The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.

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

All that we can hope from these inspirations, which are the fruits of unconscious work, is to obtain points of departure for such calculations. As for the calculations themselves, they must be made in the second period of conscious work which follows the inspiration, and in which the results of the inspiration are verified and the consequences deduced.‎

Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house.

All of mathematics is a tale about groups.

Doubting everything and believing everything are two equally convenient solutions that guard us from having to think

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.