G.H. Hardy Quotes

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

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

A chess problem is genuine mathematics, but it is in some way 'trivial' mathematics. However ingenious and intricate, however original and surprising the moves, there is something essential lacking. Chess problems are unimportant. The best mathematics is serious as well as beautiful – important if you like, but the word is very ambiguous, and 'serious' expresses what I mean much better.

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.

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.

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, because mathematical reality is built that way.

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

A painter makes patterns with shapes and colours, a poet with words. A painting may embody an ‘idea’, but the idea is usually commonplace and unimportant. In poetry, ideas count for a good deal more; but, [...] the importance of ideas in poetry is habitually exaggerated: '... Poetry is not the thing said but a way of saying it.' [In poetry,] the poverty of the ideas seems hardly to affect the beauty of the verbal pattern.

It is... natural to suppose that there is a great difference in utility between 'pure' and 'applied' mathematics. This is a delusion...

, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

Mathematicians have constructed a very large number of different systems of geometry, Euclidean or non-Euclidean, of one, two, three, or any number of dimensions. All these systems are of complete and equal validity. They embody the results of mathematicians' observations of their reality, a reality far more intense and far more rigid than the dubious and elusive reality of physics. The old-fashioned geometry of Euclid, the entertaining seven-point geometry of Veblen, the space-times of Minkowski and Einstein, are all absolutely and equally real. ...There may be three dimensions in this room and five next door. As a professional mathematician, I have no idea; I can only ask some competent physicist to instruct me in the facts.
The function of a mathematician, then, is simply to observe the facts about his own intricate system of reality, that astonishingly beautiful complex of logical relations which forms the subject-matter of his science, as if he were an explorer looking at a distant range of mountains, and to record the results of his observations in a series of maps, each of which is a branch of pure mathematics. ...Among them there perhaps none quite so fascinating, with quite the astonishing contrasts of sharp outline and shade, as that which constitutes the theory of numbers.

Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work a good deal later; Gauss's great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself.

[M]athematical reality lies outside us ...our function is to discover or observe it, and ...the theorems ...we prove, and ...describe grandiloquently as our 'creations', are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards [...]

Pure mathematics is on the whole distinctly more useful than applied. [...] For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.

It is... astonishing how little practical value scientific knowledge has for ordinary men, how dull and commonplace such of it as has value is, and how its value seems almost to vary inversely to its reputed utility. ...We live either by or on other people's professional knowledge.