Ian Stewart is a mathematical jack-of-all-trades. He has done significant research work in both pure and applied mathematics. For years he wrote a Mathematical Recreations column for Scientific American. He has written mathematics books for the general public, mathematics textbooks, and even science fiction. Early in his career, he edited Manifold, certainly the most entertaining mathematical publication ever. So he was a natural choice to write Letters to a Young Mathematician.
This book is part of a series published by Basic Books. All of the books in the series are called "Letters to a Young X", with X having stood, so far, for such things as "contrarian", "conservative", "actor", "lawyer", "Catholic", "therapist", "activist", and even "golfer". Mathematicians are the first category of academic professionals to be featured.
Stewart's book, as the series dictates, follows the traditional structure of a series of letters of advice. With the notable exception of something like C. S. Lewis's The Screwtape Letters, this is not really a recipe for great literature. What we get from Stewart doesn't really rise to the heights, but it is earnest, helpful, and interesting. And perhaps that's enough.
Stewart names his young mathematician "Meg," and then follows her career from the end of high school all the way to her being awarded tenure, trying to answer the natural questions that arise at each stage. Thus, when she is an undergraduate she gets letters about the nature of proof and the reasons for the fundamental role of proofs in mathematics. When she has just been made an assistant professor, the letter is about teaching and about the mathematical community.
The book is full of good advice, much of it direct and to the point. Given the inevitable human tendency towards going one's own way, it's hard to assess whether such advice will do any good. It does, however, paint a useful picture of what it is like to be a mathematician. The picture is slightly skewed towards the research side, perhaps. I think Stewart can't quite imagine a mathematical career that focuses entirely on teaching.
The book starts with an invocation of G. H. Hardy and his Apology, which Stewarts sets in historical context. This is, I think, a good thing. Hardy's book has been read by many non-mathematicians, and several of his stances have become indelibly etched into the cultural consensus about mathematics and mathematicians. For example, Hardy's contention that "mathematics is a young man's game" continues to be believed by far too many. Stewart points out that the world has changed, and also calls into question several of Hardy's confident assertions.
Overall, Stewart has written a useful and friendly introduction to life as a mathematician, one that many of our students (if we can get them to read it) will find helpful and motivating. Professors should read it too. After all, if your students won't read it themselves, at least you can select choice bits to quote to them.
That, I think, is what this book is for. Parents will buy it for their children. Professors will recommend it to their students. And, while it won't change the world, it may well help some young people decide to be (or not to be) mathematicians. If so, it will have achieved something worthwhile.
Fernando Q. Gouvêa is professor of mathematics at Colby College.