Philosophy of mathematics in the twentieth century: selected essays

365 Pages·2014·1.193 MB·English

by  Parsons, Charles

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