— I spent much of 2019 obsessed with the proof of this theorem, almost getting crazy over it. In the end, we were able to get an argument pinned down on paper, but I think nobody else has dared to look at the details of this, and so I still have some small lingering doubts.

(Have you ever heard somebody, especially somebody famous, admit that he or she wrote down a proof and lectured already several times about it, but is still not completely sure if the proof is correct? I think this happens more often in mathematics than we would like to admit. I am glad that Peter Scholze is brave enough to say this out loud.)

— I have occasionally been able to be very persuasive even with wrong arguments. ([…] I once had a full proof of the weight-monodromy conjecture that passed the judgment of some top mathematicians, but then it turned out to contain a fatal mistake.)

(This shows how often, even for top mathematicians, it happens that one overlooks serious mistakes or gaps in proofs. In this case the wrong argument was eventually found, but it shows that relying solely on the word of very few famous persons with respect to the validity of an argument might still be dangerous. I am referring here mainly to the notion of ‘elders’ and the discussion surrounding them which are mentioned several times in the blog post on MathOverflow that I linked in my previous post ‘Validity of results, II’.)

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