Is writing short proofs inline bad form?

Question

I'm TeXing up a paper right now and I would like to ask for a sanity test about proper form.

In the paper, there are some statements which seem too simple to be called a Proposition or Lemma, let alone a Theorem. I use the word Remark for those. However, roughly speaking, there are three kinds of Remarks:

1. Ones whose proofs are fairly easy or standard, but not completely trivial, so I put a short argument after, using the standard proof environment.
2. Ones which I think are completely obvious in context, for example they just summarise some exposition directly above, so I simply put a qed sign at the end (I sometimes do the same with Corollaries).
3. Ones which have very simple proof, which may not be entirely obvious at a glance, but I believe a minimal (say, one short sentence) explanation is enough. In this case, putting the short explanation in the proof environment seems to be a waste of space and makes the whole thing look too important. Instead, I just put the short explanation in a parenthetical comment, which I append to the body of the remark, and follow that with the qed sign.

This system seems to be logical to me, but I would like some confirmation that this is OK, particularly the last kind.

I would also be interested in alternatives, with their pros and cons (I am unlikely to change the style in this paper unless I'm convinced that there is a big problem, but I may try to do something better in the next one).

Answer 1

Maybe this is "opinion based", but here is what I prefer, and I think it is pretty standard.

I would only use the QED box 🞎 at the end of a proof environment (which started with Proof) which is preceded by a numbered Theorem / Proposition / Lemma / Corollary etc. That way, there is a clear statement of what is being proved, and the statement is clearly distinguished from the proof. Subconsciously, I think of Proof and 🞎 as matched delimiters.

I don't think "too simple" is the main consideration for whether to use Proposition or Remark. Rather, I think Remark should be used only for statements or discussions which are a digression from the main argument of the paper. If you have a statement which is important to the paper, even if the proof is very simple, make it a Proposition or Lemma or Corollary (with a number), and follow it with a Proof. (You might be able to get away with saying something like "we have the following proposition whose proof is obvious", and then omitting the Proof environment, but this can be annoying.)

In a Remark, if discussing a particular claim, you might include a short proof or sketch or outline. Generally this would be written into the prose of the paragraph and not set off as a separate proof, and I would not use the QED symbol here.

Remark. The previous result raises the question of whether every snark is a boojum. In fact, the answer is yes, and this can be seen by noting that cheese is green and the sky is mauve. However, it is still not known whether every boojum is a snark.

This would be an alternative to:

Proposition 6.37. Every snark is a boojum.

Proof. Cheese is green and the sky is mauve. 🞎

Answer 2

A Remark is a strange mathematical specimen: if you read a bunch of math papers, you will see different people using it in different ways. With that in mind, I would say that I don't think I've ever seen "Remark" used the way you do: as an especially small theorem.

Whether a remark should be done via a theorem environment at all (thus e.g. appearing italicized) is also a point of contention. I remember getting a paper rejected from a very prestigious journal, where other than "not good enough for the journal," the main comment we received was that remarks should not be italicized. But really, formatting of remarks is all over the place.

I think most people agree with @Nate Eldredge that Remarks should be "statements or discussions which are a digression from the main argument of the paper." However, exactly what that means is a bit up for grabs. For instance, some people think that you should call something a Remark if you're never going to refer to it again in the paper -- in other words, literally you could take the Remark out of the paper without disturbing anything else. Other (and more?) people think that the point of setting something off as a Remark is so that you can refer to it later: if it's really completely disposable, why not just have inline in the text, or in parentheses, or in a footnote? I think that in my mathematical writing I have come around from the former use to the latter use...I think.

Anyway, fundamentally I don't buy that something can be too simple to be either a Proposition or a Lemma but it still needs a formal statement and a proof. That's exactly what Proposition is for: like a Theorem, only smaller. Even Lemma and Corollary have other uses: a Lemma comes just before a bigger result and a Corollary comes just after it. The point of using these terms is to make the logical flow of the paper visually apparent. How does using "Remark" for "not even a Proposition" help the reader more than "Proposition" followed by a two-line proof?

The right usage of a Remark is subtle, but it's roughly contrapuntal to Theorems, Propositions and so forth. In a Remark you step aside for a moment and place something outside of the direct logical sequence you were following (though you may call on it later). Thus for instance Remarks may belong at the end of (sub...)sections as commentary or summary on what has come before.

Or so it seems to me, anyway. Again, different people do things differently here.

Answer 3

On markers immediately after statement:

Sometimes, after a number of lemmas and theorems you need to write some further remarks to highlight some things that will be useful later. To avoid breaking the flow, I found it useful to put a statement that could be referenced only at the end of that part, usually as a corollary. The problem I had was that frequently the preceding paragraphs would include the proof, but in some cases the statement was too involved and needed a separate formal proof. Although this is not standard (yet), to make clear whether the reader should expect a proof or not I choose to include a special sign (a distinctively smaller version of QED) that I would put after statements with no proof environment.

\newrobustcmd{\xsmallsquare}{%
  \text{\fboxsep=-.2pt\fbox{\rule{0pt}{4pt}\rule{4pt}{0pt}}}%
}
\declaretheorem[name=Corollary,sibling=theorem,style=definition]{corollary}
\declaretheorem[name=Corollary,sibling=theorem,style=definition,qed=${\color{black}\xsmallsquare}$]{qedcorollary}

For consistency there was a small diamond shape after every statement for which there should be no proof, like definitions. It became very useful for complex statements that included enumerations – it made really clear where these ended, while the mark was small enough to avoid distraction. It had the additional benefit for the (rare) cases where there was a paragraph of text between the theorem and its proof to explain how the proof will proceed and present some intuitions which would made no sense before the formal statement of the theorem.

I never received any negative remarks on that style, and there were some positive comments. Also, please bear in mind that many journals and some conferences have editorial guidelines to which you should conform. Furthermore, while this might look great in a bigger work (book, thesis, long tech report), in a short paper that is a part of a bigger collection it usually will be just distracting.

On remarks:

I agree with @Pete L. Clark and @Nate Eldredge. They have put it more eloquently, so I will just skip that part.

On proofs in parentheses:

Although I love using parentheses, I would strongly advise you against putting proofs in there. Note that it still might be ok to remind the reader about something in parentheses, for example a lemma that makes the statement in context a direct corollary. However, please do not do so in statements – almost always it is possible to mention that lemma immediately before or after the statement. The only non-math parentheses in statements I use are for pattern-matching like "we will call a X left-nice (right-nice) if Y is right-nasty (left-nasty)" or referenced theorems "Theorem 42 (Famous Researcher [42])."

On obviousness:

Let me quote from a really nice post by Joel David Hamkins on MathOverflow:

I don't agree that if something is obvious, then it is obvious that it is obvious. When an author declares in a mathematical exposition that a fact is obvious, or says "of course" or something with a similar meaning, then it is a signal that the reader should be able to find a very easy reason justifying the statement, rather than a complex one. This is useful information for the author to signal, and I for one as a reader have often been grateful for it.

Furthermore, including lots of trivial statements and their proofs will make the paper unreadable. Theorems and their proofs are some of the most important parts of a math paper, but one should emphasize the main results rather than simple observations which frequently are community folklore.

Finally, you should be really careful that things you deem obvious really are obvious. In practice I often do write down the proofs and only decide against including them. This way you can reduce mistakes and choose actually for better clarity (with a small useful bias because one does not want the work being wasted). It takes more time, but at least for me it is worth it.

I hope this helps ;-)