26

I've recently obtained some results which I have not found in the literature explicitly. Namely they are hinted in certain published literature, but very shortly and not in a very constructive sense.

After consulting with several people with experience in academia, I was told that it is frowned upon to publish these results more explicitly. I therefore decided to not write these results out properly.

Following this experience I was wondering whether there is a requirement when working on new projects to verify whether certain results are 'folklore'? Should one consult with an expert in the field before trying to publish results? Is it frowned upon to write something out explicitly, while trying to extensively discuss the result, and upload it to the Arxiv?

I should mention that some of my obtained results seemed not known by people in the field in my working group. I also wished I could refer to some of these properties in related topics.

1
  • Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Academia Meta, or in Academia Chat. Comments continuing discussion may be removed.
    – Bryan Krause
    Commented Jan 7 at 20:31

7 Answers 7

34

I don't think this is a problem per se, but you need to have a good reason for doing so. For example, if your paper uses folklore result X in some interesting way, then it is perfectly sensible to say "although X is well known, we were unable to find a proof in the literature and provide one for completeness".

However, writing a paper that just says "we provide a proof of the folklore result X" is a waste of time: if it's a folklore result, that means other people know how to prove it but don't think it's worth publishing on its own.

7
  • 72
    "if it's a folklore result, that means other people know how to prove it but don't think it's worth publishing on its own." ─ This reminds me of the joke about two economists who walk past a $20 bill on the ground, one saying to the other: it must be fake, or somebody would have picked it up by now.
    – kaya3
    Commented Jan 4 at 23:49
  • 20
    "that means other people know how to prove it" You forgot to add a quantifier some before the adjective "other". Sometimes that "some" is just one and occasionally that one is dead already or, in more optimistic scenario, does not remember the proof. So I would say "explain in your papers everything that you cannot quickly find a good and accessible reference for". Wasting your time is irrelevant. Spare the time of the readers, or don't communicate at all.
    – fedja
    Commented Jan 5 at 12:18
  • 2
    @fedja If it's just one person then it's not "folklore", it's "an unpublished result of X". Commented Jan 5 at 14:38
  • 6
    He-he. That is only for the first few years. Then, if it really starts circulating around, almost nobody knows about $X$ any more than you know the exact source of any particular rumor on social media. All you can say is "I got it from Agnes" (if you remember that old Tom Lehrer song).
    – fedja
    Commented Jan 5 at 14:43
  • 4
    I'd argue it's important to call out any folklore relied on in this way, as it doesn't mean anyone knows how to prove it: just that they think someone once did. How many hundred thousand people died of covid because of the folklore belief among epidemiologists that particle sizes larger than 5 microns would fall to earth rapidly and not be contagious except through contact? (see: medium.com/history-of-women/…). Commented Jan 6 at 5:23
34

I gently disagree with some of the views expressed in this thread. If a result is "folklore", that already goes to show that it is natural and relevant, others wouldn't have thought about the question before. Moreover, not everyone roams the common rooms of the seven departments where this folklore knowledge is passed on from person to person, so there is genuine value in writing such knowledge down. It also makes it available as a reference - and if this is a question it comes up often, then it is useful to have the result advertised in its own paper, rather than being hidden in a paper that uses the result along the way. And finally, only when this folklore "result" and its proof are written down do we, the mathematical community, get the chance to scrutinise it. Did a crucial assumption get omitted? Does the proof method apply more broadly? Isn't there an easier argument that also proves [well-known lemma with an intricate proof in the literature]? So in my view, there can be great value in writing down folklore "knowledge".

However, the above is just my personal opinion, which is not a consensus in the community, and maybe not even a majority view. There is certainly a good chance that a paper proving such a folklore result will get lukewarm referee reports at best. It is unlikely to get published in a very selective journal (and these days, more journals than you might think would consider themselves to be very selective). As a matter of career advice, I would nudge a junior mathematician to spend their time on work seen as original and innovative. Do you think you have a clean proof of a result that you and others may use later and that will only take you a short time to write up in detail? Go for it. Is it tangential to your work, and it'd take you months to finish the paper? Maybe focus on your main project instead.

3
  • We have additional information here that experienced people in academia who are aware of the specific results have suggested it's not worth it for this specific example; so, it seems they already got some outside advice to focus on something else.
    – Bryan Krause
    Commented Jan 5 at 13:17
  • 5
    Well, I don't understand how publishing these results could ever be "frowned upon" (unless the author learned much of the argument from others). So I probably disagree with some of this advice. We also learn that the original poster wants to use them, and that some people in his working group are unaware of them. I agree we lack context, but I doubt I would conclude that it would be improper to publish these results if I had all context. Commented Jan 5 at 22:49
  • I'd go further and say that such "folklore" unpublished results should be frowned upon. Let's see the proof and talk about it. Sometimes mistakes survive for a long time since nobody cares to re-do the work. If a result is of enough importance to be cited, and has not been published, then either it's a wrong result, or it should be out there damnit :) Commented Jan 7 at 19:19
24

What is "folklore"?

