Non-rigorous proof in an article

Question

I am writing about the analysis of a certain mathematical object. The expression for this object is obtainable in several ways:

• non-rigorous ways: they amount to just computations
• rigorous ways: they are precise and apart from the above computations, one needs to supply some proof of certain passages

I have a fully rigorous proof of the derivation of the expression for this object. I think that including it in my article is no good service for the community, because:

• there is no more space
• the proof is technical and not enlightening (no new ideas, essentially)

The only element of novelty is that the object is related to a new problem, not found in the literature, but very similar to others. And in a somewhat different formal setting than most other examples, but again, the proof outline and the key formulas/conceptual passages, are already present.

I am targeting a respectable journal where usually, facts are proved rigorously, not formally. And where proofs (formal or not) are not deferred to the supplemental material. That is the space for pictures, code description, implementation details and so on.

Should I include the lengthy proof, hence, in the supplemental material? How about including a non-rigorous proof using a quicker method in the appendix?

Otherwise, is it a good way to go just citing the various computation recipes out there, without spelling the computation (nor a proof) out in detail?

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More about "The object"

I am working in optimal control and I have to compute a derivative of a functional. A non-rigorous proof would be to form a Lagrangian and differentiate it formally, without checking that things are actually differentiable, for instance (it is a bit more complicated but this is the essence). The Lagrangian approach is non-rigorous. For a fully rigorous approach one should prove differentiability of every piece and this takes considerably more effort. Say, using the implicit function theorem in function spaces.

The non-rigorous passages are especially already present in the literature, for examples more or less.similar to mine. The rigorous passages maybe are not exactly equal to something seen before, but there are standardized recipes to obtain them (e.g. apply the implicit function theorem).

Answer 1

Sometimes there is a way to write down a proof that treats a new situation but contains many non-original elements in a very short way by referring to other papers wherever possible, omitting details. Often you can get away with this even if a reader who wanted to reconstruct the whole thing would have to put quite some work in. Ask yourself: What is the minimal way to write down how this works so that an expert reader with some time in their hand can follow the recipe, possibly involving cited literature, and convince themselves that the statement holds in this way? A reader who then doesn't succeed doing this can still contact you, and of course a reviewer can complain if they don't find this sufficient, however this kind of thing is not a reason for rejection.

In my experience, quite a bit of proof completing work is left to the interested reader in many math journals, with the effect that proofs can be presented in very limited space if they are not of central importance or interest.

Whether this approach works in your case of course I can't know.

Another option is to put a version with detailed proof on arxiv.

Answer 2

I suggest that you match the standards of the journal if they are clear. Readers of the journal, and reviewers, will look for that and expect it. You will get feedback from reviewers who may ask for changes.

The other option is to write it the way you think it seems best and submit it. Again, the reviewers may want changes.

You also need to consider whether you have chosen the right journal if you think that they way you think it should go isn't a good match. Maybe another journal that doesn't expect complete rigor would be best.

Answer 3

It’s really up to you to decide what is the best use of your time and effort to make the greatest positive contribution to your research community. I’ve also struggled with these sorts of questions and it’s not always an obvious call to make. However, I think your reasoning is influenced by some superficial and irrelevant considerations, particularly “there is no more space”.

If you don’t think the “rigorous proof”^* is useful to anyone, fine, don’t write it - that’s a totally valid reason not to write something. But if on the other hand it is actually useful, then presumably there will be a journal that is willing to publish it. Instead of fixating on a specific journal to target and lowering the usefulness of your content to satisfy the journal’s space constraints, consider writing the actual content that would make the greater impact and publishing it in a place where it will be appreciated (either instead of, or in addition to, publishing the slick non-rigorous derivation in the journal you currently have in mind).

^{* putting this is in quotes because from my point of view it’s a tautology that belongs in the department of redundancy department.}

Answer 4

Writing a paper always means weighing the reader's attention span: you are trying to present enough for your reader to, well, learn something new, without supplying so much information that your reader loses the important details.¹ I can think of some possible functional outcomes of not choosing the appropriate depth:

• Skepticism: If an important step is glossed over, the reader may find it too dubious and not accept your overall argument.
• Brittleness: If an important step is glossed over, and it explains a limitation of your argument, a reader may be tempted to expand your argument beyond its due limits.
• Rabbit-holing: If too many trivial steps are expounded, the reader may get lost in trying to prove them and forget the overall argument.
• Pigeon-holing: If too many trivial steps are expounded, the reader may think of your result as being very limited (for example due to many things being taken "without loss of generality") and not grasp its general extent.

Surely there are more. I hope you can see that I am encouraging you to consider the functional impact of what you keep in or leave out of a proof, not just whether its form fits a certain brevity or verbosity.

As an example -- I recently published a derivation of an equation, connecting certain thermodynamic ensemble averages, that I've been working on for a while. The equation began life as an equality between some esoteric integrals which I "projected down" into the thing I actually wanted. Later, while writing the paper, I reasoned that there must be a simpler way to prove it; I eventually managed it by shuffling around some integrands, but I was quite unsatisfied because I couldn't see the physical significance of what I'd done.

Eventually, when I published the paper, I only included the simpler proof in the text. But I included the original proof in the supplementary information and wrote a paragraph in the text explaining its physical significance. Alas, the simpler proof did introduce a physically-significant quantity, which I only identified when a collaborator asked me just exactly how it worked (naturally, just after the paper had been published). This is an example of me skipping an expositional step where, had I fussed over writing it down properly, I might have mined an additional insight for the paper.

Maybe you can do something similar, such as writing a shorter proof in your publication and referring the reader to an arXiv document for more details?

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¹Think of your reader as a biological large language model, and your job is to fit every important concept into the available context window.

Answer 5

First, this is mathematics, so you really need to include your rigorous proof somewhere in the article.

However, there is absolutely nothing wrong with putting the non-rigorous proof in the body of the article and deferring the rigorous but unilluminating proof to the supplemental material. Supplemental material is intended for things that the average reader of an article may not be interested in, but which may be important for readers who are experts in the subject area. The rigorous proof fits these criteria perfectly.

Answer 6

"The proof is left as an exercise to the reader" is one of the most irritating concepts in modern acadaemia.

It's the same in other fields where only vague details of methodologies are provided or source code and algorithmic implementations aren't included.

You may think your proof is rigorous, but without including this detail how can a reviewer know?

If it's too long to include in your article, publish it somewhere publicly available (it doesn't really matter where as long as it's not ephemeral), and cite it or include it as an appendix.