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.)
1 3 5 7 9 1 08 6 4 2


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 do 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 5 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


This page intentionally left blank


Contents
Preface

vii

Introduction
1. A Nominalist's View of Philosophy: The Big Picture
2. Nominalistic Reconstructions

1
1
5

1. Five Puzzles in Search of an Explanation
1. A Puzzle about Geometry
2. Different Attitudes of Practicing Mathematicians Regarding

the Ontology of Mathematics
3. The Inertness of Mathematical Objects
4. Consistency and Mathematical Existence
5. The van Inwagen Puzzle

8
8
11
12
17
19

2. Geometry and Mathematical Existence
1. The Frege-Hilbert Dispute Concerning the Axioms of Geometry
2. Some Suggestions Regarding the Nature of Mathematics
3. The First Puzzle
4. The Fourth Puzzle
5. Some Concluding Thoughts

27
27
40
41
46
48

3. The van Inwagen Puzzle
1. Structures
2. My Resolution of the Puzzle
3. The Genetics Objection

4. The Problem of Multiple Reductions
5. An Intuitive Understanding of Set
6. Structuralism?

49
49
52
56
59
60
60

4. Structuralism
1. Shapiro's Characterization of Structures
2. Mathematics Viewed as the Science of Structures
3. Mathematical and Ordinary Structures
4. Some Ways in Which My Account will Differ from
the Structuralists'
5. Ontological Aspects of Shapiro's Structuralism

62
62
63
65
65
66


xii / CONTENTS
6.

7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.

Why Believe in the Existence of Structures?
Resnik's Early Theses about Structures and Positions
Shapiro and Thesis [2]
Shapiro's Rejection of Thesis [1]
Some Problems with Shapiro's ante rem Structuralism
A Problem with Shapiro's Acceptance of Thesis [2]
The Main Problem with Eliminative Structuralism
Resnik on Structures
Resnik's No-Structure Theory
Problems with Resnik's No-Structure Theory
Resnik's Rejection of Classical Logic
The Main Problem with Resnik's Version of Thesis [2]
A Conceptual Objection to Structuralism

70
71
73

75
76
81
83
84
86
90
91
95
96

5. Platonism
1. Gödelian Platonism
2. Quine's Challenge to the Nominalist
3. Nominalistic Responses to Quine's Challenge
4. The Direct Indispensability Argument
5. The Indispensability Argument Based on Holism
6. Putnam's Version of the Indispensability Argument
7. Resnik's Version of the Argument
8. Sober's Objection to the Indispensability Argument
9. Maddy's Objections
10. Resnik's New Indispensability Argument
11. Concluding Assessment of the Various Arguments

99
99
104
107
115
119

123
126
127
136
146
149

6. Minimal Anti-Nominalism
1. The Burgess-Rosen Account of Nominalism
2. Nominalistic Reconstructions of Mathematics Reexamined

151
151
159

7. The Constructibility Theory
1. A Brief Exposition of the Constructibility Theory
2. Preliminaries of the Constructibility Theory of
Natural Number Attributes
3. Cardinal Number Attributes
4. Shapiro's Objections to the Constructibility Theory
5. Other Objections Shapiro has Raised
6. Resnik's Objections

169
170
173
177
184
195

207


CONTENTS / xiii
8. Constructible Structures
1.
2.
3.
4.
5.
6.
7.

218

A Problem for the Structuralist
Realizations without Commitment to Mathematical Objects
Constructible Realizations
Constructible Realizations of Mathematical Theories
The Natural Number Realization
Higher-Level Constructible Realizations
Realization of First-Order Theories

218
218
221
224
225
226
227

9. Applications
1. Applications of Constructibility Arithmetic
2. Peano Arithmetic
3. Putnam's If-Thenism
4. Frege and Dummett on Why Mathematical Theorems
must Express Thoughts
5. A Reexamination of Various Indispensability Arguments
6. Shapiro's Account of Applications
7. Fermat's Last Theorem
8. Applications of Analysis: Some General Considerations
9. Mathematical Modeling
10. Albert's Version of the Mathematics of Quantum Mechanics
11. The Mathematics of Quantum Physics
12. The Fundamental Theorem of the Integral Calculus
13. The Burgess "Tonsorial Question"
14. Applications of Set Theory in Logic
15. Maddy's Mystery

