Hi Jason,

On 14 Jun 2010, at 15:55, Jason Resch wrote:



On Mon, Jun 14, 2010 at 3:08 AM, Bruno Marchal <marc...@ulb.ac.be> wrote:

You have added the UTM and its variants to the pile. Any of these could be just as right as you think COMP is.

I have no idea about the truth status of Digital Mechanism, except nothing in nature suggest it to be false. On the contrary, the main startling consequence of digital mechanism (non locality, first person indeterminacy, symmetry at the bottom, etc.) are confirmed by the empirical study of nature. But I am open to the idea that comp may be false. That is why I study it: to show it falsifiable and thus scientific in Popper sense.



Bruno,

The part about symmetry at the bottom being a consequence of mechanism intrigued me. How is symmetry at the bottom related to mechanism?


It is rather technical. Let me give you a non rigorous overview. It is easier to explain this on Löbian machine "talking" first order language.

In that case, Gödel's completeness theorem insures that if the machine proves A then A will be true in all mathematical models satisfying (making true) all the beliefs of the machine (soundness), and vice versa (completeness).

So: M proves A iff A is true in all models of M (I identify a machine with the set of all its beliefs/theorem/assertion).

Now let us write Bp for the machine proves p. Bp is meant for Bew('p') and is supposed to be expressible in the language of the machine (like Gödel's showed it to be the case for a large class of theory/machine).

A first idea to tackle the first person indeterminacy would consist to say that the probability of a 'proposition' is one in case 'Bp' is true, given the Gödel's soundness/completeness theorem. This would indeed making that proposition true in all consistent extension (or models) of the machine's beliefs. But this does not work, due to incompleteness. Incompletness implies the existence of cul-de-sac world/model/state in which Bf, the provability of 'false' is true. There, Bf is trivially satisfied, 'f' is true in all extension, given that there are no extensions! This has been the main motivation for looking to intensional variant of 'B'.

So let us invent a new modal operator Bp, defined by Bp & p (the first Theatetical variant), or the weaker Bp & Dp, or Bp & Dp & p. The logic G* shows them all equivalent for any 'p' being a proposition in arithmetic (the "ontic" propositions), but, by incompleteness the machine cannot prove this facts. So, although those variant are 'truly' equivalent, they will not seem equivalent for the machine's points of view.

And finally let us restrict p to the Sigma_1 sentences (that is those with the shape ExP(x), P decidable). This is how we translate "comp" itself in the language of the machine. A machine is universal iff it proves all the true sigma_1 sentences, and a machine is Löbian if it proves p->Bp, for all p sigma_1. To be sigma_1and true means also to be accessible by the universal dovetailer.

In that case, it can be shown that the modal symmetry condition "p -> BDp" is satisfied for all theaetetical variant of provability. It means that any time you can fly from a world to another world, or from a state to another state, you can come back following either the accessibility relation (if there is one), or the neighborhood relation (if there is one). At the "star" level (under G*, the arithmetical "Noûs"), unfortunately, we don't have modal logics with accessibility relations, so we have to use more complex semantics, and the symmetry conditions is admittedly rather abstract. We have also Bp -> p, and despite the lack of necessitation, this gives a formal quantization for p: BDp. This gives a formal quantum logic, admitting an arithmetical interpretation, and it provides the logic of "measure one" statements, and it is symmetrical in the sense above.

I hope this can help to give you the idea of where that symmetry comes from. I hope also that the bold characters don't disappear! (Tell me if you don't see the difference between B and B).

I have explained on this list, a long time ago, why in the Kripke semantics for modal logic, the validity of the formula Bp -> p makes the accessibility relation reflexive, and why the validity of p -> BDp makes the accessibility relation symmetrical. You may search in the archive, perhaps with the older notation:
p -> [ ]<>p.

In some of my french writing I suggest (wrongly) that such a symmetry is impossible for the first Theatetical variant of B (Bp & p), because it is an antisymmetrical structure. So I thought the symmetry provided by the restriction to the sigma_1 sentence would make the modal logic collapse into the trivial theory where Bp is equivalent with p. But I was wrong! In particular we have p -> BDp, but we don't have BDp -> p. This may confirms the neoplatonist idea that the soul (Bp & p) has already a foot in matter (p -> BDp).

From this, you can formally isolate a formal quantum computing machinery in the neighborhoods of any (self-referentially) Löbian machine. As we can expect, to verify such a quantum machine works is very difficult. Its simulation is not tractable (a good thing, actually), so to verify this we need a subtle new argument, or ... a quantum computer.

Bruno


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



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