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.
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.
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.
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.
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.