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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'?"

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Having one's name an uncapitalized mathematical adjective is the highest honor a mathematician can get –  Dror Speiser Nov 5 '10 at 20:17
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I don't want to dictate what others do, but I myself have a consistent convention: I capitalize the lot. –  gowers Nov 5 '10 at 21:22
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If you were to capitalize Cartesian, would you also write coCartesian? –  Tunococ Nov 5 '10 at 22:29
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@Tunococ: Yes, we do. The solution is to use a hyphen. It should be co-Cartesian, pseudo-Riemannian, non-Abelian. –  Spiro Karigiannis Nov 5 '10 at 22:55
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Replacing English with Hebrew would solve all these problems. –  Yaakov Baruch Feb 4 '11 at 3:40

7 Answers 7

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.

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I was told by a liguistics 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.)

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Speaking as a geometer, we use "Riemannian". "Euclidean", and "Hermitian" all the time. Also, the adjective "Euclidean" has been around much longer than "abelian" or "artinian", so familiarity can't be the entire answer. I think it has more to do with culture of algebraists versus culture of geometers, but I may be wrong, and I'm curious to see what others think. –  Spiro Karigiannis Nov 5 '10 at 15:10
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Regardless of the reason, if there are more proper names floating around, it seems to me more likely that people will "recognize" them as proper names, and capitalize them. In fact, it was at least a year after I first saw the word "abelian" that I even realized that it was named after Abel. If geometers did not capitalize "Killing vector field" then no one would know there was a guy named Killing... –  Spiro Karigiannis Nov 5 '10 at 16:27
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I found "Killing" to be seriously confusing when I first saw it used as an adjective. –  Deane Yang Nov 5 '10 at 17:31
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Michael: it is a little silly to say a sequence can't be Cauchy. A red house is also a house that is red. A Cauchy sequence is also a sequence that is Cauchy. Someone who says that is not a good way to write should be reminded that words can have more than one meaning (a name vs. an adjective). –  KConrad Nov 5 '10 at 20:21
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@KConrad. I don't find your comment cogent. The word "red" in "a red house" is simply an adjective. The point was that the term "Cauchy sequence" can be viewed as a compound, rather than viewing "Cauchy as an attributive adjective (which of course could then also be used as a predicate adjective). I don't think that position is silly. But it may be an attempt to push back the tide with a pitchfork. –  Michael Hardy Nov 6 '10 at 18:00

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 ?

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It certainly occurred to me that the question of in which language (capitalizing or not) the adjective was first popularized might have to do with the eventual usage in English. –  Thierry Zell Nov 5 '10 at 15:31

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).

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What's the French practise? It seems that it would be of interest, given the influence of French mathematicians in the last century. –  Ketil Tveiten Nov 5 '10 at 18:28
    
I guess the french, ops, French practice is like the Italian one: they teach you in elementary school that capitalizing an adjective that comes from a noun of person (or geogrphic region, people, languages etc) is a grammar mistake. –  Qfwfq Nov 5 '10 at 22:32
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I'm not sure German is any more consistent about this than English. In any case the 'rule' isn't so simple, because for instance adjectives derived from city names are generally capitalised. –  Colin Reid Nov 5 '10 at 23:46
    
I wasn't saying German is more consistent than English. The problem in English was "abelian" as an isolated exception. (If one wants to get into complications of German usages in such matters, there are also adjectives that get capitalized when used as nouns even though they still follow adjective declensions rather than noun declensions. E.g. "der Auszubildende" = the apprentice, "ein Auszubildender" = an apprentice (the final "r" isn't there when the definite article is used), "die Auszubildende" = the (female) apprentice (without the feminine suffix one uses when it's simply a noun).) –  Michael Hardy Nov 6 '10 at 17:57
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The French rule is that no adjective should be capitalised (except of course at the beginning of a sentence). This rule is nowadays often violated, obviously under the influence of English. The rule extends to the nouns derived from adjectives ("le laplacien") with the strange exception of geographic/ethnic origins: "un ami italien" vs. "un Italien". –  Laurent Moret-Bailly Nov 7 '10 at 8:57

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?

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Calabi-Yau spaces? –  Deane Yang Nov 5 '10 at 18:11
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Or do you mean with the -ian suffix? –  Deane Yang Nov 5 '10 at 18:12
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Richard: Yes, but without a suffix. Nobody would ever capitalize the Galois in Galois group or the Bott in Bott periodicity or the Hilbert in Hilbert space. But if you called it Hilbertian space or Galoisian group then you might think about lower case. I'm not saying we don't name things after mathematicians; I'm saying that, at least in English, in recent decades we rarely if ever turn the proper name into a differently-spelled word in order to do so. –  Tom Goodwillie Nov 5 '10 at 20:47
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Funny: If you reimport a noun-made-an-adjective into the realm of nouns again it may restore its capital initial - Jacobian, Pfaffian, Hessian. –  Peter Arndt Nov 5 '10 at 21:10
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How about Yangian? en.wikipedia.org/wiki/Yangian –  Kevin H. Lin Nov 5 '10 at 23:33

A thought:

The word "abelian" is not pronounced like "Abel" + "-ian".

Hence the connection between the mathematician and the adjective is less strong.

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But I've heard British people say it differently from me: with more stress on the first syllable, and a real "ah" for the first vowel (though still "ee" for the second vowel). –  Tom Goodwillie Nov 5 '10 at 22:15
    
That abelian is not abel+ian in English is just an effect of how native English speakers (in some countries) instinctively pronounce words. The same things occurs with Hamiltonian, which in US English is not pronounced like Hamilton+ian and I would be amazed if this weren't also the case in other English-speaking countries. So I do not think stress change is an explanation of abelian being lower case. Note: Russian is like French in that, as a rule, adjectival names begin with a lower case letter, but the Russian word for abelian is pronounced essentially like abel+ian (абел+ева). –  KConrad Nov 6 '10 at 8:34
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The general rule in English is (for '[something]-ian'), the stress falls on the last syllable before '-ian', regardless of where it was in '[something]'. So, no, this is not the explanation. –  Ketil Tveiten Nov 6 '10 at 12:26

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.

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