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Primary question: What great mathematics are we missing out on because of language barriers?

Please post interesting results, pursuits, and branches in mathematics that have not been translated to English. Make sure to mention the language(s) they are accessible in!

note: this question (and the further two) are about specific theorems or branches in mathematics, not about bringing up 'general answers'. Please post the (original language) names of specific theorems, or tell about theories/branches in mathematics worked on by specific groups with the name of the group and a simple note how to contact the group (a name + university name, or www address, or email address suffice).

And as a secondary question: Please post examples of where mathematical thought is more easily expressed or more intuitive in a language other than English, rather than in English.

As a ternary question: Please post examples of mathematical theories in history that have stalled due to unnatural or awkward spoken-language associated with the theory, while counterparts in another language have flourished.

I know this question is English-centric, but if I were posting this on a German- (Russian-, Spanish-, Japanese-) speaking forum I would make it German-centric; on the other hand English is now the standard into which people translate their papers.

One historical example of a big chunk of maths that never made it to English is Grothendieck's EGA.

A co-question to this question has been asked here: What are good non-English languages for mathematicians to know? However, I disagree with the comments that languages other than English are only good for dead maths; pending research in every country happens in the local language, and that is a huge amount of knowledge to wave away.

A similar question but not directly stated in this form is here: Books you would like to see translated into English

This question is a chance for the native (as well as second-language) speakers of many beautiful languages to tell about mathematical ideas, concepts, theories and results that have not yet transcended the language they were first written in. It is also a place to reflect on expressing specific mathematical concepts in very elegant ways that you think surpass the way we are thinking of them in English. I don't know if this is against the rules, but if not, do not hesitate to post examples in the language you're talking about.

Motivation and background: The recent question on what books we would like to have translated to English has revived an idea, or question, that I had about the way we learn and propagate ideas in mathematics. What knowledge, and more specifically mathematics (just to be on topic) are we missing out on because we don't know the less-commonly-taught second languages?

One person might say that English is the main language currently in which we are publishing papers. For example Mathematics as a language (Wikipedia)

It is interesting to note that there are very few cultural dependencies or barriers in modern mathematics.

This might be true, to some extent. We all have seen those bad papers - many of us have commited the crimes - of submitting papers with broken English, explaining delicate concepts with the subtlety of a jackhammer. The lingual density of mathematics is immense and, since the spoken word is much more precise, we enhance it by using small nuances that we exaggerate (e.g. contains/consists of) to fit more information into the language that we speak. Before this paper makes it to an English-speaking journal a lot of mathematics is lost: it might be lost in translation; maybe the author didn't have enough time to translate everything; perhaps the author brought up some interesting adages that didn't work in English; maybe their first language allowed a specific 'slang' that made the concepts much easier to talk about. Most importantly, before a paper is submitted, a big, big amount of work happens - you will not learn of it before the select results are published; for one thing, it is a lot of time; for another, we all know that sometimes the most interesting mathematics stay hidden because they somehow didn't make the cut. Finally, maybe the research group did not publish to English because their work was meant to support other research in their country; or they just didn't want to bother, being happy with reaching their local environment. This is a place to bring up this sort of research.

In this question we are talking about understanding on a level much higher than 'being able to apply the theorems and formulas'. We are also talking about the 'enlightenment'. A lot of - maybe most? - mathematical thought is encoded in the every-day language being used to describe it, which can be more or less elegant. The fitness of this language to the purpose of the concept can make a big difference - compare the Newtonian school of Calculus becoming stalled because they would not want to forgo the 'dot notation' ($\dot{x}$) that is now largely abandoned and limited only to papers in mechanics. Compare Origins of Mathematical Symbols/Names

A concept can be explained in raw, dry definitions and formulas using thousands of sentences, or it can be explained in a swift, elegant way because the language has got just the right logical constructs and subtle interactions between linguistic concepts to express the logical constructs and interactions between the concepts in mathematical theory we are learning. Compare Examples of great mathematical writing

Related but orthogonal questions that might explain the nature of the problem at hand; further reading:

MO: What are some good resources for mathematical translation?

MO: What’s so great about blackboards?

MO: japanese/chinese for mathematicians?

Towards a New Model of Bilingual Mathematics Teaching: the case of China

Mathematics is Not a Universal Language - Tara N. Tevebaugh; Teaching Children Mathematics, Vol. 5, December 1998 - children are having problems understanding simple, basic mathematics across language barriers; what can be said about immensely more complex ideas that we try to juggle? Concepts can be expressed in mathematical notation; the motivation of those concepts cannot be.

Thanks!

P.S. This is one of my first few posts on MO. I welcome any comments that can make my contributions better.

