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
On Tue, Jul 12, 2011 at 11:17 AM, Bruno Marchal <marc...@ulb.ac.be> 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 :-]
>> 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 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.
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. 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); 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.
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.
>> 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?
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. If no, then I
struggle to see how invoking a 'theorem prover' is not a "and then the
magic happens" leap of faith.
>> 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.
>> 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:
> 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
> 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. 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? If no, do you think it is important to explain how
biological machines like us do have access to our beliefs? 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? Maybe this is easy for you to prove, I may be
missing that as well.
Do you have an explanation for how Löbian self-reference occurs in
biological machines? Is natural language required?
>> 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. What does ROE stand for?
>> 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
> 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
> 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
> 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.
>> 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. 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.
>> 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.
>> 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
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.
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
>> 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?
It sounds more like you are saying you have zero self-awareness. 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
>> 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. Arguments
made from introspection are always suspect.
> 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?
>> 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?
>> (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? 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.
>> 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
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!)
>> 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:
> 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
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. 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?
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