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

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

I am too good for philosophy and not good enough for physics. Mathematics is in between.

My method to overcome a difficulty is to go round it.

The future mathematician ... should solve problems, choose the problems which are in his line, meditate upon their solution, and invent new problems. By this means, and by all other means, he should endeavor to make his first important discovery: he should discover his likes and dislikes, his taste, his own line.

The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. ... Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information.

The best of ideas is hurt by uncritical acceptance and thrives on critical examination.