Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

... there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.

I am interested in mathematics only as a creative art.

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

A painter makes patterns with shapes and colours, a poet with words. A painting may embody an ‘idea’, but the idea is usually commonplace and unimportant. In poetry, ideas count for a good deal more; but, [...] the importance of ideas in poetry is habitually exaggerated: '... Poetry is no the thing said but a way of saying it.' [In poetry,] the poverty of the ideas seems hardly to affect the beauty of the verbal pattern.

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.

Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. "Immortality" may be a silly word, but probably a mathematician has the best chance of whatever it may mean.

No discovery of mine has made, or is likely to make, for good or ill, the least difference to the amenity of the world.