Re: Provable vs Computable
from George Levy, 1 Jun 2001: > A purely mechanical model no matter how complicated, including random > variables, cannot replicate the results generated by Quantum mechanics + > probability theory. This is exactly what Bell's inequality implies. In > fact Bell proved his inequality using Quantum theory and probability. > Therefore Juergens' erector (fr: meccano) set approach using pseudo-random > generators, would definitely violate Bell's inequality theorem, and would > not be phenomenally or experimentally equivalent to quantum mechanics. I do not agree, of course. Since you keep insisting on this, I suggest you clearly write down all implicit assumptions and why exactly you believe Bell's inequality is not compatible with pseudorandomness and algorithmic TOEs. http://www.idsia.ch/~juergen/everything/html.html http://www.idsia.ch/~juergen/toesv2/
Re: Provable vs Computable
John and Hal, Bruno and all everythingers, sorry for the delay guys, I was travelling and had lots of work. Bruno, I just scanned your post quickly. It seems to me we are going in the right direction but I shall need time to digest what you wrote. I shall reply to you later Let me first reply to John and Hal because it is the shortest reply. Let's go back to the original Juergens' post [EMAIL PROTECTED] wrote: > Example: a never ending universe history h is computed by a finite > nonhalting program p. To simulate randomness and noise etc, p invokes a > short pseudorandom generator subroutine q which also never halts. The > n-th pseudorandom event of history h is based on q's n-th output bit > q(n) which is initialized by 0 and set to 1 as soon as the n-th element > of an ordered list of all possible program prefixes halts. Whenever q > modifies some q(n) that was already used in the previous computation of > h, p appropriately recomputes h since the n-th pseudorandom event. > > Such a virtual reality or universe is perfectly well-defined. I replied: >Such a universe would violate Bell' inequality theorem. Quantum randomness >cannot be simulated by hidden variables. We have to move beyond >realism..to get a model of objective reality we must first develop a >model of consciousness. A purely mechanical model no matter how complicated, including random variables, cannot replicate the results generated by Quantum mechanics + probability theory. This is exactly what Bell's inequality implies. In fact Bell proved his inequality using Quantum theory and probability. Therefore, Juergens' erector (fr: meccano) set approach using pseudo-random generators, would definitely violate Bell's inequality theorem, and would not be phenomenally or experimentally equivalent to quantum mechanics. Some of his (our) choices are: 1) Quantum mechanics + probability -> Bell's inequality and give up on a mechanical hidden variable, on pseudo random generators, and more generally, on realism. 2) Something else of power equivalent to Quantum mechanics in describing natureGood Luck!!! I do not believe the route to this solution is the erector set technique. Many a 19th and early 20th century physicist has broken a tooth on that bone! George jamikes wrote: George, thanks for your reply, which is almost as convoluted and hard-to-follow as was my question. You wrote: > I am not restricting anything. I am only saying that Juergens has to choose > between violating Bell's inequality theorem and all that this implies, or not > and all that this implies. My stand is that we shouldn't. > George > So ;let me rephrase the question: is your stand that if an imaginary universe would violate eg. Bell's theorem, it should be excluded from consideration as a possibility, - or - we should rather conclude that Bell's theorem (or any other fundemntal "human" rule) has a limited validity and does not cover every possible universe? John
Re: Provable vs Computable
Hi John, Hal I have to leave on a week long trip I'll reply to your posts when I return. George
Re: Provable vs Computable
George, thanks for your reply, which is almost as convoluted and hard-to-follow as was my question. You wrote: > I am not restricting anything. I am only saying that Juergens has to choose > between violating Bell's inequality theorem and all that this implies, or not > and all that this implies. My stand is that we shouldn't. > George > So ;let me rephrase the question: is your stand that if an imaginary universe would violate eg. Bell's theorem, it should be excluded from consideration as a possibility, - or - we should rather conclude that Bell's theorem (or any other fundemntal "human" rule) has a limited validity and does not cover every possible universe? John
Re: Provable vs Computable
