While browsing through some issues of Annals of Math. a few days ago, I read the following rather spectacular example of long-forgotten mathematical terminology. It is in a paper by A. Borel, “Groupes linéaires algébriques”, Annals of Math. 64, 1956, p. 20 to 82, which is one of the foundational papers of the whole theory of linear algebraic groups. In this paper, Borel found what groups could play in algebraic geometry the role that “tori” played in the theory of Lie groups (tori being there groups isomorphic to **R**^{n}/**Z**^{n}). Reasonably enough, he decides to call them by the same name (“nous nous permettrons de leur donner le même nom”, or “we allow ourselves the right to give them the same name”). But they had been introduced also by Kolchin earlier (in papers on differential Galois theory from the later 40’s), for whom they were apparently *“connected quasicompact (commutative) algebraic groups”* (the adjective “commutative” being optional by virtue of a theorem of Kolchin). Similarly, it seems that unipotent groups were called “anticompact groups” by Kolchin.

In the same vein, it is also amusing to see Borel write what translates to “Algebraic linear groups” instead of the now dominating terminology “Linear algebraic groups”; the change of emphasis is quite interesting. But Kolchin’s earlier works spoke of “Algebraic matric groups”, which looks like a misprint, but is not. In fact, the Oxford English Dictionary (which we are lucky to have available online here at ETH…) *does* confirm that “matric” is an English word, as an adjective, meaning *“Of, or relating to a matrix or matrices”*. The earliest quotation given is from 1921, and the latest is from 1994 (in a paper in the *Rendiconti del Circolo Matematico di Palerma*; now I would have bet that this latest example was a misprint, as many occurences of the word “matric” in Mathscinet undoubtedly are, but it doesn’t seem so, since the word occurs in the title and in the body of the article. Interestingly, the OED does not gives the names of the authors for this citation (they are I. Bajo and J. Torres Lopera). I wonder (idly) how many mathematicians (in particular, how many born after 1900) have the honor of being quoted in the OED…

Beyond the example of tori (which shouldn’t be taken too seriously, and certainly not as a negative comment on Kolchin!), maybe there is some lesson in this. Often, in particular when a new theory emerges, it’s not clear which concepts should be emphasized and which are less important (or more derivative; for Kolchin, the pivotal role of algebraic tori was probably not obvious at all). However, once the picture clarifies, the terminology should also bring this into focus, and the most basic concepts should have basic (and hopefully, short) names.

For instance, one reason that modern algebraic geometry may seem appealing (for budding mathematicians of a certain kind in particular, whatever their technical inclinations might be) is that quite a lot of the vocabulary not only sounds nice (even almost poetic, of a sort), but is also well considered (from sheaves and schemes to étale and crystals…). On the other hand, I’ve tried a few times to read books on **C**^{*}-algebras, and always find the terminology painful.

There’s probably not much mystery on why bad terminology should be dominating. To find good terminology requires a particular type of creativity; to create great mathematics another type, and there’s no reason the two should usually be combined in the same persons. Except in very few cases, creating the terminology is left to the creators of the theory; if they have no ear for poetry, or no interest in spending some time looking for the right words when new mathematics calls them instead, one may well be left to deal with terrible choices. (It’s a good thing that someone – Weil maybe? or Chevalley? – decided that “valuation vector”, as used by Tate in his thesis, was an awful name for an adèle…) Of course, to avoid the worse, there is the usual escape route of naming a concept after someone, creating bad blood and long-running misunderstandings instead. Maybe Borel should have also tried hard to find a nice name for maximal connected solvable subgroups…

One may wonder why to bother at all about such terminological issues. After all, in terms of mathematical content, a torus is a torus is a connected quasicompact commutative algebraic group. I must say I can’t really explain why I find this of any importance, but I do!