On 4/5/2017 1:54 AM, Bruno Marchal wrote:
On 04 Apr 2017, at 16:47, David Nyman wrote:
I've been thinking about the Lucas/Penrose view of the purported
limitations of computation as the basis for human thought. I know
that Bruno has given a technical refutation of this position, but I'm
insufficiently competent in the relevant areas for this to be
intuitively convincing for me. So I've been musing on a more
personally intuitive explication, perhaps along the following lines.
The mis-step on the part of L/P, ISTM, is that they fail to
distinguish between categorically distinct 3p and 1p logics which,
properly understood, should in fact be seen as the stock-in-trade of
computationalism. The limitation they point to is inherent in
incompleteness - i.e. the fact that there are more (implied) truths
than proofs within the scope of any consistent (1p) formal system of
sufficient power. L/P point out that despite this we humans can 'see'
the missing truths, despite the lack of a formal proof, and hence it
must follow that we have access to some non-algorithmic method
inaccessible to computation. What I think they're missing here -
because they're considering the *extrinsic or external* (3p) logic to
be exclusively definitive of what they mean by computation - is the
significance in this regard of the *intrinsic or internal* (1p)
logic. This is what Bruno summarises as Bp and p, or true, justified
belief, in terms of which perceptual objects are indeed directly
'seen' or apprehended. Hence a computational subject will have access
not only to formal proof (3p) but also to direct perceptual
apprehension (1p). It is this latter which then constitutes the
'seeing' of the truth that (literally) transcends the capabilities of
the 3p system considered in isolation.
I don't think so. It is not direct perceptual "seeing the truth"; it is
an inference in language and depends on language. The fallacy of L/P is
they assume you can know what machine you are and therefore you can
"see" the truth of your Godel sentence, but in fact you don't know what
algorithmic machine you are.
Brent
Exact. And going a little further, that is what the Gödel-Löbian
machine already says (or say out of time and space).
If the foregoing makes sense, it may also give a useful clue in the
debate over intuitionism versus Platonism in mathematics. Indeed,
perceptual mathematics (as we might term it) - i.e. the mathematics
we derive from the study of the relations obtaining between objects
in our perceptual reality - may well be "considered to be purely the
result of the constructive mental activity of humans" (Wikipedia).
However, under computationalism, this very 'perceptual mathematics'
can itself be shown to be the consequence of a deeper, underlying
Platonist mathematics (if we may so term the bare assumption of the
sufficiency of arithmetic for computation and its implications).
Is this intelligible?
I have no critics. Your point is done by the machine through a theorem
of Grzegorczyk on one par: the fact that S4Grz, like S4, formalises
Intutionistic logic, and of Boolos and Goldblatt on another par: the
fact that the formula Grz *has to* be added to S4 to get the
arithmetical completeness of the "[]p & p". Note that this makes the
intuitionist into a temporal logic, and attach duration to
consciousness, like with Bergson and Brouwer himself.
Eventually it is amazing and counter-intuitive, because it ascribes
consciousness to all universal numbers, probably the same before they
get the differentiation along the infinitely many computations
supporting them. Needless to say that such consciousness is in a
highly dissociated state at the start, a bit like after consuming some
salvia perhaps (!).
Your analysis can be extended on the intelligible and sensible
(neo)Platonist theory of matter, but with p restricted to the sigma_1
sentences (which describe in arithmetic the universal dovetailing),
with or without the adding of "<>t", which typically transform the
notion of "belief []p" or "knowledge []p & p" into notion of
"probabilities".
In summary
p (truth, god, the one)
[]p (rational belief)
[]p & p (knowledge, intuitionist subject)
[]p & <>t (probability, quantum logic)
[]p & <>t & p (intuitionist probability, quale logic).
The quanta themselves appear to be qualia. In fact a quanta is a
sharable qualia by two universal number when supported by a same
universal number. That can be used to show that the "many worlds" of
the physicists (Everett theory) confirms Computationalism and protect
it from solipsism. The physical is indeed first person PLURAL, and its
sharableness comes from the linearity of the tensor product. At each
instant we all enter the same replication machinery. The Z logics
justifies the linearity and reversibility, but not clearly enough to
extract the unitarity and use Gleason to make the measure unique. But
this is for the next generation, hopefully (as many seem to prefer the
obscurantist statu quo alas).
Bruno
David
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