March 8, 2006

The New Math

By Scott McLemee

As a kid, my favorite book in the world was E.T. Bell’s Men of Mathematics (1937). I must have read it dozens of times by the age of 14. One afternoon, coming home from the library, I could not resist opening the book to a particularly interesting chapter — and so ended up walking into a parked bus.

With hindsight, certain problems with the book are clear. Bell’s approach to the history of mathematics was exciting, but he achieved that effect, in part, through fictionalization. We now know that embroidering the truth came as second nature to Bell, who was a professor of mathematics at the California Institute of Technology until shortly before his death in 1960. In addition to writing science fiction under a pseudonym, Bell also exercised a certain amount of creativity in telling his own life story – as his biographer, Constance Reid, found out through some detective work.

But another problem with Men of Mathematics only dawned on me recently. I hadn’t thought of the book in ages, but remembered it while reading while reading Letters to a Young Mathematician by Ian Stewart, to be published next month by Basic Books.

The author is a professor of mathematics at the University of Warwick in the U.K. The imaginary recipient of his letters is named Meg — a nice departure from the longstanding and rather self-reinforcing stereotype of math as a man’s field. The idea that no gender has a monopoly on mathematical talent seems never to have occurred to E.T. Bell. (Nor, consequently, did it cross the mind of a certain young nerd colliding with stalled vehicles during the mid-1970s.)

Fortunately that situation has started to change. And the progress is reflected, in a quiet and matter-of-fact way, in Stewart’s Letters.

A story unfolds, chapter by chapter, as Stewart corresponds with Meg. In the earliest letters, she is still in high school. By the end of the book, she has tenure. It is, in effect, a bildungsroman at one remove. The reader watches over Stewart’s shoulder as the young mathematician turns an early talent into a stable professional identity.

There’s even a moment when, in search of an interesting project to test her abilities, Meg starts trying to find a method for trisecting an angle using only a compass and an unmarked straightedge. This is one of the problems handed down from ancient geometry. People “discover” solutions to this challenge all the time, then become indignant that mathematicians don’t take them seriously. (The proof of why it is impossible involves mathematical tools going way beyond anything available in antiquity.)

But most of the guidance Stewart offers is positive — and some of it seems useful even for those of us without mathematical aspirations or gifts.

“My usual method for reading a mathematics text,” he recalls about his student days, “was to thumb through it until I spotted something interesting, then work backward until I had tracked down everything I needed to read the interesting bit. I don’t really recommend this to everyone, but it does show that there are alternatives to starting at page 1 and continuing in sequence until you reach page 250.”

The most surprising thing — at least for anyone influenced by Bell’s romanticized account of the mathematical vocation — is Stewart’s emphasis on the nuts and bolts of academic life. Letters is full of pointers on academic politics, the benefits and frustrations of collaboration, and how to avoid disaster at conferences. (“Never believe your hosts when they tell you that all the equipment will work perfectly,” he notes. “Always try it yourself before the lecture.”)

E. T. Bell told stories about mathematicians whose lives were shaped, in the final analysis, only by their own creative instincts. They might occasionally win a prize offered by a learned society, or feel driven to some breakthrough by the challenge of defeating a hated rival. But Bell’s men of mathematics were, on the whole, geniuses of the purest vintage. They had inspirations, not resumes. It is hard to imagine anyone trying to give Carl Friedrich Gauss useful career advice.

So does that mean that popularized accounts like Bell’s are something a young mathematician ought to avoid? I contacted Stewart by e-mail to ask his thoughts on the matter.

“I write a lot of books popularising math and science, so I may be biased,” he said in reply, “but when I was in high school I read all the books I could find about the history of math, about mathematicians, and about various topics in math. And those definitely had a significant effect on my interest in the subject. They made it clear that math has a long and fascinating history, that the great mathematicians were real people, not just obsessed geniuses who couldn’t tie their own shoelaces, and that there is much, much more to math than the tiny part of the subject that we are all taught at school.”

Well, that’s a relief. There’s something to be said for idealization and hero worship, after all, in their proper season. You then have your whole life to become more realistic, not to say more calculating.

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