If we have no idea why a statement is true, we can still prove it by induction.

God created infinity, and man, unable to understand infinity, had to invent finite sets.

A mathematician's work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks.

Are mathematical ideas invented or discovered? This question has been repeatedly posed by philosophers through the ages and will probably be with us forever.

The lack of real contact between mathematics and biology is either a tragedy, a scandal or a challenge, it is hard to decide which.

Stan Ulam was lazy, ... He talked too much ... He was self-centered ... . He had an overpowering personality.

Very little mathematics has direct applications - though fortunately most of it has plenty of indirect ones.

Nature imitates mathematics.

There is something in statistics that makes it very similar to astrology.

The progress of mathematics can be viewed as progress from the infinite to the finite.

THE COMPUTER IS JUST AN INSTRUMENT for doing faster what we already know how to do slower. All pretensions to computer intelligence and paradise-tomorrow promises should be toned down before the public turns away in disgust. And if that should happen, our civilization might not survive.

Mathematicians also make terrible salesmen. Physicists can discover the same thing as a mathematician and say 'We've discovered a great new law of nature. Give us a billion dollars.' And if it doesn't change the world, then they say, 'There's an even deeper thing. Give us another billion dollars.'

The pendulum of mathematics swings back and forth towards abstraction and away from it with a timing that remains to be estimated.