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Consider a statement without proof in a paper, with the following assumptions:

  • it is unknown,
  • it is unused in the paper,
  • it is not written as a theorem (or proposition, or lemma…), but just a free sentence in a paragraph,
  • the proof exists somewhere but is unpublished (say just handwritten).

The author just wants to inform about something new that might interest the reader, but whose proof would not fit in the paper. The proof may appear in a future paper, or not.

The statement cannot be considered as known without a published proof, but it also cannot be considered as open.

Question: How does the mathematical community tolerate this kind of intermediate status for a statement?

Relaxing the third assumption, one famous example in my subject is the statement 4.5 in Subfactors and classification in von Neumann algebras by Sorin Popa, about subfactor indices gap (after more than 30 years, the proof is still unpublished).

LSpice
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Sebastien Palcoux
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    Reminds me on Fermat's Last Theorem... – Karl Fabian Aug 25 '22 at 09:09
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    Commonly, authors signal using words like "We note in passing that ..." or "As a sidenote, let us observe that ..." that the statement is not load-bearing. – darij grinberg Aug 25 '22 at 09:13
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    "It can be shown that ..." – J.J. Green Aug 25 '22 at 09:40
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    If a proof is unpublished (not available), one can never be certain that the proof "exists" and that the statement is correct. So it has to be considered a conjecture. – Alexandre Eremenko Aug 25 '22 at 12:29
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    This reminds me of the exchanges between Hardy and Ramanujan (I guess everybody knows the popular book https://books.google.ca/books?id=m08ADAAAQBAJ&pg=PA92). Just like the laws of physics varies in nature from microscales to macroscales, it shouldn't be surprising that the traditions of academia applies differently to superstars & genius. Please do not interpret this as an argument against proving what one claims to be true. In any case, everyone else must abide by the requirements of the discipline and provide a proof for every claim in her/his publication, especially amateurs (like myself). – Onur Oktay Aug 25 '22 at 14:48
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    @AlexandreEremenko Not really. The proof may be easily available upon request to anyone who would bother to ask for it and to read it when received. You know at least one such example: the Bernstein L^1 inequality for non-negative polynomials :lol:. Would you argue that that one is "a conjecture"? – fedja Aug 25 '22 at 14:58
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    @fedja: "The proof may be easily available upon request to anyone who would bother to ask for it " Assuming that the author is still active, or at least alive, when you want to ask... – Jochen Glueck Aug 25 '22 at 15:16
  • @JochenGlueck That is a sufficient condition, but not a necessary one. Anyone who has a copy can share it too. But, indeed, if all proof holders die, the proof gets lost. – fedja Aug 25 '22 at 15:19
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    @fedja: The existence of living people who have a copy of the proof is, in turn, not a sufficient condition: I might have no way to find out that they exist and, if they exist, who they are. – Jochen Glueck Aug 25 '22 at 17:17
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    What is the question? If it is actually "how does the mathematical community tolerate this?", then the answer is perhaps "with eyerolls and frustration". Do you mean to ask: "why does the community tolerate this?" Or perhaps "how should the mathematical community respond?" –  Aug 25 '22 at 18:00
  • @JochenGlueck Agreed. If the initial communication chain by which the statement reached the final recipient is broken by the moment of inquiry, sending the request in the backward direction may not help much. That's why if I hear something of that kind interesting enough to communicate to other people, I always request the full details and try to go over them :-) – fedja Aug 25 '22 at 19:35
  • Isn't this exactly the state of every experimental/computational result? Many conjectures are supported by offhand remarks about (a lack of) small counter-examples, or a motivating example without most of the computation – Artimis Fowl Aug 26 '22 at 02:17

4 Answers4

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Speaking for myself, I place such a statement in one of two states: I have use of this statement now, or I will file that statement away for future reconsideration. In the latter, it is perpetually "back-burnered" until I actually need it and will do a forward search on the paper (via MathSciNet or Zentralblatt) to see if the result has since been published.

In the case where I actually do have use of the result, I go ahead and see if I can prove it myself. When I can, and need it in a paper I am publishing, I reference the situation by saying something like "In paper X, author Y asserts the following to be true. For completeness, we include our proof here." Often I reach out to the original author with this info, and have never had a problem, usually getting a "you have my permission to publish." (On one occasion, that "reaching out" resulted in mutual work, streamlining what I had done.)

On the one occasion I had where I was not able to gin-up a proof, the author sent me his proof with permission to include it in my paper, so long as I attributed his work to him. No problems.

Therefore, the direct answer to the question is: I either don't care enough about it to do anything, or I do care and work to get a proof published.

John McVey
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For this answer I assume that the proof of the result is non-trivial and "novel" in the sense that there is more to it than just repeating a proof of a very similar statement somewhere else.

