal” parts of logic. His distinction parallels that which might now be made between second-order logic and set theory. He goes on to say: The class, namely, is something derived, whereas in the concept— as I understand the word—we have something primitive (etwas Ursprüngliches). . . . We can, perhaps, regard Arithmetic as a further-developed Logic. But in that, we say that in comparison with the fundamental Logic, it is something derived.21 In speaking of classes as “derived,” Frege does not make his meaning very clear. If it is purely epistemological (that is, if the point is that the laws of classes are less evident than, or presuppose a prior knowledge of, the laws of logic in the narrower sense), then the choice of terms is strange, since the relation is not that of premise and conclusion: there will have to be distinctive axioms for classes. Although Frege contrasts classes as derived with concepts as “primitive,” he could hardly mean that the language of classes is defined in any sense compatible with Frege’s views on definitions. So Frege is perhaps maintaining that classes are derived in their ontological relation to concepts. But Frege does not develop the thought further in any text known to me. In particular, although he apparently still envisages an account of arithmetic in which numbers are construed as classes (extensions), he does not indicate what the theory of extensions is that he might have in mind. Frege seems to have concentrated on discussions that could be carried out using only the resources of “fundamental Logic.” The long dain are indeed extant, contrary to what is said above. Some are in the Russell Archives, and the remainder are in Jourdain’s notebooks in the Institut MittagLeffler, Stockholm. [These originals are published in Wissenschaftlicher Briefwechsel (cited hereafter as WB); see Essay 6 in this volume.] 20 Jourdain, “Development,” p. 251. [For Frege’s German text see WB, p. 121.] 21 Ibid. 125

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essay “Logik in der Mathematik” is a case in point.22 Neither the notion of extension nor the idea of a reduction of arithmetic to logic is mentioned. Frege takes up again the polemical discussion of others’ views on numbers, with a discussion of Weierstrass. In the Grundgesetze Frege had criticized the attempts of mathematicians to “create” objects by definition, and he claims that his axiom about Wertverläufe will serve all the purposes that such creations are intended to serve.23 This issue is not raised in “Logik in der Mathematik.” He is even noncommittal about the question whether induction needs to be a purely mathematical axiom or can be reduced to logic.24 Rudolf Carnap reports in his autobiography on three courses of lectures by Frege that he attended in the winter semester of 1910–1911, the summer semester of 1913, and the summer semester of 1914.25 The last was called “Logik in der Mathematik” and its content evidently paralleled that of the essay of that title. The first course was given at about the time at which the notes for Jourdain were written. The role of the notion of extension in the first two courses is not too clear from Carnap’s account. Concerning Russell’s paradox he writes, “I do not remember that he ever discussed in his lectures this antinomy and the question of possible modifications of his system in order to eliminate it.”26 That might suggest that Frege had simply presented the original system of Grundgesetze, which seems somewhat unlikely in view of Frege’s rigorous standards: it is hard to imagine him presenting a system he knew to be inconsistent without even mentioning the problem. Carnap believed that Frege thought some solution could be found, but here he refers to the appendix to volume 2 of the Grundgesetze, written some years before, rather than to the lectures.27

22

NS, pp. 219–270, trans. pp. 203–250. Grundgesetze, 2:§147. 24 NS, p. 219, trans. p. 203. This is surprising since the core of Frege’s previous reduction is just second-order logic. Frege is making the methodological point that an inference in mathematics should proceed by purely logical rules; any distinctively mathematical aspect of the inference should be represented by mathematical axioms. 25 Carnap, “Intellectual Autobiography,” pp. 4–6. 26 Ibid., pp. 4–5. 27 Bynum’s statement “As late as 1913–14 he was presenting and defending his  logistic programme in courses at Jena University” (Frege, Conceptual 23

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Carnap does say, “Toward the end of the semester Frege indicated that the new logic to which he had introduced us, could serve for the construction of the whole of mathematics.”28 (He is referring to the first course.) But none of the information Carnap gives about the 1913 course directly shows that a construction of numbers on the basis of extensions was part of it. The remark that Carnap cites certainly indicates that in 1911 Frege believed in such a construction, but on the whole Carnap’s recollection gives some, but not very decisive, confirmation to the view that Frege concentrated almost entirely on what could be done with “fundamental Logic” independently of the notion of extension. In “Logik in der Mathematik” Frege had emphasized the lack of agreement among mathematicians about what the objects of arithmetic are and the unclarity of their statements so long as no adequate account of these objects was given. But the matter was left there. The same point is made briefly in “Aufzeichnungen für Ludwig Darmstaedter” (1919), but there follows a series of questions, which call in question even the doctrine that numbers are objects. A statement of number is a statement about a concept which therefore applies to this

Notation, p.  48) seems not to be justified by the statements in Carnap’s autobiography. Professor Bynum has kindly sent me copies of his correspondence with Carnap, including the letter of April 4, 1967 which he mentions (Frege, Conceptual Notation, p. 48n.10). There Carnap refers to his shorthand notes on Frege’s lectures and says he could find in them no reference at all to Russell, Principia Mathematica, Russell’s paradox, or the appendix to volume 2 of the Grundgesetze. However, he reports a “vague memory” that Frege mentioned Russell in some way. Although the letter so far confirms the picture I have presented and nowhere explicitly contradicts it, Carnap expresses forcefully his belief that Frege had not given up the view that arithmetic is a branch of logic. (In fact he says Frege never gave this up.) You ask: “Why did he not give it up?” I would say “Why should he?” The fact that a flaw was found in his particular form of a system of logic did certainly not destroy his belief that there is a system of logic which has in general the features which he had envisaged, although some details would have to be changed.

Carnap says that his own view of the nature of arithmetic is “chiefly based on what I learned from Frege.” Study of Carnap’s notes should shed some further light on these issues. The above material from Carnap’s letter is included by permission of Professor Bynum. [For discussion of the now published notes see the Postscript.] 28 Carnap, “Intellectual Autobiography,” p. 5. 127

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concept a second-level concept. These second-level concepts are ordered in a series, and there is a rule which for each one will give the next one. But still we do not have in them the numbers of arithmetic; we do not have objects, but concepts. How can we get from these concepts to the numbers of arithmetic in a way that cannot be faulted? Or are there simply no numbers in arithmetic? Could the numerals help to form signs for these second level concepts, and yet not be signs in their own right?29 The notes end there. Frege evidently no longer relies on extensions as the objects of arithmetic, but still the idea of extension is not explicitly mentioned, even to question or reject it. In the most extended of the fragments of 1924–1925, expressions of the form “the extension of the concept a” are given as examples of the tendency of language to create proper names to which no object corresponds. Of the expression “the extension of the concept fixed star” he says: Because of the definite article, this expression appears to designate an object; but there is no object for which this phrase could be a linguistically appropriate designation. From this has arisen the paradoxes of set theory which have dealt the death blow to set theory itself. I myself was under this illusion when, in attempting to provide a logical foundation for the numbers, I tried to construe numbers as sets.30 In my view, this rejection of extensions and of logicism is the end of an evolution which, after the initial shock of the paradox, proceeded more or less continuously. It is the positive theory of number, the attempt to construct numbers by geometrical means, that is the more radical new departure in the last fragments. In the latter context Frege writes, “From the geometrical source of knowledge flows the infinite in the genuine and strictest sense of this 29

“Aufzeichnungen für Ludwig Darmstaedter,” NS p. 277, trans. p. 257. “Erkenntnisquellen der Mathematik und der mathematischen Naturwissenschaften,” NS pp. 288–289, trans. p. 269. Note that Frege says that “the concept fixed star” is also a proper name without reference. 30

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word.”31 That Frege here means “actual” rather than “potential” infinity is probable.32 In spite of his Kantian view of geometry, Frege does not seem in the last fragments to have come any closer to the constructivistic view of the infinite that is characteristic of other Kantian views of mathematics in the twentieth century. Some years before, Frege endorsed an argument of Cantor’s to the effect that potential infinity presupposes actual.33 In his letters to Dedekind of 1899, Cantor approaches the paradoxes by distinguishing among “multiplicities” between the “consistent” (sets) and “inconsistent.”34 A multiplicity is inconsistent if it is contradictory for all its elements to “be together.” That is to say, it cannot be consistently conceived except as a potential totality. Cantor’s proposal seems inconsistent with the view Frege endorsed. A conception of the totality of sets and other “absolutely infinite” totalities along the lines intimated by Cantor is widely held today. Cantor could adopt it in response to the paradoxes more readily than Frege because his concentration on the sequence of ordinals and cardinals brought home to him how the totality of sets must burst the bounds of any overall grasp we might seek to have of it. Set theory as such clearly did not much move Frege; his interest in the concept of extension was motivated by concerns of general logic and of the foundations of classical arithmetic and analysis.35 31

Ibid., p. 293, trans. p. 273. The German reads, “Aus der geometrischen Erkenntnisquelle fliesst das Unendliche im eigentlichen und strengsten Sinne des Wortes.” 32 As Kaulbach says in the introduction to NS, p. xxxii. 33 Review of Cantor, p. 163, in the review cited in note 4 above. Cf. Cantor, Gesammelte Abhandlungen, pp. 410–411. I have not been able to identify the precise passage of Cantor Frege has in mind. 34 Cantor, Gesammelte Abhandlungen, p. 443. 35 I do not know of any remarks by Frege on any paradox other than Russell’s, in particular on Cantor’s or Burali-Forti’s. But on ordinal numbers cf.: “We do not yet have a general view of the significance which order types would then acquire for mathematics. They would perhaps enter into an intimate connection with the rest of mathematics and exert a fertilizing influence on it (wirken befruchtend auf sie ein). I would not want to exclude this possibility” (Review of Cantor, p. 165, trans. p. 181). [It is not strictly true that Frege comments on no other paradoxes, because he does remark on what we would call the sorites paradox. But he seems to regard it as a simple fallacy.] 129

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The “genetic” point of view which leads to the consequence that the totality of sets is absolutely potential belongs, of course, to the first of the two suggestions with which we began. I do not see how to make sense of set theory without some version of it. Although it may be separable from the idea which Frege so sharply criticized, that a set is constituted by its elements, it seems equally alien to Frege. Perhaps that is why no solution to the paradoxes ever satisfied him.

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Since the two essays on Frege reprinted here were written, a lot has happened in the study of Frege and the development of his ideas. But I will limit the scope of my postscripts to developments that bear directly on what is said in these essays. The first part of the present essay is structured around two ideas of what a set may be, Frege’s conception of extension and the conception of a set as constituted by its elements. I explored such ideas further in systematically motivated writings, beginning with “What is the iterative conception of set?”1 The suggestion made above that “neither offers by itself the basis of a complete account of the nature of sets,” in particular adequate to make plausible the standard axioms of set theory, is developed and defended in chapter 4 of Mathematical Thought and Its Objects. By then I distinguished two versions of the second conception: sets as collections, in some way directly constituted by their elements, and sets as pluralities, where the idea is motivated by plural constructions in natural language. However, the second of these is not especially relevant to Frege. As should be clear from chapters 3 and 4 of the book just mentioned, I now attach less significance to conceptions of “the nature of sets” than I did in the mid-1970s. The second half of the essay is devoted to tracing the development of Frege’s view of the notion of extension from his learning of Russell’s paradox to his death. Tyler Burge filled in the story from the Foundations of Arithmetic to 1903.2 In the appendix to volume 2 of Grundgesetze, Frege writes that he has never concealed from himself the fact that his 1

Essay 10 of Mathematics in Philosophy. It was in fact written between the writing of the present essay and its publication. 2 Burge, “Frege on Extensions of Concepts.” 131

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Basic Law V is not as evident as the other basic laws of his system or as evident as must really be demanded of a logical law.3 Burge analyzes two well-known passages from the Foundations, which seem to cast doubt on Frege’s commitment to the use of the notion of extension of a concept in defining cardinal number. In the first (p. 80n.), Frege says that he believes that in the definition “for ‘the extension of the concept’ we could write simply ‘the concept’.” In the second, from his summing up of the book, Frege writes concerning his definition, In this we take for granted the sense of the expression “extension of the concept.” This way of overcoming the difficulty will not win universal applause, and many will prefer to remove the doubt in question in another way. I attach no decisive importance to bringing in the extension of a concept. (p. 117)4 Burge argues in essence that these passages do show uncertainty on Frege’s part about his reliance on extensions and on the sort of inference that later would be justified by Basic Law V. About the first and more extensive passage, however, he adopts the view that Frege is not there suggesting something substantively different from relying on extensions. I would put the matter thus: As Frege emphasized later, ‘the concept F ’ designates an object. Since Frege worked with an extensional language, ‘the concept F ’ will obey the same laws as ‘the extension of the concept F ’. Thus a basis for distinguishing them is lacking. There is some evidence that Frege did consider alternatives to relying on extensions about the time of the Foundations.5 Given what happened after the discovery of Russell’s paradox, both in Frege’s own thought and in the work of others attempting to revive and develop logicism, it is not surprising that Frege did not find another promising direction or, apparently, attempt to pursue alternatives very far. Burge’s analysis gives confirmation to a retrospective comment by Frege made in 1910. In a note to Jourdain’s article on his work, after the remarks quoted above (p. 125), Frege writes:

3

Frege refers to the preface to volume 1, p. vii, which does not explicitly say that Basic Law V is not as evident as it should be but does note that it is the place where the soundness of his system is most likely to be questioned. 4 I follow Burge’s modifications of Austin’s translations, “Frege on Extensions of Concepts,” p. 274. 5 Ibid., pp. 280–282. 132

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Only with difficulty did I resolve to introduce classes (or extensions of concepts), because the matter did not seem to me quite secure—and rightly so, as it turned out. The laws of numbers are to be developed in a purely logical manner. But numbers are objects, and in logic we have only two objects, in the first place: the two truth-values. Our first aim, then, was to obtain objects out of concepts, namely, extensions of concepts or classes. By this I was constrained to overcome my resistance and to admit the passage from concepts to their extensions.6 That might suggest a more charitable attitude toward Frege’s response to Cantor’s review of the Foundations than is adopted by William Tait, so far as I know the only person to have discussed the exchange at length.7 Tait, building on his own study of Cantor’s Grundlagen einer allgemeinen Mannigfaltigkeitslehre of 1883,8 argues persuasively that Cantor saw the fatal flaw in Frege’s approach, which came fully to light with Russell’s discovery that the system of Grundgesetze is inconsistent. The key passage (quoted by Tait) is the following: He [Frege] entirely overlooks the fact that the ‘extension of a concept’ in general may be quantitatively completely indeterminate. Only in certain cases is the “extension of a concept” quantitatively determinate. Then it has, if it is finite, a definite number, or, in the case it is infinite, a definite power.9 The cases of “quantitative indeterminacy” that Cantor had in mind were very likely, as Tait says, the totalities of cardinals and ordinals. We don’t know how carefully Frege studied Cantor’s monograph; the two citations he gives are not very informative on this point.10 A careful reader would have seen that Cantor’s view of the matter was as Tait says. But it is not at all obvious how Frege could have incorporated it into his 6

Jourdain, “Development,” p. 251n.69. For the German original see WB, p. 121. Jourdain translates Begriffsumfänge as “extents of concepts”; I have substituted the now standard “extensions.” 7 “Frege versus Cantor and Dedekind,” pp. 243–246. 8 See his “Cantor’s Grundlagen and the Paradoxes of Set Theory.” 9 Cantor, Review of Grundlagen, p. 440 in Gesammelte Abhandlungen, translation from Tait, “Frege versus Cantor and Dedekind,” p. 244. 10 Foundations, pp. 74, 97. The first refers to the definition of equality of power (in his own language, cardinal number) in terms of one-to-one correspondence, the second to Cantor’s introduction of transfinite numbers. 133

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own conceptual apparatus, particularly in 1885 when that apparatus was incompletely developed. Tait writes, “It is easy to misunderstand Cantor’s review because, for many, the primary question is to be formulated by asking whether a given totality is a set. If it is, then it has a cardinal number” (p. 245). He then argues that this was not Cantor’s point of view in 1883. That point of view led Cantor to say that the concepts of number and power had to be presupposed for a “quantitative determination” of extensions. That would have been read by anyone at the time as an objection to the procedure of giving a definition of cardinal number in terms of extensions, a claim of circularity that a number of writers closer intellectually to Cantor than to Frege, such as Dedekind and Zermelo, would have objected to, once they took the notion of extension to be doing the work of the notion of set. This is not a reproach to Cantor, who had already reached a level of insight into problems of infinity and absolute totality that did not become widespread until well into the twentieth century. But it does indicate that it is setting the bar unreasonably high to expect Frege to grasp at the time the problem that Cantor was pointing to on the basis of what Cantor wrote.11 Tait closes his discussion with the following remark: There tends to be a picture of Frege as a tragic victim of fate: by his very virtue, namely, his insistence on precision, he committed himself explicitly to a contradiction that was already implicit in mathematical thought. But in fact his assumption in the Grundgesetze that every concept has an extension was an act of reck11

However, it has often occurred to me that Halle and Jena, where Cantor and Frege lived, were not far away from each other. Why did they not meet to discuss the matter more thoroughly? I might remark that some readers of Cantor’s review have interpreted him to be saying that according to Frege the number belonging to the concept F is the extension of the concept F. That would of course be incorrect. Tait does not mention this interpretation, and it is not relevant to his point. This reading of Cantor was probably encouraged by Frege’s statement in his reply (“Erwiderung”): These remarks would fit very well, and I would recognize them as wholly justified, if it followed from my definition, for example, that the number of moons of Jupiter was the extension of the concept “moon of Jupiter.”

Frege’s conditional way of putting the matter suggests that he himself did not read Cantor in this way. (This translation from Frege’s reply is my own.) 134

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lessness, forewarned against by Cantor in 1883 and again, explicitly, in his review of 1885. (p. 246)12 One could object, as suggested above, that it was not directly the assumption that every concept has an extension that Cantor warned against. How reckless Frege’s assumption was in the context of the time is not so easy to say; a deep student of Cantor’s discussion of the transfinite would not have made it, but how many such students were there, apart from Cantor himself? Even Dedekind, in par. 66 of Was sind und was sollen die Zahlen?, gave an argument that assumed as a set something that by our lights (and already Cantor’s) was not.13 History has vindicated Cantor. If one asks what Frege should have done, had he fully understood Cantor’s point, it is hard to see that any measure would have been successful that would have preserved his logicism. The neo-Fregean solution grew out of analyses of Frege’s own arguments leading up to his definition of cardinal number. It can thus be argued that it would have been the best option for Frege himself. However, even its adherents do not claim that the main axiom of Frege arithmetic, the so-called Hume’s Principle, is a logical principle. The idea that Frege was “a tragic victim of fate” survives Tait’s analysis, even if we grant the charge of recklessness. But we should recall that according to tradition, tragic heroes are brought down by a tragic flaw in a heroic character. Frege did make logicism a precise thesis, chiefly by his development of second-order logic. He also made it a falsifiable thesis, and it was not just bad luck that his thesis was falsified. Opinions will differ about what the decisive “tragic flaw” was. Tait probably thinks that Frege’s way of reading his contemporaries, which is the main target of his paper, would be an important part of the story. In this he is probably right. To turn to the second part of the paper: One piece of evidence I relied on was Rudolf Carnap’s statements about the lectures of Frege that he attended in the period 1910–1914. In the meantime his shorthand notes have been transcribed and published. We can thus partly 12

Few today would accept the claim that the contradiction was “already implicit in mathematical thought.” But it should be remembered that such a view was rather widely held in the early twentieth century. 13 To be sure, Dedekind’s argument, attempting to prove that there is an infinite set, was not as central to his enterprise as Basic Law V was to Frege’s. 135

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resolve the puzzles that Carnap’s statements gave rise to. The third of the lecture courses Carnap attended was “Logik in der Mathematik” in the summer semester of 1914. Frege’s own text for those lectures has survived and was published in Nachgelassene Schriften, as noted above. As noted there, Frege concentrated in these lectures on what could be done with his “fundamental logic,” a version of second-order logic, without introducing the notion of extension. Thus he bypassed the whole problem of the paradox. But it may not have been especially relevant to the aim of that course. My conjecture that Frege followed the same policy in the lectures “Begriffschrift I” and “Begriffschrift II” is confirmed by Carnap’s notes. The first series deals with truth-functional logic and the beginnings of quantificational logic, and the idea that mathematics might be developed within the logic being developed is not even suggested.14 In “Begriffschrift II” Frege turns quickly to mathematical examples, starting with the continuity of a function, showing how to define them in his formal language. He gives two proofs, the second rather lengthy (of the uniqueness of the limit of a function as its argument approaches infinity), stating explicitly a number of simple mathematical theorems that are assumed. But he says nothing about how these theorems might be proved or what assumptions would be needed to prove them. As Gabriel points out in his introduction (p. v), Frege uses only Basic Laws I–III and the rules of inference from Grundgesetze, and the notion of extension is not introduced. He does, however, point out that an expression of the form “the concept . . .” designates an object, but that is only in informal remarks. Why was Carnap convinced that Frege never gave up logicism? It was not on the basis of discussion with Frege; he remarked that Frege’s lecturing style precluded discussion, and it appears that he never exchanged a word with Frege.15 Frege’s silence in the lectures about the 14

Thus the notes do not bear out Carnap’s statement quoted above (p. 127) that at the end of “Begriffschrift I” Frege indicated that the logic he had introduced could serve for the construction of the whole of mathematics. The same would be true of “Begriffschrift II.” 15 Carnap’s friend Wilhelm Flitner, who attended “Begriffschrift I” with Carnap, explicitly says this; see Erinnerungen, p. 127, quoted in Kreiser, Frege, p. 277, and in English in Reck and Awodey, Frege’s Lectures, p. 22. In 1921 Carnap did write a (now lost) letter to Frege, asking for a copy of “Über Begriff und Gegenstand.” See WB, p. 16. 136

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difficulty created by Russell’s paradox no doubt played a role. But I would guess that the more decisive reason was Carnap’s subsequent reading of Frege’s published writings and his own more mature views. In his early career he evidently thought that the view had been successfully reconstructed in Principia Mathematica. Later, for example in Logical Syntax, he assimilated mathematics to logic as playing the same role in the edifice of science, without the question of a reduction being especially important.

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The publication of this volume1 of Frege’s correspondence completes the project of publishing the Frege Nachlass, begun by Heinrich Scholz in 1935, though because of losses during the Second World War, what is published in this volume and its predecessor2 falls short of what Scholz planned. In view of the extensive searches that the custodians of the Frege Archive have made for additional letters and other materials, it seems unlikely that the Frege corpus will be much augmented in the future.3 We now have what amounts to an edition of Frege’s collected works, which compares favorably with what is available for other major figures in the formative period of modern logic and “analytical philosophy.”4 From now on, it should be standard procedure in scholarly works on 1

Gottlob Frege, Wissenschaftlicher Briefwechsel, edited with introduction and notes by Gottfried Gabriel, Hans Hermes, Friedrich Kambartel, Friedrich Kaulbach, Christian Thiel, and Albert Veraart, volume 2 of Nachgelassene Schriften und Wissenschaftlicher Briefwechsel, edited by Hans Hermes, Friedrich Kambartel, and Friedrich Kaulbach (Hamburg: Meiner, 1976). [As noted in Essay 5, note 19, hereafter cited as WB. Translations from this volume are my own, although I cite the published (not complete) translation.] 2 Nachgelassene Schriften, hereafter cited as NS. 3 See the notes on individual correspondents as well as the editors’ introduction. The most up-to-date account of the Frege Nachlass and its fate, and of attempts to uncover more materials, is Albert Veraart, “Geschichte des wissenschaftlichen Nachlasses Gottlob Freges.” [But see the Postscript below on Frege’s letters to Wittgenstein.] 4 Begriffschrift und andere Aufsätze; Die Grundlagen der Arithmetik [in particular now Thiel’s edition]; Grundgesetze der Arithmetik; Kleine Schriften, hereafter cited as KS; Nachgelassene Schriften (note 2 above), and the volume under review. The editor’s task is simplified by the fact that no publication of Frege’s was reprinted during his own lifetime. Contrast the case of Bertrand Russell, many of whose books have been in print more or less continuously. This is no doubt one 138

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Frege to cite this edition, or at least to cite in a way which makes it easy for a reader to locate a passage in this edition.5 The present volume includes all surviving letters by or addressed to Frege that in the opinion of the editors of the Nachlass are of scientific relevance. The arrangement is by correspondent (in alphabetical order) and for each correspondent chronological. All letters are given in their original languages.6 For each correspondent a complete list is given of the letters known to have been written, where possible with an indication of the content of letters whose texts are lost. This applies mainly to letters acquired by Scholz before the war, which were destroyed in bombing of the university library in Münster in 1945.7 For each correspondent of the causes of the fact that critical editions of works of Russell are almost nonexistent. [At the time this was written, no volumes of the Collected Papers of Russell had appeared. That I wrote “almost nonexistent” may indicate that I was aware of the project, but probably not of its scope.] 5 It should be remarked that Frege’s three books are in the Olms edition reproduced by photo-offset, with the original pagination. The pagination of the German text of the Grundlagen printed with Austin’s translation is the same as that of the original. The original pagination of the essays reprinted in Angelelli’s two collections is given in the margins. Since it is also given in Günther Patzig’s two collections and in some translations (including the Geach and Black collection), it may be best to cite work published in Frege’s lifetime by the original pagination. We follow that policy in the present review. It may seem pedantic to dwell on this issue. I do so partly because there seems to be an increasing tendency among American writers on historical figures in philosophy to cite only currently published translations, thus adding to the reader’s difficulty in locating the original. [Two collections published after the original publication of this essay, Collected Papers and Beaney, The Frege Reader, continue to give the original pagination. This is also done in Thiel’s edition of Grundlagen. Unfortunately this was not done in the translations of the posthumous works, Posthumous Writings and Philosophical and Mathematical Correspondence.] 6 P. E. B. Jourdain wrote in English, and several correspondents wrote in French. 7 The story is well known that Scholz deposited for safe-keeping in the university library all the original Frege papers that he possessed, and apparently also some copies that he had made. This material was all destroyed by bombs, but some copies that Scholz kept in his home for his own use survived, and it is on these that the Nachgelassene Schriften is almost entirely based. See Veraart, op. cit., pp. 62– 70. What is lost can be gathered from the catalogue appended to Veraart’s article. It does seem that Scholz chose to keep copies of papers he thought more important. For the correspondence, there are fortunately more sources, especially the materials (consisting mostly of letters addressed to Frege) deposited after Frege’s death in what was then the Preussische Staatsbibliothek (now Staatsbibliothek 139

