# The Princeton Companion to Mathematics

EDITOR Timothy Gowers
June Barrow-Green
Imre Leader
Copyright Date: 2008
DOI: 10.2307/j.ctt7sd01
Pages: 1008
https://www.jstor.org/stable/j.ctt7sd01

## Table of Contents

1. Front Matter
(pp. i-iv)
DOI: 10.2307/j.ctt7sd01.1
2. Table of Contents
(pp. v-viii)
DOI: 10.2307/j.ctt7sd01.2
3. Preface
(pp. ix-xvi)
Timothy Gowers
DOI: 10.2307/j.ctt7sd01.3
4. Contributors
(pp. xvii-xxii)
DOI: 10.2307/j.ctt7sd01.4
5. Part I Introduction
(pp. 1-76)
DOI: 10.2307/j.ctt7sd01.5

It is notoriously hard to give a satisfactory answer to the question, “What is mathematics?” The approach of this book is not to try. Rather than giving adefinitionof mathematics, the intention is to give a good idea of what mathematics is by describing many of its most important concepts, theorems, and applications. Nevertheless, to make sense of all this information it is useful to be able to classify it somehow.

The most obvious way of classifying mathematics is by its subject matter, and that will be the approach of this brief introductory section and the longer section entitled...

6. Part II The Origins of Modern Mathematics
(pp. 77-156)
DOI: 10.2307/j.ctt7sd01.6

People have been writing numbers down for as long as they have been writing. In every civilization that has developed a way of recording information, we also find a way of recording numbers. Some scholars even argue that numbers came first.

It is fairly clear that numbers first arose as adjectives: they specified how many or how much of something there was. Thus, it was possible to talk about three apricots, say, long before it was possible to talk about the number 3. But once the concept of “threeness” is on the table, so that the same adjective specifies three...

7. Part III Mathematical Concepts
(pp. 157-314)
DOI: 10.2307/j.ctt7sd01.7

Consider the following problem: it is easy to find two irrational numbersaandbsuch thata + bis rational, or such thatabis rational (in both cases one could take$a = \sqrt 2$and$b = - \sqrt 2$), but is it possible for${a^b}$to be rational? Here is an elegant proof that the answer is yes. Let$x = \sqrt {{2^{\sqrt 2 }}}$. Ifxis rational then we have our example. But${x^{\sqrt 2 }} = \sqrt {{2^2}}$is rational, so ifxis irrational then again we have an example.

Now this argument certainly establishes that it is possible foraandbto be irrational and for ab...

8. Part IV Branches of Mathematics
(pp. 315-680)
DOI: 10.2307/j.ctt7sd01.8

The roots of our subject go back to ancient Greece while its branches touch almost all aspects of contemporary mathematics. In 1801 theDisquisitiones Arithmeticaeof CARL FRIEDRICH GAUSS [VI.26] was first published, a “founding treatise,” if ever there was one, for the modern attitude toward number theory. Many of the still unachieved aims of current research can be seen, at least in embryonic form, as arising from Gauss’s work.

This article is meant to serve as a companion to the reader who might be interested in learning, and thinking about, some of the classical theory of algebraic numbers. Much...

9. Part V Theorems and Problems
(pp. 681-732)
DOI: 10.2307/j.ctt7sd01.9

The ABC conjecture, proposed by Masser and Oesterlé in 1985, is a bold and very general conjecture in number theory with a wide range of important consequences. The rough idea of the conjecture is that it is impossible for one number to be the sum of two others if all three numbers have many repeated prime factors and no two have a prime factor in common (which would then have to be shared by the third).

More precisely, one defines theradicalof a positive integernto be the product of all primes that dividen, with each distinct...

10. Part VI Mathematicians
(pp. 733-826)
DOI: 10.2307/j.ctt7sd01.10

One of the most elusive figures of antiquity, Pythagoras is famous not just for his alleged mathematical achievements: it has been claimed that he had a golden thigh and that he issued a prescription against broad beans. Few things about him can be taken as historical facts, but we can be reasonably confident that he lived in around the sixth century B.C.E. in Greek southern Italy and that he established a group of followers, the Pythagoreans, who shared not just beliefs, but also dietary habits and a code of behavior. The existence of anecdotes about splinter Pythagoreans who revealed secrets...

11. Part VII The Influence of Mathematics
(pp. 827-954)
DOI: 10.2307/j.ctt7sd01.11

Since archimedes [VI.3], and his experimental investigation (described by Vitruvius) of the proportions of gold and silver in an alloy, the solution of chemical problems has employed mathematics. Carl Schorlemmer studied the paraffinic series of hydrocarbons (then important because of the discovery of oil in Pennsylvania) and showed how their properties changed with the addition of successive carbon atoms. His close friend in Manchester, Friedrich Engels, was inspired by this to introduce the transformation of “quantity into quality” into his philosophical outlook, which then became a mantra of dialectical materialism. From a similar chemical observation, CAYLEY [VI.46] in 1857 developed...

12. Part VIII Final Perspectives
(pp. 955-1014)
DOI: 10.2307/j.ctt7sd01.12

In English the word “problem” has negative connotations, suggesting some unwanted and unresolved tension. Zinoviev’s reminder is therefore important: problems are the stuff of life—and of mathematics. Good problems focus the mind: they challenge and frustrate; they cultivate ambition and humility; they show up the limitations of what we know, and highlight potential sources of more powerful ideas. By contrast, the word “solving” suggests areleaseof tension. The juxtaposition of these two words in the expression “problem solving” may encourage the naive to think that this unwelcome tension can be massaged away by means of some “magic formula”...

13. Index
(pp. 1015-1034)
DOI: 10.2307/j.ctt7sd01.13