Le 25-mai-07, ŕ 02:39, Tom Caylor a écrit :
>
> On May 16, 8:17 am, Bruno Marchal <marc....domain.name.hidden> wrote:
>> Hi,
>>
>> I take the opportunity that the list is calm to send a first
>> approximation of a possibly extendable post which addresses the
>> beginning of the background needed for the interview of the universal
>> machine on the physical laws.
>> It also addresses some point relevant for discussing the link "formal
>> system" <======> "Computation" as in Tegmark diagram page 18 of his
>> paper "the mathematical universe" (cf a post by Mark Geddes).
>>
>> Although schematic, it could already help me if you can list the
>> points
>> for which you would like examples or more technical details, or just
>> explanations. You can also ask for *less* technical details like an
>> explanation in (pure) english, perhaps.
>>
>> I am building again from Robinson Arithmetic, but I could have use the
>> combinators or any logical description of what a universal machine (a
>> computer) can do. The adavantage of using Robinson Arithmetic (or its
>> "little" variant) is that provability in Robinson Arithmetic
>> corresponds to universal computability, but not of universal
>> provability (which does not exist).
>>
>> I am perhaps on the verge of not being able to explain the sequel in
>> informal term, but I keep hope that non expert, but computer-open
>> minded people, can learn and help me to be clearer or more
>> pedagogical,
>> without having us to study thoroughly mathematical logic.
>> Tell me perhaps if you don't understand what I call "Searle's error"
>> in
>> the comp setting below.
>>
>> ***
>>
>> 0) historical background
>>
>> ARISTOTLE: reality = what you see
>> PLATO: what you see = shadows of shadows of shadows of shadows of
>> ....
>> what perhaps could be.
>> And would that be? nobody can say, but everybody can
>> get
>> glimpses by looking inward, even (universal) machines.
>>
>> Twentieth century: two creative bombs:
>>
>> - The Universal Machine (talks bits): UM (Babbage, Post,
>> Turing,
>> Church, Suze, von Neumann, ...)
>> - The other universal machine (talks qubit): QUM (Feynman,
>> Deutsch, Kitaev, Freedman, ...)
>>
>
> Could you please expand on how these 20th century ideas extended
> Aristotle and Plato?
Aristotle is (partially?) responsible to the come back to the naive
idea that matter exist primitively, and this leads quickly to the idea
that science = mainly empirical science.
Plato's intuition is that the empirical world is but one aspect of a
bigger reality, and that intuition comes from self-introspection.
The Universal Machine can illustrate that indeed when she introspects
herself, she can discover her own limitation (Godel and Lob theorem are
provable by the machines on themselves: a point frequently missed by
those who try to use Godel against Mechanism).
> Of course the quantum part is an extension, but
> what about the universal part?
> As you may suspect, I am questioning as usual the even-more-
> fundamental assumptions which might be underneath this. Sorry I don't
> really have any time lately either, so I understand if you just want
> to get on with your description based on your assumptions.
OK. Never forget I have never defend the comp hyp. I have (less
modestly) prove that it is impossible to believe in both the comp hyp,
and the weak materialist thesis (the thesis that there exist primary
matter having a relation with the physical knowledge). With comp matter
emerges from mind which emerges from numbers.
>
>> Comp = Milinda-Descartes Mechanism in a digital version. = (also) "YES
>> DOCTOR" + CHURCH'S THESIS. (I suppress the arithmetical realism,
>> because it is implicit in CHURCH'S THESIS).
>>
>> UDA: a reasoning which shows that if comp is correct then the physical
>> laws have to be derived by a measure on states (the measure being made
>> up through their computational histories).
>>
>> Subgoal: extract QUM from UM's self-observation.
>>
>> Link with everything-list: search for the "observer moments" and the
>> relevant structure operating on them (not yet solved).
>>
>> ***
>>
>> 1) The ontic theory of everything: LRA (Little Robinson Arithmetic),
>>
>> CLASSICAL LOGIC (first order predicate logic axioms and
>> inference
>> rules)
>> AXIOMS OF SUCCESSION
>> AXIOMS OF ADDITION
>> AXIOMS OF MULTIPLICATION
>>
>> That's all. It is the "Schroedinger" equation of the comp-everything!
