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The Language of Mathematics
Making the Invisible Visible
Keith Devlin
 
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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
What is Mathematics?
1
Chapter 1
Why Numbers Count
13
Chapter 2
Patterns of the Mind
51
Chapter 3
Mathematics in Motion
95
Chapter 4
Mathematics Gets into Shape
139
Chapter 5
The Mathematics of Beauty
189
Chapter 6
What Happens When Mathematics Gets into Position
221
Chapter 7
How Mathematicians Figure the Odds
271
Chapter 8
Uncovering the Hidden Patterns of the Universe
299
Postscript
337
Index
339
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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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