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).
Bruno
http://iridia.ulb.ac.be/~marchal/
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