The only royal road to elementary geometry is ingenuity.

Any impatient student of mathematics or science or engineering who is irked by having algebraic symbolism thrust upon him should try to get along without it for a week.

Obvious is the most dangerous word in mathematics.

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

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.

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

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.