A quip of S. Lang states that “analysis is number theory at the place infinity”.  (Which implies, correctly, that analytic number theory is some particularly exalted form of number theory).

The equally quipful E. Witten goes rather further in reducing mathematics to its essentials: during the conference organized by the Mathematics Department of Princeton University in honor of the 250th anniversary of Princeton University, he said something like: “Most of 20th century mathematics is the study of the harmonic oscillator”. (This can be seen, in a slightly different and weakened form, on page 120 of the Google Book preview linked above; my memory is that he did state, during his lecture, something closer to what I wrote; but that was a while ago, so I may be over-reacting in hindsight…)

P.S. For the obligatory etymological epilogue: the word “one-upmanship” is quite recent (1952), but “quip” goes back to the early 16th century. I didn’t know about the charming derivative “quipful” before looking in the OED.

Four letter words

Here’s another funny question about the English language: what is the sequence of three letters XYZ (where X, Y and Z are among the 26 letters of the alphabet) such that the number of correct (English) words of the form TXYZ is maximal, where T is just another letter? In this, it seems also best to assume that the last letter Z is not “s”, since the latter allows for many plurals of what are really three-letter words.

I think that, depending on the dictionary used (in particular, archaic words can be quite common among short words like this, and will only appear in the OED), the answer is between 13 and 17. The dictionary on my laptop (aspell-0.60-en) gives one word with 13 (say TWR), two with 12 (say BGA and IPX), and a few ending with “s”. The ending TWR jumps to 17 when permitted to appeal to the OED, and so does BGA. However, IPX only goes to 13.

Answers next week if nobody tries his/her hand first at the solution…


What does it say of the psychology of English-speaking people that, according to the Oxford English Dictionary, they can say sympathetic in (at least) three additional languages without leaving the confines of theirs? Indeed, we read:

(1) sympathisch, a.

Also erron. sympatisch. [Ger.: see SYMPATHIC a.] =SYMPATHIQUE a.

(2) sympathique, a

[Fr.: see SYMPATHIC a.] Of a thing, place, etc.: agreeable, to one’s taste, suitable. Of a person: likeable, en rapport with one, congenial. Cf. SYMPATHETIC a. 2b.

(3) simpatico, a.

Also (fem.) simpatica. [It. or Sp.: see SYMPATHIC a.] Pleasing, likeable; congenial, understanding; sensitive, sympathetic.

(My impression was that “simpatico” is Italian rather than Spanish, but another dictionary gives “Simpático” for the Spanish translation of “sympatisch” and for the Portuguese translation, so if the accent can be omitted, this makes five languages for the price of three…)

There are of course copious supporting quotations; the best is

“There is something simpatico about Pascal; he is a kind of Central European Baron Munchausen.” (A. Huxley, 1969).

though this one is close:

“I do think, when you get to my age, dear, there is something sympathique about a wig, don’t you?” (E. Waugh, Vile Bodies).

Questions for all friends of alphabets, syllabaries and other dictionaries: Are there examples, in English or another language, of words with more translations allowed? In fact, are there any more translations of sympathetic in the O.E.D?


Here is a minor post about one of the minor pleasures of life: etymology. I realized recently that “a gossip” (as a person) and “une commère”, which are more-or-less translations of each other in French and English, have the same higher-level etymology. In other words, they evolved in the same way from words having the same meaning, that of “godmather”, or “marraine” respectively, although they do not share a common older Latin, or Greek, or other, root (the English word comes from “God” and “sib”, of teutonic origins related to “kin”; the French comes from Latin “co” and “mater”). In English, the forms “Godsibbas”, “godzybbe”, go as far back as the 11th century, but the modern meaning is attested in the OED only from the late 16th century on. In French, the original meaning is seen from the late 12th century, and the modern one from the 14th century.

(I will leave aside any discussion about the historical or psychological insights provided by this parallel evolution of godmather).

There’s not any mathematics in there, of course, except for having decided to look for the relevant information (in the OED and the Grand Robert de la Langue Française, or the Trésor de la langue française, respectively) after discussions with colleagues at ETH where the question arose of the right translation of “gossips” in French. Note that although “commérages” is in a sense obvious, the undercurrents seem to not be quite the same…

Mathematical terminology of yore…

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 Rn/Zn). 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!