I recently learned about the philosophy of constructive mathematics. In several discussions of the topic, I keep seeing a quote from G. H. Hardy's book A Mathematician's Apology;
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 a piece, but a mathematician offers the game.
I understand what Reductio ad absurdum is, but I don't understand what this quote is saying. Can someone explain how a mathematician 'offers the game' by using this argumentative technique? Perhaps give a concrete example of this being used in a proof somewhere?
Thank you,
This is mostly poetic. You can imagine a proof as a game against an adversary who is trying to argue the opposite. A reductio ad absurdum proof starts by assuming your adversary is correct (i.e. granting them victory) and arguing from those premises.