On 07 Apr 2017, at 00:11, David Nyman wrote:
On 6 Apr 2017 6:44 p.m., "Bruno Marchal" <[email protected]> wrote:
On 06 Apr 2017, at 12:02, David Nyman wrote:
On 6 Apr 2017 8:45 a.m., "Bruno Marchal" <[email protected]> wrote:
On 05 Apr 2017, at 22:51, David Nyman wrote:
On 5 Apr 2017 7:46 p.m., "Brent Meeker" <[email protected]>
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. 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.
That's an interesting point also, but I'm not sure you've quite
taken my meaning. I'm specifically making use of Tarski's
criterion of truth as correspondence with the facts. When
considering matters in the first-person, the "facts" in question
are in the first instance perceptual and hence as such directly
apprehended. Hence we have access to a second means of judging
truth, in this specific sense, over and above the restrictions of
any purely algorithmic procedure. In other words, we are able
directly to apprehend or "see" a correspondence *in concrete
perceptual terms* of an assertion with facts to which it purports
to refer. And indeed that's exactly how we are able to make the
relevant distinction: i.e. between working through a formal
procedure, which we are equally able to do, and at the same time
grasping a directly perceptible correspondence that eludes the
restrictions of that procedure. The linguistic part comes later in
justifying our judgement (to another or for that matter to
ourselves) post hoc.
Yes, that is what I said, but you put it in a much more better way
than me! Consciousness is in the truth, or in its "direct
perception through sense". Note that happens in dreams too, where
the cortex will build the []p, and the stem is bringing the "p",
which sometimes can be random letting the [] in need of some
imagination (dream weirdness).
Actually it might really have been more accurate to have said that,
rather than it being a second means, our *primary* means of judging
truth is by direct apprehension of perceptual correspondence.
Algorithmic proof is surely secondary to this.
It is secondary, like the brain is secondary to consciousness.
Yes, I think I grasp that subtlety, because of the relation between
the two logics encapsulated in Bp and p. However, I was referring
here in particular to formal analysis as actually performed within
an individual first-person perspective, being inevitability
secondary to primary apprehension within that perspective.
Yes. I ask myself if you are not tempted by founding everything from
the 1p view. Note that this can be done, but is technically much more
difficult. You would need to formalize "provability" in something like
Heyting Arithmetic (PA without excluded middle). Albert Visser has
studied this, but it is more difficult (no truth table in intuitionist
logic, so the propositional basic logic is already more complex). You
will need some formula like
([](~~[]A -> []A) -> [][]A). in Intuitionist logic ~~A does not imply A.
An interesting question would be in this context: can we get all this
from S4Grz. I think we should, but that is an open problem, like the
intuitionist Church thesis, or close constructive principle.
I understand your motivation for this. I prefer to stay in the
classical framework mainly for pedagogical reason: the basic logics
are better known.
This is a bit like the egg and the chicken. p does precede logically
[]p (the representational or algorithmic). Yet the senses are useful
only if we can re-enact the experience. You can see "p" as the fact
(like the true fact that it rains, blurred with some representation
of that fact), and []p as the building of a theory with the axiom
"it rains", which needs to be represented in some way that the
entity can re-enact the experience that it rains when needed, like
when looking for an umbrella in a room without windows (so that you
need to remember that it rains all along).
p comes first, and like Everett you can identify it with the first
perceptual judgment (to be sure that will need more basic "theory"
already in the brain, so we might add nuances on the perceptual p,
(but here the theories are trivial, like accepting that the needle
is on 4 when it is on 4) in between the truth of p and the truth of
the perceptual experience. Then []p is more for a long term memory,
building into the subject his/her conception/theory predicting/
anticipating/extrapolating/explaining the probable neighborhood.
So, as far as reality/truth/the-one is concerned, p is more primary.
But concerning the subject I would say that both p and []p are
needed, and both []p and []p & p are needed to, and collaborate
together though some (arithmetical) corpus callosum. Going from []p
& p to p is quite an out-of-body experience!
It can only be subsequent to apprehension of primary facts (which
exhaust in effect our grasp on concrete inter-subjective reality)
that we are able to deploy algorithmic methods. These latter are
applicable not to the concrete perceptual world directly but rather
to its formally abstracted "view from nowhere" idealisation.
Hmm.. the "[]" is really the body/brain. It is the local
representation of you in the languages of, say, nucleus and
electromagnetic interaction (chemistry). Some would say that it is
the "p" which is in the view of nowhere. It is delicate because it
depends from which mode we tackle the distinction. Eventally we know
that G* knows that there is no difference: all the points of view
points on the same reality (the sigma1 truth)
In that case, perhaps p is the view from everywhere.
p is the view from all points of view, but that is seen only from the
"divine intellect" view. I would say here that it is the view from
everywhere, when see from the view of nowhere. But I guess we get
closer and closer to "1004". Of course, subtle philosophy requires
eventually to add nuances in the formal apparatus.
But again, my reference was to the actual practice of algorithmic
reasoning which I'm contending leads to the L/P mis-step because of
the implicit assumption that its application is entirely restricted
to the 3p view from nowhere.