229
231
240
245
249
253
254
257
261
262
265

270
280
286
289
291

10. If-Thenism
1. The Third Puzzle
2. Brown's "Other Avenues"
3. The Second Puzzle
4. Criticisms of If-Thenism
5. The Main Stumbling Block of the Eliminative Program

293
294
297
304
307
315

11. Field's Account of Mathematics and Metalogic
1. Why I Should not be Called a "Fictionalist"
2. Field's Metalogical Theorems
3. Field's Justification for Accepting Standard Metalogical Results
4. Field's Other Justifications of his Conservation Principle

317
318
319
321

329


xiv / CONTENTS
5. Field's Arguments that Good Mathematics is Conservative
6. A Comparison between Two Views of Mathematics
7. The Fundamental Theorem Revisited

332
339
345

Appendix A. Some Doubts About Hellman's Views

349

Appendix B. Balaguer's Fictionalism

356

Bibliography

363

Index

373


Introduction

I begin this work with a personal view of philosophy: a view that is set forth
not as something to be argued for and defended with abundant quotations
from philosophical journals and the writings of great philosophers, but as a
kind of orientation piece, aimed at setting out much of the motivation for the
philosophical developments to follow.

1. A NOMINALIST'S VIEW OF PHILOSOPHY:
THE BIG PICTURE
The field of philosophy is divided into a number of specialties. Among these,
there is philosophy of language, philosophy of mind, philosophy of science,
philosophy of logic, philosophy of art, philosophy of history, philosophy of
religion, and so on. For practically any area X of intellectual study, there is a
philosophy of X. As a general rule, one can say that the philosophy of X is aimed
at achieving a kind of understanding of X that is unique to philosophy. One
might call this sort of understanding "Big Picture understanding". What one
seeks in philosophy is the really "Big Picture": what, in general and in broad
outlines, is the universe like? What, in general and in broad outlines, is our
(i.e. humanity's) place in the universe? How, in general and in broad outlines,
do we (humans) gain an understanding of the universe? And, more specifically,
how, in general and in broad outlines, does X fit into this Big Picture?
Of course, this goal of producing such a Big Picture should not mislead one
into thinking that subtle distinctions, careful and detailed examination of
conceptual matters, and lengthy and intricate reasoning about minute points
should not matter. Philosophers are concerned with very fundamental concepts upon which much rests, so that their analyses, even about apparently
small matters, have very far-reaching consequences for the Big Picture being
constructed.
In this search for the Big Picture, coherence is an essential ingredient. We
seek an understanding of X that is consistent with, and holds together with,
the other views we accept about the universe and about us. Take the
philosophy of language, for example. Here, we seek an understanding of the

nature of language and our mastery of language that is consistent with our


2 / INTRODUCTION
general scientific, epistemological, and metaphysical views, both about the
universe we inhabit and also about us as organisms with the features attributed to us by science. An account of the nature of language that made our
ability to learn a language into a complete mystery would be considered by
most philosophers of language to be in serious trouble. We seek a coherent
and comprehensive Big Picture, where all the different Xs fit together. In
general, one would not expect a contemporary philosopher's account of
language to contradict any of our prevailing views of science and scientific
knowledge without very compelling reasons.

Revolutions in science
This conception of philosophy suggests an explanation of a striking feature of
the history of philosophy. Following the development and acceptance of a
revolutionary scientific theory—a theory that undermines fundamental and
central beliefs of the well-educated elite—there tends to appear a heightened
amount of philosophical theorizing. Think of the important (and radical)
philosophical writings that appeared following the seventeenth-century scientific discoveries that undermined much of the medieval conception of the
universe. Or consider the philosophical activity that arose from the publication of Darwin's work on evolution.1 Other examples are the enormous
number of philosophical works dealing with Freud's writings on mental illness and childhood sexuality, and the lively discussions in present-day philosophy of science dealing with the remarkable implications of relativity
theory and quantum mechanics that conflict with so many fundamental
beliefs underlying Newtonian physics.
The above-noted activity of philosophers is fitting, given the conception of
philosophy I have been describing. When science undermines fundamental
and central beliefs—fractures our Big Picture of the universe—then philosophers understandably feel a pressing need to put the pieces together again, to
develop a new and coherent Big Picture of the universe.

The importance of paradoxes

Another characteristic of philosophy is its great attention to, and serious
interest in, uncovering and solving paradoxes or antinomies. A paradox is an
argument that starts with premises that seem to be incontestable, that
proceeds according to rules of inference that are apparently incontrovertible,
1

A work that emphasized the philosophical responses to these two cases, the seventeenthcentury scientific discoveries and the Darwinian theory of evolution, is Girvetz et al., 1966.


