Mathematicians are born, not made.

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

A small error in the former will produce an enormous error in the latter.

What is it indeed that gives us the feeling of elegance in a solution, in a demonstration?

A very small cause, which escapes us, determines a considerable effect which we cannot ignore, and we say that this effect is due to chance.

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?

It is often said that experiments should be made without preconceived ideas. That is impossible. Not only would it make every experiment fruitless, but even if we wished to do so, it could not be done. Every man has his own conception of the world, and this he cannot so easily lay aside. We must, example, use language, and our language is necessarily steeped in preconceived ideas. Only they are unconscious preconceived ideas, which are a thousand times the most dangerous of all.