Ouch.

Look, no. Grammars look like "the key to the gates" because of the equivalence between grammars-language-automata etc. You can write up grammars for fully Turing-complete languages, or indeed context-free languages that allow you to declare Turing-complete automata.

You can create grammars that display infinite recursion with minimal effort:

  A --> ε
  A --> Aa
  a --> <whatever you please>

So it's not like someone (cough, Chomsky) woke up one nice day and thought "blimey, I'll tell the world that grammars are a powerful tool for modelling hierarchical processes". It's that they are.

We have used grammars in practice to model complex hierarchical processes- except it's only those processes that we already know how to model, because we came up with them ourselves, like the aforementioned Turing-complete ones.

The problems begin when we try to fit a process we hardly understand to a grammar. That is not a limitation of the tool itself. It's a limitation of our ability to use it.

To me, and I think many other outsiders, putting a lot of emphasis on the equivalent of grammars-language-automata looks like mathematical naivety. I don't say this to be rude but because you (and Chomsky) claim to be able to interpret the implications of these mathematical results, but I don't think you are doing so correctly. Grammars look like a human (mathematical) invention and not some deep mathematical structure, and these results appear shallow. In the broader context, lots of mechanisms are able to do Turing complete computation.

This doesn't just apply to grammars. There is a huge array of formalisms (e.g. logics, type systems) out there and most just look like the result of someone saying "what if I did this?".

>> I don't say this to be rude but because you (and Chomsky) claim to be able to interpret the implications of these mathematical results, but I don't think you are doing so correctly.

It's alright- if I'm being naive, I'm being naive.

But- what am I missing? You're saying we're doing it wrong- how? For me the intuition that infinite generative ability flows naturally from unbounded recursion, like an egg from a hen's bottom, is kind of obvious. Is it naive? I guess it's empirical, for me at least.

Also, btw, I was introduced to the idea of language equivalence through Hopcroft and Ullman, so from the point of view of computer science, where it's been very useful, in practical terms. I guess if you're coming from a mathematical or theoretical physics background it might sound a bit silly, but it's allowed us to make a lot of progress, for instance to create a few thousand different architectures and languages... but maybe I shouldn't be bringing that up as progress...

Anyway, I don't know- how would you interpret the observation correctly? Where are we going wrong?