Wait, but I thought the question being posed wasn't about the computational complexity of any model but about the power of a given computational model relative to another (e.g. the subject of the CT thesis).
Even if we're talking about just computational complexity, if we allow Turing machines with arbitrary oracles, or (equivalently) we allow arbitrary mathematical constructions in languages, I'm still unsure as to why the complexity matters.
W.r.t the waterfall case. Sure, I agree with that statement, but I'm unsure as to how it plays into the final case---the question isn't about a specific model performing a specific task, but about a model being able to completely replicate another in terms of computational power. Anything I can do in a language (with some functions) I can do in a Turing machine with the appropriate oracles; and conversely, anything I can do in a Turing machine with some given oracles, I can do in a language.
What I get is that the question is "Does it make any sense to compare the two models, or are they just different things?" and the tentative answer is "They are different things, but one end of TOP coincides with TOC".
I'm fully convinced. Both concepts answer different questions, and that is my favorite test (instead of the OP's "does one model map into the other?").
Both models are equally powerful in the things they compute in terms of the Church-Turing Thesis. The TOP view also has other consideration. My point in the article was not to settle the debate on Aaronson's blog but to show that language models and machine models are objectively and inherently so different from one another that comparing them in terms of programming concepts (like types) is meaningless.
And it seems that most of the "my way is superior to yours" is based on setting up strawmen of "yours" to make it be addressing the same questions as "mine".
Yes, but it goes a little further. I can't say my way is superior to yours if my way means solving a harder task and paying market price for it.
I got the leather seats because I chose to pay full price for them. That they come as a standard option with the car because the car is more expensive is not an argument. You can choose to pay the same difference in price and have them installed in your car, too.
Whether those leather seats matter or not is no longer an objective question but a question of values, and here is where caring about completely different values is the point.
I see, okay; I caught that point in the article through the first part, but the latter part got me stuck back on the same point hence the question. Thank you for the clarification.
Even if we're talking about just computational complexity, if we allow Turing machines with arbitrary oracles, or (equivalently) we allow arbitrary mathematical constructions in languages, I'm still unsure as to why the complexity matters.
W.r.t the waterfall case. Sure, I agree with that statement, but I'm unsure as to how it plays into the final case---the question isn't about a specific model performing a specific task, but about a model being able to completely replicate another in terms of computational power. Anything I can do in a language (with some functions) I can do in a Turing machine with the appropriate oracles; and conversely, anything I can do in a Turing machine with some given oracles, I can do in a language.