Well I thought the whole point was to restrict the universe (that we're in) by the anthropic principle. But if the anthropic principle is to meant to include all intelligent beings, then some theory will be necessary to say in what respects the universe could differ and still produce intelligent beings. Have you read Tegmark's paper, quant-ph/9907009v2 10 Nov 1999, which shows that entangled quantum probabilities are not necessary for consciousness - only ordinary randomness. Brent Meeker "As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality." -- Albert Einstein On 22-May-01, George Levy wrote: > > > jamikes wrote: > >> "George Levy" <[EMAIL PROTECTED]> wrote >> Saturday, May 05, 2001 : >> >> (SNIP Jurgen's remark about "such a universe" whatever, my remark is >> not >> topical, rather principle:) >> >>> Such a universe would violate Bell' inequality theorem. Quantum >> randomness >>> cannot be simulated by hidden variables. We have to move beyond >>> realism..to get a model of objective reality we must first >> develop a >>> model of consciousness. >>> >>> George >> >> Can you restrict a universe according to its compliance with or >> violation of >> a theory, no matter how ingenious, or vice versa? Are WE the >> creators who >> has to perform according to some rules/circumstances of human logic >> or >> computability? >> John Mikes >>> > > I am not restricting anything. I am only saying that Juergens has to > choose between violating Bell's inequality theorem and all that this > implies, or not and all that this implies. My stand is that we > shouldn't. > > George > Regards
Re: Provable vs Computable
Dear George: I do not see how the aspect of Juergen's approach he cited at the initiation of this part of the thread causes a dilemma re Bell's inequality. As I understand it the history h is not THE history until the applicable portion of h stops changing. But p and q are both non halting. No part of h may settle in a finite time. So what? The generator q" is not "in" h. The fact that p may recompute virtually any amount of h depending on what happens to q(n) is not a non locality "in" h but rather a substitution of a new h for the old h. The old h is no longer relevant. However, p itself being global seems to me to allow the violation of Bell's inequality in its universe. My "formal" system has the same feature as does a UD IMO. As far as I know the violation of Bell's inequality was already well established in our universe. Hal At 5/22/01, you wrote: >jamikes wrote: > > > "George Levy" <[EMAIL PROTECTED]> wrote > > Saturday, May 05, 2001 : > > > > (SNIP Jurgen's remark about "such a universe" whatever, my remark is not > > topical, rather principle:) > > > > > Such a universe would violate Bell' inequality theorem. Quantum > randomness > > > cannot be simulated by hidden variables. We have to move beyond > > > realism..to get a model of objective reality we must first develop a > > > model of consciousness. > > > > > > George > > > > Can you restrict a universe according to its compliance with or > violation of > > a theory, no matter how ingenious, or vice versa? Are WE the creators who > > has to perform according to some rules/circumstances of human logic or > > computability? > > John Mikes > > > > >I am not restricting anything. I am only saying that Juergens has to choose >between violating Bell's inequality theorem and all that this implies, or not >and all that this implies. My stand is that we shouldn't. > >George
Re: Provable vs Computable
jamikes wrote: > "George Levy" <[EMAIL PROTECTED]> wrote > Saturday, May 05, 2001 : > > (SNIP Jurgen's remark about "such a universe" whatever, my remark is not > topical, rather principle:) > > > Such a universe would violate Bell' inequality theorem. Quantum randomness > > cannot be simulated by hidden variables. We have to move beyond > > realism..to get a model of objective reality we must first develop a > > model of consciousness. > > > > George > > Can you restrict a universe according to its compliance with or violation of > a theory, no matter how ingenious, or vice versa? Are WE the creators who > has to perform according to some rules/circumstances of human logic or > computability? > John Mikes > > I am not restricting anything. I am only saying that Juergens has to choose between violating Bell's inequality theorem and all that this implies, or not and all that this implies. My stand is that we shouldn't. George
Re: Provable vs Computable
"George Levy" <[EMAIL PROTECTED]> wrote Saturday, May 05, 2001 : (SNIP Jurgen's remark about "such a universe" whatever, my remark is not topical, rather principle:) > Such a universe would violate Bell' inequality theorem. Quantum randomness > cannot be simulated by hidden variables. We have to move beyond > realism..to get a model of objective reality we must first develop a > model of consciousness. > > George Can you restrict a universe according to its compliance with or violation of a theory, no matter how ingenious, or vice versa? Are WE the creators who has to perform according to some rules/circumstances of human logic or computability? John Mikes >
Re: Provable vs Computable
Dear Bruno:
At , you wrote:
>Hal Ruhl wrote:
>
> >This is particularly due to my stand that true random noise is
> >inherent in each universe within the Everything.
