As far as I know, nowadays most of the mathematical literature is written and published in English and mathematicians communicate with each other in English. Although there are certain number of books written in other languages (like EGA), but at the same time their counterparts also appear in English (like Stacks Project). However, many graduate programs still require their students to pass a language translation test in French, German or Russian (which a paper dictionary, not a dictionary app in cellphones, which seems even more ridiculous to me...). I wonder what makes it still necessary to have foreign language requirement as of 2010s.

I believe my question has different focus than this one

Mathematics Ph.D program foreign language requirement

Where the questioner specifically asked for advice for the most useful language among French, German and 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.

while I am asking why we ever need a second language for mathematical study in 2010s:

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    It can be quite helpful to know another language, as is nicely evidenced in this answer to a related question. – Anyon Oct 8 at 13:14
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    Good luck reading the SGA if you only speak English. The proficiency required is not really that much, just basic reading comprehension. – Denis Nardin Oct 8 at 17:34
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    One thing about mathematics, compared to other technical fields, is that older literature doesn't tend to become obsolete. So although most new mathematics is written in English, you will often still need to refer to older books and papers, written before English became dominant and hence likely to be in other languages. This may not happen so much in, say, computer science. – Nate Eldredge Oct 8 at 19:34
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    @NajibIdrissi are you suggesting people learn every language a math paper might be written in? Surely even if you learn two or three, that will still happen? – Kat Oct 9 at 5:47
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    @NajibIdrissi: comments are not for answers. If you think there are advantages in learning a second language, maybe you could write an answer. – Taladris Oct 9 at 9:44

Just an anecdotal answer: My mathematics PhD program (at UC Berkeley) had a requirement to pass a language exam. This involved translating to English 1-2 pages excerpted from a mathematics paper written in French, German, or Russian (student's choice). We were given several hours to do this, and the use of a paper dictionary.

I did not, and still do not, speak a word of any of these three languages (unfortunately). But I took the exam in French and passed easily thanks to the high number of cognates between mathematical French and English, context clues (being somewhat familiar with the subfield of math helps), and the fact that I had plenty of time to look up any words I didn't know in the dictionary.

The main thing I gained from this experience was confidence that with a little extra effort, I could actually read math written in French. Since then, I have at times had reason to use this skill that I didn't know I had.

I believe Berkeley math did away with its language exam requirement the year I graduated (2016), and I think this was an entirely sensible thing to do. But I would encourage all grad students to get some experience reading papers written in a language they don't speak!

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    My experience at UC San Diego was similar, except that it was 5-6 pages from a book, and I got a week to translate it, using any available dictionaries or translation tools. A classmate had gotten somewhere a "mathematical French" dictionary, about a dozen photocopied pages, that focused on words like "therefore" and "conversely" and so on; it was extremely helpful. – Nate Eldredge Oct 8 at 23:46
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    It might be worth mentioning that there are plenty of important math papers (even today!) written in French, let alone the super important historical papers like Deligne's Weil I, II, Hodge I, II, III, SGA, etc. – Denis Nardin Oct 9 at 9:56
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    This was similar to my experience in the same program (finished 1989). As it turned out, my dissertation required reading one paper in French (easy) and two in German, one by Hasse so difficult a native speaker told me it was almost incomprehensible. – Andrew Lazarus Oct 9 at 16:47
  • Indeed, this is exactly right about at least one aspect of reading mathematics not written in one's first language: it is in fact possible to arrange to understand them, although obviously more slowly. That is, more doors are open than one might have imagined... – paul garrett Oct 9 at 22:39
  • This is the same as how it works at Chicago, and I’m glad it’s still required. I believe the language list is the same as what you’ve mentioned, though there’s discussion of adding Chinese – Stella Biderman Oct 9 at 23:40

Part of it is just inertia. But it is a small part, I think. When I studied maths in the previous century there was a two language requirement. Initially it was French and German. Russian was added as a third option when it was realized that a lot of great math was being created in the USSR that wasn't available in English (or French or German). Later, one could substitute a Programming Language for one of your two languages.

However, even today, not everything that a working mathematician wants to know is available in English, so for a practical reason it is useful, still, to have language skills beyond English. Machine Translation has made great strides in the past decades, but mathematics is probably still very difficult to translate correctly. This is partly due to the smaller sample size of available texts on which to train translators.

But, I would, myself, be hesitant to drop a language requirement from a modern mathematics graduate program for a completely different reason. Consider, as I do, that language skill is a help in mathematics itself. Among other things, mathematics is a language, and it requires a certain training of the mind in order to speak it well. Mathematics uses vocabulary and structure to express deep ideas - language. So, language training of any sort, trains the brain in a certain way that may actually assist in the mathematical way of thinking.

