Not sure about mathematically rigorous "proof", but a powerful theory of grammar at least. His claim is "true" in the sense of computation, ie the lowest level Chomsky grammar[1] is just like a Turing machine and consequently can generate anything.
But the analogy with computing languages also supplies a practical insight that isnt captured by the academic theory, ie that some concepts are easier to express in one language than another. If I'm inverting a matrix, I'd reach for Python over C for example.
I think Chomsky's theories became popular in anglophone linguistic circles (and later in the rest of the world) because their math-like structure and close applicability to computer science. They've proven to be useful in some cases, but there has been no proof of a universal grammar. The fact that you can create a formal system to represent an idea, is not a proof by any stretch of that definition.
This is why some people would even go as far as classifying Chomsky's theories as pseudo-science (see one of the replies to GP). I wouldn't go as far, but considering the almost toxic disdain Chomsky himself has to every linguist who is not interested in researching his supposed Universal Grammar (he has famously compared structural and functional linguistics to "butterfly collectors"), we should view this theory with more criticism.
Well, the real issue with Chomsky is that he came into vogue during the “linguistic” turn, except he misunderstood the anglo philosophers, thinking their theories had a natural basis, and misunderstood the french, thinking that their studies of writing and discourse made them philosophers of language. But the Kantian project still stands, one must have an architectonic, before one can make a scientific approach. But Kant was always sure to set his critical project within its dialectic, and placed the basis of cognition below language and discourse. That is what the French were able to capture, and its why contemporary philosophy today is so suffuse with discourses on aesthetics (though I’m being a bit broad with what I refer to as “Philosophy,” perhaps it’d be better to say “philosophical work”).
What is logic here against the subjective internal experience you're responding to. Do I have to hold an axiom true to believe the person describing their internal experience is specifically as Chomsky proved in the language of logic?
I'm actually glad you posted this because it reminded me of a quote from Wittgenstein's page on Wikipedia [1]
> According to Wittgenstein, philosophical problems arise when language is forced from its proper home into a metaphysical environment, where all the familiar and necessary landmarks and contextual clues are removed. He describes this metaphysical environment as like being on frictionless ice: where the conditions are apparently perfect for a philosophically and logically perfect language, all philosophical problems can be solved without the muddying effects of everyday contexts; but where, precisely because of the lack of friction, language can in fact do no work at all.[259] Wittgenstein argues that philosophers must leave the frictionless ice and return to the "rough ground" of ordinary language in use. Much of the Investigations consists of examples of how the first false steps can be avoided, so that philosophical problems are dissolved, rather than solved: "The clarity we are aiming at is indeed complete clarity. But this simply means that the philosophical problems should completely disappear."
Maybe not, but most of them are expecting users to have English in mind to some degree, plus the top ten most used are definitely all procedural paradigm plus some encapsulation such as OOP, as opposed to something like prolog.
The point I was snidely trying to make is that certain languages make certain representations and tasks easier than others. Ease, convenience, and representation length are probably more useful metrics than "possible" in the context of this discussion.
But the analogy with computing languages also supplies a practical insight that isnt captured by the academic theory, ie that some concepts are easier to express in one language than another. If I'm inverting a matrix, I'd reach for Python over C for example.
[1] https://en.m.wikipedia.org/wiki/Chomsky_hierarchy