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Many mathematical properties are derived from the name of a mathematician associated with them: "Abelian," "Noetherian," "Artinian," "Frobenius (algebra, category, etc...)," "Cauchy," etc. It seems that I've given these examples more-or-less in increasing order of probability of capitalization: the locution "abelian group" is very common, while I'd expect to nearly always see "Cauchy sequence."

Is there a conventional way to determine capitalization in such situations?

The simplest solution, of always capitalizing, doesn't seem to be available, due to the situation with "abelian." It would probably be my preference to capitalize if and only if the name appears unaltered? But "Noetherian" and "Artinian" seem often to show up capitalized. Beyond that, it seems you'd get into subtler issues: perhaps it's important whether a word is an integral part of a noun phrase, as "Eilenberg-Maclane space," or rather an adjective, as the first few of my examples. Of course, then we'd probably have to say, "Every Cauchy sequence is cauchy..."

Kevin Carlson
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This variation concerns adjectives like Abelian/abelian, not nouns like Cauchy or Frobenius, which as far as I know are always capitalized.

My impression is that the general convention in English, outside mathematical circles, would be to capitalize adjectives that arise this way. Personally, I like the capitalized forms much better.

The question is: Why does math differ from the general convention? I would conjecture that the reason is because English mathematical language is heavily influenced by French, where adjectives formed from proper names are spelt with lower-case letters (as in eulérien, noethérien, arguésien). I would guess that the lower-case spelling is more common for words that have entered the language from French, or that are most commonly used by researchers working in areas of math where there are reasons to read things in French a lot.

I think this fits well for "noetherian space" and abelian versus Riemannian, for example. Cartesian comes from French, of course, but it's used commonly outside the mathematical community, so the general rule in English has been applied to it and mathematicians haven't departed from it in that case.

Edit: I've just examined the first twenty results on Google Books for "Noetherian scheme" and "Noetherian ring." For schemes, it was capitalized seven times out of twenty and for rings, seventeen times. Algebraic geometry is a field where you read stuff in French often.

user208259
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