First, because there appears to be some confusion in the comments, I think it is important to give some context. In contrast to other areas in academia, mathematical results can be proven to be true. Mathematics is not particularly empirical—we don't experiment or interview research subjects. A well-formed statement in mathematics is either true, or it is not (whether or not it can be proved one way or another question entirely... thanks, Gödel).

Published results in mathematics typically consist of one or more mathematical statements ("theorems"), along with whatever argumentation is required in order to prove that those statements are true (within whatever system of axioms and logical reasoning has been adopted for that publication). There are no "negative results", nor ethnographies, nor partial experimental results.

In this context, the culture of mathematics values novel results (i.e. statements which have never been proved before) or, barring that, novel methods (i.e. lines of argumentation which have never been used before). Proving an incremental result with standard techniques is typically not considered interesting, and will typically not merit publication.

In this setting, the term "folklore" has a fairly specific (if somewhat ambiguous or non-rigorous) meaning. The "folklore" of mathematics consists of results which are believed to be true, but which no one has bothered to fully flesh out or publish. Generally speaking, these are results that experts agree are true, and it is very possible that some of these experts have taken the time to walk through a proof, but haven't published, either because they seem too trivial or too tedious to bother with.

"Folklore" is not necessarily "common knowledge", nor is it necessarily "low-hanging fruit", though it can have connotations of either (or both). More typically, "folklore" consists of results which are assumed to be true, but which aren't particularly novel and/or appear to be amenable to standard techniques—these things typically don't rise to the level of "interesting enough to publish". These are results which are communicated primarily through word-of-mouth (e.g. from advisor to student, or within a particular working group), but which aren't published (for whatever reason).

For reference, there is a pretty good discussion of mathematical folklore on nLab (nLab is a kind of mathematical "wiki"—it is a collectively edited collection of articles which is meant to serve as a kind of "lab notebook" for mathematical reasearch—the article on "folklore" roughly reflects what (a select group of) mathematicians think about the topic).

Is it "ethical" to publish folklore?

Assuming that one has a proof of some folkloric result, there is nothing unethical about publishing that result. Indeed, I would argue that it is probably a Good Thing™ to publish these results, if for no other reason to have it written down for the sake of peer review (both in the context of having a committee of experts vet the result, and in the context of having more eyes in the community to spot bugs).

Is is a Good Idea™ to publish folklore?

*shrugs*

Ideally, if an old result is being used to prove a new result, then a proof of the old result should either be cited (if it exists in some other paper or text) or explicitly given as part of the exposition leading to the proof of the new result. If part of proving a new result includes allusions to the "folklore" of a field, then note that it is a folkloric result, give a proof, and move on.

It is also, in my opinion, reasonable to put forward a paper which seeks to do nothing more than prove a result from the folklore, though there are things to consider when going down this path.

From a CV / career standpoint, it very much depends on how advanced the author is:

  • For a student, I would think that attempting to publish a folkloric result can only be a good thing. Because of a perceived lack of novelty, it is unlikely that such a result will go into a "good" journal, but students (undergraduates and graduate students alike) typically have very few publications, and any publication is going to look good.

  • There is also nothing wrong with a late-career researcher publishing such results. They have nothing to lose (e.g. they have tenure and are not looking for further career advancement), and are potentially adding to the published record. Why not?

  • However, for an early career researcher, one should carefully consider the opportunity cost in pursuing folkloric results. Is it a good use of your time to write a paper that is probably only ever going to get into a C+/B- journal? or would it be better to work on something that is going to go into a better journal? There is also a chance that publishing this kind of result will just look like CV padding, but this is going to depend very much on whoever is reading over that CV.

From the point of view of recording human knowledge, I think that there are many good reasons to go ahead and publish folkloric results:

  • These results are sometimes wrong. If someone manages to prove that something in the folklore is wrong, then it needs to be recorded, particularly if many other results depend on the incorrect folkloric assumptions. Moreover, I would imagine that someone who can disprove a result in the folklore would have a huge leg up on the job market.

  • Even if the results are not wrong, they may be very niche. An outsider, not immersed in the culture of a particular sub-sub-sub-sub-field might not know the folklore, and might waste time on something which is already "known" (imagine the physician who reinvents calculus, but with something known only to a small number of specialists). It is helpful for the dissemination of knowledge to publish these things, even if it is not a huge career boost.

  • Many specialist fields in mathematics are opaque to students. It is helpful to publish the folklore if for no other reason than to make the barrier to entry that much lower. Publishing the results means that students can learn the material, which makes it easier for new researchers to enter the field.

Overall, my preference is that these results get proper peer review and publication. They are likely not to go into particularly prestigious journals, but I think that it is valuable to make folklore explicit—a "shadow repository" of folkloric knowledge is kind of antithetical to the ideals of academia—it makes knowledge less accessible and circumvents peer review. I am on the side of "publish".

