# Why would one number theorems, propositions and lemmas separately?

When it comes to numbering results in a mathematical publication, I'm aware of two methods:

1. Joint numbering: Thm. 1, Prop. 2, Thm. 3, Lem. 4, etc.

2. Separate numbering: Thm. 1, Prop. 1, Thm. 2, Lem. 1, etc.

Every piece of writting advice I have encountered advocates the use of 1. over 2., the rationale being that it makes it easier to find the result based on the number. It seems that 1. is more popular than 2., although 2. still exists, especially in books. I can only imagine that people using 2. must have a reason, but I have not yet to encounter one. I hope it is not too opinion-based to ask:

What is the rationale for separately numbering theorems, propositions and lemmas, like in 2.?"

• The reader may quickly count the theorems in your paper. – Fedor Petrov Aug 2 '19 at 22:32
• If a short paper has three main results, the second of which has a long proof with five lemmas, then Thm. 1, Thm. 2, Lem. 1-5, Thm. 3 makes total sense. – François G. Dorais Aug 2 '19 at 22:33
• I have always assumed that most people using method 2 haven't really thought about it and are just letting LaTeX get away with its default behavior. To make LaTeX use method 1 you have to explicitly tell it to use the same counter for all results. – Mike Shulman Aug 2 '19 at 22:56
• In the numbering method 1, maybe I would antepone the number: 1-Thm. , 2-Prop., 3-Thm., 4-Lem. , etc. – Pietro Majer Aug 3 '19 at 8:39
• Try writing a book (or long article) without a computer, and certainly without any way of auto-cross-referencing. Then you realize that you want to add a lemma, or an example. Using Method 2, the required changes are much less than if you are using Method 1. Another way to accomplish this, as is done for example in Hartshorne, is to label using A.B.C, where A is the Chapter number (in upper case roman), B is the section number, and C is the item number in that section. LaTeX, of course, has an easy way to number in that fashion. – Joe Silverman Aug 3 '19 at 20:16

This is a slight elaboration of François Dorais's comment. If you have a small number of theorems/lemmas/propositions—let's say, small enough that readers can reasonably be expected to hold all the theorems in their head at once—then the second method of numbering can help readers grasp the flow of the paper and can even serve as a mnemonic aid.

A secondary consideration, similar to what Fedor Petrov said, is that the reader may want to skim through and just look at the main theorems. If you adopt the first method of numbering, then readers might accidentally skip from (say) Theorem 8 to Theorem 17 without realizing that they missed Theorem 14.

One famous book that uses the second method of numbering is Serre's Course in Arithmetic. Serre uses the "Theorem" designation very sparsely in that book, and the numbering system helps make the Theorems stand out.

• Personally, I'm unconvinced that any such minor advantages would outweigh the resulting inability to find anything. However, I wonder how ugly and difficult it would be to use both numbering systems and get the best of both worlds? E.g. something like "2.7 Theorem 3" for the 3rd theorem which is also the 7th result in section 2. – Mike Shulman Aug 6 '19 at 1:01
• @MikeShulman: Agreed. I've also tried the "Theorem A (=Theorem 2.7)" "Theorem B (=Theorem 4.6)" approach in the past, which I still kind of like. I think changing to letters for the main results can help by not having two numbering systems, which could get confusing. (But still get the best of both worlds, as you suggest.) – Joshua Grochow Aug 6 '19 at 4:57
• Useful for counting theorems? what's the diff between theorem and proposition? just emphasis? – Jim Stasheff Aug 6 '19 at 19:45

If the paper contains three main theorems, each generalizing the previous, it is nice to be able to discuss them like this:

While the extension of Theorem 1 to Theorem 2 uses only complex analysis, in Theorem 3 we will have to employ some Ramsey theory.