Sublime

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

In both theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy.

Hardy was, in Newton’s phrase, ‘in the prime of his age for invention’, and this came in his early forties, unusually late for a mathematician.

A mathematician may still be competent enough at sixty, but it is useless to expect him to have original ideas.

There are two things at any rate which seem essential, a certain generality and a certain depth; but neither quality is easy to define at all precisely. A significant mathematical idea, a serious mathematical theorem, should be ‘general’ in some such sense as this. The idea should be one which is a constituent in many mathematical constructs, which

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.

I said that a mathematician was a maker of patterns of ideas, and that beauty and seriousness were the criteria by which his patterns should be judged.

There is something else, though, at which he was clearly superior to Einstein or Rutherford or any other great genius: and that is at turning any work of the intellect, major or minor or sheer play, into a work of art. It was that gift above all, I think, which made him, almost without realizing it, purvey such intellectual delight.