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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."

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Whichever rule you follow, the journal will insist on the opposite. – Richard Kent Jun 10 '10 at 14:20
On a related note, should one capitalize "named" theorems? Is it "the Fundamental Theorem of Calculus" or "the fundamental theorem of calculus"? When it's named after a person, is it "the Hahn-Banach Theorem" or "the Hahn-Banach theorem"? "Fatou's Lemma" or "Fatou's lemma"? – Nate Eldredge Jun 10 '10 at 14:48
What about properties like Noetherian and Euclidean, should those be capitalized? – Bart Snapp Jun 10 '10 at 15:31
I thought “abelian” was rather unique in its near universal lack of capitalization. And @Nate: I prefer “Hahn–Banach theorem” with an en dash. Thanks to having a hyphenated name myself, I am rather sensitive to the difference between hyphens and en dashes. “Who is this guy Hahn-Banach anyhow?” – Harald Hanche-Olsen Jun 10 '10 at 17:32
@Harald: "gaussian" and "euclidean" for example are also very common, but by no means as universal as "abelian". I think you may be right that it's a unique case. Interestingly, I've noticed that physicists frequently capitalize "Abelian". Also, I'd never thought of the en dash v. hyphen point; I'll try to remember to use en dashes in the future. Maybe we should all start referring to the Birch–Swinnerton-Dyer conjecture? – Mark Meckes Jun 10 '10 at 17:48
up vote 37 down vote accepted

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.

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.

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JDH: I'm sort of with you, but I'm not sure. Would you write something like, "This follows by the remarks on Page 5."? – JBorger Jun 10 '10 at 22:56
James, I don't think Page 5 is the name of page 5, and so it doesn't seem to me to be a proper noun. My wife, a philosopher who cares about writing, tells me that "section 2" is probably also not a proper noun either. So you may be correct to object, although I have always thought of Section 2 as a proper noun. I'll check the Chicago Manual of Style. Meanwhile, I am certain that I live on Fifty-seventh Street... – Joel David Hamkins Jun 11 '10 at 3:02
Ah, but as everyone knows, the press (Press?) is on 60th East. It's the library that's on 57th. :) When people quote line and verse in canonical texts, do they capitalize 'line' and 'verse'? I doubt it, because 'line 5' is short for 'line 5 of EGA' or 'the fifth line of EGA'. But then you could argue that 'theorem 90' is short for 'theorem 90 of Zlbt' or 'the 90th theorem of Zlbt'. On the other hand, maybe you could also argue similarly for 57th Street. I suspect that trying to go by general principles here, rather than by established conventions and a sense of style, will only get us so far. – JBorger Jun 11 '10 at 3:29
Also, "room 1016" or "Room 1016"? – JBorger Jun 11 '10 at 3:48
Incidentally, I found in the Chicago Manual of Style a section on manuscript preparation (section 13), which includes information about mathematical typesetting, and in 13.45 they refer to "lemma 3.3" uncapitalized. – Joel David Hamkins Jul 2 '10 at 1:21

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".

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That has always been my use, but I was not sure. It seems confirmed by your answer, the one of Joel and the upvotes. – Andrea Ferretti Jun 10 '10 at 14:52
Not too long ago, most Nouns in English Language were capitalized. This rule is still strictly observed in German. Clearly, language grammar in general and capitalization rules in particular are evolving. – Victor Protsak Jun 11 '10 at 2:38
@Victor: Diese Tatsache ist ja selbstverständlich. – Mark Meckes Jun 11 '10 at 17:17

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.

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But this does not answer the question because Knuth's examples are all numbered. – Mariano Suárez-Alvarez Nov 9 '13 at 14:24
It does answer a point disputed in discussion above, and I'm with Knuth here. I've tried upper-case (Theorem 1) and lower-case (theorem 1), and contrary to what Joel David Hamkins predicts, I still prefer the former. But when unnumbered, I'd say e.g. "the previous lemma". – Todd Trimble Nov 9 '13 at 14:28

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

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

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Your link is only available to subscribers. – Andrea Ferretti Jun 10 '10 at 19:27
Sorry, Andrea. I didn't realize that I was using a subscription. Anyway, my example above gives the gist of it. I think that "big bang theory" and "Hooke's law," or maybe "Newton's laws of motion" were the examples they used. They didn't have any theorems listed per se. I hope this helps. – Dan Margalit Jun 10 '10 at 22:46
Although if you're referring to the TV show, I think you want to capitalize Big Bang Theory. – Gerry Myerson Nov 9 '13 at 22:38

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 :-).

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I'm not a native english speaker, but I actually use a spell-checker, so I would not write "ortography" in a paper. Good to know, anyway. It is one of those words one actually never uses, bu it is so similar to italian that one does not bother check before writing on a forum... :-( – Andrea Ferretti Jun 10 '10 at 14:47
Even if you would not write "ortography", would you write "litterature"? – Gerald Edgar Jun 10 '10 at 15:12
@Gerald: ? I don't quite get your point. Are you implying that "litterature" is a word and so it wouldn't be spotted by the spell checker? Maybe something which has to do with litter? Or rather that "literature" is another word very akin to the italian translation, but different in spelling? (For those who do not speak italian, it translates "letteratura"). – Andrea Ferretti Jun 10 '10 at 15:51
@Andrea: I think Gerald's question was directed to coudy, not you. – Mark Meckes Jun 10 '10 at 17:27

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