But all of my efforts served only to make me better acquainted with the difficulty, which in itself was something.

Mathematics is the art of giving the same name to different things.

Mathematical discoveries, small or great are never born of spontaneous generation They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subconscious.

We also know how cruel the truth often is, and we wonder whether delusion is not more consoling.

All that is not thought is pure nothingness; since we can think only thoughts, and all the words we use to speak of things can express only thoughts, to say there is something other than thought is therefore an affirmation which can have no meaning.

Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. The geometer might be replaced by the "logic piano" imagined by Stanley Jevons; or, if you choose, a machine might be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in alive and come out transformed into hams and sausages. No more than these machines need the mathematician know what he does.

It is with logic that one proves; it is with intuition that one invents.