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Timothy Gowers is the Rouse Ball Professor of Mathematics at the University of Cambridge. He received the Fields Medal in 1998, and is the author of "Mathematics: A Very Short Introduction". June Barrow-Green is lecturer in 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

De la solapa interior

"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 transformations, multiplying them together, and seeing what transformation corresponds to the product. In this way, the geometrical problem has been converted into algebra and solved algebraically.

Thus, while one can draw a useful distinction between algebra and geometry, one should not imagine that the boundary between the two is sharply defined. In fact, one of the major branches of mathematics is even called ALGEBRAIC GEOMETRY [IV.4]. And as the above examples illustrate, it is often possible to translate a piece of mathematics from algebra into geometry or vice versa. Nevertheless, there is a definite difference between algebraic and geometric methods of thinking-one more symbolic and one more pictorial-and this can have a profound influence on which subjects a mathematician chooses to pursue.

1.2 Algebra versus Analysis

The word "analysis," used to denote a branch of mathematics, is not one that features at high school level. However, the word "calculus" is much more familiar, and differentiation and integration are good examples of mathematics that would be classified as analysis rather than algebra or geometry. The reason for this is that they involve limiting processes. For example, the derivative of a function f at a point x is the limit of the gradients of a sequence of chords of the graph of f, and the area of a shape with a curved boundary is defined to be the limit of the areas of rectilinear regions that fill up more and more of the shape. (These concepts are discussed in much more detail in [I.3 5].)

Thus, as a first approximation, one might say that a branch of mathematics belongs to analysis if it involves limiting processes, whereas it belongs to algebra if you can get to the answer after just a finite sequence of steps. However, here again the first approximation is so crude as to be misleading, and for a similar reason: if one looks more closely one finds that it is not so much branches of mathematics that should be classified into analysis or algebra, but mathematical techniques.

Given that we cannot write out infinitely long proofs, how can we hope to prove anything about limiting processes? To answer this, let us look at the justification for the simple statement that the derivative of [x.sup.3] is 3[x.sup.2]. The usual reasoning is that the gradient of the chord of the line joining the two points (x, [x.sup.2]) and [((x + h), (x + h).sup.3]) is

[(x + h).sup.3] - [x.sup.3]/ x + h - x,

which works out as 3[x.sup.2] + 3xh + [h.sup.2]. As h "tends to zero," this gradient "tends to 3[x.sup.2]," so we say that the gradient at x is 3[x.sup.2]. But what if we wanted to be a bit more careful? For instance, if x is very large, are we really justified in ignoring the term 3xh?

To reassure ourselves on this point, we do a small calculation to show that, whatever x is, the error 3xh+[h.sup.2] can be made arbitrarily small, provided only that h is sufficiently small. Here is one way of going about it. Suppose we fix a small positive number [element of], which represents the error we are prepared to tolerate. Then if |h| [is less than or equal to] [element of]/6x, we know that |3xh| is at most [element of]/2. If in addition we know that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then we also know that [h.sup.2] [is less than or equal to] [element of]/2. So, provided that |h| is smaller than the minimum of the two numbers [element of]/6x and [square root of [element of]/2], the difference between 3[x.sup.2] + 3xh + [h.sup.2] and 3[x.sup.2] will be at most [element of].

There are two features of the above argument that are typical of analysis. First, although the statement we wished to prove was about a limiting process, and was therefore "infinitary," the actual work that we needed to do to prove it was entirely finite. Second, the nature of that work was to find sufficient conditions for a certain fairly simple inequality (the inequality |3xh + [h.sup.2]| [is less than or equal to] [element of]) to be true.

Let us illustrate this second feature with another example: a proof that [x.sup.4] - [x.sup.2] - 6x + 10 is positive for every real number x. Here is an "analyst's argument." Note first that if x [is less than or equal to] -1 then [x.sup.4] [is greater than or equal to] [x.sup.2] and 10-6x [is greater than or equal to] 0, so the result is certainly true in this case. If -1 [is less than or equal to] x [is less than or equal to] 1, then |[x.sup.4] - [x.sup.2] -6x| cannot be greater than [x.sup.4] + [x.sup.2] +6|x|, which is at most 8, so [x.sup.4] - [x.sup.2] - 6x [is greater than or equal to] -8, which implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Also, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Finally, if x [is greater than or equal to] 2, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], from which it follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The above argument is somewhat long, but each step consists in proving a rather simple inequality-this is the sense in which the proof is typical of analysis. Here, for contrast, is an "algebraist's proof." One simply points out that [x.sup.4] - [x.sup.2] - 6x + 10 is equal to [([x.sup.4] - 1).sup.2] + [(x - 3).sup.2], and is therefore always positive.

This may make it seem as though, given the choice between analysis and algebra, one should go for algebra. After all, the algebraic proof was much shorter, and makes it obvious that the function is always positive. However, although there were several steps to the analyst's proof, they were all easy, and the brevity of the algebraic proof is misleading since no clue has been given about how the equivalent expression for [x.sup.4] - [x.sup.2] - 6x + 10 was found. And in fact, the general question of when a polynomial can be written as a sum of squares of other polynomials turns out to be an interesting and difficult one (particularly when the polynomials have more than one variable).

There is also a third, hybrid approach to the problem, which is to use calculus to find the points where [x.sup.4] - [x.sup.2] - 6x + 10 is minimized. The idea would be to calculate the derivative 4[x.sup.3]-2x-6 (an algebraic process, justified by an analytic argument), find its roots (algebra), and check that the values of [x.sup.4] - [x.sup.2] - 6x + 10 at the roots of the derivative are positive. However, though the method is a good one for many problems, in this case it is tricky because the cubic 4[x.sup.3] - [x.sup.2] - 6 does not have integer roots. But one could use an analytic argument to find small intervals inside which the minimum must occur, and that would then reduce the number of cases that had to be considered in the first, purely analytic, argument.

As this example suggests, although analysis often involves limiting processes and algebra usually does not, a more significant distinction is that algebraists like to work with exact formulas and analysts use estimates. Or, to put it even more succinctly, algebraists like equalities and analysts like inequalities.

2 The Main Branches of Mathematics

Now that we have discussed the differences between algebraic, geometrical, and analytical thinking, we are ready for a crude classification of the subject matter of mathematics. We face a potential confusion, because the words "algebra," "geometry," and "analysis" refer both to specific branches of mathematics and to ways of thinking that cut across many different branches. Thus, it makes sense to say (and it is true) that some branches of analysis are more algebraic (or geometrical) than others; similarly, there is no paradox in the fact that algebraic topology is almost entirely algebraic and geometrical in character, even though the objects it studies, topological spaces, are part of analysis. In this section, we shall think primarily in terms of subject matter, but it is important to keep in mind the distinctions of the previous section and be aware that they are in some ways more fundamental. Our descriptions will be very brief: further reading about the main branches of mathematics can be found in parts II and IV, and more specific points are discussed in parts III and V.

2.1 Algebra

The word "algebra," when it denotes a branch of mathematics, means something more specific than manipulation of symbols and a preference for equalities over inequalities. Algebraists are concerned with number systems, polynomials, and more abstract structures such as groups, fields, vector spaces, and rings (discussed in some detail in SOME FUNDAMENTAL MATHEMATICAL DEFINITIONS [I.3]). Historically, the abstract structures emerged as generalizations from concrete instances. For instance, there are important analogies between the set of all integers and the set of all polynomials with rational (for example) coefficients, which are brought out by the fact that both sets are examples of algebraic structures known as Euclidean domains. If one has a good understanding of Euclidean domains, one can apply this understanding to integers and polynomials.

(Continues...)

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