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

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

Sometimes it happens that a man's circle of horizon becomes smaller and smaller, and as the radius approaches zero it concentrates on one point. And then that becomes his point of view.

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

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.

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.

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

Begin with the simplest examples.

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

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.

One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs

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

One hears a lot of talk about the hostility between scientists and engineers. I don't believe in any such thing. In fact I am quite certain it is untrue... There cannot possibly be anything in it because neither side has anything to do with the other.

A mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution.

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

Physics is becoming too difficult for the physicists.

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

Besides it is an error to believe that rigour is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof.

The arithmetical symbols are written diagrams and the geometrical figures are graphic formulas.

We must know. We will know.

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

Is mathematics doomed to suffer the same fate as other sciences that have split into separate branches?... Mathematics is, in my opinion, an indivisible whole... May the new century bring with it ingenious champions and many zealous and enthusiastic disciples.

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

Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics.

We ought not to believe those who today, adopting a philosophical air and with a tone of superiority, prophesy the decline of culter and are content with the unknowable in a self-satisfied way. For us there is no unknowable, and in my opinion there is also non whatsoever for the natural sciences. In place of this foolish unknowable, let our watchword on the contrary be: we must know - we shall know.

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