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I am teaching in an Eastern Asian environment (precisely, teaching Mathematics using English in Korea, with Asian students) and I figured out that my reasoning is a lot based on my language proficiency in English and latin languages. As a latin languages speaker, mistakes in the difference between definite article “the” and indefinite article “a” hurt me a lot, whereas these articles do not exist in my students' mother language (at least, in Korean). I also have a quite efficient writing style for proofs, organizing my reasoning on the multiple logical connectors that Western Europe languages provide: so, then, hence, therefore, as a consequence,...; the actual content of my sentences being as simple as possible to convey the mathematical content in the most straightforward way. More elaborated sentences are used for broad introduction of ideas, and conclusions. But other languages have a completely different structure.

I would be interested in references on how the structure of a language influences the logical reasoning. For example, I believe that the importance given on existence problems and unicity problems is mostly due to the emphasis on definite and indefinite articles in Western languages. When introducing a new mathematical object in class, I feel very uncomfortable having to say “a” until unicity is proved. For practical reasons, I am mostly interested in Asian languages (mainly Korean, and also Chinese and Japanese), but references for other languages may also be useful.

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    $\begingroup$ I am a native Chinese speaker, but I don't find any considerable difference from the reasoning you describe in the question body. Maybe it is because mostly I think in English? :D $\endgroup$ – awllower Apr 23 '15 at 6:06
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    $\begingroup$ There's a big problem with your premise that uniqueness is (necessarily) connected to definite articles: in English the definite article "the" can precede a plural noun. You can write about "the zeros of the Riemann zeta function" or "the prime factors of this number". While there are no definite article in Chinese per se, the sense conveyed by definite articles certainly can be clearly conveyed in Chinese. $\endgroup$ – Willie Wong Apr 23 '15 at 11:59
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    $\begingroup$ Note that Russia (not sure whether you count that as Eastern or Western) also has a strong mathematical tradition in which existence and uniqueness problems are studied in detail, but the Russian language has no articles whatsoever. // In any case, a random Google search brought up this PhD dissertation which you may find interesting. $\endgroup$ – Willie Wong Apr 23 '15 at 12:12
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    $\begingroup$ Your premise about existence and uniqueness being related to definite or indefinite articles is bogus. As Willie wrote, Russian has no articles and also has absolutely no problem conveying ideas related to existence or uniqueness. In English we may distinguish between "the algebraic closure" and "an algebraic closure" while in Russian they just don't care that much about it (since, frankly, it's usually not as important as you might think). If there truly is a need to speak about a choice of an algebraic closure, say, then in Russian you could just use their word for "choice." $\endgroup$ – KCd Apr 24 '15 at 16:38
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    $\begingroup$ As a general remark: It may be helpful to check out a topic's Wikipage in English and then compare it to the corresponding language page of your choice. I have certainly done this for Chinese in order to see how various terms are defined and discussed. A difference in the use of (in)definite articles may influence a non-native writer's exposition, but there is certainly no insurmountable obstruction inherent to any of the languages mentioned... $\endgroup$ – Benjamin Dickman Apr 25 '15 at 1:48
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I find this to be an interesting question. It seems to propose that the math is language dependent. I believe that math itself is a language and we learn it alongside our other languages. Certain things are more precisely expressed in certain languages, and some languages have poetic simplicity yet need more clarification in explaining certain ideas not often expressed in that language. As a language, math is used without other languages guiding it, which is why when we are confronted with a problem in our language we need to translate it into mathematical sentences. For an anecdote, when I was a young girl my parents went to Japan. We were there for three weeks and I had to bring my school work with me. I was learning Algebra at the time, and there was a young college student where we were staying who noticed me struggling one day. He struggled speaking English and I only knew a few phrases in Japanese. But when he sat down to help me he had no trouble guiding me through my difficulties. I learned how to recognize when he said numbers or that I had lost my sign. It was he who stressed the importance to me of keeping my work neat and orderly marching down the page to my final conclusion.

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The idea that mathematics is natural language-dependent is a tempting one, and would be interesting to see an illustration. The particular example you give, namely an alleged relation between (1) emphasis on existence and uniqueness issues and (2) the existence of definite and indefinite articles in many Western languages, fails to convince because in Russian for example there are no definite or indefinite articles yet there is no evidence that the Russians are any less preoccupied by issues of existence and uniqueness than English speakers.

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I think the point is that quantifiers often roughly translate into more-or-less colloquial English in a way that can be expressed by choice of articles. That is, there is a not-so-bad mapping from mathematics to colloquial English. That's a good thing. However, the point that other natural languages do not have the details of English, and thus would not allow the same rough mapping to natural/colloquial language, certainly does not entail any lack of mapping.

For that matter, if we go so far as to ask about mapping of first-order logic expressions of mathematics to natural language, I think we are asking for needless trouble, insofar as it is my feeling that the mapping of mathematics to first-order logic is imprecise and does not capture everything we'd want... despite the benefits of the mapping landing us in first-order logic, which achieves (by some simplisticness, etc) a seeming clarity that has admitted appeal.

It is surely worth noting that in natural-language settings, rarely do existence and uniqueness play a significant role, because they're either irrelevant or strongly implicit. So it would be inaccurate to say that colloquial English really makes that distinction. Rather, mathematicians speaking English have overloaded the colloquial language with responsibility for making distinctions that would be incomprehensible to almost all native English speakers.

So, just on the grounds of these disconnects, one shouldn't think that natural languages with different grammar would have trouble mapping mathematics into themselves. After all, again, the alleged distinctions made by the mapping in English are incomprehensible to most native speakers. It's just that the mapping would be different.

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