On 10/22/2011 8:23 AM, Bruno Marchal wrote:

On 21 Oct 2011, at 20:34, Stephen P. King wrote:

On 10/21/2011 10:10 AM, Bruno Marchal wrote:

On 21 Oct 2011, at 15:08, Stephen P. King wrote:

On 10/21/2011 8:14 AM, Bruno Marchal wrote:

On 19 Oct 2011, at 05:30, Russell Standish wrote:

On Mon, Oct 17, 2011 at 07:03:38PM +0200, Bruno Marchal wrote:
This, ISTM, is a completely different, and more wonderful beast, than
the UD described in your Brussells thesis, or Schmidhuber's '97
paper. This latter beast must truly give rise to a continuum of
histories, due to the random oracles you were talking about.


All UDs do that. It is always the same beast.


On reflection, yes you're correct. The new algorithm you proposed is
more efficient than the previous one described in your thesis, as
machines are only executed once for each prefix, rather over and over again for each input having the same prefix. But in an environment of unbounded resources, such as we're considering here, that has no import.

Note that my programs are not prefixed. They are all generated and executed. To prefix them is usefulm when they are generated by a random coin, which I do not need to do.



So the histories, we're agreed, are uncountable in number, but OMs
(bundles of histories compatible with the "here and now") are surely
still countable.

This is not obvious for me. For any to computational states which are in a sequel when emulated by some universal UM,there are infinitely many UMs, including one dovetailing on the reals, leading to intermediate states. So I think that the "computational neighborhoods" are a priori uncoutable. That fits with the topological semantics of the first person logics (S4Grz, S4Grz1, X, X*, X1, X1*). But many math problems are unsolved there.


Hi Bruno and Russel,

I would like to better understand what "topological semantics" means. Are you considering relations defined only in set theoretical sense, ala the closed or open or clopen nature of the sets relative to each other?

I guess you know the topological semantics of intuitionist logic. Instead of interpreting the propositions by sets in a boolean algebra, you interpret it by open set in a topological space. You have soundness and completeness theorem for that. I think this came from semantics for the semantics for the modal logic S4, which mirrors well intuitionist logic.



What about the form of the axiom of choice for the set theory?

?

There are multiple versions of set theory depending on the selection of its form of axiom of choice.

There are many set theories at the start, and they are not equivalent. Set theories assumed too much, especially with respect to the mechanist assumptions.





How do you induce compactness?

Why do you want the space being compact. The point of the topological semantics is that it works for all topological spaces, like any boolean algebra can be used for classical logic.

A space needs to be compact for many reasons including, but not exclusive to, the ability to have a physics in them. Spaces that are not compact will not have, for instance, fixed points that allow for notions such as centers of mass, etc. There are also analycity reasons, and more.

UDA shows that space and time belongs to the category of mind or machine perceptions, themselves belonging to the category of number relation. To *assume* space and time in the ontology can only be misleading, with respect to the mechanist formulation of the mind-body problem.

[SPK]
I am thinking of mathematical spaces, not the physical space of experience.





How is a "space" defined in strictly arithmetic terms?

Why do you want to define it in arithmetic. With comp, arithmetic can be used for the ontology, but the internal epistemology needs much more. Remember that the tiny effective sigma_1 arithmetic already emulate all Löbian machines like PA, ZF, etc. Numbers can use sets to understand themselves.

Yes, that numbers are what the Stone duality based idea assumes, but numbers alone do not induce the "understanding" unless and until sets are defined in distinction to them.

Numbers alone are not enough, you need to make explicit the assumption of addition and multiplication. And that is provably quite enough. To understand this you need to understand that the partial recursive functions are representable in very tiny theories of arithmetic (like Robinson Arithmetic), and you need to recall that you assume that you would survive with a digital brain/body. The rest follows from the UD reasoning.

    That is not addressing my point.


This is why it cannot be monistic above the "nothing" level.

?



Tarski's theorem prevents understanding in number monist theories.


?

If arithmetic truth cannot be defined in arithmetic, how can a notion of understanding obtain. Lobian machines are not just pure arithmetic, it seems.







If we take the no information ensemble,

You might recall what you mean by this exactly.



and transform it by applying a
universal turing machine and collect just the countable output string
where the machine halts, then apply another observer function that
also happens to be a UTM, the final result will still be a
Solomonoff-Levin distribution over the OMs.

This is a bit unclear to me. Solomonof-Levin distribution are very nice, they are machine/theory independent, and that is quite in the spirit of comp, but it seems to be usable only in ASSA type approach. I do not exclude this can help for providing a role to little program, but I don't see at all how it could help for the computation of the first person indeterminacy, aka the derivation of physics from computer science needed when we assume comp in cognitive science. In the work using Solomonof-Levin, the mind-body problem is still under the rug. They don't seem aware of the first/third person description.

  S-L seems to assume 1p = 3p or no 1p at all!

I think they just ignore 1p.





This result follows from
the compiler theorem - composition of a UTM with another one is still
a UTM.

So even if there is a rich structure to the OMs caused by them being
generated in a UD, that structure will be lost in the process of
observation. The net effect is that UD* is just as much a "veil" on
the ultimate ontology as is the no information ensemble.

