Mathematics is not a deductive science - that's a cliché... What you do is trial and error, experimentation, guesswork.

The heart of mathematics is its problems.

A good stack of examples, as large as possible, is indispensable for a thorough understanding of any concept,and when I want to learn something new, I make it my first job to build one.

The most spectacular thing about Johnny [von Neumann] was not his power as a mathematician, which was great, or his insight and his clarity, but his rapidity; he was very, very fast. And like the modern computer, which no longer bothers to retrieve the logarithm of 11 from its memory (but, instead, computes the logarithm of 11 each time it is needed), Johnny didn't bother to remember things. He computed them. You asked him a question, and if he didn't know the answer, he thought for three seconds and would produce and answer.

Applied mathematics will always need pure mathematics just as anteaters will always need ants.

The beginner should not be discouraged if he finds he does not have the prerequisites for reading the prerequisites.

The best way to learn is to do; the worst way to teach is to talk.