Mathematics: A Very Short Introduction

Oxford University Press has been publishing, since 1995, a series of Very Short Introductions to various subjects. The idea is to ask an expert to introduce readers to the essential ideas of a subject. The books are small (roughly 4 x 7 inches) and short. The web site for the series says that the books are "perfect for train journeys, holidays, and as a quick catch-up for busy people who want something intellectually stimulating." The series includes some very general volumes (such as this one) and some that are much more specific (there are volumes on Egyptian Mythology and The Cold War, for example).

There is something very pleasant about holding in one's hand a small book that can serve as a serious introduction to a serious topic. At the same time, however, one must be aware of some risks. Asking a scholar to write a broad survey of his or her field is an invitation to ponder what is crucial and what is ancillary, what must be said and what must be left out. As a result, a "very short introduction" is likely to be more influenced by the author's particular "take" on the subject than a longer book would be. When it comes to a very short introduction to Atheism or to Postcolonialism, readers are likely to come to the book prepared to deal with such biases. They may not expect to find similar biases in a book on Mathematics. But they are there nevertheless.

Timothy Gowers is an eminent mathematician, a Fields medalist and currently the Rouse Ball Professor of Mathematics at the University of Cambridge. His work ranges from combinatorial number theory to Banach spaces, and he has also written several papers explaining his views on mathematics. He is absolutely the right sort of person to write a very short introduction to mathematics: as a top-notch mathematician, he can write with a deep understanding of what mathematical research is like; as someone who has thought seriously about the nature and meaning of mathematics, he can offer us a coherent view of the field. In other words, while Gowers' introduction does give a personal point of view on the nature of mathematics, it is a carefully considered and credible take on the subject.

In the introduction, Gowers explains that while he does not assume the reader has much more than a high school education in mathematics, he does assume that the reader is interested. "For this reason," he says, "I have done without anecdotes, cartoons, exclamation marks, jokey chapter titles, or pictures of the Mandelbrot set." For all of which this reader, at least, is grateful.

What Gowers chooses to focus on is abstraction. He repeatedly brings up questions like "but do complex numbers exist?", and repeatedly dismisses them with "it does not matter." His slogan is "a mathematical object is what it does." In other words, Gowers presents mathematics as essentially concerned with investigating the formal consequences of explicitly-stated axioms. Questions about what mathematical objects actually are and whether they exist can (and should) be left to the philosophers.

This stance allows Gowers to lead the reader quite a way into certain mathematical structures. For example, to explain higher-dimensional geometry he first shows how to reduce two- and three-dimensional geometry to algebra, and then uses the algebraic formulation to define what is meant by "10-dimensional space." This is indeed how mathematicians do it, and it is irrelevant to ask whether any such thing as "10-dimensional space" actually exists.

The chapter titles give a good outline of the topics discussed in the book:

1. Models
2. Numbers and Abstraction
3. Proofs
4. Limits and Infinity
5. Dimension
6. Geometry
7. Estimates and Approximations
8. Some Frequently Asked Questions

The third chapter, on proofs, discusses why mathematicians bother to prove things. The section on "Three obvious-seeming statements that need proofs" is particularly nice, and addresses something that often puzzles non-mathematicians. The seventh chapter is also welcome, since it highlights the importance of approximations in mathematics. Other chapters touch on more standard topics in books of this sort (the hyperbolic plane, for example). There is no question that non-mathematicians will learn a lot from reading this book.

On the other hand, Gowers' approach sometimes leads him to strange places. For example, when introducing the real numbers, he says that "the reason for extending our number system from rational to real numbers is similar to the reason for introducing negative numbers and fractions: they allow us to solve equations that we could not otherwise solve" (p. 29). Even if we lay aside doubts that fractions and negative numbers were really introduced "to solve equations" (this reviewer has serious doubts about that), this is a remarkable statement, since it seems to say that the only real numbers we really need are the algebraic ones! Maybe Gowers is thinking of some very general sense of "equations" here. Still, it seems strange to introduce the real numbers without once mentioning the idea of a line.

Another example occurs in a discussion of mathematical pedagogy. He suggests that at times it is better to simply tell students "that's the rule" than to try to explain the reason why certain things are the way they are. (His example, on page 133, is explaining why x^a+b is not equal to x^a + x^b.) I suspect, however, that there is far too much "that's the rule" teaching, and far too little explaining of reasons in elementary mathematics teaching. Such a focus on rules can easily lead to students having to remember a huge list of unrelated rules. I fear Gowers' suggestion here may in fact be counterproductive.

Gowers' emphasis on abstract structures leads him, I think, to shortchange two other important aspects of mathematical practice: examples and metaphors. The book is, of course, full of examples of the abstract structures he mentions, but Gowers does not always highlight how important examples are in creating mathematical intuition and posing mathematical questions.

Metaphors are also crucial. Even if we regard the real numbers as an abstract structure (say, as a complete ordered field), the metaphor of the real numbers as points on a line is bound to be a crucial part of our understanding. Such metaphors become more and more important as one moves to more advanced topics, because even the best-trained mathematicians cannot constantly keep in mind all the elaborate definitions that go into defining something like a coherent sheaf on a projective algebraic variety. Alongside the formal definitions must be (at least for most of us) a mental image, a metaphor for what such an object is.

As a mathematician who has a serious interest in history I had cause to cringe here and there. It is not that Gowers doesn't know his history; I suspect he does. Rather, he chooses to simplify things at times without really warning the reader that he is doing so. For example, he states Euclid's five axioms for geometry in a modernized form that is not quite equivalent to what Euclid actually said (or at least I think so). He also perpetuates the myth of Gauss's having measured a triangle in order to determine whether space is Euclidean. (He does acknowledge that there is some doubt that this is what Gauss was doing; I would claim that there is more than "some" doubt.)

Despite these reservations, I still feel that this is a remarkable and useful book. I strongly recommend it as supplementary reading for students who are beginning to move into more serious mathematics. For non-specialist readers, it offers a slightly different angle into what mathematics is all about than can be found in most popular mathematics books, and that's all to the good. Finally, as one might expect, there are all sorts of very interesting mathematical ideas and examples sprinkled throughout. (Do you know how to add two infinite decimals? Gowers tells us how.)

The book's final chapter is a collection of "frequently asked questions" about mathematics and mathematicians. These are handled quite well, for the most part, but the treatment here is far less deep than in the rest of the book.

Overall, this is an excellent introduction to mathematics. I would like to complement it with something that emphasized mathematical intuition a bit more, but there is no doubt that anyone who thoroughly digested Gowers' book would have an excellent grasp on what mathematicians actually do. So I would urge you to get it, read it, give it to your students.

Fernando Q. Gouvêa (fqgouvea@colby.edu) is professor of mathematics at Colby College in Waterville, ME.