Mathematical discoveries, small or great are never born of spontaneous generation.

It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better.

For a long time the objects that mathematicians dealt with were mostly ill-defined; one believed one knew them, but one represented them with the senses and imagination; but one had but a rough picture and not a precise idea on which reasoning could take hold.

Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose.

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.

[T]he different branches of Arithmetic - Ambition [G]eometry is not true, it is advantageous.

Ideas rose in clouds; I felt them collide until pairs interlocked, so to speak, making a stable combination.