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An Aristotelian Realist Philosophy of Mathematics 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 An Aristotelian Realist Philosophy of Mathematics Mathematics as the Science of Quantity and Structure James Franklin University of New South Wales, Sydney, Australia © James Franklin 2014 Softcover reprint of the hardcover 1st edition 2014 978-1-137-40072-7 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 978-1-349-48618-2 DOI 10.1057/9781137400734 ISBN 978-1-137-40073-4 (eBook) 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. Contents List of Figures ix List of Tables x Introduction 1 Part I The Science of Quantity and Structure 1 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 vi Contents 5 Necessary Truths about Reality Examples of necessity Objections and replies 67 67 71 6 The Formal Sciences Discover the Philosophers’ Stone