Wir müssen wissen. Wir werden wissen.

[We must know. We will know.]

Before beginning I should put in three years of intensive study, and I haven't that much time to squander on a probable failure.

Galileo was no idiot. Only an idiot could believe that science requires martyrdom - that may be necessary in religion, but in time a scientific result will establish itself.

I have tried to avoid long numerical computations, thereby following Riemann's postulate that proofs should be given through ideas and not voluminous computations.

Mathematics is a game played according to certain simple rules with meaningless marks on paper.

How thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which, at the same time, assist in understanding earlier theories and in casting aside some more complicated developments.

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

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.

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

Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.

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

No one shall expel us from the paradise that Cantor has created for us.

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

If one were to bring ten of the wisest men in the world together and ask them what was the most stupid thing in existence, they would not be able to discover anything so stupid as astrology.

Physics is becoming too difficult for the physicists.

Meine Herren, der Senat ist doch keine Badeanstalt.

[The faculty is not a pool changing room.]

Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?

Every mathematical discipline goes through three periods of development: the naive, the formal, and the critical.

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.

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.

A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.

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

Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts.

Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.

(On Cantor's set theory:) The finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.

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

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.