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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.

  • When I was being taught how to proof I was told two things about a the definition of a theorem, 1. A theorem is an important and true proposition. 2. The distinction between a proposition, lemma, and corollary is somewhat fluid. Recommending a change from theorem to proposition or lemma could legitimately be taken as a diminishing of it's importance, depending on their academic background. – thisischuck Jul 18 '18 at 12:32
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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.

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    It's also worth considering that the author is from a different discipline (e.g., computer science) where the word theorem doesn't have the same connotation. I've considered submitting some of my more general results to mathematical journals, and I was not aware of this connotation. As an author, I'd appreciate a reviewer bringing it to my attention (but as you suggest, without sarcasm). – Ben Hocking Jul 17 '18 at 20:48
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    @DanRomik — computer science was an e.g. (i.e., "for example"). The author I'm referring to is the one the OP is referring to. We don't know the author's discipline. In the NASA PVS libraries, you'll see theorem occasionally used for properties that are not necessarily terminal, though they are typically higher-level than lemmas, but you'll also find terminal lemmas. PVS itself enforces no hierarchy between CHALLENGE, CLAIM, CONJECTURE, COROLLARY, FACT, FORMULA, LAW, LEMMA, PROPOSITION, SUBLEMMA, or THEOREM (all are PVS keywords). See also pvs.csl.sri.com/doc/pvs-language-reference.pdf – Ben Hocking Jul 18 '18 at 1:10
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    @DanRomik, I correctly read the OP's question. He states nothing about whether the author comes from the mathematics discipline. He states that the author is doing something unusual for the mathematics discipline. I posit that it's at least worth considering that the author is not embedded in the mathematics discipline, but could be an outsider so might not be familiar with the connotation the OP refers to. It seems to me to be worth considering. Cross-discipline scholarship is not that unusual. It appears to me that you feel very strongly that it shouldn't even be considered. – Ben Hocking Jul 18 '18 at 11:58
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    @NajibIdrissi: When an author fails to follow the stylistic conventions of a field, one very common reason is that they’re coming from a different field. And considering the likely reasons for the author’s overuse of “theorem” here is very relevant for considering how the referee should respond. – PLL Jul 18 '18 at 12:11
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    @BenHocking As a computer scientist (algorithms, complexity theory, etc.), I don't see any difference in the use of the word "theorem" between field and that of mathematicians (combinatorics, graph theory, probability, etc.) – David Richerby Jul 18 '18 at 14:33
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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.

  • For the final "well-received" remark, it might not be in the journal's best interest for a referee to require a certain stylistic change that the author may refuse to do. Then the poor editor has to sort it out. – Lee Mosher Jul 18 '18 at 16:02
  • @LeeMosher if the author refuses to do something to bring their paper to publishable quality, the paper should not be published. It is not in the journal’s best interest to publish papers that are not up to its standards. – Dan Romik Jul 18 '18 at 16:24
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    Referees can be wrong, sometimes. I've had it happen that the referee's insistence on a stylistic change was just wrong, in my judgement. I explained why to the editor. – Lee Mosher Jul 18 '18 at 18:08
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    @LeeMosher of course. Referees can be wrong about more substantive things as well. In any case, if you are a referee, “referees can be wrong” is a reason to be cautious about anything that you are asking the author to do, but is not a reason to not ask the author to do things. – Dan Romik Jul 18 '18 at 18:10
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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.

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    Just for the laugh: lemmata is the real plural of lemma. :-) – Peter K. Jul 17 '18 at 15:28
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    I have never encountered anything named as a proposition which was not as rigorous as a lemma or a theorem. To me (and everyone I work with), the difference is in "importance". A lemma is mainly of interest as a stepping stone towards another result. A proposition has interest on its own, but not as much as a theorem does. – Tobias Kildetoft Jul 17 '18 at 15:57
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    @Buffy It is not Latin, it is Greek :-) – PsySp Jul 17 '18 at 18:45
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    In the wide world the term "proposition" is of course used for many things, including the usage in mathematical logic (cf. "propositional logic") mentioned above. But that doesn't make the usage of e.g. "Proposition 2.76: In a compact metric space, a sequence with exactly one partial limit converges" ambiguous in any way. Such usage is extremely common in mathematical writing. (In fact this is quoted from some notes of mine on general topology.) – Pete L. Clark Jul 17 '18 at 18:47
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    Well, it's an argument in a teapot, but in your answer you write that calling mathematical results Propositions is "nebulous" (and give an example that is over a hundred years old). I'm just mentioning that the practice that you advocate against is extremely standard in mathematical writing. No big deal, to be sure... – Pete L. Clark Jul 17 '18 at 18:53

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