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I will soon be attending a Ph.D program in mathematics. I do not know how common this practice is, but the department has a requirement that graduate students must learn a foreign language and be able to read and write mathematics in that language. The three choices are French, German, or Russian. I personally have no preference on which to learn, but I was wondering if there were other reasons that would make one language more advantageous over the others in terms of a general mathematical career. Thanks in advance in any response.

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    Perhaps surprisingly this actually depends on what subfield of math you will be specializing in, For number theory and algebraic geometry, French seems the best choice. For some aspects of representation theory, Russian might be best, and for finite group theory, German would be the choice. However, even with this in mind, French might be the best choice in general, as the sources only available in French tends to have a higher quality gap down to the English equivalents than the other languages (this is only my impression though, so it might be wrong). Mar 28 '17 at 15:32
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    @wolfgangbangerth I mean, let's just stop learning foreign languages because nothing of any consequence is done in any language other than English and if it I'd it'll be translated right? While we're at it, let's just get rid of math courses too, we already have computers to handle that. Sigh Mar 28 '17 at 18:59
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    @WolfgangBangerth: A counterexample: my dissertation was based primarily on a series of papers published only in French between 2004 and 2007, which prove some rather important results and are pretty consequential in my research area. Mar 28 '17 at 22:33
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    @WolfgangBangerth even if that were true, one does occasionally have to read papers published >25 years ago.
    – Aru Ray
    Mar 29 '17 at 1:12
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    ¨@WolfgangBangerth perhaps your opinion is less relevant here exactly because you work in applied math. It seems that here there is a significant difference between pure and applied math.
    – Rüdiger
    Mar 29 '17 at 19:58
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While it may be true that, if one had to pick a single language in which to read mathematics, English would currently be the best choice, ... it is certainly not the case (as many people do claim) that these days there's an English version of anything worthwhile. True, it is possible to "get by" reading only English, but there are little problems now and then.

In particular, for mathematics, perhaps unlike computer science, things don't really become "wrong" or obsolete (although they certainly can become less stylish), and until about WWII the cutting-edge writing in mathematics was mostly in French and German. (Depending on the epoch, Russians often wrote in French or German, as did Japanese in those times. English was not the lingua franca... :)

So, for people in mathematics of roughly my age, it was necessary to have some reading knowledge of French and German to be professionally competent in those years. And many of the classics of those times have not been supplanted by English-language versions, and are still relevant at least as subtle background to contemporary work.

Also, though I don't make a hobby of tracking this, it is my impression that many French mathematicians in number theory and automorphic forms and representation theory certainly do write in French, and at a level and substance that is not at all automagically transferred to English-language writing. And many pieces in Seminaire Bourbaki, a wonderful resource, continue to be in French.

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The decision does depend on you field. As Tobias Kildetoft mentioned above there may be original non-translated stuff which is interesting to you. Also there could be more position of field A in country B is something to consider. Then in Germany/Austria/Switzerland you may even get away in a teaching position in the beginning without speaking any German and in France this is less likely. I don't know about Russia.

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I took an oral exam to satisfy that requirement in my applied math department. I was psyched. I dusted off my French, which I hadn't used in a while. I was looking forward to the exam. To my disappointment, the page I was asked to translate was mostly symbols; the few words on the page could have been machine-translated, they were so straightforward. So in terms of satisfying your department's requirement, if your requirement is anything like mine was (and you can ask your department to give you a sample page ahead of time, to judge the expected level of fluency), pick French or German, whichever seems easier for you. (For most English speakers, that would be French.) (Avoid Russian because of the other alphabet.)

On the other hand, if your goal is to truly learn a new language, then choose from among all the languages in the world, not just those three. Choose it for more than one reason, not just to make more articles in your field accessible to you. And learn it thoroughly -- or don't bother. Speaking 200 words won't give you the benefits of knowing another language.

The benefits of speaking another language well include: - being able to communicate with a bunch of people you wouldn't be able to otherwise - understanding another culture, especially its humor, in a way you wouldn't be able to otherwise - being able to get nuanced meanings from articles and books in your field (note, these nuanced meanings will not be accessible to you without a less-than-superficial study of the language, which typically takes several years to achieve)

If you want to be able to read papers in your field, and you're not sure if knowing another language will be necessary for that, then do a thorough literature search in your area, to find out which other language(s), if any, will be needed.

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Here is my opinion. Of French, German, and Russian the former two are much more important than the latter, and you will recognize this if you look into the history of mathematics. Hilbert, Riemann, Jacobi, Noether, Weierstrass, Von Neumann, Klein and other members of the Göttingen school (arguably the most influential in the history of mathematics) published in German, while Cauchy, Fourier, Poincaré, Hadamard, Euler and others published in French. Associated with those names is the founding of so many fields of mathematics.. analysis (rigorous calculus), complex analysis, commutative algebra,.. essentially all fields that are the basic cornerstones of mathematics. One cannot say the same about Russian; there are quite a few very decent Russian mathematician, but none is of the same calibre as the mathematicians mentioned above. This makes French and German (by far) the most important languages in the history of mathematics.

In the 1930's Hitler essentially killed German mathematics; Germany's position in mathematics is nowadays merely a shadow of what it used to be. There are still some important German mathematician's, but they are few. And they mainly publish in English (Hopf, Hirzebruch, and Faltings were notable exceptions). This put French mathematics to the front, and there is a countless number of important French publications of the postwar area.. Schwartz's work on distribution theory, Serre and Borel's work in algebraic topology, Serre and Grothendieck's work in algebraic geometry and commutative algebra, Serre's work in number theory. etc. etc.

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    "there are quite a few very decent Russian mathematician, but none is of the same calibre as the mathematicians mentioned above" Maybe "none" should be "few"? I'd certainly include Kolmogorov among rarified group of people you listed. Mar 29 '17 at 14:29
  • @DaveLRenfro well he is perhaps comparable to Fourier or Klein, but certainly not to Hilbert, Riemann, or Euler.
    – Rüdiger
    Mar 29 '17 at 19:27
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    Gelfand? ....... May 6 '17 at 18:20
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    While I do not agree with this answer, I think the conclusion is right, for a different reason. During Soviet times, there were quite a lot of important works published in Russian. Maybe less so in Algebra, but definitely in some of the more applied fields. However the more important works generally had good English translations, often published with involvement of the original author.
    – mlk
    Oct 31 '17 at 18:06

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