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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."

gmvh
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Andrea Ferretti
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    Whichever rule you follow, the journal will insist on the opposite. – Autumn Kent Jun 10 '10 at 14:20
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    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
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    What about properties like Noetherian and Euclidean, should those be capitalized? – Bart Snapp Jun 10 '10 at 15:31
  • @Bart: logic suggests not to capitalize, but usage I have seen suggests otherwise. – Andrea Ferretti Jun 10 '10 at 15:46
  • @Bart: In standard English, such terms are usually considered proper adjectives and are capitalized, just like names of languages when used as adjectives. But usage varies. One very frequently sees "abelian", for example. @Andrea: your sense of logic here is probably influenced by the fact that most other European languages do not capitalize proper adjectives the way English does. – Mark Meckes Jun 10 '10 at 15:51
  • @Mark: I guess you're right. What even IS a proper adjective? I never heard about them (though I capitalize some adjectives like "French"). – Andrea Ferretti Jun 10 '10 at 16:22
  • http://en.wikipedia.org/wiki/Proper_adjective The first sentence of this article will look distinctly unhelpful, but look down at the description section. The article states that the term is only used informally, although I learned it in school (in the U.S.); for all I know the concept doesn't exist with respect to other languages. – Mark Meckes Jun 10 '10 at 17:04
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    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
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    @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
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    Also tricky with names-as-adjectives are compounds. Should it be "subriemannian," "subRiemannian," "sub-Riemannian"? – Nate Eldredge Jun 10 '10 at 18:25
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    I have heard it said, only half-jokingly, that lowercasing a proper adjective is a sign of respect: The concept has become so fundamental that it it becomes part of the furniture of mathematics and loses its connection with a specific person. I think "archimedean" is about as common as "abelian" but I've also seen "noetherian" and various others. @Mark: Your point about the Birch–Swinnerton-Dyer conjecture has been made many times before. – Timothy Chow Jun 10 '10 at 18:31
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    @Harald: regarding the punctuation of "Hahn-Banach" or "Hahn--Banach", the Wikipedia on "en dash" suggests that authorities differ. Evidently the Chicago Manual of Style prefers the hyphen. – Nate Eldredge Jun 10 '10 at 18:34
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    @Nate: That was news to me. But I notice an exception when one component is already hyphenated: So they too would go for “the Birch–Swinnerton-Dyer conjecture”, I guess. – Harald Hanche-Olsen Jun 10 '10 at 20:42
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    Regarding capitalization of proper adjectives, it's interesting to note that although German capitalizes all nouns (as pointed out by Victor in a comment on my answer below), it does not normally capitalize proper adjectives. Hence "noetherscher Ring", for example. – Mark Meckes Jun 11 '10 at 17:23
  • And a related question (not sure if it merits a new one): What about names in parenthesis after names (as in \begin{theorem}[open mapping] vs. \begin{theorem}[Open mapping] vs. \begin{theorem}[Open Mapping])? – Christian Clason May 31 '19 at 10:56
  • @NateEldredge At least, it seems that English Wikipedia consistently writes "Zorn's lemma", "the fundamental theorem of calculus", etc (which I strongly approve). Since it's systematic while I very frequently encounter such phrases with capitals, I believe that there was once a discussion in Wikipedia somewhere about this convention; it would be great to locate it. – YCor Apr 01 '21 at 11:53
  • By the way these conventions equally apply to "section", etc: "In Section 4 we recall the notation of the previous chapter, which are explained in more detail in the appendix (dependence between sections are discussed in Chapter 17 and Appendix C)". – YCor Apr 01 '21 at 11:56

7 Answers7

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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.