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there is an introduction giving a brief identification of the correspondent and information about the occasion of the exchange, with some discussion (varying in extent) of the content of the correspondence. The notes supply numerous useful references. The editing of the whole is done with exemplary thoroughness and attention to detail. With the correspondence with P. E. B. Jourdain, the book also includes the German text of Frege’s notes to Jourdain’s expository article on his work.8 It goes without saying that this publication enlarges or corrects our picture of Frege’s thought on many points. However, it contains few surprises, even for someone whose knowledge of Frege is confined to what is published. With some exceptions, these texts do not have the same importance as those collected in Nachgelassene Schriften. Some of the letters have been published previously, and published work on Frege contains considerable discussion of some of the correspondence. In particular, this is true of the exchanges with Hilbert and Russell, which are the most extensive and informative of what survives. Some of the correspondence with Hilbert was published in the 1940s by Max Steck.9 I shall not try to add here to what has been written about the “Frege-Hilbert controversy.”10 The Russell correspondence has also been previously discussed,11 and Russell’s opening letter, which announced his paradox, and Frege’s reply are well known.12 I shall add some comments about their correspondence below. Preussischer Kulturbesitz). [The latter is as of 1982; it is now Staatsbibliothek zu Berlin—Preussischer Kulturbesitz.] See Veraart, op. cit., pp. 60–61. Relevant details are given in the editors’ introduction and other editorial apparatus of WB. 8 “The Development of the Theories of Mathematical Logic and the Principles of Mathematics: Gottlob Frege.” The article is reprinted in full as an appendix to WB. The notes in English, with an indication of context, were reprinted in KS. The history of the German text is complicated (see the note in WB, pp. 114–115). However, since the Frege Archive contained two copies of a draft, Angelelli’s statement that the notes were “known only in the English version published by Jourdain” (KS, p. 334) was misleading even on the basis of information available at the time (1967). 9 See the reprints in KS, pp. 395–442. 10 For example Resnik, “Frege-Hilbert Controversy,” and Kambartel, “Frege und die axiomatische Methode.” 11 Sluga, “Frege und die Typentheorie.” 12 They appeared in English in van Heijenoort, From Frege to Gödel, pp. 124–128. 140

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The correspondents for whom there is a surviving exchange of some extent and substantive content are Hugo Dingler, Hilbert, Edmund Husserl, Jourdain, Giuseppe Peano, and Russell. In the cases of Louis Couturat, A. R. Korselt, and Moritz Pasch, only the other party’s letters are known. There are some other isolated letters of interest. Two tantalizing lost correspondences are those with Leopold Löwenheim and Ludwig Wittgenstein. About the first, Scholz and Bachmann reported in 1936 that Löwenheim had convinced Frege that a viable “formal arithmetic” could be constructed (see p. 158 of the volume under review). This correspondence might have helped to clear up some of the obscurity surrounding the development of Frege’s philosophy of arithmetic after he learned of Russell’s paradox.13 Wittgenstein and Frege corresponded in the period 1913–1919 about criticisms by Wittgenstein of Frege’s views, and about the Tractatus. Wittgenstein’s letters seem to have come to Scholz (see p. 265) but were among the materials destroyed. Wittgenstein seems to me to have been less than fully cooperative with Scholz’s efforts to obtain Frege’s letters; he wrote to Scholz in 1936 that the letters he had were of purely personal, not philosophical, content and were of no value for a collection of Frege’s writings (ibid.). But it seems quite clear from what is known about Wittgenstein’s letters that Frege wrote some substantive replies. In any case, neither the letters Wittgenstein may have been referring to nor any others have been found in his posthumous papers.14 The correspondence does a lot to fill out our picture of Frege’s relations with the scientific and philosophical world of his time. Although a number of notable figures were among the correspondents, the impression of Frege as somewhat isolated is not overcome. We even see difficulties in getting his work published; for the essay “Booles rechnende Logik und die Begriffschrift” (first published in NS) we have rejection letters from three journals (pp. 134, 254, 259). Wilhelm Koebner, publisher of the Grundlagen, offers to publish Funktion und Begriff at Frege’s expense (pp. 138–139).15 Before Russell, those who attribute 13

On the closely related question of the development of his views on the concept of extension, see my “Some Remarks on Frege’s Conception of Extension” (Essay 5 of this volume) and also below. 14 [Of course, letters of Frege to Wittgenstein were found after all. See the Postscript to this essay.] 15 Since the pamphlet was published instead by H. Pohle, perhaps Frege obtained better terms. 141

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to Frege’s writings an important influence on their own thinking are minor figures. However, Moritz Pasch, who contributed importantly to the axiomatization of geometry, acknowledges considerable kinship of outlook with Frege. Peano addresses him with great politeness and encourages him to contribute to his journal (pp. 181–186, 180, 193, trans. pp.  112–118, 111–112, 125). Frege’s only publication in the Rivista di matematica was his letter replying to Peano’s review of Grundgesetze (pp. 181–186, already reprinted in KS). It seems unlikely that Frege would have been ready to participate in Peano’s collective enterprise, so long as his own conceptions were not better understood. Peano asked Frege more than once to publish something that would help someone accustomed to his own symbolism to understand Frege’s. Frege evidently saw too many difficulties in the conceptual basis of Peano’s symbolism to do this. In the remainder of this review, I shall point out some substantive points where the correspondence gives new information and make some general remarks about the correspondence with Russell. Sense and Reference. In a letter to Husserl of May 24, 1891, Frege diagrams his theory of sense and reference, making clear that the distinction is to apply to concept words as well as proper names. This shows that the application to concept words was not an afterthought after “On Sense and Reference,” since otherwise the earliest text that is explicit on the point is “Ausführungen über Sinn und Bedeutung” (NS, pp. 128–136, trans. pp. 118–125), written after “On Sense and Reference.” The letter does not make clear, as Frege does in “Ausführungen” (NS, p. 129n., trans. p. 119n.), that the sense of a concept word is itself unsaturated. What is a proper Fregean view of multiply embedded oblique contexts has long been controversial. In a letter to Russell (December 28, 1902), Frege seems to commit himself to the interpretation of Carnap and Church. He speaks there of “indirect reference of the second degree” (p. 236, trans. p. 154). He is considering the expression the thought, that the thought, that all thoughts in the class M are true, does not belong to the class M,16

16

On the context in which this example arises, see the discussion below of the corrspondence with Russell. 142

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in which the sentence “All thoughts in the class M are true” is in a doubly indirect context.17 In two letters to Husserl in 1906, Frege makes remarks relevant to the question when two expressions have the same sense. In the first he says that “equipollent” sentences express the same thought (p. 102, trans. p. 67). Frege is commenting on a paper which Husserl had sent, in which Husserl criticized a claim of Anton Marty’s to the effect that “If A then B” and “Not both A and not B” agree in sense.18 Frege asserts that they do, on the ground of their truth-functional equivalence.19 In a text written in the same year, Frege applies the term “equipollence” to the relation of two sentences A and B that obtains when whoever recognizes the content of A as true must without further ado (ohne weiteres) also recognize that of B as true, and conversely, whoever recognizes the content of B as true must also immediately (unmittelbar) recognize that of A, where it is presupposed that there is no difficulty in grasping the contents of A and B. (NS, p. 213, trans. p. 197) Here also he makes clear that equipollent sentences express the same thought. In the second letter Frege gives another criterion: In order to decide whether the sentence A expresses the same thought as the sentence B, only the following method seems to me to be possible, where I assume, that neither of the two sentences contains a logically evident part (Sinnbestandteil). If both the assumption that the content of A is false and that of B is true, and the assumption that the content of A is true and that of B false, lead to a logical contradiction, which can be determined without knowing whether the content of A or B is true or false, and without using other than purely logical laws, then nothing can belong to the content of A, insofar as it can be judged true or false, which would not also belong to the content of B. . . . Equally, under our assumption, nothing can belong to the content of B, 17

I am indebted to Terence Parsons for pointing out to me the significance of this passage. 18 References are given in the editor’s notes 2 and 4 to this letter. The passage of Husserl occurs in Aufsätze und Rezensionen, 1890–1910, p. 255 (hereafter cited as AR). 19 Cf. “Gedankengefüge,” pp. 39, 40, 42, 45 (KS, pp. 381, 382, 383, 387). 143

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insofar as it can be judged true or false, which would not also belong to the content of A. (pp. 105–106, trans. p. 70) Note that the criterion cited from NS is epistemic, that of the letter to Husserl logical. Given the proximity in date of the two texts, it is natural to conjecture that they were meant to agree. Given the difficulty of deciding questions of logical derivability, this is very questionable, although cases where such difficulties arise might be said to be cases in which there is difficulty in grasping the contents of the sentences involved. But if Frege had that in mind, he must surely have recognized that his criterion in the passage from NS would be of limited application.20 The criterion in the letter to Husserl seems to me very difficult to reconcile with the several places in Frege’s writings where the two sides of simple arithmetical identities such as “2 + 3 = 5” are said to differ in sense. It seems not to be a matter of his having given up an earlier view. Such passages occur in the Russell correspondence in 1902–1903 (pp.  232, 235, 240, trans. pp. 149–150, 152, 157–158) and in a letter to Paul F. Linke of 1919 (p. 156, trans. p. 98). I do not believe that Frege has a consistent position about identity of sense. Clearly some of the different views that have arisen later are already suggested by him. There is no direct evidence that he saw the tensions between them. But clearly the two criteria of 1906 are formulated with some care; in the logical criterion of the letter to Husserl, Frege evidently wanted to avoid saying that if A is logically true (or known to be so), then the conjunction of A and B expresses the same thought as B, for any B. Frege and Husserl. The exchange of letters between Frege and Husserl in 1891 reveals something of how they looked at each other’s views at that time.21 However, the exchange does not directly touch on psychologism and therefore sheds little light on the most controversial question about the Frege-Husserl relationship, how far Husserl’s turn away from psychologism may have been due to Frege’s influence. Frege’s opening letter contains an exposition on sense and reference. In view of the importance in Husserl’s philosophy of a sense-reference scheme paralleling Frege’s, one could be momentarily tempted to sup20

Cf. van Heijenoort, “Frege on Sense-identity.” Of their later correspondence, in 1906, Husserl’s letters are lost (see pp.  105, 107). 21

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pose that Husserl learned the distinction from this source. This is evidently not so; Frege is commenting on a similar scheme in papers Husserl had sent him.22 Anyone who has pondered Husserl’s relation to Frege will be struck and probably shocked by the remark about Frege which Husserl wrote to Scholz in 1936 (best left untranslated): “Er galt damals allgemein als ein scharfsinniger, aber weder als Mathematiker noch als Philosoph fruchtbringender Sonderling.”23 This appears to express Husserl’s own view in his old age, but in earlier times he had expressed himself more warmly about Frege; see the praise of the Grundlagen in his 1891 letter (p. 99, trans. pp. 64–65) or the recommendation of Funktion und Begriff in a review published in 1903 (AR, p. 202). The editor of the Frege-Husserl correspondence, Gottfried Gabriel, suggests that Husserl’s remark to Scholz implies a refusal to acknowledge a significant influence of Frege on him.24 22

These included Husserl’s review of Schröder’s Vorlesungen über die Algebra der Logik and “Der Folgerungskalkül und die Inhaltslogik,” both reprinted in AR. See for example the passage from the review, AR, pp. 11–12, cited by J. N. Mohanty in his discussion of this issue in “Husserl and Frege: A New Look at Their Relationship,” p. 53. That Frege is not the source for Husserl’s making this distinction was remarked on by Dagfinn Føllesdal, “An Introduction to Phenomenology for Analytic Philosophers,” p. 421. Professor Føllesdal informs me that the same remark occurred in an earlier version of the paper published in Norwegian in 1962. 23 Cited on p. 92. For the full text of the letter, see Veraart, op. cit., p. 104. 24 However, in 1935 Andrew Osborn asked Husserl about Frege’s influence on the abandonment of psychologism; Husserl is reported to have “concurred” but also to have mentioned Bolzano. See Schuhmann, Husserl-Chronik, p. 463. I owe this reference to Føllesdal. I do not know whether the question of the reverse influence has been discussed. One could not expect much, in view of the maturity of Frege’s views when Husserl began to publish significant work. Husserl’s sending Frege his review of Schröder seems to have stimulated the latter to carry through his own plan to write a critique of Schröder (see p. 94 and Frege, “Kritische Beleuchtung einiger Punkte in E. Schröders Vorlesungen über die Algebra der Logik,” in KS). Possibly Husserl stimulated Frege to clarify his position on the issue of the time between Inhaltslogik and Umfangslogik (roughly, the intensional and extensional point of view); see “Ausführungen,” NS, pp. 128–136, trans. pp. 118–125. Husserl was basically intensionalist (see “Der Folgerungskalkül und die Inhaltslogik”); Frege gave points to both sides but is of course at bottom extensionalist. (On this aspect of Husserl, I am indebted to my student Nathaniel S. Heiner.) A more speculative question is whether Frege read the Logische Untersuchungen and whether it may have influenced his late writings. So far as I know, there is no direct evidence that he knew the book. The title and a little of the content of his 145

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Apart from the question of influence, there are undoubtedly important convergences between Frege and Husserl. But one should keep in mind their limitations. In the 1891 letter, Husserl expresses regret that he has not had time “to form a clear picture of the nature and extent of your original Begriffschrift” (p. 99, trans. p. 64). In my opinion, Husserl never shows a grasp of quantificational logic and its significance, although his program of a “pure theory of manifolds” can be read as prophetic of model theory.25 Russell’s “On denoting” of 1905 is in this respect in advance of everything Husserl wrote on the philosophy of logic. More fundamentally, in spite of his good opinion of Funktion und Begriff, Husserl could have benefited from greater appreciation of the treatment of predication that goes with Frege’s theory of functions and objects.26 last series of published essays are suggestive. “The thought” goes more deeply than Frege’s earlier writings into the relation of thoughts to ideas and the mind generally; more specifically, it takes account of indexical expressions, which Frege had not done in the earlier writings, but which Husserl discusses in the Logische Untersuchungen (1st Investigation, §26; hereafter cited as LU). However, the “Logik” of 1897 (NS, pp. 137–163, trans. pp. 126–151) contains both an extended discussion of the relation of thoughts to the subjective and some remarks about indexicals. It is evidently a prototype of Frege’s “Logische Untersuchungen.” Moreover, Frege could also have borrowed this title from a book of Trendelenburg, of whose existence he probably knew; see Sluga, Gottlob Frege, p. 49. 25 LU, Prolegomena, §§69–71. 26 Cf. the telling criticisms of Husserl’s treatment of predication in Ernst Tugendhat, Vorlesungen zur Einführung in die sprachanalytische Philosophie, esp. lectures 9 and 10. This admirable and lucid book deserves to be better known among English-speaking philosophers. [The publication of an English translation does not seem to have made it much better known.] Something like Frege’s “unsaturatedness” occurs in another place in the Logische Untersuchungen, in Husserl’s conception of nonindependent parts (3rd Investigation, §§8 ff.). This conception is applied to the theory of meaning when Husserl discusses “nonindependent meanings” (4th Investigation, §§5–6), where Husserl even uses the Fregean term ergänzungsbedürftig (vol. 2/1, p. 309). Husserl seems to miss construing such meanings as functions, because according to his general conception a “nonindependent content” is something that can only exist as part of a larger whole (ibid., p. 311). But it should be pointed out that in talking of senses Frege also used the language of whole and part. Since for Husserl a nonindependent part is connected with the other parts by a law (ibid.; cf. 3rd Inv. §10), there seems to be an intrinsic correspondence between nonindependent parts and functions. When the Polish logicians such as Ajdukiewicz came to develop Husserl’s conception of logical grammar into what is now called categorial grammar, they readily interpreted certain categories in functional terms. 146

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With respect to the theory of sense and reference, we should keep in mind that to make such a distinction was not in itself especially original or significant. Certainly the parallels between Frege and Husserl go further, but they also have their limits. For Frege, it is a fundamental postulate that the reference of a whole expression should be a function of the references of its parts. Sense and reference are connected by this, in that it requires that in oblique contexts the reference of an expression should be its ordinary sense. The latter idea does not occur clearly in Husserl, and certainly not in this systematic context. Although Husserl seems to have viewed meanings as composing functionally, he did not have the same conception of reference (for him, Gegenstand), and moreover he did not make the same intimate connection between reference and truth. The Correspondence with Russell. As one would expect already from the identity of the correspondents this correspondence is of unique value. Of Frege’s extended correspondences, it is much the best preserved.27 It begins in a dramatic manner, with the letter in which Russell informs Frege of the contradiction in his system. Frege’s immediate recognition that the paradox had shaken his system and his whole approach to the foundations of arithmetic makes the correspondence unique in another respect. It is characteristic of Frege to expound his views in letters in a somewhat magisterial fashion. Though this tone is not absent from the letters to Russell, he is here more often tentative and exploratory. The paradox and ideas for resolving it are the central theme of the correspondence. However, many of the main ideas of both are discussed. In the philosophy of logic, we have a confrontation of the mature Frege with a Russell who is taking his first steps beyond the position of the Principles of Mathematics. Frege criticizes Russell’s formulations with respect to use and mention, function and object, and sense and reference. Only about the second does he appear to convince Russell, and indeed that is the issue among the three where Russell’s mature position is closest to Frege’s. 27

Russell’s letters were among those given by Alfred Frege to the Preussische Staatsbibliothek (see note 6 above). Russell gave Frege’s letters to Scholz, who responded to Russell’s request for copies by sending him photocopies (see WB, p. 200), which then survived although the originals were lost. In keeping with its importance, the correspondence is provided with an extended analytical introduction and especially helpful notes by the editor, Christian Thiel. 147

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Russell states the paradox for classes by saying that there is no class as a whole (als Ganzes) of classes that are not elements of themselves. “From this I conclude that under certain circumstances a definable set does not form a whole” (p. 211, trans. p. 131). He is very likely thinking in terms of the distinction made in the Principles between a “class as many” and a “class as one”;28 this seems clear from what he says when he returns to the matter in the letter of July 10, 1902: A class which consists of more than one object is in the first instance not one object, but many. Now an ordinary class does form one whole; for example the soldiers form the army. But this seems to me not to be a necessity of thought; however it is essential if one is to use the class as a proper name. Therefore I think I may say without contradiction that certain classes (more exactly, those defined by quadratic forms)29 are only multiplicities (Vielheiten) and do not form wholes at all. Therefore false propositions and even contradictions arise when one views them as unities. (pp. 219–220, trans. p. 137) In the background here are surely Cantor’s informal explanations of the concept of set, for example the “definition” of 1895 of a set as “any collection M into a whole of definite, well-distinguished objects of our intuition or our thought.”30 The term Vielheiten is of course just the term that Cantor uses in his own discussion of the paradoxes in his 1899 correspondence with Dedekind,31 and Russell’s remark that paradoxical class abstracts define mere multiplicities that do not form unities parallels Cantor’s own statements in the correspondence about “inconsistent multiplicities.” It is very doubtful that in 1902 Russell knew the Cantor-Dedekind correspondence, although a couple of years later he must have learned something of its content from Jourdain. The “theory of limitation of size” discussed by Russell in 1906 is, I think, Cantor’s proposal of 1899 filtered through Jourdain’s understanding of it.32 28

Russell, Principles of Mathematics, pp. 68, 76, 102, and elsewhere. See p. 215 (trans. p. 133) and Russell, Principles of Mathematics, p. 104. 30 Cantor, Gesammelte Abhandlungen, p. 282. 31 Ibid., pp. 443–447. 32 See Russell, “On Some Difficulties.” In his discussion of the theory of limitation of size, Russell refers to papers of Jourdain which contain explicit references to Cantor’s letters. 29

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Frege’s reply (July 28) contrasts a class with a “system,” that is, a whole consisting of parts, in much the same terms as in his 1906 draft “Uber Schoenflies: Die logischen Paradoxien der Mengenlehre,” but in some respects more explicitly and vividly.33 Russell declares himself convinced by Frege’s criticism (August 8), and indeed the conception of a class as consisting of its elements does disappear from the surface of Russell’s thought on the subject. But characteristically he says: I still lack altogether the direct intuition, the direct insight into what you call Werthverlauf; it is necessary for logic, but for me it remains a justified hypothesis. (p. 226, trans. pp. 143–144) It is fairly far along in the correspondence (Frege’s letter of October 20, 1902) that Frege proposes the solution to the paradox that he presents in the appendix to volume 2 of Grundgesetze, which has come to be known as Frege’s Way Out. Russell seems to find it intuitively unconvincing, though in his first reply he says it is “probably correct” (p. 233, trans. p. 151).34 He raises some questions (pp. 233, 238, trans. pp. 151, 155) but the discussion of the proposal is not extensive. Soon Russell is pursuing another line (see below). One could perhaps sum up Russell’s unease by saying that intuitively the extension ε’ F(ε) of the concept F should have as its elements exactly those objects x for which F(x) holds, but the Way Out allows that F(ε’ F(ε)) hold, but ε’ F(ε) is never an element of itself.35 33

NS, pp. 196–197. trans. p. 181. Cf. Essay 5 of this volume, pp. 120–121. Cf. Russell, Principles of Mathematics, p. 522. 35 The editor (p. 238n.4) calls attention to a related difficulty that Frege answers in the appendix to volume 2 of Grundgesetze. This is that two concepts would under the Way Out have the same extension, and therefore the same number, although one more object (namely its extension) falls under one than the other. As concerns number, Frege replies (2:264) that he defines number as the number of the extension. He seems to say that nε’Φ(ε) is really the number belonging to the concept -ξ∩ ε’Φ(ε) and not that of Φ(ε). That seems to me to concede the objection rather than answer it, since what one can define in the system is the number of elements of the extension of a concept, and this may not be the number belonging to the concept, for example in the sense of the Grundlagen: if we were to define “the number belonging to the concept F” as nε’ F(ε), then the Way Out allows the existence of F and G which have the same number, but which are not gleichzahlig. Could this consideration have contributed to the disappearance of the Way Out from Frege’s writings after 1904? The same difficulty can arise in formulations of set theory in which the range of the first-order variables includes proper classes that cannot be elements of sets or 34

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More prominent in the letters are ideas related to the simple theory of types.36 Today, that theory seems to us a very simple and natural way of avoiding the set-theoretic paradoxes, and the interpretation of superficially different systems of set theory draws on the same hierarchical conception of sets or classes. It is perhaps something of a puzzle that the simple theory of types was so slow to emerge clearly from the research and discussion prompted by the paradoxes. The idea of it occurred to Russell very early on (letter of August 8, 1902; cf. Principles, appendix B), and not only to Russell: the same idea is set forth in a letter to Frege by Alwin Korselt in 1903 (p. 142, trans. pp. 86–87). There seems to have been some difficulty on the part of both Frege and Russell in actually envisaging a full theory on this basis. Frege interpreted the proposal as implying that classes are second-class “improper” objects, because they cannot be arguments of all first-level functions (p. 228, trans. p. 145); cf. Grundgesetze, 2:254–255, trans. pp. 128–129). Both parties seem to have had in mind at this stage what is now called a cumulative theory. Particularly when one considers functions as well as predicates, Frege found the complexity of the hierarchy daunting. Frege assumes that the distinction between a function and its course of values will be maintained in such a theory. There would then be an elaborate hierarchy of objects. Functions would have to be of different types because of the types of the objects that they take as arguments. Clearly Frege assumed that quantification over functions was still needed, so that the theory would have an additional complexity over and above that of modern formulations of the simple theory of types, which (in their extensional forms) either replace quantification of function and predicate places entirely by quantification over classes or functions-asobjects, or quantify function or predicate places directly and thus bypass and step from a concept to its extension, or from a function to its course of values. The latter type of theory could be seen as a development of Frege’s basic logic without the addition of extensions, by allowing functions of arbitrary levels, thus iterating Frege’s own step from first- to second-level functions. Such a theory recalls Russell’s later idea of a “no-class theory,” and indeed something like it is the idea that distracts Russell from Frege’s classes. For example the class {x: x is a proper class} would have number 0 although the predicate ‘x is a proper class’ is true of something. 36 Cf. Sluga, “Frege und die Typentheorie.” 150

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Way Out. On May 24, 1903, he writes that he thinks he has discovered that classes are completely superfluous (p. 241, trans. p. 158). However, Russell uses function symbols without their argument places and does not seem to have in mind the hierarchy of levels of functions that would be required. The result is that this proposal is shot down by Frege in his reply (pp. 243–245, trans. pp. 160–162), which, since it was made a year and a half later, finds Russell already convinced (p.  248, trans. p.  166). What one might call “Fregean type theory” (i.e., ωth order predicate logic, or, if one wishes, a corresponding theory of functions) does not really come to the consciousness of either correspondent.37 We have not yet considered Russell’s own reason for rejecting at this point his first proposal of a type theory. It is well known that later he thought that the introduction of the ramified theory was necessary in order to handle the semantic paradoxes. In the correspondence, he presents an interesting paradox which is also stated in the same connection in the Principles (p.  527). Let m be a class of propositions. Then “∀p(p ∈ m ⊃ p)” expresses their logical product. This proposition can belong to the class m or not. Let w be the class of propositions of this form which do not belong to the associated class m, i.e., w = {p : ∃m[p = ∀ q(q ∈ m ⊃ q). p ∉ m]}. Then if r is the proposition ∀p(p ∈ w ⊃ p), one has r ∈ w if and only if r ∉ w (p. 230, trans. p. 147). This paradox can certainly be stated in a form of the simple theory of types that allows quantification of sentence places. The latter can do

