Introducing Philosophy of Mathematics

217 Pages·2007·2.87 MB·English

by  Michele Friend

Introducing Philosophy
of Mathematics

Introducing Philosophy
of Mathematics
Michèle Friend
acumen
Dedicated to my parents, Henriette and Tony Friend.
I wish that their nobility of spirit were more commonplace.
© Michèle Friend 2007
Th is book is copyright under the Berne Convention.
No reproduction without permission.
All rights reserved.
First published in 2007 by Acumen
Acumen Publishing Limited
Stocksfi eld Hall
Stocksfi eld
NE43 7TN
www.acumenpublishing.co.uk
ISBN: 978-1-84465-060-6 (hardcover)
ISBN: 978-1-84465-061-3 (paperback)
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Designed and typeset in Warnock Pro by Kate Williams, Swansea.
Printed and bound by Cromwell Press, Trowbridge.
Contents
Acknowledgements vii
Preface ix
1. Infi nity 1
1. Introduction 1
2. Zeno’s paradoxes 2
3. Potential versus actual infi nity 7
4. Th e ordinal notion of infi nity 12
5. Th e cardinal notion of infi nity 13
6. Summary 22
2. Mathematical Platonism and realism 23
1. Introduction 23
2. Historical origins 23
3. Realism in general 26
4. Kurt Gödel 35
5. Penelope Maddy 37
6. General problems with set-theoretic realism 41
7. Conclusion 46
8. Summary 47
3. Logicism 49
1. Introduction 49
2. Frege’s logicism: technical accomplishments 52
3. Frege’s logicism: philosophical accomplishments 58
4. Problems with Frege’s logicism 63
5. Whitehead and Russell’s logicism 66
6. Philosophically, what is wrong with Whitehead and 71
Russell’s type theory?
contents v
7. Other attempts at logicism 78
8. Conclusion 78
9. Summary 79
4. Structuralism 81
1. Introduction 81
2. Th e motivation for structuralism: Benacerraf’s puzzle 83
3. Th e philosophy of structuralism: Hellman 85
4. Th e philosophy of structuralism: Resnik and Shapiro 90
5. Critique 96
6. Summary 100
5. Constructivism 101
1. Introduction 101
2. Intuitionist logic 106
3. Prima facie motivations for constructivism 113
4. Deeper motivations for constructivism 114
5. Th e semantics of intuitionist logic: Dummett 121
6. Problems with constructivism 123
7. Summary 124
6. A pot-pourri of philosophies of mathematics 127
1. Introduction 127
2. Empiricism and naturalism 130
3. Fictionalism 134
4. Psychologism 137
5. Husserl 141
6. Formalism 147
7. Hilbert 153
8. Meinongian Philosophy of Mathematics 157
9. Lakatos 163
Appendix: Proof: ex falso quod libet 167
Glossary 169
Notes 177
Guide to further reading 191
Bibliography 195
Index 201
vi introducing philosophy of mathematics
Acknowledgements
I should like to thank John Shand for suggesting that I write this book, and for
initial encouragement, and I should like to thank Steven Gerrard at Acumen
for endorsing the proposal and publishing it.
I received very helpful and careful comments from my two reviewers,
Stewart Shapiro and Alan Baker. Any mistakes that remain are my fault
entirely. I should also like to thank an anonymous reviewer for helpful com-
ments. Some colleagues have helped with the section on Husserl. Th ese were
Alena Vencovska, Jairo DaSilva and Marika Hadzipetros. I should also like
to thank Graham Priest for comments on the paper that underpins the sec-
tion on Meinongian philosophy of mathematics, and for the many audiences
to whom I have exposed papers that underscore some of the other sections.
Th ese include the philosophy departments at the University of Hertfordshire
and George Washington University, the mathematics department at George
Washington University and particularly the audience for the Logica ’05 con-
ference in the Czech Republic. I should like to thank David Backer for help-
ing with the fi nal notes and bibliography. Kate Williams edited the text and
produced the illustrations.
On a more personal front, I should like to give special thanks to my parents,
my husband, my enthusiastic seminar students and my friends, who encour-
aged me to write, although they knew not what. I should also like to thank
the philosophy department at George Washington University for academic,
personal and fi nancial support.
Michèle Friend
acknowledgements vii

Preface
Th is book is intended as an upper-level undergraduate text or a lower-level
graduate text for students of the philosophy of mathematics. In many ways
the approach taken is standard. Subjects discussed include Platonism, logi-
cism, constructivism, formalism and structuralism; others that are less often
discussed are also given a hearing.
Th is is not meant to be a comprehensive handbook or defi nitive exhaustive
treatment of all, or even any, of the ideas in the philosophy of mathematics.
Rather, this book contains a selected set of topics that are aired in such a way
as to give the student the confi dence to read further in the literature. A guide
to further reading is given at the back of the book. All the books cited are in
English, and should be available from good university libraries. Having read
this book, the student should be equipped with standard questions to bear
in mind when doing further reading. Th e arguments rehearsed in the text
are by no means the fi nal word on the issues. Many open questions reveal
themselves, inviting further investigation. Inevitably, some of my prejudices
can be detected in the text.
Most of the chapters are self-contained. Anomalous in this respect are
Chapter 1 on infi nity and Chapter 2 on Platonism. Chapter 1 is a technical chap-
ter. I believe that students of the philosophy of mathematics should have a grasp
of what the mathematician means by “infi nity”, since many of the philosophies
of mathematics either have something direct to say about it, or use the concept
implicitly. It is also an engaging technical topic and, thereby, an interesting point
of comparison between the diff erent theories. Pedagogically, it makes sense to
discuss some technical issues while the student is fresh to the work. Having
worked through some technical material, the student will have the courage to
tackle some more technical aspects of the philosophy of mathematics on her
own. Th e remaining chapters are less technical, but be warned: serious readings
in the philosophy of mathematics rarely shy away from discussing quite techni-
cal notions, so a good grounding is essential to further study. For example, it is
usual to be well versed in set theory and model theory.
preface ix