A Mathematician’s Apology
by G. H. Hardy
An excellent extract from the aforementioned:
The proof is by reductio ad absurdum, and 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 gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.
What Hardy is talking about is a method of proving mathematical statements using a proof by contradiction. First you suppose something is true; after working through some logical instances, you conclude that what you have assumed cannot be the case – you run into a dead end, meaning that you could not have taken the turning in the first place.
He then goes on to say:
…both the statements and the proofs [infinite number of primes, irrationality of root 2] can be mastered in an hour by an intelligent reader, however slender his mathematical equipment.
Long live the eccentrics.