A structural account of mathematics CHARLES s CHIHARA

A Structural Account of Mathematics Charles Chihara develops and defends a structural view of the nature of mathematics, and uses it to explain a number of striking features of mathematics that have puzzled philosophers for centuries The view is used to show that, in order to understand how mathematical systems are applied in science and everyday life, it is not necessary to assume that its theorems either presuppose mathematical objects or are even true Chihara builds upon his previous work, in which he presented a new system of mathematics, the constructibility theory, which did not make reference to, or presuppose, mathematical objects Now he develops the project further by analysing mathematical systems currently used by scientists to show how such systems are compatible with this nominalistic outlook He advances several new ways of undermining the heavily discussed indispensability argument for the existence of mathematical objects made famous by W V Quine and Hilary Putnam And Chihara presents a rationale for the nominalistic outlook that is quite different from those generally put forward, which he maintains have led to serious misunderstandings A Structural Account of Mathematics will be required reading for anyone working in this field Charles S Chihara is Emeritus Professor of Philosophy at the University of California, Berkeley This page intentionally left blank A Structural Account of Mathematics C H A R L E S S C H I H A R A CLARENDON PRESS - O X F O R D OXFORD UNIVERSITY PRESS Great Clarendon Street, Oxford ox2 6np Oxford University Press is a department of the University of Oxford It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Charles S Chihara 2004 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2004 First published in paperback 2007 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Biddies Ltd, King's Lynn, Norfolk ISBN 978-0-19-926753-8 (Hbk.) ISBN 978-0-19-922807-2 (Pbk.) 08 Dedicated to My Brother Paul whose music soars around the world with a logic all its own This page intentionally left blank Preface This work develops and defends a structural view of the nature of mathematics, which is used to explain a number of striking features of mathematics that have puzzled philosophers for centuries It rejects the most widely held philosophical view of mathematics (Platonism), according to which mathematics is a science dealing with mathematical objects such as sets and numbers—objects which are believed not to exist in the physical world Instead, it makes use of the constructibility theory of my earlier work, Constructibility and Mathematical Existence (Oxford University Press, 1990), to develop a view of mathematics that is distinct from Structuralism and yet makes use of some key ideas of Structuralism The structural view is used to show that, in order to understand how mathematical systems are applied in science and everyday life, it is not necessary to assume that its theorems either presuppose mathematical objects or are even true My previous work had a different aim: its goal was to present and develop a new system of mathematics that did not make reference to, or presuppose, mathematical objects Both works support a nominalistic point of view However, whereas the earlier book was aimed at creating a new nominalistic system of mathematics, the present work analyzes mathematical systems currently used by scientists to show how such systems are compatible with a nominalistic outlook The present work also advances several new ways of undermining the heavily discussed indispensability argument for the existence of mathematical objects made famous by Willard Quine and Hilary Putnam I also endeavor, in this book, to present a rationale for the nominalistic outlook that is quite different from those generally put forward by nominalists I this, to a great extent, because I believe that serious misunderstandings of the nominalistic outlook have been fostered by the type of rationale for nominalism that is typically discussed in the recent philosophical literature A number of criticisms that have been leveled at my constructibility theory by the Structuralists Since these criticisms have for many years been largely unanswered, they may appear to students and non-specialists to be unanswerable In this work, the criticisms will be rebutted viii / PREFACE This work grew out of some graduate seminars in the philosophy of mathematics I gave at Berkeley in the late 1990s I am grateful to the students and faculty members who participated in these seminars and were instrumental in getting this project off the ground In addition some of the ideas of this work were tried out in lectures I gave at the following conferences: The Seventh Asian Logic Conference, held in Hsi-Tou, Taiwan, in June 1999 (a revised version of the paper delivered, "Five Puzzles About Mathematics in Search of Solutions", is to be published in the proceedings of the conference); "One