I'm not sure I'd be adamant with my colleagues who wanted to replace the last language requirement in a program with something else, but I'd want to hear arguments about how that would make for better thinking. Another math course or two might be useful, but would it be better? Hard to say.

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    “Mathematics uses vocabulary and structure to express deep ideas - language. So, language training of any sort, trains the brain in a certain way that may actually assist in the mathematical way of thinking.” This is absolutely true, but it would seem that just more training in the mathematical language would help more. So far as I am aware, studies have shown that the best training for any activity is that activity; related activities may help, but not as much. Food for thought, anyway. – KRyan Oct 8 at 20:49
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    @KRyan: I'd hesitate to guess that it'd help in a different way, though. Maybe practicing A helps more with A than B does, but B might help in ways that A doesn't. And if you're going to do A anyway, then you'd get the practice you need from A, whereas you might not get what you'd need from B... – Mehrdad Oct 9 at 6:26
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    For language exams at math departments I've seen, they require such a rudimentary familiarity with the language that I don't think preparing for them could be considered "language training." – Kimball Oct 9 at 13:43
  • " trains the brain in a certain way that may actually assist in the mathematical way of thinking." -> except learning another language occupies space in the brain. And that space might've been used for math instead. – user14156 Oct 9 at 20:28
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    @JonathanReez the brain isn't about "space". It is about organization. – Buffy Oct 9 at 20:30

As a thought experiment, let us assume a hypothetical scenario in which there was a tradition for mathematics graduate programs to require their graduate students to take a cooking class. Someone would then come to academia.se and ask: why do they have this requirement? Several well-meaning, well-intentioned, extremely intelligent people would then post answers offering quite rational explanations, which might go along the following lines:

Why do mathematics graduate programs require students to take cooking classes?

  1. Mathematics is an intense activity that requires a lot of energy. Food provides energy. Hence, a mathematician who can cook well and efficiently will be more productive than one who can’t. Moreover, a mathematician who can make tasty food will be happier and free from the distraction of constantly thinking where to find tasty food. Again, they will end up being more productive and producing more and better mathematics.

  2. Mathematics is a social activity. If you can cook well, guess what? Lots of really good mathematicians will want to be your friends and collaborate with you. You’ll produce more and better research.

  3. Mathematics is a language, and cooking is also like a language. Learning how to read and execute a recipe, which is really an algorithm written in a kind of pseudocode with a particular grammar and syntax, is a skill that will carry over well to many areas of mathematics (combinatorics, logic, theoretical computer science, and much more).

  4. Cooking develops your brain. Cooking requires a substantially greater intellectual effort than ordering food at a restaurant. Naturally, having a better developed brain will make you a better mathematician.

  5. Cooking teaches you to care about the order of operations. Ever tried making a recipe and added a vital ingredient at the wrong stage, with disastrous results? I bet you’ll never put your operators in the wrong order in your next algebra paper!

Etc etc.

To summarize, the tradition of cooking classes in graduate programs is 100% logical, and should continue. It is not at all due to inertia. I’m sure graduate programs evaluate their requirements all the time, and are constantly considering the benefit provided by any requirement against the opportunity cost of not replacing that requirement by something else.

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    Also, since cooking is required by most programs, a program that drops that requirement risks damage to its reputation. – PersonX Oct 9 at 21:24
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    A language is a general purpose tool that can be used to use other tools. A meta-tool if you want. Cooking is a tool in itself, not without its own benefits, but definitely not comparable to a language. You might even need another language to understand recipes. So, I don't think this is a fair comparison. (And just to nitpick, from a linguistic point of view mathematics is not a language, because there are no native speakers) – ChatterOne Oct 10 at 9:11
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    @ChatterOne The requirement that language have native speakers seems bizarre to me. Many constructed languages have no native speakers - does this mean they aren't languages? If I raise my child to speak Klingon from birth, does Klingon suddenly become a language, when it wasn't before? Does a language lose that status when its last native speaker dies? I'm curious if you have a reference showing that this is a common view. – Alex Kruckman Oct 10 at 12:57
  • @AlexKruckman This was exactly the argument that CS PhD programs used to get rid of their foreign language requirements. "Our students already speak C like natives; why do they need to learn French?" – JeffE Oct 11 at 17:14
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    @ChatterOne your description of a “meta-tool” (which I rather like) as a tool that can be used to use other tools also fits mathematics itself. Yet for some reason I don’t see French language grad programs requiring their students to master the meta-tool that is mathematics... – Dan Romik Oct 11 at 17:38

(Remark: This answer was written while it was not clear that the question referred to graduate programs in the US only.)

