G. H. Hardy: I propose to put forward an apology for mathematics; and...

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 the notes of our observations.

G. H. Hardy

science believe lying

I remember once going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways."

G. H. Hardy

numbers two lying

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... Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating. And that I have created something is undeniable: the question is about its value.

G. H. Hardy

escaping creating differences

It is not worth an intelligent man's time to be in the majority. By definition, there are already enough people to do that.

G. H. Hardy

intelligent men knowledge

A mathematician, like a painter or a poet, is a maker of patterns.

G. H. Hardy

original-ideas patterns math

Beauty is the first test: there is no permanent place in the world for ugly mathematics.

G. H. Hardy

apology math beauty

Perhaps five or even ten per cent of men can do something rather well. It is a tiny minority who can do anything really well, and the number of men who can do two things well is negligible. If a man has any genuine talent, he should be ready to make almost any sacrifice in order to cultivate it to the full.

G. H. Hardy

sacrifice men order

All analysts spend half their time hunting through the literature for inequalities which they want to use and cannot prove.

G. H. Hardy

hunting use math

When the world is mad, a mathematician may find in mathematics an incomparable anodyne. For mathematics is, of all the arts and sciences, the most austere and the most remote, and a mathematician should be of all men the one who can most easily take refuge where, as Bertrand Russell says, "one at least of our nobler impulses can best escape from the dreary exile of the actual world."

G. H. Hardy

men science art

They [formulae 1.10 - 1.12 of Ramanujan] must be true because, if they were not true, no one would have had the imagination to invent them.

G. H. Hardy

ramanujan being-true imagination

I am interested in mathematics only as a creative art.

G. H. Hardy

creativity creative art

No mathematician should ever allow him to forget that mathematics, more than any other art or science, is a young man's game. ... 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 later; ... [but] I do not know of a single instance of a major mathematical advance initiated by a man past fifty. ... A mathematician may still be competent enough at sixty, but it is useless to expect him to have original ideas.

G. H. Hardy

science past art

If I could prove by logic that you would die in five minutes, I should be sorry you were going to die, but my sorrow would be very much mitigated by pleasure in the proof.

G. H. Hardy

sorrow would-be sorry

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.

G. H. Hardy

math silly mean

Asked if he believes in one G-d, a mathematician answered: "Yes, up to isomorphism".

G. H. Hardy

mathematician ifs believe

Real mathematics must be justified as art if it can be justified at all.

G. H. Hardy

justified real art

In [great mathematics] there is a very high degree of unexpectedness, combined with inevitability and economy.

G. H. Hardy

economy mathematics degrees

Sometimes one has to say difficult things, but one ought to say them as simply as one knows how.

G. H. Hardy

simplicity science knowledge

For my part, it is difficult for me to say what I owe to Ramanujan - his originality has been a constant source of suggestion to me ever since I knew him, and his death is one of the worst blows I have ever had.

G. H. Hardy

ramanujan suggestions blow

Good work is no done by "humble" men. It is one of the first duties of a professor, for example, in any subject, to exaggerate a little both the importance of his subject and his own importance in it. A man who is always asking "Is what I do worth while?" and "Am I the right person to do it?" will always be ineffective himself and a discouragement to others. He must shut his eyes a little and think a little more of his subject and himself than they deserve. This is not too difficult: it is harder not to make his subject and himself ridiculous by shutting his eyes too tightly.

G. H. Hardy

hard-work eye humble

I propose to put forward an apology for mathematics; and I may be told that it needs none, since there are now few studies more generally recognized, for good reasons or bad, as profitable and praiseworthy.