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Comment by modeless | Hacker News Reader

> it was absolutely clear to both me and Antoine that the proofs of the main results were of course going to be fixable, even if an intermediate lemma was false, because crystalline cohomology has been used so much since the 1970s that if there were a problem with it, it would have come to light a long time ago.

I've always wondered if this intuition was really true. Would it really be so impossible for a whole branch of mathematics to be developed based on a flawed proof and turn out to be simply false?

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As mentioned by another comment, this is a big reason that Vladimir Voevodsky started his Homotopy Type Theory and Univalent Foundations program. He had see first hand a field collapse by a mistake in “first lemma on the first page” of a foundational paper. Arguably, he initial work on UniMath and the special year at IAS ending up in the HoTT book, pushed the whole formalization of mathematics topic forward to where it is today.

People hunt for counter examples to proofs they are working on. If the basis of their work is wrong, it's quite possible for one of the counter examples to also disprove the base theorem. So building on a faulty foundation is likely to reveal faults in the foundation.

Similarly, once in a while math gets applied and is used to make predictions. When the math is wrong, those predictions are wrong. And those wrong predictions draw a lot of attention.

I think it depends how widely used that branch of maths becomes. In fact, I'd say that the word "branch" is a bit misleading - for many theories it's much more of a "knot", with influences tying them to many many other theories across the mathematical landscape. And those theories are themselves tied to others, etcetera.

It would be a very strange situation if the foundation fell apart logically without any ramifications in the rest of this "knot". A huge swathe of free-floating mathematics that's completely internally consistent but for this one error? Difficult to imagine for me in the case of cohomology from the article.

I guess strictly speaking this is more of a philosophical stance - I like to believe that a lot of the current mathematics has been discovered "naturally" in some sense =)

This has happened before, see, e.g. the biography of Vladimir Voevodsky. Spoiler: the world kept spinning.

Russell’s paradox did the same thing. They had to go back to the drawing board.

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