On 15 Jul 2011, at 19:54, Terren Suydam wrote:
Hi Bruno,
Roughly speaking, my main struggle with your wonderful arguments is
making the leap from the domain of mathematical logic to the one and
only domain we can be sure of as conscious, namely biological human
consciousness, and this without rejecting comp. Unfortunately I am
hindered by my lack of fluency in mathematical logic. See below for
comments.
On Tue, Jul 12, 2011 at 11:17 AM, Bruno Marchal <[email protected]>
wrote:
Hi Terren
Apology for commenting your post with some delay.
No worries about the delay. I play email chess and have had games over
a year old, so I am used to being patient :-]
<snip>
To be sure, the
mathematical/logical framework you elucidate that captures aspects
of
1st/3rd person distinctions is remarkable, and as far as I know, the
first legitimate attempt to do so. But if we're talking TOE, then
an
explanation of consciousness is required.
Right. But note that the notion of fist person experience already
involved
consciousness, and that we are assuming comp, which at the start
assume that
consciousness makes sense. The "explanation" per se comes when we
have
understand that physics emerge from numbers, and this in the double
way
imposed by the logic of self-reference. All logics (well, not all,
really)
are splitted into two parts: the provable and the non provable (by
the
machine into consideration).
I think the explanation of how physics emerges from the "number
theology" as you put it is a great contribution and certainly *part*
of an explanation of consciousness, especially in that it reduces the
mind/body problem to computer science, as you say.
But it is not enough to "merely" deal with the mind/body problem. The
hard problem of how qualia arise needs to be explained.
I think that the original mind-body problem is, or at least includes
the "hard problem". The hard aspect has been intentionally dismissed
by the behaviorist and positivist schools (like with the Vienna circle).
In the frame of comp, that is what AUDA should explain, and what UDA
formulates.
I know you
have identified a logical framework that is capable of distinguishing
qualia and quanta from the point of view of the lobian machine, but
again, that strikes me as a description, not an explanation.
The explanation comes from the fact that such a distinction is made
necessary. Machines encounters necessarily the "hard" mind body
problem, by the logic of self-reference. WE know, by having build a
simple correct machine that Bp and Bp & p are equivalent (proves the
same arithmetical propositions), but the machine cannot know that (by
the logic), and the two points of view are not conciliable.
Another way to put it perhaps is that such a logical framework may
well be a *necessary* condition of a machine that can experience
qualia, but not a sufficient one.
If it is not sufficient, I am not sure it makes sense to accept the
"doctor" proposition.
An example of a hypothesis that
takes this further towards an explanation is that an experiencing
machine needs to be embodied (a closed system) in some context (even
if in platonia) with a boundary that can be perturbed as a result of
that embodiment (i.e. what we think of as a sensory apparatus);
But this is automatically taken into account. You expect that the
doctor does not just copy your brain, but that it reconstitute it
relatively to your (most probable) environment. A brain has many
inputs (from eyes, and from the cerebral stem).
And the measure problem comes from the fact that the UD does also
reconstitute you in many environments.
Note also that the argument (both UDA and AUDA) does not necessitate
that consciousness supervenes only on the biological brain, the
"generalized brain" might include the environment, even the whole
physical reality. That appears at the step seven, where you can
eliminate the neurophysiological hypothesis (used only in steps 1-6
for pedagogical purpose).
and
that the machine synthesize these perturbations within the context of
a recursively updated model of "the world", grounded in the patterns
generated by those perturbations, and this model is the content of its
experience. Once the machine develops a model its world sophisticated
enough to include itself, it perhaps achieves Lobianity, although my
grasp of mathematical logic is too limited to say, unfortunately.
Löbianity is very cheap. Peano Arithmetic has an implicit "model" of
itself at the start. This is due to the fact that "provable" is an
arithmetical predicate.
Of course a complex and deep Löbian machine will have a far more
sophisticate self-representation, but this will not change the logic
of self-reference (as far as the machine is correct about that self-
representation).
This hypothesis is what I happen to believe, but I'm not attempting to
argue for it or defend it here (if I were, I'd include much more
detail!) My point here is only that I think there's an explanatory
gap that is possible to bridge, but that the self-references logics
that give rise to incommunicable beliefs don't bridge that gap....
more on this later.
The solution of the hard problem is that the machine makes the
experience of the gap, and can explain why the gap is not bridgeable.
The explanation is that there is necessarily, from the machine's
points of view, a real non bridgeable gap.
Using the descriptor Bp to signify a machine M's ability to prove
p is
fine. But it does not explain how it proves p.
It proves p in the formal sense of the logician. "Bp" suppose a
translation
of all p, of the modal language, in formula of arithmetic. Then Bp
is the
translation of beweisbar('p'), that is provable(gödel number of p).
If the
machine, for example, is a theorem prover for Peano Arithmetic,
"provable'
is a purely arithmetical predicate. It is define entirely in term
of zero
(0), the successor function (s), and addition + multiplication, to
gether
with some part of classical logic. It is not obvious at all this
can been
done, but it is "well known" by logicians, and indeed that is done
by Gödel
in his fundamental incompleteness 1931 paper.
When you say "if the machine is a theorem prover", are you referring
to a trivial machine? Something you can assign to your students?
Yes. I think that I did come back on this below. Now, the notion of
triviality is relative, and starting from a simple theory like PA, you
need to be Gödel to find it in the theory itself. That is a major
discovery in science. But if the student are familiar with the notion
of interpreter, and with a bit of logic programming, it becomes,
starting from a relatively high level programming language, a tedious
exercise.
If yes, then I struggle to see how we can relate such a machine to the
consciousness we have access to (our own), see below.
OK. It is not an easy point.
If no, then I
struggle to see how invoking a 'theorem prover' is not a "and then the
magic happens" leap of faith.
<snip>
Löbian machines are mere descriptions, absent
explanations of how a machine could be constructed that would have
the
ability to perform those operations.
Those are very simple (for a computer scientist). I give this as
exercise to
the most patient of my students.
Then as above, I struggle to see how we can interpret the biological
machines we are familiar with (namely, us) in terms of Löbian logic.
Is human language an adequate substitute for the precise logical
domain of arithmetic and Gödelian numbering of propositions? Natural
language is so messy and imprecise, but I may be missing the point.
In natural language, we confuse all modalities. We confuse easily ~Bx
with B~x (cf the confusion between atheism and agnosticism). Lucas and
Penrose, on Gödel, confuse Bp and Bp & p, and that confusion appears
also in all easy explanation of the mind-body problem. People often
confuse Bp and Dp, or tend to believe that Bp -> p, or that Bp -> Dp,
which is indeed the case for most modal logics studied before the
discovery of Löb's theorem (B(Bp -> p) -> Bp). I recall that B is put
for Gödel's beweisbar, and p is any arithmetical proposition.
So modal logic helps a lot, in both philosophy and math (provability/
consistency logics).
Taking the biological as an
example, it is self-evident that we humans can talk about and
evaluate
our beliefs. But until we have an explanation for *how* we do that
at
some level below the psychological, we're still just dealing with
descriptions, not explanations. Taking the abstract step towards
logical frameworks helps in terms of precision, for sure. But as
soon
as you invoke descriptors like Bp there's an element of "and then
the
magic happens."
The machine lives in Platonia, so I give her as much time as they
need.
Let me give a simple example. The machine can prove/believe the
arithmetical
laws, because those are axioms. They are sort of initial
instinctive belief.
axiom 1: x+0 = x
axiom 2: x + s(y) = s(x + y)
Just from that the machine can prove that 1+1 = 2 (that is, the
addition of
the successor of 0 with the successor of zero gives the successor
of the
successor of 0:
indeed:
s(0) + s(0) = s(s(0) + 0) by axiom 2 (with x replaced by s(0)
by the
logical substitution rule: the machine can do that)
but s(0) + 0 = s(0), by axiom 1 (again, it is easy to give to the
machine
the ability to match a formula with an axiom)
so s(0) + s(0) = s(s(0)), by replacing s(0) + 0 with s(0) in the
preceding
line.
Amazingly enough, with just the mutiplication axiom:
axiom 3: x * 0 = 0
axiom 4: x * s(y) = (x * y) + x
you add already prove all the sigma_1 sentences, that is, the one
having the
shape "it exists n such that P(n)", P(n) being decidable/recursive.
This is
call sigma_1 completeness, and is equivalent with Turing-
universality. That
is certainly amazing, but a bit of logic + addition and
multiplication gives
already Turing universality.
This means also that the machine, without induction, is already a
universal
dovetailer (once asked to dovetail on all what she can prove). But
such a
machine is not Löbian: it still needs the infinity of induction
axioms. That
infinity is recursively computable, so it remains a machine!
And that machine is Löbian, which technically means that not only the
machine can prove all the true sigma_1 sentences, but she can prove
for each
(fasle or true) sigma_1 sentences p that p -> Bp. In a sense, a
Löbian
machine is a universal machine which knows (in that technical
sense) that
she is universal.
But I am not a dovetailer.
You are right. My fault: the "that" in the last paragraph refers to
the universal machine + the induction axiom (and I assume the
universal machine is presented in a logic system, or I take the least
first order logical specification of the universal machine (a priori a
universal machine is not an axiomatic system, but it can easily be
transformed into one).
The universal dovetailer itself is not even a universal machine, given
that it has no inputs, nor outputs. But this is dependent on the
definition chosen for "universal machine".
Does a machine in your framework need to
dovetail on what it can prove for us to explain how it gets access to
its beliefs?
It does not need that. Usually machines and observers are not conceive
as dovetailers, except when they do explicit exploration for searching
a proof.
If no, do you think it is important to explain how
biological machines like us do have access to our beliefs?
That is crucial indeed. But this is exactly what Gödel did solve. A
simple arithmetical prover has access to its belief, because the laws
of addition and multiplication can define the prover itself. That
definition (the "Bp") can be implicit or explicit, and, like a patient
in front of the description of the brain, the machine cannot recognize
itself in that description, yet the access is there, by virtue of its
build in ability. The machine itself only identifies itself with the
Bp & p, and so, will not been able to ever acknowledge the identity
between Bp and Bp & p. That identity belongs to G* minus G. The
machine will have to bet on it (to say "yes" to the doctor).
If the
answer to that is no, are you just taking it on faith that assuming
comp, any machine that can access its own beliefs is in implementation
of a Löbian machine?
Not at all. It is a theorem. All self-referentially correct machine
are Löbian, once she is universal and can prove the induction axioms.
All recursively enumerable extensions of Peano arithmetic, or of
equivalent theories, are Löbian.
Maybe this is easy for you to prove, I may be
missing that as well.
It is not easy, but a minimal amount of familiarity with mathematical
logic makes it rather easy. It follows from standard proofs of Gödel's
theorem.
Do you have an explanation for how Löbian self-reference occurs in
biological machines? Is natural language required?
I don't think natural language is required. On the contrary, I would
say that natural language will usually entails a departure from
Löbianity, due to the confusion described above. The humans, and any
"embodied in complex reality" machines, will usually have a non Löbian
supplementary layer to handle "beliefs revision". We don't need that
to solve the mind-body problem, which is better handled with ideal
machine in empty environment (closed eyes meditation!). The non
monotonic supplementary layers is of course the crucial ingredient for
having a machine capable to go through any form of concrete life
struggle. But that's AI, not fundamental cognitive/physical science.
Believe me, I'm not expecting source code, so much as
a clarification that we don't quite have a TOE yet.
We have it. The "ontological TOE" (the ROE) is just elementary
arithmetic
(without induction). Such a theory already emulates (in "platonia")
all
machines, and this all the Löbian machines, which are considered as
the
internal observers in arithmetic. Here we have to be careful of not
doing
Searle's error, and to remember that by emulating a machine, you
don't
become that machine! (in particular your brain emulates you, but
your brain
is not you; the UD emulates all machines, but is only one
paricular, non
universal, machines).
I agree in the big picture, but I'm not sure you can say the TOE is
complete without some more explanation.
It is! In the sense of not necessitating any other axioms (than
elementary arithmetic, *without* induction). Then you need only
*definitions* to proceed. The reality = arithmetic without induction.
The observer = arithmetic with induction. The first emulate the
second, and the physics (and other modalities) are extracted from the
interview of the second, when emulated by the first.
What does ROE stand for?
Realm of Everything. It is the ontological part of the TOE. It is what
we take as existing or true independently of ourselves.
Moving on, one technical question I have is how you get the basis
for
quanta/qualia distinction - namely the property of
noncommunicability.
Unfortunately I probably won't understand the answer as the Solovay
logics are beyond me... but I hope to be able to understand how
noncommunicability manifests as a logical property of a machine.
It is consequence of what is called "the diagonalization
lemma" (Gödel
1931).
It asserts that for each arithmetical predicate P (like being
prime, being
the Gödel number of a theorem by the machine, etc.) you can find a
sentence
k such that PA (say) will prove k <-> P(k).
So for each predicate you can find a so-called fixed point. The k
above.
Now, take the predicate "provable", which Gödel has shown to be
definable in
Peano Arithmetic (or principia mathematica, whatever), that is, it is
definable in the formal language of the machine under consideration.
Now if P(n) is definable, then ~P(n) is also definable (= not P(n),
if P is
definable, the negation of P is also definable).
So by the diagonalization lemma, you can find a sentence k such
that PA will
prove:
k <-> ~P(k)
From this you can prove that if the machine is ideally correct, she
will
never prove k. Indeed, if she proves k, she will prove ~P(k), and
so will
lose self-referential correctness (and thus correctness). She will
prove k
and she will proves that k is not provable.
To be sure, Gödel assumed only omega-consistency (weaker from
correctness),
and Rosser extends the result for all simply consistent machines.
But I
don't want to go into much details, and I do assume the machines are
correct, for other reasons.
But you see that k is true also. Indeed by k <-> ~P(k), k asserts
its own
non provability, and k is indeed not provable. So k is an example
of true
but non provable, or non communicable, sentence.
That is the first incompleteness result. It is not difficult to
show a
concrete example of such a sentence k. Indeed ~Bf is such an example.
Self-consistency is incommunicable by the consistent machine. (It
is what I
like to call a protagorean virtue). f if the constant false, and t is
constant true. Or you can take f = = '0 = s(0)', and t == '0=0'.
thanks, I understand how you derive 'incommunicable' now, as the set
of propositions that are true but not provable (as in Gödel's
theorem).
OK.
More difficult to prove, is the fact that if the machine believes
also in
the induction axioms, then the machine can prove that IF she is
consistent,
then she cannot prove that she is consistent:
~Bf -> ~B~Bf
or (if you see that ~Bf = Dt):
Dt -> ~BDt; or again Dt -> DBf.
Löb will find the maximal generalization of that sentence (B(Bp ->
p) ->
Bp). With p = f, it should be easy to see that Löb generalizes
Gödel (hint:
in classical propositional logic ~p is equivaent with p -> f, so
you need
just to take p = f in Löb's formula).
So a machine that holds no contradictory beliefs cannot prove to
another that it never contradicts itself... interesting.
Indeed. It is the key, + the fact that the machine can prove that very
fact, once she take for granted some description of itself (like the
one given by the doctor).
Another concern I have is that there seems to me a lot of
imprecision
in the language used to correlate the consequences of the Löbian
machine with the folk-psychological terms we all use. For instance,
I've seen you refer to Bp in separate contexts as M's ability to
prove
p, and as M "believing" proposition p.
It is "belief" as used in cognitive science and epistemology. Not
the belief
of religion. Although there are no differences, actually, but that
is a very
hot debate. It is weird because that use of belief is very common.
It can
only shock people who believe religiously (pseudo-religiously) in the
propositions of science. But we always start from belief and get
beliefs.
Here you are using 'belief' in a way that is counter-intuitive in the
ordinary sense of the word.
But this is weird. I really use "belief" like in the belief theory.
Like in "do you believe that it will rain today?", or like in "do you
believe that Obama is the president of the US?". It seems to me that I
heard that use all the times when seeing a movie. The religious notion
of belief is used only in the religious context, but the word belief
is much more wider than that. A confusion comes from the fact that
people believes (!) that science = knowledge. But science is only
belief. The main difference with knowledge is that for knowledge we
have Bp -> p (and B(Bp -> p)). For belief we don't have Bp -> p or we
don't have B(Bp -> p). For machine: the situation is clear: G* proves
Bp -> p (trivially in the sense that we work on correct machines, like
PA or ZF), but G does not prove it (the machine does not believe it,
or does not prove it). The machine cannot know that she is correct. By
Löb's theorem, the machine knows that only on the proposition she can
actually prove.
What is amazing, and is the core of Gödel discovery, is that proving
acts like believing, and not like knowing, for the correct machine.
That makes the correct machine maximally humble and modest.
What you are saying suggests that "all
primes are odd" has the same epistemological status as "God does not
exist", or less controversially, "I am consistent". I hope we agree
that these are different kinds of beliefs, the primary distinction
involving provability. This is why invoking Bp in some contexts as
'provable' and in others as 'belief' is confusing.
It is the belief of the perfect (self-referentially correct machine)
when talking about a third person presentation of itself. Of course it
is the scientific third person self-reference. The itself is the 3-I,
or the body, or a description of the body.
Now, the epistemology is not in the proposition. So it makes no sense
to argue of the nature of the three propositions; because their
epistemological status will depend on the machine that you interview,
or of the theory that you are using.
For example, if you take the theory PA + "God exists" (a ridiculous
theory just for making my point), then "all the primes are odd" and
"God does not exist" have the same status (refutable).
"I am consistent" is true and not provable, nor refutable, in that
theory. The epistemology is in the machine/theory, not in any
proposition (I suspect you have some implicit theory in the
background; you should not).
if you define like me (and Plato) God by Truth. Then the proposition
"God does not exist" is no more expressible in the language of (any)
machines. Weakening of it will be accessible under the form of a bet
or guess.
That is confusing precisely
because proof and belief are actually opposed in certain
human-psychological contexts, such as belief in god. This concern
extends to the language you invoke in your "discourse with Löbian
machines" which I feel takes a lot of liberties with
anthropomorphizing, and sneaks in a lot of folk-psychological
concepts. Giving you the benefit of the doubt, I understand that
evangelizing these ideas means being able to make non-technical
analogies in the interest of accessibility. But it is also possible
that in one context you mean Bp to mean "prove" and in another you
mean Bp to "believe" in semantically non-identical ways,
I try not. You can feel that the theorem will apply to you and to any
machine which
1) are machine (obvious for the machine, and it is equivalent to
comp, for
the human)
2) believes in the elementary axioms of PA (so belief that x + 0 =
x, etc.).
3) are arithmetically correct (this is the "simplifying" assumption
or
studying *that* class of machine, which is motivated by
interviewing correct
machine to get the correct physical laws).
I think this gets to the core of my issues. I think we can agree that
humans that have never done any arithmetic in their lives are still
conscious (e.g. http://en.wikipedia.org/wiki/Pirah%C3%A3_people). So
(2) and (3) do not apply to humans.
Well, here I disagree. I have worked with strongly mentally disabled
people during two years. They were unable to count and most of them
could not even talk. With the help of computers I have been able to
convince external observers that they were only handicapped, and that
they were able to count, add an multiply, ... and to do induction. I
don't think it exists humans for which "2)" and "3)" does not apply,
even if the task for motivating them, and helping them to express
themselves, can be insuperable.
I am not convinced at all that the Piraha people escapes "2)" and
"3)". It seems clear that they are just not interested, like they are
not interested in canoe, nor in anything capable of changing their
life, and it is their right. But they are Löbian.
Actually I tend to believe that octopus and spider, and all
vertebrates are Löbian. Löbianity concerns believability, not actual
beliefs. Still less the ability to use or express such beliefs.
And also, Löbianity is needed only for self-consciousness, but
universality is enough for consciousness, I begin to think. More on
this below.
and this lets
you "cover more ground" in making the leap to the aspects of
consciousness that we can analogize from. In other words, imprecise
language may allow you to claim a more comprehensive mapping from
Löbianity to psychology than is actually possible.
It might be the case, but I don't think so. You might try to find a
specific
example.
OK, beyond the Bp confusion (if only in my head), another example is
making the leap from identifying a logical domain of propositions that
are true but not provable to our experience of qualia. While it is
certainly true that qualia can be considered true propositions (from
machine's 1p) that are not communicable (provable in 3p), it is not
obviously true that all such incommunicable propositions represent
qualia. Yet the AUDA routinely makes these kinds of leaps.
The true and not provable sentences are given by G* minus G, and they
does NOT represent the qualia.
For the qualia, I am using the classical theory of Theaetetus, and its
variants. So I define new logical operator, by Bp & p, Bp & Dt, Bp &
Dt & p. The qualia appears with Bp & p (but amazingly enough those
qualia are communicable, at least between Löbian entities). The usual
qualia (red, yellow, pain, pleasure) appears in the non communicable
part of the logic with the operator defined by Bp & Dt & p. Bp makes
it UD accessible, Dt makes it belonging to a "reality" (a model, a
maximal extension of a computation) and "p" makes it true. The logic
we get is close to the "quantum logic" of field perception (but works
remains to assess this, and evaluate such logics). Note that the
motivation of such classical knowledge theory in AUDA are given in the
UDA. Note also that I interview computationalist machines (not just
correct one), and this is formalized by restricting the atomic
arithmetical propositions to the sigma_1 sentences (having the shape
ExP(x) with P decidable).
As humans, we are epistemologically bound to consider abstract
arguments such as AUDA in the context of our experience. It is too
easy for us, in other words, to make those leaps with you in a
non-critical way, because we are already leaping just to comprehend
the argument. This is why the lack of precision concerns me, because
intuitively I feel that those leaps need more scaffolding, so to
speak.
The scaffolding is given by the classical theory of knowledge, that
the self-referentially correct machine is bound to find by itself when
introspecting herself. It leads to 8 (natural) hypostases, although in
reality it is 4 + 4*infinity (indeed, the weakening like "BBBp & DDt
& p" plays a role too, and seems to be necessary for some belief in
some notion of space, but again this is under development, well sleepy-
development, since a time.
I see more evidence of imprecision in your willingness to describe
your salvia experiences as totally non-personal.
To be sure I have published all my works in the 1988, except for the
dicovery of the arithmetical quantum logic, which I have published
in the
nineties, and I have discovered salvia in 2008.
The experience salvia are personal experiences.
But they lead sometimes the experiencer to a total amnesia which
makes it
feel as being a non personal experience.
OK, you are saying you (sometimes) have no memories of what happened?
They are completely inaccessible to to you?
Not really. When lucky, I can have a good memory of what happened.
When the memory comes back, I do remember that I was lacking my
memories, retrospectively.
It sounds more like you are saying you have zero self-awareness.
You can say that. Zero self-awareness, or even zero-self-
consciousness, but yet: maximal awareness, or maximal consciousness.
Memories and the self seems to make you less conscious (paradoxically).
If
that's the case, that does not mean that your (constructed) self is
gone, necessarily, only that you are not aware of it in the ongoing
experience.
OK.
Now, I have no
experience with salvia myself. However, the fact that such
experience
is available to you afterwards tells me that some aspect of your
self
is still present during the experience, regardless of how it feels.
Well, possibly so.
Contrast this with the experience of a baby, who actually has no
psychological self yet, or an extremely rudimentary one, and tell me
you are able to remember what it's like to be a baby.
Some experience are described like that. you feel becoming a baby,
or you
feel becoming what you have been before birth, or before the big
bang, or
beyond. It is just a feeling, and is reported as such by the
experiencer.
This is used for inspiration, or for doubting some prejudices only.
I was
willing to believe that consciousness and time was the construct of
the
third hypostases (Bp & p), but the salvia experience makes me feel
consciousness is more primitive than time, indeed.
So long as one can remain skeptical about the results of such
inspirations, I think such voyages away from our ordinary
consciousness can be extremely valuable. We can never forget how easy
it is to delude ourselves about what we feel, sober or not.
Yes. Those interested in consciousness are lucky that something like
salvia exists. It looks not toxic at all (even beneficious) and it can
lead to a short but quite interesting change of consciousness, which
is repeatable, and with an experience which is shared by many people
who are patient enough with the plant.
Like with sharable experiment, you learn only through it, by refuting
or doubting previous prejudices.
Arguments
made from introspection are always suspect.
They are 100% useless in the scientific endeavor. But like
consciousness, they can be the object of the scientific endeavor when
we tackle the mind body problem. Here, a lot of people confuse those
things. They understand that first person experiences are not
scientific (third person communicable), and so they induce that we
cannot talk *about* such experiences in any third person way. Of
course that is non a valid deduction, and it is a confusion of
category. In science we can talk about anything once we make our
theory clear enough. The idea that science cannot *address* some
question is obscurantism.
OK. I take the opportunity of the explanation above to explain what
is the
(Bp & p) stuff, and clarify why consciousness, or first person
self-apprehension leads to a notion which is beyond word.
Gödel's incompleteness theorem asserts Dt -> ~BDt (consistent -> non
provable consistent). So Dt, that is ~Bf, is not provable. But ~Bf is
equivalent with Bf -> f. So, in general Bp -> p is not provable. So
in
general Bp does not imply p, like a knowledge predicate or operator
should
do. So it makes sense to define, like Theaetetus, Kp (the knowledge
of p) by
Bp & p (knowledge = true (justified) belief). Of course we have Kp -
> p
(trivially given that Kp is Bp & p, and from a & b you can deduce
b). Indeed
Kp, defined in this way does follows the usual axiom of knowledge
(even
temporal knowledge) theories.
So you see that incompleteness justifies the working of the
classical theory
of knowledge for the machines.
Even more interesting is that Bp & p leads to an operator which is
not
definable in the language of the machine, and this explains a lot of
confusion in philosophy and theology, including why consciousness
cannot be
defined (only lived). The 1-I (captured by the Bp & p) has no name
from the
point of view of the machine.
You might try to define it like (Bp & Tp), with Tp put for an
arithmetical
truth predicate. But such a predicate cannot exist. Indeed, if it
exists,
then you can find a k, by applying again the diagonalization lemma
of Gödel
on ~V(n), so that PA would prove p <-> ~Vp, and from this you can
proof that
PA is inconsistent. So already Truth is not definable by the machine
(although she can define many useful approximations). Similarly, it
can be
proved that no notion of knowledge by a machine can be defined by the
machine. Classical (Theaetetical) knowledge is already like
consciousness:
we can' define it. But again, we can define the knowledge of
simpler (than
us) machine, derived the theology, and lift it on us, in a betting
way, at
our own risk and peril. We do that when we say "yes" to the doctor:
it *is*
a theological act, and people have the necessary right to say "no".
Now, we can study Bp & p logic at the modal level, and so can the
machines
too. This is a trick which makes us possible to bypass our's or the
machine's limitations.
The (Bp & p) hypostase (the first person point of view) has many of
the
feature of the "universal soul" of Plotinus (the greek mystical
inner God).
The machine lives it, but cannot give a name to it. It answers Ramana
Maharsi koan "Who am I?". The Lôbian machine's answer is "I don't
know, but
I can explain why I *cannot* know that in case I (my third person 3-
I, or
body) is a machine".
To get the logic of measure one in UD multiplication, Bp & p is not
enough,
we need a weakening and a strengthening which are given by Bp & Dt,
and Bp &
Dt & p.
You might take a look on the Plotinus paper, but to be honest, it
requires
familiarity in logic.
I can give it a shot, do you have a link?
It is on my front page of my URL. Click on the little "pdf" near the
title of the Plotinus paper, or just click here:
http://iridia.ulb.ac.be/~marchal/publications/CiE2007/SIENA.pdf
My final concern, as I've tried to elaborate on previously, is your
willingness to posit consciousness as a property of a (virgin)
universal machine. For me this is pretty counter-intuitive
For me too. That is why I have already written 8 diaries from the
salvia
experience. I see it, but can't believe it :)
It is very counter-intuitive. And I can't dismiss the experience as
a mere
hallucination, because it is the very existence of that
hallucination which
is counter-intuitive.
Why is the existence of the hallucination counter-intuitive?
Because it is an hallucination of a de-hallucination. With most
hallucinogen, you feel like dreaming or hallucinating, with some range
of lucidity. With salvia you loose completely lucidity, and feel the
experience as being realer than what you feel usually, and you feel
like awakening from an hallucination (your life) and being, at last,
really awake.
It is an hallucination that your life was an hallucination. With high
dose, you feel like your life is a vague dream and you forget it like
we usually forget dreams. With low dose, you keep the memory, but you
get disconnected from it. You feel your life as a dream, but not even
a personal dream, you can feel it as not belonging to you: you are
someone else, not even related to anything you knew.
To be honest, the experience can have other very astonishing feature,
and not all of them are easy to conciliate with comp, although that
might be possible (but then it is even more astonishing).
Another utterly counterintuitive aspect of the experience, is that,
you can feel to be conscious, yet you don't feel time going on, and
you can even forget what time (and space) are. before salvia, I was
linking consciousness and (subjective) time. I was thinking that all
qualia (like seeing red) was embedded in a time-like sensation. Even
now, I cannot imagine giving sense to any qualia, with some
subjectivity of time. With salvia people can hallucinate that time
disappear. You can be eternal for happens to be later a short instant!
It gives the mystical immortality apprehension, where immortality is
not some hope in some afterlife, but the living of eternity ... in the
past. You get the feeling you know that you are immortal, because you
have lived it. That's paradoxical and counterintuitive at the most.
Coming back from there, I am tempted to dismiss this as insanity (type
Bf, as it is most plausibly), but if I do the experience again, it is
(again) felt as the most obvious fact of life.
The hallucination existence is counter-intuitive because it seems to
imply that our consciousness is statical, and that the time is a
complex product of the brain activity (or of the existence of some
number relation). I thought that consciousness needs the illusion of
time, but salvia makes possible an hallucination which is out of time.
How could we hallucinate that? I see only one solution, we are
conscious even before we build our notion of time. Mathematically,
with comp, this invites us to consider that consciousness begins with
universality, even the statical one "living" in Platonia.
(which is
saying something because I'm with you on the UDA!).
Wow. I am very glad to hear that.
I had already come to an intuitive sense of the UDA before I
encountered your arguments, so I had already experienced that
"metaphysical vertigo" you warn about :-] Then to see that you had
actually mathematically formalized that intuition, I was pretty blown
away by that. The AUDA arguments are all new to me and that is what
I'm struggling with.
It means my
computer is conscious in some form, regardless of (or in spite of)
the
program it is running. And that for me leads to a notion of
consciousness that is extremely weak. It is why I compared it to
panpsychism previously, because panpsychism similarly attributes
consciousness to aspects of reality (assuming MAT) that lead to an
extremely weak form of consciousness that deprives it of any
explanatory potential. In your case at least it is possible in
principle to explain what it is about a universal machine that gives
rise to consciousness (and that, without any recourse to Löbianity
or
anything beyond universality).
When I read salvia reports, I was quite skeptical. I don't like the
idea
that the non Löbian machine is already conscious. But then the math
are OK.
Such machine lacks only the ability to reflect on the fact. They
believe t,
Bt, BBt, BBBt, etc. but they cannot believe Bp -> BBp. So they have
a far
simpler notion of themselves, and they lack the full self-
introspective
self-awareness of the machines having the induction axioms. Note
also that
although non löbian universal machine are in principle very simple,
they are
still far from trivial.
When you talk about the consciousness of the universal machine, you
require that it be dovetailing, in order for it believe t, Bt, etc.,
correct?
Not at all. That is again a consequence of my ambiguous use of "that"
above. The universal dovetailer is not a universal machine, and
usually, universal machine does not dovetail. "Bp is true" means "the
machine justifies or believes p", and if "Bp is asserted by the
machine", it means that the machine justifies or believes that "the
machine justifies or believes p"".
A virgin universal machine represents pure potential, and
attributing consciousness to pure potential is no different from
saying (in MAT) that all matter is conscious.
The assertion that matter is conscious does not make sense, for me.
Only a machine, or a person vehiculated by that machine, can be said
conscious.
Some years ago, I would have said that you need Löbianity to have a
person, but now, I think that the universal machine can be conscious,
and so I have to enlarge my notion of person. It is not to hard,
because a universal machine is not a completely trivial machine. Sure,
just addition and multiplication gives rise to universality, but the
whole point of Gödel & Co. is that addition and multiplication are
only apparently trivial. In fact they are not trivial at all. Number
theorists intuit this from their working familiarity of numbers (like
the quasi random primes), but it is an hard work for a logician, or
its students, to show that addition+multiplication are Turing universal.
And yes, it attaches consciousness to a potential. As I said, this is
counterintuitive, mainly because that consciousness is necessarily out
of time, space, or anything physical.
<snip>
In my view of things,
bacteria and viruses are not conscious because they lack a nervous
system that would satisfy the cybernetic organization I have in
mind.
I am interested in your proof they are universal, btw.
We agree, I think. All universal machine have a sophisticate, yet
sometimes
hidden in a subtle apparent simplicity, cybernetic organization.
Bacteria
have very complex series of regulator genes, which make it possible
to
program them for addition and multiplication (or simpler, but still
universal tasks). Viruses too, at least in combination with their
hosts.
I think also that an eukaryotic cells are already the result of a
little
bacteria colony, so that we are swarms of bacteria, somehow.
The cybernetic organization does not need neurons, it can use genes
and
"meta-genes" (genes regulating the action of other genes). In fact a
bacteria like E. Coli, is an incredibly complex structure, with
very subtle
self-regulating actions.
I see, yes, and actually I want to say I remember hearing about
research that involved programming bacteria for some task, but I could
be wrong.
What has become so appealing about cybernetics to me is that it tries
to characterize systems in terms of information flows, which may be
implemented in any kind of substrate (or none at all, as in platonia!)
Well, with UDA you should be able to see that substrate can't help. To
introduce substrate can only hinder the search of a solution to the MB
problem.
I'm also
wondering if you have an english-language explanation of the
MGA... I
recall seeing one a long time ago.
Try with this:
http://old.nabble.com/MGA-1-td20566948.html
Let me now if you have a problem.
Thanks, that is a very effective argument. The one thing I didn't
understand very well was Maudlin's argument... is there a meaty
summary of that argument somewhere? I don't get how the
counterfactuals can be dealt with by such minimal additions to the
machine.
On an unrelated (to this thread) topic, I have a question about 1p
indeterminacy. You say the universe as we experience it is a sum on
the computational histories of an infinity of programs running on the
UD.
Yes. This is the UDA conclusion.
And that what makes the universe consistently communicable from
one person to another is the "gluing properties" of such histories.
Can you explain "gluing properties"? Is there a mathematical
formalization of that concept?
Well, not yet really. I leave this for the next generation :)
As I did explain to Stephen, to formalize it in the AUDA, you need to
define a tensor product in the matter hypostases, and for this is you
need some sophisticated semantics for the Z and X logics. Progresses
have been done, but it lead toward difficult mathematical questions.
Actually this is a problem even for quantum mechanicians, and solution
already exists in the frame of some logic (by Girard, but also
Kaufmann (in knot theory!) and Abramski, linking knot theory and
quantum statistic, but it would still be treachery to use them
directly without extracting them from the self-reference logics (which
would threat the theory of qualia, which needs to extract quanta and
qualia simultaneously from self-reference).
So, I use "glueing" in the intuitive sense that you can extract from
the UDA. Basically two dreams (computations seen from inside, that is
from first person points of view/hypostases) by different subjects
will glue if there is a reality (or just locally: a larger
computation) generating those two computations, in some "natural way".
The usual instinctive root of gluing dreams, is the idea that there is
a common geometrical reality. But that simple idea is not available in
the UD, or in arithmetic, given that there are infinitely many
computations, and no primitive geometry at all. Technically it means
that we have to extract a notion of resource (linearity), and of
tensor product (interaction). The logic of the material hypostases are
very promising for doing that (or at least they show that the
impossibility of this is hard to prove, and this shows that the white
rabbits might be hunted away in the comp frame). We can come back on
this, I have to go now.
Bruno
http://iridia.ulb.ac.be/~marchal/
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