The mistakes and unresolved difficulties of the past in mathematics have always been the opportunities of its future...

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.

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

Nevertheless, the consuming hunger of the uncritical mind for what it imagines to be certainty or finality impels it to feast upon shadows in the prevailing famine of substance.

The full impact of the Lobachevskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobachevsky the Copernicus of Geometry [as did Clifford], for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought.

Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions.

Obvious is the most dangerous word in mathematics.