37

Alonzo Church in “A Formulation of the Simple Theory of Types” gives what seems to be the first precise formulation of a simple theory of types in which, for any types α and β, there is a type of functions from type α into type β. Church avoids the complication of unsaturatedness by using functional abstraction. In effect he replaces quantification over what Frege would have called higher-level functions by quantification over objects in a hierarchy of types. It is well known that in his published writings Frege confines himself to firstand second-level functions, with very few cases of individual third-level functions. However, in a draft reply to a letter of Jourdain of January 15, 1914, Frege speaks of his own, “Theorie der Funktionen erster, zweiter u. s. w. Stufe” (p. 126, trans. p. 78). The “u. s. w.” is not put into his mouth by Jourdain, who wrote of “your theory of ‘functions “erster, bzw. zweiter Stufe” ’.” But it is probable that the idea of such iteration comes from Russell; the context is a comparison of Frege’s conception of levels of functions with Principia. 151

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much of the work of a predicate of truth.38 However, to obtain the usual semantical paradoxes the language must be able to express some such notion as the relation of expression between a sentence or an utterance and a proposition. For example, one might formulate “What I am now saying is false” as “∃p(My present utterance expresses p. ∼ p).” Our present paradox is not exactly a semantical paradox and involves no such notion. To state it, all that is required is that sentences be treated as denoting objects of some type such that classes of objects of that type can be formed. Thus it seems to be the conception of sentences as standing for propositions that led Russell to formulate the paradox and to turn away at this point from the simple theory of types. It is not surprising that Frege, in replying, is led into a discussion of sense and reference (pp. 231–232, trans. pp. 149–150). But in order to obtain the contradiction, one has to assume (as Russell admits in his next letter, p. 233, trans. p. 151) that if ∀p(p ∈ m ⊃ p) is the same proposition as ∀p(p ∈ n ⊃ p), then m = n. Outside the context of a theory of propositional identity, this is not evident; Russell’s thinking it so may have rested on confusion of propositions with sentences. One can ask what happens to the paradox in a Fregean interpretation where the language is completely extensional and sentences denote truth-values. In that case m and n are just classes of truth-values, and Russell’s assumption is refuted by taking m empty and n containing only the True.39 Frege does not make this reply; he takes the more interesting interpretation where m is still a class of propositions (for him, thoughts). He points out that the phrase “the thought that all the thoughts in the class m are true” involves using “m” in an indirect context. But then what would be a constituent of the thought expressed by a sentence containing the phrase would be not the class m, but the sense of an ap38

Cf. F.  P. Ramsey’s famous remarks on truth in his “Facts and Propositions,” pp. 142–143. 39 A similar argument shows that the assumption can be refuted if we construe propositions as sets of possible worlds. In an obscure passage in his letter of May 24, 1903, Russell seems to take the assumption to be refuted by a theorem stated by Frege in the appendix to Grundgesetze, vol. 2 (p. 261, trans. pp. 132–133). It is not clear how Russell understands it, since he uses a notation for identity (between what he usually takes to be propositions) which is explained in the letter in a quite different context. It seems that the purport of Frege’s result is limited to his own interpretation of sentences as denoting truth-values and that Russell does not grasp this. 152

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propriate name of m (p.  236, trans. p.  153). Russell’s proposition r then involves a doubly embedded indirect context (see above). So Frege does not accept Russell’s formulation. Russell, with his belief that objects can be constituents of a proposition, is not much moved by Frege’s objection. It is, however, so far as I know, Frege’s most explicit comment on the possibility of quantifying into an indirect context. In this light, the subsequent history of the paradox is ironical. Russell’s assumption is plausible if one’s criterion of propositional identity is refined. Such criteria are given by Church in what he called Alternatives (0) and (1) for a logic of sense and denotation, which he constructed on the basis of Fregean views of quantifying in and embedded indirect contexts agreeing with those on which Frege’s objection is based. But then this very paradox suggested derivations of contradictions in Church’s first formulations.40 On sense and reference, it is not surprising that the correspondents did not understand each other very well. Russell already had the basic intuitions which distinguish his view of such matters from Frege’s, and he held to them in the face of Frege’s thorough criticism, but he did not yet have some of the ideas that would be needed for an effective defense. As elsewhere, Frege argues from the substitutivity of identity to the conclusion that the truth-value, and not the thought expressed, must be the reference of a sentence. To reply effectively to this argument, Russell would have to distinguish proper names from descriptions and then apply his analysis of descriptions. The former step is really already present in the Principles,41 but Russell does not really bring it to bear in the exchange with Frege, although he does argue in his last letter on the matter (December 12, 1904, p. 251, trans. p. 169) that for a “simple proper name” there is no distinction of sense and reference. But the theory of descriptions came to him only after the correspondence ended. Frege took up the issue once more, in a draft reply to a letter of Jourdain of January 15, 1914. Jourdain asked

40

Church, “A Formulation of the Logic of Sense and Denotation.” For the paradox see Myhill, “Problems Arising in the Formalization of Intensional Logic,” and Anderson, “Some New Axioms for the Logic of Sense and Denotation: Alternative (0).” Anderson shows that the difficulty still affects Church’s revised version of Alternative (0); see Church, “Outline of a Revised Formulation,” pp. 149–153. 41 In the discussion of “denoting,” ch. 5. 153

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whether, in view of what seems to be a fact, namely, that Russell has shown that propositions can be analyzed into a form which only assumes that a name has a ‘Bedeutung’ & not a ‘Sinn’, you would hold that ‘Sinn’ was merely a psychological property of a name. (p. 126, trans. p. 78) In reply Frege gives two arguments. First, in order for us to understand sentences that we have never heard, the sense of a sentence must be constructed of parts corresponding to the words. But the part of a thought corresponding to a name like “Aetna” cannot be the mountain Aetna itself. “For then each individual piece of solidified lava, which is a part of Mt. Aetna, would also be part of the thought, that Mt. Aetna is higher than Mt. Vesuvius” (p. 127, trans. p. 79). This remark is striking for the literalness with which Frege takes the idea of the parts of a thought; it is a general principle for him that a part of a part is a part of the whole.42 The second argument gives a case where two names of the same mountain have been learned in such a way that the truth of the identity statement with the two names is far from obvious. This is a case of the well-known identity puzzle43 where the terms of the identity are undoubtedly proper names. Frege uses an epistemic criterion of sense identity like that of the text of 1906 (NS, p. 213, trans. p. 197) discussed above: The sense of the sentence “Ateb is at least 5000 meters high” is different from the sense of the sentence “Afla is at least 5000 meters high.” Someone who takes the former to be true by no means has to take the latter to be true. (p. 128, trans. p. 80)44 Russell could have replied to the first argument by saying that the constituents of a proposition are not parts in as literal a sense as Frege is speaking of. The second poses what is even now one of the greatest 42

NS, p. 197, trans. p. 181; the context is a discussion of the view that a set consists of its elements. See my “Some Remarks on Frege’s Conception of Extension,” p. 268 [Essay 5, p. 120 in this volume]. 43 “Über Sinn und Bedeutung,” pp. 25–26 (KS, pp. 143–144). 44 It should be remarked that probably Frege’s discussion of this issue was never sent to Jourdain; what is probably the second draft of Frege’s reply to Jourdain’s letter (dated January 28, 1914) is devoted entirely to Frege’s difficulties with Principia Mathematica, which concerned use and mention, the notion of a variable, and the notion of a propositional function. It seems that Frege was prevented by the obscurities he found from reading very far into the book. 154

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difficulties for the view that proper names are directly referential. Russell would no doubt appeal to his view that ordinary proper names are not logically proper names. Nowadays we might begin by distinguishing an epistemic from other notions of sense.45 Other Points on Number and Extension. The publication of the German text of Frege’s notes to Jourdain’s article gives me the occasion to amplify and to some extent correct my remarks46 on Frege’s statement in the notes that “the class is something derived, whereas in the concept we have something primitive.”47 The German for “something primitive” is “etwas Ursprüngliches,” which unlike Jourdain’s English does not suggest the contrast of primitive and defined. I failed to notice a text of 1896 where Frege had already spoken of extensions as derived. Commenting on Peano’s view of classes, he says: For him, the class appears at first as it does for Boole as something primitive (etwas Ursprüngliches), which is not to be reduced further. But in § 17 of the Introduction I find a designation “x ∈ Px” for a class of objects which satisfy certain conditions, which have certain properties. The class appears here, therefore, relative to the concept as derived (das Abgeleitete), it appears as extension of a concept, and I can declare myself quite in accord with that, although I do not much like the notation “x ∈ Px”.48 That classes (extensions) are derived from concepts would certainly be implicit in Frege’s conception of them from the beginning. Apart from the above citation, it is pointed to fairly explicitly in the Grundgesetze (1:2–3; 2:150). What is added in 1910 is the emphasis on a distinction between “fundamental logic,” which does not depend on the concept of extension, and a “further-developed logic,” that is also “derived,” to which arithmetic belongs. Even this is expressed tentatively: “We can perhaps regard Arithmetic as a further developed Logic.” The sense in which classes are derived seems primarily ontological. It seems that in 1910 Frege did not have an exact conception of the

45

See for example Salmon, Review of Linsky, Names and Descriptions. See Essay 5 of this volume, p. 125. 47 Jourdain, “Development,” p. 251 (KS, p. 339, or WB, p. 286. The German is on p. 121 of WB. 48 “Über die Begriffschrift des Herrn Peano,” p. 368 (KS, p. 225). 46

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implications of this for the status of the laws of classes and therefore of the logistic thesis, but the logistic thesis is given a weaker sense than he gave it before he learned of Russell’s paradox. The picture of the late development of Frege’s views is somewhat clouded by a rather mysterious draft of a letter to Karl Zsigmondy reacting to an address given by the latter in 1918. The form is that of a genetic explanation of a conception of a cardinal number as a class of numerically equivalent sets. But Frege plays along with the idea that a number attaches to a heap, the sort of conception vehemently criticized in the Grundlagen. The notion of class as extension of a concept, and the doubts about that notion expressed about the same time in “Aufzeichnungen für Ludwig Darmstaedter,”49 are not mentioned. I am not sure what to make of this text. It is probably unfinished. My conjecture is that he is presenting somewhat ironically an explanation of an illusion, which possibly he intended to go on to expose more directly. The view Frege develops is evidently suggested to him by what is expressed by Zsigmondy in a passage of his address cited by the editor, but there are differences.50 An ironical note appears at the beginning; Frege says that his efforts to clarify the concept of number “apparently ended in complete lack of success,” which, however, caused the question not to rest in his mind although I am, so to speak, officially no longer concerned with the matter.51 And this work, which has gone on in me independently of my will, has suddenly and surprisingly shed full light on the question. (p. 270, trans. p. 176) In the next paragraph, the idea that “number is a heap” is set forth in an ironical tone. The text ends as follows: 49

NS, pp. 273–277, esp. pp. 276–277, trans. pp. 256–257. Cf. Essay 5 of this volume, pp. 127–128. 50 P. 269n.3. Where Zsigmondy talks of sets (Mengen), Frege talks of heaps (Haufen). And Zsigmondy does not take the last step of dropping the distinction between a number and a class of numerically equivalent sets. It would be instructive to confront Frege’s draft with the full text of Zsigmondy’s address, but I have not succeeded in obtaining it. [See now the Postscript to this essay.] 51 [Here Frege doubtless alludes to his retirement, which occurred officially on his 70th birthday, November 8, 1918. See Kreiser, Frege, p. 519.] 156

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What more do we know of numbers in general than that we can reidentify the same number and that we can distinguish different numbers. The same holds for our classes. Therefore we are strongly inclined to say: Our classes are numbers, and numbers are classes of heaps. Therefore we drop the distinction of the numbers from our classes. With that, do we not have everything we need? (p. 271, trans. p. 178) The last question, I suggest, is also to be understood ironically.52

52

I am indebted to Dagfinn Føllesdal and Wilfried Sieg for valuable comments on an earlier version of this review. 157

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Certainly the most important development concerning Frege’s correspondence in the period since the present essay was written is the discovery and publication of Frege’s letters to Ludwig Wittgenstein. A lot has been written about Wittgenstein’s relations with Frege and the influence on him of Frege’s work, and I will not attempt to summarize it or add to it. Wittgenstein had had more than one meeting with Frege between 1911 and 1913, although testimony differs about the time and circumstances of the first meeting.1 But the earliest of the lost letters that Scholz had acquired is dated October 22, 1913, and the earliest of the surviving letters of Frege to Wittgenstein is dated October 11, 1914, when Wittgenstein was already in the Austrian army and serving at the front in Poland. I will not comment on these letters in any detail. Those written during the war bring to light Frege’s nationalism and support of the German and Austrian war effort, as well as his pleasure that Wittgenstein was able to carry on some scientific work under the conditions of fighting in a war.2 The later letters express Frege’s reaction to the manuscript of the Tractatus. He evidently had difficulty making his way past the opening sentences of that work; he misses any argument, does not find their sense at all clear, and attempts to explicate them using his own conceptual apparatus.

1

See McGuinness, Wittgenstein, pp. 73–76. On the first point, it is known that Frege was up through the end of the war a supporter of the National Liberal party, a conservative but mainstream party in Imperial Germany. Frege’s political views in his last years, expressed in his notorious diary, are another matter. 2

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What is striking to a reader who knows Frege almost entirely through his writings is the great respect and friendship that Frege shows toward Wittgenstein. It is likely that during their conversations before the war Frege recognized Wittgenstein’s gifts, and he could not have been unaware of his aristocratic origin. But it was surely significant to him that Wittgenstein took his philosophical work seriously, and that they were able to discuss it at length.3 That was probably a rare occurrence for Frege. He may have viewed Wittgenstein as something like a pupil, who might carry on his own work. Consider the following quite moving passage from a letter of September 16, 1919: I hold that the prospect of our coming to understand one another in the domain of philosophy is not so slight as you seem to. I combine with that the hope that you will one day come to the defense of what I believe I have come to know in the domain of logic. In long conversations with you, I have become acquainted with a man who, like me, has searched after truth, partly on other paths [from mine]. And just that lets me hope to find in you something that will amplify what I have found and perhaps correct it. Thus while I try to teach you to see with my eyes, I expect to learn to see with your eyes. I don’t give up so easily the hope of an understanding with you. The present essay ends with some comments on a draft of a letter to Karl Zsigmondy, apparently in response to Zsigmondy’s inaugural 3

Wittgenstein’s forwardness contrasts with the attitude that Rudolf Carnap expressed to Günther Patzig in 1967. Patzig had asked if Carnap, when he returned to Jena after the war and after reading Frege’s principal writings, had sought Frege out and, in particular, let him know how important he found his writings. Carnap replied that this had not occurred to him and that he would have felt it as presumption for an “unknown doctoral student” to visit a Herr Geheimrat “and as it were tap him on the shoulder and say how important he found his works.” That was just not done. (From a letter of Patzig to Lothar Kreiser, November 15, 1988, quoted in Kreiser, Frege, p. 277n.5.) In fact Frege had the title Hofrat, not the more prestigious Geheimrat.) To judge from his reported response to Patzig, Carnap may have misjudged Frege’s character. The eminent scholar Gersom Scholem attended Frege’s “Begriffschrift” lectures a few years later and was much impressed by Frege’s “completely unpompous manner” and its contrast with that of the philosopher Rudolf Eucken. (See Scholem, Walter Benjamin, p. 66, quoted in Kreiser, op. cit., p. 469.) 159

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address as Rector of the University of Vienna in October 1918.4 At the time I was unable to obtain Zsigmondy’s text. Reading it since has not led me to change what is written above. However, I will note that the genetic style of Frege’s discussion, a style that is otherwise uncongenial to him, could well have been prompted by the fact that Zsigmondy engages in a brief speculative discussion of the origin of the number concept.5 However, he puts such considerations within what he calls the “psychological standpoint” and distinguishes that from the “mathematical-logical standpoint,” which is the context in which he sets his somewhat Cantorian explication of the notion of cardinal number that seems to be the more direct object of Frege’s ironic commentary.6

4

“Zum Wesen des Zahlbegriffs und der Mathematik.” I wish to thank Professor Friedrich Kambartel for providing me with a copy of this text. It has occurred to me that Frege may never have intended to send an actual letter. 5 Ibid., pp. 43–44. 6 Ibid., pp. 47–48; cf. the quotation in WB, p. 269n.3. 160

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1. Introduction It is well known that Brentano classified “psychical phenomena” as presentations, judgments, and phenomena of love and hate. Presentations are presentations of objects, although their objects may not exist. One might say roughly that presentations are the vehicles of content, but a presentation is not propositional in form and does not embody any stance of the subject toward the content in question. Judgments are affirmations or denials of presentations. Thus they are based on presentations but are not a species of them. It is of course judgments that are true or false. Phenomena of the third class are also based on presentations, and like judgments also embody a stance of the subject toward the content in question. Brentano sometimes characterizes this as Gefallen oder Mißfallen, which might be rendered roughly as a proor con-attitude. Such attitudes can also be correct or incorrect, an idea that is the starting point of Brentano’s ethics. However, phenomena of love and hate will play almost no role in what follows. The threefold classification is presented in Psychology from an Empirical Standpoint in 1874 and Brentano held to it for the remainder of his career. The common-sense idea of a judgment is that it is an instance of someone judging something; where what is at issue is truth or falsity, the agent comes to a belief one way or the other.1 It should follow that a judgment would incorporate what Frege called force, in this case the agent’s stance toward the truth or falsity of the proposition judged to be one or the other. But it follows that many sentences that occur as parts 1

It is this case that Brentano calls judgment, although in ordinary language judging is often appraisal as to value, as for example the judging of figure-skating or other performances. 161

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of other sentences, for example antecedents of conditionals, do not express judgments. Suppose that Smith judges: (1) If it rains tomorrow, the game will not be played. In a typical case, where Smith is uncertain about tomorrow’s weather, he does not judge that it will rain tomorrow; even if it happens that he does, (1) does not express such a judgment. Frege’s view of this situation was an early version of a view that became standard in the twentieth century, although it has been subjected to many challenges. According to him, one should distinguish judgments from what he calls thoughts, which are roughly what is commonly called propositions. A thought does not embody any force; to say that a sentence expresses a certain thought says nothing about whether someone uttering it takes that thought to be true. In a suitable context, (1) combines two thoughts, that it will rain tomorrow and that the game will not be played, in order to form a single compound thought. Smith judges that thought to be true, but he makes no judgment at all concerning the two thoughts of which it is composed. By a “propositional object” I mean an object that (according to one or another theory) is expressed or designated by a sentence. Judgments might be taken as one kind of such objects. Frege’s thoughts and the propositions of the early Russell and of many other English-language writers are another. One might add states of affairs (Sachverhalte) or situations, as well as facts. In much logical literature from early modern times into the twentieth century, judgments are the principal propositional object, but the term has significant ambiguities. The suggestion derived from common sense is that there is a judgment only if an agent judges something. That would suggest viewing a judgment as an event and thus doubtfully a propositional object at all. But logical writers used the term to do the work of the term “proposition,” with the effect of detaching the idea of a judgment from judging or assertion. In contrast, Brentano holds consistently to the conception of a judgment as the outcome of an actual judging and thus as embodying a commitment as to truth or falsity. Judgments are thus clearly distinguished from the thoughts or propositions that, on another view, might be their constituents but about whose truth or falsity the agent takes no stance, such as the antecedent and consequent of (1). Judgments appear to be the only propositional objects Brentano admits. 162

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Brentano differs in this respect from some of his principal pupils, in particular Marty, Meinong, and Husserl.2 In later writings, written after he had adopted the position called reism, according to which an object of thought has to be a Reales or a thing (something concrete), Brentano argues frequently against propositions or states of affairs. However, as we shall see in §6, he did accept them from the 1880s until his adoption of reism. In his late phase Brentano is probably best interpreted as rejecting even judgments as propositional objects, in the sense of objects expressed by sentences. What he admits are subjects who affirm or deny presentations.3 However, we will for much of our discussion abstract from Brentano’s later reism. It will be discussed in §5. Brentano argues for his view that judgment is a distinctive form of mental phenomenon, and thus a distinctive intentional relation to an object, in chapters 6 and 7 of the 1874 Psychology. Much of the argument is directed at theories of judgment current at the time, in particular the idea that goes back to Aristotle that judgment consists of combination or separation of presentations. Brentano’s underlying idea is that the object of a presentation can be the object of a judgment affirming or denying it. Since a presentation need not be a combination or separation, judgments, such as simple existential judgments, affirming or denying presentations that are not are counterexamples to the Aristotelian account.4 According to Brentano, judgments are affirmative or negative, so that negation belongs to the judgment and not to the structure of the presentation judged. This is another place at which Brentano disagrees with Frege, where Frege’s view has become the received view in later times. Brentano’s is a traditional view, and against it Frege argued forcefully that negation is not a mode of judgment but belongs to the content, so that a sentence like “it will not rain tomorrow” expresses a thought that is the negation of the thought expressed by “it will rain 2

For a wide-ranging treatment of judgment in the Brentano school, see Mulligan, “Judgings.” 3 It might seem that the idea of judgments as events, i.e., someone’s judging, would be congenial to Brentano’s reism. However, I have not found a place where he admits events as Realia. 4 Brentano summarizes his argument in §8 of chapter 7 (Von der Klassifikation, pp. 64–65, trans. pp. 221–222). I am indebted here to Kai Hauser. 163

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tomorrow.”5 A judgment that it will not rain tomorrow does not differ in force from a judgment that it will rain tomorrow; where they differ is in the thought that is judged to be true. In Brentano’s view, in contrast, “rain tomorrow” might well express a certain presentation; the judgment that it will rain tomorrow affirms this presentation, while the judgment that it will not rain tomorrow denies it.6 To carry through Brentano’s view, it would be necessary to represent all complexity of content as belonging to the presentation judged. Brentano’s theory of judgment can be viewed as a brave attempt to carry through a view of this kind. Much of his effort in discussion of judgment is in attempts to do justice to the various forms of complexity that arise from the complex logical form of sentences. In its original form, Brentano’s view of judgment implies that in a sense all judgments are existential judgments or negations of existential judgments. This peculiarity of his view of judgment influenced his thought on truth at an early point and led to a particular line of questioning of the traditional idea of truth as adaequatio rei et intellectus, the root of what has come to be called the correspondence theory of truth, already adumbrated in the 1889 lecture that is the opening essay in the compilation Wahrheit und Evidenz. Brentano was not the only or even the most influential philosopher to question the correspondence theory at the time, but his criticisms had distinctive features. In late writings he sketched as a positive view an epistemic conception. The discussion below of Brentano’s views on truth will concentrate on these aspects.

