A Mathematician's Apology

by G.H. Hardy

He wasn’t a great genius, as Einstein and Rutherford were. He said, with his usual clarity, that if the word meant anything he was not a genius at all. At his best, he said, he was for a short time the fifth

best pure mathematician in the world.

he always made the point that his friend and collaborator Littlewood was an appreciably more powerful mathematician than he was, and that his protégé Ramanujan really had natural genius in the sense (though not to the extent, and nothin...

“Well, I have done one thing you could never have done, and that is to have collaborated with Littlewood and Ramanujan on something like equal terms.”’

Both were gifted and mathematically inclined. In his case, as in that of most mathematicians, the gene pool doesn’t need searching for.

He was demonstrating a formidably high I.Q,. as soon as, or before, he learned to talk. At the age of two he was writing down numbers up to millions (a common sign of mathematical ability).

Unlike Einstein, who had to subjugate his powerful ego in the study of the external world before he could attain his moral stature, Hardy had to strengthen an ego which wasn’t much protected.

The scholarships have been there all right, if one knew how to win them. There was never the slightest chance of the young Hardy being lost—as there was of the young Wells or the young Einstein. From the age of twelve he had only to survive, and his talents would be looked after.

He never went near Winchester after he had left it: but he left it, with the inevitability of one who had got on to the right tramlines, with an open scholarship to Trinity.

The first was theological, in the high Victorian manner. Hardy had decided—I think before he left Winchester—that he did not believe in God. With him, this was a black-and-white decision, as sharp and clear as all other concepts in his mind.

It was all very English. It had only one disadvantage, as Hardy pointed out with his polemic clarity, as soon as he had become an eminent mathematician and was engaged, together with his tough ally Littlewood, in getting the system abolished: it had effectively ruined serious mathematics in England for a hundred years.

This coach knew all the obstacles, all the tricks of the examiners, and was sublimely uninterested in the subject itself.

He was Fourth Wrangler in 1898. This faintly irritated him, he used to confess. He was enough of a natural competitor to feel that, though the race was ridiculous, he ought to have won

In many senses, then, he was unusually lucky. He did not have to think about his career. From the time he was twenty-three he had all the leisure that

man could want, and as much money as he needed. A bachelor don in Trinity in the 1900’s was comfortably off.

will anticipate now what I shall say later. His life remained the life of a brilliant young man until he was old: so did his spirit: his games, his interests, kept the lightness of a young don’s. And, like many men who keep a young man’s interests into their sixties, his last years were the darker for it.

In 1911 he began a collaboration with Littlewood which lasted thirty-five years. In 1913 he discovered Ramanujan and began another collaboration.

he was thinking about his old age. Oxford colleges, in many ways so human and warm, are ruthless with the old: if he remained at New College he would be turned out of his rooms as soon as he retired, under the age limit, from his professorship. Whereas, if he returned to Trinity, he could stay in college until he died. That is in effect what he managed to do.

‘Young men ought to be conceited: but they oughtn’t to be imbecile.’ (Said after someone had tried to persuade him that Finnegans Wake was the final literary masterpiece.)

Some egotism of this sort is inevitable, and I do not feel that it really needs justification. Good work is not done by ‘humble’ men.

If a man is in any sense a real mathematician, then it is a hundred to one that his mathematics will be far better than anything else he can do, and that he would be silly if he surrendered any decent opportunity of exercising his one talent in order to do undistinguished work in other fields. Such a sacrifice could be justified only by economic necessity or age.

which I will call ‘mathematical reality’; and there is no sort of agreement about the nature of mathematical reality among either mathematicians or philosophers. Some hold that it is ‘mental’ and that in some sense we construct it, others that it is outside and independent of

A man who could give a convincing account of mathematical reality would have solved very many of the most difficult problems of metaphysics. If he could include physical reality in his account, he would have solved them all.

I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations’, are simply our notes of our observations.