UD*, or sigma_1 arithmetic, can be seen as an effective (mechanically defined) definition of a zero information. It is the everything for the computational approach, but it is tiny compared to the first person view of it by internal observers accounted in the limit by the UD.

  How do we define this notion of size?

There are many ways. Tiny in the sense of the ordering of sets of provable proposition of arithmetic, for example. Robinson theory is tiny compared to PA which is tiny compared to ZF, in that sense.


Tiny as opposed to ???

To big! Or strong. ZF proves much more arithmetical propositions than PA.

   Oh, the number of independent but mutually necessary axioms?

This would not been a good measure of complexity or strongness, given that you can find theories with many independent and mutually necessary axioms which can be forlaised with the use of very few (different) axioms. It is less ambiguous to measure the "force" of a theory by the amount of arithmetic theorem they are able to prove. Note that ZF and ZFC (ZF + axiom of choice) have the exact same force. The axiom of choice has no bearing on the arithmetical reality. Of course, some proofs of arithmetical theorem can be shorter, but all proof using the axiom of choice can be done without using the axiom choice.

    I wish I could find a broader discussion of that claim.









Unless I'm missing something here.




Lets leave the discussion of the universal prior to another post. In a nutshell, though, no matter what prior distribution you put on the "no information" ensemble, an observer of that ensemble will always see
the Solomonoff-Levin distribution, or universal prior.

I don't think it makes sense to use a universal prior. That would
make sense if we suppose there are computable universes, and if we
try to measure the probability we are in such structure. This is
typical of Schmidhuber's approach, which is still quite similar to
physicalism, where we conceive observers as belonging to computable
universes. Put in another way, this is typical of using some sort of
identity thesis between a mind and a program.

I understand your point, but the concept of universal prior is of far more general applicability than Schmidhuber's model. There need not be
any identity thesis invoked, as for example in applications such as
observers of Rorshach diagrams.

And as for identity thesis, you do have a type of identity thesis in
the statement that "brains make interaction with other observers
relatively more likely" (or something like that).


yes, by the duplication (multiplication) of populations of observers, like in comp, but also like in Everett.

But we also need something that acts to code the "no preferred reference frame" of GR. Everett does not solve this problem, it only compounds it. :-(

I am not sure I follow you. Everett makes clear that the choice of base is a false problem.

How so? Please point to a discussion of this! Everett is explicitly non-relativistic....

I suggest you read the original long text by Everett, in the DeWitt and Graham book. The fact that Everett shows this without assuming anything about relativity makes the case even stronger. But I don't think this is relevant on the topic. The digital mechanist hypothesis is neutral on physics. And the conclusion is that the whole of physics is a number "illusion".


Yes, and that is its failing. It takes the physical world to be epiphenomena. Why does the physical world even need to exist at all?







There has to be some form of identity thesis between brain and mind
that prevents the Occam catastrophe, and also prevent the full retreat
into solipsism. I think it very much an open problem what that is.

This will depend on the degree of similarity between between quantum mechanics and the comp physics, which is given entirely by the (quantified) material hypostases (mainly the Z1* and X1* logics). Open but well mathematically circumscribed problem.



From what I have studied so far, the (static) relationship between QM and COMP physics is the relationship between Boolean algebras and Orthocomplete lattices.

Let us hope. Even just that has not been proved. Open problem.


I think that a partial solution might require a weakening of the definition of a countable model of arithmetic and application of the Tennenbaum theorem (that would allow for a variational principle, something like q-deformed theories where q is a measure of the relative recursiveness of the model of the theory).

?

I am still studying the math on this, the problem that I have is that there do not seem to be words for the idea that I see. The closest concept is how a fixed point is defined on a manifold... Crudely, think of a manifold (not a space per se) where every point is a model of arithmetic, the fixed point is the one that is countable and recursive.



The idea is that every 1p would observe itself, in the Lob sense, to be recursive.

?

   How does a Lobian machine recognize its properties?

Which properties. I'm sorry but you are losing me.

Any of its properties. How does a Lobian machine know what it is, even if incompletely?






The proof would require showing that a Lobian machine on a non-standard model of arithmetic would *not* be able to "see" its non-standardness and thus it would bet that only it is recursive, thus it's Bp&p would be 1p and not 3p truth.

? (Bp&p) is the definition of 1-p in, well not really in arithmetic, but in terms of arithmetic. We cannot define "p" (p is true) in any effective theory.

   This is part of the incompleteness of your result :-(

It is part of Tarski undefinability result, and it concerns *all* effective theories, and *all* machines interested in searching truth. And actually, it is a big chance for machine's theology, given that such undefinability is what makes truth behaving like a machine's god (like in Plato) and the knower like an inner God (like in Plato, or like with Plotinus' universal soul).

Bruno

http://iridia.ulb.ac.be/~marchal/

I wish we could dispense with the idea of entities what are impossible to exist. An entity cannot both be *all-knowing* and have an existence apart from the rest of the Totality. Theologies seriously need to be sure that their entities are not self-contradictory.

Onward!

Stephen

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