Re: Computational irreducibility and the simulability of worlds
Hal Finney wrote: How about Tegmark's idea that all mathematical structures exist, and we're living in one of them? Or does that require an elderly mathematician, a piece of parchment, an ink quill, and some scribbled lines on paper in order for us to be here? It seems to me that mathematics exists without the mathematician. And since computer science is a branch of mathematics, programs and program runs exist as well without computers. Ok, but real computers are "math with motion". You have to have the program counter touring around through the memory in order to make a narrative sense of anything "happening". Mathematics, being composed of our symbols, is an abstract "re-presentation". I think what Tegmark must be saying is that "something" exists which is amenable to description by all self-consistent mathematical theories (logical sentence sets) , and by no inconsistent theories. To me, this is just equivalent to saying that "all possible configurations of differences exist" and that any SAS that represents its environment accurately (e.g. via abstract mathematics) is constrained, by its own being part of the information structure, to only perceive self-consistent configurations of differences as existing. Self-consistency of mathematical theory, as it translates from the representation level to the represented level, just means that things "perceived" can only be one way at a time, and that's the kind of thing that a consistent mathematical theory describes.
Re: Computational irreducibility and the simulability of worlds
Eugen, an outsider thought to your interesting attachment: We know about two parallel worlds (wit languages?): A. the 'physos'-observable one - som call material reality (I don't), B. mathematics I extend A into all white elephant/rabbit versions we can 'talk' about. B exists in the mind of mathematicians (including simpler levels existing in simpler minds one would not call 'a mathematician'. E.g. me. The problem starts when scientists start to apply one to the other, mostly B to A, forcing connections between the parallels. It leads to omissions, unnatural conclusions, I call it reductionism into those cases where it was (successfully???) done. I know this was not what you intended. John Mikes PS to your interesting "Rock" post: that is what your human mind says. Ask the rock, you may be surprised. - J - Original Message - From: "Eugen Leitl" <[EMAIL PROTECTED]> To: "Hal Finney" <[EMAIL PROTECTED]>; <[EMAIL PROTECTED]> Sent: Saturday, April 17, 2004 4:25 AM Subject: Re: Computational irreducibility and the simulability of worlds
Re: Computational irreducibility and the simulability of worlds
On Sat, Apr 17, 2004 at 01:03:03AM -0700, Hal Finney wrote: > How about Tegmark's idea that all mathematical structures exist, and we're > living in one of them? Or does that require an elderly mathematician, > a piece of parchment, an ink quill, and some scribbled lines on paper in > order for us to be here? That wouldn't quite do. Just simulating this planet takes a lot of hardware. Just because you can write down Navier-Stokes it doesn't mean rivulets, streams and oceans spring into being. A little more work is required for that. > It seems to me that mathematics exists without the mathematician. To me it seems the opposite is true. As long as it's an unfalsifyable prediction, there's not much point to pursue it further. > And since computer science is a branch of mathematics, programs and > program runs exist as well without computers. While I'm open to existence of a metalayer, built from information or otherwise, I'm very much opposed to mysticism. -- Eugen* Leitl http://leitl.org";>leitl __ ICBM: 48.07078, 11.61144http://www.leitl.org 8B29F6BE: 099D 78BA 2FD3 B014 B08A 7779 75B0 2443 8B29 F6BE http://moleculardevices.org http://nanomachines.net pgp0.pgp Description: PGP signature
Re: Computational irreducibility and the simulability of worlds
Eric Hawthorne writes: > So does that mean we just say "think of the substrate of the universe as > being a turing machine equivalent", > any old turing machine equivalent. Ok, but still, you have to admit that > every "easy to think of" instantiation > of a turing machine (e.g. a PC with a lot of time on its hands) is a > terribly implausible universe substrate. > For heavens sake, the PC with a lot of time on its hands presupposes > time (and space (i.e. different localities, > with notions of adjacency), in which to write the tape). Classic > chicken and egg problem. > > Does anyone know the way out of that particular conceptual pickle? How about Tegmark's idea that all mathematical structures exist, and we're living in one of them? Or does that require an elderly mathematician, a piece of parchment, an ink quill, and some scribbled lines on paper in order for us to be here? It seems to me that mathematics exists without the mathematician. And since computer science is a branch of mathematics, programs and program runs exist as well without computers. Hal Finney
Re: Computational irreducibility and the simulability of worlds
We're just doing models or thought experiments here when we postulate that a universe is (could be) a simulation in a computer running a cellular automaton, are we not? Whatever explanation we do come up with eventually is going to have to explain 1. the "memory" cells themselves, and 2. what a "state transition of which the dance of shiva is composed" is, in itself. If we can't explain those things, we just get one of those "I don't know, but it's turtles all the way down" kind of unsatisfactory theories. This seems a hard problem to me. There seems to be a fundamental connection between information and representation of state. But we are after an explanation of "physical state" itself, i.e. the presence of state, and not just after the secondary thing; the "re-presence" of state. Can we come down on the side of saying "There's really nothing but information"? I find it better to say "No no, information is a RE-PRESENTATION of differences." What there really is is some kind of medium with a capacity for being described as having (arbitrarily combinable) differences. Given such a medium, information theory and computability theory is probably sufficient to ensure that everything observed with certainty will be logically consistent with everything else observed with certainty, by any co-existing observers in the medium. But what the heck is the medium (the "aether" of old)? I guess comp sci tells us it could be any turing machine equivalent. So does that mean we just say "think of the substrate of the universe as being a turing machine equivalent", any old turing machine equivalent. Ok, but still, you have to admit that every "easy to think of" instantiation of a turing machine (e.g. a PC with a lot of time on its hands) is a terribly implausible universe substrate. For heavens sake, the PC with a lot of time on its hands presupposes time (and space (i.e. different localities, with notions of adjacency), in which to write the tape). Classic chicken and egg problem. Does anyone know the way out of that particular conceptual pickle? Hal Ruhl wrote: Hi Stephen: Observers: Accepting as a starting point the earlier argument that our universe is an interpretation of a collection of the automaton cells considered there then going further: What can the collection look like in order to have an interpretation compatible with our universe? It is entirely reasonable that collections of cells extending beyond nearest neighbors have state histories that are coordinated to some degree. I have called these coordinations dances since the particular state progressions can move from cell to cell without the cell itself moving - the cell does not change nearest neighbors. Some dances look like this. For example what we call a photon can be such a dance moving through the grid of cells. Dark matter could be just a very large scale dance somewhat kept in place by smaller dances within and in turn influencing these internal dances. Further it must be remembered that the progression of states in any cell is a succession of discontinuities. One state shifts to another with no continuity of states between. In this venue dances interact and change each other discontinuously by mutual collision or by exchanging smaller dances. How then does a human differ in kind from a rock? Should we expect them to differ in kind? Yours Hal
Re: Computational irreducibility and the simulability of worlds
Hi Stephen: Observers: Accepting as a starting point the earlier argument that our universe is an interpretation of a collection of the automaton cells considered there then going further: What can the collection look like in order to have an interpretation compatible with our universe? It is entirely reasonable that collections of cells extending beyond nearest neighbors have state histories that are coordinated to some degree. I have called these coordinations dances since the particular state progressions can move from cell to cell without the cell itself moving - the cell does not change nearest neighbors. Some dances look like this. For example what we call a photon can be such a dance moving through the grid of cells. Dark matter could be just a very large scale dance somewhat kept in place by smaller dances within and in turn influencing these internal dances. Further it must be remembered that the progression of states in any cell is a succession of discontinuities. One state shifts to another with no continuity of states between. In this venue dances interact and change each other discontinuously by mutual collision or by exchanging smaller dances. How then does a human differ in kind from a rock? Should we expect them to differ in kind? Yours Hal
Re: Computational irreducibility and the simulability of worlds
Dear Hal, In general I am in agreement with your argument here but do not understand how it generalizes to the case where we consider a plurality of observers, each with their own sets of boundaries. Kindest regards, Stephen - Original Message - From: "Hal Ruhl" <[EMAIL PROTECTED]> To: "Stephen Paul King" <[EMAIL PROTECTED]> Sent: Wednesday, April 14, 2004 7:59 PM Subject: Re: Computational irreducibility and the simulability of worlds > Hi Stephen: > > What I am basically saying is that you can not define a thing without > simultaneously defining another thing that consists of all that is "left > over" in the ensemble of building blocks. I suspect that usually the "left > over" thing is of little practical use. > > However, this duality also applies to the "Nothing" and its left over which > is the "Everything". A look at this pair allows the derivation that the > boundary between them [the definition pair] can be represented as a > "normal" real and can not be a constant if zero info is to be maintained. > > Thus, given the dynamic, this boundary's representation as I said in the > last post can be modeled as the output of a computer with an infinite > number of asynchronous multiprocessors. A cellular automaton with > asynchronous cells. Universes are interpretations of this output. > > Sort of a left wing proof that we are "in" a massive computer. > > The Hintikka material you pointed me to is far too imbedded in mathematical > language symbols for me to understand. > > Yours > > Hal > > At 12:03 AM 4/13/2004, you wrote: > >Dear Hal, > > > > I will have to think about this for a while. Very interesting. Meanwhile > >I ask that you take a look at the game theoretic semantic idea by Hintikka. > > > >Kindest regards, > > > >Stephen
Re: Computational irreducibility and the simulability of worlds
Hi Stephen: What I am basically saying is that you can not define a thing without simultaneously defining another thing that consists of all that is "left over" in the ensemble of building blocks. I suspect that usually the "left over" thing is of little practical use. However, this duality also applies to the "Nothing" and its left over which is the "Everything". A look at this pair allows the derivation that the boundary between them [the definition pair] can be represented as a "normal" real and can not be a constant if zero info is to be maintained. Thus, given the dynamic, this boundary's representation as I said in the last post can be modeled as the output of a computer with an infinite number of asynchronous multiprocessors. A cellular automaton with asynchronous cells. Universes are interpretations of this output. Sort of a left wing proof that we are "in" a massive computer. The Hintikka material you pointed me to is far too imbedded in mathematical language symbols for me to understand. Yours Hal At 12:03 AM 4/13/2004, you wrote: Dear Hal, I will have to think about this for a while. Very interesting. Meanwhile I ask that you take a look at the game theoretic semantic idea by Hintikka. Kindest regards, Stephen
Re: Computational irreducibility and the simulability of worlds
Dear Stephen, snip > [BM] > Giving that I *assume* that arithmetical truth is independent > of me, you and the whole physical reality (if that exists), "I" do have > infinite resources in that Platonia. Remember that from the first person > point of view it does not matter where and how, in Platonia, my > computational states are represented. Brett Hall just states that > the proposition "we are living in a massive computer" is undecidable > (and he adds wrongly (I think) that it makes it uninteresting), but > actually with my hypotheses physics is a sum of all those > undecidable propositions ...(Look again my UDA proof if you are not > yet convinced, but keep in mind that I assume the whole > (un-axiomatizable by Godel) arithmetical truth, which I think you > don't. [SPK] This is very unsettling for me as it seems to claim that we can merely postulate into existence whatever we need to make up for deficiencies in our theories. This can not be any kind of science. But Mendeelev discovered new atoms by that method. I am not sure what you mean. But I can put that complaint aside. It is what is missing in this "Platonia" that bothers me: how does it necessitate an experienciable world. It necessitates the experienciable truth, and "worlds" emerge from that. The fact that I experience a world must be explained, even if it is merely an illusion. It must be necessitated by our theories of Everything. Sure. I tend to think of the "truth" in Arithmetic Truth (and any other formal system) to be more of a notion that is derived from game theoretics (http://www.csc.liv.ac.uk/~pauly/Submissions/mcburney.ps and http://staff.science.uva.nl/%7Ejohan/H-H.pdf) than from hypostatization. "arithmetical truth" is not (by Godel, Tarski, ...) formally definable in any formal arithmetic. This, of course, degenerates the notion of "objective truth", but I have come to the belief that this notion is, at best self-stultifying. What sense does it make to claim that some statement X is True or that some Y "exists" independent of me, you and the whole of physical reality when X and Y are only meaningful to me, you, etc.? I know you dislike arithmetical realism, but it is hard for me to believe that the primality of 317 is contingent, or even remotely linked to us. We can claim that anything at all is True, so long as it is not detectable. This entire argument of "independence" teeters on the edge of indetectability. I don't understand. You should put your cart on the table. What are your presupposition? > >[SPK] > > I agree with most of your premises and conclusions but I do not > >understand how it is that we can coherently get to the case where a > >classical computer can generate the simulation of a finite world that > >implies QM aspects (or an ensemble of such), for more than one observer > >including you and I, without at least accounting for the appearance of > >implementation. But I do. See the ref to the everything-list in my url. > >[BM] > A non genuine answer would be the following: because the solutions > of Schroedinger equations (or Dirac's one, ...) are Turing-emulable. > This does not help because a priori we must take into account all > computation (once we accept we are turing-emulable), not only > the quantum one (cf UDA). [SPK] A priori existing UDA, Platonia, whatever, how is this more than mere hypostatization? Because those are well defined arithmetical object. UD is a well defined program. Again I am reminded of Julian Barbour's notion of best matching. He himself discussed the difficulty of running the computations to find best matchings among a small (finite!) number of possibilities, and yet, when faced with an infinity of possibilities the complexity is hand waved away by an appeal to "Platonia"! Even if we assume that Platonia has "infinite Resources", the kind of computation that you must run takes an Eternity to solve. Yes, but our first person experiences rely on that infinity just because we cannot be aware of any delay in the UD processing, so that we must take into account the infinite union of all initial segment of the whole processing of the UD. It is like a Perfectly Fair game: it takes forever to verify its fairness and, once that infinity has passed, it is a game that never ends. Is our 1 person experience a trace of this game? Not exactly. It is less false to consider it as a "partial view" on an infinity of traces, giving that we cannot distinguish the infinity of version of that trace. > [BM] > A priori > comp entails piece of non-computable "stuff" in the neighborhood, > but no more than what can be produced by an (abstract) computer > duplicating or differentiating all computational histories. [SPK] Surely, but "all computational histories" requires at least one step to be produced. In Platonia, there is not Time, there is not any way to "take that one step". There is merely a Timeless Existence. That is t
Re: Computational irreducibility and the simulability of worlds
Dear Hal, I will have to think about this for a while. Very interesting. Meanwhile I ask that you take a look at the game theoretic semantic idea by Hintikka. Kindest regards, Stephen - Original Message - From: "Hal Ruhl" <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Sent: Monday, April 12, 2004 9:34 PM Subject: Re: Computational irreducibility and the simulability of worlds > Hi Stephen and Bruno: > > I only managed to jump into the list and read the last two posting on this > subject so I hope this effort to contribute is of interest in areas such as: > > [Cut and pasted out of context:] > > > >[SPK] > > > I agree with most of your premises and conclusions but I do not > > >understand how it is that we can coherently get to the case where a > > >classical computer can generate the simulation of a finite world that > > >implies QM aspects (or an ensemble of such), for more than one observer > > >including you and I, without at least accounting for the appearance of > > >implementation. > > > > [SPK] > Surely, but "all computational histories" requires at least one step to > be produced. In Platonia, there is not Time, there is not any way to "take > that one step". There is merely a Timeless Existence. How do you propose > that we recover our experience of time from this? Perhaps I need to learn > French... > > > As Alastair indicated awhile back, he and I are having a discussion in the > "Agreed Fundamentals Project" re something/nothing. > > The following is a rework of my most recent response in that > discussion. Definitions extracted from the module are below. > > xxx > > Here is sort of a very short form of the module. > > There are, it seems, three information content possibilities for the system > that could be the basis of our universe and these are: > > 1) The system contains no information. > 2) The system contains some information. > 3) The system contains all information. > > The second seems unsatisfactory since you could tune the information > content to fit your purpose. > > All I really do is to assume what is actually (I think) a bundle of no > information - my "Nothing (N)" or #1 in the above list, and a bundle of all > information - i.e. all complete sets of cf-counterfactuals - i.e. my > "Everything (E)" or #3 in the above list simultaneously. > > I then show - I think - that they are fundamentally not independent. I now > call such interdependence an example of a definitional pair. [ Whenever a > definition is made there are actually definitions of two things being > forged simultaneously - Whatever the thing you are defining is and and > another thing that is all that is left over.] > > If "all complete sets of cf-counterfactuals" is the same as all bit > strings, then as I see it the above is the same as saying that "N" contains > no number at all and that "E" contains a "normal" real number. > > Further if all information is equivalent to having no information then "all > sets of cf-counterfactuals" results in "no potential to divide" i.e. no > cf-information. So we note an odd thing: we have a definitional pair that > define two forms of the same thing - the net absence of a potential to > divide - no cf-information. > > The dynamic I develop in the module [from: only cf-counterfactuals allowed > in "E"] says that any such pair can not be static or have a fixed > evolution. In other words the boundary - no number opposed to a particular > normal real number - between the two must be dynamic and therefore > represent a sequence where "E" contains a series of normal real numbers in > random order. And because of this dynamic our universe's current state > which is a particular decode [interpretation] of a particular string in > that number will always be present and will eventually come into proper > juxtaposition [also necessarily a dynamic] with all those strings that > represent encoded possible next states - evolutionary trees - during the > dynamic. > > Now the final point I have interest in in the module is: Can there be a > fixed number of these evolutionary trees [all, some fixed fraction, none] > that have at least one path that is free of external true noise? No > because any such number would represent a cf-factual not a > cf-counterfactual. Therefore all paths eventually experience an external > noise event since "none" must randomly be the right number. > > One view of the dynamic is a comp
Re: Computational irreducibility and the simulability of worlds
Hi Stephen and Bruno: I only managed to jump into the list and read the last two posting on this subject so I hope this effort to contribute is of interest in areas such as: [Cut and pasted out of context:] > >[SPK] > > I agree with most of your premises and conclusions but I do not > >understand how it is that we can coherently get to the case where a > >classical computer can generate the simulation of a finite world that > >implies QM aspects (or an ensemble of such), for more than one observer > >including you and I, without at least accounting for the appearance of > >implementation. > [SPK] Surely, but "all computational histories" requires at least one step to be produced. In Platonia, there is not Time, there is not any way to "take that one step". There is merely a Timeless Existence. How do you propose that we recover our experience of time from this? Perhaps I need to learn French... As Alastair indicated awhile back, he and I are having a discussion in the "Agreed Fundamentals Project" re something/nothing. The following is a rework of my most recent response in that discussion. Definitions extracted from the module are below. xxx Here is sort of a very short form of the module. There are, it seems, three information content possibilities for the system that could be the basis of our universe and these are: 1) The system contains no information. 2) The system contains some information. 3) The system contains all information. The second seems unsatisfactory since you could tune the information content to fit your purpose. All I really do is to assume what is actually (I think) a bundle of no information - my "Nothing (N)" or #1 in the above list, and a bundle of all information - i.e. all complete sets of cf-counterfactuals - i.e. my "Everything (E)" or #3 in the above list simultaneously. I then show - I think - that they are fundamentally not independent. I now call such interdependence an example of a definitional pair. [ Whenever a definition is made there are actually definitions of two things being forged simultaneously - Whatever the thing you are defining is and and another thing that is all that is left over.] If "all complete sets of cf-counterfactuals" is the same as all bit strings, then as I see it the above is the same as saying that "N" contains no number at all and that "E" contains a "normal" real number. Further if all information is equivalent to having no information then "all sets of cf-counterfactuals" results in "no potential to divide" i.e. no cf-information. So we note an odd thing: we have a definitional pair that define two forms of the same thing - the net absence of a potential to divide - no cf-information. The dynamic I develop in the module [from: only cf-counterfactuals allowed in "E"] says that any such pair can not be static or have a fixed evolution. In other words the boundary - no number opposed to a particular normal real number - between the two must be dynamic and therefore represent a sequence where "E" contains a series of normal real numbers in random order. And because of this dynamic our universe's current state which is a particular decode [interpretation] of a particular string in that number will always be present and will eventually come into proper juxtaposition [also necessarily a dynamic] with all those strings that represent encoded possible next states - evolutionary trees - during the dynamic. Now the final point I have interest in in the module is: Can there be a fixed number of these evolutionary trees [all, some fixed fraction, none] that have at least one path that is free of external true noise? No because any such number would represent a cf-factual not a cf-counterfactual. Therefore all paths eventually experience an external noise event since "none" must randomly be the right number. One view of the dynamic is a computer [Turing?] moving along an infinite string as data and outputing the original