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.

If indeed, as Hilbert asserted, mathematics is a meaningless game played with meaningless marks on paper, the only mathematical experience to which we can refer is the making of marks on paper.

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