American Scientist

Letters to a Young Mathematician. Ian Stewart. xii + 210 pp. Perseus Books, 2006. $22.95.

Many recent popular books on mathematics paint a beautiful portrait of the discipline as a whole (Mathematics: A Very Short Introduction, by Fields Medalist William Timothy Gowers) or of a particular problem (recent books on the Riemann hypothesis) or even of a particular number (Gamma: Exploring Euler's Constant, by Julian Havil). Ian Stewart's latest book, Letters to a Young Mathematician, is quite unusual in that it focuses on what mathematicians do and why that is worth doing, and in general deals in a very practical way with the question of what it means to be a mathematician.

The recent popular portrayal that is perhaps closest in spirit to Stewart's book is the television show Numb3rs, with its entertaining depictions of mathematics being used to solve crimes. But the program's hero, Charlie Eppes, a math professor at a university that resembles Caltech, seems to be able to devote most of his time to helping the FBI, without having to worry about committee work or teaching undergraduates or even research.

G. H. Hardy's A Mathematician's Apology (1940) was a brilliant attempt at answering the sorts of questions Stewart addresses here, although some of Hardy's views have become outmoded. Indeed, Stewart says in his preface that Letters to a Young Mathematician is meant to be an updating of Hardy's book. Although his prose lacks some of Hardy's eloquence, Stewart gives a fuller picture of what it means to be a mathematician today.

Stewart's book is written as a series of 21 letters to a young woman named Meg as she progresses through high school, university, graduate school, a postdoctoral position and a tenure-track job, finally obtaining tenure. Each letter is a short essay addressing a single topic ("Fear of Proofs," "Why Do Math?" and so forth). The style is pleasant and easygoing, and the author's gentle humor shines through—you may not laugh out loud, but there are plenty of reasons to smile. Without ever seeming heavy-handed or condescending, Stewart manages to give much good advice on a variety of subjects, including how to learn math and how to teach it, what to look for in a thesis adviser, how to be effective on committees, how to manage career issues and the need to balance work on big problems with work on more manageable ones.

Although there are no equations in this book, it does contain much interesting mathematics. Stewart has a wonderful knack for communicating the spirit of a mathematical idea by means of apposite examples. To illustrate the idea of proof, he uses the "SHIP-DOCK" theorem, demonstrating that if one wishes to change the word SHIP to the word DOCK by interposing a series of other words, each of which differs from the preceding word by only one letter (as in the sequence SHIP, SHOP, CHOP, COOP, COOK, COCK, DOCK), the series must include at least one word with two vowels in it. He has a very nice explanation (again without equations) of why angles cannot be trisected using straight-edge and compass constructions. The book also offers overviews of Andrew Wiles's work on Fermat's last theorem and Grigori Perelman's proof of the Poincaré conjecture (recently adjudged by Science to be the breakthrough of the year for 2006).

I was fascinated by "the sausage conjecture," of which I had been unaware. If you plan to wrap a number of tennis balls in plastic film and want to know which arrangement would consume the least amount of film, the curious answer is that for 56 balls or fewer, the best arrangement is all in a line, like a sausage, but for more than 56 balls the best arrangement is like that of potatoes in a sack. For tennis balls in five or more dimensions, "a sausage" is supposedly always the answer; this has been proved in at least 42 dimensions, but the question is still open!

On the whole, I found Letters to a Young Mathematician to be engaging and a fun read. It is a worthy updating of Hardy's classic but is quite different from it. As much as I love A Mathematician's Apology, I now find it a little sad. Hardy begins by saying that "it is a melancholy experience" to find himself writing about mathematics instead of doing it, and some of this melancholia manifests itself in the book. Stewart is much more positive and lively, and is convinced of the value not only of doing mathematics, but also of encouraging others to pursue it.

Stewart and Hardy also differ notably in their attitudes toward "pure and applied mathematics." Hardy's belief that only dull mathematics is useful is no longer viable, and Stewart gives many nice examples where the distinction between pure and applied mathematics is blurred. His answer to the question of which of these Meg should choose is, in effect, "Both!"

I knew from the time I was a high school student that I wanted to be a mathematician. My schoolteachers had put me in contact with R. Balasubramanian, a number theorist in Madras, and I learned from him what a career in mathematics would be like. I remember asking him how he decided to be a mathematician, and the short answer was "purely by chance." He was good at math but had had no concept that one could make a living at it. Somehow he ended up in graduate school and became a mathematician. As for Hardy, being a mathematician meant for him "a Fellowship at Trinity College, Cambridge," along with "port and walnuts in the Senior Combination Room."

Stewart's book may be heartily recommended to any young person considering a future in the field, and indeed to anyone with more than a passing interest in mathematics or mathematicians.