> On 20 Jun 2019, at 00:26, Lawrence Crowell <[email protected]> > wrote: > > On Tuesday, June 18, 2019 at 6:02:54 AM UTC-5, Bruno Marchal wrote: > >> On 18 Jun 2019, at 02:14, Lawrence Crowell <[email protected] >> <javascript:>> wrote: >> >> The stochastic aspects of QM emerge in measurement, where the modulus square >> of amplitudes are probabilities and there are these random outcomes. The >> measurement of a quantum state is not a quantum process, but has stochastic >> outcomes predicted by QM. Based on the Hamkin's work where I only looked at >> the slides and not yet the paper, it seems possible to do this with quantum >> computer. >> >> http://jdh.hamkins.org/computational-self-reference-and-the-universal-algorithm-queen-mary-university-of-london-june-2019/ >> >> <http://jdh.hamkins.org/computational-self-reference-and-the-universal-algorithm-queen-mary-university-of-london-june-2019/> >> slides: >> http://jdh.hamkins.org/wp-content/uploads/Computational-self-reference-and-the-universal-algorithm-QMUL-2019-1.pdf >> >> <http://jdh.hamkins.org/wp-content/uploads/Computational-self-reference-and-the-universal-algorithm-QMUL-2019-1.pdf> >> I wrote a couple of elementary Python codes for the QE machine IBM has to >> prepare states and run then through Hadamard gates. The thought occurred to >> me that this Quining could be done quantum mechanically as a set of Hadamard >> gates that duplicate a qubit or an bipartite entangled qubit. This is a part >> of my ansatz that a measurement is a sort of Gödel numbering of quantum >> states as qubit data in other quantum states. >> Quantum computations are mapped into an orthomodular lattice that does not >> obey the distributive property. The distributive law of p and (q or r) = (p >> and q) or (p and r) fails. The reason is due to the Heisenberg uncertainty >> principle. Suppose we let p = momentum in the interval [0, P], q = position >> in the interval [-x, x] and r = particle in interval [x, y]. The proposition >> p and (q or r) is true if this spread in momentum [0, P] is equal to the >> reciprocal of the spread of position [-x, y] with >> P = ħ/sqrt(y^2 + x^2). >> The distributive law would then mean >> P = ħ/|y| or P = ħ/|x| >> which is clearly false. This is the major difference with quantum logic and >> Boolean classical logic. These lattices of quantum logic have polytope >> realizations. >> This is in fact another way of realizing that QM can't be built up from >> classical physics. If this were the case then quantum orthomodular lattices, >> which act on convex sets on L^p spaces with p = ½ would be somehow built >> from lattices acting on convex sets with p → ∞. This is for any >> deterministic system, whether Newtonian physics or a Turing machine. It is >> this flip between convex sets that is difficult to understand. With p = ½ >> and the duality between two convex sets as 1/p + 1/q = 1 the dual to QM also >> has L^2 measure. This is spacetime with the Gaussian interval. For a p → ∞ >> the dual is q = 1 which is a purely stochastic system, say an idealized set >> of dice or roulette wheel with no deterministic predictability. >> The point of Quining statements quantum mechanically is that this might be a >> start for looking at a quantum measurement as a way that quantum states >> encode qubit information of other quantum states. It is a sort of Gödel >> self-reference, and my suspicion is the so called measurement problem is not >> solvable. The decoherence of states is then a case where p = ½ → 1 with an >> outcome. That is pure randomness. > > With mechanism, that randomness is reduced into the indeterminacy in > self-multiplication experience. It come from the many-histories internal > interpretation of arithmetic, in which all sound universal numbers converges. > The quantum aspect of nature is just how the (sigma_1) arithmetical reality > looks like from inside. This explains where the apparent collapse comes from, > in a similar way than Everett, but it explains also where the wave comes > from. Eventually quantum mechanics is just a modal internal view of > arithmetic, or anything Turing equivalent. The math, and quantum physics > confirms computationalism up to now, where physicalism and materialism are > inconsistent, or consciousness or person eliminative. > > > Thanks for addressing this. > > I guess in a way I do not entirely understand this. The above illustration is > the main difference between Boolean and quantum logic.
OK. I have no problem with this. I agree and understand that quantum logic cannot be embedded or extended into a classical logic. This is related to the fact that there is no local hidden variable theory compatible with the quantum experiments. But this does not mean that quantum logic cannot have a classical explanation. In fact the quantum formalism is by itself a classical description, even local and deterministic, but hard to interpret in any local realistic way. Assuming the mechanist hypothesis, we have a similar (to QM) form of indeterminacy, due to the fact that we can be duplicated, and in that case the person who is duplicated cannot predict with certainty which of the copies she will feel to be, as both will be right to say that they have survived in the place where they are reconstituted. We can come back on this if you want to know more. That leads to the problem that no machine can know which computations (which exists in arithmetic as we know since Gödel-Turing 1930s papers) support her, and we know that there is an infinity of such computations in arithmetic: this eventually rediuce physics (the art of predicting the observable) into a relative statistics on all computations in arithmetic. In fact with mechanism, we have a canonical “many-world” interpretation of elementary arithmetic. And with mechanism, it should explain the existence and persistence of the physical laws (and indeed up to now this is confirmed, notably by the Everett formulation of QM). > It is not clear to me in what way quantum mechanics is σ_1 arithmetic viewed > from the "inside." I guess I am not sure what is meant by σ_1 arithmetic. The sigma_1 arithmetical sentences are the sentences provably equivalent (in PA, say) with sentences having the shape “ExP(x), with P a decidable or recursive (sigma_0) predicate. Turing-completeness or Turing-universality is equivalent sigma_1 completeness, i.e. the ability to prove all true sigma_ sentences. Intuitively it is obvious that you and me, all humans, and in fact all computers, are sigma_1 complete. If is true that ExP(x), and if P is decidable, then by testing 0, 1, 2, … we will eventually find that x, and be able to verify it satisfies p. The reverse is true also: if something can prove all true sigma_1 sentences, then it can emulate all computations, and it provides “one more” formal definition of computation, and one more universal machine. A normal form theorem by Kleene makes it possible to identify halting computations and true sigma_1 sentence. The set of all true sigma_1 sentences is more or less equivalent with the universal dovetailing (a procedure which generate all programs and execute them all). It has been shown that RA, or SK are Turing-complete theories, and thus constitute universal machine or machinery. RA is classical logic + the seven axioms: 1) 0 ≠ s(x) 2) x ≠ y -> s(x) ≠ s(y) 3) x ≠ 0 -> Ey(x = s(y)) 4) x+0 = x 5) x+s(y) = s(x+y) 6) x*0=0 7) x*s(y)=(x*y)+x SK is theory (without logic!): Rules: 1) If A = B and A = C, then B = C 2) If A = B then AC = BC 3) If A = B then CA = CB Axioms: 4) KAB = A 5) SABC = AC(BC) > > The space of computation for quantum computers is not clear. Aaronson showed > the space is a bounded quantum polynomial space, which contains P and now > appears to extend into NP. The measure of quantum computing is PSPACE is as > yet not known. For my “mind-body” interest, we need only to know that quantum digital machines do not violate the Church-Turing thesis. It seems to me that David Deutsch has already shown that the universal quantum Turing machine emulates all machines polynomially, so Aaronson is correct. But of course, we can expect this is false if we put a rounded polynomial measure on the computations. Typically, we can expect an exponential slow-down when a classical machine emulates a quantum algorithm, although this has not been yet proved. Most people believe in this conjecture, and that motivates the research in quantum computation. > > Quantum logic are in nondistributive orthomodular lattices of p = ½ convex > functions, classical probability systems p = 1 and deterministic systems > without a definable measure. We do not think of deterministic classical > systems, or for that matter Turing machines as having a measure over which > one integrates a density. The classical probability system and deterministic > system are in a dual relationship, as are quantum mechanics and spacetime > physics with L^2 measure. OK. > How QM flips from a p = ½ system to a p = 1 system is unknown. Indeed. It is the problem. Now, this is less mysterious when we abandon the collapse, as this makes the quantum indeterminacy a particular case of the first person indeterminacy, and the math confirms that we do find a quantum logic there. I do not claim that this solves all interpretation problem; but with Mechanism, we have no choice: we must reduce physics into a statistics on the first person view distributed on all computations. If I did not get a non boolean quantum logic there, I would probably believe that Mechanism (as an hypothesis in cognitive science) is refuted, or made implausible. > There was a recent paper that demonstrated how a quantum system about to > enter decoherence exhibited some behavior, which means there may be some > process involved whereby a quantum deterministic system transforms into a set > of classical probabilities. This process may have some analogues I think with > singular perturbation theory. I would need more on this to evaluate if this is consistent with digital mechanism or not. Then, I might need to progress more on the “arithmetical quantum logic” related to that first person statistics calculus. Bruno > > LC > > >> Now of course we can ask what we mean by random, and that is undefinable. >> Given any set of binary strings of length n there are N = 2^n of these, and >> in general for n → ∞ there is no universal Turing machine which can compress >> these into any general algorithm, or equivalently the Halting problem can't >> be solved. A glance at this should indicate that N is the power set of n and >> this is not Cantor diagonalizable. Chaitin found there is an uncomputable >> Halting probability for any subset of these strings. Randomness is then >> something that can't be encoded in an algorithm, only pseudo-randomness. >> The situation is then similar to the fifth axiom of geometry. In geometry >> one may consider the 5th axiom as true and remain within a consistent >> geometry. One may similarly stay within the confines of QM, but there is >> this nagging issue of decoherence or measurement. One may conversely assume >> the 5th axiom is false, but now one has a huge set of geometries that are >> not consistent with each other. Similarly in QM one may adopt a particular >> quantum interpretation. > > > QM cannot be invoked except as a toll to test Mechanism (computationalism). > > Bruno > > >> LC >> >> -- >> You received this message because you are subscribed to the Google Groups >> "Everything List" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to [email protected] <javascript:>. >> To view this discussion on the web visit >> https://groups.google.com/d/msgid/everything-list/504fa0ed-686e-4e17-bbdc-68dfa609008f%40googlegroups.com >> >> <https://groups.google.com/d/msgid/everything-list/504fa0ed-686e-4e17-bbdc-68dfa609008f%40googlegroups.com?utm_medium=email&utm_source=footer>. > > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected] > <mailto:[email protected]>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/8819a3ce-6e7d-443c-ba3a-2555fccac0d1%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/8819a3ce-6e7d-443c-ba3a-2555fccac0d1%40googlegroups.com?utm_medium=email&utm_source=footer>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. 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