Hi Terren
Apology for commenting your post with some delay.
On 06 Jul 2011, at 19:54, Terren Suydam wrote:
Hey Bruno,
Thanks for your comments... I'm a little clearer now on your stance on
consciousness and intelligence, I think. I have a few more questions
and concerns.
Regarding consciousness, my biggest concern is that you're not really
explaining consciousness, so much as describing it.
Yes. That is true. I think computationalism can explain consciousness,
except for a remaining gap, but that it can explain why such gap is
unavoidable. So in a sense I do think that comp explains as completely
as possible consciousness. I will try to convey this in my further
commenting below. It does not really describe it, though, because the
explanation rule all description for it. See below.
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).
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.
Ditto for the
induction axioms;
Those are simply the scheme of axioms:
[P(0) and for all n (P(n) -> P(s(n)))] -> for all n P(n)
You cannot prove that [for all n and m, n+m = m+n] without it.
Of course, such a machine talk only on numbers, so to define
provability *in* the language of the machine, you have to represent
the formula and the finite piece of proofs by numbers (like we have to
represent by strings in some language to communicate them);
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.
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.
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).
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'.
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).
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.
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).
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.
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.
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.
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.
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.
(which is
saying something because I'm with you on the UDA!).
Wow. I am very glad to hear that.
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.
I realize you're not saying for certain that universal machines are
conscious, and that this is somewhat informed by your salvia
experiences. But for where my head is right now, consciousness ought
to be explainable in terms of some kind of cybernetic organization
that goes well beyond "mere" universality.
Good idea to put "mere" in quote. It is just a flabbergasted fact that
addition and multiplication already leads to Turing universality, and
that is not trivial at all to prove. But afterwards it shows that
universality is cheap. It explains why nature recurrently build
universal structure.
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'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.
Apologies for the length of this response!
You are welcome.
Bruno
http://iridia.ulb.ac.be/~marchal/
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