Mathematics, however, is, as it were, its own explanation; this, although it may seem hard to accept, is nevertheless true, for the recognition that a fact is so is the cause upon which we base the proof.

Mathematics is not only real, but it is the only reality.

Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.

Logic is the hygiene the mathematician practices to keep his ideas healthy and strong.

Mathematics is concerned with "all possible worlds."

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.

Where there is matter, there is geometry.

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

In the company of friends, writers can discuss their books, economists the state of the economy, lawyers their latest cases, and businessmen their latest acquisitions, but mathematicians cannot discuss their mathematics at all. And the more profound their work, the less understandable it is.

Mathematics is the art of giving the same name to different things.

A mathematician's reputation rests on the number of bad proofs he has given.

Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.

Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colours or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.

How happy the lot of the mathematician. He is judged solely by his peers, and the standard is so high that no colleague or rival can ever win a reputation he does not deserve.

It should seem that it is easier to square the circle than to get round a mathematician.

A mathematician is a blind man in a dark room looking for a black cat which isn't there.

Mathematics is not a deductive science - that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.

Mathematics consists in proving the most obvious thing in the least obvious way.

There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.

The only way to learn mathematics is to do mathematics.

Perfect numbers like perfect men are very rare.

God used beautiful mathematics in creating the world.

The mathematician's patterns, like the painter's or the poet's, must be beautiful.

A mathematician is a device for turning coffee into theorems.

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