While I don't have statistical data available, I know several colleagues who are annoyed by this practice. From my perspective the bottom line is this:

Why would your readers be interested in learning that something is true, but not in learning why it is true?

[Note: The first version of the answer, to which the first comments below refer, ended here. I later extended the answer with the following content in order to better explain my point.]

Of course, you can send the proof to people who ask you for it, but this has a number of disadvantages:

  • You make a permanently available claim, but do not make sure that the proof is also permanently available. (As an author you will die at some point, or become unavailable earlier than that, and people who would like to see the proof after this point might have no way to know who has a copy of your proof.)

  • Announcing results and distributing proofs only in private considerably contributes to the generation of "folklore knowledge" in a field, i.e., results which are known to some experts in the field but are not properly documented anywhere. In other words, knowledge about this result will only be circulated either in private documents or orally. Such folklore knowledge tends to make it very difficult for non-experts (and people who yet wish to become experts) in the field to access what is actually known.

    Note that the argument that "one can simply write the author and ask for the proof" is a bit besides the point here, besides it only considers one single proof of one single result. The problem is rather that announcing results without proofs contributes to establishing an entire "folklore culture", which then creates serious accessibility problems for non-experts.

  • The assumption in the question that "the proof exists somewhere (maybe handwritten)" is rather vague. One might assume for the sake of a question that the quality in which this proof has been written and checked by the author is similar to what one would expect from a published proof. But such an assumption is not particularly useful, since experience shows that it is unrealistic.

    So, realistically, even if there is some proof in some drawer, it will probably be less polished, and thus leave it less clear to non-experts whether there are any gaps — thus again contributing to the aforementioned folklore culture with very limited accessibility for non-experts.

  • Finally, there is also an ethical issue, especially towards younger researchers: When you annouce a result in a published paper, some people will interpret this in the sense that you essentially claim priority for it. Now if you never publish a proof, but only distribute some handwritten notes, it might happen (see above) that the situation becomes quite unclear to the community concerning what is actually proved and what is not (and what is difficult about it and what is not).

    But at the same time, it's nearly impossible for other mathematicians who might have taken it upon themselves to establish a complete and clearly written proof of the result to actually publish this, since editors and referees are likely to just say "Well, but this is already known, although it's not written down anywhere." So you essentially took the credit for something without providing all the necessary explanations, but at the same time prevented others from providing those explanations, since they would be unlikely to get any credit for it.

Disclaimer. Those things said, I have to plead guilty that I have also, on occasions, announced results in papers without provding a proof. As you can infer from above, I now consider this a mistake.

