THE second quality which I demanded in a significant idea was depth, and this is still more difficult to define. It has something to do with difficulty; the ‘deeper’ ideas are usually the harder to grasp: but it is not at all the same. The ideas underlying Pythagoras’s theorem and its generalizations are quite deep, but no mathematician now would

Their answers, if they are honest, will usually take one or other of two forms; and the second form is merely a humbler variation of the first, which is the only answer which we need consider seriously. (I) ‘I do what I do because it is the one and only thing that I can do at all well.

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.

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

I cannot remember ever having wanted to be anything but a mathematician. I suppose that it was always clear that my specific abilities lay that way, and it never occurred to me to question the verdict of my elders. 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

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.

A man’s first duty, a young man’s at any rate, is to be ambitious. Ambition is a noble passion which may legitimately take many forms; there was something noble in the ambition of Attila or Napoleon: but the noblest ambition is that of leaving behind one something of permanent value—

He was the classical anti-narcissist. He could not endure having his photograph taken: so far as I know, there are only five snapshots in existence.

There are many highly respectable motives which may lead men to prosecute research, but three which are much more important than the rest. The first (without which the rest must come to nothing) is intellectual curiosity, desire to know the truth. Then, professional pride, anxiety to be satisfied with one’s performance, the shame that overcomes any