In the future, as in the past, the great ideas [of mathematics] must be simplifying ideas, the creator must always be one who clarifies, for himself, and for others, the most complicated issues of formulas and concepts.

When a branch of mathematics ceases to interest any but the specialists, it is very near its death, or at any rate dangerously close to a paralysis, from which it can be rescued only by being plunged back into the vivifying source of the science.

What is crucial is there be laws.

Every mathematician worthy of the name has experienced . . . the state of lucid exaltation in which one thought succeeds another as if miraculously . . . this feeling may last for hours at a time, even for days. Once you have experienced it, you are eager to repeat it but unable to do it at will, unless perhaps by dogged work. . . .

Rigour is to the mathematician what morality is to men.

First rate mathematicians choose first rate people, but second rate mathematicians choose third rate people.

Intuition makes much of it; I mean by this the faculty of seeing a connection between things that in appearance are completely different; it does not fail to lead us astray quite often.

God exists since mathematics is consistent, and the Devil exists since we cannot prove it.