*The Great Mathematical Problems: Marvels and Mysteries of Mathematics*, Ian Stewart (Profile Books 2013)

If you asked me what the Riemann hypothesis was, I would be able to say that it was a very important mathematical problem that had something to do with prime numbers and zeros on a line. And that would be it. There’s an entire chapter in this book about the hypothesis, but with me as a reader Stewart might as well have been “teaching Urdu to a marmoset” (as Laurence Olivier said about trying to teach Marilyn Monroe to act). Stewart is a great popularizer of advanced mathematics, but he faces a big problem: if you understand the profundity, you’ll understand the popularization of it. If you don’t, you won’t.

So I don’t understand what the Riemann hypothesis is about. Or the Poincaré conjecture. Or the Hodge conjecture. Or the Birch–Swinnerton-Dyer conjecture (note long hyphen: it’s named after two mathematicians, not three). So more than half of the chapters in this book lost me almost immediately. The other chapters lost me later, because it’s easy to understand the questions behind the Goldbach conjecture, the four-colour problem and the three-body problem. Is every even integer greater than 2 equal to the sum of two primes? Can we colour all flat two-dimensional maps with four colours or fewer? Can we write an equation to predict the gravitational interaction of three celestial bodies?

The questions are easy to understand, but the answers are very difficult. In fact, only the four-colour problem has been solved. The mathematicians who solved it famously used computers to do so, and their solution can’t be held in or followed by a single human mind. That was something new and it raised interesting questions about the nature of mathematical proof. Stewart discusses them here and supports the idea that computer proofs are legitimate. He ends the book with a list of newer, less famous but perhaps, in some cases, even more important problems. Again, some of them are easy to understand, some aren’t.

So you can get a good sense of the size, scope and complexity of mathematics from this book. And the difficulty. I found a lot of it incomprehensible, but if Ian Stewart can’t explain it to me, no-one else could. And there’s fun amidst the befuddlement:

According to a time-honoured joke, you can tell how advanced a physical theory is by the number of interacting bodies it can’t handle. Newton’s law of gravity runs into problems with three bodies. General relativity has difficulty dealing with two bodies. Quantum theory is over-extended for one body and quantum field theory runs into trouble with

nobodies – the vacuum. (ch. 8, “Orbital Chaos: Three-Body Problem”, pg. 136)

There are also ideas to explore for yourself, like Langton’s ant, because maths is like a mountain range. Even if you can’t get to the peaks, you can enjoy climbing some of the way. There are gentle slopes before the sheer, ice-sheened cliffs. Ian Stewart doesn’t get to the cliffs, but there’s some tough climbing here and I quickly fell off. A lot of amateurs will do much better and this book is worth trying anyway. Being baffled teaches you something both about a subject and about yourself.