On 05 Apr 2017, at 20:46, Brent Meeker wrote:
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.
?
It is not an inference, but the recognition of a fact, like when a
smoke detector detect smoke. There is an implicit assumption of being
awake, or not dreaming, but still no inference, nor does it use
language, at least not necessarily. The smoke detector detects smoke
through it senses, and so believe in some representational sense that
there is smoke (the [](smoke)), and ... there is smoke (the p of []p &
p).
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,
They assumed they know that they are correct. Not that they are
machine, which is indeed what they want prove to be impossible.
Bruno
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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