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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  • the proof exists somewhere but is unpublished (say just handwritten).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in von Neumann algebras by Sorin Popa, about subfactor indices gap (after more than 30 years, the proof is still unpublished).

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    $\begingroup$ Reminds me on Fermat's Last Theorem... $\endgroup$ Aug 25 at 9:09
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    $\begingroup$ 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. $\endgroup$ Aug 25 at 9:13
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    $\begingroup$ "It can be shown that ..." $\endgroup$
    – J.J. Green
    Aug 25 at 9:40
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    $\begingroup$ If a proof is unpublished (not available), one can never be certain that the proof "exists" and that the statement is correct.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. $\endgroup$ Aug 25 at 12:29
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    $\begingroup$ 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?" $\endgroup$
    – Matt F.
    Aug 25 at 18:00

4 Answers 4

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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.

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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.

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    $\begingroup$ 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. $\endgroup$
    – LSpice
    Aug 25 at 14:03
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    $\begingroup$ 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. $\endgroup$
    – John McVey
    Aug 25 at 14:32
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    $\begingroup$ 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. $\endgroup$
    – John McVey
    Aug 25 at 14:34
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    $\begingroup$ 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. $\endgroup$
    – John McVey
    Aug 25 at 14:47
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    $\begingroup$ 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. $\endgroup$
    – John McVey
    Aug 25 at 15:06
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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.

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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.

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    $\begingroup$ 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. $\endgroup$
    – LSpice
    Aug 26 at 1:08
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    $\begingroup$ @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. $\endgroup$ Aug 26 at 1:35
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    $\begingroup$ Very interesting papers, thanks for sharing! Let me mention here that there is also a response to the responses: doi.org/10.1090/S0273-0979-1994-00506-3 $\endgroup$ Aug 26 at 7:51

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