Abstractness, sometimes hurled as a reproach at mathematics, is its chief glory and its surest title to practical usefulness. It is also the source of such beauty as may spring from mathematics.

Poincaré was a vigorous opponent of the theory that all mathematics can be rewritten in terms of the most elementary notions of classical logic; something more than logic, he believed, makes mathematics what it is.

Galois read the geometry from cover to cover as easily as other boys read a pirate yarn.

I have always hated machinery, and the only machine I ever understood was a wheelbarrow, and that but imperfectly.

Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos.

Science makes no pretension to eternal truth or absolute truth.

Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness.