Hi Stephen,
On 01 Oct 2010, at 08:22, Stephen P. King wrote:
Hi Bruno,This is very exciting to me as it gives me some hope that the Stone duality idea will work as it is envisioned. I found several papers that discussed Scattered Topologies and the definition fits the idea that a dust is the dual of a Boolean algebra. Such as:www.jstor.org/stable/2274571 www.normalesup.org/~cornulier/CGPab.pdf www.phil.uu.nl/preprints/preprints/PREPRINTS/preprint282.pdf
That paper by Beklemishev is a very interesting paper, thanks. Ir pursues the work by Esakia on GL (G) on GLB which is a multimodal generalization of G. In GLB you can axiomatize some dialog between ZF and PA. Also interesting are the self-reference logics of the autonomous progression (a "line" of consistent extensions, studied by Beklemishev, notably. This should also admit 'scattered topological semantics'.
The complement of the diagonal of the Julia set defines the Mandelbrot set, I think. The Mandelbrot set is another structure which classify dusts from connected spaces(*), it would be nice to find a semantics of G/G* related to this. The diamond is the derivative, d is del, the B little schroedinger equation p -> BDP, which makes possible a notion of quantization on the sigma_1 proposition, would be a sort of derivative!
I will try to find some more time to read your long post, and (most probably) comment it. Thanks for making my poor brain boiling up! :)
Have a good week-end, Bruno (*) http://www.youtube.com/watch?v=E2pzEN_4pug
www.mrlonline.org/jot/2010-063-002/2010-063-002-012.pdfMy intension is to see if we can run Pratt's idea to get a way to model dynamics in this way. Think of dust scattered in empty space as moving randomly; how does the change in the relative distances and so forth of the dusts dualize to transitions from one form of a Boolean algebra to another? Pratt gives us a toy model that I need to generalize. He wrote in http://boole.stanford.edu/pub/ ratmech.pdf :"When we unravel the primitive causal links contributing to secondary causal interaction we find that two events, or two states, communicate with each other by interrogating all entities of the opposite type. Thus event a deduces that it precedes event b not by broaching the matter with b directly, but instead by consulting the record of every state to see if there is any state volunteering a counterexample. When none is found, the precedence is established. Conversely when a Chu space is in state x and desires to pass to state y, it inquires as to whether this would undo any event that has already occurred. If not then thetransition is allowed."We then need to work out the kinds of transitions that could occur following this method and the invariants under those transitions to recover the symmetry groups that relate to those transitions and invariances. The eventual goal is to show that all of the symmetry groups that we associate with physical systems, even general covariance of General relativity, can be derived this way. There is a lot of hard work to do.BTW, I think that it might be a mistake to assume that it is necessary to have a tensor product for the quantum orthoalgebras as discussed in the quote below. I think that all that might be necessary is some finite approximation to it over a large collection of "semi-separate" quantum systems. My idea for that involves finding a way to define something like strata or layers within which the notion of "separate systems" is valid to some finite approximation. I will get back to this when we talk about the problem of matter (and space!) that you mentioned in your reply to my paper post.Onward! Stephen P. King -----Original Message-----From: everything-list@googlegroups.com [mailto:everything-list@googlegroups.com ] On Behalf Of Bruno MarchalSent: Thursday, September 30, 2010 12:53 PM To: everything-list@googlegroups.com Subject: Re: As to the problem of tensor products for quantum logic Hi Stephen,I see you mention the work of my friend Diederik Aerts. We discussed those problems a very long time ago. Actually it was the difficulty of quantum logic which motivated me in part for searching a non physical origin of the quantum, and this leads to the intensional variants of self-reference (through UDA-like reasoning). Now it is almost fun that the quantum logics of self-reference (S4Grz1, X1*, Z1*) seems to have very similar problems, notably with the notion of subsystem, composite systems, etc. This is due to the easy to prove indeterminacy, non locality, and non clonability of 'matter' from the first person views (assuming DM). (I recall that S4Grz1 = S4Grz + 1, and 1 is "p -> Bp", the "1" is for Sigma_1. All true Sigma_1 sentences are provable by Sigma_1 complete entities, and löbianity is characterized by being able to prove that p->Bp for all Sigma_1 sentences. As for the covering laws that you mention, I just cannot translate them in arithmetic, despite dovetailing gives an intuitively simple way to encode parallelism and interaction. This cannot be lifted easily on the "consistent extensions".Btw, I was wrong in a previous post about the Esakia logic wK4. It is not the logic G in that paper, although Esakia studied relation between wK4 and G. wK4 is K + ((p & Bp) -> BBp), and the usual modal rules (modus ponens, necessitation). This can be used to prove completeness of G with respect to Cantor scattered topological ordinals. But that's a technical result a bit beyond the topic, I think.- Bruno On 27 Sep 2010, at 17:21, Stephen P. King wrote:Hi Folks, I have been researching the notion of tensor products in quantum logic and found the following for your consideration and comment: http://plato.stanford.edu/entries/qt-quantlog/ *** 7. Composite Systems Some of the most puzzling features of quantum mechanics arise in connection with attempts to describe compound physical systems. It is in this context, for instance, that both the measurement problem and the non-locality results centered on Bell's theorem arise. It is interesting that coupled systems also present a challenge to the quantum-logical programme. I will conclude this article with a description of two results that show that the coupling of quantum- logical models tends to move us further from the realm of Hilbert space quantum mechanics. The Foulis-Randall Example A particularly striking result in this connection is the observation of Foulis and Randall [1981] that any reasonable (and reasonably general) tensor product of orthoalgebras will fail to preserve ortho- coherence. Let A5 denote the test space {{a,x,b}, {b,y,c}, {c,z,d}, {d,w,e}, {e,v,s}} consisting of five three-outcome tests pasted together in a loop. This test space is by no means pathological; it is both ortho- coherent and algebraic. Moreover, it admits a separating set of dispersion-free states and hence, a classical interpretation. Now consider how we might model a compound system consisting of twoseparated sub-systems each modeled by A5. We would need to construct a test space B and a mapping ⊗ : X × X → Y = ∪B satisfying, minimally,the following; a. For all outcomes x, y, z ∈ X, if x⊥y, then x⊗z ⊥ y⊗z and z⊗x ⊥ z⊗y, b. For each pair of states α, β ∈ ω(A5), there exists at leastone state ω on B such that ω(x⊗y) = α(x)β(y), for all outcomes x, y ∈X. Foulis and Randall show that no such embedding exists for which B is orthocoherent. Aerts' Theorem Another result having a somewhat similar force is that of Aerts [1982]. If L1 and L2 are two Piron lattices, Aerts constructs in a rather natural way a lattice L representing two separated systems,each modeled by one of the given lattices. Here “separated” means thateach pure state of the larger system L is entirely determined by thestates of the two component systems L1 and L2. Aerts then shows that L is again a Piron lattice iff at least one of the two factors L1 and L2is classical. (This result has recently been strengthened by Ischi [2000] in several ways.) The thrust of these no-go results is that straightforward constructions of plausible models for composite systems destroy regularity conditions (ortho-coherence in the case of the Foulis- Randall result, orthomodularity and the covering law in that of Aerts' result) that have widely been used to underwrite reconstructions of the usual quantum-mechanical formalism. This putsin doubt whether any of these conditions can be regarded as having theuniversality that the most optimistic version of Mackey's programmeasks for. Of course, this does not rule out the possibility that theseconditions may yet be motivated in the case of especially simple physical systems. *** What does this imply? Bruno wrote that " If *you* can demonstrate that arithmetical quantum logic have no tensor product enough "coherent" for allowing concurrency, then you have refuted DM." but given the results discussed here I think that there might be some way to wiggle out of this by attacking the notion of "separated". Onward! Stephen P. King--You received this message because you are subscribed to the Google Groups "Everything List" group.To post to this group, send email to everything-l...@googlegroups.com.To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com . For more options, visit this group at http://groups.google.com/group/everything-list?hl=en .
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