Quite Right QUITE RIGHT The Story of Mathematics, Measurement, and Money norman biggs Emeritus Professor of Mathematics London School of Economics 1 3 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom 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. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Norman Biggs 2016 The moral rights of the author have been asserted First Edition published in 2016 Impression: 1 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, by licence 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 work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2015944379 ISBN 978–0–19–875335–3 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work. PREFACE My aim is to explain how mathematical ideas evolved in response to the growing levels of organization in human societies, from pre-historic times to the present day. The traditional approach to the history of mathematics focussed on its relationship to the current activities of professional mathematicians, and this resulted in a somewhat narrow view. In recent decades there has been a welcome movement towards looking at history from a broader viewpoint. Combined with new discoveries of artefacts and documents, this approach leads to a better understanding of the part played by mathematics in the human story, and (rather surprisingly) the nature of mathematics itself. In the spirit of this wider view, the book aims to display the power and beauty of mathematical concepts, which often belie their utilitarian origins. The twin paradigms of logical justification and algorithmic calculation recur throughout. Another recurring theme is the relationship between mathematics and measure- ment of all kinds, including the measurement of value in monetary terms. So we begin with the measurement of length and area in ancient times, continue with the arithmetical problems of using money in the middle ages, and conclude with the contemporary idea that information itself can be measured. Many friends and colleagues have contributed in many ways. Perhaps my great- est debt is to those historians of mathematics who set the standard for me: John Fauvel and Jeremy Gray for their splendid creation of the Open University’s course MA290, and Jacqueline Stedall for her wise and insightful writings. I thank spe- cifically Bob Burn, Leo Rogers, Adam Ostaszewski, and Robin Wilson, who read draft chapters and provided helpful comments. The participants in the monthly Oxford Forum on the history of mathematics listened patiently to some of my early attempts to tell the story. At the Oxford University Press, Keith Mansfield, Dan Taber, Harriet Konishi, Eilidh McGregor, and Victoria Mortimer provided professional support and advice. Christine and Juliet also provided support and advice, as only family can. Norman Biggs October 2015 CONTENTS 1. The Unwritten Story 1 The story of mathematics 1 Fairness in the age of the hunter-gatherers 2 Counting without numbers 3 The origins of money 5 Mysterious moonshine 7 2. The Dawn of Civilization 11 Writing and counting 11 Operating with numbers 13 Fair division in the age of arithmetic 15 Measuring length and area 18 Measuring quantities 21 The origins of units of measurement 24 The elements of finance 26 3. From Tax and Trade to Theorems 29 The coming of coins 29 Practical mathematics 31 The new mathematics 35 Problems of measurement 38 Proving results about whole numbers 42 4. The Age of Algorithms 45 How to do arithmetic 45 Advances in arithmetic 48 The uses of arithmetic 50 The north-west frontier 52 The art of al-jabr 54 The origins of symbolic algebra 57 5. The End of the Middle Ages 59 Merchants and mathematicians 59 Interest upon interest, and more . . . 64 Solving equations 66 viii | CONTENTS Counting and arranging 69 How to keep a secret: the medieval way 73 6. A New World of Mathematics 79 Measurement and calculation 79 To the limit 82 Algebra and geometry combine 84 Back to the future: Archimedes 87 Fermat and the new algebra 88 One small step . . . 90 More about series 93 7. Mathematics Ascending 95 Calculus, coinage, and controversy 95 Logarithms and exponents 98 Numbers, perfect and prime 102 New kinds of numbers 106 All kinds of wonderful things 108 To the limit, carefully 110 8. Taking a Chance 113 Great expectations 113 Games and strategies 118 The law of large numbers 121 Statistics 124 9. Modelling and Measuring 129 Mathematics in context 129 Measuring value and utility 130 The problems of democracy 135 The measure of the world 137 10. Mathematics and Money in the Age of Information 141 Financial speculation 141 A model of how prices change 143 Forms of money 145 Money, information, and uncertainty 147 Measuring information 150 11. Can Mathematics Keep Us Safe? 155 The search for security 155 Some things are easy, but some are not 156 How to keep a secret: the modern way 160 Can mathematics keep us safe? 162 Notes 163 Index 173 CHAPTER 1 The Unwritten Story Imagine that you have to solve your daily problems without the aid of writing or arithmetic. Not so long ago all human beings were in that state. The fact that the human race survived, despite its blissful ignorance, indicates that our prehistoric ancestors were able to cope with problems like dividing the carcass of a slain animal and deciding when to prepare for winter. Such mundane tasks were the seeds from which there emerged the universal problem-solving method that we now call mathematics. The story of mathematics This is a story without a clear beginning, and without an end. It tells how some sim- ple ideas were developed, refined, and applied, so that they pervade almost every aspect of life in the twenty-first century. To make our story intelligible we shall tell it in roughly chronological order. But ideas march to the beat of a different drum, the force of logic, which is often at odds with the arrow of time. Further complication is created by the geographical spread of events: ideas arose independently in different parts of the world, at dif- ferent times, and were developed in different ways. Globalization is a very recent phenomenon. We must set aside almost the entire history of our planet, which, we are told, has been in existence for over four billion years. Human-like beings have lived here for only about two million years, and for most of that time they managed to survive without writing down anything about their activities. The written record, what we might properly call history, covers only the last 5000 years or so, and we cannot attach much meaning to the word mathematics before that time. Even if we look for clues a little further back, it is only within the last 30000 years that we can detect even the faintest hints of mathematics, or the related concepts of measurement and money. This means that we are interested almost entirely in the activities of just one species, Homo sapiens. To the best of our knowledge there have been no significant evolutionary changes in the mental capacity of our