I am familiar with the mechanism of proof by contradiction: we want to prove $P$, so we assume $¬P$ and prove that this is false; hence $P$ must be true.
I have the following devil's advocate question, which might seem to be more philosophy than mathematics, but I would prefer answers from a mathematician's point of view:
When we prove that $¬P$ is "false", what we are really showing is that it is inconsistent with our underlying set of axioms. Could there ever be a case were, for some $P$ and some set of axioms, $P$ and $¬P$ are both inconsistent with those axioms (or both consistent, for that matter)?