INTRODUCTION / 3
but that ends in a conclusion that appears to be obviously false. In many
cases, a paradox ends in a conclusion that is downright self-contradictory.
From the earliest beginnings of philosophy in Classical Greece (think of
Parmenides, Heraclitus, and Zeno), paradoxes have played an important role
in motivating and developing philosophical theories.
Consider what is undoubtedly one of the most striking cases of philosophical fervor brought about by the discovery of paradoxes: the discovery of
the various paradoxes of mathematics and set theory in the late nineteenth
and early twentieth centuries. These paradoxes stimulated much research in
the foundations of logic and mathematics.2 They led Frege eventually to
abandon his logicism.3 They also stimulated Poincare to come up with his
vicious-circle principle.4 The paradoxes figured in Zermelo's defense of his
axiomatization of set theory.5 Russell, who discovered the paradox that bears
his name, was led to develop his Theory of Types and his "no-class" theory
during the many years he spent searching for a solution to the paradoxes.6
Hilbert motivated certain aspects of his formalist philosophy of mathematics
by the need to make certain that such paradoxes would never again be
produced in mathematics.7
Reasons for the importance philosophers attach to paradoxes are not hard
to find, given the above view of philosophy. An antinomy starts from premises that appear to be obviously true and proceeds according to principles of
inference that seem to be clearly valid. These premises and principles may be

fundamental to our belief system (some may belong to a body of beliefs
classified as "folklore"). An antinomy may show that some such beliefs and/or
principles clash with recent developments in science, mathematics, or logic or
are simply inconsistent.
Alfred Tarski once wrote: "The appearance of an antinomy is for me a
symptom of disease" (Tarski, 1969: 66). What is diseased, evidently, is a body
of beliefs or system of principles that had been largely unquestioned or taken
2
Cf. Raymond Wilder's assessment that "symbolic logic itself had its beginnings long before the
discovery of the contradictions ... There can be little doubt, however, of the great impetus given to
its development by the logical contradictions" (Wilder, 1952: 56).
3
Near the end of his life, Frege completely abandoned his logicism and came to the conclusion
that the source of our arithmetical knowledge is what he called "the Geometrical Source of
Knowledge". See Frege, 1979a.
4
See Chihara, 1973: ch. 1, sect. 1 for a discussion of Poincare's reasoning.
5
See Bach, 1998: 21-2 for supporting arguments.
6
Chihara, 1973: ch. 1, contains a detailed discussion of Russell's attempt to solve the paradoxes
and his development of the Theory of Types, as well as of his "no-class" theory.
7
In Hilbert, 1983: 190-1.


4 / INTRODUCTION
for granted. In those cases, philosophers take it upon themselves to try to heal
the body or system. In the case of the mathematical and set-theoretical
paradoxes, it was thought that the basic principles of mathematics or logic

were shown to be diseased.8 It is no wonder that the discoveries of these
paradoxes brought about so much unease and disquiet.9 One can see why
foundationalists—mathematicians, logicians, and philosophers—put so
much effort into an attempt to put the mathematical house in order. Since a
paradox is a symptom that our body of beliefs and principles is not coherent,
the above account of one of the principal goals of philosophy allows us to see
why philosophers feel the need to attempt to refashion our beliefs and
principles into a more coherent Big Picture in which the paradoxes can no
longer be constructed.

Philosophy of mathematics
Philosophy of mathematics may seem, at first sight, not to fit the overall
scheme I have sketched above. Some philosophers of mathematics frequently
seem not at all to be interested in fitting mathematics into the sort of Big
Picture I have been discussing. Many researchers classified as philosophers of
mathematics seem to be primarily mathematical logicians intent upon
proving theorems and only vaguely aware of questions of epistemology or
metaphysics. Big Picture questions seem to be too far removed from their
expertise to be of any interest to them. Indeed, some philosophers of
mathematics make no attempt to understand our actual mathematical practices at all, but rather set about constructing a new kind of mathematics.
A striking case in point is the philosophy of mathematics known as
"intuitionism" that was advanced by L. E. J. Brouwer, Herman Weyl, and
Arend Heyting. Heyting tells us that he simply 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 developing a new kind of mathematics: "intuitionistic mathematics". Intuitionists have thus appeared to some mathematicians and philosophers to be revolutionaries, seeking to overthrow the established
8
Cf. Russell's attitude toward the paradoxes. He was convinced that logic itself needed to be
reformed. See Chihara, 1973: 1.
9

Frege was dismayed because the very foundations of his system of arithmetic were shaken by
Russell's paradox. Russell, at first, thought that some relatively trivial error was responsible for the
paradoxes. It was only later that he came to think some radical changes in logic were necessary to
resolve the paradoxes. See in this regard Chihara, 1973: ch. 1.
10
See Chihara, 1990: 21-2 for references and more details.


