He who seeks for methods without having a definite problem in mind seeks in the most part in vain.

No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.

We do not master a scientific theory until we have shelled and completely prised free its mathematical kernel.

The art of doing mathematics consists in finding that special case which contains all the germs of generality.

If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?

In mathematics ... we find two tendencies present. On the one hand, the tendency towards abstraction seeks to crystallise the logical relations inherent in the maze of materials ... being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency towards intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations.

The further a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separated branches of the science.