Reddit reviews Informal Lectures on Formal Semantics (Suny Series in Linguistics)

Noam Chomsky, following in the footsteps of logicians/mathematicians like Emil Post and Yehoshua Bar-Hillel, conceived of language as a formal object: a language is simply a set of structures, namely the set of all (and only) the grammatical structures of that language. On this view, English can be equated with a set which contains Alice runs, Alice sees Bob, etc., but which does not contain Runs Alice, Sees Bob, etc.

Indeed, it's easy to construct a "language" (under this definition). For instance, L = {a} is a language. It has just one grammatical structure, namely a. Not a very interesting language, but a "language" (under this definition) nonetheless.

As Chomsky pointed out, and as did others before him, like Wilhelm von Humboldt and Gottlob Frege, human languages are infinite: there is no limit to the number of grammatical sentences you can construct in a human language. So, English (Spanish, Arabic, ...) is not just a set, but an infinite set of grammatical structures. This means we cannot simply list all (and only) the grammatical structures of English (the list would never end). Instead, Chomsky argues, we need to specify a generative grammar, i.e. a device that generates exactly the (infinite) set of grammatical English sentences. (Chomsky believes that linguistic competence involves generative grammar at the cognitive level, i.e. that we all have generative grammars in our heads.)

Indeed, it's even easy to construct generative grammar that generates an infinite "language". Here's one (using recursion):

1. a is in L.
   
2. For any x, if x is in L, then xb is in L.
   
3. Nothing else is in L.
   
   So, L contains things like a, ab, abb, abbb, and so on (infinitely), but does not contain b, aa, aba, and so on.
   
   Now you can start to see that mathematics is clearly a language under this conception of language, for exactly the reasons you mention: it has a fixed syntax, etc. So 3 + 2 = 5 is grammatical, while = 3 2 5 + is not. And indeed, one can specify a generative grammar for whatever fragment of math one is interested in.
   
   Of course, (human) language is more than just grammar. It's more than just a set of structures. Language has meaning, too. But so do mathematical and logical expressions: a > b has a very different meaning than a < b. The logician Alfred Tarski was the first to give a real semantic analysis of logical language, but he doubted that his analysis of logical language could be successfully applied to human language. His student, Richard Montague, showed that in fact, a formal, logical analysis of human language is possible. In doing so, Montague effectively created the field of formal semantics as we know it today.
   
   As the linguist Emmon Bach succinctly put it, in his book Informal Lectures on Formal Semantics:
   
   > Chomsky's thesis was that natural languages can be described as formal systems. Montague added to this the idea that natural languages can be described as interpreted formal systems.
   
   tl;dr Yes, depending on your definition of "language", mathematics (and logic, etc.) is absolutely a language — one with meaning, no less. Both artificial (mathematical, etc.) and natural languages can be described as interpreted formal systems — a revolutionary idea put forth by Chomsky and Montague (and others) in the 1950s-60s that spawned the entire field of formal linguistics as we know it today. (So, I clearly (and strongly) disagree with /u/raendrop, who instead prefers a circular definition of language, on the basis of which nothing counts as language except, by definition, human language.)