The aim of science is not things themselves, as the dogmatists in their simplicity imagine, but the relation between things.

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?

In the old days when people invented a new function they had something useful in mind.

All that we can hope from these inspirations, which are the fruits of unconscious work, is to obtain points of departure for such calculations. As for the calculations themselves, they must be made in the second period of conscious work which follows the inspiration, and in which the results of the inspiration are verified and the consequences deduced.‎

The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.

Mathematicians do not deal in objects, but in relations between objects; thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only.

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