2. The Problem of Compound Judgments Presentations as Brentano conceives them are what in traditional logic was expressed by terms, singular and general. Since the object of a presentation need not exist, singular as well as general presentations can be either affirmed or denied. What we would express as someone’s judg-

5

For example “Die Negation,” pp. 152–155. There is no reason to think that Brentano individually is Frege’s target; he is not referred to in Frege’s extant writings. 6 Apparently Brentano does not distinguish terminologically between affirming a presentation and affirming its object, so that affirming rain tomorrow and affirming the presentation are expressed by the same word, generally anerkennen. 164

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ing that Pegasus does not exist would be in Brentano’s language his denying or rejecting Pegasus; the case is exactly parallel to that of unicorns. The difficulty an account such as Brentano’s faces is how to represent judgments that involve compounding, particularly sentential combination such as that embodied by (1). This issue already arises in Brentano’s first development in the 1874 Psychology, where he sketches an explanation of the syllogistic forms. Brentano’s view immediately gives a distinctive place to existential statements, “A exists,” where A is a term, since to judge that is just to affirm A. Thus his view immediately removes the temptation to treat “exists” in such statements as a predicate, even a “logical” but not “real” predicate, as Kant did.7 The most direct way of looking at the syllogistic forms from the point of view of modern logic yields the result that categorical propositions are equivalent either to existential propositions or negations of such, since we have: ‘All A are B’ is equivalent to ‘There are no As that are non-Bs’. ‘No A are B’ is equivalent to ‘There are no As that are Bs’. ‘Some A are B’ is equivalent to ‘There are As that are Bs’ or ‘There are ABs’. ‘Some A are not B’ is equivalent to ‘There are As that are non-Bs’ or ‘There are A non-Bs’. These readings can go directly into Brentanian terms: To judge that all A are B is to deny As that are non-Bs; to judge that no A are B is to deny As that are Bs; to judge that some A are B is to affirm As that are Bs; to judge that some A are not B is to affirm As that are non-Bs. Essentially these readings are given by Brentano in Psychology.8 He draws a number of conclusions that modern logicians have drawn, such as that the inferences from A to I and from E to O are not valid, 7

Brentano credits Herbart with treating existential propositions as distinct from categorical subject-predicate propositions (Von der Klassifikation (hereafter cited as KPP), p. 54, trans. p. 211). Kai Hauser has suggested (in correspondence) that treating affirmative judgment as judgment of existence may have arisen from Brentano’s reflection on Aristotle; cf. the remark that Aristotle recognized that the concept of existence is obtained by reflection on affirmative judgment (Wahrheit und Evidenz [hereafter cited as WE] p. 45, trans. p. 39). 8 KPP 56–57, trans. pp. 213–214. 165

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and that certain traditionally accepted syllogisms are not valid, although they become so if an existential premise is added.9 Second, the readings make clear that already at this level Brentano’s account requires some principle for the combination of terms or presentations. The first is basically conjunction, so that given A and B we have ‘As that are B’. A second would be negation applied to terms: as they stand, the readings involve an “internal” negation in addition to the negation embodied in negative judgment, i.e., denial. Some of the neatness of the theory is lost by admitting term negation in addition to denial. Brentano does not address this issue in Psychology, but as we shall see he was uncomfortable with term negation and did develop some ideas for eliminating it. Brentano in one place at least admits disjunctive terms, so that we can also allow judgments that affirm or deny A-or-Bs.10 At any rate, if term negation is applicable to compound terms, then any truth-functional combination of terms can be expressed as a term. Two problems would remain before Brentano’s theory could yield the expressive power of first-order logic. First, one would have to accommodate truth-functional combination of closed sentences. If we make the assumption about terms of the last paragraph, that would be sufficient to generate a logic with expressive power equivalent to that of monadic quantificational logic, since in monadic logic nested quantification can be eliminated. Second, one would have to have a treatment of many-place predicates and polyadic quantification. If Brentano had developed the second, he would have been one of the founders of mathematical logic, which he neither was nor claimed to be. The question whether this can be done in the framework of a Brentanian theory of judgment is one external to Brentano himself. Term logics that are equivalent to first-order logic have been developed, but they involve devices that were not thought of in Brentano’s time even by mathematical logicians. It would have been necessary for Brentano to consider manyplace predicates on the same footing as one-place predicates. His remarks 9

See Simons, “Judging Correctly.” In reading categorical propositions in this way, Brentano was anticipated by Boole. An elegant decision procedure for syllogisms so interpreted was devised in the 1880s by Charles Peirce’s student Christine LaddFranklin. In response to criticism by J.  P.  N. Land, Brentano admitted that one might read the categorical propositions as presupposing the nonemptiness of the subject concepts. 10 Kategorienlehre, p. 45, trans. p. 42. 166

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on relations take in only binary relations, and there he holds the unusual view that only the first place of a binary relation is direct or referential (modo recto in Brentano’s terminology); on this subject see §4 below. We can remain closer to Brentano in considering how the first question might be addressed. This has been treated in some detail by Roderick Chisholm.11 Consider first the simplest case, judging that p and q. One might say that S judges that p and q if he (simultaneously) judges that p and judges that q. But as Chisholm points out, that would not be sufficient, since S might not put the two together. Suppose first that both judgments are affirmative, so that S accepts A and accepts B. Brentano admitted conjunctive objects, objects consisting of an A and a B. Call them A-and-Bs. S’s accepting A-and-Bs has the requisite property of committing S both to As and to Bs in a single judgment. One might object that S is committed to more, to another object, precisely the Aand-B. That would be so if we think of it as a set having an A and a B as elements. If these objects are distinct non-sets, then the pair set must be distinct from both of them. Brentano did not think of conjunctive objects as sets, at least not as set theory has come to think of them. It is well known that given either the empty set or a single individual, one can generate an infinite sequence of sets by successive application of the forming of pair sets. Brentano considers and rejects an argument for such generation beginning with two apples. A key step that he rejects is that a pair of apples is something “in addition” to the original two apples: Someone who has one apple and another apple does not have a pair of apples in addition, for the pair which he has simply means the one apple and the other taken together. So what people wanted to do was to add the same thing to itself, which is contrary to the concept of addition. . . . The pair is completely distinct from either of the two apples which make it up, but it is not at all distinct from both of them added together.12 Particularly the last remark suggests that Brentano thinks of the pair as the mereological sum, and some of his remarks about pluralities parallel 11

Chisholm, “Brentano’s Theory of Judgment.” KPP 253, trans. p.  352. Brentano reveals that the example of the two apples comes from Cantor, who is said to have claimed before a meeting of mathematicians to generate an infinity of objects starting with two apples. 12

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claims made by defenders in later times of mereological sums. That would serve to block the generation of an infinite sequence out of only one or two individuals. However, elsewhere Brentano writes in connection with the question of the relation of such a whole and its constituents that “there are things that compared with others have revealed themselves neither as wholly the same nor as wholly other, that are partially the same” (Kategorienlehre, p.  50, trans. p.  46). Mereology plays a larger role in Brentano’s work, so that he could claim that the introduction of conjunctiva in the present context is not ad hoc. Now consider the disjunction of two affirmative judgments, again one affirming As and one affirming Bs. Admitting disjunctive terms, one can render the judgment as one that affirms (A or B)s. We would say that this works because ‘∃xAx v ∃xBx’ is equivalent to ‘∃x(Ax v Bx)’. It is for that reason that the solution is simpler than that concerning conjunctions of affirmative judgments. This simple solution is also available for the case of conjunction of two negative judgments. To judge that there are no As and that there are no Bs would be simply to deny (A or B)s. The idea used for conjunctions of affirmative judgments will clearly work for disjunctions of negative judgments. Judging that either there are no As or that there are no Bs would be to judge that there are no (A-and-B)s. For let a be an A and b be a B. Then a and b “taken together” constitute an A-and-B. So if there are no A-and-Bs, then either there are no As or there are no Bs. Conversely, since any A-and-B has an A as a part, if there are no As, then there cannot be any A-and-Bs, and likewise if there are no Bs. There remains the problem of binary combination of an affirmative with a negative judgment. How might Brentano analyze the judgment that either there are no As or there are Bs? Chisholm’s proposal is that such a judgment would reject As that are not part of A-and-Bs.13 For suppose that judgment is true, and it is likewise true that there are As. Then any such A must be part of an A-and-B, and so there are Bs. Hence either there are no As or there are Bs. Conversely, suppose there are no As. Then clearly there are no As that are not part of A-and-Bs. Suppose that there are Bs. Then let b be such. If there are As, then any such A will combine with b to form an A-and-B and hence is part of an Aand-B. So if there are Bs, then there are no As that are not part of 13

Clearly this paraphrase involves a negative term. 168

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A-and-Bs. The symmetry of disjunction implies that we can handle in the same way a judgment that either there are As or there are no Bs. Consider now the case of a mixed conjunction, a judgment that there are As and there are no Bs. Chisholm proposes that such a judgment be viewed as accepting As that are not part of (A-and-B)s, and this is evidently correct since it is equivalent to ‘It is not the case that either there are no As or there are Bs’. One might also ask about conditional judgments, such as the judgment that if there are As, then there are Bs. Brentano’s suggestion about hypothetical judgments seems to me to amount to reading the conditional in the now familiar truth-functional way.14 Thus this case is reduced to cases already considered. ‘If there are As then there are Bs’ is the mixed disjunctive judgment ‘Either there are no As or there are Bs’. It thus appears that the judgments Brentano is able to handle are closed under truth-functional combination and, assuming the truthfunctional interpretation of the conditional, under the formation of conditionals. The price of this, however, is high. To handle simple conjunction, he needs to introduce mereological sums or some other conjunctive objects, thus introducing possibly contestable ontology in order to handle one of the simplest logical operations. To handle mixed binary compounds he needs in addition the notion of being part of an A-and-B. This in fact generates a more serious problem. Clearly the statement ‘x is part of an A-and-B’ means that x is part of some A-and-B. Thus there is an implicit quantifier that seems not to be captured by Brentano’s reduction of existential quantification to affirming a presentation, universal quantification to denying one. We shall consider in §3 how Brentano might deal with this without accepting the idea of being a part of some A as simply primitive.

3. Can One Eliminate Term Negation? Let us now step back and consider how Brentano might avoid admitting negative terms and so reduce all negation to denial. In order to address this issue, we turn to his conception of double judgment. A double judgment affirms an object and then affirms or denies something of it. Brentano characterizes them as judgments that “accept something 14

See Die Lehre vom richtigen Urteil (hereafter cited as LRU), pp. 122–123. 169

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and affirm or deny something of it.”15 In the essay “On Genuine and Fictitious Objects,” added to the 1911 edition of Psychology, Brentano deploys this idea to analyze the categorical forms of judgment.16 With respect to our problem about negation, it offers a solution to the problem of the O form. ‘Some S is not P’ affirms an S and denies of it that it is P.17 Brentano also proposes that a psychologically more accurate rendering of the I form would also view it as a double judgment, affirming an S and affirming of it that it is P.18 However, the notion of double judgment has the limitation that it affirms an S (for some S or other) and then affirms or denies some predicate P of it. There is no negative counterpart. Indeed, it is hard to see what sense it could make to deny an S and then affirm or deny something of it. Thus, while the notion of double judgment elegantly eliminates the negative term from the O form, it does not seem to solve the corresponding problem about the A form. Thus Chisholm, who claims about as much as could be claimed for Brentano on this issue, seems to give up at this point on trying to eliminate term negation from Brentano’s theory.19 The notion of double judgment might be applied to a problem we encountered concerning truth-functional combination. For example, a mixed conjunction, affirming As and denying Bs, was analyzed as an affirmation of As that are not part of (A-and-B)s. That would be represented as a double judgment affirming an A, and denying an A-and-B of which it is a part. We have, however, simply exploited the strategy for dealing with the O form, and the same problem that we met with in connection with the A form prevents us from extending this to other cases, in particular that of mixed disjunctions, which are in Brentanian terms negative judgments. 15

KPP 194. Translation, from Origin, p. 107, modified. This remark occurs in a footnote added in 1889 to “Miklosich über subjektlose Sätze” (1883). 16 So far as I know Brentano does not address directly the problem how to understand simple judgments of the form ‘there are non-As’ or ‘there are no non-As’. The obvious idea is to take them as judgments of the form ‘there are [are no] things that are non-As’. Then in the negative case, the elimination of the term negation would pose the same problem as that noted in the text for the A form. 17 KPP 165–166, trans. p. 296. 18 Ibid., 165, trans. p. 295. 19 Chisholm, “Brentano’s Theory of Judgment,” p. 24. 170

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The weakness of double judgments for Brentano’s purposes is that they do not have straightforward negations. In particular, if they are introduced in order to handle truth-functional combination, the iteration that such combination involves will not be available. A device that Brentano uses in order to give analyses in accord with his later reism is to introduce the idea of someone thinking of an A, for some A, or someone making a judgment with respect to As. That suggests another solution to the problem of the A form. Brentano writes: If the O form means the double judgment ‘There is an S and it is not P’, then the proposition ‘Every S is P’ says that anyone who makes both of these judgments is judging falsely. I think of someone affirming S and denying P of it, and say that in thinking of someone judging in this way, I am thinking of someone judging incorrectly.20 It is not clear that this is offered as a way of eliminating the negative term in the rendering of the A form. Still, we might, following Peter Simons, derive from it the paraphrase of ‘Every S is P’ as ‘Whoever affirms S and denies P of it judges incorrectly’.21 Simons states that this is still in the A form and so does not advance the case. But it is perhaps better viewed as of the E form ‘No one who affirms S and denies P of it judges correctly’ and so as denying a correct acceptor-of-S-denying-Pof-it. Still, introducing what is effectively the concept of truth, and applied to a double judgment, seems a very questionable move in order to analyze one of the simplest and most traditional logical forms. The notion of double judgment itself raises some questions. First of all, for a given presentation S, to affirm an S is not in general to affirm any particular S; for example one can believe that there are cows without there being any particular cow in whose existence one believes. This is particularly true on Brentano’s scheme, since he thinks of existence as tensed. To accept cows is to accept cows as existing now. But suppose I have not been near a farm for a number of years. I’m confident that there are cows, but the only ones I can point to are from the past. I can’t rule out the possibility that all of them have by now died, even though the supply of milk in the supermarket assures me that if so, they 20 21

KPP 168–169, trans. p. 298. Simons, “Brentano’s Reform of Logic,” p. 43 of reprint. 171

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have been replaced by others. So there’s no particular cow that I accept. However, it seems that, say, judging that some cows are not white involves accepting a cow and denying of that cow that it is white. How can that be if there is no particular cow that I accept, and so a fortiori none that I judge not to be white? We could render such a double judgment as affirming an x that is a cow and denying of x that it is white. The x would have to be in some way indeterminate. Brentano does not put the matter this way, and I am not sure that it accords with his views; for example it represents even the subject term in such a judgment as a predicate. What he says that bears on the question is obscure, as for example this explanation of the I form: Looked at more closely, it signifies a double judgment, one part of which affirms the subject, and, after the predicate has been identified in presentation with the subject, the other part affirms the subject which had been affirmed all by itself in the first part, but with this addition—which is to say that it ascribes to it the predicate P.22 What is it for the predicate to be “identified in presentation with the subject”? It appears that Brentano means what is explained in his last dictation, included in the 1924 edition of Psychology. There he states that there are presentations which are unified only through a peculiar kind of association, composition, or identification, as, for example, when one forms the complex concept of a thing which is red, warm, and pleasant-sounding.23 A little later he elaborates by saying, “When we say, ‘a red warm thing’, the two things presented in intuitive unity are not totally identified but identified only in terms of the subject.”24 What seems to be needed is some version of the content-object distinction: In a double judgment, the predicate is identified with the subject in being affirmed or denied 22

KPP 165, trans. p. 295. Ibid., 206, trans. p. 316. 24 Ibid., 207, trans. p.  317. It is puzzling that Brentano speaks here of intuitive unity, since the case is essentially the one that on the previous page he has contrasted with intuitive unity. Kraus appends to “intuitive unity” a note, “Read: presented things.” This is not very clear, but it is likely that he thought “intuitive unity” in the quotation in the text a slip. 23

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of an object that the subject is presupposed to apply to. But that would restate the formulation of the last paragraph and not clarify it. We have concluded that Brentano’s ideas for reducing negation to denial and thus for avoiding Frege’s conclusion that negation belongs to the content of a judgment rather than being a mode of judgment itself are inadequate for the purpose and not entirely clear in themselves. Before leaving the subject I will comment on some remarks about term negation in the same essay from the 1911 Psychology that we have been considering. If negative terms are admitted, then it seems that negation is simply allowed as an operator on terms. Nonetheless Brentano regards term negation as introducing a kind of fiction, the fiction of “negative objects.” He seems to think such a fiction involved in the everyday understanding of negative terms: This fiction . . . is a commonplace to the layman; he speaks of an unintelligent man as well as an intelligent one, and of a lifeless thing as well as of a living thing. He looks on “attractive thing” and “unattractive thing, “red thing”, and “non-red things”, equally, as words which name objects.25 One might well ask, why not? In the sense in which “red thing” names anything, it names those things that are red, and then surely “non-red thing” names those things that are not red. Brentano does not give an argument, but it is very likely that “red thing” names a general presentation, and he may think that such a general presentation as would be named by “non-red thing” would be a negative object. The general background is discussed in §5 below.

4. Modes of Presentation A quite different aspect of Brentano’s treatment of complex judgments belongs actually to his account of presentations. That is that he distinguishes modes of presentation (Modi des Vorstellens).26 The major distinctions subsumed under these headings are what he calls temporal modes and the distinction between direct and oblique (modus rectus and 25

Ibid., 169, trans. p. 298. The English phrase reminds one of Frege, but Frege’s term is Art des Gegebenseins, and it should be clear from the text that the meaning is quite different. See “Über Sinn und Bedeutung,” p. 26.

26

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modus obliquus). The latter, although it is applied in the first instance to presentations rather than to linguistic contexts, is essentially the distinction that is familiar to us. A simple, straightforward presentation will represent its object in modo recto; in particular, if a judgment affirms such a presentation, it commits one to the existence of the object. He says that the direct mode “is never absent when we are actively thinking.”27 Oblique reference arises primarily in two cases: where one is thinking of a “mentally active subject,” where a thought of such a subject in recto will involve thought of the objects of his thought in obliquo. Thus a presentation of Kant thinking of the pure intuition of space will present Kant in recto and the pure intuition of space in obliquo. That is what we would expect since thought of an object is a “referential attitude” in contemporary terminology. Brentano allows that something thought of in recto might be identified with something thought of in obliquo: as for example when I have a presentation in recto of flowers and of a flower-lover who wants those flowers, in which case flowers are thought of both in recto and in obliquo and are identified with one another.28 The other case is more surprising: “Besides the fundament of the relation, which I think of in recto, I think of the terminus in obliquo.”29 In other words, in a thought to the effect that aRb, only a is presented in recto, so that the second term of the relation is an oblique context. I don’t know of an argument Brentano gives for this somewhat strange view. He does distinguish relations where if the first term of the relation exists, the relation implies that the second does as well; his example is ‘taller than’.30 Cases of this kind are not as frequent as one might think. But the reason for this lies in Brentano’s view of temporal modes. Brentano holds that the existence and properties of objects are essentially tensed. So he denies that being past, present, or future represents differences in the objects. A presentation thus has a temporal mode of presentation, in the simplest case present. To say that something exists, without qualification, is to say that it exists now; therefore Brentano says of figures from the past that they do not exist. It also follows that 27

KPP 145, trans. p. 281. Ibid., 147, trans. p. 282. 29 Ibid., 145, trans. p. 281. 30 Ibid., 218, trans. p. 325. 28

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a relation like ‘earlier than’ does not require the existence of both terms.31 Of course it follows that it doesn’t require the existence (now) of either. The battle of Blenheim was earlier than the battle of Waterloo, although both are past and so do not exist on Brentano’s view. What is relevant to his view of judgment is that a temporal mode is an additional complication to the logical form of a judgment. If I judge that the battle of Waterloo occurred, I affirm it in a past mode. If I judge that the presidential election of 2004 will occur, I affirm it in a future mode. Clearly much more complex combinations are possible. However, it is only affirmation of present existence that is affirmation “in the strict sense.”32 He seems to hold that other temporal modes are varieties of the oblique mode. I will not, however, pursue the question how Brentano develops or might have developed the conception of temporal modes.

5. General Presentations and Reism As is well known, shortly after the turn of the century Brentano abandoned the whole idea of objects other than things except as sometimes useful fictions, adopting the view called reism, according to which an object of thought must be a Reales or thing. This raises a question how Brentano would understand general terms or predicates occurring in judgments, even the simplest ones affirming or denying P, where ‘P’ replaces a general term. If a judgment affirms horses, it would naturally be taken as, in our terms, making reference to horses, that is, the animals with which we are familiar, and not to anything further such as a property or attribute of being a horse. We must ask, however, what the presentation is that is affirmed in such a case. What we might expect from Brentano’s reism is that he would hold that a general horse-presentation would have many objects, just those that are objects of individual horse-presentations. However, Brentano distinguishes sensory from noetic or intellectual consciousness; the latter includes what we would describe as the exercise of concepts. He seems rather firmly to reject the view I have suggested: A term can only be called general, if there is a general concept that corresponds to it. If we deny this and say that a term is general

31 32

Ibid., 218–219, trans. pp. 325–326. Ibid., 221, trans. p. 327. 175

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if many individual presentations are associated with it, then we would misinterpret the difference between ambiguity and generality, and would fail to see that the statement that many individual presentations are associated with one and the same term, in itself expresses a general proposition concerning these individual presentations.33 The beliefs that we cannot think of universals, and that so-called general terms are only associated with a multitude of individual presentations, have also been refuted.34 In fact, Brentano’s view is that all presentations are in a way general, that none can by virtue of its content fully individuate an object, although in some cases, such as presentations of inner perception referring to the self, it can be argued that they can have at most one object (SNB 98, trans. 72). Although he makes a distinction of intuitions and concepts parallel to Kant’s, he denies that intuitions have a content that individuates their objects (KPP 199–200, 204, trans. 311–312, 315). In the first of these texts (supplementary essay XII to Psychology) he justifies this by a rather intricate argument concerning perception and space. That need not concern us here; the question is how this view comports with his reism (which is in evidence in this text and even more in the following one). An answer is suggested by some passages in Die Lehre vom richtigen Urteil, which, however, often does not give the ipsissima verba of Brentano. Brentano often speaks of the use of language as introducing fictions; many of his examples are mathematical, and some are logical (e.g., KPP 215, trans. 322–323). In LRU 41, it is explicitly stated that concepts are fictions; however, in one place (§29), the language clearly comes from Kastil, and in the other (beginning of §30), this also ap33

Sinnliches und noetisches Bewusstsein (hereafter SNB), p. 89, trans. p. 63. Ibid., 89, trans. 65. The second of these passages undoubtedly comes from Brentano’s reistic period, and although the editor of SNB is not explicit about its date, it seems very likely that the first does as well, since nearly all the texts in the volume for which he gives dates are from the last years of Brentano’s life. The mention of “association” suggests that Brentano’s target is a view like Berkeley’s. Deborah Brown argues that Brentano’s rejection of the view I suggest rests in considerable part on identification of medieval nominalism with views like Berkeley’s. See her “Immanence and Individuation,” pp. 36–38. 34

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pears to be the case.35 However, the view of general thought presented is plausibly Brentanian. Thinking of something as a man, a human being, and a living thing are increasingly general ways of thinking of a thing. But the thing referred to is an individual, even though thinking of it in any of these ways fails to single it out as an individual. Brentano himself says elsewhere that a thing (Reales) is always determinate, but is object of a presentation “in a now more, now less differentiated way, without therefore ceasing right away to be thought of in a certain way generally and indeterminately.”36 In this passage he uses ‘concept’ without any comment but denies that universals are things. “Every such universally thought thing is, if it is, completely individualized.” A less Brentanian way of putting the point is that thought of something as, say, a man is the thought of an x that is a man. What we have said in §3 about double judgments indicates that some such perspective is essential for Brentano’s treatment of rather simple judgments. We can’t eliminate the x by taking the thought as of a definite particular object, which the thought represents as being a man. That would run afoul of Brentano’s claim that the content of our thought never yields a genuinely individual representation, and furthermore in the cases considered in his treatment of syllogistic, the x is bound by a quantifier. It is somewhat awkward because, if one takes seriously the doctrine that all presentations are general, it implies that all presentations have in some sense the form of predicates. I am not at all sure that that is a consequence that Brentano would have embraced. And it is undoubtedly uncomfortably close to nominalism, even from the point of view of the later Brentano.37

6. Questions about Truth as Correspondence Brentano’s substantial publication on truth during his lifetime was a lecture of 1889, “On the Concept of Truth,”38 reprinted in the posthu35

See notes 36 and 37, LRU 312. Note 37 intimates that §30 comes from supplementary essay XII of Psychology, but that is accurate only for the last part. 36 Die Abkehr vom Nichtrealen, p. 348. 37 For a historically informed and much more detailed treatment of Brentano’s views on individuation and his relation to nominalism, see Brown, “Immanence and Individuation.” 38 Section numbers in the text below refer to this essay; this will enable the reader to locate a passage either in the German WE or the English. 177

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mous Wahrheit und Evidenz. It shows a characteristic of much of his reflection on truth. His point of departure is the traditional characterization of truth as adaequatio rei et intellectus. His inclination is to defend it but much of the discussion concerns what it means, and some points are made that suggest real criticisms of the correspondence theory as it developed at the time and later. The line of thought then inaugurated leads him to be more definitely critical of the traditional formula in the later writings first published in Wahrheit und Evidenz. But even late, he shows some reluctance to abandon it altogether. Brentano’s thought on truth develops out of his thought on judgment, in particular the central role that (affirmative and negative) judgments have in his view and his criticism of a traditional view of judgment as a combination of presentations. The discussion of truth in the 1889 essay begins with a formula of Aristotle: He who thinks the separated to be separated and the combined to be combined has the truth, while he whose thought is in a state contrary to that of the objects is in error.39 After some discussion of subsequent history and examples, Brentano offers a corrected version: A judgment is true if it attributes to a thing something which, in reality, is combined with it, or if it denies of a thing something which, in reality, is not combined with it. (§33) He makes no difficulty about the case of affirmative subject-predicate judgments. But he immediately asks about judgments of existence: What is combined if I judge that a dog exists?40 Clearly, on Brentano’s view such a judgment affirms a dog, so that dog is the only presentation involved. A little later he says that in the case of a negative existential judgment like “There is no dragon” there is no object to which the judgment corresponds if it is true. It could not be a dragon, since ex hypothesi dragons do not exist. “Nor is there any other real thing which could count as the corresponding reality” (§42). 39

Metaphysics 1051b 3, translation by W. D. Ross quoted in §11 (in the translation). 40 Brentano states that Aristotle too recognized that this was not a case of combination. 178

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Brentano goes on to find a similar difficulty in negative predications. Suppose I say, “Some man is not black”. What is required for the truth of the statement is, not that there is black separated from the man, but rather that on the man, there is an absence or privation of black. This absence, this non-black, is clearly not an object; thus again there is no object given in reality which corresponds to my judgment. (§43) At this point one might well expect him to reject the correspondence theory or at least to admit that it has significant exceptions. He introduces a contrast between things (Dinge) and “objects to which the word ‘thing’ should not be applied at all” (§44). As examples he mentions “a collection of things, or . . . a part of a thing, or . . . the limit or boundary of a thing, or the like” (§45). He also mentions things that have perished long ago or will only exist in the future as well as “the absence or lack of a thing,” an impossibility, and eternal truths. Because none of these are things, “the whole idea of the adaequatio rei et intellectus seems to go completely to pieces” (§45). That is, however, not the conclusion that Brentano draws. Instead he says that we must distinguish between the concept of the existent and that of thing, and so he says: A judgment is true if it asserts of some object that is, that the object is, or if it asserts of some object that is not, that the object is not. And this is all there is to the correspondence of true judgment and object about which we have heard so much. To correspond does not mean the same as to be similar; but it does mean to be adequate, to fit, to be in agreement with, to be in harmony with. (§§51–52) Brentano’s formulation is reminiscent of another much-quoted Aristotelian formulation: To say of what is that it is not, and of what is not that it is, is false; to say of what is that it is, and of what is not that it is not, is true.41 41

Metaphysics 1011b 26–27, Ross trans. 179

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Aristotle, however, undoubtedly has the ‘is’ of predication in mind, while Brentano is thinking in terms of his early doctrine that all judgments are (affirmative or negative) existential judgments. Brentano has saved a version of the traditional formula, but apparently at the cost of introducing “objects” that are not “things.” He does think that in cases where the presentation underlying a judgment does not have a thing as its object, in cases other than judgments of necessity and possibility, there is an indirect dependence on things (§55). He also suggests that there is something trivial about the definition (§57) but responds that it still offers useful conceptual clarification. Brentano does not make clear here how far he is prepared to go in admitting objects that are not things, what he later calls irrealia. Without more explicitness, it is not clear that he has answered even his first sharp question about the traditional version: To what object does a negative existential truth like “There are no dragons” correspond? He suggests that he would admit absences or privations as objects, but this is clearer in the case of absences relating to things, such as the absence of black in a man who is not black. Alfred Kastil reports Brentano as having said in 1914 that he had thought he had to extend the adaequatio rei et intellectus to negative judgments, “as if in this case as well an objective correlate corresponded to the judgment, the nonbeing of what is correctly rejected” (WE 164, trans. 142). In later writings reflecting his turn to reism, he frequently criticizes the claim that if it is true that there are no As, then there must be “the nonbeing of As.” This would, apart from other objections to it, introduce a new kind of object to correspond to a true judgment, a state of affairs or perhaps fact.42

