Pure logic could never lead us to anything but tautologies; it can create nothing new; not from it alone can any science issue.

Hypotheses are what we lack the least.

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

To invent is to discern, to choose.

Mathematicians do not study objects, but the relations between objects.

I entered an omnibus to go to some place or other. At that moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with non-Euclidean geometry.

A sane mind should not be guilty of a logical fallacy, yet there are very fine minds incapable of following mathematical demonstrations.