>
>Remember that true random noise appear in the UDA because we don't know
>in which computation ("universe") we belong. So random noise does
>not need to be added. It is something we cannot avoid from our first
>person point of view of embedded observer.
>It is the point where Schmidhuber disagrees so much. We will come back
>on this question soon.
My position is a bit different - I think - in that I do not see observers
as essential to the presence of true random noise in a universe. The UD,
IMO - after a concatenation of its output strings into one long one that is
then altered at the knit points in some pattern - is within itself a single
valued elegant cascade. This would hold even if the activity around the
knit points is spread out along a region of the string as in Juergen's
model [If I have it right]. It is one out of an infinite number of similar
cascades that form a sub set within the more general isomorphism tree I
described.
My position is that all these attempted single valued elegant cascades run
into the complexity limit imposed by Chaitin's incompleteness in such a way
as to be unable to halt absent contradiction. The contradiction can only
be cured by an increase in complexity of the FAS governing the
cascade. The added information can only come from outside the
cascade. Not from outside the isomorphism tree.
If I have it right your approach is that the UD has an inherent true random
noise content if some of its sub strings define universes sufficiently rich
in logic to be subject to other incompleteness mechanisms especially if
observers are present. Actually I think you may be saying that the entire
UD is rich enough in any manifestation to necessarily posses this
characteristic. With this I would agree if a minimum possible complexity
UD can be shown to meet the threshold of these incompleteness mechanisms.
If you rely on the scanner transporter duplicator for this demonstration I
consider it to be the same as my E/N alternation.
However, in my approach the UD [all of them] are a small part of the
Everything.
> Well, I hope, because for teachers may and june are
>terrible :-(.
Perhaps this will allow me a chance to catch up with your logic posts.
Yours
Hal
Re: Provable vs Computable
Hal Ruhl wrote:
>This is particularly due to my stand that true random noise is
>inherent in each universe within the Everything.
Remember that true random noise appear in the UDA because we don't know
in which computation ("universe") we belong. So random noise does
not need to be added. It is something we cannot avoid from our first
person point of view of embedded observer.
It is the point where Schmidhuber disagrees so much. We will come back
on this question soon. Well, I hope, because for teachers may and june are
terrible :-(.
Bruno
Re: Provable vs Computable
Dear Juergen:
My little isomorphism tree did not transmit very well.
The second column heading should read: "horizontal on this page isomorphic
links"
The third should read: "the isomorphically linked string"
All the binary strings should be under the third column.
Yours
Hal
*<
::: > * =0
{*} =
1
{*,{*}} =
10
{*,{*},{*,{*}}} =
11
: :
:
: :
:
=
11001110111...0110
||
active
and inactive
page
"vertical" isomorphic
links
to this string
:
:
=
11100110111...0011
||
active
and inactive
page
"vertical" isomorphic
links
to this string
:
:
etc.
The page horizontal isomorphic links on the right hand side are the usual
ones.
The page vertical isomorphic links are the ones I have used in my
model. Each of these vertical isomorphic links [there can be more than one
per string] uses a portion of the string to which it links to define its
self contained FAS. The rest of the string determines the associated state
of the vertical isomorphism. The current state of a vertical isomorphism
is the active link for that isomorphism. All inactive links are either
past or future states of isomorphisms. The self contained FAS of a
particular vertical isomorphism determines which links are acceptable
immediate successor states of that isomorphism. Depending on the nature of
the FAS there could be more than one acceptable immediate successor state
for that isomorphism.
The symbol "< ::: >" indicates that the isomorphism tree structure - "The
Everything" - vanishes upon occasion and the anisomorphic null set - "The
Nothing" - resumes. This is the E/N alternation. Neither the anisomorphic
null set nor the isomorphism tree since they both contain no information
can internally address the unavoidable question of their own
durability. This bilateral incompleteness drives the E/N
alternation. The alternation since it destroys any record of the previous
mix of active/inactive vertical isomorphic links causes a new random
active/inactive mix each time the isomorphism tree resumes. This avoids a
"selected" structure to The Everything.
In order for a vertical isomorphic link to transfer to another string both
the current link and an acceptable successor link must be simultaneously
active. The transfer inactivates the prior link. Vertical isomorphic
links driven active or inactive by the E/N alternation absent an active
acceptable successor is simply in stasis. For a transfer to take place
both the current and acceptable successor link must be simultaneously
active. It is the transfer that is an "event" to an isomorphism.
