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.

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

There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn't say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann.

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.'

If there is a problem you can't solve, then there is an easier problem you can't solve: find it.

Success in solving the problem depends on choosing the right aspect, on attacking the fortress from its accessible side.

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