in the case of mathematics, is there really such a huge volume of past publications that they cannot be all translated into English? Or is this language requirement anachronistic? Or has it long since been abolished?
I don’t think many schools still require a PhD student to know a foreign language. That went away a while back.
Reputedly by whom?
I’m just about to start my fifth year of the mathematics PhD program at UNC. I can tell you that there was a requirement to demonstrate profiency in two foreign languages when I arrived here that has since been reduced to one.
For the most part I feel the requirement is anachronistic but on occasion I and my colleagues have run across papers, generally newer ones, that either have no English translation or an English translation that isn’t very good. This will generally be corrected after some time, but some times you can’t wait.
When my friend did his math PhD, he took a French course. I don’t know if it was required, but maybe it was suggested. I know that he read some papers in French, though; maybe some by Alain Connes?
Think of it the other way: Are all mathematics papers so popular it’s worth the time and effort to translate every single one into English?
There’s also the advantage that you can hack your way through a foreign mathematics paper with a considerably less then masterful command of the language in question. Equations, theorem names and a decent chunk of the technical terms aren’t really different, and what words are different are usually from a small subset of the foreign language. The word “L’anana” (pineapple) is probably not going to come up in a French Mathematics paper, even though its part of the French language, so you don’t really need to know what it means.
So if you get your mathematics students to take a semester of French, it might actually be useful for them, even though they won’t be able to sit down and read Huis Clos in its native language.
I remember my friend asking for some help translating the term “noyau”. We all agreed that it meant the pit of a fruit (like a cherry or a peach); it took a while to figure out (from context) that it meant “kernel” (of a function or operator).
In that case, knowing math language was more important than knowing French. 
As a practical matter, I’m sure the answer is yes. Of course they could be, but the expense would be too large for anyone to undertake it.
Even when this requirement was the standard, it usually consisted of being able to read one mathematical paper (or even just one page from one paper) in the language of your choice, dictionary allowed. The majority of the page is likely to filled up with epsilons and deltas and equals signs that need no translation, and you weren’t expected to translate it, just understand the math being written about. Very different from learning a language.
Mathematische Annalen has been published continuously since 1868. As of this date, it has published 347 volumes of mathematical research, each one containing anywhere from 700 to 1000 pages, at least half of which (historically) has been published in German. This means that there have been somewhere north of 100,000 pages of German mathematics articles published in this journal alone, or (at 400 words per page) on the order of forty million words. For comparison, the complete works of William Shakespeare run to about 900,000 words, or less than 3% the size.
So yeah. Go ahead and get started on that. When you’re done, don’t forget to translate the other 8-10 long-lived mathematics journals published in the three languages you mentioned, as well as dozens of minor ones. Oh, and of course it’s highly unlikely that you’ll be compensated for any of this in any way, since 99% of it is in the “long tail” that is only sporadically used nowadays. (Sure, when someone happens to need it they really need it, but that’s rare. You’re not going to turn a profit that way.) Get back to me when you’re done, if either of us are still alive.
Especially math, a universal language all it’s own.
What makes you think that?
In addition to University of North Carolina which I mentioned above I checked a couple of other schools (University of Illinois and University of Washington) and they both required at least one foreign language for a mathematics PhD.
Do you have a school in mind that doesn’t require this?
I very much suspect that most professional translators, and even most people who are completely bilingual, would be quite unable to translate a mathematics research paper adequately. Indeed, even most well educated native speakers of English would be incompetent to paraphrase a math paper written in English. Very likely, even many professional mathematicians cannot fully understand many mathematics research papers written in their own language, but dealing with issues outside of their particular sub-specialty.
In general, a mathematician with a rudimentary grasp of a foreign language, and access to a dictionary, has a much better chance of understanding a research paper written in that language than has an expert translator with only a layman’s knowledge of mathematics. And if you can’t understand something, you can’t translate it reliably.
There is also the circumstance that a theoretician thinking in French, German, or Russian may well be making nuanced statements intelligible to an English-speaking specialist in that discipline who can read the language that would totally escape a translator. Two good examples:
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Werden in German translates as “become” but carries a connotation of “makes itself” that the English construction does not convey. I have no idea how one says “X becomes Y” with the implication “Z converts X into Y while X remains the inert, unresisiting patient of Z’s action,” but werden is not the proper verb. It is likely a translator would miss this subtlety.
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The two-word Russian definition of a prime number, «Фактора нет», is literally untranslatable. Both “it has no factor(s)” and “there are no factors” are close but not quite on the mark. It has an ineluctible meaning in Russian that cannot be rendered simply in English; “there is nothing of factor-ness about it” is probably as close as one can come.
Translators make a point of knowing how more general terms change meaning subtly when in specific contexts such as mathematics. While it is true that a translator who has either not specialized in mathematics or who has no previous experience in translating mathematical material would probably miss this subtlety, it is hardly accurate to say that such subtleties are beyond the grasp of translators as a whole.
The correct Russian translation of ‘prime number’ is простое число. While фактора нет might be part of the dictionary definition of простое число (and would be properly translated as you point out, or even acceptably as ‘no factors’), you wouldn’t see that phrase used in Russian when saying ‘prime number’.
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An’ another thing. (I’m sorry, Polycarp, but your statement about how to translate ‘prime number’ is sticking in my craw a little.) The phrase ‘X нет’ is very common in Russian and is hardly “literally untranslatable”. Essentially it conveys the meaning of the absence of X and can be used to mean “we’re out of X”, “I/you have (he/she has) no X”, or in the case of people, “X isn’t here”. So, in the example you cited, фактора нет can be translated as “It has no factor.” (Факторов нет would be “It has no factors.”)
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And I honestly think njtt is selling the translation profession far short here. Translators need to acquire a working level of knowledge of several subjects in order to translate reliably, in addition to having expert knowledge of the language pair they’re working in. While a mathematician with two or three years of college-level Russian and little prior translation experience would probably easily understand a mathematical article in that language, I’d argue he’d have more difficulty translating the article into readable English than an experienced translator with a working knowledge of mathematics would.
Not true. Reading modern mathematical articles is a highly specialized skill that takes training beyond the master’s level, and even then an expert isn’t going to be able to figure out anything too far outside of their narrow subfield.
Sure, for some value of “working knowledge of mathematics”, but that’s probably going to be more than two or three years of college-level math courses. I’m pretty sure I’ve had more college-level math than almost any professional translator would have, and I don’t think I could translate most math papers from English into English.
Why translate everything into English? Why not translate everything into French and make that the official language of mathematics? Or maybe honor the mathematical work done in India and translate all mathematics articles into Sanskrit.
Because every one understands English if you speak loudly and slowly while gesticulating in a manner unrelated to what you are saying.