Jochen Glueck
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    I can imagine this perspective as a reader, but, with my referee's hat on, if I were refereeing an enormous paper that included an unnecessary proof of a result as tangential as the poster is positing, then I would definitely suggest removing the proof. – LSpice Aug 25 '22 at 14:03
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    One reason for this is that "not all readers have the same opinion on what is relevant." Hence, this then provides a forum for determining if the result is worth publishing. Said another way, your question assumes homogeneity among the readers. – John McVey Aug 25 '22 at 14:32
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    As for another example, my advisor published a paper classifying groups that have Property X as being among 6 examples. He then made a statement along the lines of "in the third case, we can actually show something stronger, that indeed property Y holds." It wasn't needed for the "if and only if" proof, so was extraneous to the paper at hand. It was very extraneous to the particular task at hand, and only some of the audience would care about the additional info. But there were some who were interested. But not ALL were interested. – John McVey Aug 25 '22 at 14:34
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    In my previous comment, I forgot to mention... Adding the sentence didn't even add 2 lines to the paper. Including the proof would've added at least 2 pages. – John McVey Aug 25 '22 at 14:47
  • @JohnMcVey: Thanks for your comments! "But there were some who were interested." And those people were not interested to learn about the proof, but only about the statement of the result? – Jochen Glueck Aug 25 '22 at 14:58
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    As I recall, interested people asked him offline for the general shape of the proof. This resulted in some back-and-forths that eventually turned into stronger results and co-authorship of subsequent research that included the result as a special case of something more general. – John McVey Aug 25 '22 at 15:02
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    Had the proof been included originally, it is not clear the subsequent exchanges would have occurred. Indeed, it's not clear the first paper would've been published, akin to LSpice's comment at the start of this thread. – John McVey Aug 25 '22 at 15:06
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    @LSpice: Could you elaborate on your reasons to suggest the removal of the proof when you are a referee? I'm also not sure why you distinguish so much between the readers' and the referee's perspective: Whenver I referee a paper my major concern, besides from the suitability for the journal, is the question whether the paper will probably be useful to its readers. – Jochen Glueck Aug 25 '22 at 17:50
  • @JohnMcVey: Yes, I understand that this might sometimes result in a fruitful exchange of ideas. But given that there are numerous ways to establish such fruitful exchange, I don't see how this is a good justification of a practice which has a number of negative consequences. (Please see the extended version of my answer for a description of these negative consequences.) – Jochen Glueck Aug 25 '22 at 17:54
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    @JochenGlueck, as a referee, I know that it is not my job to certify the correctness of the paper, but still I would like at least to believe in the correctness before I recommend publication. Papers in my field can run very long—80 pages is a bit long, but not uncommon—and, at that length, just keeping the whole structure in my head is so exhausting that anything that I can remove helps me be more sure that I've understood the resulting structure. – LSpice Aug 25 '22 at 18:19
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    "When you announce a result, you essentially claim priority for it". No, no, and once more an emphatic no. Those two actions are completely unrelated, IMHO. The priority is being claimed by a formal publication without a reference and only by that. Otherwise everybody is free to communicate their observations and ideas privately or publicly without creating any formal rights or obligations. – fedja Aug 25 '22 at 19:49
  • Would you be willing to mention in the body of your proof that it was considerably expanded? My initial comment refers to "this perspective", but only the one in your revision 1; I did not mean to respond to your revision 2. – LSpice Aug 26 '22 at 01:10
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    @LSpice: Sure. Done. – Jochen Glueck Aug 26 '22 at 12:17
  • @fedja: Thanks for your response. I think I did not phrase this well - I intended to say: If you announce a result in a published paper, (without a proof), this will often make it difficult for other mathematicians to publish the proof, since many people will either interpret the situation in the way that you have priority for the result, or will at least find the priority situation messy and don't want to offend you by accepting a paper by someone else which publishes the proof. (In principle I agree with your opinion, but my concerns are about what will actually happen.) – Jochen Glueck Aug 26 '22 at 12:23
  • @fedja: What might be easier to resolve is the following situation: If somebody publishes a paper with a different focus, but needs your unpublished proof as an ingredient, then you can (and, I think we both agree on this, should) prevent any trouble for this person by simply allowing them to include the proof in their new paper. I think the issue I pointed out above mainly occurs when the proof is considered one of the major insights in the new paper. – Jochen Glueck Aug 26 '22 at 12:27
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    @JochenGlueck Funny, but I just did exactly that 2 days ago. Moreover, such permission is there by default as far as I'm concerned: no need to ask for it. – fedja Aug 26 '22 at 15:51
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Placing this in future work section seems as a reasonable idea. The important issue is that the topic of the proof is indeed relevant to the topic of the paper where it is being mentioned. The author should make sure to choose correct wording to emphasise that she/he was able to show the claimed result. Then, if the author manages to publish (non necessarily peer reviewed) version of the proof before the final version of the paper where it was announced a citation is welcome/necessary.

Marko Rajkovic
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I have voted to close on the grounds that the question as stated is unclear or opinion-based. However, assuming the community decides that the question should stay open, let me link to some previous discussions of this topic.

Jaffe and Quinn called this sort of thing theoretical mathematics. They spelled out some of its disadvantages, and suggested some mitigations. Their paper generated many responses.

Kevin Buzzard has expressed doubts that such statements constitute rigorous mathematics, and suggests that, by contrast, proof assistants give us rigorous mathematics. Buzzard's talks have inspired an MO question which illustrates, by example, that the community sometimes accepts such statements at face value despite the lack of proof.

Timothy Chow
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    I think that the use of "theoretical mathematics" in JQ is a little different from what the questioner is proposing. The question is not, I think, about results with proofs that might be insufficiently rigorous, but rather about results whose proofs cannot be judged at all, because the community has no immediate, direct access to them while reading the paper. A proof in the paper can be theoretical in JQ's sense, and a proof omitted from the paper can be fully rigorous; I think these concerns are nearly orthogonal. – LSpice Aug 26 '22 at 01:08
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    @LSpice I see the distinction you're making, but I think in practice there is substantial overlap. Consider a nontrivial statement that is asserted with no explicit acknowledgment that anything further needs to be said. The reader is left wondering, is the author claiming that this is a theorem? If so, is the proof supposed to be "easy and left to the reader" or "nontrivial and will be published eventually" or "nontrivial but won't be published"? This would seem to be a candidate for "intermediate status" (as the OP calls it) and it is also characteristic of theoretical mathematics. – Timothy Chow Aug 26 '22 at 01:35
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    Very interesting papers, thanks for sharing! Let me mention here that there is also a response to the responses: https://doi.org/10.1090/S0273-0979-1994-00506-3 – Sebastien Palcoux Aug 26 '22 at 07:51