HI Bruno Thank you so much for your answers to my queries so far. I really need to do some more thinking about all that you have said so far and to understand why I am having difficulty replacing a real physical universal machine existing in the future (like Tipler suggests) or a great programmer existing now (like schmidhuber suggests) with your arithmetical realism. I also need to search some previous posts to make use of past discussion topics that are relevant. Perhaps my background makes me a physicalist who can currently accept a milder form of comp. However, I want to explore your position because I think it makes sense in so far as I think it is less vulnerable to the threat of infinite regressions like in Schmidhuber’s great programmer (or even the greater programmer that programmed him). Your version of computationalism would still be valid if either or both of the two options above were true. Herein lies its appeal to me (both fundamental and universal). I would like to read up on logic and computation as you suggest. I have read about all the books you recommend . However, can you suggest topic areas within these texts which I can focus on to help me get up to speed with the problems I have regarding arithmetical realism with the UDA? There is much that could perhaps be left out on a first reading and to my untrained eyes, it’s difficult to know what to omit (for example what would godels arithmetisation technique come under? (Googling it brings not much up). Sorry but I haven’t ordered any books yet so I can’t look into them. Is there an English translation of your Ph.D. thesis yet? Sorry but I can’t do French. My thanks and best wishes.
Nick On Dec 31 2009, 6:10 pm, Bruno Marchal <marc...@ulb.ac.be> wrote: > On 30 Dec 2009, at 17:51, Nick Prince wrote: > > > > > > > Hi Bruno > > >>> If the UD was a concrete one like you ran then it would start to > >>> generate all programs and execute them all by one step etc. But are > >>> you saying that because the UD exists platonically all these > >>> programs > >>> and each of their steps exist also and hence, by the existence of a > >>> successor law they have an implicit time order? > >> Yes. The UD exist, and is even representable by a number. UD*, the > >> complete running of the UD does not exist in that sense, because it > >> is > >> an infinite object, and such object does not exist in simple > >> arithmetical theories. But all finite parts of the UD* exist, and > >> this > >> will be enough for "first person" being able to glue the > >> computations. > >> For example, you could, for theoretical purpose, represent all the > >> running of the UD by a specific total computable function. For > >> example > >> by the function F which on n gives the (number representing the) nth > >> first steps of the UD*. Then you can use the theorem which asserts > >> that all total computable functions are representable in Robinson > >> Arithmetic (a tiny fragment of Pean Arithmetic). That theorems is > >> proved in detail, for Robinson-ile arithmetic, in Boolos and Jeffrey, > >> or in Epstein and Carnielli. In Mendelson book it is done directly in > >> Peano Arithmetic. > > >> It is because our "3-we", our bodies, or our bodies descriptions, are > >> constructed within these steps. But our first person are not, and no > >> finite pieces of the UD can give the "real experience". This is a > >> consequence of the first six steps: our next personal experience is > >> determined by the whole actual infinity of all the infinitely many > >> computations arrive at our current state. (+ step 8, where we abandon > >> explicitly the physical supervenience thesis for the computational > >> one). > > This “glueing” idea reminds me of David Deutsch’s attempt to explain > > how time is an illusion in “The Fabric of Reality”. I never have got > > this one! > > I can follow your argument but it seems to put a very special status > > on the ist person experience. You say that our “3-person”/ bodily > > descriptions are contained as subprograms in the (infinite) programs > > which collectively provide Observer Moments for them. > > OK. > I rephrase for myself. If you meant things differently, just tell me. > By comp assumption, I survive if some "machine" goes through a > computation, that is, a sequence of computational states related by > some universal machine: s0, s1, s2, s3, s4, s5, s6, s7, ... > The bodily description are, strictly speaking defined by the doctor > choice of level of my description. They are third person sharable, you > can send them by mail attachment, in principle (a lot of giga!). > But the computation itself is defined by the logical relation between > those steps, and by digitality those steps, and their sequencing (made > by a universal machine) are definable in arithmetic, and the existence > of the steps, the states, the finite piece of computations, and (in a > slightly different sense for technical reason) the infinite > computations are all described completely in the elementary relations > between number (or between combinators, or whatever is your favorite > universal inductive structure, say). I take the number because they > are taught in school (I think). > > So, all the statements asserting that there are machines x accessing > state i and (may be) 'outputing' j, are arithmetical true statement > (when true), and actually, with Church thesis, they are theorems of > any Sigma_1 complete