We must know. We will know.

{Speech in Königsberg in 1930, now on his tomb in Göttingen]

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

[On why he didn't try to solve Fermat's last theorem]

Quoted in E T Bell *Mathematics, Queen and Servant of Science* (New York 1951).

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.

Quoted in H Eves *Mathematical Circles Squared* (Boston 1971).

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

*Report on Number Theory*, 1897.

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

Quoted in N Rose *Mathematical Maxims and Minims* (Raleigh N C 1988).

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.

Quoted in N Rose *Mathematical Maxims and Minims* (Raleigh N C 1988).

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.

Quoted in N Rose *Mathematical Maxims and Minims* (Raleigh N C 1988).

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

Quoted in H Eves *Mathematical Circles Revisited* (Boston 1971).

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

Quoted in H Eves *Mathematical Circles Squared* (Boston 1971).

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

Quoted in J R Newman, *The World of Mathematics* (New York 1956).

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.

Quoted in D MacHale, *Comic Sections * (Dublin 1993)

Physics is becoming too difficult for the physicists.

Quoted in C Reid, *Hilbert * (London 1970)

*Meine Herren, der Senat ist doch keine Badeanstalt.*

The faculty is not a pool changing room.

[On the proposed appointment of Emmy Noether as the first woman professor.]

Quoted in A L Mackay, *Dictionary of Scientific Quotations * (London 1994)

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?

Opening of his speech to the 1900 Congress in Paris.

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

Quoted in R Remmert, *Theory of complex functions* (New York, 1989).

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.

*Geometry and the imagination* (New York, 1952).

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.

Quoted in E Maor, *To infinity and beyond* (Boston, 1987).

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.

JOC/EFR February 2006

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