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I would like to extend a previous question on the capitalization of theorem names . This time, it's not about numbered theorems like "Lemma 1.1", but about theorems that carry a name.

Question: Which of the following variants is to be preferred?

  1. "By Zorn's lemma, we can conclude..."
  2. "By Zorn's Lemma, we can conclude..."

Additionally, I would like to know how to deal with the corner case where the name of the person after which the theorem is named starts with a lowercase letter

Question: Which of the following variants is to be preferred?

  1. "The proof of van der Waerden's theorem can be extended to ..."
  2. "The proof of Van der Waerden's theorem can be extended to ..."
  3. "The proof of van der Waerden's Theorem can be extended to ..."
  4. "The proof of Van der Waerden's Theorem can be extended to ..."
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closed as off topic by Charles, Suvrit, quid, Andy Putman, Chandan Singh Dalawat May 24 '12 at 6:59

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This is a question about Dutch spelling practice. Note that Dutch-speaking Belgians have different rules, and then there's the French... – Chris Godsil May 23 '12 at 12:31
Downcasing ("Zorn's lemma", "van der Waerden's theorem") is increasingly popular. "Zorn's Lemma" is still acceptable if you prefer it, though. But really, this is a question about English rather than math. – Charles May 23 '12 at 13:20
In Dutch, the proper way of writing the name would be "Van der Waerden", and to sort the name under the "W". This does not mean that you should do that when communicating with non-Dutch people though. I don't know how this should be done in Flemish, French or German (von Dyck or Von Dyck?). – Jan Jitse Venselaar May 23 '12 at 14:33
If this question is inappropriate, I apologize, but I do not believe that consulting, say, a generic style manual will help me adhere to the conventions that are actually practiced by the mathematical community. It won't be much of a problem if I refer to "Van Der Waerden's Theorem" in my thesis, but I would like to dot all i's and cross all t's. – Greg Graviton May 23 '12 at 19:11
@Greg Graviton: first, personally I like this type of question, but in principle agree it is not on topic. You say you will not be able to know the conventions of the math community by consulting standard style guides. But look, either there is a set convention then it will be ease to figure it out by just emulating what is done in the majority of (high quality) source you use, or there is none. Actually I think the later will be the case. Eg with the right crowd you can have a good discussion whether it is Abelian or abelian. – user9072 May 23 '12 at 20:24

If you are writing for a journal or book series, follow their style guide. Both versions

Zorn's Lemma

Zorn's lemma

can be found recommended in different places. Sometimes you could imagine more than two possibilities:

Chinese Remainder Theorem

Chinese Remainder theorem

Chinese remainder theorem

And also

Triangle Inequality

Triangle inequality

triangle inequality

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I'd say that in general, such questions about standardization in mathematics (not only in mathematics, of course) are quite important today, because they make easier the use of search engines. So, it is relevant that we all agree to call "Zorn's lemma" a certain result, instead of e.g. "Chain's Upper Bound Theorem", much more relevant than in the past.

In particular, spelling of names is an important issue, of course, and puts a problem, for several different transliterations from other alphabets are used: e.g., Чебышёв is romanized as Chebyshev (English), Tchebychev (French), Tschebyschow (German), Chebyshov (Spanish), Tsjebysjev (Dutch), Čebyšëv (ISO 9),...&c. Here, we should at least care to write correctly the names in their original form, which is also a point of respect (yet we meet wrong forms like Lebesque or Holder or Erdös even here at MO).

On the contrary, the issue of upper vs lower case does not seem that relevant, as search engines are usually case-insensitive. Also, Google does not care so much about the Saxon genitive, even for queries into commas.

In conclusion, as to your questions, for your thesis I would mainy care to respect an internal coherence, and of course, the correct spelling of personal names.

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