>> The reason is that LRA is already as powerful as a universal machine.
>> LRA proves all verifiable sentences with the shape ExP(x), with P(x)
>> decidable. It is equivalent with the universal dovetailer.
>>
>> Now we have to do with LRA what Everett has done with QM. Embed the
>> observer in the ontic reality.
>> For this we have to "modelize" the observer/knower/thinker.
>>
>> ***
>>
>> 2) The epistemic theory, or the generic observer theory: PA (the
>> lobian
>> machine I will interview).
>>
>> CLASSICAL LOGIC (first order predicate logic axioms and
>> inference
>> rules)
>> AXIOMS OF SUCCESSION
>> AXIOMS OF ADDITION
>> AXIOMS OF MULTIPLICATION
>> AXIOMS OF INDUCTION
>>
>> Note: the observer extends the ontic reality! It extends it by its
>> beliefs in the induction axioms. They are as many as they are formula
>> F(x), and they have the shape:
>>
>> [F(0) & Ax(F(x) -> F(x+1))] -> AxF(x)
>>
>> ***
>
> OK. Would you say that LRA plays the part of Arisotle and PA the part
> of Plato here?
No, why? LRA is a version of the universal dovetailer (the ontic
reality) in the form of a subset of the beliefs of the lobian machine
like Peano Arithmetic.
I recall often the difference between Aristotle and Plato, because it
corresponds to two diametrically different conception of reality. With
Aristotle, roughly speaking, there is mainly a physical world, and
explanations are supposed to be of a naturallistic type. With Plato,
and the mystics (those who search the truth by looking inward) the
physical world is the interface (to borrow Rossler's vocabulary)
between us and something else: there is a deeper reality, a priori not
of a naturalistic type.
>
>>
>> OBVIOUS IMPORTANT QUESTION: How to interview PA when we dispose
>> ontologically only of LRA?
>>
>> NOT OBVIOUS SOLUTION: just try to obviate the fundamental SEARLE ERROR
>> (cf Mind's I, Hofstadter -Dennett describe it well) in front of the
>> LRA
>> theorems.
>>
>> I explain: Searle's goal consisted in arguing against mechanism, that
>> is arguing we are not machine, and in particular that a simulator is
>> not the real thing. He accepts the idea that in principle a program
>> can
>> simulate a chinese speaker. Knowing the program, Searles accepts he
>> can
>> simulate it, and this without understanding chinese. He concludes that
>> we have to distinguish between speaking chinese and simulating
>> speaking
>> chinese.
>>
>> True! but with comp you have to distinguish between the simulated
>> chinese speaker and the simulator of the chinese speaker! By being
>> able
>> to simulate the chinese speaker, Searle can have a conversation with
>> the chinese speaker (well assuming that the chinese speaker can talk
>> english, or that Searle knows chinese).
>>
>> This is particularly important in our setting. LRA has the power of a
>> universal turing machine, so it has universal computability power, and
>> can act as a universal simulator. In particular LRA can simulate PA,
>> and any recursively enumerable theory/machine. But compute or simulate
>> are not similar to believing or proving (or talking in some genuine
>> personal way).
>>
>> LRA provides a view of computability as a very particular and quasi
>> debilitating case of provability. LRA can prove almost only the true
>> Sigma1 sentences (which is enough to run the UD). For example, LRA
>> cannot prove (about its ontic reality or intended interpretation) that
>> Ax (x + y = y + x). PA can. PA is already a sort of Ramananujan, a
>> total genius compared to LRA, despite the fact that LRA is already
>> universal for computation. For the notion of provability there is no
>> universal notion. There are as many notion of provability than there
>> are machines (human included).
>>
>
> So the induction power of PA brings in an infinity, since we are
> saying "for all x in N" (N=natural numbers). However, doesn't LRA
> already bring in an infinity at the ontic level?
Yes sure. LRA is UD*. LRA concerns already all accessible truth, common
to all universal machine, about natural numbers.
LRA is, like PA, under the godelian limitation joug. Only, PA knows it!
Lobian machine, like PA or ZF, are godel-limited, but they are aware of
their limitation. OK?
> This is because even
> the statement 1+2=2+1 is a Plato-like statement. The Aristotle
> verification would be to take 1 object and then take 2 more objects
> and count the group as a whole. Then take 2 objects and then 1 object
> and count the group as a whole. But, first of all, there are at least
> conceptually a (at least potentially) infinite number of objects you
> could use for this experiment, and you could do the experiment as an
> observer from an infinite number of angles/perspectives. Plus, a
> difference in perspective could make it so that you are taking the
> objects in a different order and so invalidate the experiment. I
> don't know what the implication is here other than there are very
> fundamental philosophical assumptions to deal with here. This is even
> without bringing multiplication into the picture. It seems, if you
> are going to base your reality on math, that these kinds of questions
> aren't unimportant because they remind me of the fundamental problems
> at the base of the quantum versus relativity paradox.
I cannot comment because it is a bit vague for me. Normally I can not
address physical question before getting the comp-physics.
Bruno
>
> Tom
>
>> If you want, LRA is gifted in proving the existence of number having
>> verifiable property, and giving that PA can be defined, by Godel
>> arithmetizability, in the language of LRA , although LRA has no
>> opinion
>> on the induction axioms (believed by PA), LRA can prove that PA proves
>> this or that. And with strong-AI (that is Searle's weak Turing notion)
>> LRA will prove "and Bruno says this or that". In this setting, Searle
>> would confused LRA and PA, or LRA and Bruno.
>>
>> Now, I will not interview the brunos, but the PAs, as they have much
>> less prejudices, if only that. By the "PAs" I mean the many recurring
>> proofs by LRA that PA proves something (you can see that as the many
>> simulations of PA by LRA). Well, the main advantage of interviewing PA
>> is that anyone just a bit reasonable, believes in PA's beliefs.
>>
>> The PAs, are Löbian. In a nutshell, if LRA is indeed universal, PA is
>> not only universal, but knows, in some weak sense, that she is
>> universal.
>> LRA has universal "existential provability" power about the verifiable
>> propositions, but LRA has almost no universal "universal provability"
>> power. LRA's talks go like this: oh a prime number! oh a proof by ZF
>> that PA is consistent, oh ...", but LRA cannot prove the infinity of
>> the primes, nor can LRA prove any reasonable result in number theory
>> begining by a "Ax" (for all x) quantification. The induction axioms,
>> in
>> which PA believes, provide her with, not only a tremedous
>> generalization power, but confers to PA the (modal-logically-viewed)
>> maximal indepassable self-referential power (the one which are
>> axiomatisable by the modal logics G, G*).
>>
>> Being universal LRA is under the joug of the incompleteness
>> phenomenon.
>> But this is not saying a lot given that we already know LRA's
>> provability power are so weak. PA, being universal too, is obviously
>> under that joug too, and that is more astonishing a priori, because,
>> as
>> I said, PA is already quite a genius with tremendous provability
>> power.
>> Now, by its powerful self-referential knowledge inherited by that
>> provability power, knowing her universality, PA *knows* that she is
>> under the joug of the incompleteness phenomenon.
>>
>> This makes PA, and all her correct (with respect to the usual model
>> (N,+,x)) recursively enumerable extensions, incredibly modest! (if
>> not
>> a bit naive). But it makes also PA "aware" of intensional
>> distinctions,
>> already described in the Theatetus, which provide then the
>> arithmetical
>> interpretations of Plotinus Hypostases. Including his "Matter"
>> hypostases, making them comparable with quantum logic/quantum
>> computations.
>>
>> OK, that last paragraph was quick. I have to explain more on logic
>> (godelian, modal, but also the weak logics like intuitionistic logic
>> or
>> (mainly!) quantum logic, ...). Asap.
>>
>> Does this help a bit?
>>
>> Bruno
>>
>> http://iridia.ulb.ac.be/~marchal/
>
>
> >
>
http://iridia.ulb.ac.be/~marchal/
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Received on Fri May 25 2007 - 09:57:22 PDT