Kevin Buzzard's slides (PDF version) at a recent conference have really unsettled me.
In it, he mentions several examples in what one would imagine as very rigorous areas (e.g., algebraic geometry) were the top journals like Annals and Inventiones have published and never retracted papers which are now known to be wrong. He also mentions papers relying on unpublished results taken on trust that those who announced them indeed have a proof.
He writes about his own work:
[...] maybe some of my work in the p-adic Langlands philosophy relies on stuff that is wrong. Or maybe, perhaps less drastically, on stuff which is actually correct, but for which humanity does not actually have a complete proof. If our research is not reproducible, is it science? If my work in pure mathematics is neither useful nor 100 percent guaranteed to be correct, it is surely a waste of time.
He says that as a result, he switched to formalizing proofs completely, with e.g. Lean, which guarantees correctness, and thus reusability forever.
Just how widespread is the issue? Are most areas safe, or contaminated? For example, is there some way to track the not-retracted-but-wrong papers?