42

Peter Simons comments that the admission of such objects was an innovation in the 1880s. Since it was abandoned with the turn to reism, it would be characteristic only of the middle period of Brentano’s thought. It should be noted that the “problem of nonbeing” to which Brentano responded at this point is one concerning judgment (or on other theories propositions), roughly the problem how something could be true without there being anything in virtue of which it is true. It should thus be distinguished from the problem posed by presentations of objects that do not exist, which led to Meinong’s theory of objects. Cf. Jacquette, “Brentano’s Concept of Intentionality.” 180

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7. Virtual Abandonment of the Correspondence Formula The discussion of the last section should show that with the adoption of reism (see §5) Brentano effectively gave up the basis of his continuing to defend a conception of truth as correspondence. And that is indeed what one finds in the late letters and essays in Wahrheit und Evidenz. However, he seems still to have been reluctant to abandon the formula. Thus much of Brentano’s letter to Marty of September 2, 1906 is devoted to arguing against admitting such states of affairs as “the being of A” as had been accepted by Marty and, as we have just seen, earlier by Brentano. Against the idea that they are useful, Brentano writes: Where someone might say, “In case there is the being of A, and someone says that A is, then he is judging correctly”, I would say, “In case A is and someone says that A is, he judges correctly”. Similarly instead of “If there is the non-being of A and someone rejects A, he judges correctly”, I would say “If A is not and someone rejects A, he judges correctly”, and so on. (WE 94, trans. 84) Thus he seems to think states of affairs not necessary to state basic truthconditions. He also offers a regress argument against them: Suppose someone wishes to judge with evidence that A is. But he could not affirm A with evidence unless he could also affirm the being of A. Otherwise “he would be unable to know whether his original judgment corresponds with it.” But then by parity of reasoning he would also have to be able to affirm the being of the being of A, and so on (WE 95–96, trans. 85–86). This argument might be generalized to an argument against any form of correspondence theory: Suppose that its being true that p consists in the correspondence of p with something, call it P. Then to determine whether it is true that p, it would be necessary to determine whether p corresponds with P. But the correspondence theory implies that that consists in a correspondence of the proposition that p corresponds with P with something, call it P’. Then the same question arises again.43 One might reply that to judge that p, or determine whether p, is one thing, to judge that it is true that p or determine whether it is true that p is another. If the sentence ‘p’ is, say, ‘Tame tigers exist’, it 43

Such an argument is intimated by Frege, “Der Gedanke,” p. 60. 181

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does not refer to a proposition, thought, or judgment, whereas ‘it is true that tame tigers exist’, in the sense that is being interpreted by correspondence, does so refer since it predicates truth of one of these entities. To determine whether tame tigers exist we do not have to investigate judgments or other propositional objects. If we find that tame tigers exist, then some logical principle leads us to the conclusion that it is true that tame tigers exist, but only then is reference to a propositional object introduced. Thus we can reject Brentano’s claim that to accept tame tigers, we must simultaneously accept the being of tame tigers. However, it seems likely that even if Brentano had accepted this objection, he would still have objected to the infinite sequence that is generated by passage from ‘p’ to ‘it is true that p’. Whatever the conclusion about the regress argument, the conception against which it is directed, that of truth as correspondence to a state of affairs, seems unmotivated unless a sentence designates a state of affairs, or at least a true sentence does. But Brentano, both in the 1889 essay and later, offers characterizations of the truth of a judgment without any such assumption. And he seems to be rejecting this suggestion even if states of affairs are admitted when he writes: But if we were to suppose that the non-being of the devil is a kind of thing, it would not be the thing with which a negative judgment, denying the devil, is concerned; instead it would be the object of an affirmative judgment, affirming the non-being of the devil. (WE 134, trans. 117) At the end of the dictation (of May 11, 1915) from which this passage comes, Brentano says that “we may stay with the old thesis” (WE 136, trans. 119). But his reading of it is clearly deflationary. The next item in the compilation, a dictation from two months earlier, makes this deflationary reading more explicit, by emphasizing not only the kind of example with which he has raised difficulties previously but also bringing up oblique, modal, and temporal contexts. If I judge that an event took place 100 years ago, “the event need not exist for the judgment to be true; it is enough that I who exist now, be 100 years later than the event” (WE 138, trans. 121). He concludes that the thesis [that truth is adaequatio rei et intellectus] tells us no more nor less than this: Anyone who judges that a certain thing exists, or that it does not exist, or that it is possible, or impossi182

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ble, or that it is thought of by someone, or that it is believed, or loved, or hated, or that it has existed, or will exist, judges truly provided that the thing in question does exist, or does not exist, or is possible, or is impossible, or is thought of . . . etc. (WE 139, trans. 121–122) From our own perspective, we might summarize what Brentano says as that someone who judges that p judges truly if and only if p. Brentano lacks two things in order to come up with the familiar truth schema: some sort of general schema for judgment and seeing the predicate ‘true’ as a device of disquotation applied to linguistic items. Brentano was far from being the only philosopher of his time to question the correspondence theory of truth. After all, the coherence theory was a staple of British idealism, whose main exponents were contemporaries. And the pragmatists’ distinctive ideas about truth were advanced during Brentano’s lifetime, even though it was late in Brentano’s career that William James’s views on truth led to considerable debate. Nonetheless Brentano’s line of questioning seems to me of continuing interest, and the ideas discussed above have more in common with those of Alfred Tarski and his successors than with those advanced in the debates on truth at the turn of the century. His coming close at least to the propositional form of the now standard truth schema is not duplicated by another writer of the time known to me except Frege. Frege went further than Brentano in claiming in a few texts that the thought that p is true is just the same thought as p. That claim is bound up with Frege’s particular conception of judgment; he would reject the idea advanced above in connection with the regress argument, that ‘the thought that p is true’ introduces content additional to that of ‘p’, namely reference to the thought that it expresses. Although what appears to be a regress argument by Frege has been criticized, once the context in Frege’s theory of judgment is recognized it may be defensible. Where Brentano comes a little closer to Tarski is in suggesting the idea that the condition for the truth of a judgment should parallel its structure. To be sure, Frege does in explaining the language of Grundgesetze give compositional truth conditions that are more rigorous than anything Brentano offers, but he does not make the connection that Brentano does with the explanation of the notion of truth. Just what the connection should be between compositional truth conditions and explanations or definitions of truth has continued to be a disputed matter in our own day. 183

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8. Truth and Evidence If Brentano had stopped his account of truth with remarks like the last one quoted, he might count as an ancestor of what is nowadays called deflationism. But instead he continues and offers a characterization of truth in terms of evidence, that is, in terms of evident judgment. If a judgment is evident, then it constitutes certain knowledge. Evidence is therefore clearly a much stronger notion than truth. Although judgments of inner perception can be evident, and they would count as empirical for Brentano, his concept of evidence is for practical purposes rational evidence, since if a judgment is evident no reason can override it. Although he is critical of Descartes’s particular formulation (WE 61–62, trans. 52–54), Descartes’s clear and distinct perception seems to have provided a model for Brentano’s conception of evidence. In his late writing evidence seems to have been treated as a more basic notion than truth. Thus he follows his deflationary rendering of the import of the adaequatio formula with what reads as a definition of true judgment in terms of evident:44 Truth pertains to the judgment of the person who judges correctly—to the judgment of the person who judges about a thing in the way in which anyone whose judgments were evident would judge about the thing; hence it pertains to the judgment of one who asserts what the person whose judgments are evident would also assert. (WE 139, trans. 122, emphasized in the German) Thus, if an agent x affirms A with evidence, and an agent y affirms A, whether or not with evidence, then y judges truly. Brentano held that an evident judgment is “universally valid”; in particular no other evident judgment can contradict it. Thus any other evident judgment with respect to A will agree with x’s, so that the truth-value of y’s judgment is uniquely determined. If a third agent z denies A, then z judges falsely, as one would expect. Evidently this definition requires the possibility of comparing the content of the judgment of different agents or of agents of different times; it must make sense to say of y that his judgment affirms or denies what x’s judgment affirms or denies. 44

Oskar Kraus, Brentano’s disciple and editor, clearly reads this as a reductive definition; see WE xxiii–xxv, trans. xxiv–xxv. I would wish for more evidence before taking it that way, but for convenience I will refer to it as a definition. 184

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The definition faces a pretty obvious difficulty, which was pointed out by Christian von Ehrenfels.45 Suppose that an agent y affirms A. If it is possible for there to be an agent x who judges with evidence with regard to A, then by the above there is at most one possible result of his judgment, and if it is affirmative then y judges truly; if it is negative then y judges falsely. But suppose that it is not possible for an agent to judge with evidence with regard to A. Then it seems that Brentano’s characterization does not give an answer as to whether y’s judgment is true. Or, if one holds that a vacuous contrary-to-fact conditional is true, then both the affirmation of A and the denial of A will be true. Brentano’s disciple and editor Oskar Kraus offers another formulation: y’s affirmation of A is true if no possible evident judgment can contradict it, that is, deny A (WE xxvi–xxvii, trans. xxv). But, as Ehrenfels seems to have pointed out, if no evident judgment is possible one way or the other with respect to A, it seems that by Kraus’s criterion both a judgment affirming A and a judgment denying A will be true. To this objection Kraus replies that supposing that A exists, then even if knowledge about A were possible, it could not be negative (i.e., an evident negative judgment). But an evident affirmative judgment is impossible only because it is assumed that the existence of A is unknowable. This does not seem to me to avoid the conclusion that according to the definition, a negative judgment with regard to A is true. This type of objection touches Brentano particularly, because according to him the scope of evident judgment (for humans at least) is limited to the deliverances of inner perception and analytic judgments. Hence even simple common-sense statements about the outer world have the property that neither they nor their negations can be affirmed with evidence. The above remark expressing an epistemic criterion of truth was dictated by Brentano some years after Husserl had already published in the Logische Untersuchungen an account of truth in which there is an internal connection of truth and evidence.46 Husserl’s account is embedded in his intention-fulfillment theory of meaning and thus has a quite different context from Brentano’s. It would be distracting to engage in a detailed comparison of the two accounts. However, it is 45

See WE xxvii, trans. xxv–xxvi. Logische Untersuchungen VI, ch. 5; cf. Prolegomena (i.e., volume 1), §§49–51. The page references given will fit either the first or the second edition. 46

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instructive to see how Husserl deals with problems similar to those posed by Ehrenfels’s objection to Brentano. In the Prolegomena he asserts an equivalence between ‘A is true’ and ‘It is possible that someone should judge with evidence that A’ (§50, I 184). But he denies that they mean the same. More relevant to our present problem is that he insists that the possibilities in question in such statements are ideal possibilities, so at least many examples that come to hand of statements we cannot know to be true or false become irrelevant, as is presumably the case with the example in Kraus’s discussion of the existence of a diamond weighing at least 100 kilograms. Husserl is willing to assert the ideal possibility of knowledge of a solution to a problem in a case where the reason for thinking there is one is purely mathematical and he concedes that to find it may be beyond human capabilities; the example he gives is the general n-body problem of classical mechanics (I 185). In the fuller discussion of truth in the Sixth Investigation, Husserl discusses in general terms what he calls the ideal of final fulfillment (§37). An act is fulfilled to the extent that its content is presented in intuition.47 Final fulfillment involves the presence in intuition of the object, complete agreement of intuition with what is intended, and in addition the absence of any content in the fulfilling act that is an intention that calls for further fulfillment. Thus in final fulfillment the object itself is given, and given completely. Husserl illustrates these ideas by means of perception, although he insists that fulfillment by outer perception is always incomplete. That, however, serves his purpose in bringing out that in general final fulfillment is an ideal. The concept of evidence applies to “positing acts” of which judgments would be an instance (although Husserl also regards normal perception as positing its object).48 In the case of judgments, the object is a state of affairs (Sachverhalt); Husserl’s view about propositional objects is closer to that of the pupils with whom Brentano disagreed than to that of the later Brentano. The epistemologically significant concept of evidence applies to positing acts that are adequate in 47

Or “represented” in imagination; however, this case is excluded by the idea of final fulfillment. 48 Husserl’s positing acts correspond to Brentano’s affirmative judgments, in which an object is posited in Husserl’s language, affirmed or accepted in Brentano’s. Brentano regarded perception as involving a judgment. Husserl denied this, but the issue is at least initially terminological: according to Husserl, the simple positing of a perceived object is not yet a judgment. 186

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the sense of leaving no unfulfilled components, in which, again, the object is given completely (§38).49 Such evident positing has an objective correlate, which he says is “being in the sense of truth” (Sein im Sinne der Wahrheit), an echo of Aristotle that is no doubt derived from Brentano. This reliance on a strong concept of evidence to explain the notion of truth makes Husserl vulnerable to the type of objection made by Ehrenfels. What his reponse to it amounts to is that with respect to any positing act final fulfillment (or cancellation through conflict between what is intended and what is given) is “in principle” possible. Husserl’s own view of outer perception created a difficulty for this view. Even in the Logische Untersuchungen his position was that outer perceptions always contain unfulfilled intentions, because in perception the object is always incompletely given. At the time he seems to have thought that the impossibility of complete fulfillment of outer perception was only impossibility for us, and that in an appropriately ideal sense complete fulfillment is possible. By the time of Ideen I in 1913, he had changed his mind, and he states there that it belongs to the essence of outer objects that they can be given only from a perspective and thus incompletely (§§43–44); not even God could overcome the inadequacy of outer perception. Nonetheless he writes that complete givenness of the object is “predelineated as an Idea in the Kantian sense” (§143); complete givenness is approached as a kind of limit by an infinite continuum of perceptions of the same object in harmony with one another. It seems that truth itself will have to be adjusted to the fact that evidence in the strong sense also has the character of a Kantian idea.50 Let us return to Husserl’s statement of Prolegomena §50 that ‘A is true’ is equivalent to ‘It is possible that someone should judge with evidence that A’. This formulation is somewhat more perspicuous than the formulations of Brentano and Kraus. If we accept that it might be impossible to judge with evidence either that A or that not-A, then what we have is a violation of the law of excluded middle. Since the intuitionist challenge to classical mathematics of L. E. J. Brouwer, of which the first steps were taken during Brentano’s lifetime, the idea that the 49

In the same section Husserl allows that evidence admits of levels and degrees, but this applies to what he calls the more lax and less epistemologically significant concept of evidence. 50 We do not deal here with the later evolution of Husserl’s views on these matters, which move further from the view of the Logische Untersuchungen. See Føllesdal, “Husserl on Evidence and Justification.” 187

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law of excluded middle might be given up or qualified has become familiar to us, and it is one of the possibilities that has to be considered in developing an epistemic conception of truth. The most straightforward way of carrying this out would be to adopt something like the Husserlian formulation and declare that, if it is not possible to judge with evidence with regard to A, then A is neither true nor false. If evidence is interpreted as entailing the degree of certainty that Brentano takes it to, and we measure possibility by the actual capabilities of the human mind, that will lead to a counterintuitive result, for example that ordinary empirical judgments are neither true nor false. The development of epistemic conceptions of truth in the twentieth century has proceeded differently. Intuitionism, which offers the most rigorous and thorough development, is primarily a view about mathematics. We could translate Brouwer’s view into Brentano’s language by saying that A can be said to be true only when one judges with evidence that A. Unlike Brentano, Brouwer does not think it makes sense to talk about truth with regard to “blind” judgments. But rather than allow truth-value gaps, Brouwer interprets negation so that one can judge that not-A if one knows that an absurdity results from the supposition that one has a proof of A, that is, that one can judge with evidence that A.51 It follows that it is impossible for neither A nor not-A to be true, but it does not follow that either A or not-A is true. Although the idea has been advanced of extending the intuitionistic approach to logic and truth in general, this program has not been carried out, and the problem of certainty that we have been discussing is a serious obstacle to it. In intuitionism, possession of a proof of A guarantees the truth of A. But in most domains of knowledge even very strong evidence for a statement A might be called in question by additional evidence. The result is that although epistemic conceptions of truth have been found attractive by many philosophers, there is no canonical development of it for the empirical domain corresponding to intuitionism for the mathematical. Many writers have, following Charles Sanders Peirce and Husserl, taken what is true to be what is evident under highly idealized conditions. 51

Curiously, Kraus’s rendering of Brentano’s criterion for the truth of A amounts in Brouwerian terms to the truth-condition for not-not-A. That is not surprising given Brentano’s tendency to paraphrase judgments apparently not involving negation by negative judgments. 188

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In Brentano, the epistemic characterization of truth is offered after a deflationary reading of the correspondence formula. In the writing about truth in our own time, some writers have been led to some version of an epistemic conception by what is nowadays called deflationism, the view that the equivalence of ‘“p” is true’ and ‘p’ represents the whole content of the concept of truth, and perhaps in addition that the concept of truth serves no purpose beyond that of “disquotation,” that is, of passing from statements in which linguistic items are mentioned to statements in which they are used, and perhaps of generalization as in statements like “Everything Dean says about Watergate is false.” Although Brentano’s meditation on the adaequatio formula led in a deflationary direction, it would be overinterpretation to describe him as a deflationist in contemporary terms. He does not explain the transition from his deflationary remarks to the epistemic criterion. But he evidently thought that there is a connection, and in this respect he is a precursor of one strand of contemporary deflationism.52

52

I am indebted to Dagfinn Føllesdal, Kai Hauser, Peter Simons, and the editor for helpful comments. 189

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The study of the history of analytical philosophy generally begins with Frege. As a consequence, Edmund Husserl stands in some significant relation to that history almost from its beginning. Husserl and Frege exchanged letters in 1891; Husserl’s first book, Philosophie der Arithmetik (1891), contained critical comments on Frege’s Die Grundlagen der Arithmetik (1884); Frege reviewed Husserl’s book; and they corresponded again in 1906. The relation between Frege’s views and Husserl’s, particularly in Husserl’s Logische Untersuchungen1 (1900–1901), and the possibility of a significant influence of Frege on Husserl’s decisive turn away from psychologism in the late 1890s have been extensively explored. Husserl also enters the history at later points, in particular in the early period of the Vienna Circle. Influence of Husserl on Carnap is in evidence at least as late as Der logische Aufbau der Welt (1928),2 but already then Carnap’s philosophical direction is in many ways opposed to Husserl’s. Schlick wrote a widely read criticism of Husserl’s particular version of the synthetic a priori.3 My purpose is not to explore these or other historical relations, but rather to discuss some aspects of Husserl’s relation to analytical philosophy in a more philosophical way, following the example of Michael Dummett in his recent Origins of Analytical Philosophy.4 Dummett is interested not only in the origins of analytical philosophy, but 1

Cited hereafter as LU. I will give page references to the second German edition and to J. N. Findlay’s translation of that edition (cited as F), which I will quote with some modifications. The differences from the first edition, though important for many purposes, play no role in my discussion. 2 This was pointed out to me by Abraham Stone. 3 “Gibt es ein materiales Apriori?” 4 Cited hereafter as O. 190

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also in the origins of the “gulf” between analytical and so-called continental philosophy. From this double point of view Husserl is clearly of particular interest. In his early period his thinking was close enough to Frege’s so that they could at least have exchanges with one another. Yet Husserl was the founder of the phenomenological movement, at one time the paradigm of continental philosophy at least in the eyes of Englishspeaking philosophers, and which is certainly a major source of subsequent continental philosophy. Dummett locates the beginning of the gulf in Husserl’s transcendental turn of 1905–1907 and its published manifestation in Ideas I in 1913.5

I Dummett’s Origins is guided by a particular conception of what is fundamental to analytical philosophy, a conception which frames his assessment of Husserl’s significance for the history of analytical philosophy and his more detailed discussions of Husserl. It also frames Dummett’s more extensive and, as one would expect, more sympathetic discussion of Frege. Dummett’s starting point is a thesis concerning what he calls the philosophy of thought; he says that what distinguishes analytical philosophy is “the belief, first, that a philosophical account of thought can be attained through a philosophical account of language, and, secondly, that a comprehensive account can only be so attained” (O, p. 4). He doesn’t even attempt to propose an explanation of the term “thought” that wouldn’t be tendentious between the different analytical philosophers adhering to this view. Instead he relies heavily on Frege, whose use of “thought” has roughly the meaning of “proposition” in Englishlanguage philosophy. I shall make only a few remarks about the question of how accurate Dummett’s characterization of analytical philosophy is, with reference to the different periods of its history.6 And I shall distinguish two ways of objecting to it. First, Dummett holds that what has long been called the linguistic turn is the essence of analytical philosophy. Second, he 5

I will give page references to the original German edition; they are included in the two Husserliana editions and in F. Kersten’s translation. My quotations will largely follow that translation. 6 With respect to early analytical philosophy (by which I mean roughly the period from Frege through the publication of Wittgenstein’s Tractatus), see Hylton’s review of Origins. 191

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offers a very specific statement about what the linguistic turn is, a statement dependent on his conception of a “philosophical account of thought,” the search for which is a program he himself has followed and has found inspiration for in Frege. Some counterexamples to Dummett’s characterization would impugn only the latter, more specific formulation, not the more general idea that the linguistic turn is the fundamental move distinguishing analytical philosophy, however difficult it might be to give an adequate general statement of what the linguistic turn is.7 In fact, the idea that a certain kind of reflection on language is fundamental to much of philosophy does in my view characterize quite well one important period in the history of analytical philosophy, that of its rise to dominance in the English-speaking world, roughly from the early 1930s to the early 1960s.8 But the critical discussions of Dummett’s book have argued rather convincingly that his characterization does not fit the wider history.9 Dummett contends that Husserl exemplified a philosophical development essential to the prehistory of analytic philosophy, namely “the extrusion of thoughts from the mind.” According to Frege, thoughts are not constituents of the stream of consciousness; they exist independently of being grasped by a subject (O, p. 22). A similar view was held earlier by Bernard Bolzano, whose influence Husserl acknowledges. Just this step is taken by Husserl, first in his polemic against psychologism in the first volume (1900) of the Logische Untersuchungen. The result is what has been called a platonist theory of meaning. Evidently Dummett considers this theory a fundamental step on the road to ana7

Thus Herman Philipse questions whether Wittgenstein, not only a paradigm analytical philosopher but one to whom Dummett appeals, would embrace the idea of a comprehensive philosophical account of thought; see “Husserl and the Origins of Analytical Philosophy,” p. 167. In commenting on my APA paper, Dagfinn Føllesdal remarked that Quine, surely an exemplar of the linguistic turn, is skeptical about the very idea of thought as Dummett conceives it; cf. Føllesdal, “Analytic Philosophy,” p. 195. 8 The terminus a quo is chosen in part because the 1930s saw the beginning of the Oxford tradition of analytical philosophy as well as the emigration of leading logical positivists to the United States. Around 1960 the idea that “analysis of language” should displace “metaphysics” began to lose its hold. Another development of that time was the growing influence of Rawls, which ended analytical moral philosophers’ almost exclusive concentration on metaethics. 9 On early analytical philosophy, see Hylton, Review of Origins, and more generally Philipse, “Husserl and the Origins.” 192

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lytical philosophy. The reason is apparently that the “ontological mythology” that such a view involves gives rise to dissatisfaction that leads naturally to the linguistic turn. According to Dummett, One in this position has therefore to look about him to find something non-mythological but objective and external to the individual mind to embody the thoughts which the individual subject grasps and may assent to or reject. Where better to find it than in the institution of a common language? (O, p. 25) Dummett projects a highly idealized picture of how analytical philosophy originated, first through the extrusion of thoughts from the mind and then by the step just indicated to the linguistic turn. Husserl took the first of these steps but not the second. Dummett sees in this a respect in which Husserl has positive importance for the history of analytical philosophy. But he sees Husserl’s failure to take the second step as one of the roots of the separation between continental and analytic philosophy. Now had Dummett said nothing more of a positive nature about Husserl’s relevance to the history of analytical philosophy, then Peter Hylton would be justified in finding Dummett’s claim for Husserl’s importance seriously overstated.10 Dummett, however, implicitly makes another claim, with which I entirely agree. This is, roughly, that Husserl is of great interest as an object of comparison. The point is not to issue a call for an exercise in comparative philosophy. Rather, Frege and Husserl worked at a time when there was no such schism as the later analytical-continental one, and the problems faced by each were similar (O, p. 4). Although the actual debates between them were limited, they might have been much greater.11 I would, somewhat speculatively, enlarge Dummett’s case in the following way. There were two late nineteenth-century scientific developments that had very great importance for the development of 10 Hylton, op. cit. Hylton writes as if the issue were whether Husserl is a “precursor of analytic philosophy,” a claim he attributes to Dummett. I think that frames the question of Husserl’s relevance too narrowly, at least if one works with a conception like Dummett’s of what analytical philosophy is, or even with Hylton’s contrasting understanding of what is essential in early analytical philosophy. 11 For example if Frege had been a little younger when LU appeared and had not gone through the period of greatest discouragement in his life in the years just afterward.