Hundred Years of Russell's Paradox", the International Conference in Logic and Philosophy, held at the University of Munich, June 2001 (where I discussed Shapiro's objections to my earlier work); the symposium in the philosophy of mathematics, Pacific Division, American Philosophical Association, held in Seattle in March 2002 (where Mark Wilson responded to my paper on the van Inwagen puzzle); and the Hawaii International Conference on Arts and Humanities, held in Honolulu in January 2003 (where I discussed nominalism) Ideas from the present work found their way into philosophy lectures I gave at the following institutions: Massachusetts Institute of Technology, Cambridge, November 1999; Institut fur Philosophic, Logikund Wissenschaftstheorie at the University of Munich in November 2000; University of Saarlandes in November 2000; and the Logic Colloquium of the University of California, Berkeley, October 2001 I am grateful to those who raised interesting objections or made helpful suggestions at these lectures (some of whom are mentioned later in footnotes) A number of philosophers have aided the writing of this book Some were kind enough to read and comment on parts of preliminary versions of this work; others have responded to my queries or requests for prepublications or references I am especially grateful to Susan Vineberg for her careful study of Chapters and 11 and for providing me with many very helpful objections to early versions of these chapters; Paul Teller and Guido Bacciagaluppi for their helpful comments on my discussion of the mathematics of quantum mechanics; Alan Code for his many useful insights and references pertaining to Greek mathematics and philosophy; Paolo Mancosu for his help in improving my discussions of the history of geometry; John MacFarlane for his careful reading and criticisms of an early version of the chapter on structuralism; Ellery Eells and Elliot Sober for their many helpful comments on the sections dealing with the holistic version of the indispensability argument; Geoffrey Hellman for his detailed comments on my early objections to his modal structuralism; Stewart Shapiro for his many responses to my queries about his version of structuralism; Richard Zach for his helpful replies to my PREFACE / ix queries about Hilbert; and Penelope Maddy for looking over the sections dealing with her criticisms of the indispensability argument Two mathematicians should also be thanked for their assistance: Theodore Chihara for his helpful comments on the section dealing with Fermat's Last Theorem and James T Smith for providing me with a useful list of references pertaining to the Hilbert-Frege dispute I also wish to thank two budding philosophers, Jonathan Kastin and Jukka Keranen, for allowing me to read prepublications of their papers on Shapiro's structuralism Many thanks also to two unnamed readers for OUP for their many genuinely useful suggestions My Ph.D student William Goodwin has ably served as my research assistant for this work, reading the whole of the manuscript, making useful suggestions and corrections, and constructing the index For this, I am most grateful Thanks also to Angela Blackburn, for her excellent editorial assistance As always, my beloved wife Carol has aided me in a variety of ways throughout the writing of this work, but she has been especially helpful by serving as my in-house specialist dealing with the many problems that arose involving the computer and also by serving as my consultant on all matters pertaining to biology and genetics Finally, I would also like to express my deep thanks to Drs Lolly Schiffmann and Paul Li of Kaiser Permanente for extending the time I have left for the kind of productive research needed to complete this work NOTATIONAL CONVENTIONS The notational conventions I use in this work are those of my earlier works, Chihara, 1990 and Chihara, 1998 Briefly, double quotation marks are used for direct quotation and as scare quotes Single quotation marks are used to refer to linguistic items such as words and symbols Greek letters are used as meta-variables The primitive symbols of an object-language discussed are frequently used autonymously For additional explanations, with examples, of the conventions I use, see Chihara, 1998: pp x-xi Throughout this work, I use 'iff as an abbreviation for 'if and only if C S C Berkeley, California June 2003 .. .A Structural Account of Mathematics Charles Chihara develops and defends a structural view of the nature of mathematics, and uses it to explain a number of striking features of mathematics. .. Indispensability Arguments Shapiro 's Account of Applications Fermat 's Last Theorem Applications of Analysis: Some General Considerations Mathematical Modeling 10 Albert 's Version of the Mathematics of Quantum... does not understand classical mathematics and that, for him, statements of classical mathematics have no clear sense.10 So he, with his other intuitionist colleagues, set themselves the task of