One extra element I'd add is that what counts as an error in a proof is different than what counts as a bug. In pure math, an error is a strict failure of logic

One extra element I'd add is that what counts as an error in a proof is different than what counts as a bug. In pure math, an error is a strict failure of logic that breaks the proof. In programming, a bug doesn't need to only refer to something that stops the code from working. I've also heard it referred to as something that stops the code from working properly or efficiently, for instance. Mathematicians don't care about that because they have a different goal than a programmer, that is, to prove the truth of something (though some are motivated to produce more "elegant" proofs).