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I get a repeated feeling of insecurity when I write an article that needs to contain hierarchical organizations among its theorems. It isn't uncommon that one article contains a few important theorems whose proofs require using other, less important, theorems. I resolve this conflict by differentiating theorems with a hierarchical organization in order of importance: Lemma < Proposition < Theorem.

Some of these higher-importance theorems have great consequences (and hence a "theorem" denomination), and are used a lot after being proved to show their direct consequences. Some of these consequential theorems might have to be proved using lesser theorems, which gets us stuck again with the hierarchy problem.

I could, in principle, finish the article there and make another showing applications of the important theorems. But, I've noticed that this could also be resolved by "naming" the main theorems, as in "Pythagoras' Theorem".

  1. Is it consensually correct to "name" your own theorems?
  2. Are there standardised or incorrect ways to do so?
  3. Suppose I take the first approach and end an article to start another for implications of the main theorems of the previous article. Then, how are the previous article's theorems referenced?
  4. What if the referenced theorems are "named theorems"?

These are my first publications, so I don't want them to have bad reception because of trivialities.

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    I don't really get the connection between your introduction and the actual question. I don't get which "problem" mentioned earlier (if there even is one) is solved by naming. What's wrong with just having Theorem 1, Lemma 2, Proposition 3 etc.? (Of course the question about naming theorems is still legitimate without connection to the other 2/3 of your text, but...) Commented Jan 5 at 23:19
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    @ChristianHennig think about it: you have a set of (demonstrated) theorems that you want to publish, but the content is too much for an article to contain, and you need to split the thing in two or more articles...then how do you go about it? For the naming part, it is easier for the to reader to refer to "the polynomial factorising theorem" rather than to the "theorem 2.6.18"... Specially if you use it a lot. Taking all these in account, it shouldn't be difficult to find the overall onnection of the question. Commented Jan 5 at 23:32
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    Fair enough. I don't necessarily agree with this, but it's an interesting question anyway. (+1) Commented Jan 5 at 23:44
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    I do not understand why do you see any of these as problematic. Commented Jan 6 at 0:26
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    As a Coq user, I have to name my theorems Commented Jan 6 at 15:56

2 Answers 2

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I don't think there's a consensus, but my own opinion is that giving both precise numbering, and a descriptive name, at least for more substantive results, is optimal. And referring to them by both.

An extreme case of really-unhelpful referencing purely by numerical labels is Bourbaki's textbooks, where, instead of saying something like "by the uniform boundedness theorem", the reference is something like "volume IX, chapter VII, theorem 3.11.5". :)

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  • Thanks! What about the referencing of theorems in previous articles? That has to have at least some form of standard structure. Commented Jan 6 at 3:06
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    I don't think there's a standard... Perhaps it's not so important to have a standard structure, as to have a practical/usable structure. Even if a prior paper, by someone else, does not name their larger results, but just numbers them, I'd prefer to refer to the number and have a short phrase that tells what the result is... if at all possible... rather than just a bare "by 4.3.12 in [45]", ... which drives me crazy. :) Commented Jan 6 at 5:32
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    I remember a teacher who occasionally said things like "And in this situation we can apply the theorem we saw 3 weeks ago at 10:37"
    – Stef
    Commented Jan 6 at 11:53
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Personally, I find it extremely helpful as a reader for an author to add a descriptive name for an important theorem they are presenting to me. A clear descriptive name assists in referencing the theorem in prose and it also gives a prompt as to the substance of the result. Note that in cases where a theorem is named after a person (e.g., the Neyman-Pearson lemma) the name is attributed later by others, and indeed, it would be a bit on-the-nose for an author to name their theorem with their own name.

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    Completely agree. By all means name your theorems, but naming it after yourself should be left to others who came after you and referred to your work.
    – niemiro
    Commented Jan 6 at 15:24
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    Yes, a bit later i noticed that many of the "arrogance" arguments may have stemed from the concern of people naming theorems after themselves. The example i chose about "Pythagoras' theorem" doesn't exactly diminishes that concern. Commented Jan 6 at 21:22
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    I would go on to say that "naming it after the author" probably evolves via citation. "As a result of CitedAuthor[N] Theorem X.Y.Z, ..." eventually becomes (if the result is important enough) "By CitedAuthor's Theorem, ..." Commented Jan 7 at 2:10
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    @R..GitHubSTOPHELPINGICE Which only happens if it doesn't have a useful descriptive name. So if you want a particular theorem to be named after you, give descriptive names to every theorem in the paper except that one. :-)
    – Ray
    Commented Jan 8 at 17:57

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