THE contrast between pure and applied mathematics stands out most clearly, perhaps, in geometry. There is the science of pure geometry1,

Each of these geometries is a model, a pattern of ideas, and is to be judged by the interest and beauty of its particular pattern. It is a map or picture, the joint product of many hands, a partial and imperfect copy (yet exact so far as it extends) of a section of mathematical reality.

that there is one thing at any rate of which pure geometries are not pictures, and that is the spatio-temporal reality of the physical world.

It is plain, first, that the truth of the theorems which I prove is in no way affected by the quality of my drawings. Their function is merely to bring home my meaning to my hearers, and, if I can do that, there would be no gain in having them redrawn by the most skilful draughtsman. They are pedagogical illustrations,

The study of that pattern, and of the general pattern of physical reality, is a science in itself, which we may call ‘physical geometry’. Suppose now that a violent dynamo, or a massive gravitating body, is introduced into the room. Then the physicists tell us that the geometry of the room is changed, its whole physical pattern slightly but definitely distorted. Do the theorems which I have proved become false?

nonsense to suppose that the proofs of them which I have given are affected in any way. It would be like supposing that a play of Shakespeare is changed when a reader spills his tea over a page.

and ‘pure geometries’ are independent of lecture rooms, or of any other detai...

pure mathematician. Applied mathematicians, mathematical physicists, naturally take a different view, since they are preoccupied with the physical world itse...

We may be able to trace a certain resemblance between the two sets of relations, and then the pure geometry will become interesting to physicists; it will give us, to that extent, a map which ‘fits the facts’ of the physical world.

there is probably less difference between the positions of a mathematician and of a physicist than is generally supposed, and that the most important seems to me to be this, that the mathematician is in much more direct contact with reality.

went on to say that neither physicists nor philosophers have ever given any convincing account of what ‘physical reality’ is, or of how the physicist passes, from the confused mass of fact or sensation with which he starts, to the construction of the objects which he calls ‘real’.

but this need not prevent us from understanding roughly what a physicist is trying to do. It is plain that he is trying to correlate the incoherent body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme which he can borrow only from mathematics.

It may be that modern physics fits best into some framework of idealistic philosophy—I do not believe it, but there are eminent physicists who say

Pure mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way.

THESE distinctions between pure and applied mathematics are important in themselves, but they have very little bearing on our discussion of the ‘usefulness’ of mathematics.

The great modern achievements of applied mathematics have been in relativity and quantum mechanics, and these subjects are, at present at any rate, almost as ‘useless’ as the theory of numbers.

Time may change all this. No one foresaw

and it may be that some of the ‘highbrow’ applied mathematics will become ‘useful’ in as unexpected a way; but the evidence so far points to the conclusion that, in one subject as in the other, it is what is commonplace and dull that counts for practical life.

Eddington giving a happy example of the unattractiveness of ‘useful’ science.

It appeared that the members (whether citizens of Leeds or not) wanted to be entertained, and that’ heavy wool’ is not at all an entertaining subject. So the attendance at these lectures was very disappointing; but those who lectured on the excavations at Knossos, or on relativity, or on the theory of prime numbers, were delighted by the audiences that they drew.

We must also remember that a reserve of knowledge is always an advantage, and that the most practical of mathematicians may be seriously handicapped if his knowledge is the bare minimum which is essential to him; and for this reason we must add a little under every heading.

general conclusion must be that such mathematics is useful as is wanted by a superior engineer or a moderate physicist;

roughly the same thing as to say, such mathematics as has no partic...

curious conclusion emerges, that pure mathematics is on the whole distinctly more useful than applied. A pure mathematician seems to have the advantage on the practical as well as on the aesthetic side. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.

But is not the position of an ordinary applied mathematician in some ways a little pathetic?

he must work in a humdrum way, and he cannot give full play to his fancy even when he wishes to rise to the heights. ‘Imaginary’ universes are so much more beautiful than this stupidly constructed ‘real’ one; and most of the finest products of an applied mathematician’s fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts.