Dear Bruno, Interleaving. ----- Original Message ----- From: "Bruno Marchal" <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]>; <[EMAIL PROTECTED]> Cc: <[EMAIL PROTECTED]>; <[EMAIL PROTECTED]> Sent: Friday, January 23, 2004 9:42 AM Subject: Re: Is the universe computable

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> Dear Stephen, > > At 12:39 21/01/04 -0500, Stephen Paul King wrote: > >Dear Bruno and Kory, > > > > Interleaving. > > > >----- Original Message ----- > >From: "Bruno Marchal" <[EMAIL PROTECTED]> > >To: <[EMAIL PROTECTED]> > >Sent: Wednesday, January 21, 2004 9:21 AM > >Subject: Re: Is the universe computable > > > > > > > At 02:50 21/01/04 -0500, Kory Heath wrote: > > > >At 1/19/04, Stephen Paul King wrote: > > > > > Were and when is the consideration of the "physical resources" > > > > > required for the computation going to obtain? > > > > > Is my question equivalent to the old "first cause" question? > > > >[KH] > > > >The view that Mathematical Existence == Physical Existence implies that > > > >"physical resources" is a secondary concept, and that the ultimate ground > > > >of any physical universe is Mathspace, which doesn't require resources of > > > >any kind. Clearly, you don't think the idea that ME == PE makes sense. > > > >That's understandable, but here's a brief sketch of why I think it makes > > > >more sense than the alternative view (which I'll call > > > > "Instantiationism"): > > > > > > [SPK] I should respond to Kory's ME == PE idea. In PE we find such things as "thermodynamic entropy" and "temporality". If we are to take Kory's idea (that Mathspace doesn't require resources) seriously, ME does not. This seems a direct contradiction! Perhaps Kory has a paper on-line that lays out his thesis of "Instantiationism". > >[SPK] > > > > Again, the mere postulation of existence is insufficient: it does not > >thing to inform us of how it is that it is even possible for us, as mere > >finite humans, to have experiences that "change". We have to address why it > >is that Time, even if it is ultimately an "illusion", and the distingtion > >between past and future is so intimately intetwined in our world of > >experience. > >[BM] > Good question. But you know I do address this question in my thesis > (see url below). I cannot give you too much technical details, but here is a > the main line. As you know, I showed that if we postulate the comp hyp > then time, space, energy and, in fact, all physicalities---including the > communicable (like 3-person results of experiments) as the uncommunicable > one (like qualie or results of 1-person experiment) appears as modalities > which are > variant of the Godelian self-referential provability predicates. As you know > Godel did succeed in defining "formal provability" in the language of a > consistent machine and many years later Solovay succeeds in formalising > all theorems of provability logic in a couple of modal logics G and G*. > G formalizes the provable (by the machine) statements about its own > provability ability; and G* extends G with all true statements about the > machine's ability (including those the machine cannot prove). [SPK] In my thinking all 1st person experiences are "best possible simulations". The problem I find is that we can not use the modern equivalent to Leibniz' "preordained harmony", whether in the form of a "universal prior" or "modelization" of some modal logic, since the "list" of all possible interactions is not enumerable. This is the aspect that I have tried to address by referencing Wolfram on the computational intractibility of some key aspects of "physicality". There is also the seperate issue of how does one aspect of a logic "address" some other? We have the example of a Turing Machine that considers a "tape" and a "head": there are separate in that one can "move" relative to the other all the while the transitions of the state of the head and the "spot" on the tape change. I do not see how some form of Monism can explain this. Additionally, there is the problem of "simulating" QM using formal logics. I have reference the Calude et al paper on this and you have said that it is good, but you seem to not have actually read it and let its implications "set in". ;-) > [BM] > Now, independently, temporal logicians have defined some modal > systems capable of formalizing temporal statements. Also, Brouwer > developed a logic of the conscious subject, which has given rise to a whole > constructive philosophy of mathematics, which has been formalize > by a logic known as "intuitionist logic", and later, like the temporal logic, > the intuitionist logic has been captured formally by an modal > extension of a classical modal logic. Actually it is Godel who has seen > the first that Intuitionist logic can be formalised by the modal logic S4, and > Grzegorczyk makes it more precise with the extended system S4Grz. > And it happens that S4Grz is by itself a very nice logic of subjective, > irreversible (anti-symmetric) time, and this gives a nice account too of the > relationship Brouwer described between time and consciousness. > Now, if you remember, I use the thaetetus trick of defining > (machine) "knowledge of p" by "provability of p and p". Independently > Boolos, Goldblatt, but also Kusnetsov and Muravitski in Russia, showed > that the formalization of that form of knowledge (i.e. "provability of p > and p") > gives exactly the system of S4Grz. That's the way subjective time arises > in the discourse of the self-referentially correct machine. [SPK] Does S4Grz solve the "true concusseny" problem? If so, how? Again, we have to be able to represent more than one self-referential system simultaneously if we are going to have a model that allows for the existence of more than one observer. Otherwise we have a solipsism of a single mind that must believe that all other appearences of minds are "zombies". It is an all or nothing situation! > [BM] > Physical discourses come from the modal variant of provability given > by "provable p and consistent p" (where consistent p = not provable p): > this is justified by the thought experiment and this gives the arithmetical > quantum logics which capture the "probability one" for the probability > measure on the computational histories as seen by the average consistent > machine. [SPK] This is where you lose me! Again, please point me to the "thought experiment". It is this part of your model that most intrigues me! ;-) > [BM] > Physical time is then captured by "provable p and consistant p and p". > Obviously people could think that for a consistent machine > the three modal variants, i.e: > > provable p > provable p and p > provable p and consistent p and p > > are equivalent. Well, they are half right, in the sense that for G*, they > are indeed equivalent (they all prove the same p), > but G, that is the self-referential machine > cannot prove those equivalences, and that's explain why, > from the point of view of the machine, > they give rise to so different logics. To translate the comp hyp > into the language of the machine, > it is necessary to restrict p to the \Sigma_1 arithmetical > sentences (that is those who are accessible by the Universal Dovetailer, > and that step > is needed to make the physicalness described by a quantum logic). [SPK] I still do not see how "provable p" tells us anything about how some self-referential system seperates it's model of itself from itself! This, is similar to the situation that I mentioned above about turing machine. I am reminded of Searl's (?) discussion of a book containing a complete map of Einstein's brain... Have you truly looked at the requirements involved in representing a sufficient quantum logic? It does not seem that you have! > [BM] > The constraints are provably (with the comp hyp) enough to defined all > the probabilities on the computational histories, and that is why, if ever > a quantum computer would not appear in those logics, > then (assuming QM is true!) comp > would definitely be refuted; and that is all my point. [SPK] Something about this stinks! Are you saying that "If a quantum computer does not appear in these logics" = "comp is falsified"? Is this not equivalent to saying that "If a white crow is not found" = "all crows are black"? THis is NOT falsification! BTW, I take QM to be true to the extent that it has yet (!) not been falsified, so I guess that I am hoisted upon my own petard! > > > [BM] > > > OK. Just to cut the hair a little bit: with Church thesis "computational > > > realism" is equivalent to > > > a restricted form of arithmetical realism. Comp. realism is equivalent to > > > Arith. realism restricted to the Sigma_1 sentences, > > > i.e. those sentence which are provably equivalent > > > (in Peano arithmetic, say) to sentences of the form "it exists x such that > > > p(x)" with p(x) a decidable (recursive) predicate. > > > This is equivalent to say that either a machine (on any argument) will > > > stop or will not stop, and this independently of any actual running. [SPK] NO! You can not toss away the fact that some kind of implementation MUST occur of the Sigma_1 sentences, or at least, be allowed. We are back to where we started! One must have some "running" in order for the notion of "stop" and "not stop" to be meaningful! This is my problem with the use of Platonic realms. One must have some have to extract temporal modality from timeless existence. Prof. Kitada does this in his theory, but it seems that he requires that the "machinery" of QM must have ontological reality and not just be a Model, ala Lowenheim-Skolem. The "Model" is the world of experience... > > > [BM] > > > Indeed, sometimes I say that > > > (Sigma_1) arithmetical realism > > > is equivalent to the belief in the excluded middle principe (that is "A or > > > not A) applied to > > > (Sigma_1) arithmetical sentences. (Sigma_1 sentences plays a prominant > >role > > > in the derivation > > > of the logic of the physical propositions from the logic of the > > > self-referential propositions). Actually > > > the Universal Dovetailing is arithmetically equivalent with an enumeration > > > of all true Sigma_1 sentences. The key feature of those sentences is that > > > their truth entails their provability (unlike > > > arbitrary sentences which can be true and not provable (by Peano > > > arithmetic, for exemple). > > > > >[SPK] > > > > Bruno, I do not understand why you use so weak a support for your very > >clever theory! If we are to take the collection of a "true Sigma_1 > >sentenses" to have "independent of implementation" existence, why not all of > >the endless hierarchy of Cantor's Cardinals? I have never understood this > >Kroneckerian attitute. > >[BM] > This is related to the Skolem theorems or to the so-called "Skolem Paradox", > Cantorian cardinals are relative notions (in set theory). There is no sense > to talk about an absolute implementation of those cardinals. I do believe > in the Cantorian theory but high cardinal are measure on our high ignorance; > I don't see how we could extend a "set realism" for them, unlike what we can do > with numbers (but also with all ordinals less than the first non constructive > ordinal of Church and Kleene, by using an extended version of Church > thesis, the > so-called hyper-arithmetical Church thesis; but that step would be a step > toward a weakening of comp and the \Sigma_1 sentences should be replaced > by something else ...). [SPK] Please take a look at the following paper: http://www.kitada.com/timeXIV.html It addresses the notion of a non-constructive ordinal. Your critique is most welcome! > Hoping having not been too technical. A good book is Rogers' one, see the ref > in my thesis or papers, and of course a must is the Boolos 1993 book. > Even in my french thesis I have not give all technical details. I have done > that > in my "brussel's thesis" which I will soon put on my web page, but then it > is 800 pages of technical details ... in french (I did not choose it). [SPK] I can only dream of the day that I can either read French or an English (or Spanish!) translation is made of your paper. ;-) > Remember that I am not a defender of comp. My only goal is to show that comp > is enough precise so that it can be refuted; or put in another way, my goal is > to show comp is a scientific theory in the popper sense of the word; a > little like > John Bell succeeded in showing that the Einstein Podolski Rosen paper > leads to verifiable propositions, and was not philosophy (as Bohr implicitly > argued, and as so many physicist did believe (without thinking on the matter). > [SPK] But its refutation seem to be one that can not occur in finite time, so I can only take your word or "by an act of blind faith" as the justification of a belief in it. ;-) Kindest regards, Stephen > Best Regards, and have a nice week-end, > > Bruno > >