Why is "abelian" infrequently capitalized? Posted with input from meta for improvement. I usually read, e.g. "Gaussian integers" and "Riemannian metrics", and occasionally "euclidean" or "cartesian" or even "lorentzian space", but the latter examples are relatively uncommon. I have also seen, e.g. "artinian" (and it has been commented that "algorithm" might be interpreted in a similar context, though the "al-" prefix seems to me to militate against it). I can't imagine that the relative standing of the individual has much effect, because, hey: Gaussian. Perhaps algebraists have collectively decided that spaces with some structure are fair game for the lower-case treatment?
Scott Carnahan pointed out that some people apparently subscribe to consistent conventions, e.g. "don't capitalize when the name is modified to form an adjective. These seem somewhat inconsistent with current practice, though." As Harry Gindi put it: "I think that if you can say X is (term) without the noun following it, it is generally left uncapitalized. So, notice, for instance, 'the ring X is noetherian', 'the ring X is artinian', 'the group X is abelian', 'the square F is cartesian', 'the square F is cocartesian', etc."
So, to take Deane Yang's formulation (note already the discrepancies in capitalization!): "Is there really a rational explanation why it's 'abelian', 'noetherian', and 'artinian' but 'Euclidean', 'Riemannian', and 'Lorentzian'?"
 A: In English one normally capitalizes both nouns and adjectives that refer to languages, peoples, religions, geographic regions etc., Thus "A Catholic priest spoke in French about his Antarctic exploration at a German university."  Writing "abelian" with a lower-case initial "a" while "Euclidean" has a capital "E" is inconsistent usage and can be justified only by the fact that it's conventional.  Sometimes one sees "euclidean" with a lower-case initial "e", but making some of these conventionally lower case while others are conventionally capital would lead to needless complication.  I am philosophically opposed to needless complications.
German follows a different rule: all nouns have a capital initial letter; all adjectives have a lower-case initial except when capitalized for some other reason such as being at the beginning of a sentence or in signs in which all letters are capital.  Thus:
"Die euklidische Geometrie" (de.wikipedia.org/wiki/Euklidische_Geometrie).
"Eine riemannsche Fläche" (de.wikipedia.org/wiki/Riemannsche_Fläche).
A: A simple rule for this is that, if the adjective has a precise mathematical meaning, it should be lower-cased: for example, abelian (= commutative) group, euclidean metric, or gaussian (= normal) distribution. If the meaning is vaguer, referring to the method, style, or approach generally associated with the originator, then capitalizing is appropriate: for example, Bayesian statistics. If the word does not carry an adjectival ending—e.g. Hilbert (space)—it should always be capitalized.
A: I don't think that there's a good answer to this. The following episode at least shows that people have been thinking about the problem of capitalizing Abelian for quite a while.
In her letter to Hasse dated Oct. 29, 1932, Emmy Noether informed Hasse that Ferdinand Springer had demanded that adjectives like "abelian" and "galois" had to be capitalized
in all journals published by Springer. On  Dec. 9, 1932, she wrote that she had passed on "Hasse's declaration of war" via Blumenthal (editor of the Math. Annalen) to Springer.
Apparently Hasse had demanded that this should remain the author's decision. 
A: A thought:
The word "abelian" is not pronounced like "Abel" + "-ian".
Hence the connection between the mathematician and the adjective is 
less strong.
A: As a Springer author, I have accepted that all these adjectives formed from scientists names be capitalized. Therefore, I speak (infinitely many times) of "Hermitian matrices". I even came to think that that was a rule in English. On the contrary, the rule in French is that these adjectives are not capitalized. Could it be that variations ("cartesian", but "Gaussian") have their origin in strong local tradition, which eventually impose themselves everywhere ?
A: I was told by a linguistics professor that de-capitalisation happened when the word was sufficiently common to become 'everyday', so to speak. So, perhaps we have 'abelian' rather than 'Abelian' because we use the word a lot. Certainly, we use 'abelian' and 'noetherian' all the time, whereas say, 'Cohen-Macaulay' doesn't appear quite so often.
It seems like this happens more often to 'algebra words' (abelian, noetherian, artinian) than, say 'geometry words' (Euclidian, Riemannian). Could it be that algebraists use the name-words more in 'everyday' speech than geometers? Maybe they even have sloppier attitudes to grammar-pedantry, or like to save their shift keys; who knows?
(Of course, this is a very tentative hypothesis and I expect to see it demolished by counterexamples.)
A: Hasn't the practice of making mathematical adjectives this way (capitalized or not) more or less come to an end? "Artinian" seems like a late example. I guess "Tannakian" is even later. What else?
