Table Of ContentPage iii
The Language of Mathematics
Making the Invisible Visible
Keith Devlin
Page iv
Text Design: Diana Blume
Illustrations: Mel Erikson Art Services, Publication Services,
Ian Warpole, Network Graphics
Library of Congress Cataloging-in-Publication Data
Devlin, Keith J.
The language of mathematics: making the invisible visible / Keith
Devlin.
p. cm.
Includes index.
ISBN 0-7167-3379-X (hardcover)
ISBN 0-7167-3967-4 (paperback)
1. Mathematics—Popular works. I. Title.
QA93.D4578 1998
510—dc21 98-38019
CIP
© 1998, 2000 by W. H. Freeman and Company. All rights reserved.
No part of this book may be reproduced by any mechanical, photographic, or electronic process, or in the
form of a phonographic recording, nor may it be stored in a retrieval system, transmitted, or otherwise
copied for public or private use, without written permission from the publisher.
Printed in the United States of America
First paperback printing 2000
W. H. Freeman and Company
41 Madison Avenue, New York, NY 10010
Houndmills, Basingstoke RG21 6XS, England
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Contents
Preface vii
Prologue 1
What is Mathematics?
Chapter 1 13
Why Numbers Count
Chapter 2 51
Patterns of the Mind
Chapter 3 95
Mathematics in Motion
Chapter 4 139
Mathematics Gets into Shape
Chapter 5 189
The Mathematics of Beauty
Chapter 6 221
What Happens When Mathematics Gets into Position
Chapter 7 271
How Mathematicians Figure the Odds
Chapter 8 299
Uncovering the Hidden Patterns of the Universe
Postscript 337
Index 339
Page vii
Preface
This book tries to convey the essence of mathematics, both its historical development and its current breadth.
It is not a 'how to' book; it is an 'about' book, which sets out to describe mathematics as a rich and living part
of humankind's culture. It is intended for the general reader, and does not assume any mathematical
knowledge or ability.
The book grew from an earlier volume in W. H. Freeman's Scientific American Library series, titled
Mathematics: The Science of Patterns. Written for what is generally termed a 'scientifically literate'
audience, that book proved to be one of the most successful in the Scientific American Library series. In
conversation with Jonathan Cobb, my editor on that project, the idea arose of a 'spin-off' book aimed at a
much wider audience. The new book would not have the high-gloss finish and the masses of full-color
artwork and photographs that make the Scientific American Library series so special. Rather, its aim would
be to get essentially the same story across in a format that made it accessible to a much wider collection of
readers: the story that mathematics is about the identification and study of patterns. (Like its predecessor,
this book will show just what counts as a 'pattern' for the mathematician. Suffice it at this point to observe
that I am not just talking about wallpaper patterns or patterns on shirts and dresses—though many of those
patterns do turn out to have interesting mathematical properties.)
In addition to completely rewriting much of the text to fit a more standard 'popular science book' format, I
have also taken advantage of the change of format to add two additional chapters, one on patterns of
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chance, the other on patterns of the (physical) universe. I had wanted to include both topics in the original
book, but there was not enough space in the Scientific American Library format.
Fernando Gouvea, Doris Schattschneider, and Kenneth Millett offered comments on all or parts of the
manuscript for the original Scientific American Library book, and their helpful advice has undoubtedly
found its way into this new book. Ron Olowin provided helpful feedback on Chapter 8, which along with
Chapter 7 is brand new to this book. Susan Moran was the ever-vigilant copy editor for the Patterns book.
Norma Roche copy-edited this new book.
Historically, almost all leading mathematicians were male, and that is reflected in the almost complete
absence of female characters in the book. Those days are, I hope, gone forever. To reflect today's reality, this
book uses both 'he' and 'she' interchangeably as the generic thirdperson pronoun.
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Prologue:
What Is Mathematics?
It's Not Just Numbers
What is mathematics? Ask this question of persons chosen at random, and you are likely to receive the
answer "Mathematics is the study of numbers." With a bit of prodding as to what kind of study they mean,
you may be able to induce them to come up with the description "the science of numbers." But that is about
as far as you will get. And with that, you will have obtained a description of mathematics that ceased to be
accurate some two and a half thousand years ago!
Given such a huge misconception, there is scarcely any wonder that your randomly chosen persons are
unlikely to realize that research in mathematics is a thriving, worldwide activity, or to accept a suggestion
that mathematics permeates, often to a considerable extent, most walks of present-day life and society.
In fact, the answer to the question "What is mathematics?" has changed several times during the course of
history.
Up to 500 B.C. or thereabouts, mathematics was indeed the study of numbers. This was the period of
Egyptian and Babylonian mathematics. In those civilizations, mathematics consisted almost solely of
arithmetic. It was largely utilitarian, and very much of a 'cookbook' nature ("Do such and such to a number
and you will get the answer").
The period from around 500 B.C. to A.D. 300 was the era of Greek mathematics. The mathematicians of
ancient Greece were primarily concerned with geometry. Indeed, they regarded numbers in a geomet-
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ric fashion, as measurements of length, and when they discovered that there were lengths to which their
numbers did not correspond (irrational lengths), their study of number largely came to a halt. For the Greeks,
with their emphasis on geometry, mathematics was the study of number and shape.