Some graduate programs may come with additional requirements, e.g. teaching duties in the local language or come with a working contract and this may require a certain visa and the visa requirements are not controlled by the graduate programs.

Also, living in a foreign country is much easier if you have basic knowledge of the local language, e.g. for communicating with landlords, offices, or the university administration. And no, English is not always enough (e.g. in France or Germany, you will encounter situations where English is not very helpful).

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    These are valid considerations, but I don't think the language exam addresses either; such exams typically focus only on one's ability to read mathematics in the target language(s). This is next to useless for any sort of non-mathematical communication. Even the grammar one learns is quite limited. – Nate Eldredge Oct 8 at 13:51
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    Learning a language is quite useful generally. However, based on this answer, it seems that PhD programs in _________ (fill in blank with any subject) should require second language requirements. After all, it could be quite useful to be a botanist and know Portuguese if living in Brazil. It could also be useful to know about local foods and traditions. But none of this is required knowledge to get a PhD in botany. – Vladhagen Oct 8 at 17:44
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    I heard a story about a mathematician who tried to extend mathematical language knowledge to general communication, and as a result would say things like "Let us consider a bottle of wine". – Nate Eldredge Oct 8 at 19:30
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    @NateEldredge: I've known mathematicians to talk that way in their native languages. – Mark Meckes Oct 10 at 17:29

Not all math papers were written in the 21st century, and not all have been translated into English.

When I did a Math research paper in college, I found that all the books and papers I needed were written in the late 1800's, in French. Many were not translated into English, and most of the ones that had been were lent out.

My professor was mildly amused to see that my bibliography contained only French original sources. (I finally got some mileage out of AP French.)

The era in which the work you're interested in was done will dictate whether it was most likely written in French, German, Russian, or English.

  • It sounds like you did more of a math history project than graduate level research in math, no? Perhaps that isn't the best reason for a graduate program to make a foreign language mandatory for all their students. – Anyon Oct 9 at 21:56
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    @Anyon, I think your comment/question does accurately reflect the viewpoint of many grad students... but not so many faculty, insofar as the state of mathematics in the late 19th century was very close to our contemporary viewpoint (quite in contrast to other sciences, apparently). If I can have research students who have really assimilated the research-level mathematics prior to WWII (up to which time only a tiny fraction of serious mathematics was written in English) I'd be wildly happy. :) – paul garrett Oct 9 at 22:43
  • @paulgarrett I'm probably just biased by my non-maths perspective, but it certainly seems unusual to me that all the references would be that old. – Anyon Oct 9 at 22:56
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    @Anyon, maybe not all, but it is conceivable that many would be. Many serious, and still-not-resolved, issues were raised then. The only reason we don't see more references to those days is that very many people are not aware... apparently thinking that nothing worthwhile happened before they were born, etc. :) – paul garrett Oct 9 at 22:58
  • I venture to suggest that Latin should be in the list of languages in the last sentence. – Peter Taylor Oct 10 at 8:18

To answer the literal question at the end of the post: yes, I think the optimal second language after English is French, ... and my own students in number theory, automorphic forms, representation theory, etc, would be disadvantaged at not being able to read Sem Bourb and many other things. Yes, the "classic" number theory stuff (Hecke, especially, but also to some degree Siegel) has been rewritten in English... but not everything, and as expected with some "lossiness".

So, sure, one can seemingly "survive" English-only, but I'd feel awfully claustrophobic if I had no idea what those dang not-English-writing French people were doing these days, ... not to mention Germans. Perhaps ironically, it appears that most Russian writing has an immediate English equivalent. (Not at all the case some decades ago.)

It's a question of how to best spend one's time, sure.

Back in the OLD days, there was no internet. You went to the LIBRARY and read BOOKS. For budding mathematicians, a lot of those books were in French, German or Russian. So, you would learn enough of one or more of those languages to learn from those books.

I, personally, took a year of German. Helped with my Master's dissertation, didn't hurt and thought I could use it when I went to Europe.

So, should the budding mathematician take a foreign language? My answer is yes. German preferably, if you can pronounce the word, you can spell it. I could neither get all the various accent marks of French, nor the Cyrillic alphabet of Russian.

  • Whatever your considerations of learning or not learning one of those languages may be, the differences in alphabets shouldn't be among them. I tried teaching Russian to some foreign acquaintances of mine (not seriously, just to have a little fun) - it took me mere minutes to teach them the alphabet, and they could already sort of read (with horrific accents and without understanding a word, of course, but still). And then I had no idea where of how to even start about explaining all those inflection rules. – Headcrab Oct 11 at 7:57

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