theory. > > When true, they are true independently of you and me, and when they > are proved in a theory, that fact is true independently of me and you. > Theories and machines are mathematical object, and the fact that a > theory or a machine proves a theorem is a mathematical truth. That is > independent of you, me, but also of time and space. > > Up to this, we did not mention first person experiences. Just all > machine's histories, described by numbers relations. > > The "problem" of the first person view of the machine, is that a > machine cannot know which machines "it" is, nor which computations > emulate it. He can bet for a continuum (with the rule Y = II, > bifurcation of "futur" retrospect on the "path"). > > > But I think you > > saying that our 1-person experience (frog view) is emergent from the > > collective (infinite) computations which are consistent with this > > emergent experience which is elaborated in your steps 1-7. It seems > > to make this ist person experience somewhat mystical as to why it is > > “experienced” at all. > > I think you are right. But here the amount of mysticism needed, is the > amount needed to say "yes" to the doctor. The belief in the > possibility (in principle) of technological reincarnation. > And then, the math explain why this, which is our consciousness, has > to seem completely mysterious at first sight. > But that mystery is no more mysterious than our awakening in the > morning. > > Consciousness is the most basic mystical state, somehow. > > > Some people wonder why we cannot see the other > > worlds in QM but I am often amazed that we experience one at all! > > Me too! > But there is a case that it should be amazing indeed, and it is the > result of a truth fixed point of a universal transformation. > > > Anyway all of what you say seems consistent with the many worlds > > picture (which it should be). > > I think so. > > > > > > > > >>> Time is not difficult. It is right in the successor axioms of > >>> arithmetic. > > I’ll come back to this > >>> Here again you confirm the invocation of the successor axioms. > >> Yes. It is fundamental. I cannot extract those from logic alone. No > >> more than I can define addition or multiplication without using the > >> successor terms s(-) : > >> for all x x + 0 = x > >> for all x and y x + s(y) = s(x + y) > >> You have to understand that all the talk on the phi_i and w_i, > >> including the existence of universal number > >> (EuAxAy phi_u(<x,y>) = phi_x(y)) can be translated in pure first > >> order > >> arithmetic, using only s, + and *. > >> I could add some nuances. "To be prime" is an intrinsic property of a > >> number. To be a universal number is not intrinsic. To define a > >> universal number I have to "arithmetize" the theory. The theory uses > >> variables x, y, z, ..., so I will have to represent "to be a > >> variable" > >> in the theory. The theory "understands" only numbers. I can decide to > >> represent the variables by even numbers (for example). "Even(x)" can > >> be represented by "Ey(x = s(s(0)) * y)". So "variable(x)" will be > >> represented by the same expression. Then I will represent "to be a > >> formula", "to be an axiom", to be a proof", "to be a computation", > >> using Gödel's arithmetization technic (which is just a form of > >> programming in arithmetic). This will lead to a representation of > >> being a universal number. > > > Where can I find out about this arithmetization technique > > The arithmetization technique is explained in all serious textbook in > mathematical logic. > I have explained this in this list once, using exponentiation. > The original Gödel 1931 (now in the Dover book of Davis 1965) is a > quite clear exposition. > I may explain it, or give the idea. It is really an encoding of the > syntax of a theory in term of numbers, and addition, and > mutiplication. A highly non trivial part of metamathematics can be > translated into mathematics, indeed arithmetic. > > > and what do > > you mean by a “universal number”? > > I mean a universal machine. It is a machine which can imitate all > other machine, from a description of that other machine. > > Consider the phi_i. The number u is universal if for all x, and y > phi_u(<x,y>) = phi_x(y). with <x,y> a bijective pairing of x and y. > > > > > > > > >> Now, would I decide to represent the variable in some other way (by > >> the odd numbers, for example), the preceding universal number will > >> still be in a universal number (intrinsically), but I will not been > >> able to see it, or to mention it explicitly. But here, you have to > >> just realize (cf the first six step of uda) that the first person > >> experience depends on all universal numbers, in all possible sense/ > >> arithmetical-implementations. > >> In particular "you here and now" are indeed implemented in arithmetic > >> in both the universal numbers based on (variable(x) = even(x), and > >> variable(x) = odd(x)). *ALL* universal numbers will compete below > >> your > >> substitution level. > >> The fact that elementary (Robinson) arithmetic is already (Turing) > >> universal is an impressive not obvious fact. But it is no more > >> astonishing than the fact that Conway Game of Life is already Turing > >> universal, or that the combinators S and K are Turing universal, etc. > > > Could you explain what your definition of Turing Universal is? Sorry > > but my background reading is not