> On 21 May 2020, at 21:43, Jason Resch <jasonre...@gmail.com> wrote: > > > > On Thu, May 21, 2020 at 1:33 PM Bruno Marchal <marc...@ulb.ac.be > <mailto:marc...@ulb.ac.be>> wrote: > >> On 20 May 2020, at 18:45, Jason Resch <jasonre...@gmail.com >> <mailto:jasonre...@gmail.com>> wrote: >> >> >> >> On Wed, May 20, 2020 at 7:05 AM Bruno Marchal <marc...@ulb.ac.be >> <mailto:marc...@ulb.ac.be>> wrote: >> >>> On 19 May 2020, at 05:20, Jason Resch <jasonre...@gmail.com >>> <mailto:jasonre...@gmail.com>> wrote: >>> >>> I recently wrote an article on the size of the universe and the scope of >>> reality: >>> https://alwaysasking.com/how-big-is-the-universe/ >>> <https://alwaysasking.com/how-big-is-the-universe/> >>> >>> It's first of what I hope will be a series of articles which are largely >>> inspired by some of the conversations I've enjoyed here. It covers many >>> topics including the historic discoveries, the big bang, inflation, string >>> theory, and mathematical realism. >> >> >> >> It has not been proved that the decimal expansion of PI contains all (finite >> codes of all) sequences. >> >> I understand that Pi is proven to be normal, > > > (Oops I meant to say "Pi is not proven to be normal" somehow I deleted the > not while refactoring the sentence)
OK. > > > But that is not the case. Pi win all experimental test, but the normality of > basically all irrational numbers are open problems. It is generally > conjectured that they are all normal. > For the Champernow number, the normality is easy to prove, but it has been > build that way. > > > >> but is it true for the irrational numbers (Pi, e, sqrt(2), etc.) that >> probabilistically the chance of not finding a given finite sequence of >> digits goes to zero? > > Most would bet that this is indeed the case, but that is unsolved today. > > > >> Is it correct to say that almost surely >> <https://en.wikipedia.org/wiki/Almost_surely> any sequence can be found? > > Hmm… “almost” has already a technical meaning in computer science. It means > for all but a finite number exceptions. It existential dual is “there is > infinitely many …”. > > Then, I don’t want to look like pick nicking, but “almost” and “sure” seems a > bit antinomic. > > Some intuition of infinite decimal series, and of irrational numbers (which > have no infinite repetition, etc.) gives a feeling that it would be quite > astonishing that it is not the case, even for sqrt(2), and we can say that > this has been experimentally verified, but mathematicians ask for proof, and > some ask for an elementary proof (not involving second order arithmetic or > analysis). > > > > >> If it does not hold for Pi, are there other numbers that would be better >> examples for the type of analogy I am making? > > > The Champernowne Number > > https://mathworld.wolfram.com/ChampernowneConstant.html > <https://mathworld.wolfram.com/ChampernowneConstant.html> > > > > >> I want to show why statistically an infinite space leads to near certainty >> of repetitions of material arrangements assuming some kind of infinite >> uniformity, just like the infinity of random-looking digits of an irrational >> number leads to infinite repetitions among any finite sequence. > > > You get this with Champernowne number. It is normal, despite extraordinarily > compressible. It is about equal to 0.123.., but all kids can easily write > the decimals without ending! It is obviously normal, as it goes through all > the numbers, and thus all the sequences. > > But the universe appears more random than something so well structured like > the Champernowne constant. I doubt this. Most subsequence of the Champernowne number are completely random, and *very* long. Only the tiny initial segment does not look random, when you know the algorithm to generate it. It can be proved that most natural number have incompressible sequences. The number of compressed algorithm grows much less that the numbers of number (for each finite length). > What about Chaitin's Omega? Hasn't Chaitin proved a certain randomness for > that digits of that constant? Up to see constant related to the universal machine used to make that number precise, it can be shown that indeed, that number (Omega) is random and incompressible. But all the finite subsequences of Omega appears in the Champernowne number, and only for that last one have we a proof of normality. Omega is so compressed that it has no useful pattern in it. Much more interesting is the Post number, which is 0,0001101111010101001… with 1 (res 0) at the nth place if phi_n converges (or not), where phi_i is an enumeration of the programs without arguments. Post number is compressible (indeed Chaitin’s Omega is Post number when maximally compressed: both gives the halting oracle). But post number illustrates the needed redundancy that we need to get the pattern from which the physical laws can evolve. It is an “interesting” number (in the sense of Bennett). > > > It has not be confused with a universal dovetailing which is a computation > which happens to execute all computations, which are peculiar number > relations. > > The problem is that each of us (us, the universal number) are implemented in > many computations, and indeed, below our substitution level, we get > infinitely many computations). Physics, conceptually, becomes a statistical > measure on uncertainty on which are our most probable computations, as “seen > from inside”. Here the mathematical logicians have a tool which lacks to the > physicalists, which is “transparent” mathematical theory of self-reference, > indeed, they get both the machines’ own theory (G) and the true theory (G*), > and the difference (G* minus G) which is so important to get the difference > between the quanta and the qualia. > > > > > >> >> >> It is easy to fiw, as you can take the number of Champernow, which trivially >> contain all sequences: >> >> C = 0,12345678910111213141516…. >> >> OK? >> >> Now, this is different from the universal dovetailing, which *executes* >> (semantically) all computations, and makes unavoidable that to solve the >> mind body problem, we have to extract the believes in bodies from the >> statistics on the first person continuation determined by all computations. >> It is here that it is crucial to distinguish between a computation (a notion >> involving counterfactuals) and a description of a computation, which does >> not. >> >> Indeed. To be clear I am not making the case here that our universe is >> contained within Pi, only showing that infinity leads to repeats so long as >> the description is finite, be it a volume of matter and energy, or a finite >> length of decimal digits. > > As long as you don’t assume simultaneously Mechanism and some “physical > universe” (making it or its elements primitive), there is no (logical) > problem. > > With mechanism, the laws of physics emerges from the statistics on the > dreams/computations of the natural number. > > The “god” of the universal Löbian machine, G*, provides the truth, the > believable, the knowable, the observable, and the one which feels. And this, > modulo Mechanism at the metalevel, assuming only two equations, like Kxy = x, > and Sxyz = xz(yz). > > > > > > >> >> >> With Mechanism, physics is reduced to number psychology or theology, and >> theology is reduced to arithmetic (through the Gödel-Löb-Solovay theorems). >> >> I am working on a post now which will get more into this, about why there is >> something rather than nothing. How to bootstrap reality and universes from >> arithmetical truth will be part of that. :-) >> I appreciate your comments. Thank you. > > > The discovery of the universal machine, and especially the Löbian one, is an > event more important than the Big Bang. Of course, that discovery is made an > infinite number of times in very elementary arithmetic. > > The Lôbian machines are the universal machines which knows (even in a rather > weak sense) that they are universal, and they know the complicated > consequences that happens, especially if they want to remain universal …They > oscillate easily between freedom and security in a not entirely vain attempt > to fill the gap between G* and G and their necessary intensional variants. > > With universality, you get free will, but you need löbianity, to get > responsibility. All universal machine, like RA, with enough induction axioms, > like PA, is Löbian. > > Their weakness? They are credulous, and hallucinate easily, like seeing far > away galaxies, sun, moon, and Higgs bosons…, but eventually they can explain > the why and the how of all sharable aspect of their experiences, and detect > possible oracles, who knows. > > The G*/G gap is really the difference between Computer Science (where there > is no hallucinations) and Computer’s Computer Science, which can contains > many hallucination, like notions of some absolute harwdare. > > With Mechanism, the laws of physics does not depend on the universal > machinery chosen for the ontology. The choice of a universal machinery, is > equivalent with the choice of a base for the recursive enumeration of all > partial computable functions. > > Thanks to QM, Nature fits well with the most startling aspect of mechanism > (or self-multiplication at the basic level). > The theology of machine will not replace physics, on the contrary, it > predicts that larger and larger part of mathematics will be “known” > “experimentally” (betting). > > Concerning our local cosmos, I am fascinated by the black holes, but very > ignorant, I see it implies multiverses of different kinds, super-imposed to > the Everett-Omen-Griffith entangled consistent histories. With Mechanism, it > is an open problem to just define a notion of a singular physical universe, > without mentioning the complex “intermediate histories” between Earth and > Heaven … > > I read this lately, and found it very interesting: > https://cse.buffalo.edu/~rapaport/111F04/lloyd-ng-sciam-04.pdf > <https://cse.buffalo.edu/~rapaport/111F04/lloyd-ng-sciam-04.pdf> Black hole are very interesting, including for the role they give to quantum information. But, from a quick look at the paper this is till “digital physicalism”, and it refutes itself. Indeed, it entails computationalism, but computationalism entails that the physical universe is not simulable exactly by a computer. Mechanism (aka computationalism) entails that even to simulate a nanometer^3 of vacuum, you need to run instantaneously the entire universal dovetailing, and compute the probabilities from there, which is not possible. As far as the paper is physically sound, if Mechanism ic correct, it can only be an approximation. It is interesting for physics, but does not address the fundamental question, like where there is a an appearance of a physical universe, and where does all appearances come from. > > Among some of the most interesting conclusions: quantum mechanics/Planck's > constant imposes an upper bound on the speed of computation, general > relativity/Newton's constant imposes an upper bound on the density of > computation. There are various intermediate possibilities of parallel vs. > sequential computing, but the maximum sequential information processing speed > is reached only for black holes. There the number of bits that can be > processed per step is given by Bekenstein's bound, and the "clock cycle" is > the amount of time it takes light to cross the diameter of the black hole. > > Another fascinating consequence: given that the matter-energy density of the > universe as a whole is right at the cusp of gravitational collapse, the total > mass of the observable universe is exactly equal to the density of a black > hole of the same volume of the observable universe. The estimated number of > bits within the universe is also exactly equal to the total number of bit > operations that have occurred in the universe since the big bang. In other > words: for every one of the 10^120 bits in the universe, each has been > processed (flipped) exactly once (on average) in the time since the big bang. That looks interesting, but if this is not derivable from Kxy = x + Sxyz = xz(yz), it will have to eve abandoned. > > There are incredible relations between fundamental physics and computation > which amaze me. Honestly, how could that been amazing? If we assume mechanism in cognitive science, the physical universe is entirely explainable in term of a statistics on *all* computations. In mathematics, “all computation” is the only place where “all” is well defined, thanks to the “miracle” of the Church-Turing thesis. Have you understand that all computations are run in arithmetic? Here “in arithmetic” can be replaced by “in all models of arithmetic” or “in the standard model of arithmetic” or “provable in RA”, or provable in all combinatory algebra, etc. We don’t need to postulate a physical universe, nor even induction axioms, to explain where the quantum computations come from, and why we tend to trust the induction axioms. But in theology (aka philosophy of mind, metaphysics) the situation is worst than that/ We just cannot postulate a physical universe, if we want it to be related to any conscious first person experience by machines. > > > Did you know that contrary to some myth (that I were “almost sure” about), > even quarks can maintain the social distancing, if you provide enough energy! > That is what happen in the gluon-quark plasma! I guess that is very hot. > > I read recently > <https://frankwilczek.com/Wilczek_Easy_Pieces/342_Origin_of_Mass.pdf> that > it's estimated 90% of our mass comes from the relativistic speed of quarks > and other particles inside the nucleus. Interesting. > If you could somehow still that motion, we'd weigh only a few pounds. > Something to ponder next time we step on a scale. :-) Is there some mass which is not kinetic energy in disguise? If yes, I will have to revise my understanding of the Higgs-Englert-Brout boson ... 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