Mathematics in Philosophy [1 ed.] 0801414717

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Mathematics in Philosophy SELECTED ESSAYS

Cornell University Press· ITHACA, NEW YORK

Cornell University Press gratefully acknowledges a grant from the Andrew W.

Mellon Foundation that aided in bringing this book to publication.

Copyright © 1983 by Cornell University Press

All rights reserved. Except for brief quotations in a review, this book, or parts

thereof, must not be reproduced in any form without permission in writing from

the publisher. For information, address Cornell University Press, 124 Roberts Place,

Ithaca, New York 14850.

First published 1983 by Cornell University Press.

Published in the United Kingdom by Cornell University Press, Ltd., London.

International Standard Book Number 0-8014-1471-7

Library of Congress Catalog Card Number 83-45153

Printed in the United States of America

Librarians: Library of Congress cataloging information

appears on the last page of the book.

The paper in this book is acidjru and meets the guidelines

for permanence and durability of the Committee on Production

Guidelines for Book Longevity of the Council on Library Resources.

To the memory of my father

Contents

Preface

Introduction

PART ONE. 1.

9 15

MATHEMATICS, LOGIC, AND ONTOLOGY

Ontology and Mathematics

Appendix

58; Postscript

37 60

63

2. A Plea for Substitutional Quantification 3. Informal Axiomatization, Formalization, and the

Concept of Truth

PART TWO.

71

INTERPRETATIONS

4. Infinity and Kant's Conception of the "Possibility

of Experience"

5. Kant's Philosophy of Arithmetic

Postscript

95 110

142

6. Frege's Theory of Number

Postscript

150

173

7. Quine on the Philosophy of Mathematics PART THREE.

176

SETS, CLASSES, AND TRUTH

8. Sets and Classes 9. The Liar Paradox

Postscript 10.

209 221

251

What Is the Iterative Conception of Set?

7

Contents 1 1.

Sets and Modality

298

Appendix I. Free Modal Logic and Modal Logic with Scope

Operators 33 I; Appendix 2. Relation of Theories of

Potential Sets to Standard Set Theory 336

Bibliography Index

8

34 2

359

Preface

This book contains the most substantial philosophical papers I wrote for pUblication up to 1977, with one new essay added.! , I distinguish philosophical work from work in mathematical logic; I have not included any of my technical' papers.2 The collection is unified by a common point of view underlying the essays and by certain problems that are approached from different angles in different essays. Most are directly concerned with the philosophy of mathematics, and even in those that are not, such as Essay 9 on the liar paradox, the connection between issues discussed and math­ ematics is never far from the surface, and the issues are approached "from a mathematical point of view." In the Introduction I artic­ ulate some elements of my general point of view and point out some connections between the essays. As I try to make clearer there, I hope that the essays reveal not just a specialist's concern with problems about the foundations of mathematics but also get across my sense of the central role of mathematical thought in our thought in general. Along the way in the Introduction, something will be said about the importance I attach to certain historical figures. I will also offer the reader some guidance and assistance with the technical back­ ground of some of the papers. Of the essays in Part One of the book, the first two concentrate on problems of ontology. the first in relation to elementary and IThe collection does not include reviews, short conllllenling paj>er~, encyclopedia articles, and two essays: "Was ist eine mug-liche Welt?" alld "Some Rel\liIrks on Frege's Conception of Extension." More recent work. notably "Objects and Logic" and "Mathematical Intuition," is intended to form another book when combined with other work in progress or projected. 'This principle is compromised to some extent in Essay 11, in which I report some technical work that grew directly from the philosophical work of Es.~ay w.

9

Preface constructive mathematics, the second in relation to substitutional quantification. The third discusses a number of issues about in­ formal and formalized axiomatic theories, Godel's theorem, and Tarski's theorem on the indefinability of truth. It could serve to introduce the themes of Part Three. Part Two consists of essays on two important historical figures, Kant and Frege, and one contemporary, W. V. Quine. Part Three consists of essays on the notions of set, class, and truth, notions among which I find crucial connections. The dis­ tinction between the notion of set and the notion of class is prob­ lematic; that is the main subject of Essay 8. The connection of the notion of class with satisfaction and truth, remarked on in Essay 8, looms large in Essay 9 on the liar paradox, which is also con­ cerned with showing connections between the semantical and set­ theoretic paradoxes. Essay 10, mainly concerned with a commonly used intuitive explanation of the universe of sets, returns to the notion of class in set theory and to the problem of speaking about all sets. The ideas of Essay 10 raise questions about reference to sets in modal contexts; some of the loose ends of the discussion are pursued in Essay 11, which also examines some axiomatic set the­ ories suggested by the ideas of Essay 10 and some differently mo­ tivated modal set theories. With some minor exceptions noted below in opening footnotes, the essays have not been revised, even where I now think their arguments unclear or mistaken. I have added to the footnotes in some places, either to clear up purely technical confusions or be­ cause I was not able to resist the temptation to add new comments or references. Such additions are enclosed in brackets. Postscripts have been added to Essays 1, 5. 6, and 9. Essay 5 generated some controversy about Kant's conception of intuition. Especially since J have never replied to Hintikka's criticism,' I now comment on that controversy and on some other issues about Kant's philosophy of arithmetic. The publication of Essay 9 was followed by important technical work on semantical paradoxes by others, notably Saul Kripke. Here the main purpose of the Postscript is to say how that work affects the viewpoint of my own paper on the subject. In editing the published texts, 1 have put the references into a uniform style and updated some of them. Citations are given only

