The result of the mathematician’s creative work is demonstrative reasoning, a proof, but the proof is discovered by plausible reasoning, by GUESSING.

The world is anxious to admire that apex and culmination of modern mathematics: a theorem so perfectly general that no particular application of it is feasible.

You should not put too much trust in any unproved conjecture, even if it has been propounded by a great authority, even if it has been propounded by yourself. You should try to prove it or disprove it...

Euclid ’s manner of exposition, progressing relentlessly from the data to the unknown and from the hypothesis to the conclusion, is perfect for checking the argument in detail but far from being perfect for making understandable the main line of the argument.

John von Neumann was the only student I was ever afraid of.

Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements.

What is the difference between method and device? A method is a device which you use twice.

I am intentionally avoiding the standard term which, by the way, did not exist in Euler’s time. One of the ugliest outgrowths of the “new math” was the premature introduction of technical terms.

Look around when you have got your first mushroom or made your first discovery: they grow in clusters.

Hilbert once had a student in mathematics who stopped coming to his lectures, and he was finally told the young man had gone off to become a poet. Hilbert is reported to have remarked: ‘I never thought he had enough imagination to be a mathematician.’