The hunt for memorable proof

No one drifts into being a mathematician. On the contrary, it is a pursuit from which even the talented are too easily turned away." This quote from Ian Stewart's new book strikes me as exactly why something like Letters to a Young Mathematician is needed. I believe that many young (and not-so-young) people will enjoy this highly personal but up-to-date account of the merits of a mathematical life. They will almost certainly get encouragement - and possibly inspiration - from it. For while a number of recent "popular" books give, at last, some idea of what mathematics is really all about, finding out what it is like to be a mathematician is quite another matter.

The book is written as a series of letters to a fictional recipient, "Meg", who starts by studying mathematics at university, then proceeds to a doctorate and eventually ends up as a fully fledged academic mathematician. Although I cannot say that I ever really believed in Meg as a real character, the letter-writing works well as a purely rhetorical deviceand lends a certain crispness to the whole book.

The emphasis at first is on the general nature of mathematics, and some of Stewart's observations about the way in which the subject is full of surprises certainly struck a chord with me. He tackles head on, too, why mathematicians are obsessed by proof, and I particularly like one of his more striking observations on this topic: "If a proof is a story, then a memorable proof must tell a ripping yarn."

It is only when Stewart seems to suggest that the place for proof is university, not school, that I find myself in disagreement - if I am not simply taking him too literally. I should have thought that the sooner a school pupil sees a truly "ripping yarn" in a proof of, say, Pythagoras'

theorem, the better.

As Meg's career progresses, we learn what life as a mathematician is like and some of the more down-to-earth and practical decisions that have to be made. Stewart urges, in particular, broad-mindedness on the relative merits of "pure" and "applied" mathematics ("Keep your mind open, but not so open that your brains fall out."). This is in stark contrast to the views expressed in G. H. Hardy's famous essay, "A mathematician's apology", but good advice nonetheless. I would like to see a still greater generosity of spirit between different kinds of mathematician, along with more recognition that big mathematical discoveries are occasionally made by scientists who do not really think of themselves as mathematicians at all.

Stewart lets rip in a particularly amusing fashion when it comes to some of the characters we must expect to meet if we pursue mathematics beyond first-degree level. The most colourful of these figures are the GOP (grand old person) and the MG (maverick guru), although Stewart advises steering well clear of both when it comes to choosing a thesis supervisor.

Later still in Meg's career, Stewart offers good advice about the whole matter of teaching and research (which can otherwise degenerate destructively into teaching versus research). Although he spells out how strongly our current university system can sometimes be geared towards research, he claims unashamedly that "a good teacher is worth his or her weight in gold". All in all, Stewart is on cracking form, and this is, in some ways, a brave book. (Perhaps we expect no less from someone who once gave a television lecture standing next to a live tiger.) Finally, I should not give the impression that the book is intended only for prospective professional mathematicians. There is much here that will be of interest to those who never got on with mathematics at school and who have never really been able to get inside the mathematical "mindset". And my own view, for what it is worth, is that the public understanding of mathematics is one of the great educational problems of our time.

I recognise, of course, that the solution, if it exists, will not be easy.

We may have a very long way to go indeed, and this is rather indicated, I am sorry to say, by one of Stewart's own anecdotes: a mathematician at a famous university went to look around the new auditorium, and when she got there she found the dean staring at the ceiling and muttering to himself, "45, 46, 47". Naturally she interrupted the count to find out what it was for. "I'm counting the lights," said the dean.

The mathematician looked up at the perfectly rectangular array of lights and said: "That's easy, there are 12 that way, and eight that way. Twelve eights are 96." "No, no," said the dean impatiently. "I want the exact number."

David Acheson is a fellow in mathematics, Jesus College, Oxford.