TITLE: Extent of “unscientific”, and of wrong, papers in research mathematics
QUESTION [98 upvotes]: This question is cross-posted from academia.stackexchange.com where it got closed with the advice of posting it on MO. 

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) where 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 He also mentions papers relying on unpublished results taken on trust that those who announced them indeed have a proof.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?

The answer I accepted on academia.stackexchange before the closure gives a useful general purpose method, but I'd really appreciate more detailed area-specific answers. For example, what fraction of your own papers do you expect to rely on a statement "for which humanity does not actually have a complete proof" ?

REPLY [44 votes]: As Kevin Buzzard himself admits in his answer, he somewhat exaggerated his point for effect. 
However, I'd submit that if you were unsettled by his talk, then that's a good thing.  I don't think that the proper reaction is to look for reassurance that mathematics really is fine, or that the problems are restricted to some easily quarantined corner.
Rather, I think the proper reaction is to strive for a more accurate view of the true state of the mathematical literature, and refuse to settle for comforting myths that aren't based on reality.  Some of the literature is rock-solid and can stand on its own, much more of it is rock-solid provided you have access to the relevant experts, and some of it is gappy but we don't really care.  On the other hand, some small percentage of it is gappy or wrong and we do care, but social norms within the mathematical community have caused us to downplay the problems.  This last category is important.  It is a small percentage, but from a scholarly point of view it is a serious problem, and we should all be aware of it and willing to acknowledge it.  If, every time someone brings it up, we try to get them to shut up by repeating some "propaganda" that makes us feel good about mathematics, then we are not solving the problem but perpetuating it.
Some related concerns were raised over 25 years ago by Jaffe and Quinn in their article on Theoretical Mathematics.  This generated considerable discussion at the time.  Let me quote the first paragraph of Atiyah's response.

I find  myself  agreeing with  much  of  the  detail  of  the  Jaffe–Quinn  argument, especially the importance of distinguishing between results based on rigorous proofs and  those  which  have  a  heuristic  basis.   Overall,  however,  I  rebel  against  their general tone and attitude which appears too authoritarian.

My takeaway from this is that Jaffe and Quinn made many valid points, but because this is a sensitive subject, dealing with societal norms, we have to be very careful how we approach it.  Given the way that the mathematical community currently works, saying that someone's work has gaps and/or mistakes is often taken to be a personal insult. I think that if, as a community, we were more honest about the fact that proofs are not simply right or wrong, complete or incomplete, but that there is a continuum between the two extremes, then we might be able to patch problems that arise more efficiently, because we wouldn't have to walk on eggshells.
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