Well, when there's a mistake found in a proof, the mistake needs to be rectified before the proof can be accepted. There have certainly been proofs that X is t

Well, when there's a mistake found in a proof, the mistake needs to be rectified before the proof can be accepted. There have certainly been proofs that X is true, where X has turned out not to be true based on the fact that there's a mistake in the logic. While it's possible a mistake won't be found, it's a much different problem than running empirical experiments and getting different results.