Are UD's or other such string generating machines vertical isomorphic
links? I think so if I understand them correctly. Simply concatenate all
the output strings of a UD so that successor vertical link shifts are just
based on very localized regions of the overall string.
I also think this process is similar to "observer moments" if "observer
moment" means active links.
Since the null set just "is" then the process may satisfy the idea that the
Everything just "is".
It is the isomorphic tree that undergoes change.
Now I see nothing wrong with FAS that have a "do not care" content to their
rules determining acceptable successor links.
My goal is to try to show that universes [vertical isomorphic links] that
can support SAS have at least some "do not care" content in the rules. To
do this I turn to Chaitin and explore the viability of deterministic
cascades. By deterministic I mean each state has only one possible prior
and one possible successor - t
Re: Provable vs Computable
Dear Bruno: At , you wrote: >Juergen Schmidhuber wrote > > >Which are the logically possible universes? Max Tegmark mentioned > >a somewhat vaguely defined set of ``self-consistent mathematical > >structures,'' implying provability of some sort. The postings of Bruno > >Marchal and George Levy and Hal Ruhl also focus on what's provable and > >what's not. > >You know that Hal Ruhl doesn't distinguish computability and provability, >so >it is open for me if his approach is nearer your's or mine. Awhile ago I did identify computability and provability, but now see provability as a subset of the computable. But is computable enough and computable where [inside/outside a particular universe considerations]? I place myself nearer your work Bruno and also that of Russell Standish. This is particularly due to my stand that true random noise is inherent in each universe within the Everything. This does not mean that I consider the isomorphic tree I described in another post to contain randomly generated strings [the horizontal isomorphisms will be a list of all strings] but rather that isomorphic links to these strings necessarily IMO have some random nature in the determination of acceptable successor links - the evolution of the active/inactive mix I described. Also IMO the logics we may find in our universe derive from this foundation not the reverse. If one looks at the frequency of events versus event size in the "large" event region we find that some experiments in our universe produce slopes close to but slightly higher than -1. The field observations I have seen produce somewhat higher valued slopes going towards -2. A slope of -1 would as I see it indicate a deterministic cascade. However, one must also examine the "small" event part of the spectrum. Here a slope of +1 IMO indicates a deterministic cascade but the two ends of the spectrum need not be symmetric. Our logic IMO is in the "large" event part of the spectrum and a thought experiment as well [not even a "lab" let alone the "field"] and therefore likely to be almost deterministic in our universe. IMO the incompleteness types we have discovered do not have a large impact on our everyday use of our reasoning methods. Yours Hal
Re: Provable vs Computable
Dear Juergen:
I am not so much interested in provability as I am in whether or not the
"noise" in a universe's evolution is pseudorandom or random and forging an
Everything that was as free of information [selection] as possible. I try
to use incompleteness in various forms to show that as far as an individual
universe is concerned the noise comes from outside that universe.
I believe that many proposals on this list are more in agreement than we
might at first glance think. Below I use the approach of producing numbers
using the null set to try to demonstrate this.
Using the symbol * to represent the null set and {} to represent sets with
elements:
horizontal
the
on this
page isomorphically
isomorphic
linked
anisomorphic links string
*<
::: > * =0
{*} =
1
{*,{*}} =
10
{*,{*},{*,{*}}} =
11
: :
:
: :
:
=
11001110111...0110
||
active
and inactive
page
"vertical" isomorphic
links
to this string
:
:
=
11100110111...0011
||
active
and inactive
page
"vertical" isomorphic
links
to this string
:
:
etc.
The page horizontal isomorphic links on the right hand side are the usual
ones.
The page vertical isomorphic links are the ones I have used in my
model. Each of these vertical isomorphic links [there can be more than one
per string] uses a portion of the string to which it links to define its
self contained FAS. The rest of the string determines the associated state
of the vertical isomorphism. The current state of a vertical isomorphism
is the active link for that isomorphism. All inactive links are either
past or future states of isomorphisms. The self contained FAS of a
particular vertical isomorphism determines which links are acceptable
immediate successor states of that isomorphism. Depending on the nature of
the FAS there could be more than one acceptable immediate successor state
for that isomorphism.
