On 05 Apr 2017, at 12:54, David Nyman wrote:
On 5 Apr 2017 9:54 a.m., "Bruno Marchal" <[email protected]> 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.
I confirm. My brain focused on "formal" which is at the antipode of
the 1p.
Bruno
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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