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philosophy. One was the beginning of modern logic and (more broadly but a little less directly) the nineteenth-century transformation of mathematics, both decisive for early analytical philosophy in ways by now well known. The other was the development of scientific psychology, originally institutionally united with philosophy, but gradually emancipated from it. Many of the important founders of experimental psychology were psychologist-philosophers, the exemplary and most influential case being Wilhelm Wundt. The development of experimental psychology went hand-in-hand with the development of a more sophisticated philosophical psychology. Brentano’s contribution was mainly here, although he was a strong proponent of the growth of experimental psychology and through the work of pupils exercised a strong indirect influence on it as well. Husserl was perhaps the only major figure in philosophy who was formed intellectually by both the mathematical and the psychological currents of the time, as is illustrated by the fact that his principal mentors were Weierstrass and Brentano.12 Unlike Frege, he was able to see the issues surrounding “psychologism” from both sides. Although, at least in the Logische Untersuchungen, he does in a way “extrude thoughts from the mind,” he never at any time separates the issues concerning the nature of thoughts from the philosophy of mind. What Frege says about such matters combines rather traditional elements, such as a conception of “ideas” hardly differing from that of classical empiricism, with elements derived from or worked out in connection with his logic. Although Frege has the notion of grasping a thought (or, more generally, a sense), he says little about what this is. Husserl, for better or for worse, always connects what he has to  say about meaning with a much larger story about mind and consciousness. Although I am not qualified to engage seriously in the enterprise myself, I applaud the efforts of recent scholars such as Kevin Mulligan and Barry Smith to give developments in psychology an important place in the history of philosophy in the late nineteenth and early twentieth centuries. The attempt to develop a philosophical psychology by a 12

Husserl himself confirmed as much at a celebration of his seventieth birthday in 1929. See Schuhmann, Husserl-Chronik, p. 345. (Thanks to Dagfinn Føllesdal for pointing this out.) 194

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method that could be called scientific was, I think, another source of the standards of argument and analysis associated with analytical philosophy, although its influence was not especially marked on the figures of early analytical philosophy.13

II I provide these historical remarks as stage-setting for what is our proper concern, themes in Husserl that relate him in an interesting way to analytical philosophy as Dummett characterizes it. Our focus will be Dummett’s question, Why did Husserl not take the linguistic turn? And more generally, What separates Husserl from analytical philosophy, in particular in Ideas I? Dummett’s answer to the first question is that Husserl’s introduction of the noema, which Dummett sees as involving the generalization of the notion of meaning to all acts, made the linguistic turn impossible.14 This answer poses a difficulty for Dummett’s historical picture, since the essentials for the generalization of meaning to all acts are already present in the Logische Untersuchungen. Acts are intentional experiences. And intentional experiences are distinguished by the peculiarly intentional relation to an object that for Brentano was distinctive of “mental phenomena.” A point Dummett himself emphasizes is that linguistic expressions, on actual occasions of use, are meaningful by virtue of accompanying “meaning-conferring acts” on the part of the speaker. The meaning on that occasion of the expressions the speaker uses is a function of these acts, which themselves have semantical properties. The Fifth Investigation is devoted to exploring these matters for acts in general. All acts have matter and quality, which are analogous 13

These considerations would also suggest that the anti-psychologism of Frege, Husserl, and other figures of the turn of the century should be studied with close attention to the views of those they were criticizing. Much valuable work of this kind has been done by Eva Picardi, herself a former student of Dummett who played a role in the origins of Origins (O, p. vii). 14 Dummett’s reading of Husserl is clearly much influenced by Dagfinn Føllesdal. That is also true of my own. It would be interesting to see the issues considered here discussed by a commentator who disputes Føllesdal’s theses concerning the noema (see “Husserl’s Notion of Noema”). Philipse is apparently such a commentator (see O, p.  71), but the discussion of Husserl in “Husserl and the Origins” takes another direction. 195

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to sense and force in Frege’s scheme. For present purposes, it is matter that is important, since it is matter that determines the relation to an object, not only to what object an act is directed, but how it is directed to it. The matter, therefore, must be that element in an act that first gives it reference to an object, and reference so wholly definite that it not merely fixes the object meant in a general way, but also the precise way in which it is meant. (LU, 5th Investigation, §20, II/1 415, F 589; emphases added in the second edition) Shortly thereafter Husserl characterizes the matter as the “sense of the objectual interpretation [Auffassung]” (II/1 416). Now the matter is, according to Husserl, a moment of the act, whereas according to him meanings are ideal. In the Logische Untersuchungen, they are “species,” that is, universals instantiated by something concrete. But what instantiates them is the matter of meaning-conferring acts.15 Husserl introduces this species conception of meaning explicitly only for expressions. Matter and quality together constitute what he calls the intentional essence of an act. In the special case of acts “that function or can function as meaning-conferring acts for expressions,” he talks of the semantic essence (bedeutungsmäßiges Wesen) of the act. “Its ideating abstraction gives rise to the meaning in our ideal sense” (LU, 5th Investigation, §21, II/1 417, F 590).16 Husserl makes clear, however, that different acts of other kinds, for example perception, even of different subjects, can share intentional essence and matter in particular, and he ends a discussion of different types of acts by saying that something analogous holds for acts of every kind (II/1 420). The motivation for Husserl’s introducing ideal meanings only for expressions is probably his concern to give an account of the meaning of linguistic expressions, and not to confine talk of meaning to the case of linguistic expressions alone. What, then, is the difference made by the introduction of the noema? In the terms of Ideas I, the earlier concepts of matter and quality describe aspects of noesis; the matter of an act is a genuine moment of it, 15

Compare Simons, “Meaning and Language,” p. 114. Husserl says he will have to investigate later whether all acts can serve as meaningconferring acts. 16

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so that it is what Husserl calls reell. I’m not enough of a Husserl scholar to give a full account of why Husserl became dissatisfied with the conception of ideal meanings as species. Clearly, he thought of the correlation of noesis and noema as more intimate than that between the matter of an act and the ideal meaning it instantiates. On Husserl’s account, different noeses, that is, different acts, with exactly the same noema differ only “numerically,” or only as events in conscious life; intentionally they are the same. However, the equivalence involved is far more refined than what we would ordinarily recognize as sharing a species. Indeed, in §94 of Ideas I, Husserl makes it clear that the correlation of the noemata to acts of judgment is more refined than the assignment of meanings that concerns logic, what we might call the assignment of the proposition expressed. Thus, in the case of linguistic expressions, the move from the species conception of ideal meaning to the noema conception introduces a more refined way of distinguishing among meaning-conferring acts. There remains the question of how equivalences among acts that are not meaning-conferring should be determined. The Logische Untersuchungen had suggested the possibility of applying the less refined species account here. Husserl’s move to the noema yields a more fine-grained account of act equivalences in this case as well. In §94 of Ideas I, Husserl brings the notion of noema to bear on perceptual judgments. He says that, in the case of an object presented in a certain way, that mode of presentation of that object enters into the noema of the act of judgment (p. 194). Suppose I perceive an apple tree before me and judge that it is in bloom. I might express this by saying, “That apple tree is in bloom.” On this view, however, the noema of the judgment would incorporate the noema of the perception of the tree, which already on the level of sense would be far richer than what is communicated in the reference to “that apple tree.” The hearer may understand the latter with the help of his own perception of the tree, the perspective of which will differ from the speaker’s, so that this perception will have a distinguishable noema. Which apple tree is referred to may of course also be determined in some other way, so that the hearer does not need to perceive the tree in order to understand what apple tree is being said to be in bloom. Husserl’s focus in this passage is on the sense of the judgment as an experience (Urteilserlebnis). We should perhaps think of the question as being, first of all, What is the full sense of the judgment when it is 197

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made privately, in response to the perception?17 Husserl explicitly refrains from bringing in at this point the complications of expressing the judgment verbally. The contrast Husserl makes between the full noema that is at issue when we “take ‘the’ judgment exactly as it is conscious in this experience” and the judgment that concerns formal logic implies, for the reasons just given, that we should not expect full identity of sense between them (pp. 195–196). The contrast Husserl explicitly makes, however, is not one of sense but one drawing on other dimensions of the noema. Before going further, we have to consider the connection of the concept of noema with Husserl’s transcendental idealism. That the introduction of the noema coincided with the transcendental turn is, for Dummett, a reason for locating the beginning of the gulf between analytical and continental philosophy in the development leading to Ideas I. This could not be because idealism as such is alien to analytical philosophy; it is not. But it can hardly be disputed that Husserl’s version of idealism is alien to early analytical philosophy. Even those who dispute the interpretation (held by Dummett) of Frege as a thoroughgoing realist will agree that there is no place in Frege’s philosophy for a transcendental ego and its “constitution,” whatever that elusive Husserlian term means. And of course Russell and Moore explicitly reacted against British idealism. Although there are echoes of transcendental philosophy in Wittgenstein’s Tractatus, here too the upshot is quite different from that in Husserl, as for example in Wittgenstein’s statement that solipsism in the end coincides with pure realism (5.64). Thus we need to ask, How far is Husserl’s conception of the noema bound up with idealism? It is certainly explained in a way that presupposes the phenomenological reduction, at least in §88 of Ideas I, where Husserl uses the example of perceiving with pleasure a blooming apple tree. The explanation of the conception includes the equation of the perceptual sense (noematic sense) with “the perceived as such,” of judging with “the judged as such,” and so on (p.  182), equations that have given rise to much controversy among Husserl’s interpreters. Husserl wants to describe the fact that, when the positing of the world and of particular objects in a perception or a thought has been bracketed, it 17

We will consider later the problem of perception as attributing properties to an object. If I see a blooming apple tree, its being in bloom is plausibly already part of the noematic sense of the perception. 198

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still remains a perception of, or a thought of, its objects. In his example of perceiving with pleasure a blooming apple tree, the “transcendent” tree itself is bracketed. “And yet, so to speak,” Husserl writes, everything remains as of old. Even the phenomenologically reduced perceptual experience is perception of “this blooming apple tree, in this garden, etc.,” and likewise the reduced liking is a liking of this same thing. (Ideas I, pp. 195–196) In the natural attitude, when I see the tree, I take it for granted that it is really there; in Husserl’s terms from the Logische Untersuchungen, “positing” belongs to the quality of my act. In Ideas I, Husserl uses the term “thetic character.” It belongs to my perceptual consciousness of the tree to take it to be really there. This is to say both more and less than that I believe the tree to be really there: more because it is part of perceptual consciousness; less because, although my perception may posit the tree, I may because of other knowledge distrust it and believe the tree is not really there. Since this positing is a moment of the perception itself, it does not disappear with the reduction; it is just “put out of action.”18 But what Husserl emphasizes at this point is that what he is calling the sense of the perception is not bracketed.19 It is not in any case posited in the act itself but, rather, in the phenomenologist’s reflection, despite his not being entitled to make any positing regarding the outer world. Since it is the sense of a perception, it must be the sense that the perception has independently of whether its positing is bracketed, and independently of what judgments are made on the basis of it. (If there are such judgments, they too are potential fodder for phenomenology, although in that case what is put out of action is an essential element of what makes them judgments as opposed to propositional acts of other kinds.) On my reading, it is clearly not necessary to undertake the phenomenological reduction in order to talk of the meaning of acts, and in the passage that has concerned me Husserl says explicitly that “obviously the perceptual sense belongs to the phenomenologically unreduced perception (perception in the sense of psychology)” (Ideas I, §89, p. 184). 18

“As phenomenologists we abstain from all such positings. But on that account we do not reject them by not ‘taking them as our basis,’ by not ‘joining in’ them. They are there; they belong essentially to the phenomenon” (Ideas I, §90, p. 187). 19 I ignore the fact that phenomenology also involves an eidetic reduction. 199

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For this reason, I think that Husserl’s purpose in bringing in the reduction at this point is to emphasize that the sense of our acts survives it, and the reduction makes it possible to engage in reflections having as objects only objects that either are really immanent in consciousness or are meanings of them (in the broad sense including thetic character as well as sense, but not including reference). The conception of the noema is thus at least to a certain degree independent of the reduction and of transcendental idealism. Husserl in Ideas I is, to be sure, more distant from analytical philosophy than he was in the Logische Untersuchungen. What is responsible for this is not, I think, the generalization of meaning to all acts, which I have argued is already present in the Logische Untersuchungen. Nor is it the further development of this generalization in Husserl’s theory of the noema. Instead it is, I propose, the Cartesianism underlying the transcendental reduction. There is a step from the generalization of meaning to the reduction, but it requires a highly contestable assumption about meaning. Roughly, this assumption is that it is possible to express and to explicate the meaning of our acts, even on a quite global level, without making any presuppositions about reference. In §89 of Ideas I, Husserl describes statements about external reality as undergoing through the reduction a “radical modification of sense” (p. 183). Bringing to bear Frege’s theory of indirect reference,20 we could describe this reduction as consisting in our putting our whole description of the world into one big intensional context, where what is designated is not the ordinary reference of the words but their sense. This description must assume, however, that these senses do not presuppose, for their very existence and identity, reference to external reality. In particular, it must be assumed that there are no “Russellian” or “object-dependent” thoughts about external reality, which by their very nature involve reference to particular objects, often in the immediate environment. Another sort of assumption I have in mind, however, is even stronger than the rejection of such thoughts. For meaning might be dependent on external reference in a more global or diffuse way. For example, it might be that we could not entertain the thoughts we do without an existing external world. Or, short of the nonexis20

In fact Husserl echoes Frege’s theory in this passage, though probably not consciously, in using words such as “plant” and “tree” in quotes to indicate the modification of their meaning (p. 184). 200

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tence of the external world, it might be that we could not entertain the thoughts we do about the world if they were radically false. Such a more global dependence of meaning on reference does not imply the existence of Russellian thoughts as they are usually understood. But it is incompatible with the contestable assumption about meaning that leads from Husserl’s generalization of meaning to his reduction. The Cartesian tenor of Husserl’s justifications of the reduction in Ideas I as well as in other texts, such as his Cartesian Meditations, clashes with at least the most characteristic views among analytical philosophers. At the time of Ideas I, Husserl’s transcendental idealism probably also clashed with more widely held views in British and American philosophy; that was after all a time of reaction against idealism and the revival of realism.21 I would suggest, however, that it is only later developments that make this clash a step on the way to the gulf between analytic and continental philosophy. As regards Husserl’s own thought, such a gulf is always limited by his adherence to rather traditional scientific ideals. I would further suggest that we can’t very meaningfully speak of “continental” philosophy in anything like the sense current since the Second World War before Heidegger’s Sein und Zeit (1927) and other work of the 1920s, such as that of Jaspers.22 Moreover, we must consider that Husserl’s transcendental idealism did not find wide acceptance and was not maintained in anything very close to Husserl’s form by the most influential later phenomenological philosophers.

21

Husserl gave lectures in London in 1922. There does not seem, though, to have been much understanding between him and the British philosophers he met. See Spiegelberg, “Husserl in England.” 22 Consider Husserl’s own comment, referring to his preface to the first English translation of Ideas I, which appeared in 1931: No account is taken, to be sure, of the situation in German philosophy (very different from the English), with its philosophy of life [Lebensphilosophie], its new anthropology, its philosophy of “existence,” competing for dominance. Thus no account is taken of the reproaches of “intellectualism” or “rationalism” which have been made from these quarters against my phenomenology, and which are closely connected with my version of the concept of philosophy. In it I restore the most original idea of philosophy, which, since its first definite formulation by Plato, underlies our European philosophy and science and designates for it a task that cannot be lost. (“Nachwort zu meinen Ideen,” p. 138 of reprint; my translation) 201

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III On Dummett’s reading, Frege parallels Kant in distinguishing between sensibility and understanding, between the faculty of sensation and that of thought. Where Frege takes the linguistic turn, he applies it to the study of thoughts. He has quite a bit to say about ideas (Vorstellungen), taking as prominent examples ideas which Kant would have called sensible, in particular sense-impressions. But Frege makes no use of a connection between ideas and language to get at the structures of ideas. This is not only, though, because ideas have subjects as bearers, for so do propositional attitudes, but Frege’s writings contain serious suggestions as to how to understand the structure of propositional attitudes by way of an analysis of sentences expressing them. This simple observation is relevant to the question whether Husserl’s generalization of meaning precluded the linguistic turn. For the generalization, that is, the extension of the notion of meaning beyond its application to language, is most in evidence when it is applied in domains whose relation to a domain of thought is not simple or straightforward. Husserl repeatedly brings up examples from either perception or imagination. Dummett evidently believes that attributing something like a sense to perceptions is incompatible with the linguistic turn (O, p. 27). The question is, Why? An inadequate answer would be that a philosopher who believes that perception involves something fundamentally different from thought could not take the linguistic turn. For Frege and a large number of subsequent analytic philosophers, including Dummett himself, who certainly do take the linguistic turn, also accept the Kantian distinction between perception and thought.23 In any event, the acceptance of this distinction does not obviously go against Dummett’s axiomatic characterization of the linguistic turn: that thought can and must be analyzed in terms of language. So we must seek further to see where and how Husserl might have violated Dummett’s axioms. Thoughts as Frege understood them are propositional, and Frege’s steps toward the linguistic turn are thus bound up with the context principle. Translated into the terms of an inquiry into thought, the principle says that “there is no such thing as thinking of an object save in the course of thinking something specific about it” (O, p.  5). One 23

Dummett explicitly affirmed this view in his reply to my APA paper. 202

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might say that, at least in the domain of thought, intentionality is fundamentally propositional. As for perception, according to Frege, something “non-sensible” is necessary for perception to represent an outside world. Discussing Frege’s view of perception, Dummett argues that this “non-sensible” must be a complete thought and, at least in most cases, a judgment (O, p. 97). That would give a handle to the linguistic turn, though not one developed by Frege. We are, however, still left with the sensible element, in Frege’s case the sense-impressions. For Frege himself that remained an obstacle, because in his view ideas are incommunicable. The notion that there is something incommunicable in sensory experience dies hard, as is shown by contemporary controversies about qualia. But is it clear that every philosophical view about such incommunicability is incompatible with Dummett’s axioms of analytical philosophy? To show this, we would have to show that sense-impressions or qualia or whatever either belong to the domain of thought or else do not exist. However this may be, Dummett’s claim that it is Husserl’s generalization of meaning that precludes him from taking the linguistic turn raises other issues than those about sense-impressions.24 Let us pursue the matter of Husserl’s view of the perceptual noema. Dummett attributes to Husserl the view that the noematic sense of acts in general is expressible in language, a view developed by Føllesdal’s pupils, particularly Smith and McIntyre in Husserl and Intentionality. It seems that such expression should give us the same kind of handle on the noematic sense of perceptions as we have on the structure of thoughts. That would call in question Dummett’s claim that Husserl’s attribution of sense to perceptions precludes him from adopting the twin axioms of the analytical tradition. Husserl describes the noematic sense of a perception as “the perceived as such”; one way of saying what this involves would be to say that it is the sense that would be expressed by the subject in saying what he perceives. Clearly any one statement would express this sense very incompletely. So the sense would have to be taken to be expressible in the sense that the subject is able to express, through more and more detailed description, everything contained in it. Full expression could be an infinite task. Moreover, there is a criterion of the accuracy 24

In fact, Dummett is almost silent on Husserl’s notion of hyletic data and does not rest any of his case on it. 203

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of an expression: what is reported should be only what is perceived and not more, although it can and should include what is illusory, provided that it is illusory perception and not a mistaken judgment of some other kind. This may be a difficult distinction to make, but Husserl’s conception of horizon is sensitive to the facts involved. The difficulty, related to other difficulties about meaning discussed in the analytic tradition, is how to separate what belongs to the perception itself from what belongs to the background the subject brings to it and the inferences he makes from it. Dummett admits that noematic senses generally are expressible. But why does he nonetheless think that Husserl’s theory of the senses of perceptions—or of acts generally—makes the resources of an analysis of language unavailable to him? One reason seems to me to point to something important about perception, though it does not get to the heart of the issue. Dummett refers to two additional components of Husserl’s noema beyond the noematic sense, components he says are not expressible. The first such aspect of the noema plays a role like that of Frege’s force; an example is the positing involved in normal perception. The second aspect is perhaps not really a dimension of meaning at all; it is what makes an act the particular kind of act that it is—a perception, imagination, or judgment. If there is enough correspondence between language and other embodiments of meaning, we can capture noematic sense and the first of these aspects of the noema by using words of the right sense and force. But how could words express the second additional aspect? Words can describe it, as when we say that an act is a perception. And perhaps words could express it in a broader sense of “express,” as when we talk of expressing emotion, or when Wittgenstein talks of the natural expression of pain. But these questions of expression, interesting though they are, are not an issue between Husserl and analytical philosophy as Dummett characterizes it. For they concern what distinguishes perception from thought. The second, more fundamental reason why Dummett thinks Husserl’s conception of the noema of acts like perceptions violates his axioms of analytical philosophy is expressed in the following telling comment: We should expect the veridicality of the perception or memory, the realization of the fear or satisfaction of the hope, and so on, 204

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to be explicable as the truth of a judgment or proposition contained within the noematic sense; but we do not know how the constituent meanings combine to constitute a state of affairs as intentional object, since they are not, like Frege’s senses, by their very essence aimed at truth. (O, p. 116) Perception, according to Husserl, is an act directed to the object perceived; if we can attribute to it sense and reference, the reference, if it exists, will be just the object perceived. It thus seems that what the sense would have to “aim at” is reference to this object, something quite different from truth. Husserl has a reply to Dummett’s objection, a reply drawing on a dimension of his philosophy that Dummett does not treat in Origins or elsewhere, though it has some relevance to his own views. There is something a meaning-intention aims at, what Husserl calls “fulfillment,” which is achieved when the object of the act is given. The schema of intention and fulfillment is central to Husserl’s account of meaning, in particular in application to nonlinguistic cases like perception. In external perception the object is given, leibhaft gegeben in Husserl’s famous phrase. That case has, however, a special complexity because external perception always contains unfulfilled intentions toward aspects of the object that are not properly speaking perceived, such as the back and the inside of an opaque object. A full description of the meaning of a perception would have to describe both what is “bodily present” and what would fulfill the unfulfilled intentions in the perception. The intention-fulfillment schema generalizes not the relation of propositions to truth, but their relation to verification. In fact, in Husserl’s discussion of truth, much of what he says suggests a verificationist view.25 This is of interest because there is a line of descent from Husserl to Heyting’s explanation of the intuitionistic meaning of the logical connectives, and from there to much of what Dummett himself has written about an anti-realist program in the theory of meaning. It seems to me that, to be consistent with his own views, Dummett has to take the difficulty with Husserl’s generalization of the notion of meaning to lie 25 See LU, 6th Investigation, §§36–39. These sections treat complete verification, however, as only an ideal possibility, and even that possibility is later called into question by the thesis of Ideas I that the inadequacy of perception of transcendent objects is essential to them. These issues are instructively discussed in chapter 3 of Gail Soffer, Husserl and the Question of Relativism.

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in the manner of its generalization to categories other than sentences, propositions, or judgments, rather than in Husserl’s replacement of the notion of truth with the more directly epistemic notion of fulfillment. In his reply to my APA paper, Dummett raises another point, namely that Husserl does not give a compositional theory in his discussions of meaning. This can’t really be quarreled with: though Husserl did have ideas for the program of giving such a theory, even in application to linguistic meaning his position is far less developed than Frege’s. It is also the case that Husserl does not hold a principle like Frege’s context principle; for Husserl, terms are at least as basic units as sentences. But this is not a fatal obstacle, as is indicated by the existence of formalized languages based on the λ-calculus and their application to the semantics of natural language. I suspect that what Dummett sees as fatal to Husserl’s taking the linguistic turn is his generalization of the notion of meaning to a domain where a compositional theory is not possible. That that might be the case for perception is not wildly unlikely. But since perception is not thought, the implications of such a  conclusion for the linguistic turn as Dummett conceives it are not obvious.

