> It's worth noting that math and programming do not appear to be considered "languages" by much of the academic

There is math term "formal language", and both math and programming perfectly fit into it: https://en.wikipedia.org/wiki/Formal_language

Yes, calling an algorithm, a formal description of a generative function "a language" is clearly wrong. I don't think we need an academic dissertation on this.

I take it you disagree with the conclusions from TFA then?

> Sometimes a natural language such as English is a good choice, sometimes you need the language of mathematical equations, or chemical equations, or musical notation, and sometimes a programming language is best. Written language is an amazing invention that has enabled human culture to build over the centuries (and also enabled LLMs to work). But human ingenuity has divised [sic] other notations that are more specialized but very effective in limited domains.

Huh, interesting. Programming languages were devised with Chomsky’s foundational theory of formal languages in mind, and they’re one of the few actual implementations of it. I read your comment and it seems your main thrust is that arithmetic activity lights up different brain regions than communicative activity, which I don’t personally see as a compelling basis for a definition of the word “language”.

Of course, this is what Chomsky calls a “terminological dispute”, so I mean no offense and you’re ofc free to stand your ground that the only language is what appears in human brains! But if mathematical notation and programming languages aren’t languages, what are they…? Protocols? Recursively generative patterns? Maybe just grammars?

The best move in any terminological dispute is “my terms are more useful”, so this seems like a good reason to keep language as it’s defined by the generative linguistics. Or, more broadly:

  Saussure approaches the essence of language from two sides. For the one, he borrows ideas from Steinthal and Durkheim, concluding that language is a 'social fact'. For the other, he creates a theory of language as a system in and for itself which arises from the association of concepts and words or expressions. Thus, language is a dual system of interactive sub-systems: a conceptual system and a system of linguistic forms. Neither of these can exist without the other because, in Saussure's notion, there are no (proper) expressions without meaning, but also no (organised) meaning without words or expressions. 

https://en.wikipedia.org/wiki/Theory_of_language

I don't actually take any strong positions on the matter so long as the sense in which a term is used is clearly defined and agreed upon from the outset. This is merely an observation regarding what the literature considers "language" and this narrow definition's basis in brain structure contrasted with other forms of mental activity.

But if I must, I suppose I am indeed assuming a stance by taking potshots at this narrower (albeit more precise) use of the word language by (obliquely) pointing to counterexamples that could be considered languages in their own right; the sweeping claim that "language is not essential for thought" seems far broader than the narrow sense in which the term is construed in the actual paper.

The confusion is in a sense intentional as the whole point of Chomsky's program (Universal Grammar) is that natural languages essentially work like formal languages.

I don't think that this is actually true at all (and modern neuroscience, sociolinguistics, etc. disagree pretty heavily with Chomsky), but it is impressive how far you can get with modelling natural language grammar with formal methods.