From: Allan Adler Subject: Re: Reading math in unfamiliar languages Date: 28 Sep 1999 04:05:12 -0400 Newsgroups: sci.math Keywords: [missing] dtreu@yahoo.com writes: > What languages would be most useful to an aspiring mathematician? > (German, French, English are indispensable, I understand. Are there > others?) It depends on what mathematicians you want to read. Archimedes wrote in ancient Greek and Gauss wrote in Latin. There is a source book on the parallel postulate entitled Axiome de Paralleles (if I remember right) that I saw in the library at MIT a few decades ago. It is a bilingual edition of important sources on hyperbolic geometry, bilingual in the sense that each source is presented in its original language and in Interlingua. Thus, to read Bolyai from this source, you have a choice of learning Interlingua or learning Hungarian. A lot of 19th century algebraic geometry is written in Italian and there is still reason to read it. One of the articles in Weil's collected works is in Italian. Are ALL of Gromov's old articles available in English or are some still only available in Russian? Some of Fredholm's articles, as they appear in his collected works, are in Swedish. Even if something happens to be available in English translation, as many things are these days, an edition in the original language might be much cheaper. So there is an economic advantage in being able to read other languages, even when it isn't absolutely necessary. I found it necessary to read an article of Landazurri in Spanish several years ago. Kuga's book Garowa-no Yume was published in English translation several years ago under the title Galois' Dream (which is what Garowa-no Yume means) but this is not really a translation. Two chapters were omitted, Kuga's carefully chosen bibliography was replaced (rather than supplemented), among other atrocities. The new edition would be better called Kuga's Nightmare. If you want to read what Kuga really wrote, you have to learn Japanese. This is not a bad idea anyway since sometimes Japanese articles are circulated as Japanese preprints before being translated into English; this was the case with a paper of Kurihara that I studied a while back. Other stuff never gets translated: as far as I know, Nagata's algebraic geometry book still exists only in Japanese. Newgebauer published a book entitled Mathematical Cuneiform Texts. To read it critically, one probably needs to know a little about reading Akkadian and Sumerian on clay tablets, including the Akkadian and Sumerian languages. There is a Sanskrit mathematical treatise entitled Lilivati. Omar Kahayyam, in addition to writing the Rubiyat, wrote a mathematical treatise on problems in Euclidean geometry. The only edition I know was published in Teheran in Persian and Arabic. Some of the books of Diophantos were recovered from Arabic translations. There is an edition of these, including the Arabic, published by Springer-Verlag. The collected works of Euclid, as published by Teubner, have not only Greek but Arabic. Some of the bilingual editions of Greek mathematics, e.g. Heiberg's edition of Archimedes, are Greek and Latin. They were written so that the average educated person (who, of course, knew Latin, but not necessarily Greek) could read them. Heiberg's commentary on the mathematics is also in Latin. There is no end to it... but the cool thing is that once you can read mathematics in other languages, you can also contemplate reading other things in those languages and that opens up some rich territory indeed. > How well do I need to know these languages in order to read math texts? That also depends. Sometimes mathematics is written with so much standard notation and equations that one almost doesn't need to know the language at all to read the paper. When I wanted to read some papers of Giambelli and other Italian mathematicians, I got the papers and an Italian-English dictionary and ploughed through them without having studied the language (I did, however, know French and some Latin). One of the lasting effects of having done this is that I acquired a passive knowledge of the language. I cannot speak Italian to this day but I can get a sense of what something means when I read it. The same thing happend with Spanish after reading Landazurri's article. I would actually go further and suggest that a mathematician has an advantage that others who learn a language usually don't possess: knowledge of mathematics allows him/her to decipher the language to some extent. Assuming the paper is correct, if you want to know whether it is asserting some theorem, you can try to prove the theorem yourself. If you find it is true, you gain some confidence in your intuitions about the language as well. A tremendous amount of information is contained in format as well: when I was in India, someone showed me a page of a magazine written in, I think, Telegu. I know only a few words of Telegu and never learned to read it, but I just knew from looking at the page that it was your typical horoscope page; it was all in the format. One other advantage that a mathematician has is that the vocabulary of mathematics is quite limited and the prose is formulaic and repetitive in many respects. That means that instead of focusing so much on vocabulary, a mathematician learning to read mathematics in another language can instead focus on grammar and syntax. Here, the advantage of the mathematician is quite unique, since the ordinary constructions of mathematical locution are more complicated than those of more informal writing and the mathematician is already used to them. There are grammatical constructions that occur in the mathematics of Archimedes that occur much more rarely in other Greek writing. By being able to insist that the mathematics make sense, the mathematician can overrule his reference materials: if a theorem becomes false unless a certain word has a certain meaning, then to hell with the dictionary (which was probably written by non-mathematicians and based on non-mathematical writings), you've got a good case for the meaning indicated by the context. There is also some mathematics written in turgid prose rather than in equations and notation. For such stuff, you really need to know the language well. Introductions may be harder to read than the rest of a paper for that reason. I might add that there might be some advantage in not having perfect mastery of a language as spoken by our contemporaries. I've noticed that when I show someone from Germany some prose by Felix Klein or when I showed someone from Italy some of the libretto of Il Trovatore, and asked for some help with it, they were both unaccustomed to reading stuff like that and were to some extent thrown by the archaic form of the language. But for someone who is glad to make any sense at all of what he/she is reading and doesn't necessarily expect to, such difficulties are nonexistent. > How do I teach myself these languages? Really, I'm not interested in > being able to communicate well. Just read well enough to follow math. I would suggest that you just plunge in. Choose some mathematician whose work you particularly admire and move heaven and earth to read it. I got better at French by reading Sierpinski's Hypothese du Continu as an undergraduate, even though my goal was only to read his book which looked so fascinating. There is no harm in reading books on or synopses of grammar as well, since the more you know the better, but it is also good to develop a kind of "street smarts" about languages. Fortunately, languages fall into some families and by learning one in a family, you have a real advantage in trying to learn another one. Donald Knuth says in The Art of Computer Programming that anyone serious about computer programming should expect to learn at least one new computer language a week. I would recommend that a mathematician approach natural languages in the same spirit. I might add that there are some specialized works that are quite helpful. For example, the book of McIntyre and Witte, "A German-English Mathematical Vocabulary" (now out of print) really helped me get started reading mathematics in German. Allan Adler ara@altdorf.ai.mit.edu **************************************************************************** * * * Disclaimer: I am a guest and *not* a member of the MIT Artificial * * Intelligence Lab. My actions and comments do not reflect * * in any way on MIT. Morever, I am nowhere near the Boston * * metropolitan area. * * * ****************************************************************************