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When working on a research project, one tries to spend their time answering questions that have not yet been answered. There enters the terminology of "known" versus "unknown" results, which we generally take to mean whether a problem has already been solved. On the other hand, we know that mathematics is always a work in progress, including instances of "known" facts that have turned out to be wrong.

The proofs of some results are quite esoteric, requiring extreme specialization in the topic to be able to understand. It is feasible that a paper might be peer reviewed, accepted by the community, and its theorems entered into mathematical canon, only for everyone capable of following the arguments to then pass away leaving no apt descendants to maintain the knowledge. My question is whether those results are still considered "known." The deeper question is about the value of finding new and more accessible proofs for such results, such that they may be more widely known in the literal sense.

To make the question less subjective, let's focus on the etiquette of using this terminology. For a mathematician to publicly proclaim that something is "known," does it require them to have read and understood the proof, to know of someone who has read and understood the proof, and if the latter, must that person be alive? On the other hand, does "known" merely mean that a proof has been published in a peer-reviewed journal at some time in history, no matter how long ago?

Post-closing comment

Just as food for thought, notice how closing this question as opinion based is to say that there is no precise universal definition for the term "known," or at least, that there are some ambiguities to it. I find this interesting.

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j0equ1nn
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    To me, "known" refers to the result being known (by statement) by people in the field, not that anyone knows the proof. If it has been proven by someone, I pretty much don't care whether other people know the proof. An example is the classification of finite simple groups. It is a known example (I think everyone working in group theory has heard of it), but I believe there is no single person who understands the full proof. – Wojowu May 11 '19 at 18:19
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    @Wojowu The question does not ask about how many people must know the result, but rather whether anyone alive must know the result. – j0equ1nn May 11 '19 at 18:21
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    You specifically ask about whether anyone alive must know the proof, and from my comment you can infer that in my opinion the answer is "no". Regarding whether anyone has to know about the result itself, then I would say yes, but this kind of follows trivially, because the person stating the result (and claiming it's known) knows the result. – Wojowu May 11 '19 at 18:24
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    Related: https://mathoverflow.net/questions/176425/ (Rediscovery of lost mathematics) – YCor May 11 '19 at 18:29
  • @Wojowu Right, I should have said "proof" rather than "result" in my prior comment. I find it interesting that you would say a result being known does not require "that anyone knows the proof." Surely someone must have known the proof at some point, or it wouldn't be a "result." – j0equ1nn May 11 '19 at 18:31
  • I repeat my example of CFSG. I don't think there has been or there is a person out there who knows the proof in its entirety. Yet I think few would argue that this is not a known result. – Wojowu May 11 '19 at 18:33
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    "does «known» merely mean that a proof has been published in a peer-reviewed journal at some time in history": this is a sufficient, but not necessary condition. It might be spread by other means (e.g., a paper left on arxiv, etc). Also there are results for which the tricky part is to observe it/ introduce a clever definition, while the proof is an immediate exercise (e.g., that the free Jonsson-Tarski algebra on 1 and 2 generators are isomorphic, just to convey my latest fad). – YCor May 11 '19 at 18:35
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    @Wojowu Hmm, yeah that is an interesting example. I see what you mean. I guess I was also reacting to the dismissive nature of the comment. "Known" is a term we use a lot but I don't think we always think about what it means, and its meaning is entirely unclear to younger people, or newer students. I find it a worthwhile topic for pedagogical purposes. – j0equ1nn May 11 '19 at 18:41
  • @YCor Thanks for that link to the other post. That's a better way to delve into the topic. I might just delete this actually... – j0equ1nn May 11 '19 at 18:50
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    If a published proof is generally accepted, but there are a few people alive who are aware of slight gap ( let's say, which they know to be fixable for he sake of argument), but do not disseminate that fact, then I suppose the status of the result becomes murky when those few people die.. – Geoff Robinson May 11 '19 at 18:51
  • I don't think it's a duplicate of the linked question. However, the link provides some concrete substance to think about your question. – YCor May 11 '19 at 19:07
  • @YCor Okay, I'll leave this up for a while and see what happens. I am now fearing some silly arguments resulting from it though. – j0equ1nn May 11 '19 at 19:09
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    referring to the question in title I would say such a result would be "not known and not new". But if nobody knows that once it was known, I would say it is almost new, say good quality second hand – Pietro Majer May 11 '19 at 19:30
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    On my opinion, "known" means "has been published" (no matter how long ago) and still available. The existence of a live person who knows it is unnecessary. – Alexandre Eremenko May 12 '19 at 00:06
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    @Wojowu There is something of substance to be said, and potentially concerned about, that the number of people who understand meaningful swathes of the classification is diminishing with time; and the ability to build a group of mathematicians collectively understanding everything would also seem to be diminishing. It increasingly becomes a black box, the validity of which becomes little different from a piece of religious dogma. See Geoff's comment, for example: if there is a known or yet undetected gap within the proof, this might disappear from sight indefinitely. – zibadawa timmy May 12 '19 at 06:11
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    There are several good recent examples of maths that was “almost lost” and then revived: for years very few people understood the proof of Hilbert’s 5th problem, but recent contemporary proofs have reestablished it as “known”. Likewise, Friedman’s proof the 4-dimensional Poincaré conjecture was fully understood by increasingly few mathematicians, but has been reestablished by a forthcoming book of Behrens et al. – HJRW Mar 18 '21 at 09:05
  • @HJRW Interesting! I came upon something this week related to the Friedman result you mentioned, but for unrelated reasons. It's something I'm hoping to read once I have time. Can you provide more info on the forthcoming book? I wasn't able to find anything doing a naive search. – j0equ1nn Mar 19 '21 at 19:31
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    @j0equ1nn: the book is called the The 4-dimensional disk embedding theorem. At the moment, the only evidence on the internet that google can find seems to be on Behrens' web page: https://www.math.uni-bielefeld.de/~sbehrens/ . However, a comprehensive manuscript was circulared in late 2020, so it certainly exists, and hopefully should be ready for publication soon! (By the way, apologies, I got the spelling gone: it's Freedman.) – HJRW Mar 20 '21 at 10:52
  • @HJRW Thanks! I'll keep a lookout for that. I should've caught the typo in Freedman myself. (I came upon it by sheer coincidence out of curiosity about Frank Quinn's writings on math education from around 2011. So many common interests there, I need to send an email to the folks involved once I get a chance to sit down with the stuff.) Thanks again! – j0equ1nn Mar 20 '21 at 23:14

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