Über die Autorin bzw. den Autor

Timothy Gowers is a Fields Medal–winning British mathematician. A professor at the Collège de France, he also teaches at the University of Cambridge and is the author of Mathematics: A Very Short Introduction. June Barrow-Green is professor of the history of mathematics at the Open University. Imre Leader is professor of pure mathematics at the University of Cambridge.

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"This is a wonderful book. The content is overwhelming. Every practicing mathematician, everyone who uses mathematics, and everyone who is interested in mathematics must have a copy of their own."--Simon A. Levin, Princeton University

"The Princeton Companion to Mathematics fills a vital need. It is the only book of its kind."--Victor J. Katz, professor emeritus, University of the District of Columbia

"I think that this is a wonderful book, completely different from anything that has been written before about mathematics and mathematicians."--Endre Süli, University of Oxford

"The Princeton Companion to Mathematics is a much needed--and will become a much used--reference work. In fact, it will stand alone as the reference work in mathematics."--John J. Watkins, Colorado College

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"This is a wonderful book. The content is overwhelming. Every practicing mathematician, everyone who uses mathematics, and everyone who is interested in mathematics must have a copy of their own."--Simon A. Levin, Princeton University

"The Princeton Companion to Mathematics fills a vital need. It is the only book of its kind."--Victor J. Katz, professor emeritus, University of the District of Columbia

"I think that this is a wonderful book, completely different from anything that has been written before about mathematics and mathematicians."--Endre Süli, University of Oxford

"The Princeton Companion to Mathematics is a much needed--and will become a much used--reference work. In fact, it will stand alone as the reference work in mathematics."--John J. Watkins, Colorado College

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The Princeton Companion to Mathematics

Princeton University Press

Contents

Preface.......................................................ixContributors..................................................xviiIndex.........................................................1015

Chapter One

I.1 What Is Mathematics About?

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 a definition of 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 SOME FUNDAMENTAL MATHEMATICAL DEFINITIONS [I.3]. However, it is not the only way, and not even obviously the best way. Another approach is to try to classify the kinds of questions that mathematicians like to think about. This gives a usefully different view of the subject: it often happens that two areas of mathematics that appear very different if you pay attention to their subject matter are much more similar if you look at the kinds of questions that are being asked. The last section of part I, entitled the general goals of mathematical research [I.4], looks at the subject from this point of view. At the end of that article there is a brief discussion of what one might regard as a third classification, not so much of mathematics itself but of the content of a typical article in a mathematics journal. As well as theorems and proofs, such an article will contain definitions, examples, lemmas, formulas, conjectures, and so on. The point of that discussion will be to say what these words mean and why the different kinds of mathematical output are important.

1 Algebra, Geometry, and Analysis

Although any classification of the subject matter of mathematics must immediately be hedged around with qualifications, there is a crude division that undoubtedly works well as a first approximation, namely the division of mathematics into algebra, geometry, and analysis. So let us begin with this, and then qualify it later.

1.1 Algebra versus Geometry

Most people who have done some high school mathematics will think of algebra as the sort of mathematics that results when you substitute letters for numbers. Algebra will often be contrasted with arithmetic, which is a more direct study of the numbers themselves. So, for example, the question, "What is 3 x 7?" will be thought of as belonging to arithmetic, while the question, "If x + y = 10 and xy = 21, then what is the value of the larger of x and y?" will be regarded as a piece of algebra. This contrast is less apparent in more advanced mathematics for the simple reason that it is very rare for numbers to appear without letters to keep them company.

There is, however, a different contrast, between algebra and geometry, which is much more important at an advanced level. The high school conception of geometry is that it is the study of shapes such as circles, triangles, cubes, and spheres together with concepts such as rotations, reflections, symmetries, and so on. Thus, the objects of geometry, and the processes that they undergo, have a much more visual character than the equations of algebra.

This contrast persists right up to the frontiers of modern mathematical research. Some parts of mathematics involve manipulating symbols according to certain rules: for example, a true equation remains true if you "do the same to both sides." These parts would typically be thought of as algebraic, whereas other parts are concerned with concepts that can be visualized, and these are typically thought of as geometrical.

However, a distinction like this is never simple. If you look at a typical research paper in geometry, will it be full of pictures? Almost certainly not. In fact, the methods used to solve geometrical problems very often involve a great deal of symbolic manipulation, although good powers of visualization may be needed to find and use these methods and pictures will typically underlie what is going on. As for algebra, is it "mere" symbolic manipulation? Not at all: very often one solves an algebraic problem by finding a way to visualize it.

As an example of visualizing an algebraic problem, consider how one might justify the rule that if a and b are positive integers then ab = ba. It is possible to approach the problem as a pure piece of algebra (perhaps proving it by induction), but the easiest way to convince yourself that it is true is to imagine a rectangular array that consists of a rows with b objects in each row. The total number of objects can be thought of as a lots of b, if you count it row by row, or as b lots of a, if you count it column by column. Therefore, ab = ba. Similar justifications can be given for other basic rules such as a(b + c) = ab + ac and a(bc) = (ab)c.

In the other direction, it turns out that a good way of solving many geometrical problems is to "convert them into algebra." The most famous way of doing this is to use Cartesian coordinates. For example, suppose that you want to know what happens if you reflect a circle about a line L through its center, then rotate it through 40 counterclockwise, and then reflect it once more about the same line L. One approach is to visualize the situation as follows.

Imagine that the circle is made of a thin piece of wood. Then instead of reflecting it about the line you can rotate it through 180 about L (using the third dimension). The result will be upside down, but this does not matter if you simply ignore the thickness of the wood. Now if you look up at the circle from below while it is rotated counterclockwise through 40, what you will see is a circle being rotated clockwise through 40. Therefore, if you then turn it back the right way up, by rotating about L once again, the total effect will have been a clockwise rotation through 40

Mathematicians vary widely in their ability and willingness to follow an argument like that one. If you cannot quite visualize it well enough to see that it is definitely correct, then you may prefer an algebraic approach, using the theory of linear algebra and matrices (which will be discussed in more detail in [I.3 3.2]). To begin with, one thinks of the circle as the set of all pairs of numbers (x,y) such that [x.sup.2] + [y.sup.2] [is less than or equal to] 1. The two transformations, reflection in a line through the center of the circle and rotation through an angle [theta], can both be represented by 2 x 2 matrices, which are arrays of numbers of the form ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). There is a slightly complicated, but purely algebraic, rule for multiplying matrices together, and it is designed to have the property that if matrix A represents a transformation R (such as a reflection) and matrix B represents a transformation T, then the product AB represents the transformation that results when you first do T and then R. Therefore, one can solve the problem above by writing down the matrices that correspond to the...