[cyphar]

> a wrong mathematical statement can still yield useful results in practice

Yes and no. For some basic conjectures this might be the case, but one of the greatest utilities of certain theorems is how they can be applied to completely new fields. If a theorem used in this manner is not actually true, then the proof is invalid -- and it might be the case that (several proofs down the line) the conjecture you've ended up proving is almost entirely invalid.

There's also the slippery slope of "Principle of Explosion"-type problems with accepting "mostly right" proofs -- it means you will be able to prove any given statement (even ones which are outright false).