Is it acceptable for a referee to suggest changing "theorem" into "proposition"?

Question

In mathematics, it is usual to call terminal results "theorems" and intermediate results "propositions" or even "lemmas" depending on importance and place in the overarching proof. Suppose that one is refereeing a paper where the authors have decided to call almost all their results "theorems", making a paper with a large number of "theorems" that even emeritus professors don't usually reach by the end of their career. (Such theorems include computation that could conceivably given to as end of year exams to master students. Not to diminish the importance of the paper, the actual theorems are good, but the 20 others are not theorems. There are more theorems than pages.)

Would it be acceptable and well-received to suggest toning it down? Or would it be overstepping and rude? This is not just a philosophical question: I truly believe that it makes the paper harder to reader, because it is difficult to separate the wheat from the chaff, so to say. A reader does not know what is important and what is not.

Answer 1

I think we should begin by acknowledging that it is the author's choice whether to label results Theorem, Lemma, Proposition, Remark, etc., so a referee should not insist on changing this as they should insist on the correction of an actual error or gap.

Suppose that one is refereeing a paper where the authors have decided to call almost all their results "theorems", making a paper with a large number of "theorems" that even emeritus professors don't usually reach by the end of their career.

Don't say that to the author, by the way. It's snarky and not really helpful -- you seem to suggest that the author is somehow cheating their way to "too many theorems." That's not a thing.

Referees can make stylistic suggestions, of course. How much they should do this is quite a judgment call. I think a good referee should make stylistic suggestions when these suggestions impact the readability of the paper or affect its appraisal by readers. Will it look a bit weird to many readers to have multiple theorems on every page? Yes, I think so, and perhaps the author deserves to know. However, if by having "too many theorems" the author makes it hard for the reader to see what are the important results of the paper -- and, as a statement about human cognition rather than mathematical achievement, a paper simply cannot have, say, 50 important results -- then that's a much bigger deal.

If I were you, I would lead with the latter: explain that you had trouble isolating the important results of the paper because so many results are being presented in the same way, then suggest that changing some theorems to propositions or lemmas might be helpful in alleviating this. (Then, if you like, say that having fewer results called theorems might make a better impression on the reader.) I think this is putting things in the right order and does the best possible job at getting your concerns addressed.

But it is possible to address the main concern -- that the important results not get drowned out -- while still calling every result a "Theorem." If the author pulls that off...okay. Up to them, really.

Answer 2

I’ll work my way to answering your question by going down in decreasing levels of generality.

First, at the most general level, it is acceptable to require the author to make any changes to their paper that you believe in good faith to be warranted to bring the paper up to whatever level of clarity and readability that is your and the journal’s standard for publishability. In fact, it’s not just acceptable - it’s your job as a referee to do exactly that.

Second, it is perfectly reasonable if some of those changes are purely stylistic. If the paper has horribly formatted equations, or uses highly nonstandard terminology or notation (e.g., using a swastika as a mathematical symbol), or any number of other stylistic issues are present, it is perfectly acceptable to make the author correct them to make the paper readable by the community the journal is targeting. Someone might object and say “but authors should get to choose their notation!” (analogously to Pete Clark advocating in his answer that authors get to decide how to label a mathematical claim). Well, yeah, they 100% get to choose that, and the journal gets to choose whether to publish their paper once they’ve made their choice. You can’t “make” the author do anything, but you can certainly set conditions for recommending acceptance of their paper.

Finally, on the specific matter that you bring up, it seems pretty clear from your description that the paper does not adhere to standard conventions about publishing in mathematics. “(x+y)^2=x^2+2xy+y^2” is not a “theorem” in any context other than a middle school algebra textbook, and calling it (or something like it) a theorem in a research paper obfuscates the content of the paper and makes life difficult and unpleasant for the readers in exactly the same way as any of the other weird stylistic issues that referees are tasked with calling out. So, in my opinion it would be completely appropriate to ask the author to fix this issue.

As for whether it would be well-received, that’s anybody’s guess.

Answer 3

I think you are probably correct, but it is also a matter of style, not formal definition. I think, also, that the paper would be strengthened by more carefully separating what is supportive (lemmas) from what is fundamental (theorems).

The main difference (personal view) is whether a proved result has "legs" to prove many other things (a theorem) or is primarily just a support lemma for something that is more generally useful. If you can envision a "proposition" as being generally useful in itself, call it a theorem. Otherwise probably a lemma.

The term "proposition" is a bit more nebulous. Sometimes it is something that is the subject of exploration with no known proof or counterexample. The "Four Color Problem" was a proposition until it became the "Four Color Theorem", for example. Personally, I'd stick with "lemma" and "theorem" as described above unless presenting statements that might be provable (or not).

And, of course, a big part of a referee's job is to help an author improve a work. Some of what you say may well be subjective, but that doesn't make it invalid. The author remains the author, of course.