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    $\begingroup$ This question as posed is just fodder for debate. You might have better luck with a more focused question, such as: "What are examples of important recent mathematical research which are published in Chinese and not translated into English?" $\endgroup$
    – Charles Rezk
    Commented Apr 11, 2010 at 21:17
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    $\begingroup$ The primary question is a big-list target for which I can see little point, really... The premise of the secondary question (that mathematical ideas are more easily expressed or more intuitively in some languages than in others) is of dubious standing, and in any case is the subject of a question which is not mathematical at all! The ternary question, a request for examples of instances of the secondary one, has the same problem. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Apr 11, 2010 at 21:49
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    $\begingroup$ Cheater, you said you'd welcome comments to improve your contribution. I just sharpened up the formatting a bit, so that your three main questions leap out at people, and the discussion following them is separated off. $\endgroup$
    – Tom Leinster
    Commented Apr 11, 2010 at 22:39
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    $\begingroup$ I cannot see the point of your primary request-for-a-list because I cannot tell it apart from either (A) asking for the listing of all books and papers which have not been translated into English, since they all, in principle, were esteemed worthy of publication, hence interesting or (B) asking people to enumerate things they are interested in and which have not been translated into English, resulting in a list which will converge to the first one. Neither of the two cases seems useful. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Apr 11, 2010 at 23:15
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    $\begingroup$ Arnold mentions in his books a lot of results rediscovered much later than they were originally published. Most of those rediscoveries were due to language barrier. $\endgroup$
    – TT_ stands with Russia
    Commented Feb 3, 2014 at 4:11

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Addressing just the primary question, I think there are clear examples where English-speaking mathematicians have been slow to catch up with developments well-known to German or Russian speakers. One I would mention is the result that surface mappings are generated by twists, published by Dehn in 1938, but rediscovered by Lickorish in 1963. The big advances in the theory of surface mappings due to Thurston in the 1970s were, I think, somewhat slowed by the fact that he had to rediscover many results of Nielsen published in German or Danish in the 1920s.

At a more trifling level, I was once embarrassed to learn that a result I published in 1987 was well-known to Russian mathematicians.

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    $\begingroup$ I once got a paper rejected from a journal because "it was well known to Russian mathematicians", even though no published account of the result existed at the time! This contributed to my changing research interests at one point. $\endgroup$
    – José Figueroa-O'Farrill
    Commented Apr 12, 2010 at 0:15
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    $\begingroup$ I was once told that Selberg published an important paper in Norwegian so that he would be able to milk the consequences before anybody else. Can anyone confirm this? $\endgroup$
    – François G. Dorais
    Commented Apr 12, 2010 at 0:22
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    $\begingroup$ "Those that are well-known to Russian mathematicians" sounds like an entry in Borges' fanciful taxonomy: en.wikipedia.org/wiki/… $\endgroup$
    – Steve Jessop
    Commented Sep 11, 2015 at 9:01
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    $\begingroup$ Another canonical example of this is the notion of NP completeness, which was developed independently in the Soviet Union roughly concurrently to or a couple of years before it was found in the west, depending on your particular timeline. $\endgroup$
    – Steven Stadnicki
    Commented Mar 20, 2018 at 3:07
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    $\begingroup$ @FrançoisG.Dorais, I realize it was almost a decade ago that you asked about that anecdote,,, but as far as I know the only paper of Selberg that was in Norwegian was his paper on what's now called the Selberg Integral. He published his result in a journal read mostly by high school teachers and his own claim is that he did this because he assumed the result was just a rediscovery. (Then again...) See arxiv.org/abs/0710.3981 for more, including the history. $\endgroup$
    – Brad Rodgers
    Commented Mar 20, 2018 at 6:05
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Not sure if this quite fits the criteria, but . . .

The Ramanujan-Nagell equation was introduced in 1913 by Ramanujan in the J. Indian Math. Soc. (Vol.5):

Question 464: $2^n - 7$ is a perfect square for the values $3$, $4$, $5$, $7$, $15$ of $n$. Find other values.

The question was asked independently in 1943 by W. Ljunggren in Norsk Mat. Tidsskr. (Vol.25), and answered by T.Nagell five years later in the same journal ("Løsning till oppgave nr 2"). Since that journal is in Norwegian, most of the community did not know of the result until 1961 when Nagell republished it in Ark. Mat. (Vol.30, "The Diophantine equation $x^2 + 7 = 2^n$").

The result is that there are no solutions other than the five that Ramanujan listed.

