Gowers Timothy - The Princeton Companion to Mathematics

The Princeton Companion to Mathematics

EDITOR

Timothy Gowers

University of Cambridge

ASSOCIATE EDITORS

June Barrow-Green

The Open University

Imre Leader

University of Cambridge

Princeton University Press
Princeton and Oxford

Copyright 2008 by Princeton University Press

Published by Princeton University Press,

41 William Street, Princeton, New Jersey 08540

In the United Kingdom: Princeton University Press,
6 Oxford Street, Woodstock, Oxfordshire OX20 1TW

All Rights Reserved

Library of Congress Cataloging-in-Publication Data

The Princeton companion to mathematics / Timothy Gowers, editor;

June Barrow-Green, Imre Leader, associate editors.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-691-11880-2 (hardcover : alk. paper)

1. MathematicsStudy and teaching (Higher) 2. Princeton University.

I. Gowers, Timothy. II. Barrow-Green, June, date-III. Leader, Imre.

QA11.2.P745 2008

510dc22 2008020450

British Library Cataloging-in-Publication Data is available

Grateful acknowledgment is made for permission
to reprint the following illustrations in part VI:

Page 739. Portrait of Ren Descartes taken from Pantheon
berhmter Menschen aller Zeiten (Zwickau, 1830). Courtesy of
Niederschsische Staats- und Universittsbibliothek Gttingen.

Page 742. Portrait of Isaac Newton. By permission of
the Master and Fellows, Trinity College Cambridge.

Page 744. Copy after a portrait of Gottfried Leibniz by Andreas
Scheits (1703). Courtesy of Gottfried Wilhelm Leibniz
BibliothekNiederschsische Landesbibliothek Hannover.

Page 748. Portrait of Leonhard Euler by J. F. A. Darbs (inv. no.
1829-8). Copyright: Muse dart et dhistoire, Ville de Genve.

Page 756. Portrait of Carl Friedrich Gauss. Courtesy of
Niederschsische Staats- und Universittsbibliothek Gttingen.

Page 775. Portrait of Bernhard Riemann. Courtesy of
Niederschsische Staats- und Universittsbibliothek Gttingen.

Page 786. Portrait of Henri Poincar. Courtesy of
Henri Poincar Archives (CNRS, UMR 7117, Nancy).

Page 788. Portrait of David Hilbert. Courtesy of
Niederschsische Staats- und Universittsbibliothek Gttingen.

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Bertrand Russell, in his book The Principles of Mathematics, proposes the following as a definition of pure mathematics.

Pure Mathematics is the class of all propositions of the form p implies q, where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form. In addition to these, mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth.

The Princeton Companion to Mathematics could be said to be about everything that Russells definition leaves out.

Russells book was published in 1903, and many mathematicians at that time were preoccupied with the logical foundations of the subject. Now, just over a century later, it is no longer a new idea that mathematics can be regarded as a formal system of the kind that Russell describes, and todays mathematician is more likely to have other concerns. In particular, in an era where so much mathematics is being published that no individual can understand more than a tiny fraction of it, it is useful to know not just which arrangements of symbols form grammatically correct mathematical statements, but also which of these statements deserve our attention.

Of course, one cannot hope to give a fully objective answer to such a question, and different mathematicians can legitimately disagree about what they find interesting. For that reason, this book is far less formal than Russells and it has many authors with many different points of view. And rather than trying to give a precise answer to the question, What makes a mathematical statement interesting? it simply aims to present for the reader a large and representative sample of the ideas that mathematicians are grappling with at the beginning of the twenty-first century, and to do so in as attractive and accessible a way as possible.

The central focus of this book is modern, pure mathematics, a decision about which something needs to be said. Modern simply means that, as mentioned above, the book aims to give an idea of what mathematicians are now doing: for example, an area that developed rapidly in the middle of the last century but that has now reached a settled form is likely to be discussed less than one that is still developing rapidly. However, mathematics carries its history with it: in order to understand a piece of present-day mathematics, one will usually need to know about many ideas and results that were discovered a long time ago. Moreover, if one wishes to have a proper perspective on todays mathematics, it is essential to have some idea of how it came to be as it is. So there is plenty of history in the book, even if the main reason for our including it is to illuminate the mathematics of today.

The word pure is more troublesome. As many have commented, there is no clear dividing line between pure and applied mathematics, and, just as a proper appreciation of modern mathematics requires some knowledge of its history, so a proper appreciation of pure mathematics requires some knowledge of applied mathematics and theoretical physics. Indeed, these areas have provided pure mathematicians with many fundamental ideas, which have given rise to some of the most interesting, important, and currently active branches of pure mathematics. This book is certainly not blind to the impact on pure mathematics of these other disciplines, nor does it ignore the practical and intellectual applications of pure mathematics. Nevertheless, the scope is narrower than it could be. At one stage it was suggested that a more accurate title would be The Princeton Companion to Pure Mathematics: the only reason for rejecting this title was that it does not sound as good as the actual title.

Another thought behind the decision to concentrate on pure mathematics was that it would leave open the possibility of a similar book, a companion Companion so to speak, about applied mathematics and theoretical physics. Until such a book appears, The Road to Reality, by Roger Penrose (Knopf, New York, 2005), covers a very wide variety of topics in mathematical physics, written at a level fairly similar to that of this book, and Elsevier has recently brought out a five-volume Encyclopedia of Mathematical Physics (Elsevier, Amsterdam, 2006).

The word companion is significant. Although this book is certainly intended as a useful work of reference, you should not expect too much of it. If there is a particular mathematical concept that you want to find out about, you will not necessarily be able to find out about it here, even if it is important; though the more important it is, the more likely it is to be included. In this respect, the book is like a human companion, complete with gaps in its knowledge and views on some topics that may not be universally shared. Having said that, we have at least aimed at some sort of balance: many topics are not covered, but those that are covered range very widely (much more so than one could reasonably expect of any single human companion). In order to achieve this kind of balance, we have been guided to some extent by objective indicators such as the American Mathematical Societys classification of mathematical topics, or the way that mathematics is divided into sections at the four-yearly International Congress of Mathematicians. The broad areas, such as number theory, algebra, analysis, geometry, combinatorics, logic, probability, theoretical computer science, and mathematical physics, are all represented, even if not all their subareas are. Inevitably, some of the choices about what to include, and at what length, were not the result of editorial policy, but were based on highly contingent factors such as who agreed to write, who actually submitted after having agreed, whether those who submitted stuck to their word limit, and so on. Consequently, there are some areas that are not as fully represented as we would have liked, but the point came where it was better to publish an imperfect volume than to spend several more years striving for perfect balance. We hope that there will be future editions of