A History of Mathematics This page intentionally left blank A History of Mathematics From Mesopotamia to Modernity Luke Hodgkin 1 3 Great Clarendon Street, Oxford OX2 6DP 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 NewYork Auckland CapeTown 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 © Oxford University Press, 2005 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2005 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 Antony Rowe Ltd., Chippenham, Wiltshire ISBN 0–19–852937–6 (Hbk) 978–0–19–852937–8 13579108642 Preface This book has its origin in notes which I compiled for a course on the history of mathematics at King’s College London, taught for many years before we parted company. My major change in outlook (which is responsible for its form) dates back to a day ten years ago at the University of Warwick, when I was comparing notes on teaching with the late David Fowler. He explained his own history of mathematics course to me; as one might expect, it was detailed, scholarly, and encouraged students to do research of their own, particularly on the Greeks. I told him that I gave what I hoped was a critical account of the whole history of mathematics in a series of lectures, trying to go beyond what they would find in a textbook. David was scornful. ‘What’, he said, ‘do you mean that you stand up in front of those students and tell stories?’ I had to acknowledge that I did. David’s approach meant that students should be taught from the start not to accept any story at face value, and to be interested in questions rather than narrative. It’s certainly desirable as regards the Greeks, and it’s a good approach in general, even if it may sometimes seem too difficult and too purist. I hope he would not be too hard on my attempts at a compromise. The aims of the book in this, its ultimate form, are set out in the introduction; briefly, I hope to introduce students to the history, or histories of mathematics as constructions which we make to explain the texts which we have, and to relate them to our own ideas. Such constructions are often controversial, and always provisional; but that is the nature of history. The original impulse to write came from David Robinson, my collaborator on the course at King’s, who suggested (unsuccessfully) that I should turn my course notes into a book; and providentially from Alison Jones of the Oxford University Press, who turned up at King’s when I was at a loose end and asked if I had a book to publish. I produced a proposal; she persuaded the press to accept it and kept me writing. Without her constant feedback and involvement it would never have been completed. I am grateful to a number of friends for advice and encouragement. Jeremy Gray read an early draft and promoted the project as a referee; the reader is indebted to him for the presence of exercises. Geoffrey Lloyd gave expert advice on the Greeks; I am grateful for all of it, even if I only paid attention to some. John Cairns, Felix Pirani and Gervase Fletcher read parts of the manuscript and made helpful comments; various friends and relations, most particularly Jack Goody, John Hope, Jessica Hines and Sam and Joe Gold Hodgkin expressed a wish to see the finished product. Finally, I’m deeply grateful to my wife Jean who has supported the project patiently through writing and revision. To her, and to my father Thomas who I hope would have approved, this book is dedicated. This page intentionally left blank Contents List of figures xi Picture Credits xiv Introduction 1 Why this book? 1 On texts, and on history 2 Examples 5 Historicism and ‘presentism’ 6 Revolutions, paradigms, and all that 8 External versus internal 10 Eurocentrism 12 1. Babylonian mathematics 14 1. On beginnings 14 2. Sources and selections 17 3. Discussion of the example 20 4. The importance of number-writing 21 5. Abstraction and uselessness 24 6. What went before 27 7. Some conclusions 30 Appendix A. Solution of the quadratic problem 30 Solutions to exercises 31 2. Greeks and ‘origins’ 33 1. Plato and the Meno 33 2. Literature 35 3. An example 36 4. The problem of material 39 5. The Greek miracle 42 6. Two revolutions? 