An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure PDF

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This file is to be used only for a purpose specified by Palgrave Macmillan, such as checking proofs, preparing an index, reviewing, endorsing or planning coursework/other institutional needs. You may store and print the file and share it with others helping you with the specified purpose, but under no circumstances may the file be distributed or otherwise made accessible to any other third parties without the express prior permission of Palgrave Macmillan. Please contact [email protected] if you have any queries regarding use of the file.

An Aristotelian Realist Philosophy of Mathematics

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Also by James Franklin CORRUPTING THE YOUTH: A HISTORY OF PHILOSOPHY IN AUSTRALIA PROOF IN MATHEMATICS: AN INTRODUCTION (with A. Daoud) THE SCIENCE OF CONJECTURE: Evidence and Probability before Pascal WHAT SCIENCE KNOWS: And How It Knows It

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An Aristotelian Realist Philosophy of Mathematics Mathematics as the Science of Quantity and Structure James Franklin University of New South Wales, Sydney, Australia

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© James Franklin 2014 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No portion of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, Saffron House, 6–10 Kirby Street, London EC1N 8TS. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The author has asserted his right to be identified as the author of this work in accordance with the Copyright, Designs and Patents Act 1988. First published 2014 by PALGRAVE MACMILLAN Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited, registered in England, company number 785998, of Houndmills, Basingstoke, Hampshire RG21 6XS. Palgrave Macmillan in the US is a division of St Martin’s Press LLC, 175 Fifth Avenue, New York, NY 10010. Palgrave Macmillan is the global academic imprint of the above companies and has companies and representatives throughout the world. Palgrave® and Macmillan® are registered trademarks in the United States, the United Kingdom, Europe and other countries ISBN: 978–1–137–40072–7 This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. Logging, pulping and manufacturing processes are expected to conform to the environmental regulations of the country of origin. A catalogue record for this book is available from the British Library. A catalog record for this book is available from the Library of Congress.

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Contents List of Figures

ix

List of Tables

x

Introduction

1

Part I 1

The Science of Quantity and Structure

The Aristotelian Realist Point of View The reality of universals Platonism and nominalism The reality of relations and structure ‘Unit-making’ properties and sets Causality Aristotelian epistemology

11 11 12 15 16 17 18

2 Uninstantiated Universals and ‘Semi-Platonist’ Aristotelianism Determinables and determinates Uninstantiated shades of blue and huge numbers Possibles by recombination? Semi-Platonist Aristotelianism

21 22 23 25 26

3 Elementary Mathematics: The Science of Quantity Two realist theories of mathematics: quantity versus structure Continuous quantity and ratios Discrete quantity and numbers Discrete quantity and sets Discrete and continuous quantity compared Defining ‘quantity’

31 31 34 36 38 44 45

4 Higher Mathematics: Science of the Purely Structural The rise of structure in mathematics Structuralism in recent philosophy of mathematics Abstract algebra, groups, and modern pure mathematics Structural commonality in applied mathematics Defining ‘structure’ The sufficiency of mereology and logic Is quantity a kind of structure?

48 48 49 51 54 56 59 63

v

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Contents

5 Necessary Truths about Reality Examples of necessity Objections and replies

67 67 71

6 The Formal Sciences Discover the Philosophers’ Stone A brief survey of the formal or mathematical sciences The formal sciences search for a place in the sun Real certainty: program verification Real certainty: the other formal sciences Experiment in the formal sciences

82 83 89 92 95 98

7 Comparisons and Objections Frege’s limited options The Platonist/nominalist false dichotomy Nominalism Constructions in set theory Avoiding the question: what are sets? Overemphasis on the infinite Measurement and the applicability of mathematics The indispensability argument Modal and Platonist structuralism Epistemology and ‘access’ Naturalism: non-Platonist realisms

101 101 104 106 108 110 111 113 114 117 121 122

8 Infinity Infinity, who needs it? Paradoxes of infinity? ‘Potential’ infinity? Knowing the infinite

129 130 134 136 140

9 Geometry: Mathematics or Empirical Science? What is geometry? Plan A: multidimensional quantities What is geometry? Plan B: the shapes of possible spaces The grit-or-gunk controversy: does space consist of points? Real non-spatial ‘spaces’ with geometric structure The space of colours Spaces of vectors The real space we live in Non-Euclidean geometry: the ‘loss of certainty’ in mathematics?