In fact, it was only with the Greeks that mathematics came into being as an area of study, and ceased being a
collection of techniques for measuring, counting, and accounting. The Greeks' interest in mathematics was
not just utilitarian; they regarded mathematics as an intellectual pursuit having both aesthetic and religious
elements. Thales introduced the idea that the precisely stated assertions of mathematics could be logically
proved by a formal argument. This innovation marked the birth of the theorem, now the bedrock of
mathematics. For the Greeks, this approach culminated in the publication of Euclid's Elements, reputedly the
most widely circulated book of all time after the Bible.
Mathematics in Motion
There were no major changes in the overall nature of mathematics, and hardly any significant advances
within the subject, until the middle of the seventeenth century, when Newton (in England) and Leibniz (in
Germany) independently invented the calculus. In essence, the calculus is the study of motion and change.
Previous mathematics had been largely restricted to the static issues of counting, measuring, and describing
shape. With the introduction of techniques to handle motion and change, mathematicians were able to study
the motion of the planets and of falling bodies on earth, the workings of machinery, the flow of liquids, the
expansion of gases, physical forces such as magnetism and electricity, flight, the growth of plants and
animals, the spread of epidemics, the fluctuation of profits, and so on. After Newton and Leibniz,
mathematics became the study of number, shape, motion, change, and space.
Most of the initial work involving the calculus was directed toward the study of physics; indeed, many of the
great mathematicians of the period are also regarded as physicists. But from about the middle of the
eighteenth century there was an increasing interest in the mathematics itself, not just its applications, as
mathematicians sought to understand what lay behind the enormous power that the calculus gave to
humankind. Here the old Greek tradition of formal proof came back into ascendancy, as a large part of
present-day pure mathematics was developed. By the end of the nineteenth century, mathematics had
become the study of number, shape, motion, change, and space, and of the mathematical tools that are used
in this study.
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The explosion of mathematical activity that took place in the twentieth century was dramatic. In the year
1900, all the world's mathematical knowledge would have fitted into about eighty books. Today it would
take maybe a hundred thousand volumes to contain all known mathematics. This extraordinary growth has
not only been a furtherance of previous mathematics; many quite new branches of mathematics have sprung
up. In 1900, mathematics could reasonably be regarded as consisting of about twelve distinct subjects:
arithmetic, geometry, calculus, and so on. Today, between sixty and seventy distinct categories would be a
reasonable figure. Some subjects, such as algebra and topology, have split into various subfields; others,
such as complexity theory or dynamical systems theory, are completely new areas of study.
The Science of Patterns
Given this tremendous growth in mathematical activity, for a while it seemed as though the only simple
answer to the question "What is mathematics?" was to say, somewhat fatuously, "It is what mathematicians
do for a living." A particular study was classified as mathematics not so much because of what was studied
but because of how it was studied—that is, the methodology used. It was only within the last thirty years or
so that a definition of mathematics emerged on which most mathematicians now agree: mathematics is the
science of patterns. What the mathematician does is examine abstract 'patterns'—numerical patterns, patterns
of shape, patterns of motion, patterns of behavior, voting patterns in a population, patterns of repeating
chance events, and so on. Those patterns can be either real or imagined, visual or mental, static or dynamic,
qualitative or quantitative, purely utilitarian or of little more than recreational interest. They can arise from
the world around us, from the depths of space and time, or from the inner workings of the human mind.
Different kinds of patterns give rise to different branches of mathematics. For example:
• Arithmetic and number theory study patterns of number and counting.
• Geometry studies patterns of shape.
• Calculus allows us to handle patterns of motion.
• Logic studies patterns of reasoning.
• Probability theory deals with patterns of chance.
• Topology studies patterns of closeness and position.
To convey something of this modern conception of mathematics, this book takes eight general themes,
covering patterns of counting, patterns
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of reasoning and communicating, patterns of motion and change, patterns of shape, patterns of symmetry and
regularity, patterns of position, patterns of chance, and the fundamental patterns of the universe. Though this
selection leaves out a number of major areas of mathematics, it should provide a good overall sense of what
contemporary mathematics is about. The treatment of each theme, while at a purely descriptive level, is not
superficial.
One aspect of modern mathematics that is obvious to even the casual observer is the use of abstract notation:
algebraic expressions, complicated-looking formulas, and geometric diagrams. The mathematician's reliance
on abstract notation is a reflection of the abstract nature of the patterns he studies.
Different aspects of reality require different forms of description. For example, the most appropriate way to
study the lay of the land or to describe to someone how to find their way around a strange town is to draw a
map. Text is far less appropriate. Analogously, line drawings in the form of blueprints are the most
appropriate way to specify the construction of a building. And musical notation is the most appropriate way
to convey music, apart from, perhaps, actually playing the piece.
In the case of various kinds of abstract, 'formal' patterns and abstract structures, the most appropriate means
of description and analysis is mathematics, using mathematical notation, concepts, and procedures. For
instance, the symbolic notation of algebra is the most appropriate means of describing and analyzing the
general behavioral properties of addition and multiplication. The commutative law for addition, for example,
could be written in English as
When two numbers are added, their order is not important.
However, it is usually written in the symbolic form
m + n = n + m.
Such is the complexity and the degree of abstraction of the majority of mathematical patterns that to use
anything other than symbolic notation would be prohibitively cumbersome. And so the development of
mathematics has involved a steady increase in the use of abstract notation.
Symbols of Progress
The first systematic use of a recognizably algebraic notation in mathematics seems to have been made by
Diophantus, who lived in Alexan-