INTRODUCTION / 5
institutions of mathematics in order to replace them with their own brand of
mathematics.11 This will appear strange when compared to what philosophers of science, religion, music, and language do. We do not, for example,
find many philosophers of science claiming not to understand actual science
and setting out to develop a new kind of science. Philosophers of religion, in
general, do not complain of their inability to understand the religions now
being practiced and set out to form a new kind of religion. How many philosophers of language have decided that they do not understand any of the
languages now being spoken and, as a result, have set out to develop a new
kind of language that they can understand?
Of course, there is nothing in my view that precludes there being individual
scholars who are classified as philosophers of mathematics even though they
do not seem to be pursuing research that is directed at contributing to the
kind of Big Picture described earlier. An individual philosopher of mathematics, especially one whose background is primarily mathematical, might
very well undertake research aimed at achieving the kind of results judged
successful by the criteria appropriate to the field of mathematics (for example,
credit for theorems).12 Furthermore, despite appearances to the contrary, I
believe that one can understand the researches of even those philosophers of
mathematics who are primarily interested in theorem-proving, as contributing towards the long-range goal of developing the kind of Big Picture I have
been discussing. Such a strategy is apparent in the technical work of the
"reconstructive nominalist" to be discussed in the next section.

2. NOMINALISTIC RECONSTRUCTIONS

In the latter part of the twentieth century, there have been many attempts to
develop either an alternative kind of mathematics or an alternative kind of
mathematical physics that is classified as "nominalistic". (A number of such
attempts will be discussed in great detail throughout this work.) Since no one
advocates replacing our mathematical and physical theories with such
Hilbert is quoted as saying: "I believe that as little as Kronecker was able to abolish the irrational numbers ... just as little will Weyl and Brouwer today be able to succeed. Brouwer is not, as
Weyl believes him to be, the Revolution—only the repetition of an attempted Putsch, in its day more
sharply undertaken yet failing utterly, and now, with the State armed and strengthened ..., doomed
from the start!" (Reid, 1970: 157). Frank Ramsey wrote of preserving mathematics "from the
Bolshevik menace of Brouwer and Weyl" (Ramsey, 1931: 56).
12
William Thurston, commenting on Jaffe and Quinn, 1993, writes: "Jaffe and Quinn analyze the
motivation to do mathematics in terms of common currency that many mathematicians believe in:
credit for theorems."


6 / INTRODUCTION
alternatives, some philosophers have questioned just what point there is to
such "reconstructions".13 What, it may well be asked, do these nominalistic
versions of mathematics and/or physics contribute to our understanding of
the mathematics and/or physics that mathematicians and scientists actually
use? Later, I shall provide detailed explanations of what some of these
reconstructive nominalists were attempting to accomplish and also how the
nominalistic reconstructions they were devising contribute to the task of
fitting our actual mathematical theories into the Big Picture discussed above.
But at this point, the reader needs to have some idea of the basic philosophical orientation of the kind of nominalist I have in mind.
As I view philosophy of mathematics, philosophers of mathematics hope to
achieve for mathematics what philosophers of language hope to achieve for
language: they seek to produce a coherent overall general account of the
nature of mathematics (where by 'mathematics' I mean the actual mathematics practiced and developed by current mathematicians)—one that is

consistent not only with our present-day theoretical and scientific views
about the world and also our place in the world as organisms with sense
organs of the sort characterized by our best scientific theories, but also with
what we know about how our mastery of mathematics is acquired and tested.
Given this description of the aims of philosophy, all of the technical work
published by the reconstructive nominalists can be judged by its contribution
to the goal of producing such an account.
What is a nominalistic reconstruction of mathematics? To answer this
question, one needs to know what a nominalist is. The kind of nominalist
I have in mind is an anti-realist (or anti-Platonist). Thus, in order to understand nominalism in the philosophy of mathematics, one needs to understand the realist's (or Platonist's) view of mathematics that the nominalist
opposes.
What is realism (Platonism)?
In the philosophy of mathematics, the realist maintains that mathematical
objects exist; the nominalist takes the opposing position that there are no
such things (or that we have no compelling reasons to believe such things
exist). Thus, realists base much of their view of mathematics on the
hypothesis that such things as numbers, sets, functions, vectors, matrices,
13
The pun in the title of the Burgess-Rosen book, A Subject with No Object, suggests that there is
no object (goal) to the nominalistic reconstructions. I shall examine the Burgess-Rosen criticisms of
nominalism in Chapter 6.


