Conjecture vs open question on a paper

Question

Suppose that you are concluding a paper and writing the open questions (or further research directions) section, and suppose that you have several claims related to the work in the paper that your intuition is telling you they are true. Maybe you are also able to give heuristic reasons why you think the claims are likely true, but you are not yet able to prove them; or you may have some ideas on how to attack them, but that would bring you off-topic from the main content of the paper (or maybe it would make the paper unnecessarily long).

I was curious to hear what sort of criteria you use in deciding whether to give the claims as open question or rather as conjectures (or maybe not give them at all) in your paper?

PS. I have never asked a soft question on MO, but I have read many, so I hope it's okay if I post one. Please let me know if this is too much off-topic.

Answer 1

I doubt you'll get a definitive answer, so I'll frame my answer as appeal to expertise (that I happen to agree with), knowing that others may disagree. Most essays about mathematical writing encourage you to find your own voice and to think about the reader as a guiding principle. I'd add that it's wise to think about the state of your research area. Including conjectures and open questions in your papers gives other researchers something to work on. Phrasing them as conjectures also helps your field, because when some young person proves a Conjecture, it can help their career. They'll be more likely to get a job, win a grant, get tenure, etc. This is part of the argument Clark Barwick made in his essay on The Future of Homotopy Theory. In item 3 he writes (with reference to the field of homotopy theory):

We do not have a good culture of problems and conjectures. The people at the top of our field do not, as a rule, issue problems or programs of conjectures that shape our subject for years to come. In fact, in many cases, they simply announce results with only an outline of proof - and never generate a complete proof. Then, when others work to develop proofs, they are not said to have solved a problem of So-and-So; rather, they have completed the write-up of So-and-So's proof or given a new proof of So-and-So's theorem. The ossification of a caste system - in which one group has the general ideas and vision while another toils to realize that vision - is no way for the subject to flourish.

For me, the take-away is to include named Conjectures when I'm pretty sure the result is true, include Open Questions when I'm not that sure, and at all costs avoid Remarks where I claim things are true that I have not actually carefully proven. I think a Remark where one sketches a proof idea is fine, but in the interests of young people in the field, it's important to be clear that the Remark is not a complete proof and (ideally) to include the statement to be proven as a Conjecture. In my early papers, I sometimes used Remarks to advertise future papers. I'm not going to do that any more, because there are plenty of examples where someone did this and then never wrote the future paper, leading to the kind of issue Clark raises above. I'm grateful that Clark gave us something to aspire to, so we can make our field better for young people.

As for when to make something a Conjecture vs an Open Problem, Clark Barwick answers that, too, in his Notes on Mathematical Writing. On page 3, he defines

A conjecture is an assertion that meets all the following criteria.

• It is precise and unambiguous.
• The author strongly suspects that it is the case.
• The author considers the assertion interesting or difficult.
• The author has seriously attempted to prove it.
• Nevertheless, the author does not know how to prove it.

Anything that satisfies the final condition but not all of the others is a Question or a Problem. Do not fear formulating plenty of Questions and Problems.

Answer 2

As David White says, you're unlikely to get a definitive answer, since to some extent, it is a matter of taste and personal preference. There are many people who use conjecture to mean the five criteria in David's answer. There are others who use conjecture as a way to get people's attention, treating it almost as a provocation: "See if you can prove me wrong". The former have been known to write things such as "the evidence for conjecture XXX is more voluminous than it is convincing" for conjectures made primarily on numerical experiments, or even to refer to the "so-called conjecture YYY" when there is actually quite a bit of theoretical evidence. Some assertions should clearly be posed as questions, e.g., when you are unsure yourself whether you believe it is true or false; while others are very reasonably posed as conjectures because you've hit all five bullet points. For the ones in between, especially where you're basing the assertion partly on intuition gleaned from experience in your field, I'd propose this as a guiding principle: Pose it as a conjecture if you wouldn't mind seeing a paper entitled "A counter-example to a conjecture of Moreschi." But if you'd find that embarrassing, then pose it as a question!