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I think this question is based on an incorrect premise. In my experience (limited to European languages), it is relatively easy to learn enough of a language to read mathematics in it. Certainly much easier than, say, reading the newspaper. I don't think mathematical thought is necessarily expressed better or differently in this or that language.

Having said that, reading classical Italian algebraic geometry is great fun. But the language is just a small part of it, it's mostly the style.

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    $\begingroup$ I can't say I agree with the first part: people have differing aptitude for foreign languages. You are talking to a guy who knows numerous languages at a level sometimes surpassing that of the native speakers, but I know that 1. many aren't that good at it 2. there simply isn't time to spend translating papers when you're slow at doing so. Your answer also doesn't address the ability to use natural language for swift talk about maths. Finally, I don't think someone who only knows a language on 'I want eat now' level can understand elaborate motivation behind a complicated concept. $\endgroup$
    – cheater
    Commented Apr 11, 2010 at 22:07
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    $\begingroup$ As someone who knows 8 languages, and did exactly as you say with maths books, I disagree for two reasons. First, picking up a language to understand maths is harder than you say. As you continue reading a book, higher level concepts crop up that require advanced use of a language. It's easy to unknowingly misunderstand something. On top of that, the cognitive load is immense and prevents you from learning maths. You may have read a chapter or two, but not a whole book. Second, the language barrier fragments communities, disabling the network effects that propagate ideas across a social graph. $\endgroup$
    – cheater
    Commented Mar 23, 2018 at 12:02
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    $\begingroup$ It bears mentioning that the language barrier fragments communities, disabling the network effects that propagate ideas across a social graph. $\endgroup$
    – cheater
    Commented Mar 23, 2018 at 12:04
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    $\begingroup$ I agree with cheater. Not knowing Arabic or Hebrew or Japanese, I cannot imagine trying to read mathematics in any of those languages. The initial part of the learning curve is simply too high. $\endgroup$
    – Timothy Chow
    Commented Apr 13, 2018 at 21:35
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    $\begingroup$ A conjecture both deep and profound/Is whether a circle is round/In a paper of Erdös/Written in Kurdish/A counterexample is found. $\endgroup$
    – Gerry Myerson
    Commented Jun 10, 2020 at 6:42
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Reviel Netz, a scholar who has done work on the Archimedes Palimpsest, states in The Archimedes Codex that too few scholars read [ancient] Arabic.

He makes a suggestion that there may be more unpublished works of Archimedes and other Hellenistic Greeks that are to be found in some hidden Arabic library of the late first/early second millennia.

Netz argues, however, that the Arabic version would likely be far more removed from the original Greek, because copyists in the Arab world of that time were more competent/more willing make corrections/changes to the text for clarity in-situ, as opposed to the European copyists/monks who would copy verbatim, carrying mistakes and difficult readings forward down the manuscript tree.

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The "we" in the question/premise is not homogeneous. E.g., mathematicians of my age or even a bit younger in the U.S. would have been required to demonstrate some nominal reading proficiency in French and German, with Russian as a possible alternative. Latin, French, and German were taught in most high schools in the U.S. then, as well as Spanish, and it was no secret that studying Latin, French, and German was a good idea for anyone interested in the sciences, or possibly philosophy, and other things that were valued in the high-culture of Western Europe in the late 19th and early 20th centuries.

Indeed, "when I was a kid", many of the primary sources were not in English, and no one thought to object, or even remark upon it.

What "we" in those times often did "miss" were developments in Russia, written in Russian. Most Japanese writers pre-WWII wrote in some European langage, German, French, or English.

Many Russians (pre-revolution) wrote in French or German.

Quite a few people from the U.S. wrote in French or German in those times.

Nowadays, for many things there are indeed English-language versions. But this is not absolutely true, and it is invariably convenient to be able to read some French or German. But also possible to not do so...

What are "we" missing? Well, probably not much so far, because there is still an older generation familiar with languages other than English... but there indeed may be something forgotten a bit later...