4
  • Why do you write "Good Thing" and "Good Idea" with capitals and a "TM" up there? (I have a suspicion why you do it but (a) I'm not 100% sure and (b) if it's nothing more than what I believe it is, I recommend to remove it, because it doesn't serve your answer but can potentially confuse, particularly non-native speakers.) Commented Jan 5 at 23:13
  • 4
    @ChristianHennig It is both a bit of a joke (a small nod in the general direction of Martha Stewart), which is meant to indicate a little more strongly that these are opinions, and that they are put forward in a light-hearted manner. They are kind of important to the tone that I am trying to convey, hence my preference is to leave them alone. Commented Jan 5 at 23:31
  • too tedious to bother with Ah, the famous "I didn't bother with it" notation on the margin? "Too tedious to bother with" screams "publish or shut up" to me :) Commented Jan 7 at 19:21
  • @Kubahasn'tforgottenMonica Possibly, bit it is also possible that the proof is a long and tedious, but otherwise routine computation. And it may be that someone once tried to get it published, but the reviewers responded with "there's nothing novel here". That doesn't mean that folk shouldn't try to get these things into the literature, only that the culture (both of mathematics, and of academia more broadly) can make it difficult. Commented Jan 7 at 19:58
16

Some math journals do not mind publishing new proofs of old results and proofs of "folklore theorems"; one of the best (in my opinion) is "L'Enseignement Mathématique". The quality of papers they publish tends to be quite high.

In agreement with other answers: Publishing proofs of "folklore theorems" is a valuable service to the community but is easier said than done. It is easier to accomplish if a "folklore theorem" is needed for a proof of another result in your paper. Then you can try to publish a proof as a part of a paper (possibly, as an appendix). Another option is to publish such proofs as a part of a book. I have done both several times.

0
13

A mathematician Olivia Caramello [complained][1] of people saying that a lot of what she wrote was "folklore". I am not in a position to decide who was right or wrong in the dispute, but what she wrote about it looks interesting and relevant.

EDIT (1/5/2024): I forgot that I recently read an article that also seems "interesting and relevant" :-) : Synthese, Volume 197, pages 3875–3904, (2020) (see also the [preprint][2]).

Abstract:

We investigate how epistemic injustice can manifest itself in mathematical practices.

We do this as both a social epistemological and virtue-theoretic investigation of mathematical practices. We delineate the concept both positively—we show that a certain type of folk theorem can be a source of epistemic injustice in mathematics—and negatively by exploring cases where the obstacles to participation in a mathematical practice do not amount to epistemic injustice. Having explored what epistemic injustice in mathematics can amount to, we use the concept to highlight a potential danger of intellectual enculturation.

They also discuss the "Caramello case". I don't want to provide long quotes here. [1]: https://www.oliviacaramello.com/Unification/InitiativeOfClarificationResults.html [2]: http://philsci-archive.pitt.edu/15133/1/Epistemic%20Injustice%20in%20Mathematics%20%28to%20appear%29.pdf

4
  • 10
    Upvoted. One sentence that attracted my attention: "It is clearly unfair to pretend that a young researcher presents his/her results, which he/she had discovered on his/her own without being told about them by anyone, as non-original on the grounds that “experts knew them but never wrote them down (or publicly communicated them)”."
    – Stef
    Commented Jan 5 at 15:38
  • 9
    It's not rare in applied mathematics to use results from pretty advanced branches of abstract mathematics, but since applied mathematicians are not "experts" in these advanced branches, I suspect there's a bunch of "folklore results" that have never been written down even though they could be pretty useful. Arguing that some results don't need to be written down because "experts know them" is gate-keeping and elitist, and borders on obscurantism.
    – Stef
    Commented Jan 5 at 15:39
  • 5
    I agree. A statement without a peer-reviewed proof is a conjecture, no matter how simple or folklore it may be. Commented Jan 5 at 17:47
  • 5
    @SpencerKraisler: Not necessarily. For instance, you will not find a proof of the statement 1234 * 5678= 7006652 a peer-reviewed literature. Nevertheless, it is not a conjecture, but a routine calculation which nobody in a right mind will be verifying in a refereed paper. Commented Jan 6 at 12:43
12

In math, insight is king. If you publish "interesting" proofs of such theorems and those proofs give general insight into a class of important problems, then they are worth the effort of publication. If the proofs are pedestrian, then the opposite would be true.

But it is the reviewers and editors of reputable journals that can make the judgement about the value of this (or any) work. If it has the necessary aspects of novelty and importance then they might want to publish it. But you have to contribute something to the literature.

It is possible, of course, that some "well known facts" in math are actually wrong. It doesn't happen too often, but it can. What is "obvious" might actually be incorrect. Only an exploration can guarantee correctness. But even then, pedestrian work may not be publishable.

7

I don't think there's an ethical problem here. Of course any publication of this kind would need to acknowledge that this is implied in the literature already, and justify giving a proof by lack of availability and use for the community. As a reviewer I'd ask myself whether I could prove it myself in an hour or so if I had to, and whether I can find any publication where enough is said that a reader just needs to connect the elements in a straightforward fashion to arrive at the proof. In these cases I don't think it'd be worth a journal publication, although it could still make a worthwhile addition to a textbook that uses the result or provides the tools, and this thing can be given as example.

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .