Hi Bruno,
Pretty freaking cool post! A few comments...
-----Original Message-----
From: Bruno Marchal
Sent: Saturday, July 02, 2011 7:27 AM
To: [email protected]
Subject: Re: consciousness
On 01 Jul 2011, at 13:23, selva kumar wrote:
> Is consciousness causally effective ?
>
> I found this question in previous threads,but I didn't find a answer.
[Bruno Marchal]
Was it in the FOR list (on the book Fabric of reality by David
Deutsch) ? I thought I did answer this question, which is a very
important and fundamental question.
It is also a tricky question, which is very similar or related to the
question of free-will, and it can lead to vocabulary issue. I often
defend the idea that consciousness is effective. Indeed the role I
usually defend for consciousness is a relative self-speeding up
ability. Yet the question is tricky, especially due to the presence of
the "causally", which is harder to grasp or define than
"consciousness" itself.
Let me try to explain. For this I need some definition, and I hope for
some understanding of the UDA and a bit of AUDA. Ask precision if
needed.
The main ingredient for the explanation are three theorems due to Gödel:
- the Gödel completeness theorem (available for machine talking first
order logic or a sufficiently effective higher order logic). The
theorem says that a theory or machine is consistent (syntactical
notion, = ~Bf) iff the theory has a model (a mathematical structure in
which it makes sense to say that a proposition is true). I will
rephrase this by saying that a machine is consistent if and only if
the machine's beliefs make sense in some reality.
- the Gödel second incompleteness theorem ~Bf -> ~B(~Bf): if the
machine is consistent, then this is not provable by the machine. So if
the beliefs are real in some reality, the machine cannot prove the
existence of that reality. This is used in some strict way, because we
don't assume the machine can prove its completeness (despite this has
shown to be the case by Orey). This entails that eventually, the
machine can add as new axiom its own consistency, but this leads to a
new machine, for which a novel notion of consistency appears, and the
'new' machine can still not prove the existence of a reality
"satisfying its beliefs. yet that machine can easily prove the
consistency of the machine she was.
***
[SPK]
The "machine can easily prove the consistency of the machine she was"!
Woooo! "Cogito ergo sum" - I think therefore I was, just as Pratt explained in
his paper and another author explains here.
***
[Bruno Marchal]
This can be reitered as many times
as their are (constructive) ordinals, and this is what I describe as a
climbing from G to G*. The modal logic of self-reference remains
unchanged, but the arithmetical interpretation of it expands. An
infinity of previously undecidable propositions become decidable,
and ... another phenomenon occurs:
- Gödel length of proof theorem. Once a machine adds an undecidable
proposition, like its own consistency, as a new axiom/belief, not only
an infinity of (arithmetical) propositions become decidable, but an
infinity of already provable propositions get shorter proofs. Indeed,
and amazingly enough, for any number x, we can find a proposition
which proofs will be x times shorter than its shorter proof in the
beliefs system without the undecidable proposition. A similar, but not
entirely equivalent theorem is true for universal computation ability,
showing in particular that there is no bound to the rapidity of
computers, and this just by change of the software (alas, with finite
numbers of exceptions in the *effective* self-speeding up: but
evolution of species needs not to be effective or programmable in
advance).
***
[SPK]
What would prevent this speed-up from going to infinitely fast? In my idea
the speed up (and regress) is limited by the quantity of physical resource that
is available to the machine, so the existence of a physical world – even if it
is emergent and not primitive – is necessary. I suspect that the Stone spaces –
dual to the logical algebras (which I am identifying with your notion of
machine)- will have the necessary concreteness to met this requirement but this
remains to be proven.
***
[Bruno Marchal]
Now I suggest to (re)define consciousness as a machine (instinctive,
preprogrammed) ability to bet on a reality. This is equivalent
(stricto sensu: the machine does not need to know this) to an ability
of betting its own consistency (excluding that very new axiom to avoid
inconsistency). As a universal system, this will speed-up the machine
relatively to the probable local universal system(s) and will in that
way augment its freedom degree. If two machines play ping-pong, the
machine which is quicker has a greater range of possible moves/
strategy than its opponent.
So the answer to the question "is consciousness effective" would be
yes, if you accept such definition.
Is that consciousness *causally* effective? That is the tricky part
related to free will. If you accept the definition of free will that I
often suggested, then the answer is yes. Causality will have its
normal "physical definition", except that with comp such physicalness
is given by an arithmetical quantization (based on the material
hypostase defined by Bp & Dp): p physically causes q, iff something
like BD(BDp -> BDq). I recall Dp = ~B ~p. But of course, in God eyes,
there is only true (and false) number relations. The löbian phenomenon
then shows that the consciousness self-speeding up is coupled with the
building of the reality that the machine bet on. At that level, it is
like if consciousness is the main force, perhaps the only original
one, in the physical universe! This needs still more work to make
precise enough. There is a complex tradeoff in between the "causally"
and the "effective" at play, I think.
**
[SPK]
From what I have studied so far the löbian phenomenon is captured well in
my bisimulation algebra, but there is no notion of box or diamond in it that I
can find in the expressions of relations “between” machines. It could be that
we can only bet on the reality of others in terms of our own internal notion of
reality, as in the case of, informally stated, “I bet that your reality is
similar to mine”. If it is true then the appearance of an interaction will
occur. If it is false no appearance will obtain. So only the successful bets
will yield realities that have a continuance in the “time” sense of an
arbitrarily long sequence of realities.
This line of thinking dovetails nicely with the Observer Moment idea as has
been discussed and seems to be remarkably free of White Rabbits and other Harry
Potterisms because sequences could only continue if the new reality does not
introduce any new aspect that contradicts content of the prior machines. It is
like an endless Surprise 20 Questions game where “all answers are allowed
except those that would produce a contradiction” is the rule. it is similar to
the notion of stare decisis in jurisprudence.
**
[Bruno Marchal]
I hope this was not too technical. The work of Gödel plays a
fundamental role. This explanation is detailed in "Conscience et
Mécanisme", and related more precisely to the inference inductive frame.
**
[SPK]
It is the best post that I have seen on this subject from you. Thank you
for it!
***
[Bruno Marchal]
To sum up: machine consciousness, in the theory, confers self-speeding
up abilities to the machine with respect to the most probable
continuation/universal-machine.
**
[SPK]
Could you elaborate a bit more on the part where you say "...self-speeding
up abilities to the machine with respect to the most probable
continuation/universal-machine"? What defines the "most probable"? From what I
can tell from my analysis using the bisimulation idea, there is no notion of
"most probable" in a strict sense but we obtain something like it my using the
bisimularity relation as a metric of sorts. One would take the “I bet that
your reality is similar to mine” idea and sums over (not sure if that is the
proper operator) all of the simulations of one system/machine of some other.
The ones that are most similar would add and the ones that are more different
would subtract, like waves interfering.
This new twist of a speed-up is something that I have to account for... But
it seems similar to the telescope property where:
A = A~A and ~ is the bisimulation relation.
A ~B~C~B~A = A~B~A = A;
The telescope can start off arbitrarily large (but finite AFAIK) but will
still collapse to the singleton iff the bisimularity of the sequent exist. This
property looks exactly like the shortening of the proofs that you mentioned
previously if one considers the equivalent of the symbols making up the proof
as being capable of being machines themselves.
BTW, I use the term “monad”, as per Leibniz’ idea, instead of Machine
because it seems to me that these entities are far more like Leibniz’ Monads
than the idea of a machine that most people have. One thing though: Leibniz’
postulate of a “pre-ordained harmony” is impossible because for it to exist it
will require the solving of an uncountable infinite NP-Hard problem in less
than 1 step. Even with infinite computational resources this is impossible! I
solve that fatality by not requiring the Harmony to be computed in advance.
Instead it is proposed that the computation is perpetual and ongoing, ala your
UD.
**
[Bruno Marchal]
It is obviously something useful for
self-moving creature: to make them able to anticipate and avoid
obstacles, which would explain why the self-moving creatures have
developed self-reflexive brains, and become Löbian (self-conscious).
Note that here the role is attributed to self-consciousness, and not
really to consciousness. But you need consciousness to have self-
consciousness. Consciousness per se has no role, like in pure
contemplation, but once reflected in the Löbian way, it might be the
stronger causally effective force operating in the 'arithmetical
truth', the very origin of the (self) acceleration/force.
Note that the Gödel speed-up theorem is not hard to prove. There is a
simple proof of it in the excellent book by Torkel Franzen "Gödel's
theorem: An Incomplete Guide To Its Use and Abuse" which I recommend
the reading (despite it is more on the abuses than the uses). The
original paper is in the book by Davis: the undecidable (republished
in Dover), and which I consider as a bible for "machine's theology".
Bruno
http://iridia.ulb.ac.be/~marchal/
***
[SPK]
This is beautiful stuff Bruno! Bravo, Bravo! Now if I could learn enough of
the math to write up the Stone duality aspect of this! Working on it! ;-)
Onward,
Stephen
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