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I'm looking for an example of the following situation:

  • Proof of some statement that was widely accepted for a noticeable amount of time (say, at least 10 years).
  • The proof was later refuted (i.e. the proof turned out to be incorrect), and the refutation was similarly widely accepted.
  • The error in the proof was due to incorrect reasoning, not due to wrong assumptions. E.g. I'm not interested in a case when some proof relied on the laws of Newton's mechanics but is incorrect in the settings of special relativity.
  • This happened in the last, say, 150 years.

I'm also not interested in the case when some result had an error that was quickly found and fixed in the follow-up work.

Somewhat related to Is there a single example of an outsider considered a "crank" publishing a ground-breaking result that was found to be correct (in the last 30 years)?

@CrimsonDark gives a good reference to a question with some good (as far as I can tell) examples: https://math.stackexchange.com/questions/139503/in-the-history-of-mathematics-has-there-ever-been-a-mistake

Dmitry
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    I suggest to ask this question on another SE, for example, History of Science & Mathematics, Maths Over Flow. – Neuchâtel Dec 20 '22 at 16:45
  • @Pikachu피카츄, right when I was going to delete the question, it was answered. What should I do? – Dmitry Dec 20 '22 at 16:53
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    Have a look at https://math.stackexchange.com/questions/139503/in-the-history-of-mathematics-has-there-ever-been-a-mistake – CrimsonDark Dec 20 '22 at 16:53
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    @Dmitry Then just leave it here. – Neuchâtel Dec 20 '22 at 16:54
  • Probably if you publish a correct ground breaking result, you aren't, by definition, a crank, though I suppose it is possible to be so considered up to then. I think that the physics establishment of the day considered Einstein to be a crank, even after the publication of special relativity. They couldn't let go of the aether, and it took a generation to overcome that. – Buffy Dec 20 '22 at 17:21
  • The heliocentric theory of Copernicus was still being condemned in Europe after about a century. Galileo suffered for it. Neither were "cranks" though the church considered them so. (Not within the last 30 years, of course, back in my youth.) – Buffy Dec 20 '22 at 17:29
  • There is a quote (approx) from Einstein: "If the theory of relativity is proven correct, the Germans will call me German, the Swiss will call me Swiss, and the French will call me a great scientist. If it is proven false, the French will call me Swiss, the Swiss will call me German, and the Germans will call me a Jew." – Buffy Dec 20 '22 at 17:34
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    Looks like a duplicate of https://mathoverflow.net/q/35468/78525 – Dan Romik Dec 21 '22 at 00:22

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I don't have an example at hand, so this isn't technically an answer, but yes it happens in math. I once worked in a field (classical analysis) that had the reputation of all proofs being wrong, but the results still accepted. Lots of proofs in analysis are of the ε, δ variety as is, say, the definition of the derivative. The problem is that such proofs can be layered with several levels. Once you go beyond about three levels the proofs are devilishly difficult to keep straight in your mind and you can get it wrong. (See the magic number seven, plus or minus two for a possible underlying reason.)

This has been done by celebrated mathematicians. A vaguely remembered example was from around the 1920's, corrected maybe by the 70's.

Buffy
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  • Thank you for the answer. "reputation of all proofs being wrong, but the results still accepted" sounds very weird to me (and, in particular, probably means that "accepted" means something different), but I guess it makes more sense to people in the field. It would be great if you could remember the example. – Dmitry Dec 20 '22 at 17:01
  • I think that the reason was the people believed that the errors were "merely" technical and "obviously" overcome-able. In many cases that was true, but who can say. The original IIRC was (perhaps) published in Analytic Functions by Zygmund and Saks. Too long ago to remember, but I was the one that found the error. Both of those folks were giants. – Buffy Dec 20 '22 at 17:11
  • Note that the theorem was older than the book and (faulty memory here) was by another person. – Buffy Dec 20 '22 at 17:16