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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

dfeuer
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generic properties
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  • 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. – Thomas Andrews Nov 04 '13 at 15:29
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    http://math.stackexchange.com/questions/4526/mathematical-concepts-named-after-mathematicians-that-have-become-acceptable-to – HJ32 Nov 04 '13 at 15:31
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    I've heard it said that only those truly influential have their names immortalised as an adjective without capitalisation. – Dan Rust Nov 04 '13 at 15:33
  • While I have usually seen "Cantorian" (the concept from NF studies) capitalized, I have seen authors simply let it stay lowercase. I suspect it comes from laziness when you're typing the word repeatedly in almost every document you write :p – Malice Vidrine Nov 04 '13 at 15:36
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    I spell all such terms with an upper-case letter: Abelian, Cartesian, Latin square, Boolean algebra, etc. – Brian M. Scott Nov 04 '13 at 15:40
  • As a non-native speaker, actually I don't get that subtle difference between them. Does it really feel different to read an upper-cased one and a lower-cased one? – generic properties Nov 04 '13 at 15:43
  • @BrianM.Scott The book I learned combinatorics from was very specific about "latin squares," but I see from a bit of searching that this is not universal. – Thomas Andrews Nov 04 '13 at 15:45
  • @dielectric: For me, at least, seeing the capitalization very much emphasizes the crediting of a person, while a lowercase form has the feel of "just a thing people say". – Malice Vidrine Nov 04 '13 at 15:47
  • @dielectric Without the capital letter, "abelian" doesn't imply there is a person for whom the property is named. Abelian group implies there is some proper word root - either a person, or a school, or a nationality, or some other proper name. That isn't a hard and fast word. The English word "quixotic" comes from the fiction character Don Quixote, for example. – Thomas Andrews Nov 04 '13 at 15:49
  • @Thomas: Sounds to me like someone trying to make a rule out of his prejudices. – Brian M. Scott Nov 04 '13 at 15:54
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    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. – Stefan Hamcke Nov 04 '13 at 15:55
  • @Stefan: THAT's interesting... It poses me a question:Would it be an influence from English or an independent liguistic phenomenon of German? – generic properties Nov 04 '13 at 16:01
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    @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. – Hagen von Eitzen Nov 04 '13 at 16:04
  • @DanielRust Then you seem to be very influential, especially in the field of decaying metals :) – Hagen von Eitzen Nov 04 '13 at 16:05
  • @HagenvonEitzen: Oops, I was wrong, as you say, it's indeed either Noether'sch or noethersch. But even constructions as linksnoethersch (left-Noetherian) are possible in our language. – Stefan Hamcke Nov 04 '13 at 16:14
  • @StefanH On the other hand, at least when I was studying, it was not uncommon to read or write "Sei $X$ hausdorffsch." – Daniel Fischer Nov 04 '13 at 16:15
  • @DanielFischer: I even use Hausdorff as an adjective, but if I'm really lazy, then I just write $T_2$ :-) – Stefan Hamcke Nov 04 '13 at 16:16
  • @HagenvonEitzen I think rustian would need to be an established adjective, but the sentiment is warmly accepted :). – Dan Rust Nov 04 '13 at 16:22
  • @DanielRust: You never know: fustian is a noun! (I wonder a bit about your hæmoglobin, though ...) – Brian M. Scott Nov 04 '13 at 16:26
  • @HagenvonEitzen That's Neue Deutsche Rechtschreibung with the apostrophes? It sure looks wrong to me. – Daniel Fischer Nov 12 '13 at 20:56

2 Answers2

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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 teaching 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.

Per Erik Manne
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  • This is a much better answer than anything my professors have ever made up. Thank you! – Eric Stucky Nov 05 '13 at 18:22
  • Nice answer! It is pretty surprising that Lang and Mac Lane played important role. – generic properties Nov 05 '13 at 18:23
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    Since this is a language question, I hope I may mention that, in usual English, one would say that Artin taught Lang algebra, and Lang learned algebra from Artin, but not that Artin learned Lang algebra. – LSpice Apr 09 '23 at 19:13
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    @LSpice Thanks for the correction! I should of course know this - my only excuse is that Scandinavian languages lack the distinction between teaching and learning. Lang is clear in his preface, though: "Since Artin taught me algebra, my indebtedness to him is all-pervasive." – Per Erik Manne Apr 12 '23 at 11:32
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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.

Michael Angelo
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