keeping up with my probingl > > Turing universal has been shown equivalent with Church universal, Post > universal, and all modern programming language are Turing universal. > Computer, or general purpose computer are such machine, as are brain, > cells (I would argue, even bacteria), but also many physical > (mathematical physics) system (the exact many body problem), modular > functor, quantum topologies. The global theory of all that is computer > science. > > Turing theorem is the theorem above where the phi_i are given by an > enumeration of Turing machine (mathematical machine). It means that > among all Turing machines, there is a universal one. That machine can > compute phi_x(y) from a description of x and y on its band. > > > > > So to sum up so far: (Please correct me on anything if I have got it > > wrong) > > > We begin with arithmetic realism so 0 and the positive numbers exist > > in arithmetical platonia. > > Yes. The first order logical formula > > Ex(x = 0) is true (and provable in most theories!), same for > > Ex(x = s(0)) > > Ex(x = s(s0))) > > > > > From the properties of these it is possible to accept the inherent > > idea of a successor function > > from which operations (maps), like addition and multiplication can be > > inferred or proved to similarly exist. > > Well, not really. From classical logic alone, the successor laws are > not enough, you have to define addition, and mulitiplication. With > addition only, you get Presburger Arithmetic. It is very rich, but > decidable (and thus not universal). > With multiplication only you have a system studied by Skolem, and also > decidable (and thus not universal). > > The creative explosion occurs when you mix addition and mutiplication. > You don't have to define 0 nor the successor, you can define them from > + and * (difficult exercise, at least for me!). > > > > > From these can also be inferred (as logical deductions) that many > > other things exist in the arithmetical platonia including a universal > > machine and a UD ( these “exist” like pi or root of 2 etc.). > > Yes. And there, lobian machine exist and develop theories, notably > theories far more porwerful than elementary arithmetic. > > > > > Similarly, finite subsets of the outputs of all the infinity of > > possible programs in UD* can also be inferred to exist. > > Well, having the beast in front our our mind, the scientist can use > its mental scalpel, and study the thing. > > > Hence > > because we are described from 3 person point of view by finite > > subsets of UD* then we can be said to exist in arithmetic platonia. > > If you associate your mind state to a computational state S and > universal machine U, you know that a UD would access those states an > infinity of times. Now, with Church thesis, all those events are > described by true (and provable) arithmetical sentences. Arithmetic > describe already the block mindscape of he universal machines, and you > in particular, assuming comp. > > > Now our ist person perceptions, along with 1 indeterminacy arise as a > > result of the way many consistent extensions of us compete at each of > > the program steps ( perceived as time increments) which exist because > > a temporal order is intrinsic - due to there being program steps - but > > the matching (gluing) of possible consistent extensions is not exact- > > hence 1-indeterminancy. We can never be sure which sheaf of > > computations we will go through – however limited in steps that > > particular sheaf is before the next indeterminate transition occurs. > > OK. I'm not sure the notion of steps make any sense from the first > person point of view. I think, and have also formal reason, that the > first person is already a continuum. > > > Now it seems that “concrete” reality is an illusion. In a way this > > points to our world being a kind of mental construct. ( Is platonia > > a bit like the mind of a supreme sentient being or does the buck stop > > with the label?). > > Hmm... Plotinus could side with you on this, but it is difficult. It > may just be simpler to take the number as there are, that is the > simple and usual intepretation of the axioms I gave, so that we can > describe a mental construct by a universal machine (number) construct. > > You can consider the number as mental construct; They occur and > reoccur all the times as constructs in the mind of the universal > numbers ... > > > Now I can accept all this quite reasonably apart from the one stage – > > the so called gluing bit. I’m still stuck with this one. > > For example suppose I write down on a piece of paper the list 1, > > 3,2,4,6,5,7a, 7b, 8a, 8b….. > > These have an intrinsic order namely 1,2,3,4,5,6,7a, 8a….. or > > 1,2,3,4,5,6,7b, 8b….. However much I jumble them up, this order is > > still apparent. They could represent steps in a persons experience as > > he approaches the splitting world at step 7. One branch goes to a > > (Moscow) the other to b (Washington). I could write down a function or > > algorithm on the same piece of paper (platonic paper!) which would > > indicate how the steps were to be traversed in the right order, but I > > just don’t get how all this can give us the feeling of experiencing > > each of these steps as observer moments. How do we become dynamically > > emergent in this static representation. > > I really don't understand that problem. I mean there is always a > probable universal number near