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.

Time makes fools of us all. Our only comfort is that greater shall come after us.

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

The only royal road to elementary geometry is ingenuity.

Poincaré [was] the last man to take practically all mathematics, pure and applied, as his province. ... Few mathematicians have had the breadth of philosophic vision that Poincaré had, and none in his superior in the gift of clear exposition.

The longer mathematics lives the more abstract - and therefore, possibly also the more practical - it becomes.

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.