There are no solved problems; there are only problems that are more or less solved.

If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living

Most striking at first is the appearance of sudden illumination, a manifest sign of long unconscious prior work.

One geometry cannot be more true than another; it can only be more convenient.

A first fact should surprise us, or rather would surprise us if we were not used to it. How does it happen there are people who do not understand mathematics? If mathematics invokes only the rules of logic, such as are accepted by all normal minds...how does it come about that so many persons are here refractory?

All great progress takes place when two sciences come together, and when their resemblance proclaims itself, despite the apparent disparity of their substance.

If we wish to foresee the future of mathematics, our proper course is to study the history and present condition of the science.