Mathematics has two faces: it is the rigorous science of Euclid, but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself.

A GREAT discovery solves a great problem but there is a grain of discovery in any problem.

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.'

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

A mathematician who can only generalise is like a monkey who can only climb up a tree, and a mathematician who can only specialise is like a monkey who can only climb down a tree. In fact neither the up monkey nor the down monkey is a viable creature. A real monkey must find food and escape his enemies and so must be able to incessantly climb up and down. A real mathematician must be able to generalise and specialise.

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

If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it.