... up to invoking their feeling that they know their own correctness,
or that mechanism asks for knowing which machine we are. But this
cannot be known by machine. Of course would Penrose be able to prove
that he is a correct entity, then his proof on non-mechanism would be
correct. But he has not done that, and indeed, it is hard to imagine
how that would be possible. self-correctness requires some faith in a
reality. His proof relies on an intuition. But his idea that machine
cannot have a similar intuition relies on the fact that for simple
machine like PA we can be confident that []p = []p & p. PA cannot know
this, but no machine can know this about themselves. Charitably,
Penrose proves in this case, that he is not PA. But of course we knew
that beforehand.
but G* knows that the subject, and in any of its mode, is unable to
grasp the G* truth, making it trapped in the illusion (of life,
physics, ...). That illusion is "important" to survive on the
terrestrial plane. Now in that plane "important" is a difficult
matter by itself (the meaning of life question).
Hence it is in the last analysis hardly surprising that this
secondary abstraction
It is the little ego. We might need to get rid of it to get
enlightenment, but what is enlightenment for if we cannot come back
and help the others, and this needs the little ego, and its body/
brain/machine so that it can manifest its knowledge/consciousness
with respect to its peers.
Nobody needs a body, but everyone needs a body to manifest itself to
anything else which is not her/him.
Now what I just said, applies to itself. The "[]p" would grasp
noting if it was not accompanied by its semantic p. Somehow, the
meaning would get trivial at the deterministic level. Why did Deep
Blue win? Because of this boolean net configuration and the laws of
NAND? Why did Adolph killed all the kids? Because Adolph got a
quantum body following the quantum laws, etc. That lack of meaning
is lifted to all level of 3p description, but the "truth" of the
elementary relation lift the meaning of the higher level
description. For the sigma_1 we get the "enlightenment: "p <-> []p",
a sigma_1 proposition is true (nobody knows what that really means)
if and only if the machine can prove (syntactical procedure,
arithmetical relation). the machine does not fall in the blaphesm,
because despite she knows she is sigma1 complete (Universal, in the
sense of Church, Turing), that is: she knows p -> []p (for p sigma1-
arithmetical, shape "it-exists x (s(x) + s(x) = s(s(x))), she does
not know []p -> p, even for p sigma1. Indeed she does not know that
[](0=1) -> (0=1), because that would be knowing ~(0=1), by
propositional calculus, and she would proves its own consistency.
That can be shown to be true and knowable, but still not
communicable, because the "[]p" has no name/description for "[]p &
p". It leads to the idea that in the ideally correct machine the
corpus callosum should be a one way road, which I think is not the
case, or some hemisphere lies, leading to self-conspiracy
theories ... Well, I stop here.
fails to bridge the gap to all the truths primarily accessible in
terms of direct perceptual correspondence.
It fails, and the part of us which bridge the gap stay mute, or
become inconsistent.
Scientific theology is the part of science which study the part of
truth which extends science. With computationalism it is computer's
science minus computer's computer science.
From the non experiential to the non memorizable, ... to the not
describable, to the non justifiable, to the infeasible, to the
feasible, eventually to the feasible respecting the deadline.
Even for simple machine, the full theology is quite "out of
science", but there is a non trivial core common to all
arithmetically correct machine.
Propositional theology, or meta-theology, is decidable. It is
platonist only by accepting that a sigma1 sentence is either true,
or false, which is equivalent with saying that a program stop or
does not stop.
The amazing thing with computationalism, and thanks to
incompleteness, is that the proposition that truth extends science
is part of that common core of provable "scientific" statement, in
the conditional form, like <>t -> ~[]<>t. If I am consistent (if
that belongs to truth) then I can't prove/justify/communicate-
rationally that I am consistent. Of course, in the arithmetical
interpretation of [], this is the second incompleteness theorem.
Hoping not boring you too much with the technicalities, but it is
the interest of computationalism that the study is a part of
mathematics.
No indeed, I appreciate the rigour that you add. I always read with
interest and do my best to understand.
Thanks for telling me.
It's just that I have insufficient command of the technicalities to
satisfy my intuition purely by this means.
It is useful when we are on the highly counter-intuitive fringe of
computationalism. To develop an intuition there is like to develop a
taste for the bizarre. The best introduction is probably "Alice in
Wonderland". As Liz said: we must train ourself to believe at least
five impossible things before breakfast!
Hence my "grandmother" versions. Thank you as ever for your
helpful responses.
You are welcome. I thank you for giving me the opportunity to try to
handle interesting subtle points in the available formalism.
Computationalism is the diplomate which invites the mystics and the
rationalists at the same table. Not always easy those days. But when
we look at the antic greeks, we can see that it all started by that
form of open-mindness, in between the 1p mysticism/experience and the
3p-description-reasoning/sharing-understanding.
Bruno
David
Bruno
David
A remark on entheogen:
I think that with cannabis, you blur the "p" in "[]p & p", and
with salvia you blur the "[]p" in "[]p & p". (with the surprise
that you still remain as a sort of conscious person).
Oops I have to go. Before I fall in the machine's blasphem ... More
on this later most probably.
Bruno
David
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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