141 143 146 150 153 154 155 156

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Contents

Part II 10

11

12

13

14

vii

Knowing Mathematical Reality

Knowing Mathematics: Pattern Recognition and Perception of Quantity and Structure The registering of mathematical properties by measurement devices and artificial intelligence Babies and animals: the simplest mathematical perception Animal and infant knowledge of quantity Perceptual knowledge of pattern and structure

165 167 172 173 176

Knowing Mathematics: Visualization and Understanding Imagination and the uninstantiated Visualization for understanding structure The return of visualization, and its neglect Why visualization has been persona non grata in the philosophy of mathematics The mind and structural properties: the mysteriousness of understanding The chiliagon and the limits of visualization

180 180 181 185

Knowing Mathematics: Proof and Certainty Proof: a chain of insights Symbolic proof ‘versus’ visualization: their respective advantages Proof: logicist, ‘if-thenist’ and formalist errors Axioms, formalization and understanding Counting Knowing the infinite

192 192

Explanation in Mathematics Explanation in pure mathematics How do pure mathematical explanations fit into accounts of explanation? Geometrical explanation in science Non-geometrical mathematical explanation in science Aristotelian realism for explanatory success

207 208

Idealization: An Aristotelian View Applied mathematics without idealization Approximation with simple structures, not idealization Negative and complex numbers, ideal points, and other extensions of ontology

222 224 225

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185 188 191

196 197 200 202 203

212 215 217 220

229

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viii

15

Contents

Zero The empty set

234 238

Non-Deductive Logic in Mathematics Estimating the probability of conjectures Evidence for (and against) the Riemann Hypothesis The classification of finite simple groups Probabilistic relations between necessary truths? The problem of induction in mathematics

241 242 245 250 254 257

Epilogue: Mathematics, Last Bastion of Reason

260

Notes

263

Select Bibliography

302

Index

305

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List of Figures I.1 3.1 3.2 4.1 4.2 4.3 4.4 5.1 5.2 5.3 5.4 5.5 6.1 8.1 8.2 9.1 9.2 9.3 9.4 9.5 10.1 10.2 10.3 10.4 11.1 12.1 12.2 13.1 13.2 13.3 14.1 14.2

There are six different pairs in four objects Squares whose intersection is another square Why 2 × 3 = 3 × 2 The bridges of Königsberg Graphs of the same relation between different quantities Exponential growth curve Combinatorics with six points Tiling of the plane by squares Tiling of the plane by regular hexagons Regular pentagons cannot tile the plane Escher’s Waterfall A figure vertically, horizontally, and centrally symmetric The Königsberg bridges again ‘Approximation’ of the diagonal by a path of many steps Approximating the circumference by inscribed and circumscribed polygons Ordered set in two dimensions Cross product of totally ordered set with itself, with natural two-dimensional ordering Pappus’s Theorem The seven-point plane The Munsell version of the colour sphere Simple neural net to predict y from x Line of best fit to a set of points Square, diamond, and ‘diamond’ with context suggesting a square Three realizations of the same tree structure Why 2 × 3 = 3 × 2 Combinatorics with six points again Why (2k)2 = 4k2 Sum of odd numbers is a square The Bridges of Königsberg again Watermill gearing, Worthing, Norfolk, 1876 Dilation of the plane by a factor r The number line, with zero included

1 37 43 48 54 55 60 68 68 68 70 71 96 132 133 145 145 148 149 154 169 170 177 178 182 193 195 210 215 216 231 231

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List of Tables 3.1 Possible outcomes of four coin tosses 11.1 Array with i,j’th entry equal to min(i,j) 15.1 Early calculations of roots of the Riemann zeta function 15.2 Later calculations of roots of the Riemann zeta function 15.3 First few values of the Möbius function

39 184 246 246 248

x

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