by, and they come always with a notion > of clock (the stepping). It is more the symmetry of time in the > physical world which is intriguing, and the conservation principle, > and things like that, which become harder to justify. But > consciousness, subjective time, the difference between quanta and > qualia, all that is "explained" by computer science and the self- > reference logics. > > Look at humanity at the fourth dimension. It is a fractal with male > bourgeon attached to female bourgeon attached to female bourgeon etc. > You could also ask where the dynamic come from, looking at the static > humanity (or some portion of it). But there is no dynamic. But even > with materialism and the identify thesis, we could explain the feeling > of dynamic by the computation statically described in the tube-brain > of he person involved. > > With comp, it is even simpler, consciousness is attached only to the > logical arithmetical relation defining those computations. It concerns > universal machine interacting with each other, sharing deep > computational histories, leading to rare state, nevertheless super > (linearily) multiplied by their indeterminate histories, etc. > > > > > The block universe idea with world lines would have us think that our > > world was deterministic The block multiverse is also deterministic but > > gives us i-indeterminacy in the form of branching paths. What I’m > > struggling with here is how the platonia enables us to perceive the > > world from defined numbers and relationships. A computer does this in > > a first person game. The virtual world is presented to the player in > > some order at different times which gives the illusion of a reality > > albeit virtual. Program steps give an illusion of “becoming” but the > > screenshots of a video game collected (in order) don’t have this > > property of becoming. > > That depend on the video game. If it is a program, its steps are > determined by the interpreter, and the computer clock, itself > implemented in the "physical law" (bet on our most probable universal > machine history). > > > > > You see what I am getting at? There seems to be something dynamic > > missing. Even if you define a function which maps things, the > > function won’t do anything of itself it’s surely just more numbers on > > the platonic paper. > > I have some difficulty, to be honest. I would say that the dynamic is > more easy to explain from a digital block universal structure, where > we can see many different slef-representation of the whole structure > everywhere, than in a continumm block structure like a space-time or a > multiversial space time. > > Time is an illusion. But the illusion of time is NOT an illusion. It > is a cousin of consciousness. > Illusion of time comes from the extreme complexity the universal > machine are confronted with, in the average. > > > > > I’m sorry if I am so stuck with this one and I appreciate your > > patience. Funny thing is though, I sense it is me that’s got the > > wrong viewpoint here. I’m not trying to disprove anything. Rather I > > want get my philosophical thinking correct. If platonia was > > intrinsically timeful rather than timeless I’d probably feel better. > > It is weird. Group theory and the kind of mathematical structures > which play a big role in physics are more timeless than the super- > ordered arithmetical structure, which gives a simple notion of > (static) time on a plateau 0, 1, 2, 3, ... It is the elementary time > block of the digital universe. > > >>> Time is not difficult. It is right in the successor axioms of > >>> arithmetic. > > If I could grasp this statement of yours it would help. Could you > > elaborate on how this would be done. A successor function is applied > > by a (sentient) being. > > No, we are realist on the numbers. It is the theory. We take all > elementary arithmetical propositions as being true independently of us . > > Ex( x = s(4545454567676767)) has a successor, independently of me > thing to it or not. It would have a successor, even if life would have > not appear on this planet. > Even without the big bang, there is no biggest prime. The fact that a > number is prime or universal does not depend on anything physical. If > it is, show me the dependence. > > > Just writing it down next to its arguments > > does not seem to me to be the same as applying it. Or knowing what it > > is to be applied to. Any algorithm on how this would be done is not > > the same as it being done. > > The same problem occurs for any theory. Who interpret the Schroedinger > equation? God? The physicist? But what is a physicist? A nuage of > particle obeying Schroedinger equation? > > Physicist postulates the natural numbers (disguised under the form of > vibration nodes, or complex exponentiation, but the physicist does > postulate the timeless elementary arithmetic, no more no less than the > digital mechanist). It is just that if we assume comp, we need no more > than numbers. Worst, uda shows that we cannot use more than numbers. > > Don't hesitate to ask question. I guess it is too late to ask a logic > book for Xmas, but that could help, especially if you are interested > in the auda part. > > Happy 2010, > > Bruno > > http://iridia.ulb.ac.be/~marchal/- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text - -- 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. 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