Philosophy of Mathematics An Introduction

Philosophy of Mathematics: An Introduction provides a critical analysis of the major philosophical issues and viewpoints in the concepts and methods of mathematics - from antiquity to the modern era. Offers beginning readers a critical appraisal of philosophical viewpoints throughout history Gives a separate chapter to predicativism, which is often (but wrongly) treated as if it were a part of logicism Provides readers with a non-partisan discussion until the final chapter, which gives the author's personal opinion on where the truth lies Designed to be accessible to both undergraduates and graduate students, and at the same time to be of interest to professionals

David Bostock has been a Fellow and Tutor in Philosophy at Merton College, and Lecturer in Philosophy at the University of Oxford. His recent publications include Intermediate Logic (1997), Aristotle's Ethics (2000), and Space, Time, Matter, and Form: Essays on Aristotle's Physics (2006).

Preface	p. viii
Plato versus Aristotle	p. 1
Plato	p. 1
The Socratic background	p. 1
The theory of recollection	p. 3
Platonism in mathematics	p. 7
Retractions: the Divided Line in Republic VI (509d-511e)	p. 9
Aristotle	p. 15
The overall position	p. 15
Idealizations	p. 17
Complications	p. 20
Problems with infinity	p. 23
Prospects	p. 28
From Aristotle to Kant	p. 33
Mediaeval times	p. 33
Descartes	p. 35
Locke, Berkeley, Hume	p. 38
A remark on conceptualism	p. 43
Kant: the problem	p. 44
Kant: the solution	p. 51
Reactions to Kant	p. 62
Mill on geometry	p. 62
Mill versus Frege on arithmetic	p. 69
Analytic truths	p. 74
Concluding remarks	p. 85
Mathematics and its Foundations	p. 87
Geometry	p. 87
Different kinds of number	p. 90
The calculus	p. 93
Return to foundations	p. 97
Infinite numbers	p. 103
Foundations again	p. 111
Logicism	p. 114
Frege	p. 115
Russell	p. 122
Borkowski/Bostock	p. 136
Set theory	p. 140
Logic	p. 152
Definition	p. 160
Formalism	p. 167
Hilbert	p. 168
Gödel	p. 175
Pure formalism	p. 180
Structuralism	p. 183
Some comments	p. 191
Intuitionism	p. 197
Brouwer	p. 197
Intuitionist logic	p. 201
The irrelevance of ontology	p. 215
The attack on classical logic	p. 218
Predicativism	p. 224
Russell and the VCP	p. 224
Russell's ramified theory and the axiom of reducibility	p. 235
Predicative theories after Russell	p. 244
Concluding remarks	p. 253
Realism and Nominalism	p. 261
Realism	p. 262
Gödel	p. 263
Neo-Fregeans	p. 266
Quine and Putnam	p. 275
Nominalism	p. 282
Reductive nominalism	p. 282
Fictionalism	p. 289
Concluding remarks	p. 304
References	p. 310
Index	p. 325
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Philosophy of Mathematics An Introduction

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