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.

  • 2
    $\begingroup$ 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. $\endgroup$
    – Wojowu
    May 11, 2019 at 18:19
  • 5
    $\begingroup$ 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. $\endgroup$
    – Wojowu
    May 11, 2019 at 18:24
  • 7
    $\begingroup$ 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.. $\endgroup$ May 11, 2019 at 18:51
  • 3
    $\begingroup$ 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 $\endgroup$ May 11, 2019 at 19:30
  • 4
    $\begingroup$ @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. $\endgroup$ May 12, 2019 at 6:11


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