Table Of ContentQuite 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