string and a computed new string as it went. Behind that would be two more and behind each of those two more etc. These computers would all have randomly constructed rules and be asynchronous [the external true noise]. The result to me seems to be a dynamic of all possible universes evolving to all possible next states. --- From the module - more or less -- I see no difference between cf-information [a term defined in the module - see below] and the usual idea of information and intend none. [1def] cf-Information: The potential to divide as with a boundary. An Example: The information in a Formal Axiomatic System [FAS] divides true statements from not true statements [relevant to that FAS]. [2def] cf-Factual: A particular potential to divide. Used as a noun. An example: The FAS known as Arithmetic. [4def] cf-Counterfactual: A cf-factual [cf-factual B] that to some degre
Re: Computational irreducibility and the simulability of worlds
Dear Bruno, Thank you for your wise and patient reply. Interleaving. - Original Message - From: "Bruno Marchal" <[EMAIL PROTECTED]> To: "Bruno Marchal" <[EMAIL PROTECTED]> Cc: <[EMAIL PROTECTED]> Sent: Saturday, April 10, 2004 12:12 PM Subject: Re: Computational irreducibility and the simulability of worlds snip > [BM] > Giving that I *assume* that arithmetical truth is independent > of me, you and the whole physical reality (if that exists), "I" do have > infinite resources in that Platonia. Remember that from the first person > point of view it does not matter where and how, in Platonia, my > computational states are represented. Brett Hall just states that > the proposition "we are living in a massive computer" is undecidable > (and he adds wrongly (I think) that it makes it uninteresting), but > actually with my hypotheses physics is a sum of all those > undecidable propositions ...(Look again my UDA proof if you are not > yet convinced, but keep in mind that I assume the whole > (un-axiomatizable by Godel) arithmetical truth, which I think you > don't. [SPK] This is very unsettling for me as it seems to claim that we can merely postulate into existence whatever we need to make up for deficiencies in our theories. This can not be any kind of science. But I can put that complaint aside. It is what is missing in this "Platonia" that bothers me: how does it necessitate an experienciable world. The fact that I experience a world must be explained, even if it is merely an illusion. It must be necessitated by our theories of Everything. I am reminded of an idea by Jaakko Hintikka where he criticizes the notion in game theory that every player has complete knowledge of the possible moves of the other players. He goes on to explain how "imperfect information" games are more realistic. http://www.maths.qmw.ac.uk/~wilfrid/kingsfeb00.pdf I tend to think of the "truth" in Arithmetic Truth (and any other formal system) to be more of a notion that is derived from game theoretics (http://www.csc.liv.ac.uk/~pauly/Submissions/mcburney.ps and http://staff.science.uva.nl/%7Ejohan/H-H.pdf) than from hypostatization. This, of course, degenerates the notion of "objective truth", but I have come to the belief that this notion is, at best self-stultifying. What sense does it make to claim that some statement X is True or that some Y "exists" independent of me, you and the whole of physical reality when X and Y are only meaningful to me, you, etc.? We can claim that anything at all is True, so long as it is not detectable. This entire argument of "independence" teeters on the edge of indetectability. > >[SPK] > > I agree with most of your premises and conclusions but I do not > >understand how it is that we can coherently get to the case where a > >classical computer can generate the simulation of a finite world that > >implies QM aspects (or an ensemble of such), for more than one observer > >including you and I, without at least accounting for the appearance of > >implementation. > >[BM] > A non genuine answer would be the following: because the solutions > of Schroedinger equations (or Dirac's one, ...) are Turing-emulable. > This does not help because a priori we must take into account all > computation (once we accept we are turing-emulable), not only > the quantum one (cf UDA). [SPK] A priori existing UDA, Platonia, whatever, how is this more than mere hypostatization? Again I am reminded of Julian Barbour's notion of best matching. He himself discussed the difficulty of running the computations to find best matchings among a small (finite!) number of possibilities, and yet, when faced with an infinity of possibilities the complexity is hand waved away by an appeal to "Platonia"! Even if we assume that Platonia has "infinite Resources", the kind of computation that you must run takes an Eternity to solve. It is like a Perfectly Fair game: it takes forever to verify its fairness and, once that infinity has passed, it is a game that never ends. Is our 1 person experience a trace of this game? > [BM] > A priori > comp entails piece of non-computable "stuff" in the neighborhood, > but no more than what can be produced by an (abstract) computer > duplicating or differentiating all computational histories. [SPK] Surely, but "all computational histories" requires at least one step to be produced. In Platonia, there is not Time, there is not any way to "take that one step". There is merely a Timeless Existence. How do you propose that we recover our experience of time from this? Perhaps I need to learn French... > [BM] > Remember that if we are m