IV Now let us consider the delicate question of whether fulfillment of a perception (or perhaps of any act) can properly be considered to be, in Dummett’s terms, the verification of “judgments or propositions contained in the noematic sense” (O, p.  116, quoted above). Husserl’s view was that perceptions are “nominal” and not “propositional” acts; an expression in language of their senses would, I have suggested, be given by saying what is perceived. That would be done more faithfully to Husserl’s intention by using noun phrases rather than sentences. Furthermore, Husserl distinguishes the positing involved in perception from that in judgment. The former positing might be compared to using a singular noun phrase with the presupposition that it designates something, though we should not rush to the conclusion that some proposition to the effect that the phrase designates something, or of the form “P exists,” where P is the phrase in question, is part of the noema of the act. Still Husserl seems to regard perception as attributing properties to the perceived object. 206

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It is instructive to consider a passage in Ideas I, §124, the same section Dummett adduces to justify attributing to Husserl the thesis that noematic senses are expressible (O, p. 114). Husserl writes: For example: an object is present to perception with a determined sense, posited monothetically in the [thus] determined fullness. As is our normal custom after first seizing upon something perceptually, we effect an explicating of the given and a relational positing which unifies the parts or moments singled out perhaps according to the schema, “This is white.” This process does not require the minimum of “expression,” neither of expression in the sense of verbal sound, nor of anything like a verbal signifying. But if we have “thought” or asserted, “This is white,” then a new stratum is co-present, unified with the purely perceptual “meant as meant.” In the next paragraph of §124 (quoted by Dummett), Husserl writes, “ ‘Expression’ is a remarkable form, which allows itself to be adapted to every ‘sense’ (to the noematic ‘nucleus’) and raises it to the realm of ‘logos,’ of the conceptual and thereby of the universal.” The “new stratum,” evidently conceptual, must be what prompts Dummett’s comment that the noematic sense “can be expressed linguistically, but is not, in general, present as so expressed in the mental act which it informs” (O, p. 114). In the passage I have quoted, Husserl does not use noun phrases to express the sense, as I have suggested he might have done; rather he uses a sentence. That seems to me, however, not the essential point. It seems that neither the sentence, “This is white,” nor a noun phrase like “this white thing” gives quite accurately even that part of the meaning of the perception it is meant to render. On Husserl’s conception, nominal acts are simpler than propositional acts; nominal acts simply intend an object, whereas a synthesis connecting such references is necessary for judgment. Moreover, it is by expression that the “conceptual” and “universal” are brought in. The reference to explicating “parts or moments” also suggests that it may be Husserl’s view that what is meant perceptually is the object’s particular moment of whiteness, not that it is white.26 If that is so, then the 26 This is the view taken by Kevin Mulligan in his rich and illuminating article “Perception.” His interpretation refers, however, to the Logische Untersuchungen

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expression in language does not quite give the perceptual sense, since that aspect is not explicitly preserved in the linguistic expression. But elsewhere (for example at the end of §130), Husserl does say that the noema contains “predicates.” That seems to be his dominant view in Ideas I. If there is equivocation, it is in response to a genuine philosophical difficulty, which, though its particular formulation may be an artifact of Husserl’s apparatus and commitments, also arises in other philosophical discussions of perception. The difficulty is how perceptual consciousness is related to belief and judgment. One is reminded of the debate of recent years about whether there is a “nonconceptual” content of experience, with Gareth Evans and Christopher Peacocke taking the affirmative side and John McDowell the negative.27 More simply put, Does the statement that someone sees that this is white report what he sees, or rather report a judgment he makes on the basis of what he sees? It is not clear to me how Husserl reconciles the view that nominal acts are inherently simpler than propositional acts with the view of perception as attributing properties to the object and therefore as presumably involving the subject in something that, if not exactly judgment, at least has the content that x is F. And the source of my unclarity is not only, I think, the limitations of my knowledge of Husserl. Let me first consider the view that Mulligan finds in Husserl’s earlier writings. In fact, it is not directly inconsistent with the interpretation of Ideas I that I have favored, according to which the noema of an act attributes properties to the object. For it is a view about the objects of perceptual acts. According to this view, perception of a white object will contain a perception of its color moment. If the subject’s attention is directed to the color moment, however, things will be in a way reand Husserl’s 1907 lectures, Ding und Raum, so to texts earlier than the Ideas. Still, that Husserl continued to hold this view in later years is indicated by his account of the genesis of perceptual judgment in Erfahrung und Urteil; see below. 27 See Evans, Varieties, ch. 5; Peacocke, Study of Concepts; and McDowell, Mind and World, lecture 3. The discussion in Origins of the consciousness of animals seems to be responding to this debate, and Dummett mentioned McDowell’s view in his reply to my APA paper. I have found it difficult to place Husserl’s position on these issues. Mulligan clearly interprets the earlier Husserl as being on Evans’s side, and the conception of “pre-predicative experience” in Erfahrung und Urteil does look to tend in that direction. But the fact that the noema is very much in the background in that work makes it difficult to draw any definitive conclusion. 208

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versed: the perception of the color moment will, as a perception of a moment of a certain object, “contain” a perception of the object, but now relegated a little bit into the background. It is important to realize that these remarks concern the object and not the noematic sense. But the implication seems to be that an act directed to the moment of whiteness will have its own noematic sense. It seems that we could not rule out different acts, or even different perceptions, having the same moment of whiteness as their object but differing in noematic sense. In what could this difference consist? At least one possible (no doubt partial) answer would take us back where we were before: that different acts would attribute to the moment different properties. That seems to be the answer implicit in Ideas I, and I am not sure what other answers are available. I confess I also have difficulty understanding what the moments corresponding to properties and relations are. Can I understand what an object’s moment of whiteness is without understanding what it is for it to be white? Husserl might concede that I cannot but reply that neither is necessary in order to see the object’s moment of whiteness. But how is seeing a moment of whiteness different from seeing a white object whose color is visible? That there is some consciousness of the color of an object that is more primitive than applying the specific concept white to it will probably be accepted by all parties to such disputes. But if the moment is not derivative from the concept or property, why is its specific description helpful in understanding how perception of a white object can ground the judgment that it is white?28 It seems as difficult to get from a perception of a white color-moment to the judgment that the object is white as to see that the object is white to begin with. If the perception of the moment is thus derivative, have we really captured the greater primitiveness of the consciousness of color? It seems to me that an appeal to perception as perception of moments of properties does not resolve our difficulty. Another point is that it is not at all clear how Husserl conceives the role of such moments where relations are concerned. In introducing the conception of a property-moment, Husserl says, in his Third Investigation, 28

The view that the perceptual moment is not derivative from the property seems to me more plausible in itself and probably as an interpretation of Husserl. Consider an object that is red in a particular way, say one that is scarlet. If its color moment derives from the property, then it seems it will need to have both a moment of redness and a moment of scarletness, and these would have to be distinguished. But I do not find any phenomenological basis for such a distinction. 209

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that “every non-relational ‘real’ predicate therefore points to a part of the object which is the predicate’s subject” (LU, 3rd Investigation, §2, II/1 228, F 437; emphasis mine). So far as I have determined, though, he does not say here whether something analogous holds for relational predicates of two or more places. In perceptual cases, he might well say that to relations correspond certain unity-moments of what is perceived as a whole; certainly it was part of his view that there are such moments. Outside the perceptual context, however, that line of argument is highly strained. In the account of the genesis of judgments of the form “S is p” in §§50–52 of Erfahrung und Urteil, Husserl treats perception of the object S and of its p-moment. And in §53 he discusses the corresponding issue of simple relational judgments. I don’t find his treatment very clear, but at least he avoids claiming that, if A is greater than B, there is something “in” A that is the individual manifestation of its being greater than B. Instead the text seems to favor the interpretation according to which, in general, relations do not have corresponding to them moments of the objects they relate in the way that monadic properties do. For example, Husserl summarizes the discussion with the remark: Accordingly, we must distinguish: 1. Absolute adjectivity. To every absolute adjective corresponds a dependent moment of the substrate of determination, arising in internal explication and determination. 2. Relative adjectivity, arising on the basis of external contemplation and the positing of relational unity, as well as the act of relational judgment erected on it. (Erfahrung und Urteil, §53, p. 267; trans. pp. 224–225) In cases where the noema of a perception attributes a monadic property (say, whiteness) to the object, it is reasonable to suppose that the perception is, among other things, of a moment corresponding to that property. A perceptual noema will, however, also attribute all sorts of relations; Husserl’s view on the extent to which these too are based on perception of moments is not clear to me. Husserl’s reluctance to extend such a view to even the most basic relational judgments casts doubt on the attribution to him of the suggestion (to me very implausible) that every relation between objects A and B holds by virtue of a moment of some complex consisting of A and B. 210

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Let us attack again the distinction between perception and perceptual judgment. We can certainly distinguish between seeing a white object, say a white sheet of paper, and seeing of the sheet that it is white, or seeing that there is a sheet of white paper present. Now I may see a white sheet of paper without in any way identifying it as such, for example if the lighting conditions deceive me as to its color and for some other reason I also do not detect its being paper. In that case only another person, or I myself in the light of later knowledge, can say that I see (or saw) a sheet of white paper. For this reason, we normally take “x sees y” to be a straightforward predicate, with whatever replaces “y” as purely referential. But in the normal case, our perception is of a sheet of white paper in an intentional sense; on the interpretation we have been following, we could use the phrase “a sheet of white paper” to render part of the noematic sense of the perception. In this situation we see it as a sheet of white paper. For Husserl, that the conception of the noema as attributing properties such as these does not imply that we judge that the paper is white, as perhaps we do when we express our perception by making a remark to that effect, should be clear from the above-quoted passage from §124 of Ideas I. Reserving the locution “see that the paper is white” for the case where there is a judgment would preserve Husserl’s view of perception as a nominal and not a propositional act.29 I offer these observations in order to clarify the distinction between the noematic sense of a perception and the content of a perceptual judgment. But I still have to consider the question of the simplicity of perception. Our inclination would be to think of predicates in a more or less Fregean way, as sentences with empty argument places, so that, if our perception has the content “a white sheet of paper,” that perception would presuppose “x is white” and “x is a sheet of paper.” But we should not assume that Husserl thought of predicates in this way. In his account in Erfahrung und Urteil, the clearest difference between the “pre-predicative” level of perceptual experience and the level at which predicative judgment emerges is that the attribution of properties to the object at the former level is implicit and only becomes 29

The suggestion of using a distinction between seeing as and seeing that in this connection was made to me in conversation by Pierre Keller, to whom I am much indebted here. 211

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explicit upon both singling out certain properties and, by a synthesis giving rise to a judgment, formulating judgments of the form “S is p.” This account would allow Husserl to hold, as Evans did, that there is a level of experience that involves attributing properties to objects but does not require having the concepts that enter into the judgments. Thus, for example, seeing something as white might be reflected in behavior in various ways without the judgment that it is white being formulated and, in particular, without undertaking the commitment that such a judgment involves.30 On this account, the perceptual judgment that x is white, for example, makes explicit something implicit in the perception. Husserl clearly thinks that even the most primitive judgment applies to the object a general concept (Erfahrung und Urteil, §49, pp. 240–241, trans. p. 204), though the point is obscured in his account of the genesis of a monadic judgment by his emphasis on attending to moments. For him, there is always then still implicit in the judgment a reference to the general essence, say whiteness. Husserl does not tell us, though, how the generality arises. Since he makes clear that something of the kind is already present in pre-predicative experience, however, it too could be a making explicit of what was implicit. Our problem reduces, finally, to an independent difficulty, namely how a propositional act arises by a “synthesis” of subject and predicate and how it is thus founded on prior nominal acts. Husserl’s view seems to me bound to leave mysterious how the generality of the predicate arises. We can agree with Husserl that there is a level where predication remains implicit while also agreeing with Frege that what is thus implicit is something of propositional form, what I have expressed as that x is F. On this strategy, propositional acts are indeed founded on nominal acts, but in the following way: acts with definite propositional content are seen to arise from the making explicit of contents of perception that are already propositional, though implicitly so. This making explicit, by singling out one particular predicate, obviously leaves out much else that is part of the content of the perception. But a simple judgment does have the property of being founded on prior nominal acts, since it is clearly founded on the perception of the object 30

One difference between Husserl’s discussion and the contemporary one is that he does not emphasize what does or does not belong to the “space of reasons,” though I think the question is not entirely absent from his work. 212

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involved.31 Such is what happens when the noema of a perception is expressed. Whatever we think about the adequacy of Husserl’s analyses, it is important to see that his problems with the relation of the noema of a perception to its expression concern not thought itself but how perception relates to thought. The idea that perception has a sense does not, then, make the linguistic turn impossible for Husserl. It is true that the separation between “thought itself” and what is centrally related to it will seem too neat. But then we see a problem with the linguistic turn: the expressibility of the sense of a perception leaves, as both Husserl and Dummett point out, an unavoidable distance between the perception and the expression of it. This is, however, not obviously an artifact of the idea of the noema. For perception and perceptual judgment are not the same thing. Rather the dependence of thought on perception implies that something important for the study of thought has to be approached by other methods. This might indeed be a reason for not giving the linguistic turn quite the central role many analytic philosophers have given it. The result need not be the adoption of a method like Husserl’s phenomenological method, but some method is needed, perhaps an appropriation and analysis of the results of empirical psychology. Husserl’s thinking has another feature that separates him from the mainstream of analytical philosophy. However, it was present in Husserl’s thought from the beginning and is not a product of the period of his transcendental turn. That is that for him the basic concept is that of intentionality, where intentionality is consciousness of an object. In spite of the fact that he attributes something like force to all acts, nothing like Frege’s context principle ever occurs to Husserl. To the contrary, he searches in much of his philosophizing for a level of meaning more basic than anything that takes propositional form. I would see this as the fundamental obstacle to Husserl’s taking the linguistic turn. It might well be argued that his treatment of questions clearly within the philosophy of thought as Dummett conceives it suffers as a result. But his explorations of perception and time-consciousness are not obviously part of that domain, and it would take a great deal of argument 31

Dagfinn Føllesdal suggested in conversation that the greater simplicity of perception is a matter of its thetic character. I think these remarks express some of what he had in mind; I would have liked, however, to pin the idea down more precisely. 213

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to show that there too the linguistic turn would provide the key. Ironically, although Husserl’s philosophy of perception may be the part of his work that has most attracted analytical philosophers,32 it is a domain where the linguistic turn as Dummett formulates it seems to encounter limits.33

32

That is certainly true of Føllesdal and his pupils and also of Mulligan. The present essay is descended from one written for a symposium on Michael Dummett’s Origins of Analytical Philosophy at the meeting of the Central Division of the American Philosophical Association in Pittsburgh on April 26, 1997, with Richard Cartwright as co-symposiast. That essay concentrated on what Dummett had to say about Husserl. The further work leading to the present essay owes much to Dummett’s constructive and interesting reply on that occasion and to comments by Jason Stanley. Dagfinn Føllesdal also commented in detail on a presentation of the same essay at the University of Oslo, and he has made other helpful suggestions. I am indebted to Pierre Keller both for written comments on an intermediate version and for a helpful discussion. Much of the writing of the present version was done during a visit to the University of Oslo, to which I am indebted for hospitality and support, in particular again to Dagfinn Føllesdal. I am grateful to the editors for the many improvements they have proposed. 33

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INDEX

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In general, this bibliography and the references in the various essays follow the same conventions as in my previous books, Mathematics in Philosophy and Mathematical Thought and Its Objects. Works are cited by author and title only, the latter sometimes abbreviated. The following principles govern citations: 1. Works of an author are listed alphabetically by title rather than chronologically, to facilitate locating an entry. 2. In the essays, books are cited in the latest edition listed in this bibliography, unless explicitly stated otherwise. 3. For articles, reprintings in collections of the author’s own papers are listed, but reprintings in anthologies are not. Where there is a collection of the author’s papers, reference is to the reprint in the collection. 4. For two authors I observe special conventions: In the case of Kant, see the Note to Part I. In the case of Frege, as explained in Essay 5, writings published in his lifetime are cited in the original pagination. Allison, Henry E. The Kant-Eberhard Controversy. Baltimore: Johns Hopkins University Press, 1973. ———. Kant’s Transcendental Idealism. New Haven, Conn.: Yale University Press, 1983. 2nd ed., revised and enlarged, 2004. Anderson, C. Anthony. “Some New Axioms for the Logic of Sense and Denotation: Alternative (0).” Noûs 14 (1980), 217–234. Awodey, Steve, and Carsten Klein (eds.). Carnap Brought Home: The View from Jena. Chicago and La Salle, Ill.: Open Court, 2004. See also Reck and Awodey. Baum, Manfred. “The B-Deduction and the Refutation of Idealism.” Southern Journal of Philosophy 25 (supplement) (1987), 89–107. Beaney, Michael (ed.). The Frege Reader. Oxford: Blackwell, 1997. Beck, Lewis White. “Can Kant’s Synthetic Judgments Be Made Analytic?” KantStudien 47 (1956), 168–181. Reprinted in Studies in the Philosophy of Kant (Indianapolis: Bobbs-Merrill, 1965). Beiser, Frederick C. The Fate of Reason: German Philosophy from Kant to Fichte. Cambridge, Mass.: Harvard University Press, 1987. ———. “Mathematical Method in Kant, Schelling, and Hegel.” In Domski and Dickson, pp. 243–258. 217

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Benacerraf, Paul. “Mathematical Truth.” The Journal of Philosophy 70 (1973), 661–679. Beth, E.  W. “Über Lockes ‘allgemeines Dreieck’.” Kant-Studien 48 (1956–1957), 361–380. Bolzano, Bernard. Beyträge zu einer begründeteren Darstellung der Mathematik. Prague: Caspar Widtmann, 1810. Reprinted with an introduction by Hans Wussing, Darmstadt: Wissenschaftliche Buchgesellschaft, 1974. Translated by Steven Russ in Ewald, From Kant to Hilbert, vol. 1, pp. 176–224. ———. Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwei Werten, die ein engegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege. Prague: Gottlieb Haase, 1817. Translated by Steven Russ in Ewald, From Kant to Hilbert, vol. 1, pp. 227–248. ———. Wissenschaftslehre. 4 vols. Sulzbach, 1837. Boolos, George. “The Iterative Conception of Set.” The Journal of Philosophy 68 (1971), 215–231. Reprinted in Logic, Logic, and Logic. ———. Logic, Logic, and Logic. Edited by Richard Jeffrey. With Introductions and Afterword by John P. Burgess. Cambridge, Mass.: Harvard University Press, 1998. Brentano, Franz. Die Abkehr vom Nichtrealen. Edited by Franziska MayerHillebrand. Bern: A. Francke, 1966. ———. Kategorienlehre. Edited by Alfred Kastil. Leipzig: Meiner, 1933. Translated by Roderick M. Chisholm and Norbert Gutterman as The Theory of Categories. The Hague: Martinus Nijhoff, 1980. ———. Die Lehre vom richtigen Urteil. Edited by Franziska Mayer-Hillebrand. Bern: A. Francke, 1956. ———. The Origin of the Knowledge of Right and Wrong. Translated by Roderick M. Chisholm and Elizabeth Schneewind. London: Routledge and Kegan Paul, 1969. ———. Psychologie vom empirischen Standpunkt. Leipzig: Duncker & Humblot, 1874. 2nd ed. edited by Oskar Kraus. Leipzig: Meiner, 1924. Translation of 2nd ed., Psychology. ———. Psychology from an Empirical Standpoint. Edited by Linda L. McAlister. Translated by Antos C. Rancurello, D. B. Terrell, and Linda L. McAlister. London: Routledge and Kegan Paul, 1973. 2nd ed., with introduction by Peter Simons. London: Routledge, 1995. ———. Vom sinnlichen und noetischen Bewusstsein. Psychologie vom empirischen Standpunkt III. Edited by Oskar Kraus. Leipzig: Meiner, 1928. 2nd ed. edited by Franziska Mayer-Hillebrand. Hamburg: Meiner, 1968. Translation, Sensory and Noetic Consciousness, translated by Linda L. McAlister and Margarete Schattle. London: Routledge and Kegan Paul, 1981. ———. Von der Klassifikation der psychischen Phänomene. Leipzig: Duncker & Humblot, 1911. 2nd ed. as Psychologie vom empirischen Standpunkt, 2. Band. Edited with introduction, notes, and additional texts from the Nachlaß by Oskar Kraus. Leipzig: Meiner, 1924. Translation in Psychology. ———. Wahrheit und Evidenz. Edited by Oskar Kraus. Leipzig: Meiner, 1930. Translated as The True and the Evident. Edited by Roderick M. Chisholm.

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229

COPYRIGHT ACKNOWLEDGMENTS

Essay 1 appeared in Paul Guyer (ed.), The Cambridge Companion to Kant (Cambridge: Cambridge University Press, 1992), pp.  62–100, copyright © 1992 by Cambridge University Press, reprinted by permission of the publisher and editor. Essay 2 appeared in Topoi, vol. 3 (1984), pp. 109–121, copyright © 1984 by D. Reidel Publishing Company, and is reprinted by kind permission of Springer Science and Business Media. Essay 3 appeared in Allen W. Wood (ed.), Self and Nature in Kant’s Philosophy (Ithaca, N.Y.: Cornell University Press, 1984), copyright © 1984 by Cornell University, and is reprinted by permission of the editor and Cornell University Press. Essay 4 appeared in Mary Domski and Michael Dickson (eds.), Discourse on a New Method: Reinvigorating the Marriage of History and Philosophy of Science (Chicago and La Salle, Ill.: Open Court Publishing Company, 2010), copyright © 2010 by Carus Publishing Company, and is reprinted by permission of the editors and publisher. Essay 5 appeared in Matthias Schirn (ed.), Studien zu Frege I: Logik und Philosophie der Mathematik (Stuttgart: Fromann-Holzboog, 1976), pp. 265–277, and is reprinted by permission of the editor and publisher. Essay 6 appeared as “Review Article: Gottlob Frege, Wissenschaftlicher Briefwechsel,” in Synthese, vol. 52 (1982), pp.  325–343, copyright © 1982 by D. Reidel Publishing Company, and is reprinted by kind permission of Springer Science and Business Media. Essay 7 appeared in Dale Jacquette (ed.), The Cambridge Companion to Brentano (Cambridge: Cambridge University Press, 2004), pp. 168–196, copyright © 2004 by Cambridge University Press, and is reprinted by permission of the publisher and editor. Essay 8 appeared in Juliet Floyd and Sanford Shieh (eds.), Future Pasts: The Analytic Tradition in Twentieth-Century Philosophy (Oxford: Oxford University Press, 2001). Copyright was retained by the author. The essay is reprinted by permission of the editors and Oxford University Press. 231

INDEX

Absolutist versus relationist conceptions of space and time, 18; argument from “incongruent counterparts,” 19–20; Kant, Von dem ersten Grunde des Unterschiedes der Gegenden im Raume (“Regions in Space”), 12 Acceleration, 73, 78 Actuality, 44. See also Existence Aedequatio rei et intellectus, 164; Brentano’s deflationary reading of, 182–183. See also Truth, correspondence theory of Agglomeration (Aggregat), 118–119. See also Number; Set and element Algebra, 66, 85, 89, 107, 110; as general arithmetic, 108; objects of, 107–108; synthetic a priori judgments in, 108 Alteration. See Change Analytical philosophy, 190–195; “gulf” with continental philosophy, 191, 201 Analytic of Principles, 75–76 Analytic-synthetic distinction. See Judgment (Kant) Anschauung. See Intuition Antimonies of Pure Reason, 33 Appearances, 29, 33, 37, 38, 45; of outer sense, 29 A priori, 5, 71–76; a priori concepts, 6, 31, 93; a priori intuition, 5, 6, 11, 30–32, 93, 95–96; a priori judgments, 5, 6; a priori knowledge, 6, 41, 71, 75–76; a priori knowledge of objects, 45; a priori procedure, 75; a priori representation, 6; a priori synthesis, 93; Bolzano on a priori intuition, 91–92, 94, 97; Kitcher on (see

Kitcher, Philip); nonpure a priori knowledge, 70; pure, 69; “quasi a priori” knowledge, 73; representation of space is a priori (see Representation) Aristotle, 163; on truth, 178 Arithmetic, 28, 29, 42, 55, 57, 64–67, 85, 89–90, 95, 102, 107–110, 155; arithmetical propositions, 112; arithmetic has no axioms, 84, 87, 97, 109–110, 114; axiomatization of arithmetic, 98, 135; conception of as a realm of “mere concepts,” 67; content of arithmetic, 112; Frege on arithmetic, 125, 128, 129; general propositions in arithmetic, 114; necessity of construction for arithmetic, 61; proofs of arithmetical identities, 65n, 93–94, 112–114; relation to space (see Space); relation to time (see Time); Schultz on arithmetic, 83–84, 110; as synthetic and dependent on intuition, 58; synthetic a priori judgments, 108 Associativity, 84–90, 113–114. See also Bolzano, Bernard Axioms, 25–27, 80, 83–84, 87–89, 90, 96–98, 102–103, 106, 125–126; existence axioms, 25, 90; immediately certain, 106. See also Arithmetic; Geometry; Set theory Axioms of Intuition, 45, 48, 50–52, 58 Baumgarten, Alexander Gottlieb: Metaphysica, 51; whole/part relation (see Whole and part) Beck, J. S., 7, 98 Beck, Lewis White, 102

233

INDEX

Benacerraf, Paul, 31 Beth, E. W., 26, 80–81, 93, 102 Bolzano, Bernard, 80, 91–99; on a priori intuition, 91–92, 94, 97; on associative law of addition, 94–95; Beiträge zu einer begründeteren Darstellung der Mathematik, 91, 95–96; on construction of number in time (see Construction); on the distinction between representation and what is represented (see Representation); on intuition (see Intuition); logical platonism, 95; on necessary judgments, 92; on necessity as a property of judgments (see Necessity); on pure intuition (see Intuition); Rein analytischer Beweis, 96; on representation (see Representation); on the role of pure intuition in mathematics (see Intuition); ‘7 + 5 = 12,’ 93, 192; on time and arithmetic, 93–94 Brentano, Franz, 161–189, 194, 195; on concepts, 175–176; modes of presentation, 173–175; reism, 163, 175; on truth, 175–189. See also Evidence, Brentano’s view of; Judgment (Brentano); Truth, correspondence theory of Brittan, Gordon, 71, 102 Brouwer, L. E. J., 188–189 Buchdahl, Gerd, 71 Burge, Tyler, 131–132 Calculation, 67 Cantor, Georg, 51–53, 62, 118, 133–135, 148; on consistent and inconsistent multiplicities, 129; correspondence with Dedekind, 129, 148; Grundlagen einer allgemeinen Mannigfaltigkeitslehre, 133; “quantitative determination” of extensions, 134; Tait on Cantor, 133–134; on the totality of sets, 129, 133–134. See also Set and element Cardinality. See Number; Set and element Carnap, Rudolf, 142, 190; on Frege, 126–127, 135–136; Logical Syntax, 137 Carson, Emily, 105 Categories, 45, 72, 76, 78; categories of quantity (see Quantity); categories of

relation, 77; objective reality of the categories (see Objective reality); pure categories, 54, 111; schematized categories, 58, 75; Table of Categories, 50 Categories of quantity. See Quantity Cauchy, Augustin-Louis, 96 Cause, 38, 73; category of causality, 78; concept of causality, 75; principle of causality, 73, 78 Change (Veränderung), 6, 28, 75, 78; alteration of the state of a substance, 73; Cambridge change, 73; change of state, 73, 78; as an empirical concept, 72; objective change, 73; objective reality of the concept of alteration, 73–75 Chisholm, Roderick M., 167, 168 Church, Alonzo, 142, 153 Classes, 148, 150–152, 155; Frege on, 120–123, 125, 133, 142–143, 149, 155–157. See also Extension; Russell, Bertrand Closure property, 87 Cognitive faculties, 78–79 Commutativity, 84–85, 89–90, 113–114; Leibniz on, 97 Composition. See Whole and part Compositum. See Whole and part Concepts (Frege), 121–124, 127–128, 133, 155; empty, 119; as functions, 125; number as attaching to, 60; second-level, 127–128 Concepts (Kant), 31, 45, 49, 53, 56, 78, 83; abstract conception of whole, part, and quantity (see Whole and part); “actuation in the concrete,” 57, 60; analysis of concepts, 22; concept formation in geometry, 83; construction of concepts (see Construction); “containment” in another concept, 22, 91; derivative, 53–54; empirical, 70–72; intellectual, 60; mathematical, 6, 45; necessary connection of concepts and judgment (see Judgment [Kant]); of object in the A deduction, 38; pure, 43; pure, but derived concepts of understanding, 72; singular, 7. See also Kitcher, Philip Conservation of matter. See Matter