The symbol "< ::: >" indicates that the isomorphism tree structure - "The
Everything" - vanishes upon occasion and the anisomorphic null set - "The
Nothing" - resumes. This is the E/N alternation. Neither the anisomorphic
null set nor the isomorphism tree since they both contain no information
can internally address the unavoidable question of their own
durability. This bilateral incompleteness drives the E/N
alternation. The alternation since it destroys any record of the previous
mix of active/inactive vertical isomorphic links causes a new random
active/inactive mix each time the isomorphism tree resumes. This avoids a
"selected" structure to The Everything.
In order for a vertical isomorphic link to transfer to another string both
the current link and an acceptable successor link must be simultaneously
active. The transfer inactivates the prior link. Vertical isomorphic
links driven active or inactive by the E/N alternation absent an active
acceptable successor is simply in stasis. For a transfer to take place
both the current and acceptable successor link must be simultaneously
active. It is the transfer that is an "event" to an isomorphism.
Are UD's or other such string generating machines vertical isomorphic
links? I think so if I understand them correctly. Simply concatenate all
the output strings of a UD so that successor vertical link shifts are just
based on very localized regions of the overall string.
I also thi
Re: Provable vs Computable
>> Such a virtual reality or universe is perfectly well-defined. > Such a universe would violate Bell's inequality theorem. Quantum randomness > cannot be simulated by hidden variables. We have to move beyond > realism..to get a model of objective reality we must first develop a > model of consciousness. > George Levy. The picture seems even more fuzzy. There are, also, classical dynamical systems, and classical fields, violating Bell's inequality. And we can realize quantum entanglement ... by classical computers (experiment done). And, above all, can we use (as Bell did) the classical probability theory in the quantum domain? - scerir -- quant-ph/0007019 Non-locality and quantum theory: new experimental evidence Luigi Accardi, Massimo Regoli Starting from the late 60's many experiments have been performed to verify the violation Bell's inequality by Einstein-Podolsky-Rosen (EPR) type correlations. The idea of these experiments being that: (i) Bell's inequality is a consequence of locality, hence its experimental violation is an indication of non locality; (ii) this violation is a typical quantum phenomenon because any classical system making local choices (either deterministic or random) will produce correlations satisfying this inequality. Both statements (i) and (ii) have been criticized by quantum probability on theoretical grounds (not discussed in the present paper) and the experiment discussed below has been devised to support these theoretical arguments. We emphasize that the goal of our experiment is not to reproduce classically the EPR correlations but to prove that there exist perfectly local classical dynamical systems violating Bell's inequality. -- quant-ph/0007005 Locality and Bell's inequality Luigi Accardi, Massimo Regoli We prove that the locality condition is irrelevant to Bell in equality. We check that the real origin of the Bell's inequality is the assumption of applicability of classical (Kolmogorovian) probability theory to quantum mechanics. We describe the chameleon effect which allows to construct an experiment realizing a local, realistic, classical, deterministic and macroscopic violation of the Bell inequalities. -- quant-ph/9606019 A Proposed Experiment Showing that Classical Fields Can Violate Bell's Inequalities Patrick Suppes (Stanford University, USA), J. Acacio de Barros (Federal University at Juiz de Fora, Brazil), Adonai S. Sant'Anna (Federal University at Parana, Brazil) We show one can use classical fields to modify a quantum optics experiment so that Bell's inequalities will be violated. This happens with continuous random variables that are local, but we need to use the correlation matrix to prove there can be no joint probability distribution of the observables. -- quant-ph/0007044 The Violation of Bell Inequalities in the Macroworld Diederik Aerts, Sven Aerts, Jan Broekaert, Liane Gabora We show that Bell inequalities can be violated in the macroscopic world. The macroworld violation is illustrated using an example involving connected vessels of water. We show that whether the violation of inequalities occurs in the microworld or in the macroworld, it is the identification of nonidentical events that plays a crucial role. Specifically, we prove that if nonidentical events are consistently differentiated, Bell-type Pitowsky inequalities are no longer violated, even for Bohm's example of two entangled spin 1/2 quantum particles. We show how Bell inequalities can be violated in cognition, specifically in the relationship between abstract concepts and specific instances of these concepts. This supports the hypothesis that genuine quantum structure exists in the mind. We introduce a model where the amount of nonlocality and the degree of quantum uncertainty are parameterized, and demonstrate that increasing nonlocality increases the degree of violation, while increasing quantum uncertainty decreases the degree of violation. --