INTRODUCTION / 7
spaces, and so on truly exist: they generally assert, for example, that the
theorems of set theory are true statements that tell us what sets in fact exist
and how these mathematical objects are related to one another by the
membership relationship. Now mathematical entities are not supposed to be
things that can be seen, touched, heard, smelled, tasted, or even detected by
our most advanced scientific instruments. So a problem for the realist is to

explain how mathematicians have been able to gain knowledge of such
things. Realists, however, have backed their belief in the existence of such
mathematical entities with a variety of philosophical arguments (to be discussed in great detail in Chapter 5). Nominalists have found these arguments
far from convincing, and remain skeptical of the realists' philosophical
accounts of mathematics. Thus, nominalists have attempted to develop an
account of mathematics that does not require the existence of mathematical
objects which the mathematician is able somehow to discover. The
"nominalistic reconstructions" of mathematics are all supposed to contribute
to such an account.
Puzzles about the nature of mathematics
The focus of the following chapter will be on a number of puzzles about the
nature of mathematics. I look upon these puzzles much as the early
twentieth-century foundationalists looked upon the set-theoretical paradoxes:
as furnishing us with valuable clues and guides as to whether or not one's
view of mathematics is at all accurate. Thus, I am convinced that no philosopher attempting to account for the nature of mathematics can afford to
ignore such puzzles.


1
Five Puzzles in Search of an Explanation
I shall describe five puzzles about mathematics, each in need of investigation
and solution. In succeeding chapters, I shall offer analyses and explanations
of these puzzles, in the course of which my own views on the nature of
mathematics will emerge.

1. A PUZZLE ABOUT GEOMETRY
Consider the first three postulates of Euclid's version of plane geometry:
Postulate 1: A straight line can be constructed from any point to any point.
Postulate 2: A straight line can be extended indefinitely in a straight line.
Postulate 3: A circle can be constructed with its center at any point and with

any radius.1
1
The first four postulates of Euclid's geometry are translated by Thomas Heath (in Heath, 1956:
154) as follows:
Let the following be postulated:

1.
2.
3.
4.

To draw a straight line from any point to any point.
To produce a finite straight line continuously in a straight line.
To describe a circle with any centre and distance.
That all right angles are equal to one another.

But he understands Postulate 1 to be "asserting the possibility of drawing a straight line from one
point to another", Postulate 2 to be "maintaining the possibility of producing a finite straight line
continuously in a straight line" (196), and in the case of Postulate 3, he tells us that "Euclid's text has
the passive of the verb: 'a circle can be drawn'" (199). Commenting on Postulate 4, Heath notes that
according to Proclus, "Geminus held that this Postulate should not be classed as a postulate but as an
axiom, since it does not, like the first three Postulates, assert the possibility of some construction but
expresses an essential property of right angles" (200). E. G. Kogbetliantz (in Kogbetliantz, 1969: 554)
gives the straightforwardly modal translation of Euclid's postulates as follows: Postulate 1: A straight
line can be drawn from any point to any point. Postulate 2: A straight line may be produced, that is,
extended indefinitely in a straight line. Postulate 3: A circle can be drawn with its center at any point
and with any radius. Peter Wolff explains Euclid's first three postulates as follows: "The root
meaning of the word 'postulate' is to 'demand'; in fact, Euclid demands of us that we agree that the
following things can be done: that any two points can be joined by a straight line; that any straight
line may be extended in either direction indefinitely; that given a point and a distance, a circle can

be drawn with that point as center and that distance as radius" (Wolff, 1963: 47-8).


