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

317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is, because mathematical reality is built that way.

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.

Mathematical fame, if you have the cash to pay for it, is one of the soundest and steadiest of investments.

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.

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

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

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

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

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.

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

In these days of conflict between ancient and modern studies, there must surely be something to be said for a study which did not begin with Pythagoras, and will not end with Einstein, but is the oldest and the youngest of all.

A chess problem is an exercise in pure mathematics.

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.

The public does not need to be convinced that there is something in mathematics.

The study of mathematics is, if an unprofitable, a perfectly harmless and innocent occupation.

What we do may be small, but it has a certain character of permanence and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men.

Young men should prove theorems, old men should write books.

I hate teaching [but] I love lecturing.

Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not.

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.

I am interested in mathematics only as a creative art.

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

Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.

Exposition, criticism, appreciation, is work for second-rate minds.

For any serious purpose, intelligence is a very minor gift.

A person’s first duty, a young person’s at any rate, is to be ambitious, and the noblest ambition is that of leaving behind something of permanent value.

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.

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.

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

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

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

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

The seriousness of a theorem, of course, does not lie in its consequences, which are merely the evidence for its seriousness.