Re: Provable vs Computable
scerir wrote: >Juergen Schmidhuber wrote: >> Which are the logically possible universes? Max Tegmark mentioned >> a somewhat vaguely defined set of "self-consistent mathematical >> structures'' implying provability of some sort. The postings of Bruno >> Marchal and George Levy and Hal Ruhl also focus on what's provable >> and what's not. >> Is provability really relevant? Philosophers and physicists find >> it sexy for its Goedelian limits. But what does this have to do with >> the set of possible universes? > >Many people think that if a formal statement is neither provable nor >refutable, then it should be considered neither true, nor false. >But it is not that way that we - normally - use the term "true". >Somebody wrote: "Suppose that I have a steel safe that nobody >knows the combination to. If I tell you that the safe contains 100 >dollars - and it really does contain 100 dollars - then I'm telling the >truth, whether or not anyone can prove it. And if it doesn't contain 100 >dollars, then I'm telling a falsehood, whether or not anyone can prove it." >(A multi-valued logics can deal with statements that are either definitely >true or definitely false, but whose actual truth value may, or may not, >be known, or even be knowable.). That is basicaly the difference between classical logic with gap between proof and truth, and intuitionistic logic, or constructive logic, which equivote truth and provability. And that is something which will be translated in the language of the machine ... It is part of the proof I explain currently. I have begin the explanations of logic with classical logics. But other logics are fundamental in the derivation (mainly intuitionist and quantum logics). Bruno
Re: Provable vs Computable
Juergen Schmidhuber wrote >Which are the logically possible universes? Max Tegmark mentioned >a somewhat vaguely defined set of ``self-consistent mathematical >structures,'' implying provability of some sort. The postings of Bruno >Marchal and George Levy and Hal Ruhl also focus on what's provable and >what's not. You know that Hal Ruhl doesn't distinguish computability and provability, so it is open for me if his approach is nearer your's or mine. The difference relies more between averaging on the (local) set of consistent extensions defined in the whole UD*, or finding a priori defining universes probabilities, or Universal prior. I communicate with the sound lobian machine because I agree (in arithmetic) with the laws of exclude middle, and all classical logic. Provable plays the role of thoroughly verifiable "scientific" communication. (BTW *you* were the guy asking for formalisation!). Now we are working at the metalevel (as George aptly remarks) and we will interview the machine on its self-reference abilities. Recall the goal consists in translating UDA in a language interpretable by a sound UTM. And UDA is a self-referential thought experiment. It will happen that the incompleteness phenomenon will force us to take into account the nuance between []p and ([]p & p) and ([]p & <>p) in the discourse of the machine. They will correspond to provable p, knowable p, probability(p)=1. Knowable will give rise to intuitionist logic and probability 1 will give quantum logic. Probability(p) 1 will really be no more that 1) there is consistent extension, 2) p is true in all those consistent extension. (Only in an ideal frame we have []p -> <>p, remember that in the cul-de-sac world []p is always true). >Is provability really relevant? Philosophers and physicists find >it sexy for its Goedelian limits. But what does this have to do with >the set of possible universes? Wait and see. Remember I told George we have not yet really beging the proof. The hard and tiedous thing is to arithmetise the provability predicate. I will define knowledge and "observable" (in the UDA sense) *in* the language of the machine and I will show that the observable propositions obeys some quantum logic. I will only consider the case observable with a probability one. This will give a concrete purely arithmetical interpretation of a quantum logic. The probabilities are taken on the set of relative consistent (UD accessible) extensions, and by consistent I just mean "-[]-" with [] Goedel's provability predicate. (so you can guess the role of provability). The UD will be translated in the form of the set of all (true) \Sigma_1 sentences. >Is provability really relevant? Philosophers and physicists find >it sexy for its Goedelian limits. But what does this have to do with >the set of possible universes? It has to do with the origin of the belief in universe(s) once we bet we do survive digital substitution. >I believe the provability discussion distracts a bit from the >real issue. If we limit ourselves to universes corresponding to >traditionally provable theorems then we will miss out on many formally >and constructively describable universes that are computable in the >limit yet in a certain sense soaked with unprovable aspects. Actually, provability is just a step in my derivation (and we have still not begin to discusse it! nor to define it). We have just seen some modal logic which have a priori nothing to do with provability. You are still anticipating. It is a good thing you are open to unprovable aspects, and it makes weirder you are not open to uncomputable aspects. (Although