Capitalization of theorems

[Fredrik Johansson]

Prime number theorem or Prime Number Theorem?

[Greg Woodhouse]

I believe it should be Prime Number Theorem. It's not just a theorem that happens to be about prime numbers, but a theorem with an established name. It's essentially a proper noun.

[Fredrik Johansson]

Well, I like "prime number theorem". Since it's always prefixed with a definite article in text, I don't think there's any risk of confusion. I don't see a convincing line between the "theorems" and the "Theorems" (I've never actually seen anyone write "Natural Logarithm") and always using lowercase would agree with Citizendium's (not coincidentally also Wikipedia's) general policy of lowercasing article titles whenever lowercasing is correct; this makes everything more consistent. And personally, I think Random Capitalization in Running Text is Ugly.

Also snapped up on Talk:Intelligent design:

Gareth, I tend to agree with you that the name of the theory should be capitalized because it is a name, and capitalizing it is a good way to mark the fact that it's a theory. But consider that the names of many other theories are written out lowercase. I think we need to consult the Chicago Manual of Style--anyone have a copy on hand? --Larry Sanger 09:33, 19 March 2007 (CDT)

Chicago Manual of Style, 15th Ed., rule 8.157: "names of laws, theories, and the like are lowercased, except for proper names attached to them." Examples: The big bang theory; the general theory of relativity (or Einstein's general theory of relativity); Boyle's law. Brian Dean Abramson 00:08, 6 May 2007 (CDT)

I believe lowercase is more common in modern mathematical literature as well. From a quick arXiv search, "prime number theorem" outnumbers "Prime Number Theorem" by 3 to 1. I saw a few hits for things that were apparently not *the* prime number theorem; but they were called "XX prime number theorem" or "prime number theorem for XX", not "the prime number theorem". A couple were inconsistent; I even saw an article that called the (classical) prime number theorem the "prime number theorem" and a different prime number theorem a "Prime Number Theorem" (due to the latter being mentioned in a title ). In the case of "fundamental theorem of algebra", there's no competition: lowercase outnumbers uppercase 10 to 1.

Even if we need to disambiguate between different prime number theorems, I strongly believe capitalization is not the right way to do it. "Prime number theorem" and "Prime Number Theorem" should redirect to the same page; if necessary, a disambiguation message could point to "Generalizations of the prime number theorem" or whatever.

[gerg]

I've been using capitalized versions in my CZ work, but Fredrik's well-argued post above is very convincing. I'm willing to weigh down on the let's-not-capitalize side of the debate.

[Greg Woodhouse]

There is also tradition to consider. As far as I can tell, it is traditional to capitalize the name of the theorem, and insisting on not doing so would seem to be self-consciously iconoclastic. This isn't necessarily a bad thing, but I have to wonder, what the point?

[Derek Harkness]

The capitalisation is to use lower case except for proper nouns. What is a proper noun? 'Proper nouns are nouns representing unique entities (such as London or John), as distinguished from common nouns which describe a class of entities (such as city or person)'

(I've never actually seen anyone write "Natural Logarithm")

'Natural logarithms' are a class of things not the actual thing. This is a common noun and so should not be capitalised.

The question as to the capitalisation of 'Prime Number Theorem' falls on whither this is a name for a unique entity or a group of entitles. Is it a proper noun or a common noun?

An easy test (but not fool proof) is to look at the article that precedes the entity.
- If you would start the article, 'A prime number theorem is a...' with an indefinite article, then it's probably a common noun and is lowercase all the way.
- If you would start the article, 'The Prime Number Theorem is...' with a definite article, then it's probably a proper noun and should be capitalised.

This can get confusing because sometime words can be both proper and common nouns. The decision of which it is depends on the content in which it is used. For example 'prime number' is a common noun. This is a group, a collective term for many entities. 'Theorem,' on it's own, is also a common noun, referring to all theorems. However, 'The Prime Number Theorem' might be a common noun.

[Greg Woodhouse]

An easy test (but not fool proof) is to look at the article that precedes the entity.
- If you would start the article, 'A prime number theorem is a...' with an indefinite article, then it's probably a common noun and is lowercase all the way.
- If you would start the article, 'The Prime Number Theorem is...' with a definite article, then it's probably a proper noun and should be capitalised.

Right. And in this case, it is The Prime Number Theorem.

[Jitse Niesen]

There is also tradition to consider. As far as I can tell, it is traditional to capitalize the name of the theorem, and insisting on not doing so would seem to be self-consciously iconoclastic. This isn't necessarily a bad thing, but I have to wonder, what the point?

It is indeed traditional (one may also say old-fashioned) to capitalize names of theorems. However, the current literature leans towards not capitalizing; at least that's my impression. I think it's a bit of a stretch to consider following current literature as self-consciously iconoclastic.

[Greg Woodhouse]

It is indeed traditional (one may also say old-fashioned) to capitalize names of theorems. However, the current literature leans towards not capitalizing; at least that's my impression. I think it's a bit of a stretch to consider following current literature as self-consciously iconoclastic.

Fair enough.

[gerg]

I've been bold and started the CZ article http://en.citizendium.org/wiki/CZ:Formatting_mathematics, the purpose of which is to gather together all debates and policies related to formatting mathematics in Citizendium articles.''' I've put enough content there to suggest the format I have in mind: one main article with the statements of policies, proposed policies, and issues under discussion; and for each item, a subsidiary article with a detailed discussion of the issue, so that people will know why the policies that are (eventually) decided upon are the way they are (I fleshed out one such discussion article, http://en.citizendium.org/wiki/CZ:Formatting_mathematics/Theorem_capitalization, to give a sense of what I have in mind). I hope that this creation has value for our workgroup! If so, please feel free to develop the skeleton I put there.