234

INDEX

Constructibility, 44, 46; as Kantian version of mathematical existence, 44 Construction, 44, 58, 66, 81, 90; conditions of the construction of concepts, 96; construction of concepts in pure intuition, 24–25, 47, 91, 93, 95 108, 112; in Euclidean geometry, 24, 48, 66, 89, 104, 105, 108; intuitive construction, 108; necessity for arithmetic (see Arithmetic); ostensive construction, 47, 66; ostensive construction of numbers, 58; plural constructions in natural language, 131; of quantities (see Quantity); series of numbers, 61; symbolic construction, 27, 47, 48, 65, 107 (see also Shabel, Lisa); time is involved in mathematical (ostensive) construction, 65, 67, 112 Constructivism, 68 Continuum, 54–56 Copernican hypothesis, 32, 37 Correspondence theory of truth: Brentano’s early questioning of, 164, 178–179; effort to save, 179–180; virtual abandonment of, 181–183 Couturat, Louis, correspondence with Frege, 141 Dasein, 47, 94 Dedekind, Richard, 63, 134; Frege on Dedekind, 118–120; on systems (see Set and element); Was sind und was sollen die Zahlen?, 135 Definite descriptions, 7 Deflationism about truth, 189 Demonstratives, 9 Description theory of names, 9 Dingler, Hugo, correspondence with Frege, 141 Direct reference, 8, 32 Discipline of Pure Reason in its Dogmatic Employment (Use), 45, 102; on geometric proof (see Geometry) Distortion Picture (view), 34, 37 Divisibility of quanta. See Quantity Dummett, Michael, 190–193, 207, 213; conception of analytical philosophy, 191–193 Dynamical Principles (Dynamics), 71, 73, 76

Eberhard, J. A., 44 Ehrenfels, Christian von, 185 Empirical content, 72 Equality of action and reaction, 71, 77 Erscheinungen, 34. See also Appearances Euclid, 82, 97; Euclidean constructions (see Construction); Euclidean geometry (see Geometry); Euclidean space (see Space) Euler, Leonhard, on space, 19 Evans, Gareth, 208, 212 Evidence, Brentano’s view of: characterization of truth in terms of, 184; difficulties of the characterization, 185 Existence, 44; existence at a definite time, 44; existence statements, 81; Wirklichkeit, 44, 47 Experience, 28, 72, 73, 78; analogies of experience, 44; formal conditions of experience, 45; objects of experience, 38; outer experience, 75 Extension, Frege’s conception of, 117–137; as derived in relation to concepts, 121, 155–156. See also Predicate; Russell, Bertrand; Set and element Extensionality, 119–120, 124 Feature (Merkmal), 7 Finite, 63; finite iteration (see Time); finite ordinals, 63; successive repetition, 62, 63 First Analogy, 77. See also Quantity Føllesdal, Dagfinn, 195n, 213n Forces, distribution of, 73 Form of appearances of outer sense, 29 Form of intuition. See Intuition Frege, Gottlob, 63, 64, 117–160, 190, 202–203, 212; on arithmetic (see Arithmetic); “Aufzeichnungen für Ludwig Darmstaedter,” 127, 156; “Ausführungen über Sinn und Bedeutung,” 142; Basic Laws, 132, 136; Begriffschrift, 124–125, 136; “Booles rechnende Logik und die Begriffschrift,” 141; cardinal number (see Number); Carnap on Frege, 126–127, 135–136; on classes (see Classes); on concepts (see Concepts [Frege]); context principle, 213; on Dedekind (see Dedekind, Richard); Die Grundlagen der

235

INDEX

Frege, Gottlob (continued) Arithmetik, 118–120, 131–132, 141, 145, 156; equipollence, 143; on extension (see Extension, Frege’s conception of); Fregean type theory, 151; “Frege-Hilbert controversy,” 140; on function and object, 147, 150; Funktion und Begriff, 141, 145–146; on geometry, 129; Grundgesetze der Arithmetik, 118, 123–124, 126, 131–132, 134–136, 142, 149, 150, 155; “Hume’s Principle,” 135; indirect reference of the second degree, 142; on infinity, 128–129; on logic (see Logic); logicism, 128, 135, 136; “Logik in der Mathematik,” 126, 136; Nachgelassene Schrifien, 123, 136, 140; on negation, contrast with Brentano, 163–164, 173; notion of force, 161–162, 204; on the null set (see Set and element); on number (see Number); on objects (see Objects); on quantification (see Quantification); on Schoenflies (see Schoenflies, Arthur Moritz); on second-order logic (see Logic); “On Sense and Reference,” 142 (see also Sense and reference); on sets (see Set and element); on set theory (see Set theory); Tait on Frege, 134–135; on types, 150; Way Out, 123, 124, 149, 151; on Weierstrass, 126; Wertverläufe, 120, 122, 123, 126, 133, 149; on whole and part, 154. See also Couturat, Louis; Dingler, Hugo; Hilbert, David; Husserl, Edmund; Jourdain, Philip E. B.; Korselt, A. R.; Löwenheim, Leopold; Pasch, Moritz; Peano, Giuseppe; Russell, Bertrand; Wittgenstein, Ludwig; Zsigmondy, Karl Friedman, Michael, 26, 81, 84, 87, 89, 93, 96, 100, 102–110; on geometry, 103–104; on intuition, 81; on Kant on addition, 89 Fulfillment, 205–206 Function symbols, 90, 151 Gabriel, Gottfried, 136, 145 General Observations (§8 augmented in B), 11, 36, 39–40 General principles, 90 General terms, 8

Geometry, 14–18, 21, 22–30, 35, 37, 65, 82, 85, 99, 102–108; applied geometry, 23; a priori, 102; argument from the necessity of geometry, 39; axiomatization of geometry, 17, 25, 142; axioms of geometry, 21, 26–27; concept formation in geometry (see Concepts [Kant]); as dependent on intuition, 102; Euclidean constructions, 24, 85; geometrical construction (see Construction); geometrical knowledge, 36, 106; geometrical reasoning, 24–27; geometric proof, Kant’s analysis of, 102; infinity of space prior to geometry (see Space); its objects as quanta, 107; as synthetic, 21, 102 Goodman, Nelson, calculus of individuals, 119 Guyer, Paul, 35–37, 39–40 Herder, Johann Gottfried von, 113–114 Heyting, Arend, 205 Hilbert, David, 25; correspondence with Frege, 140–141 Hintikka, Jaakko, 8, 11, 26, 80–81, 93, 102; on the concept of intuition, 81; on the immediacy of intuition, 100 Homogeneity, 53–54. See also Space; Whole and part Howell, Robert, 8, 101 Husserl, Edmund, 141–147, 190–214; correspondence with Frege, 144–147; generalization of meaning to all acts, 195–196; intentionality as basic, 213; lack of context principle, 213; meaning as species, 196; on psychologism, 144; on pure theory of manifolds, 146; on sense and reference (Gegenstand), 147; on truth and evidence, 186–188 Hylton, Peter, 193 Idealization in science. See Science Identity puzzle, 154 Image (Bild), 59–60; distinguished from schema, 59; space as a pure image of quanta (see Space); spatio-temporal image of a number (see Number); time as a pure image of quanta (see Time)

236

INDEX

Imagination, 32, 45; (transcendental) synthesis of imagination (see Synthesis) Immediacy. See Intuition Inaugural Dissertation, 28, 56–58, 103, 114; Longuenesse on, 111; number in, 59, 60, 63, 111 Indirect contexts, 152–153 Induction, mathematical, 90–91, 126 Intellect, 57 Intensional view of appearances and things in themselves, 38–39, 40 Intuition (Anschauung), 6, 23, 24, 25, 26, 29, 30, 31, 33, 45, 47, 57, 58, 67, 82–83, 91, 93, 95, 100; a priori intuition (see A priori); Bolzano on intuition, 94–95; Bolzano on pure intuition, 91, 96; Bolzano on the role of pure intuition in mathematics, 92; constructions in intuition (see Construction); empirical intuition, 102; forms of intuition, 29, 40–41, 65, 112; in general, 112; imagination is immediate (see Imagination); immediacy condition, 7, 8, 9, 10–11; immediacy of intuition, 30, 82, 101–102; intellectual intuition, 10, 83; intuition of motion, 75; and magnitude, 106–107; in mathematical inference, 80, 96, 106; outer intuition, 30, 75, 77; pure intuition, 22, 29, 45, 102; sensible intuition, 83; singularity of intuition, 82; as a singular representation, 100; space and time are forms of intuition, 31; successive, 64. See also Friedman, Michael; Hintikka, Jaakko; Longuenesse, Bêatrice Iteration: finite iteration (see Time); indefinite iteration (see Time); principle of iteration, 123 Jourdain, Philip E. B., 148; correspondence with Frege, 124, 126, 132, 140–141, 153–154 Judgment (Brentano), 161–175; as affirmation or denial of a presentation, 163; categorical, 165; combination of terms in, 166–168; double, 166–173, 177; existential, 165, 178 Judgment (Kant), 5–8; analytic and synthetic, 22, 23, 98; arithmetical, 94;

logical form of, 8; mathematical, as synthetic, 23; necessary connection of concepts and judgment, 101; quantity of, 49; synthetic a priori, 94; universal form of judgment, 53. See also Algebra; Arithmetic Kant, Immanuel, 3–114, 202; arithmetic has no axioms (see Arithmetic); conception of construction of concepts in intuition (see Construction); Reflections attached to Baumgarten’s Metaphysica, 51; on ‘7+5=12,’ 43, 47, 85, 88–89, 93–94, 97, 109–110, 113. See also Whole and part Kant’s lectures on Metaphysics. See Number; Whole and part Kastil, Alfred, 180 Kitcher, Philip, 69–76; on a priori knowledge, 75; “conceptual legitimacy,” 72; “empirically legitimized concepts,” 73 Knowledge: a priori (see A priori); geometrical (see Geometry); immediate knowledge of space (see Space); independent of experience, 76; mathematical, 45, 46; nonpure a priori knowledge (see A priori); objective, 41; of objects, 46, 47, 64, 71–72; of objects as they appear, 38; of objects themselves, 38; of outer things, 30, 33; “quasi a priori” knowledge (see A priori); relativity of knowledge, 40; sensitive faculty of knowledge, 57; synthetic a priori knowledge concerning space, 30 Koebner, Wilhelm, 141 Korselt, A. R., 141; on types, 150 Lambert, Johann Heinrich, 97, 98 Law of inertia, 71, 76, 78 Laz, Jacques, 91–92 Leibniz, Gottfried Wilhelm, 12, 13; on commutativity (see Commutativity); proofs of arithmetical identities (see Arithmetic) Leonard, Henry S., calculus of individuals, 119 Lésniewski, Stanisław, 119 Linke, Paul F., 144

237

INDEX

Logic, 6; first-order quantificational logic, 80; Frege on fundamental Logic, 125, 127, 136, 155; Frege on logic, 125, 129; logical derivability, 144; monadic logic, 96; ωth order predicate logic, 151; polyadic logic, 96–97; propositional logic, 90; quantificational logic, 136, 146; second-order logic, 123, 125, 135, 136; truth-functional logic, 136 Logicism, 68, 156. See also Frege, Gottlob Longuenesse, Béatrice: on immediacy of intuition, 101; on the Inaugural Dissertation (see Inaugural Dissertation); on proofs of arithmetical identities, 90n, 114; on the Schematism (see Schematism of the categories) Löwenheim, Leopold, correspondence with Frege, 141 Magnitude, 15–17, 54, 60, 62–64, 84–85, 95, 106–108, 110; pure image of all magnitudes, 60; pure schema of magnitude, 60; relation to intuition (see Intuition). See also Number Manifold of intuition, 9 Martin, Gottfried, 81n, 83, 97–98 Marty, Anton, 143, 163 Mathematical Antinomies. See Antinomies of Pure Reason Mathematical concepts. See Concepts (Frege); Concepts (Kant) Mathematical construction. See Construction Mathematical demonstration, 45, 46. See also Geometry Mathematical existence, 44 Mathematical knowledge. See Knowledge Mathematical objects. See Objects Mathematical possibility. See Possibility Mathematics, 42, 64, 77, 94–99; of continuity, 55; mathematical induction (see Induction); mathematical inference, 90; mathematical judgments, 22; mathematical proof, 82, 90, 102; mathematical reasoning, 112; mathematical statements, 81; mathematical synthesis, 54, 57, 109–110; is necessary

(B14–15), 21, 37; pure mathematics, 108; role of intuition, 101–103, 106; rules of inference, 90, 136. See also Arithmetic; Geometry; Proof; Schultz, Johann Mathematizability of phenomena, 78 Matter, 70, 73, 78; a priori content of the notion of matter, 72; conservation of matter, 71, 77; empirical concept of matter, 70, 72; as the movable in space, 72, 78; objective reality of matter (see Objective reality); quantity of matter, 77; as substance in space (see Substance) McDowell, John, 208 Meinong, Alexius, 47, 163; theory of objects, 180n Metaphysical Exposition of the Concept of Space, 11–17, 27, 28, 82, 102–103, 106 Metaphysical Exposition of the Concept of Time, 11, 28 Metaphysical Foundations of Natural Science, 69–78 Metaphysics of nature. See Nature, metaphysics of “Metaphysics of transcendental idealism,” 32 Moments, 208–210; as derivative from properties, 209; of properties, 208–209; of relations, 209–210 Momentum, 77 Motion, 6, 28, 72–78; communication of motion, 77; of a point in space, 75; real possibility of physical motion, 75; substance as subject of motion (see Substance) Mulligan, Kevin, 194, 207–208 Multiplicity. See Set and element Naturalistic epistemology, 75 Nature, metaphysics of, 70; metaphysics of “corporeal or thinking” nature, 75; “projected order of nature,” 78; “transcendental part of the metaphysics of nature,” 75. See also Analytic of Principles Necessity, 5, 6, 18, 35, 36, 37; argument from the necessity of geometry (see Geometry); necessity of mathematics (see Mathematics); and strict universality, 5

238

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Newton, Isaac, 12, 13; laws of motion, 71 Noema, Husserl’s conception of, 194–198; of judgment, 197–198; of perception, 197, 203–204; of perception as attributing properties to the object, 206–208; thetic character of, 213n Nonconceptual content of experience, 208 Number, 42–43, 47–48, 52, 55–59, 63, 65, 95, 109, 111; attaching to an agglomeration of things (Aggregat), 118–119; cardinal number, 56, 58, 62, 108, 132, 134–135, 156; Frege on number, 118–119, 123, 127–128, 133, 156–157; infinite number, 62; intellectual conception of number, 64, 111; intellectualist view of the concept of number, 111; in Kant’s lectures on Metaphysics, 59; “ontological commitment to numbers,” 48; ordinal number, 58, 62, 63 (see also Finite); ostensive construction of numbers (see Construction); as the pure schema of magnitude, 60; pure units, 109; rational numbers, 48; relation to space (see Space); relation to time (see Time); as a schema, 61; as the schema of quantity, 58; in the Schematism (see Schematism of the categories); science of number, 64–65; as set, multitude, or plurality, 118; singular propositions about numbers, 48, 109; spatio-temporal image of a number, 60; structure of numbers, 67; in terms of pure categories, 59, 111; thought of number, 59; the unity of the synthesis of the manifold of a homogeneous intuition in general, 60; whole number, 58–59 Objective reality, 6, 45, 72–74; objective reality of matter, 72; objective reality of the categories, 75 Objects, 43, 47, 56, 73, 85, 95, 117, 119, 121, 150; existence of mathematical objects, 44; of experience, 40, 43, 47; extended objects (relation to spatial parts) (see Set and element); Frege on objects, 122, 127–128, 132–133, 136; individuation of objects, 40; logical objects, 123; mathematical objects, 43,

46, 58, 66, 100, 107; nonexistent objects, 47; of outer sense, 74; second-class objects (uneigentliche Gegenstände), 123, 150; spatio-temporal object, 43, 53–54 Parametric reasoning, 90 Parsons, Charles: “Kant’s Philosophy of Arithmetic,” 84, 97, 100, 102–104; Mathematical Thought and Its Objects, 131; “What is the iterative conception of set?,” 131 Pasch, Moritz, 141–142 Peacocke, Christopher, 208 Peano, Giuseppe, 141–142, 155; correspondence with Frege, 141–142; Rivista di matematica, 142 Peirce, Charles Sanders, 188 Perceptual judgment, 211–212 Phenomenological presence of an object, 32 Phenomenological reduction, 199–201 Philosophy of mathematics, 42, 80 Physical motion. See Motion Plaass, Peter, 72–75 Pluralities, 131 Plurality. See Quantity Possibility, 44, 45, 46; mathematical possibility, 46 Postulates, 23, 25, 46, 83–84, 87–90, 96–98, 103, 107, 110, 147 Prauss, Gerold, 39, 40 Predicables, 72; pure, but derived concepts of understanding (see Concepts [Kant]) Predicate, 8, 22, 77, 91–92, 117, 122, 150, 152. See also Extension, Frege’s conception of; Set and element Presentations, 161; as general, 176–177 Principia Mathematica, 137 Principle of contradiction, 26 “Progression of intuitions” (A25), 17 Prolegomena, 30, 31, 36, 39, 69 Proof, 24–27, 99; demonstrative proof, 45. See also Arithmetic; Geometry; Mathematics Proper names, 50 Propositional identity, 153 Propositional objects, 162, 182

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Pure intuition. See Intuition Pure propositions, 69

sensible representation, 82; singular representation, 6, 7, 25, 67, 81, 93. See also Synthesis Rules of inference in mathematics. See Mathematics Russell, Bertrand, 102–103, 121, 140–142, 144, 147–149; on classes, 148, 151; on concepts, 121; on extensions, 121, 149; on the Grundgesetze, 133; on multiplicities (see Set and element); no-class theory, 150; on objects, 153; Principles of Mathematics, 147–148, 150–153; propositional function, 121–122; range of significance, 121; Russell’s original theory of types, 122; theory of descriptions, 153; theory of limitation of size, 148; types, 151–152 Russell-Myhill paradox, 151–153 Russell’s paradox, 118, 123, 125, 126, 137, 140, 147–148, 156

Quantification, 50, 150–152. See also Logic Quantifiers, 46, 47, 96, 122–123 Quantity, 48, 50, 58, 63–64, 66, 77, 85, 95, 106, 109; categories of, 42, 60; construction of, 106; continuous, 54; discrete, 54–56, 62; discrete quantity per se, 56; extensive, 51; finite, 62; general theory of (allgemeine Größenlehre), 108; intensive, 77; irrational, 66; judgments of (see Judgment [Kant]); less, 62; logical, 50; number as the schema of quantity (see Number); plurality, 50, 52; pure schema of quantity, 52; rule of numeration (Zählen), 66; schema of quantity, 42; schematized categories of quantity, 50, 52; Schultz on quantity, 86–88; totality, 50, 52. See also Whole and part Quantum, 54–55, 56; divisibility of quanta, 55; quantum discretum, 56. See also Quantity; Whole and part Quine, W. V., 117n Recursion condition for addition, 113 Reference. See Sense and reference Refutation of Idealism, 66, 92n Rehberg, August Wilhelm, 48, 64–66, 112 Relationism. See Absolutist versus relationist conceptions of space and time Representation, 6, 7, 13, 37, 38, 56, 66, 91–92; Bolzano on the distinction between representation and what is represented, 95–96; Bolzano on representation, 95; distinguished from empirical objects, 40; general representation, 6, 7; immediate representation, 30, 82; “intellectual” representation “I think,” 67; intuition as a singular representation (see Intuition); intuitive representation, 82; of number (see Number); “original representation” of space (B40), 14; outer representation, 30; reflected representation, 6; representation of space, 15, 16, 54, 82–83, 105; representation of space is a priori, 72; representation of time, 28;

Schema, 59; as distinguished from image (Bild) (see Image); schema of quantity (see Quantity) Schematism of the categories, 42, 47, 78; Longuenesse on the Schematism, 111; number, 59, 64, 95 Schoenflies, Arthur Moritz, 118, 120; Frege on Schoenflies, 123 Scholz, Heinrich, 138–139, 140, 145, 158 Schultz, Johann, 44, 47, 64, 65, 80–91, 93, 96–99; on addition, subtraction, multiplication, 85–91; Anfangsgründe der reinen Mathesis, 86, 88, 98–99; on arithmetic (see Arithmetic); Erläuterungen über des Herrn Professor Kant Critik der reinen Vernunft, 83; on general and “special” mathematics, 85–86; on induction (see Induction, mathematical); Prüfung der kantischen Kritik der reinen Vernunft, 81–83, 86, 97; on quantity (see Quantity); on representations, 83; ‘7 + 5 = 12,’ 88–90, 94, 97, 108–110, 112–113; on the Transcendental Deduction, 84 Science: a priori science, 78; idealization in science, 72; pure natural science, 69–78 Second Analogy, 77–78

240

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Semantic paradoxes, 151–152 Sense and reference, 142–147, 152–154; sense identity, 144, 154 Sensibility, 9, 11, 64 Set and element, 51–53, 117–122, 150; agglomerations and sets, 118–121; as collections, 131; Dedekind on systems, 119; extended objects (relation to spatial parts), 52; Frege on null sets, 119; mereological sum, 52, 56; multiplicities (Vielheiten), 148; multiplicity (Menge), 51–52, 55, 56, 58, 59–61, 66, 120; as pluralities, 131; set-theoretic notion of cardinality, 62; totality of sets, 130; unit set, 120. See also Cantor, Georg; Extension, Frege’s conception of; Frege, Gottlob; Predicate Set-theoretic paradoxes, 150. See also Russell, Bertrand Set theory, 42–43, 117–118, 122–125, 128–131, 150; axioms of set theory, 117–118, 124, 131; Frege on set theory, 128; and mereology, 119, realism about, 68 Shabel, Lisa, on Kant on algebra, 85, 107 Shamoon, Alan, 50 Singular terms, 8, 25, 49 Smit, Houston, 101 Smith, Barry, 194 Space, 5, 6, 11, 30, 34, 35, 57, 65–66, 72, 77–78; absence of space, 13; a priori character of, 12–14, 72; as boundless, 16, 17, 104; as condition of outer experience, 45; as a continuum, 55; Euclidean space, 104; as form of outer intuition, 5; homogeneity of the spaces occupied by parts of an object, 54; immediate knowledge of space, 15; infinity of space, 15–16, 103, 105; as an intuition, 14; necessary to the determinate representation of a number, 66; “original representation” of space (B40) (see Representation); prior to appearances/prior to objects in space, 13, 17; relation to arithmetic, 65; representation of a single space is prior to that of spaces, 103; as “subjective condition of sensibility,” 30; substance in space

(see Substance); as unique, 17; uniqueness of space, 15. See also Unity Steck, Max, 140 Strawson, P. F., “metaphysics of transcendental idealism,” 32 Subjectivist view, 33–38 Substance, 77; category of substance, 78; matter as substance in space, 77; as subject of motion, 77–78; substance in space (Descartes’s extended substance), 77, 78 Successive addition. See Time Successive enumeration. See Time Successive repetition. See Time Sutherland, Daniel, 106, 108–111 Syllogisms, 165–166; Brentano’s modern view of, 165–166 Symbolic construction. See Construction; Shabel, Lisa Synthesis, 9; a priori synthesis (see A priori); figurative synthesis, 63; of a given manifold of intuition in general, 63; of imagination, 66, 93; intellectual synthesis, 63–65, 111–112; mathematical (see Mathematics); transcendental synthesis of imagination, 93 Table of Categories, 50 Table of Judgments, 49 Tait, William, 108, 133–134. See also Cantor, Georg; Frege, Gottlob Term negation, 166–173 Things in themselves, 29, 30, 32, 33, 34, 37, 38, 39, 40; “neglected alternative,” 33, 34, 35; things in themselves are not spatial or temporal, 32 Third Analogy, 77 Thompson, Manley, 46 Thought, relation to perception as a problem for the linguistic turn, 213 Time, 5, 6, 11, 27, 30, 52, 56, 57, 63, 66, 72, 78, 95; as a continuum, 55; finite iteration, 63; as form of inner intuition, 5; indefinite iteration, 104; as an intuition, 95; intuition of quantities taken up successively (see Intuition); necessary to the determinate representation of a

241

INDEX

Time (continued) number, 66; as a pure image of quanta, 60; relation to arithmetic, 58, 65; relation to number, 58; schematization of the category of substance in terms of time, 77; as a subjective condition of apprehension, 58; succession in time, 95; successive addition, 60, 104, 114; successive enumeration, 66; successive repetition, 62, 63; temporal conditions (conditions of time), 57, 66; temporal content of the notion of number (see Number); unity of the manifold of space and time (see Unity) Totality. See Quantity Transcendental Deduction, 45, 63, 84 Transcendental Exposition of the Concept of Space, 11, 21, 27, 30, 103 Transcendental Exposition of the Concept of Time, 11, 28, 103 Transcendental idealism, 33, 35, 36, 40, 41, 58, 61; Husserl’s conception of and contrast with early analytical philosophy, 198–199. See also Distortion Picture; Intensional view of appearances and things in themselves; Subjectivist view Triangle, 20, 24–26, 36, 43–45, 59, 107, 109 Truth, correspondence theory of, 164; Brentano’s early questioning of, 178–180; virtual abandonment of, 181–183 Types: cumulative theory of types, 150; Frege on types (see Frege, Gottlob); Korselt on types (see Korselt, A. R.); Russell on types (see Russell, Bertrand); simple theory of types, 150–151

Understanding, 38, 91, 110; action on sensibility, 93; logical use of, 47, 101 Unities, 148 Unity, 50; of a number, 66; unification, 54; unity of the manifold of space and time, 93; unity of the synthesis of the manifold of a homogenous intuition in general, 95. See also Quantity Universality, 5, 6, 50; universal validity, 24 Use and mention, 147 Variable, 90; free variables, 46, 96 Walker, Ralph, 74–75 Whole and part, 49, 50, 51, 53, 56; abstract conception of whole, part, and quantity, 60; and categories of quantity, 50–51; composition, 57, 66; compositum, 58; compositum, quantum, totum, 52–53; Frege on whole and part, 154; “homogeneity” of parts, 53; in Kant’s lectures on Metaphysics, 51; Kant’s Reflections attached to Baumgarten’s Metaphysica, 51, 53, 55; multiplicity (see Set and element); parts of space and time, 55. See also Set and element Wirklichkeit. See Existence Wittgenstein, Ludwig, 141; correspondence with Frege, 158–159; Frege’s respect and friendship for, 159 Wolff, Christian, 12 Young, J. Michael, 47, 48 Zermelo, Ernst, 134 Zsigmondy, Karl, 156, 159–160; on cardinal numbers, 160

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