On 24 Aug 2012, at 12:04, benjayk wrote:

But this avoides my point that we can't imagine that levels, context and ambiguity don't exist, and this is why computational emulation does not mean
that the emulation can substitute the original.

But here you do a confusion level as I think Jason tries pointing on.
A similar one to the one made by Searle in the Chinese Room.

As emulator (computing machine) Robinson Arithmetic can simulate exactly Peano Arithmetic, even as a prover. So for example Robinson arithmetic can prove that Peano arithmetic proves the consistency of Robinson Arithmetic. But you cannot conclude from that that Robinson Arithmetic can prove its own consistency. That would contradict Gödel II. When PA uses the induction axiom, RA might just say "huh", and apply it for the sake of the emulation without any inner conviction.

With Church thesis computing is an absolute notion, and all universal machine computes the same functions, and can compute them in the same manner as all other machines so that the notion of emulation (of processes) is also absolute.

But, proving, believing, knowing, defining, etc. Are not absolute, and are all relative to the system actually doing the proof, or the knowing. Once such notion are, even just approximated semi- axiomatically, they define complex lattices or partial orders of unequivalent classes of machines, having very often transfinite order type, like proving for example, for which there is a branch of mathematical logic, known as Ordinal Analysis, which measures the strength of theories by a constructive ordinal. PA's strength is well now as being the ordinal epsilon zero, that is omega [4] omega (= omega^omega^omega^...) as discovered by Gentzen).

It is not a big deal, it just mean that my ability to emulate einstein (cf Hofstadter) does not make me into Einstein. It only makes me able to converse with Einstein.

If you avoid gently the level confusion for the human person, there is no reason to avoid it for the machines. It is not because universal machine can do all computations, that they can do all proofs, on the contrary, being universal and consistent will limit them locally, and motivate them to change themselves, relatively to their most probable universal histories.

Such infinite progression of self-changing machines have already been programmed, by Myhill, and myself, notably. In my more technical work, I use Becklemishev results which extends the soundness of G and G* on such machines, and prove a completeness theorem for the corresponding multimodal logic, with the provability parametrized on the ordinal (I say this for those interested and open to computer science as it is natural in the frame of the comp hypothesis).



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