G. H. Hardy - A Mathematician's Apology (1940)

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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.

G. H. Hardy
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
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.

G. H. Hardy
Chess problems are the hymn-tunes of mathematics.

G. H. Hardy
A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.

G. H. Hardy
Young men should prove theorems, old men should write books.

G. H. Hardy
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
I do not know an instance of a major mathematical advance initiated by a man past fifty

G. H. Hardy
I do not remember having felt, as a boy, any passion for mathematics, and such notions as I may have had of the career of a mathematician were far from noble. I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively.

G. H. Hardy
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.

G. H. Hardy
As history proves abundantly, mathematical achievement, whatever its intrinsic worth, is the most enduring of all.

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

G. H. Hardy
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 it 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 of age.

G. H. Hardy
A man who sets out to justify his existence and his activities has to distinguish two different questions. The first is whether the work which he does is worth doing; and the second is why he does it (whatever its value may be).

G. H. Hardy
The Babylonian and Assyrian civilizations have perished; Hammurabi, Sargon and Nebuchadnezzar are empty names; yet Babylonian mathematics is still interesting, and the Babylonian scale of 60 is still used in Astronomy.

G. H. Hardy
The primes are the raw material out of which we have to build arithmetic, and Euclid's theorem assures us that we have plenty of material for the task.

G. H. Hardy
A science or an art may be said to be "useful" if its development increases, even indirectly, the material well-being and comfort of men, it promotes happiness, using that word in a crude and commonplace way.

G. H. Hardy
I was at my best at a little past forty, when I was a professor at Oxford.

G. H. Hardy
The "seriousness" of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects.

G. H. Hardy
"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.

G. H. Hardy
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.

G. H. Hardy
I will say only that if a chess problem is, in the crude sense, 'useless', then that is equally true of most of the best mathematics; that very little of mathematics is useful practically, and that the little [that is] is comparatively dull.

G. H. Hardy
It is never worth a first class man's time to express a majority opinion. By definition there are plenty of others to do that.

G. H. Hardy
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
We may say, roughly, that a mathematical idea is "significant" if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas.

G. H. Hardy