Philosophy of mathematics in the twentieth century: selected essays PDF
Preview Philosophy of mathematics in the twentieth century: selected essays
PHILOSOPHY OF MAT HE MAT ICS IN THE TWENT IETH CE N TU RY PHILOSOPHY OF MAT HE MAT ICS IN THE TWENT IETH CE N TU RY SELECTED ESSAYS Charles Parsons HARVARD UNIVERSITY PRESS Cambridge, Massachusetts London, En gland 2014 Copyright © 2014 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– [Essays. Selections] Philosophy of mathematics in the twentieth century : selected essays / Charles Parsons. pages cm Includes bibliographical references and index. ISBN 978- 0- 674- 72806- 6 (alk. paper) 1. Mathematics—Philosophy. 2. Logic, Symbolic and mathematical. I Title. QA9.2.P372 2014 510.1—dc23 2013030762 For Solomon Feferman, Wilfried Sieg, and William Tait CONTENTS Preface ix Introduction 1 Part I: Some Mathematicians as Phi los o phers 1 The Kantian Legacy in Twentieth-C entury Foundations of Mathematics 11 2 Realism and the Debate on Impredicativity, 1917–1 944 40 Postscript to Essay 2 62 3 Paul Bernays’ Later Philosophy of Mathematics 67 4 Kurt Gödel 93 5 Gödel’s “Russell’s Mathematical Logic” 103 Postscript to Essay 5 123 6 Quine and Gödel on Analyticity 127 Postscript to Essay 6 147 7 Platonism and Mathematical Intuition in Kurt Gödel’s Thought 153 Postscript to Essay 7 188 Part II: Contemporaries 8 Quine’s Nominalism 199 9 Ge ne tic Explanation in The Roots of Reference 220 CONTENTS 10 Hao Wang as Phi los o pher and Interpreter of Gödel 243 11 Putnam on Existence and Ontology 267 12 William Tait’s Philosophy of Mathematics 290 Bibliography 321 Copyright Ac know ledg ments 343 Index 345 viii PREFACE The present volume is the second of two volumes of selected essays about other phi los o phers, the fi rst being From Kant to Husserl (2012). The division is essentially chronological, although the present volume is also more focused on philosophy of mathematics. The writings discussed range in date from the fi rst de cade of the twentieth century into the fi rst de cade of the twenty- fi rst. The arrangement is roughly chronological with respect to subject matter. I will not undertake h ere to sketch the con- tent of individual essays. The Introduction that follows will undertake to do that at least to a certain extent, while placing them in the context of the broader developments of which the matters discussed in these essays are a part. Although I have followed more of the development of the founda- tions of mathematics in the twentieth century than of the (vast) de- velopment of philosophy from Kant to Husserl, this volume, like its pre de ces sor, consists of essays and leaves out much of the relevant history considered as a whole. Important fi gures whose contribution was mainly mathematical, such as Zermelo, Skolem, Herbrand, and many post-w ar fi gures, are left out because the essays concern philoso- phy of mathematics. I have also been selective in the parts of the philo- sophical history I have discussed. The most signifi cant omission from the early part of the century is undoubtedly Bertrand Russell. I have also not undertaken to write about Wittgenstein, in part because so many other able phi los o phers, some of them close to me personally, have devoted a major part of their energy to understanding his work. Furthermore, Wittgenstein’s philosophy of mathematics, what ever its intrinsic interest might be, stands apart from the main development. Another omission is that of the Vienna Circle and especially Rudolf Carnap. That omission is more serious because I have written about two critics of their views, ix
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