If you have ever found yourself next to a mathematician at a dinner party, you have probably, out of politeness, or perhaps desperation, asked what he or she works on. If you do not have a mathematics degree, you will almost certainly have received a disappointing answer such as, "I work in Iwasawa theory, but it would take too long to explain to you what that is."
If you are genuinely curious to know what it is like to be a mathematician, there is now a better way of satisfying your curiosity: read Finding Moonshine by Marcus du Sautoy.
Du Sautoy is well known for his performances on radio and television, and for his extremely successful book, The Music of the Primes, which about half the candidates I interviewed in December for admission to Cambridge claimed to have read.
The attitude of many professional mathematicians to the earlier book was ambivalent. Although they were pleased that du Sautoy was promoting mathematics, they were not always convinced by the way that he did it.
I myself expected to have a similar attitude to Finding Moonshine, but du Sautoy surprised me: he has pulled off that rare feat of writing in a way that can entertain and inform two different audiences - expert and non-expert - at the same time.
Part of the reason for this is his subject matter.
The main goal of The Music of the Primes was to describe the Riemann hypothesis, the most famous unsolved problem in mathematics. Unfortunately, a proper statement of that problem needs some fairly advanced university-level mathematics, so du Sautoy tried to convey it by means of extended analogies (one suggested by the title of the book). These did not really explain the essence of the problem, but it may have appeared to some readers that they did. In this respect, Finding Moonshine is a more truthful book. The mathematics du Sautoy describes is from his own area, group theory, and he gives elementary but accurate explanations of the basic concepts, groups and symmetry, that he is concerned with.
These mathematical explanations are an important part of the book, but there is much more besides. One of the great strengths of Finding Moonshine is its organisation, which mixes the mathematics with two other strands.
One is a glimpse - in fact, rather more than just a glimpse - of what it is like to be du Sautoy himself. He tells us about the problem that has obsessed him for years, about why it is interesting, about other people also trying to solve it, about the emotions that accompany progress, or lack of it, or progress made by other people, about the rest of his life, and about the different personal and mathematical styles of the colleagues he meets at conferences. Du Sautoy's mathematical problem is nothing like as famous as the Riemann hypothesis, and for this reason is much more representative of the problems that mathematicians typically work on: another way in which the book gives a truer picture of mathematics.
The other strand is a history of group theory, from its origins in the work of Henrik Abel and Evariste Galois in the early 19th century to the classification of finite simple groups, completed in the 1980s.
Abel made the astonishing discovery that there was no formula for the solution of a general quintic equation (that is, no analogue of the familiar formula for the solution of a quadratic), and died at the age of 26. Not to be outdone, Galois then worked out a systematic way of determining which polynomials could be solved and which could not, founding group theory in the process, and died at 20. Both had huge difficulty getting their work noticed: indeed, Galois's work was not properly appreciated until well after his death.
The classification of finite simple groups is group theory's crowning achievement: a complete description of the basic building blocks that can be used, in various combinations, to produce all groups. It is not obvious that it should be a hard problem to determine these, but the building blocks turned out to include a collection of 26 highly exotic "sporadic" groups, and the proof that there was not a th waiting to be discovered was a combined effort by many authors that ran to thousands of pages.
What makes the organisation of the book so successful is that it allows du Sautoy to develop one part of his story and then leave you in suspense while he moves on to another. This is of course a well-known literary device, and du Sautoy is very good at using it: he knows just how much information to reveal to keep you interested. The result is a gripping book with a strongly novelistic flavour.
There are a few imperfections, as with any book. Du Sautoy does rather splash his metaphors about; some of them hit the spot, but others are tired, or overworked, or, like the ones in this sentence, mixed. Slightly more serious is that he applies the theme of symmetry too widely at times. Some topics, such as the patterns on the walls of the Alhambra, are genuinely illuminated by the mathematics of symmetry. Others, such as the "mirror neurons" that allow a baby to imitate its parents, are not, although they are interesting in themselves. At one point du Sautoy confesses to not particularly liking Bach's Goldberg variations. He is entitled to his opinion, but his suggested reason for it - that they are too symmetrical - is unconvincing. One of the great joys of Bach is that the patterns he puts into his music do not have a distorting effect: they just add another dimension that one is free to ignore. But these are tiny blemishes, if they are even that, on a book that is otherwise pure pleasure.
The author Timothy Gowers is Rouse Ball professor of mathematics at the University of Cambridge
Timothy Gowers is Rouse Ball professor of mathematics at the University of Cambridge
"Mathematics has beauty and romance," says Marcus du Sautoy, professor of mathematics at the University of Oxford. "It's not a boring place to be, the mathematical world. It's an extraordinary place; it's worth spending time there."
In 2006, he became only the third mathematician to present the Royal Institution's Christmas Lectures, some 28 years after being in the audience as a 13-year-old for the first lecture by a mathematician, Sir Christopher Zeeman.
"I became a mathematician because the generation above me made the effort to excite the general public about mathematics," du Sautoy has said.
His passion is not limited to mathematics. He is a keen amateur footballer, turning out for a Sunday-league team, Recreativo FC, where, at away matches, all team members wear shirts bearing prime numbers.
Finding Moonshine: A Mathematician's Journey through Symmetry
By Marcus du Sautoy
Published 4 February 2008