> On 15 Aug 2019, at 23:22, Jason Resch <jasonre...@gmail.com> wrote: > > > > On Sun, Aug 11, 2019 at 12:24 PM Bruno Marchal <marc...@ulb.ac.be > <mailto:marc...@ulb.ac.be>> wrote: > >> On 10 Aug 2019, at 20:34, Jason Resch <jasonre...@gmail.com >> <mailto:jasonre...@gmail.com>> wrote: >> >> >> >> On Fri, Aug 9, 2019 at 10:20 AM Bruno Marchal <marc...@ulb.ac.be >> <mailto:marc...@ulb.ac.be>> wrote: >> >>> On 9 Aug 2019, at 13:09, Jason Resch <jasonre...@gmail.com >>> <mailto:jasonre...@gmail.com>> wrote: >>> >>> <snip> >>>> >>>> Bruno, >>>> >>>> Forgive me if I have asked this before, but can you elaborate on the >>>> how/why the math suggests negative interference? >>>> >>>> I currently have no intuition for why this should be. >>>> >>>> I recall reading something on continuous probability as being more natural >>>> and leading to something much like the probability formulas in quantum >>>> mechanics. Is that related? >>> >>> >>> It is not intuitive at all. With the UDA, we can have have the intuition >>> coming from the first person indeterminacy on all all computational >>> continuation in arithmetic, but in the AUDA (the Arithmetical UDA), the >>> probabilities are constrained by the logic of self-reference G and G*. So >>> the reason why we can hope for negative amplitude of probability comes from >>> the fact that modal variant of the first person on the (halting) >>> computations, which is given by the arithmetical interpretation of: >>> >>> []p & p >>> >>> or >>> >>> []p & <>t >>> >>> or >>> >>> []p & <>t & p >>> >>> With, as usual, [] = Beweisbar, and p is an arbitrary sigma_1 sentences >>> (partial computable formula). >>> >>> They all give a quantum logic enough close to Dalla Chiara’s presentation >>> of them, to have the quantum features like complimentary observable, and >>> what I have called a sort of abstract linear evolution build on a highly >>> symmetrical core (than to LASE: the little Schroeder equation: p -> []<>p, >>> which provides a quantisation of the sigma_1 arithmetical reality. >>> >>> It is mainly the presence of this quantisation which justify that the >>> probabilities behave in a quantum non boolean way, but this is hard to >>> verify because the nesting of boxes in the G* translation makes those >>> formula … well, probably in need of a quantum computer to be evaluated. But >>> normally, if mechanism (and QM) are correct this should work. >>> >>> This is explained with more detail in “Conscience et Mécanisme”. >>> >>> Bruno >>> >>> >>> Thank you Bruno for your explanation and references. >> >> Y’re welcome. >> >> >>> Regarding “Conscience et Mécanisme”, is there a web/html or English version >>> available? Unfortunately my browser cannot do translations of PDFs but can >>> translate web pages. If not don't worry, I can copy and paste into a >>> translator. >> >> Yes, There is no HTML page for the long text. But you can consult also my >> paper: >> >> Marchal B. The Universal Numbers. From Biology to Physics, Progress in >> Biophysics and Molecular Biology, 2015, Vol. 119, Issue 3, 368-381. >> https://www.ncbi.nlm.nih.gov/pubmed/26140993 >> <https://www.ncbi.nlm.nih.gov/pubmed/26140993> >> >> You will still need some background in quantum logic, like the paper by >> Goldblatt which makes the link between minimal quantum logic and the B modal >> logic. >> >> There is also a paper by Rawling and Selesnick which shows how to build a >> quantum NOT gate, from the Kripke semantics of the B logic. It is not >> entirely clear if this can be used in arithmetic, because we loss the >> necessitation rule in “our” B logic. Open problem. A positive solution on >> this would be a great step toward an explanation that the universal machine >> has necessarily a quantum structure and can exploit the “parallel >> computations in arithmetic” in the limit of the 1p indeterminacy.. >> >> Rawling JP and Selesnick SA, 2000, Orthologic and Quantum Logic: Models and >> Computational Elements, Journal of the ACM, Vol. 47, n° 4, pp. 721-T51. >> >> Ask question, online or here. It *is* rather technical at some point. >> >> Bruno >> >> >> >> I've been reading those references, and have found a few more which might be >> related and of interest. Effectively, they provide arguments for the >> quantum probability theory based on the requirement for continuous >> reversible operations, or the juxtaposition between finite information-carry >> capacity and smoothness. >> >> >> Lucien Hardy's "Quantum Theory From Five Reasonable Axioms" >> https://arxiv.org/abs/quant-ph/0101012 >> <https://arxiv.org/abs/quant-ph/0101012> >> >> The usual formulation of quantum theory is based on rather obscure axioms >> (employing complex Hilbert spaces, Hermitean operators, and the trace rule >> for calculating probabilities). In this paper it is shown that quantum >> theory can be derived from five very reasonable axioms. The first four of >> these are obviously consistent with both quantum theory and classical >> probability theory. Axiom 5 (which requires that there exists continuous >> reversible transformations between pure states) rules out classical >> probability theory. If Axiom 5 (or even just the word "continuous" from >> Axiom 5) is dropped then we obtain classical probability theory instead. >> This work provides some insight into the reasons quantum theory is the way >> it is. For example, it explains the need for complex numbers and where the >> trace formula comes from. We also gain insight into the relationship between >> quantum theory and classical probability theory. >> >> and Jochen Rau's "On quantum vs. classical probability" >> https://arxiv.org/abs/0710.2119v2 <https://arxiv.org/abs/0710.2119v2> >> >> The key (and novel) technical result, on the other hand, will pertain to the >> second objective: I will show that the single distinguishing property of >> quantum theory is the juxtaposition of finite information-carrying capacity >> and smoothness, where the concept of smoothness will be carefully defined >> and motivated. The mathematical derivation of this result will involve close >> inspection of the symmetry group, with successive constraints leading >> unequivocally to the unitary group of transformations in complex Hilbert >> space. As for the final objective, I will provide arguments why there is >> likely no further probabilistic theory that satisfies basic physical >> desiderata. > > > Interesting papers, but I agree with the second that the first assume too > much, from the continuum, the states, the tensorial structure, etc. > > > I am glad you find them interesting. Regarding you comment about the first > one assuming too much, I just learned that Markus Muller put out a paper > similar to Lucien Hardy's but without assuming the simplicity axiom: > http://arxiv.org/abs/1004.1483 <http://arxiv.org/abs/1004.1483> I haven't > had a chance to go through it yet, I am doing so now.
I will take a look. > > > > Then both assumes more or less explicitly some physical reality, and are > unaware of the need to derive it from the “universal machine’s consciousness > theory”, if relevant for relating coherently the quale logic with the quantum > logic. > > Such paper gives hope for making easier the last step of the derivation of > physics from arithmetic though. I did not know the second one, which seems > very interesting, but I read it only very quickly. It is has lady in common > the necessity of the continuum, some quantum logic which could not be > expanded for physics (but perhaps for “psychology”!). > > > >> >> Would you say these properties are inherent in the computations of the UD? > > As far as they are relevant to the correct physics, those properties have to > be derived from the right mixture of the 3p structures on all computations, > or the UD*, and the relative first person (plural) indeterminacy for the > average universal numbers with respect to all universal numbers running them. > Yes, that has to be the case … as far as both Mechanism in the cognitive > science, and the Quantum principles (Hilbert Space, or von Neuman Algebra). I > might appreciate also to derive the unitary group from few principles. I > suspect braids and Temperely-Lieb algebra, coming from the grade strcuture of > the material modes: > > I know nothing of Braids nor Temperely-Lieb algebra. In doing some searching > I came across this paper ( https://arxiv.org/abs/quant-ph/0601050 > <https://arxiv.org/abs/quant-ph/0601050> ) which claims to link the two with > quantum phenomenon, including quantum computation. That is a very good paper. The best introduction to braids and knots is Kauffman’s book “Knot and Physics”. That is a chef d’oeuvre of pedagogy. > > > []p & <>t (&p) > > becoming > > []^n p & <>^m t with n < m > > Which gives different but related quantum logic. Some sorts of dualities > between the quantisations []<>p and its dual <>[]p should “braid" the > “material mode” and I suspect space and time, or space-time, to start from > this, or similar. > > The infinities of universal systems under “our” substitution level might be a > universal topological braiding, a sort of universal quantum dovetailer. > > > > > >> In so far as any computational thread representing an observer or a system >> the observer interacts with is finite in its information carrying capacity, >> but all the threads of similar indistinguishable computations for a >> continuum? > > Right. > > > > >> Is there a reason to suppose operations are reversible (could this be due >> to some conservation of information principal in non-halting programs?). > > > We can cheat, and say that as Mechanism imposes the existence of a measure, > we impose symmetry (and continuity) to have a nice rich group structure with > know rich Measure theory (and then compact Lie groups + exceptional > structure) can pave the way. > > I can only pray of this to happen, but the material mode suggest this makes > sense by showing that the (true) sigma_1 sentences do impose symmetry at the > bottom, as the three first person (plural) modes imposes the "Brouwersche > axiom of symmetry”: p -> []<>p (when you get p, you can get p back from any > world in the neighbourhood. That introduces symmetry, a notion of > perpendicularity, a proximity relation of the type of a scalar product, if > not necessarily its square. > > With the combinators I like to sum up physics by “No Kestrels! No Starling!”. > We cannot eliminate things/information, and we cannot duplicate > things/information … at the bottom. The core of the physical reality is a > BCI-algebra (Bxyz = x(yz), Cxyz = xzy, Ix = x). You can compose/apply things > and permute them, at the bottom. Note that such a “bottom” is not Turing > universal, but the relative breaking of the symmetries are brought by what > needed to be added here, which is easy for the mind (add just K and S!), but > hard for the physical (why a tensor, why space-time waves/strings, why vertex > operator, etc.). > > > > >> >> Is the appearance of complex numbers in the quantum probability sufficient >> to get interference? > > > > Embed the real line in the plane, then a multiplication of numbers, or of a > couple of numbers, by -1, becomes a rotation of 180°, so to get (-1) = i^2, a > rotation of 90° provides a natural interpretation, and 1 and i becomes > perpendicular, which is is the key notion in the type of probabilities we > could hope to make sense in physics. > > a+ bi = re^it = cos(t) + i*sin(t), t real, the complex numbers are just > little waves at the start, they interfere all the time, so to speak, it is > more the interference which suggest the use of the complex numbers, then, > crazily enough, nature seems to be “complex” (wave like) at the bottom. > > Maybe this is due to the fact that the first order theory of the real is not > Turing universal, but the first order theory of the complex numbers is! (A > wave is a continuum trick to get the natural numbers, as you can define the > numbers by where the sinus get null (up to even multiple of pi)). > The limit on the first person indeterminacy on all computations, is expected > to be Turing universal and continuous, that might be the simplest reason. > > Very interesting. The fact that there is a universal Diophantine polynomial is rather extraordinary. It means that all proofs that some machine do something can be verified in less than one hundred operations (of addition and multiplication). >From this it can be shown that all (halting) computations can be done in less than one hundred operations. This means that the “dynamical content of a computation” can be limited to 100 operations, and all the rest will belong to statical description of (expectedly) very gigantic numbers and very long multiplications. There has been some period where I thought this could refute computationalism. I don’t think it is a threat, even if that is weird. It is weird that you can test x^(y ^(z^(t^…..^r))))…) = r with 10^1000 nested exponentiations by only 100 additions and multiplications using only much less variable/numbers. Some notorious logicians thought that this would just be impossible, and took some time to discourage the search for a diophantine polynomial computing (just with addition and multiplication) the exponential (Julia Robinson already showed that this would lead to the solution). This theorem suggests also that consciousness is mainly in the number relations, not in the operation emulating the computation, but this we already knew: it makes it more striking. The theorem is proved in the quite remarkable presentation by Martin Davis, in the Scientific American (I think) of the Putnam-Davis-Robinson-Matiyazevic's theorem (the universality of diophantine polynomials) . It has been reprinted in an appendice in the Dover edition of its “Computability and Unsolvability” book. I will, soon or later, make a summary of all the “concrete” universal machinery (the phi_i and w_i) that we have encountered (mainly, Boolean Graph + Clock/Delay, Elementary Arithmetic, Diophantine Polynomial Equation, Turing Machine, Register Machine (coffee-bar), LISP, lambda-expression and the combinators). Each have their own phi_i and w_i, but all phi_i and w_i obeys the same fundamental “computational” laws, largely captured by the combinatory algebras and the Models of Lambda Calculus). But to get the intensional nuances, the simplest way consists in using the phi_i and w_i directly. Basically everything follows from two facts, here below, about all “acceptable” universal machinery (enumeration of the partial computable function. Note that a total (everywhere defined) function is a particular case of a partial function): - 1) it exist a computable function s such that for all number x, y, and i, phi_i (x,y) = phi_s(i, x) (y) (s parametrises x on i) It is the SMN theorem (here the simplest S21 theorem) - 2) it exist a universal number u such that for all number x, y phi_u(x,y) = phi_x (y). A lot can be deduced from this. I build a self-regeretaig program (planaria) using a generalisation by John Case, of the Recursion theorem of Kleene, which can be proved in five line from just the SMN theorem. The whole logic of self-reference comes from the fact that PA (ZF, …) can prove those theorems in arithmetic. That richness has some price, and the universal machine brings a lot of mess in in (Arithmetical) Platonia, but that mess is also highly structured, which help when deriving a measure on the computational histories. Bruno > > Thank you. > > Jason > > > Note that the parallel worlds are given by perpendicular states. They should > be called the perpendicular universes. Once two “universes/histories" are not > perpendicular they can interfere “statistically”, and they are > inter-reachable “probabilistically” through appropriate > measurements/interactions. That imposes also some symmetries. > > > Bruno > > > > -- > 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 everything-list+unsubscr...@googlegroups.com > <mailto:everything-list+unsubscr...@googlegroups.com>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/CA%2BBCJUhL6we1S5Uiq9x_hJAN3PCih_M5-HZBTqgUEDav_1rMcQ%40mail.gmail.com > > <https://groups.google.com/d/msgid/everything-list/CA%2BBCJUhL6we1S5Uiq9x_hJAN3PCih_M5-HZBTqgUEDav_1rMcQ%40mail.gmail.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 everything-list+unsubscr...@googlegroups.com. 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