Capitalization of theorem names

Question

I hope this question is suitable; this problem always bugs me. It is an issue of mathematical orthography.

It is good praxis, recommended in various essays on mathematical writing, to capitalize theorem names when recalling them: for instance one may write "thanks to Theorem 2.4" or "using ii) from Lemma 1.2.1" and so on.

Should these names be capitalized when they appear unnumbered? For instance which of the following is correct?

"Using the previous Lemma we deduce..." versus "Using the previous lemma we deduce..."

"The proof of Lemma 1.3 is postponed to next Section." versus "The proof of Lemma 1.3 is postponed to next section."

Answer 1

In English, proper nouns are capitalized. The numbered instances you mention are all usages as proper nouns, but merely refering to a lemma or corollary not by its name is not using a proper noun, and so is uncapitalized.

Thus, for example, one should write about the lemma before Theorem 1.2 having a proof similar to Lemma 5, while the main corollary of Section 2 does not.

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Edit. Well, I've become conflicted. The Chicago Manual of Style, which I have always taken as my guide in such matters, asserts in item 7.136 that "the word chapter is lowercased and spelled out in text". And in 7.141 they favor act 3 and scene 5 in words denoting parts of poems and plays. This would seem to speak against Section 2 and possibly against Theorem 1.2. In 7.135 they say that common titles such as foreward, preface, introduction, contents, etc. are lowercased, as in "Allan Nevins wrote the foreward to...". This may also be evidence against Theorem 1.2. But in 7.147 they favor Piano Sonata no. 2, which may be evidence in favor of Theorem 1.2. But they don't treat mathematical writing explicitly, and now I am less sure of what I have always believed, above. I do note that the CSM text itself refers to "fig. 1.2" and "figure 9.3", and not Figure 1.2, which would clearly speak against Theorem 1.2. So I am afraid that I may have to change my mind about this.

Answer 2

No. "Theorem 2.4" is a title, hence capitalized. But the word "lemma" in "the previous lemma" is simply a non-proper noun, hence uncapitalized in English. In both your examples I would use the second version.

Although I wrote the above paragraph as though it were definitive, it is of course only my own opinion. There's (almost) no such thing as universally accepted usage.

Also, although I would write "The proof is postponed to Section 4," I'm not too bothered by "... to section 4." I have no good reason for being less rigid about this than about theorem capitalization. I've also been known to be inconsistent about things like "the Hahn-Banach theorem".

Answer 3

Donald Knuth et. al. give a clear rule for capitalization in Mathematical Writing:

19. Capitalize names like Theorem 1, Lemma 2, Algorithm 3, Method 4.

Answer 4

I've wondered about this as well. Here is an excerpt from the Chicago Manual of Style that seems appropriate:

http://www.chicagomanualofstyle.org/ch08/ch08_sec157.html

If I understand correctly, they are advocating the "Brouwer fixed point theorem" approach.

Answer 5

Really, it should not bug you.

The majority of english writing mathematicians are not native english speakers. And they often capitalize according to the rules they are accustomed to. Hence the many choices you can find in the literature. Just pretend you are correct with respect to some rules, and there is probably some place where these rules are in fact conventions. Instead bother about orthography, which is more important, in my humble opinion :-).

Answer 6

The beauty of (La)TeX is that you can, for example,

\usepackage{xspace}
\providecommand{\Thmref}[#1]{Theorem~\ref{#1}\xspace}
\providecommand{\thmref}[#1]{theorem~\ref{#1}\xspace}
%%\providecommand\thmref\Thmref

and choose between the last two according to your editor's whims (remember to use the capitalized macro at the beginning of sentences, or wherever capitalization is required by other reasons)

Even easier, consider

\usepackage{cleveref}

Answer 7

The following link from the MAA is not precisely on topic, but may be useful to people coming to this question. https://www.maa.org/sites/default/files/pdf/pubs/General_Editorial_Guidelines.pdf

Answer 8

The rules for whether to capitalize the words "theorem", "lemma", etc. are the same as for a word like "uncle".

For example, "My uncle is a firefighter". Here "uncle" is not used as part of a proper noun---that is, a title or a name. However, "Did you know that Uncle John is a firefighter?" Here, the word "Uncle" has become appropriated into a single title for a person. It is therefore part of a proper noun and is now capitalized.

Hence, "This theorem is very deep", but "Understand that Theorem 1 is very deep".

Regarding words like "section", I also am conflicted like the above accepted answer. I recall learning in my mathematics graduate education that "Section 5" should be capitalized, but I also sympathize with the Chicago style guide's apparent discouragement of this.