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Update: When I asked this question I hadn't considered that most mathematical texts (in the Western tradition) were written in Latin from the [14th?] century until the [19th?] century. This probably influenced the development of the "modern" (post 1850s-ish?) language of proof-writing. Additionally, the use of symbolic logic, particularly quantifiers, is a product of the late 19th/early 20th century, so the development of rules for inserting and manipulating logical formulae in text probably occurred around then.

While a study of the historical development of the language of mathematics isn't a study of the language itself, it is likely to provide important information. As such, this is now also a question about math history.

Also, I found this:

The language of mathematics computational, linguistic and logical aspects. Sprache und Datenverarbeitung.

So if there isn't a definitive "Compendium Linguae Mathematicae", at least I there's a good starting point for further research.

I've decided to wear my anthropologist hat today, and it has come to my attention that, from a linguistic standpoint, the language used to write proofs (in English) is not English. Though it shares a high degree of mutual intelligibility with English, the dialect - though I would argue that it is a distinct pidgin language - which mathematicians use has several features which distinguish it from the English language.

For instance, there are distinct grammatical moods which are used in mathematical writing that are not present in English. In the language of mathematics, it is incorrect to introduce a new subject using the declarative, as in "$x$ is..." (actually "$x$ is..." can be used only if $x$ names a kind of thing rather than a specific thing, but this usage distinguishes it from the declarative mood, anyway.) Instead, there is a grammatical mood which superficially resembles the English imperative which must be used to declare a subject prior to its use in a declarative sentence, as in "let $x$ be..." and "suppose that $x$ is...." I say "superficially" because although the English sentence "Let $x$ be [object]." is imperative, the mathematical sentence of the same form does not relay an instruction to the listener; a response of "No, I don't think I will" is nonsensical in mathematics - this is not the case for the corresponding English sentence.

There are many other more-than-just-jargon examples of of mathematics deviating from the English language. There are even features of mathematical language that more closely resemble those of a typed programming language than any natural language, as evidenced by the rules governing the use of keywords like "fix," "choose," "by," and "at."

So, if mathematical English is not English, what is it?

Are there any books, articles, preprints, documentaries, or Reddit threads that discuss the language of mathematics in detail (without assuming prior fluency in mathematics)?

My initial search has turned up shockingly few results, none of them peer-reviewed, and all of them based on an underlying assumption that the language of mathematics is just English with some additional technical vocabulary (though a few webpages highlight that there are syntactic differences without ever addressing them in a systematic way.) If this were true, then native English speakers would be able to pick up proof writing given a week's time with a vocabulary list. Clearly this is not the case; it seems to take students 1 to 2 years to become fluent in mathematics, which is about the length of time that I would expect for a native English speaker to learn Krio.

About the tags: I'm not sure what to tag this with. Strictly speaking, this is a question about linguistics, not mathematics; but the context is so narrow (there are no people whose primary language spoken at home is proof-writing) that it is only relevant within the mathematics community. This might be relevant to mathematics education, since educators (or at the very least, textbook authors) [seemingly] assume that the language of mathematics is inherently understood by native English speakers without introduction; but then I'm not much of an educator. All I know is that within my limited experience as a tutor many students in introductory classes attempt to write proofs in English, only to be told that they "can't say it that way" with little or no explanation. In fact, I only came to the realization that mathematics is a separate language from English when I noticed an ESL student inserting formulae into English sentences in a way that was consistent with the grammar of Mandarin (and grammatically correct, albeit unusual, in English), but mathematically "wrong" in a way that I could not explain at the time.

R. Burton
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    One thing to consider: mathematicians also use a purely symbolic language, a language so "pure" that computers can understand it. We do occasionally say those symbols out loud ("for all", "there exists", "consider x member of") and so on, but that's not really friendly to non-English speakers. – Barry Carter Jul 14 '22 at 16:41
  • @barrycarter I had thought about that. In some cases, such as quantifiers, there are distinct grammatical roles played by the English and symbolic expressions of the same kind (it is incorrect to say "It follows that $x$ is at most $5$, $\forall x$."), while in others they can be used intergangeably ("It follows that $x\le 3\delta$" is equivalent to "It follows that $x$ is less than or equal to $3\delta$".) – R. Burton Jul 14 '22 at 16:51
  • translating English to Spanish I dont see some strange thing in the mathematical texts. It means that if for English-speaking persons there is something strange in the way that mathematics is written probably it must be because the used expressions are closer to Latin. And this can be explained because it was common to use Latin to write anything in the middle age, despite the fact the writer was German, English or Swiss – Masacroso Jul 14 '22 at 18:55

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The reason mathematics is difficult for the untrained to understand is obviously not because of the language but because of the logical reasoning involved. One must have a sufficiently good grasp of basic FOL (semantics and deductive rules) in order to be able to follow mathematical arguments with ease. Ultimately, that dependency on FOL is what makes mathematical writings (especially more rigorous ones) look more like formal languages rather than natural language!

