Proof by contradiction follows from the law of excluded middle ((p or not-p) is universally true), the characteristic law of classical logic. The basic reasoning of proof by contradiction is: in order to prove p, we prove a contradiction from ~p (not-p).
“G. H. Hardy described proof by contradiction as “one of a mathematician’s finest weapons”, saying “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.” ” Wikipedia on proof by contradiction
Interestingly the United States has become known for using exactly this type of logic ever since the nuclear era began. Of course we have the Cuban missile crisis where Kennedy offered ‘the game’ in this case ‘the destruction of the world’ as a possibility to Khrushchev.
Destruction by climate change has increasingly entered into our calculations, so that during the 2016 elections we were faced with a choice between the republican party (who are blocking the Paris Agreement from really taking off) and Hillary Clinton. And the result seems to be that life in America is so bad that they are willing to try the alternative.
And logic can help, not classical logic, but constructive logic also rejects the Law of Excluded Middle and proof by contradiction. Changing our minds (our reasoning) is a very important ingredient in changing our politics. Constructive logic takes the perspective of building things as more important, more fundamental, than universal or global laws. Because of this, the “world” of a constructive reasoner is only this or that construction, and the progressive stages of continuing construction. In this sense the geography of construction is “incomplete.”
This incompleteness is fundamental and draws the rest of mathematics into reformulation (Bishop 1967). Things like space and time need a new kind of real numbers to describe. If our mathematics were different, the upgrading of nuclear weapons and their resultant destructive power by a factor of 3 would not be considered “an incredible feat of human intelligence” (Noam Chomsky) because we would have a different understanding of intelligence (reasoning power), and empty-headed “intelligent” weapons developers, along with coldly calculating CIA heads would be following different rules. Offering the game would no longer be considered “one of our finest weapons”.