FIVE P U Z Z L E S / 9
Now compare those postulates with the first three axioms of Hilbert's version
published in his Foundations of Geometry.2
Axiom 1: For every two points A, B there exists a line L that contains each of
the two points A, B.
Axiom 2: For every two points A, B there exists no more than one line that
contains each of the points A, B.
Axiom 3: There exist at least two points on a line. There exist at least three
points that do not lie on a line.
Notice that Hilbert's axioms are existential in character: they assert the
existence of certain geometric objects, that is, points and lines. Euclid's postulates, on the other hand, do not assert the existence of anything. Rather,
what is asserted is the possibility of making some sort of geometric construction.3 This constructive aspect of Euclid's geometry is fundamental, for as
Ernest Adams suggests:
Euclid was concerned with applications in a way that modern pure geometry tends
not to be. Many of the propositions of The Elements resemble recipes for doing various
things, e.g., for constructing equilateral triangles. Of course there is much more than
that to The Elements, since many if not most of its propositions are theorems that
state facts, such as the Pythagorean Theorem ... but persons seeking to understand
the applications of Geometry are not ill-advised to look to its older and more technologically oriented formulations. (Adams, 2001: 38)

2

Hilbert, 1971: 3-4.
Cf. Mueller's comments (in Mueller, 1981:14): "Hilbert asserts the existence of a straight line for
any two points, as part of the characterization of the system of points and straight lines he is
treating. Euclid demands the possibility of drawing the straight-line segment connecting the two
points when the points are given. This difference is essential. For Hilbert geometric axioms characterize an existent system of points, straight lines, etc. At no time in the Grundlagen is an object

bought into existence, constructed. Rather its existence is inferred from the axioms. In general
Euclid produces, or imagines produced, the objects he needs for a proof... It seems fair to say that in
the geometry of the Elements there is no underlying system of points, straight lines, etc. which
Euclid attempts to characterize." My colleague Paolo Mancosu has informed me that his researches
indicate that some Greek mathematicians read the modal Euclidean postulates as having an existential commitment to eternal objects and that, even at the time of Euclid, there were disputes about
the "ontological commitments" of the postulates of geometry. This is not surprising, if Reviel Netz
is right in claiming that, in cultural settings such as that of the Greeks, "polemic is the rule, and
consensus is the exception" (Netz, 1999: 2). In any case, it would seem that the historical facts
are more complicated than I have indicated above, but my basic point is that there is a way of
understanding the postulates of Euclid's geometry which is such that no commitment to mathematical objects is presupposed by them and that many mathematicians and philosophers understood them in that way.
3


10 / FIVE PUZZLES
In contrast to Hilbert's existential geometry, then, Euclid's geometry is modal
in character, asserting what it is possible to construct.4 For over two thousand
years, geometry was understood and developed by many mathematicians as a
modal theory, but for some reason, by the twentieth century, geometry had
became straightforwardly existential.
Hilbert was by no means the first mathematician to think and reason about
geometrical objects (such as points and lines) in terms of existence rather
than constructibility. Mathematicians had begun to shift to the existential
mode of expressing geometrical theorems hundreds of years before Hilbert
had written on the topic. This shift in geometry from making constructibility
assertions to asserting existence raises some fundamental questions:
(a) No one seems to have made a fuss about the change that took place or
even to have taken note of it. No one believes that an ordinary existential statement such as "There are buildings with over three hundred
stories" is equivalent to a modal statement of the form "It is possible to
construct buildings with over three hundred stories". Why weren't there
serious debates among mathematicians and philosophers over the

validity of making such an apparently radical ontological change in the
primitives of one of the central theories of mathematics?
(b) The applications to which geometry was put evidently did not change as
a result of the described shift. How is it that, in our reasoning about
areas, volumes, distances, and so on, it seems to make no difference
whether the geometry we use asserts what it is possible to construct or
whether it asserts the existence of mathematical entities?
A completely adequate answer to these questions would require a detailed
investigation into the history of mathematics spanning many hundreds of
years—something that is well beyond the scope of this work. So, although I
shall give some indications in later chapters how I would answer the above
puzzles, I shall concentrate my investigation on the following (restricted but
closely related) puzzle about Hilbert's view of geometry:
In the introduction to his Foundations of Geometry, Hilbert writes: "The
establishment of the axioms of geometry and the investigation of their
relationships is a problem ... equivalent to the logical analysis of our perception of space" (Hilbert, 1971: 2). And he goes on to say that his axioms
4
What does it mean to say that an assertion or theory is "modal"? For those unfamiliar with the
notion of modality, an explanation will be given below in Chapter 5, sect. 3.