Re: Computational irreducibility and the simulability of worlds
At 00:35 10/04/04 -0400, Stephen Paul King wrote: > BM: I agree with this. There is no embedding of QM in a Boolean representation, > if by embedding we mean a injective function which preserves the value of > the observable. But ... [SPK] Ok. Well please help me how does my argument not follow? I am trying to understand how my claim fails. You seem to understand this, I need to understand this, help me please. BM: My feeling is that we just have different fundamental hypotheses. More in the sequel. BM: > >it follows > >that it is impossible to fully simulate a QM system on a classical computer > >unless we allow for some rather exotic special conditions. > > I disagree. Unless "fully" means "in real time" ? Not only a classical > computer can compute all "quantum computable functions", but if you > allow the classical computer to simulates the system consisting of > "you + a quantum computer", then the classical computer will, relatively > to you, be able to simulate all quantum processes (and not only the function). [SPK] I have one situation in mind where your conclusion follows but it seems to be a Very Special case, the case where we have infinite resourses available for the classical systems to simulate the QM systems, all of the possible. Giving that I *assume* that arithmetical truth is independent of me, you and the whole physical reality (if that exists), "I" do have infinite resources in that Platonia. Remember that from the first person point of view it does not matter where and how, in Platonia, my computational states are represented. Brett Hall just states that the proposition "we are living in a massive computer" is undecidable (and he adds wrongly (I think) that it makes it uninteresting), but actually with my hypotheses physics is a sum of all those undecidable propositions ...(Look again my UDA proof if you are not yet convinced, but keep in mind that I assume the whole (un-axiomatizable by Godel) arithmetical truth, which I think you don't. I agree with most of your premises and conclusions but I do not understand how it is that we can coherently get to the case where a classical computer can generate the simulation of a finite world that implies QM aspects (or an ensemble of such), for more than one observer including you and I, without at least accouting for the appearence of implementation. A non genuine answer would be the following: because the solutions of Schroedinger equations (or Dirac's one, ...) are Turing-emulable. This does not help because a priori we must take into account all computation (once we accept we are turing-emulable), not only the quantum one (cf UDA). A priori comp entails piece of non-computable "stuff" in the neighborhood, but no more than what can be produced by an (abstract) computer duplicating or differentiating all computational histories. Remember that if we are machine then we should expect our "physical reality" NOT to be a machine. Indeed at first sight we should expect all "nearly-inconsistent" histories (white rabbits). But the godelian constraints add enough informations for defining a notion of normality, that is a beginning of an explanation of why coherent and sharable realities evolves from the point of view of the observers embedded in Platonia. Most of Alan and David critics of comp works fine for Schmidhuber form of comp (where physics comes from a special program) or Tegmark where physical reality is a mathematical structure among all mathematical structures. I provide arguments showing that if we belong to a mathematical computation then our future/past (that is our physics) depends on an infinity of (relative) computations (all those going through our relative states). How is it that we necessarily experience an asymmetrical flow of time given the assumption that all 1st person experiences are assumed to be merely algorithms that exist a priori in Platonia? Your phrasing is a little bit misleading here I'm afraid. The first person experiences are knowledge states. If you agree with the usual axioms for knowledge (that is : I know A implies A, I know A implies that I know that I know A, I know (A -> B) entails that if I know A then I know B, plus the traditional modal inference rules, then with comp that knowledge states are completely captured by the S4Grz modal logic which has nice semantics in term of antisymmetrical knowledge states evolution. What is absolutely nice is that from the machine point of view that knowledge cannot ever be defined. Only meta-reasoning based on comp makes it possible to handle it. You can read the appendice (in english!) in "Conscience et Mecanisme" by the Russian logician Sergei Artemov which provides an argument for identifying the notion of informal (and even un-formalizable) provability by the conjonction of formal provability and truth. By Godel, that *is* different from just formal provability: http://iridia.ulb.ac.be/~marchal/bxlthesis/Volume4CC/6%20La%20th
