This is just a question for fun:

As far as I know, frequently it is considered to be customary to denote an additive commutative group as 'abelian group' in lowercase, although the term is named after the mathematician Abel.

And also as far as I know the reason is:

Among mathematical adjectives derived from the proper name of a mathematician, the word "abelian" is rare in that it is often spelled with a lowercase a, rather than an uppercase A, indicating how ubiquitous the concept is in modern mathematics.

…as Wikipedia says.

But it seems quiet funny to me, since there are other many ubiquitous concepts throughout the whole mathematics that are named after many other mathematicians, but they are still spelled in uppercases (like 'Gaussian').

So my question is:

  1. How did the term 'abelian' started to be spelled in this way? How was such custom absorbed to the mathematician society?

  2. Are there other terms named after mathematicians that have custom of lowercase spelling in mathematics?

Maybe it more a social science question than a mathematics one, but it might be funny to know the root of the term that I have to use every day. :D

  • $\begingroup$ Another example is "latin squares," where typically in English you'd call them "Latin." Thought not a person's name, still the same case of non-capitalization of a proper name. $\endgroup$ – Thomas Andrews Nov 4 '13 at 15:29
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    $\begingroup$ math.stackexchange.com/questions/4526/… $\endgroup$ – HJ32 Nov 4 '13 at 15:31
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    $\begingroup$ I've heard it said that only those truly influential have their names immortalised as an adjective without capitalisation. $\endgroup$ – Dan Rust Nov 4 '13 at 15:33
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    $\begingroup$ In German it's the same, we write abelsche Gruppe instead of Abelsche Gruppe (Note that a noun always begins with an uppercase in German, so there is nothing special about the "G ruppe"). But then we write Hausdorff-Raum or Noetherscher Ring. $\endgroup$ – Stefan Hamcke Nov 4 '13 at 15:55
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    $\begingroup$ @StefanH In German there are in fact rules that abelsch is correct, otherwise one would write Abel'sch; also it should be either Noether'scher Ring or noetherscher Ring. I'm not sure if the English morphologoy has similar rules, e.g. for -ian vs. -otic. $\endgroup$ – Hagen von Eitzen Nov 4 '13 at 16:04

Traditionally, "Abelian" would be capitalized, as one finds, for instance, in the writings of Burnside and Ledermann.

In 1937, Emil Artin, who had a Jewish wife, had to leave Germany, and settled in the United States. He wrote the very influential Galois Theory in 1942, where "abelian" was used in lower case throughout the book. It is possible that Artin was unaware of the nuances in capitalization rules between German and English. It is also possible that his proof readers, if he had any, were unaware that "abelian" referred to a person.

Serge Lang credits Artin with learning him algebra, and he followed his use of lower case "abelian" in his Algebra (1965). Birkhoff and Mac Lane also used lower case in their Algebra (1967). With such influential text books setting the pace, it was natural for later generations to follow up.

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  • $\begingroup$ This is a much better answer than anything my professors have ever made up. Thank you! $\endgroup$ – Eric Stucky Nov 5 '13 at 18:22
  • $\begingroup$ Nice answer! It is pretty surprising that Lang and Mac Lane played important role. $\endgroup$ – generic properties Nov 5 '13 at 18:23

In general (or at least in Dutch spelling), you write words with a lowercase letter when 'most people' don't really know or recognise the person after whom it was named anymore. For example, in official Dutch spelling they have recently changed 'Cauchy sequence' to lowercase as well. This hasn't been very well adopted by the mathematics community, as far as I know, though.

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