sawyl | A Mathematician's Apology (Reply)

My third book of the year is an old favourite: G.H. Hardy's A Mathematician's Apology, which comes with a substantial biographical introduction by C.P. Snow.

Snow's introduction provides a great deal of information about Hardy's character, from his love of cricket to his realistic view of his own place in the mathematical firmament: Hardy considers himself to have been, "for a short time the fifth best pure mathematician in the world", and modestly admits that he considers both of his major collaborators, J.E. Littlewood and S.A. Ramanujan, to be his betters. After covering Hardy's great late — for a mathematician — period at Oxford in 1920s and the return to Cambridge in the 1930s, Snow candidly describes how their friendship became strained during the second world war.

Introducing the Apology proper, Snow then make the the point that for that Hardy manages to give a exuberant account of what it means to be a mathematician, the book is ultimately a melancholy work; the product of someone whose gifts have failed them. This is a point that Hardy himself stresses: he writes because he can no longer do.

In his Apology, Hardy sets out a logical prospectus in which he examines whether his life as a mathematician has value. Along the way, he considers why people choose particular lines of work, the role of intellectual pride and the permanence of mathematical discovery. He then considers the nature of mathematical puzzles, how they differ in nature from simple chess puzzles, and what it is that makes one puzzle superior to another. In demonstrating his points about puzzles, Hardy slips in the only two pieces of maths in the book: Euclid's proof of the existence of an infinity of prime numbers; and Pythagoras' proof of the irrationality of the square root of two.

Incidentally, when introducing the two ancient Greek puzzles to demonstrate his points, Hardy includes a particularly telling remark about his intellectual forebears which says a great deal about the mindset of the time:

The Greeks were the first mathematicians who are still "real" to us to-day. [Babylonian] mathematics may be an interesting curiosity, but Greek mathematics is the real thing. The Greeks first spoke a language which modern mathematicians can understand; as Littlewood said to me once, they are not clever schoolboys or "scholarship candidates", but "Fellows of another college".

Having established the delights and depths and the seriousness of mathematics, Hardy then goes on to characterise the difference between the pure and applied branches of the subject both of which, he claims, are equally useless and impractical. He also discusses the problem of what mathematics really is real or ideal — Hardy comes out as a firm Platonic realist — and its relationship to the physical world.

In the final chapters Hardy, then in his early sixties and aware of the loss of his creative spark, poignantly reflects on his achievements:

It is plain now that my life, for what it is worth, is finished, and that nothing I can do can perceptibly increase or diminish its value. It is very difficult to be dispassionate, but I count it a success; I have had more reward and not less than was due to a man of my particular grade of ability...

The case for my life, then, or for that of any one else who has been a mathematician in the same sense in which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any of the other artists, great or small, who have left some kind of monument behind them.

This last sums up the Apology. It is not about mathematics at all, but about what it means to live a rewarding life; what it means to be a creative artist and what it means when that creative talent is lost.