Intuition is more important to discovery than logic.

One would have to have completely forgotten the history of science so as to not remember that the desire to know nature has had the most constant and the happiest influence on the development of mathematics.

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.

Mathematicians do not study objects, but relations among objects; they are indifferent to the replacement of objects by others as long the relations don't change. Matter is not important, only form interests them.

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.

Mathematics has a threefold purpose. It must provide an instrument for the study of nature. But this is not all: it has a philosophical purpose, and, I daresay, an aesthetic purpose.

All the scientist creates in a fact is the language in which he enunciates it. If he predicts a fact, he will employ this language, and for all those who can speak and understand it, his prediction is free from ambiguity. Moreover, this prediction once made, it evidently does not depend upon him whether it is fulfilled or not.