Geometry, like arithmetic, requires for its logical development only a small number of simple, fundamental principles. These fundamental principles are called the axioms of geometry.

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.

I didn't work especially hard at mathematics at school, because I knew that's what I'd be doing later.

One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it

The infinite! No other question has ever moved so profoundly the spirit of man.

However unapproachable these problems may seem to us and however helpless we stand before them, we have, nevertheless, the firm conviction that their solution must follow by a finite number of purely logical processes.

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