On 5 Apr 2017 9:54 a.m., "Bruno Marchal" <marc...@ulb.ac.be> 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.
Exact. And going a little further, that is what the
Gödel-Löbian machine already says (or say out of time and space).
I'm pleased we agree. By the way, on re-reading my text above I notice
that I had written "any consistent (1p) formal system of sufficient power",
when I had actually meant to write (3p). In other words, I meant a formal
system defined extrinsically in the ordinarily accepted sense. However I
think that, despite my typo, you understood my meaning.
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).
I agree with you that all this is indeed amazing, and the amazingness
consists in the sudden apprehension of the existence of a categorically
distinct internal logic, hitherto concealed within the extrinsic, that
opens up a Vast spectrum of perceptual truth of every possible description.
Furthermore, the amazingness increases exponentially with the realisation
that such truths are in fact coterminous with the "objects" of our shared
physical environment. If truth is indeed correspondence with the facts,
then facts don't get much more "concrete" than these.
It is as if this truth had been hidden because it had been encrypted with a
key having both a public and a private component, in which the former was
always publicly inspectable but the latter was inaccessible except *from
the point of view of the system in question*. In the desire to show that
there could be no privileged position from which judgement could be made,
it simply been missed that the position from which any judgement whatsoever
could be made was already thus privileged, and that consequently the "view
from nowhere" could only be an idealisation within that position. For our
purposes here, assuming computationalism, we may consider such an
idealisation in the form of an abstract axiomatic system on the basis of
which we seek to infer the origins of our shared concrete experience. In
this sense, to employ an aphorism of which I have become rather fond, the
concrete might be understood as the subjective reflection of the abstract.
By the way, I've recently been trying to discuss some of this in a couple
of relevant FB discussion groups. Some of the response has been favourable
and it at least helps me work out some of the bugs in my own understanding.
They say if you really want to learn something try to teach it. Anyway, for
a fuller and more rigorous understanding I always refer them to your papers.
David
Bruno
David
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