S.'Kantian Intuitions." 10

Preface by author and title. the latter often abbreviated. Full bibliographical information is given in the Bibliography at the end of the book. Collecting the work of nearly twenty years offers an occasion to thank some of the many people who have offered me instruction, stimulation, and helpful criticism over the years. I must begin with my teachers at Harvard University. The intense activity in philos­ ophy and logic at Harvard led me into both subjects. Burton Dre­ ben, my principal teacher in mathematical logic. influenced my philosophical development in many ways, perhaps first by wisely urging me to postpone my ambitions in the philosophy of math­ ematics and do research in mathematical logic. His historical un­ derstanding of logic and of the analytic tradition in philosophy and his skeptical questioning of philosophical ideas have offered me both education and challenge. . It should be obvious to my readers that lowe much to the writings and teaching of W. V. Quine. Most apparent is my debt to Quine's philosophy, even where I disagree. From him I also first learned logic. In that subject he provided a model of clarity and philo­ sophical conscience, although my own logical research has been in a different tradition. For my generation, especially at Harvard, he has been a model of a philosopher-logician. Although Hao Wang was my teacher at Harvard for only one of my graduate student years, he did provide indispensable guidance into the work of the Hilbert school in proof theory, at that time . little known in America. Over the years I have had valuable dis­ cussions with him on many subjects. Though he was not formally my teacher, and our interchanges mainly concerned technical matters, I also learned much from Georg Kreisel. Proof theory, an important background for my philo­ sophical work, owes its transplantation from Germany to the En­ glish-speaking world in large measure to him. Others from whom I have learned much about this subject are W. W. Tait, W. A. Howard. Nicolas D. Goodman, Warren D. Goldfarb, and Solomon Feferman. The philosophy of Kant, phenomenology (especially Husserl), and Brouwer's intuitionism were formative influences on my phil­ osophical work which did not come from teachers at Harvard. Though my study of Kant was begun with A. C. Ewing, my real education on the subject came during my graduate student years at Harvard from fellow students, who were former students of C. I. Lewis (by then retired). The "Kant group" ofS. F. Barker, Hubert II

Preface Dreyfus, Samuel Todes, Robert Paul Wolff, and later Ingrid Stadler has already been chronicled in print. 4 To Dreyfus, Todes, and Dagfinn F~llesdal lowe what little understanding I have of Husserl and phenomenology. The stimulus to study Brouwer came from Margaret Masterman in Cambridge in 1955; Richard Braithwaite was a helpful guide in my first study of his difficult writings. Since 1965 Columbia University has been my academic home, and my work owes much to the environment provided by colleagues and students there. Deserving of special mention are Sidney Mor­ genbesser and Isaac Levi for their friendship, their knowledge of many things I do not know, their philosophical acumen, and their readiness for discussion. Among others at Columbia (now or pre­ viously) with whom I have discussed matters connected with these essays, I should thank George Boolos, Raymond Geuss, Dieter Hen­ rich, James Higginbotham, Ernest Nagel. Wilfried Sieg, Howard Stein, and Mark Steiner. The secretarial staff of the department, presided over by Sheila Farrelly Sheridan, has been of great assistance. In particular, Mi­ chael Laser did much of the typing and photocopying for this book. In the academic year 1979-1980 I spent a sabbatical year at Oxford as Visiting Fellow of All Souls College and as a Fellow of the National Endowment for the Humanities. Though no word of this book was written during that time, the conception of it resulted from the project on which I was then working, and the new material (particularly the Introduction) owes something to that work. Ac­ cordingly, acknowledgment is due to these institutions. At an early stage, my work benefited crucially from the freedom and stimulation provided by a Junior Fellowship of the Society of Fellows at Harvard, which I held from 1958 to 1961. Without the illterest alld assistance of Benlhard Kendler of Cor­ nell University Press, this volume would not have come into being. I also thank Allison Dodge, who has seen the book through the Press, alld Richard Tieszen. who did most of the work on the index. To my wife, Maljorie, and my children, Jotham and Sylvia, I owe more than can be said, not least for their efforts to instill in me that "robust sense of reality" which, according to Russell, is needed even \{Ji' "the most abstract speculatiolls." This book is dedicated to the memory of my father, Professor Talcott Parsons. He was for me the first and most significant model of "science as a vocation," and he and my mother provided much 'Wolff, Kant's Theory of Mental Activity, p. x. [2