Many educators themselves are woefully ignorant about the issues. For instance, they say "can't say it that way" instead of "it is not logically permissible" because they do not even know what precisely is logically permissible. And most people cannot figure this out on their own; one needs to be taught a deductive system for FOL such as this one. (The situation is the same as with programming; most people cannot construct a programming language all by themselves without knowing any existing programming language.) It also does not help that many educators and mathematicians alike use misleading words like "let" for creating a ∀-subcontext when there actually are appropriate words with the correct meaning like "given [any]" / "take any" (for more explanation see this post).

user21820
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  • I'm not sure that the distinction between "language" and "logic" is a meaningful one. The identification of meaning in any language is a process of identifying when it is permissible to replace a given string with another - a process which is indistinguishable from the application of inference rules in a deductive system. To say that "'the sky is blue' means 'the colour of the sky is blue'" is to say that either can be inferred from the other. – R. Burton Jul 15 '22 at 19:57
  • Given the inherent difficulty in translating informal proofs to formal ones, I would think that this suggests, if anything, that the logic of natural language mathematics is not the same as that of formal FOL to begin with. – R. Burton Jul 15 '22 at 19:58
  • @R.Burton: That's incorrect. Any logician has no trouble translating any sufficiently rigorous proof into a formal one in a reasonably practical deductive system. One cannot blame FOL for difficulty of usage if one does not even know a Fitch-style system. – user21820 Jul 15 '22 at 20:12
  • I'm not sure what you mean. I make a habit of trying to write Fitch-style proofs where I can. Even simple theorems can have proofs spanning hundreds of lines, and the process of disambiguating a few English sentences can take hours, especially as the number of available terms increases (think "supremum," "topological vector space," "product of finitely many," etc.). If the process of creating formal proofs from informal ones were a straightforward matter of transcription, I would write a program to do it for me! (or, more realistically, someone else would have done it ~40 years ago.) – R. Burton Jul 15 '22 at 20:44
  • @R.Burton: Based on your comments, I am pretty sure you have never actually asked any logician and found that they "can take hours [to] disambiguate a few English sentences". It's impossible for a rigorous proof to give a logician any difficulty in formalization other than boredom. Firstly, I think you are unfamiliar with all the components that a practical deductive system for FOL requires. This includes definitorial expansion. – user21820 Jul 16 '22 at 16:24
  • Secondly, formalization is not magic; it cannot turn wishful thinking or unsound handwaving into valid reasoning. You won't be able to point to any part of any rigorous text (e.g. Rautenberg's "Concise Introduction", Jech's "Set Theory") that cannot be easily formalized. Taking hours for one sentence is a clear symptom of some problem either in the original source (i.e. not really valid reasoning) or in the person attempting the formalization (i.e. unfamiliar with a practical full foundational system). – user21820 Jul 16 '22 at 16:27
  • Thirdly, even non-rigorous mathematics retains a shadow of FOL reasoning, which explains the programming-style structure to mathematical writings. The disconnect between non-rigorous mathematics and formalized versions in FOL-based foundational systems does not imply that mathematical language isn't heavily influenced by FOL reasoning. On the contrary, FOL is inextricably linked to mathematical thought and writing at least from the 19th century onwards. – user21820 Jul 16 '22 at 16:32
  • Let us continue this discussion in chat. – R. Burton Jul 16 '22 at 18:28
  • @R.Burton: Um, you didn't respond to my replies in chat (which pinged you). So, do you still have a problem? – user21820 Aug 03 '22 at 20:32
  • Apologies. I think the Naproche project as a whole is what I am looking for. It is true that mathematical writing contains elements of FOL and of natural language, but "a mixture of FOL and English" is far from complete description of mathematical language. – R. Burton Aug 04 '22 at 16:47
  • In any case, the answer provided here does not address the original question, which is a request for studies of the language that mathematicians actually use, not a search for the cause of students' difficulties with proof writing or an explanation of the elements of formal languages which mathematicians could use but don't. – R. Burton Aug 04 '22 at 16:47
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    @R.Burton: Oh, I thought you were focused on why native English speakers cannot quickly pick up mathematical language with just a vocabulary list, and the answer is that vocabulary obviously fails to cover the quantifier structure in proofs. However, if your focus is about how to actually describe mathematical language, then you could post your own answer about Naproche. But I would warn you not to have too much hope that such projects would actually work on the bulk of actual mathematical writings, because mathematicians don't have a closed language. =) – user21820 Aug 04 '22 at 17:20