Real mathematics must be justified as art if it can be justified at all.

No one should ever be bored. … One can be horrified, or disgusted, but one can’t be bored.

The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colours or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.

A science or an art may be said to be "useful" if its development increases, even indirectly, the material well-being and comfort of men, it promotes happiness, using that word in a crude and commonplace way.

Most people can do nothing at all well

In [great mathematics] there is a very high degree of unexpectedness, combined with inevitability and economy.

A chess problem is simply an exercise in pure mathematics.