A Mathematician’s Apology

by G.H. Hardy

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.

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.

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.

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

No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game.

A man’s first duty, a young man’s at any rate, is to be ambitious.

but the noblest ambition is that of leaving behind something of permanent value—

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.

‘I cannot satisfy myself that there are any such things as poetical ideas.… Poetry is no the thing said but a way of saying it.’

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

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.

The proof is by reductio ad absurdum, and reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons5. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

whoa!!! The proof that the series P = 2, 3, 5, 7, 11, 13… is infinite.