On May 25, 6:55 am, Bruno Marchal <marc....domain.name.hidden> wrote:
> Le 25-mai-07, à 02:39, Tom Caylor a écrit :
> > On May 16, 8:17 am, Bruno Marchal <marc....domain.name.hidden> wrote:
> ...
> >> 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.
>
I should respond to your response. I'm in a busy pensive state
lately, reading Theaetetus (as you suggested on the Incompleteness
thread) along with Protagoras and some Aristotle (along with the dozen
other books I'm always reading...) in the little time I have.
> 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.
>
But you do make assumptions as part of the comp hypothesis, including
assumptions about numbers.
> ...
> >> ***
> >> 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.
>
LRA looks to be about the particulars of arithmetic. PA, with
induction, is trying to generalize to come up with some universal
truths about 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.
But LRA has access to only one particular truth at a time, with no
awareness of generalities/universals. I'm just trying to see if I
understand correctly what you are getting at.
> 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?
>
So PA has this "awareness", by *definition*. It is a useful *tool* in
mathematics, but you are assuming it is a part of reality at the
deepest level. This is part your Arithmetic Realism part of the comp
hyp, is it not?
>
> > 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.
>
> I cannot comment because it is a bit vague for me. Normally I can not
> address physical question before getting the comp-physics.
>
> Bruno
>
The above does not require physical reality, but only concepts that we
can think about looking inward (eyes closed view). But even though it
is "only" conceptual, my point is that we are taking a "leap of faith"
even when we talk about 1+1=2, classifying an infinite number of cases
into one equivalence class.
Perhaps at the core of this issue is whether things like "+" are
prescriptive or descriptive. Is it possible that there are universes
with mathematical "white rabbits" such that when you take 1 thing and
1 other thing ("physical" or not) and associate them in any way,
including just thinking about them, then you don't necessarily get 2
things (e.g. sometime you get 1 or 3 or 0)?
Tom
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Received on Fri Jun 08 2007 - 14:17:31 PDT