G.H. Hardy Quotes

[S]cience works for evil as well as for good (...particularly ...in time of war); and... mathematicians may be justified in rejoicing that there is one science... their own, whose ...remoteness from ordinary human activities should keep it gentle and clean.

[A] good deal of elementary mathematics... 'elementary' in the sense in which professional mathematicians use it... [e.g.,] knowledge of the differential and integral calculus—has considerable practical utility. These... are... rather dull... the parts which have least aesthetic value. The 'real' mathematics of the 'real' mathematicians... of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly 'useless' (...as true of 'applied' as of 'pure' mathematics). It is not possible to justify the life of any genuine professional mathematician on the ground of... 'utility'...

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.

I am interested in mathematics only as a creative art.

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.

Chess problems are the hymn-tunes of mathematics.

, 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.

The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.

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.

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.

Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. "Immortality" may be a silly word, but probably a mathematician has the best chance of whatever it may mean.

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.

If a man has any genuine talent, he should be ready to make almost any sacrifice in order to cultivate it to the full.

… there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.

I am obliged to interpolate some remarks on a very difficult subject: proof and its importance in mathematics. All physicists, and a good many quite respectable mathematicians, are contemptuous about proof. I have heard Professor Eddington, for example, maintain that proof, as pure mathematicians understand it, is really quite uninteresting and unimportant, and that no one who is really certain that he has found something good should waste his time looking for proof.