A Mathematician's Apology

“Apology” here means a defence, not an expression of regret, although by the end the book reads a little like both. Hardy wrote it in his early sixties, after his creative powers had faded, and it is part argument and part farewell.

The argument

Hardy wants to justify a life spent on pure mathematics, and his justification is beauty. Good mathematics, in his view, is serious, deep, general and economical. It is closer to poetry than to engineering. He openly does not care whether it is useful, and he says he is glad that most of his own work never was.

He illustrates “serious” mathematics with two proofs from ancient Greece: Euclid’s proof that there are infinitely many primes, and the proof that the square root of two is not a fraction. Both are given in full. I had seen both before but never thought about why they feel so satisfying. Hardy’s answer is a mixture of inevitability and surprise: the argument is unexpected, but once you see it, it could not have gone any other way.

The irony

The famous irony is that Hardy picked number theory as his example of a beautifully useless subject. Today number theory sits underneath public-key cryptography, so every secure web page I load depends on the kind of mathematics he was proud of being useless. He also contributed a result in population genetics, the Hardy–Weinberg principle, that he seems to have thought little of. History did not cooperate with his thesis.

I don’t think that makes him wrong, though. The point he is really making is that you should not choose mathematics for its usefulness, because you cannot predict which parts will be useful, and the motivation that produces deep work is something else.

The sad part

The book is also melancholy. Hardy insists that mathematics is a young man’s game and that a mathematician past fifty has little left to give. He is writing about himself. It is uncomfortable to read, and I am not sure the claim is even true, but the honesty is striking.

The edition I read has a long foreword by C. P. Snow, who knew Hardy. It is almost as good as the essay itself, and it fills in what Hardy leaves out: his warmth, his love of cricket, and his collaboration with Ramanujan, which Hardy himself treats as the one romantic episode of his life.

Verdict

Short, opinionated and dated in places, but I have not read a better account of why anyone would want to do mathematics at all.