Joel David Hamkins
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  • That has always been my use, but I was not sure. It seems confirmed by your answer, the one of Mark and the upvotes. – Andrea Ferretti Jun 10 '10 at 14:52
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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
  • Roger Howe once pointed out to me that for the same reason, "Theorem 1.2" does not require an article, whereas "the main corollary" does. Framing it as a proper noun issue has been very helpful to me (I am comfortable with theoretical grammar, but there are no articles in my native language). – Victor Protsak Jun 11 '10 at 02:24
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    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 03:02
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    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 03:29
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    Also, "room 1016" or "Room 1016"? – JBorger Jun 11 '10 at 03:48
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    After consulting the Chicago Manual of Style, I've become conflicted, and have updated my answer accordingly. – Joel David Hamkins Jun 11 '10 at 03:52
  • I'm not sure I agree with your interpretations of the CMS. To me 7.135 seems to support referring to "the previous theorem" but does not directly bear on the issue of "theorem 1.2" v. "Theorem 1.2". Likewise 7.147's "Piano Sonata no. 2" is the title of an entire stand-alone work, more analogous to an entire paper or book than to a theorem. But I agree that their own use of "figure 9.3" suggests that if CMS were to explicitly treat mathematical writing, they would probably favor "theorem 1.2". – Mark Meckes Jul 01 '10 at 13:35
  • On a different subject, does CMS endorse your placement of commas and periods outside of quotation marks? I thought I'd been told that they don't. – Mark Meckes Jul 01 '10 at 13:35
  • Mark, yes, I agree with what you say, but I think that "figure 9.3" suggests very strongly against the "Theorem 1.2" position. So I just don't know what to do now. And I apologize for my misplaced commas---they are typos, since I usually put them inside, and I believe that the CMS agrees with you about this. – Joel David Hamkins Jul 01 '10 at 20:07
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    Actually, CMS doesn't agree with me about the commas. I prefer the style of putting them outside (except where they are part of the actual quotation) as more logical, and to my aesthetic, nicer looking as well. I bring this up partly by way of suggesting that I am not necessarily personally swayed by CMS's likely position on the original issue either. There are of course many other style manuals, too. – Mark Meckes Jul 02 '10 at 00:07
  • I looked it up, and the CMS is very clear (see 5.67 and related) about commas inside quotation marks, but outside parentheses and brackets, which seems to be a near-universal rule, at least in the US. (The British seem far more lax punctuation-wise.) Mathematicians often object on the logical grounds you mention, and although I admit to sympathy with that position, I prefer to follow the established usage. (I don't happen to agree with your aesthetic remark.) There may be other style guides, but none seem to approach the influence, thoroughness and consistency of the Chicago Manual of Style. – Joel David Hamkins Jul 02 '10 at 01:17
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    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 02 '10 at 01:21
  • Now transferring from the other question: regarding "Page 5" I think the reason against you gave above seems still relevant (same for Space 23); noone named anything here it feels more analogous to a description of a location than a name (the place five meters to my left). While for a theorem/lemma/proposition somebody actively named something. That the things are numbered, and the numbers are interchangable and insignificant (but not even this is always true) does not really alter the fact that a part was conciously named while some other parts were not. Yet that being said... –  Mar 07 '13 at 01:13
  • ...I have to agree that this is not an overly coherent reasoning but more a verbalisation of my feeling. And, an example where I have actual difficulty to reconcile my reasoning with my practise is 'Section 2' Here I cannot really say something was named 'Section 2' the name of the section is (perhaps say) 'Some preliminary results' and it happens to be the second one, just like the page. Tricky. What I found mainly suprising about your original comment was the idea of the invoking a general effort against unnecessary excessive capitalization (or something like this)... –  Mar 07 '13 at 01:19
  • ...since then the question is what capitalizations are necessary and why not, say, the chicago manual of style or john or i. But perhaps there are reasons, and now we are likely really off-topic. And, certainly I am not well-placed to discuss capitalization in English. In summary, you somehow undermined my conviction that of course it is Theorem 3, but I doubt I will change my practise of doing so (as I still can see some sense in it, I think it is more wide-spread, and with quite less than good English I rather not entertain taking on somewhat unusual even if perhaps correcter habits)... –  Mar 07 '13 at 01:30
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    ...Perhaps though I will try avoid writing both Section 2 and section 2, and declare the question obsolete for me :) But it is quite interesting. And, also in general I quite like the Chicago Manual of Style. Once upon a time I even bought a copy and thought I will study this all in detail at some point. But...wonder where I even have it. Sorry for the long commentary, which got a bit off-topic; perhaps it is better to remove it? –  Mar 07 '13 at 01:36
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    It is nice to see you going through the same process I went through when I first posted this answer---and obviously I had changed my mind halfway. First I thought of course it is Theorem 5.2, and now I am firmly in favor of theorem 5.2. Try it in your papers; I think you'll see that you like it better. One more example: in many citations, people write Journal of Such And Such, vol. 5, no. 4, p. 12--39. But "volume" and "number" are often not capitalized. But "volume 5" would seem to have essentially the same status as a proper noun or not as "theorem 5.2", no? – Joel David Hamkins Mar 07 '13 at 01:59
  • Perhaps, but then the role of volume is also not so unlike that of page, in particular for a journal. Really, tricky. I think I will have to let this sink in a bit and think it over later. In any case, thank you for the interesting new perspetive on the subject! –  Mar 07 '13 at 02:35
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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".

Mark Meckes
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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
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    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 02:38
  • @Victor: Diese Tatsache ist ja selbstverständlich. – Mark Meckes Jun 11 '10 at 17:17
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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.

Dario Seidl
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    But this does not answer the question because Knuth's examples are all numbered. – Mariano Suárez-Álvarez Nov 09 '13 at 14:24
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    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 09 '13 at 14:28
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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.

Dan Margalit
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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
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    Although if you're referring to the TV show, I think you want to capitalize Big Bang Theory. – Gerry Myerson Nov 09 '13 at 22:38
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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 :-).

coudy
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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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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}
Oskar Limka
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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

Jeremy Orloff
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