[butokai]

This document misses the point in a way that very commonly arises when mathematicians (as opposed to logicians) discuss proof by contradiction. The examples in this document all revolve around assuming a fact, showing that it would lead to an absurd, and thus establishing that that fact can’t be the case: there is no rational equal to sqrt(2), there is no finite listing of all the primes. They are not using proof by contradiction at all, and on the contrary these proof are fully constructive: if one where to give us what they believe is a finite list of all the primes, the proofs gives us a method to construct a new prime.

Proof by contradiction, on the other side, deems that we derive a contradiction from the assumption that a statement does not hold. Then, by contradiction, we may state that is true because it is impossible for it to be false.

This is why it is rejected by the intuitionists and constructivists: there is no way to extract an explicit procedure from such a proof, since it only states that what can’t be false must me true.