Preface

encouragement, support, and advice. More particulady, it was through him, beginning even with the books on his shelves in my boyhood, that I first sensed the importance of the German intel­ lectual tradition and especially of Kant. My own concern with classic figures such as Kant, Frege, and Russell no doubt owes inspiration to his dialogue with Weber, Durkheim, Freud, Pareto, and the classical economists. I greatly regret that I did not have this book, or another, ready to present to him in his lifetime. CHARLES PARSONS

New York, New York

IJ

Introduction

Most of the essays in this volume concern the philosophy of mathematics, and issues about mathematics figure prominently in the rest. Obviously a direct attraction to mathematics and its foun­ dations has motivated much of my work. I have also held the con­ viction that mathematical thought is central to our thought in general and that reflection on mathematics is accordingly central to philos­ ophy. Few would dispute its importance in the history of philos­ ophy. In some of the well-known arguments of Plato and Aristotle, for example, mathematics plays a strategic role. The scientific rev­ olution of the seventeenth century consisted in the first instance in the creation of a mathematical science of the physical world, in­ comparably deeper than the pre-Copernican mathematical astron­ omy, in that it went beyond description of the phenomena to a theory of underlying structure and dynamics. Of course Euclidean geometry, one of the pillars of this structure, was inherited from the Greeks. In the history of modern philosophy, both rationalism and empiricism have supported their points of view by interpre­ tations of this science. Seventeenth-century science and its historical successors are at once mathematical and empirical; in general the former characteristic was the basis for the rationalist's case and the latter the basis for the empiricist's. This meant neither that the rationalist could ignore experience nor that the empiricist could ignore mathematics. But the example of mathematics was used through much of the history of philosophy to support rationalistic views, and much argumentation by empiricists has been negative, either against the rationalist's conception of mathematics itself or against the inferences about other domains that he made with the help of a conception of mathematics. IS

Introduction I

Kant was expressing conventional wisdom when he maintained that mathematics is necessary and that it is a priori. He was expressing "intuitions" about mathematics that any philosophy has to take into account. It is certainly part of my own view that these intuitions cannot be ignored. The increasing rigor and abstraction of math­ ematics in the late nineteenth century brought to light another feature of mathematics that the rationalist could appeal to: its gen­ erality of content and application. So long as Euclidean geometry was the philosopher's paradigm of a mathematical. theory and was interpreted as a theory of physical space, this generality could not have its due weight in philosoplW. It came to be strongly empha­ sized by Frege as it was used by him to support his logicism. I will call this thesis, so far rather vaguely defined, the formality of math­ ematics; the idea this term conveys, that mathematics is if not part of logic, at least importantly like logic, is intended. The traditional conception of mathematics as necessary and a priori, and the Fregean thesis of formality, had the effect of dis­ tinguishing mathematics sharply from empirical science. This tra­ ditional view has survived in twentieth-century philosophy, for example in the Vienna Circle's characterization of mathematics as analytic and science as synthetic. But this distinction of mathematics from science has been sharply challenged by Quine'S critique of the analytic and the a priori. Quine presents a picture of mathe­ matics as deriving its evidence from belonging to a body of theory which is tested by experience as a whole, so that in this testing either the mathematics or the specifically scientific part of the the­ ory might be modified if experience proves recalcitrant. The line between logic and mathematics on the one side and natural science on the other is not sharp, either in itself or with respect to their basic epistemological or metaphysical character" This assimilation of mathematics to natural science is in some respects pressed even further by Hilary Plltnam. 2 The traditional view that there is an important distinction be­ tween mathematics and empirical science is one of the guiding ideas of much of my writing. The line between pure mathematics and the special sciences seems to me to be clear enough.!! But of course the real question is whether it is significant. A number of consid­ tFor example. Quine. Phi/l)wplt, of Logic, i>i>. 98-wo.

2See eSJleci,dly "What Is Mathematical Truth?"

'See Essay 7. section V.

16

Introduction

erations developed in these essays indicate to me that it is. First. in direct relation to Quine, I defend the necessity of mathematics and specificaHy the greater generality of mathematical over "physical ~)r natural" modality (Essay 7). A second point is the intimate con­ flection between the fundamental concepts of pure mathematics, such as number, set, and function, and the concepts of formal logic (Essay 6; Essay 7, section VI; Essays 8, 10). A third is the character of modern mathematics as a theory of structures of different types, where the theory is quite general with respect to the "inlernal constitution" of the objects and even of the relations making up the structure (Essay 1, section II; Essay 7. section IV; Essay 6, sections IV_V).4 Scientific theories can be mathematical in form and can thus discern a mathematically describable structure in the real world. for example, a certain geometry in space-time. But the question whether such an attribution is right is different from the "internal" questions about the structure or the question whether it "exists" in the mathematically relevant sense. A fourth considera­ tion is the gross difference in procedures of justification between mathematics and empirical science. in particular the role of proof in mathematics and of experiment and observation in science. This l