When writing a paper, what do you do with statements that you know are probably true, but you have not checked the details?

Question

The background is mathematics.

There are a few results I think are highly likely to be true, but proving them would take up some length, and they are not central to the paper I am writing. They also are not important enough to become an independent paper, so it is unlikely someone will come and prove them later.

So what should I do? Is it appropriate to say "the result A should be true, and one can prove it using method B"?

Edit: those statements are generalizations to a proposition and are not needed for the proof of the main result.

Answer 1

You can say it in pretty much any format you want provided that you convey the meaning exactly as it is, which, if I understood you correctly is "It is quite possible that method B gives a slightly more general theorem A but the details have not been verified and this generalization is not needed to prove any results that are claimed to be proved in the paper".

I would use a wording that puts it as a challenge to the reader (having primarily a student reader in mind, but not exclusively). It would read something like the following.

Remark: It would be interesting to see if our method can also yield a slightly more general statement, namely that /insert the formulation of A/. We believe that it may be quite likely but, since this more general statement would not give us any advantage as far as the proofs of the results in the present paper are concerned, we will not pursue this issue any further here and leave it to the interested reader instead.

If somebody later would like to figure it out and publish something, he would be able to refer to that as "a question raised in [X]", which is, probably, just what you are looking for. Claiming, like Fermat, that you know a proof of something but it is too long to be included into the text would be rather impolite in a published paper, IMHO, since you are not annotating your personal copy of some book but give a detailed public presentation instead.

Answer 2

I don't think "conjecture" is exactly the right word for what you have in mind, because that would imply you don't know how to prove it. If I understand the question, you're asking about small improvements that you have convinced yourself you could prove, but aren't important enough to include the full details in the paper.

In this case, I usually see people use language like "We expect Proposition X could be extended to the more general setting Y, at the cost of more careful technical arguments." Possibly followed by a few words about the specifics of the necessary arguments, if you can sum up the idea concisely. But there are a few caveats here:

• You need to be able to back this up with details if someone (such as a referee) asks.

• It's best to do this sparingly (perhaps once or twice per paper, at most) because at a certain point, if you're devoting so much verbiage to this extra unproven stuff, it should be important enough to actually prove some of it formally. To say it another way, there's a limit to how much anyone cares about mathematical claims you don't prove.

• It's important to word these statements carefully, to make it 100% clear that you aren't claiming to have proven the thing in question. ("We expect one can prove it using X method" is better than "one can prove it using X method.")

Answer 3

If the result is proven to be true (or claimed to be true, ideally in more than one place) by some reliable source(s), you can cite that, especially if the proof would he straightforward. Of course, if the main result of your paper would have relied on this result, you would have to cite a source with a detailed proof or provide one yourself.

Otherwise you could consider the following options

1. You don't claim the result to be true (call it a conjecture or similar) and treat is as such.

2. (My preference) Provide a proof for the result, if it distracts from the remainder of the paper, put it in an appendix. You will do others a service, as they now have something to cite for the result. I am in theoretical CS, and have seen some cases of people doing this in similar situations.

This is assuming the statement really is too small to be in its own (short) paper.

Answer 4

If they are proved elsewhere, search the references and just cite them. If these statements are new in the sense that the available literature does not contain their proofs, you can state them as conjectures and remember that the validity of your arguments that make use of these conjectures will be subservient to the truth value of these conjectures.

Answer 5

I completely agree with the first paragraph of fedja's answer: the only crucial point is that you need to be clear that these assertions are not needed for the proofs of any of the main / formally stated results of the paper.

In order to figure out what to do with these "probably true but not fully checked" assertions, I suggest that you think about your purpose for including them. From your comments, it sounds like these are stronger forms of intermediate results, where "stronger" means stronger than you need to prove your main results. What to do here is an interesting problem that may (or may not; I am curious) be relatively specific to the academic field of mathematics. One aspect of this to address is:

Why not include the stronger forms of these results?

In my opinion, all other things being equal it is a good thing rather than a bad thing if one's lemmas are stronger than strictly needed for the theorems they are used to prove. Virtually all academic work is done with the hope that it will be useful to other academics in the field, and this hope should include methods rather than just results. If the idea of proof of Lemma X also proves more general Lemma Y, it may well be more efficient in the long run to state Lemma Y from the outset, as otherwise there may be future mathematicians having to state and prove Lemmas Y.1, Y.2 and so forth. (Now these mathematicians may simply say "The proof of Lemma Y.n is similar to that of Lemma X and is omitted," but I don't really like this and I am not the only one.) This is especially true if you plan to make use of Lemma Y later on. For you it sounds like that is not the case.

However it is not always a good idea to follow the above procedure. If you have 20 pages of lemmas to prove a theorem when in fact 5 pages of lemmas would prove that theorem, then a different kind of disservice is being done to the reader (and one that referees are especially likely to object to). Thus I think that realistically you have to evaluate the risk/reward in every case. However I will make the following suggestion: If at all possible, you should write out a precise statement and proof and then decide whether to include it. I hope it doesn't sound paternalistic when I say that this is the kind of thing that sounds very onerous until you get into the habit of doing it, and then it is often not so bad. Another perspective on this is: you are never going to have a quicker and easier time writing down the proof of Lemma Y than when you have just written the proof of Lemma X.

This is a SE answer rather than an essay, so I will have to stop soon, but I think there are yet other interesting aspects of this question. Here is a hint of one: how valuable it is to "write down the most general version of your lemma" seems surprisingly field-dependent. I am a number theorist who is very algebraically minded (I am happy to make active use of commutative algebra), but often my work gets combined with other work in a more analytic style. It seems to me that in algebra there often is an "optimal result that the idea of proof of Lemma X actually proves" whereas in analysis this may really not be the case. There are other branches of mathematics too...