Real mathematics must be justified as art if it can be justified at all.
A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than their, it is because they are made with ideas.
A mathematician, like a painter or a poet, is a maker of poems. If his patterns are more permanent than theirs, it is because they are made with ideas.
It is not worth an intelligent man’s time to be in the majority. By definition, there are already enough people to do that.
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.
If a man has any genuine talent he should be ready to make almost any sacrifice in order to cultivate it to the full.
The best mathematics is serious as well as beautiful—‘important’ if you like, but the word is very ambiguous, and ‘serious’ expresses what I mean much better
The play is independent of the pages on which it is printed, and ‘pure geometries’ are independent of lecture rooms, or of any other detail of the physical world.
In these days of conflict between ancient and modern studies, there must surely be something to be said for a study which did not begin with Pythagoras, and will not end with Einstein, but is the oldest and the youngest of all.
It (proof by contradiction) is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.