A Mathematician's Apology

by G.H. Hardy

THERE are then two mathematics. There is the real mathematics of the real mathematicians, and there is what I will call the ‘trivial’ mathematics,

We have still one more question to consider. We have concluded that the trivial mathematics is, on the whole, useful, and that the real mathematics, on the whole, is not; that the trivial mathematics does, and the real mathematics does not, ‘do good’ in a certain sense; but we have still to ask whether either sort of mathematics does harm.

paradoxical to suggest that mathematics of any sort does much harm in time of peace, so that we are driven to the consideration of the

effects of mathematic...

Real mathematics has no effects on war. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years.

perhaps hard to call them ‘ trivial’, but none of them has any claim to rank as ‘real’. They are indeed repulsively ugly and intolerably dull; even Littlewood could not make ballistics respectable, and if he could not who can?

mathematics is, as I said at Oxford, a ‘harmless and innocent’ occupation.

The first and the most obvious is that the effect of science on war is merely to magnify its horror,

It may also be urged (though this was not one of Haldane’s theses) that the equalization of risks which science was expected to bring would be in the long run salutary; that a civilian’s life is not worth more than a soldier’s, nor a woman’s than a man’s; that anything is better than the concentration of savagery on one particular class; and that, in short, the sooner war comes ‘all out’ the better.

do not know which of these views is nearer to the truth. It is an urgent and a moving question, but I need not argue it here. It concerns only the ‘trivial’ mathematics, which it would be Hogben’s business to defend rather than mine.

Mathematics is not a contemplative but a creative subject; no one can draw much consolation from it when he has lost the power or the desire to create; and that is apt to happen to a mathematician rather soon. It is a pity, but in that case he does not matter a great deal anyhow, and it would be silly to bother about him.

still say to myself when I am depressed, and find myself forced to listen to pompous and tiresome people,‘Well, I have done one thing you could never have done, and that is to have collaborated with both Littlewood and Ramanujan on something like equal terms.’

It is to them that I owe an unusually late maturity: I was at my best at a little past forty, when I was a professor at Oxford.

A mathematician may still be competent enough at sixty, but it is useless to expect him to have original ideas.

Is there any sense in which I can claim that my life has been less futile than theirs? It seems to me again that there is only one possible answer: yes, perhaps, but, if so, for one reason only.

I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. I have helped to train other mathematicians,

I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating.

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 memorial behind them.