44 7. Drowning in the sea of Non-identity 45 8. On modernization and reconstruction 47 9. On ratios 49 Appendix A. From the Meno 51 Appendix B. On pentagons, golden sections, and irrationals 52 Solutions to exercises 54 viii Contents 3. Greeks, practical and theoretical 57 1. Introduction, and an example 57 2. Archimedes 60 3. Heron or Hero 63 4. Astronomy, and Ptolemy in particular 66 5. On the uncultured Romans 69 6. Hypatia 71 Appendix A. From Heron’s Metrics 73 Appendix B. From Ptolemy’s Almagest 75 Solutions to exercises 76 4. Chinese mathematics 78 1. Introduction 78 2. Sources 80 3. An instant history of early China 80 4. The Nine Chapters 82 5. Counting rods—who needs them? 85 6. Matrices 88 7. The Song dynasty and Qin Jiushao 90 8. On ‘transfers’—when, and how? 95 9. The later period 98 Solutions to exercises 99 5. Islam, neglect and discovery 101 1. Introduction 101 2. On access to the literature 103 3. Two texts 106 4. The golden age 108 5. Algebra—the origins 110 6. Algebra—the next steps 115 7. Al-Samaw’al and al-K¯ash¯i 117 8. The uses of religion 123 Appendix A. From al-Khw¯arizm¯i’s algebra 125 Appendix B. Th¯abit ibn Qurra 127 Appendix C. From al-K¯ash¯i, The Calculator’s Key, book 4, chapter 7 128 Solutions to exercises 130 6. Understanding the ‘scientific revolution’ 133 1. Introduction 133 2. Literature 134 3. Scholastics and scholasticism 135 4. Oresme and series 138 5. The calculating tradition 140 6. Tartaglia and his friends 143 7. On authority 146 Contents ix 8. Descartes 149 9. Infinities 151 10. Galileo 153 Appendix A 155 Appendix B 156 Appendix C 157 Appendix D 158 Solutions to exercises 159 7. The calculus 161 1. Introduction 161 2. Literature 163 3. The priority dispute 165 4. The Kerala connection 167 5. Newton, an unknown work 169 6. Leibniz, a confusing publication 172 7. The Principia and its problems 176 8. The arrival of the calculus 178 9. The calculus in practice 180 10. Afterword 182 Appendix A. Newton 183 Appendix B. Leibniz 185 Appendix C. From the Principia 186 Solutions to exercises 187 8. Geometries and space 189 1. Introduction 189 2. First problem: the postulate 194 3. Space and infinity 197 4. Spherical geometry 199 5. The new geometries 201 6. The ‘time-lag’ question 203 7. What revolution? 205 Appendix A. Euclid’s proposition I.16 207 Appendix B. The formulae of spherical and hyperbolic trigonometry 209 Appendix C. From Helmholtz’s 1876 paper 210 Solutions to exercises 210 9. Modernity and its anxieties 213 1. Introduction 213 2. Literature 214 3. New objects in mathematics 214 4. Crisis—what crisis? 217 5. Hilbert 221 6. Topology 223 x Contents 7. Outsiders 228 Appendix A. The cut definition 231 Appendix B. Intuitionism 231 Appendix C. Hilbert’s programme 232 Solutions to exercises 232 10. A chaotic end? 235 1. Introduction 235 2. Literature 236 3. The Second World War 238 4. Abstraction and ‘Bourbaki’ 240 5. The computer 243 6. Chaos: the less you know, the more you get 246 7. From topology to categories 249 8. Physics 251 9. Fermat’s Last Theorem 254 Appendix A. From Bourbaki, ‘Algebra’, Introduction 256 Appendix B. Turing on computable numbers 256 Solutions to exercises 258 Conclusion 260 Bibliography 263 Index 271 List of figures Introduction 1 1. Euclid’s proposition II.1 5 Chapter 1. Babylonian mathematics 14 1. A mathematical tablet 15 2. Tally of pigs 16 3. The ‘stone-weighing’ tablet YBC4652 18 4. Cuneiform numbers from 1 to 60 23 5. How larger cuneiform numbers are formed 23 6. The ‘square root of 2’ tablet 25 7. Ur III tablet (harvests from Lagash) 28 Chapter 2. Greeks and ‘origins’ 33 1.