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    $\begingroup$ The question is vague by design to allow people to share things that they think some people might miss out on due to language barriers. There is no reason to nit pick or to bring up the fact that a small population might know a certain language, therefore there's no language barrier to that language for that small population. $\endgroup$
    – cheater
    Commented Mar 23, 2018 at 11:57
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    $\begingroup$ @cheater, my point is that some "barriers" of all sorts are self-created, and or self-allowed, and avoidable. Maybe the "small population" is exactly the most serious, dedicated professionals? The true "novelty" in comtemporary attitudes is the (dubious) notion that being monolingual is feasible in a professional scholarly environment. $\endgroup$
    – paul garrett
    Commented Mar 23, 2018 at 12:28
  • $\begingroup$ is the fact that monolinguality is "novel" the reason why the most respected journals only accepted work in Latin until recently? $\endgroup$
    – cheater
    Commented Mar 23, 2018 at 13:08
  • $\begingroup$ the original question states things as they are. You may wish upon a star that everyone knows 10 foreign languages, but that will not change the fact that, in reality, there are languages that few people know outside that language group, and that because of this, everyone else misses out on knowledge shared in those languages. The original question asks for examples of such works, not for discussion of whether th phenomenon of language barriers is desirable or can be overcome. $\endgroup$
    – cheater
    Commented Mar 23, 2018 at 13:11
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    $\begingroup$ @cheater, it is simply not true that "until recently" respected journals only accepted submissions in Latin, etc. The very notion of "journal" did not really have any sense until the 19th century, etc. The seeming "obstacle" in the premise is a "straw man", is my point. That is, "lots of people" will miss "lots of things" by being monolingual and/or semi-literate (and the latter is worse than the former, probably). But not everyone will miss things... The "we" is very heterogeneous. $\endgroup$
    – paul garrett
    Commented Mar 23, 2018 at 13:41
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There is a paper in French published in the last 10 years that I read in detail, and I subsequently wrote a paper in which ideas found in this paper played a significant part. There are several colleagues who have told me they don't read French when I suggested they look at this paper.

Then again, I can't say with much certainty that language is much of a barrier here. Most papers are only read in that level of detail by two or fewer people anyway.

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    $\begingroup$ I don't think reading in detail is the main issue. I will only read the paper if the abstract/introduction/review is of interest to me. Some multilingual environments have interesting solutions, in some Canadian journals it is traditional to have the abstract in French and the article in English or vice versa. $\endgroup$
    – François G. Dorais
    Commented Apr 12, 2010 at 2:11
  • $\begingroup$ Alexander, thanks. Can you tell us what the paper was? What did it talk about? $\endgroup$
    – cheater
    Commented Apr 12, 2010 at 11:51
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I imagine the questions are more suitable for linguistics than for mathematics. At the core of them is the ability (or lack thereof) to express "things" to an appropriate degree of subtlety or granularity. If I were an Eskimo or one who studied dynamics of materials, I might want 57 different words for snow. Also, I might not understand algebraic geometry as well as one raised in France, but that's not the main reason I am not an algebraic geometer.

I suggest your question would be better approached from such a standpoint. If there are mathematicians out there who also know some linguistics, they may be able to explain why some concepts were developed earlier in certain cultures than in others. My guess is that a particular problem or subarea of research develops because someone is interested in it, and that the culture of the developer(s) sometimes plays a role in how that development is expressed. In other words, if I really wanted to be an algebraic geometer, being Texan shouldn't also be a hindrance, and might bring some (more) flavor to the subject.

Something that might be revealing and helpful is the following: what languages are easier (or preferred or more natural) to use to motivate, teach, enlighten people about calculus? Or the (Skolem?) paradox that in set theory there is a countable model of the reals? Or algebraic geometry?

Gerhard "Ask Me About System Design" Paseman, 2010.04.11

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    $\begingroup$ en.wikipedia.org/wiki/Eskimo_words_for_snow $\endgroup$
    – S. Carnahan ♦
    Commented Apr 12, 2010 at 5:12
  • $\begingroup$ At least I stopped short of claiming that Eskimos had 57 words for snow. I am willing to hear an Eskimo (or material dynamicist) comment on the desirability of having that many words for snow. Gerhard "Missed It By That Much" Paseman, 2010.04.11 $\endgroup$
    – Gerhard Paseman
    Commented Apr 12, 2010 at 5:49
  • $\begingroup$ At most a comment - a welcome and revealing one (thanks!), but not an answer: it does not contain mentions of any specific objects, or theorems, or theories, by specific mathematicians. The question is about results (specific theorems you can quote here) that are not known broadly in the English language space because they have not been translated to English. It is very much a question about mathematical objects, only one minor qualifier is socio-linguistic ('was not translated to English'). The more important qualifier is that what is brought up needs to be an 'interesting' result in maths. $\endgroup$
    – cheater
    Commented Apr 12, 2010 at 10:53
  • $\begingroup$ At reputation level 15, one must first provide an answer, and then one can comment on that answer only. Even so, I believe that the questions I provided will help in answering your questions. In particular, the name-use confusion that often arises in metamathematics (e.g. the Skolem and other paradoxes) I think may be more pronounced in some languages than others. I wish you luck in finding the examples you desire. Gerhard "Ask Me About System Design" Paseman, 2010.04.12 $\endgroup$
    – Gerhard Paseman
    Commented Apr 12, 2010 at 13:46

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