I know provability is relative and computability is absolute (Church's Thesis) Do you really believe than one of us limit "universes" to sets of provable theorems. I am myself just defining the local *discourse* of a machine-scientist. >Such a virtual reality or universe is perfectly well-defined. At some >point each history prefix will remain stable forever. Even if we know p >and q, however, in general we will never know for sure whether some q(n) >that is still zero won't flip to 1 at some point, because of Goedel etc. >So this universe features lots of unprovable aspects. I have no problem with that. As you should know from our earlier discussion. Remember that the big role in my work comes from (G* minus G), which is a logic of the *unprovable* statements. G ang G* will be defined formally soon, but you can also consult Solovay 1976 or Boolos ... By logic here I mean a well defined set (of formulas) logically closed for modus ponens. >Note also that observers evolving within the universe ... The UDA shows that such an expression has no meaning. The movie graph (or Maudlin's Olympia) illustrates how non trivial the "mind-body" problem is with comp. I am aware there is something very hard to swallow here. But it is a consequence of comp. > ...may write >books about all kinds of unprovable thing
Re: Provable vs Computable
George Levy wrote: > > > [EMAIL PROTECTED] wrote: > > > Example: a never ending universe history h is computed by a finite > > nonhalting program p. To simulate randomness and noise etc, p invokes a > > short pseudorandom generator subroutine q which also never halts. The > > n-th pseudorandom event of history h is based on q's n-th output bit > > q(n) which is initialized by 0 and set to 1 as soon as the n-th element > > of an ordered list of all possible program prefixes halts. Whenever q > > modifies some q(n) that was already used in the previous computation of > > h, p appropriately recomputes h since the n-th pseudorandom event. > > > > Such a virtual reality or universe is perfectly well-defined. > > Such a universe would violate Bell' inequality theorem. Quantum randomness > cannot be simulated by hidden variables. We have to move beyond > realism..to get a model of objective reality we must first develop a > model of consciousness. > > George I disagree. Hidden variables are indeed excluded, but that doesn't mean that deterministic models proposed by Jürgen or 't Hooft are in conflict with Bell's theorem. In the case of the model proposed by 't Hooft, you have a universe that is very chaotic. Quantum mechanics arises in a statistical description of the theory. Particles such as electrons, photons etc. don't describe the degrees of freedom of the original deterministic theory, but rather they arise only in the statistical description of this theory. In other words: Mach was right in not believing that atoms exist. In the case of the two slits experiment, a hidden variable theory would tell you through what particular slit an elecron travelled, and this is not possible. Okay, but does the electron exist in the first place? I think not. The electron is just a mathematical tool that allows you to calculate probabilities and is unphysical, just like virtual particles and ghosts in Feynman diagrams. Why believe in electrons, but not in the Fadeev-Popov ghost? Saibal
Re: Provable vs Computable
[EMAIL PROTECTED] wrote: > Example: a never ending universe history h is computed by a finite > nonhalting program p. To simulate randomness and noise etc, p invokes a > short pseudorandom generator subroutine q which also never halts. The > n-th pseudorandom event of history h is based on q's n-th output bit > q(n) which is initialized by 0 and set to 1 as soon as the n-th element > of an ordered list of all possible program prefixes halts. Whenever q > modifies some q(n) that was already used in the previous computation of > h, p appropriately recomputes h since the n-th pseudorandom event. > > Such a virtual reality or universe is perfectly well-defined. Such a universe would violate Bell' inequality theorem. Quantum randomness cannot be simulated by hidden variables. We have to move beyond realism..to get a model of objective reality we must first develop a model of consciousness. George
Re: Provable vs Computable (post not done)
Sorry, some how my mailer decided I wanted to send this. Clearly it is not done. Dear Juergen: I am not so much interested in provability as I am in whether or not the "noise" in a universes history is pseudorandom or random and forging an . At 5/4/01, you wrote: >Which are the logically possible universes? Max Tegmark mentioned >a somewhat vaguely defined set of ``self-consistent mathematical >structures,'' implying provability of some sort. The postings of Bruno >Marchal and George Levy and Hal Ruhl also focus on what's provable and >what's not. > >Is provability really relevant? Philosophers and physicists find >it sexy for its Goedelian limits. But what does this have to do with >the set of possible universes? > >I believe the provability discussion distracts a bit from the >real issue. If we limit ourselves to universes corresponding to >traditionally provable theorems then we will miss out on many formally >and constructively describable universes that are computable in the >limit yet in a certain sense soaked with unprovable aspects. > >Example: a never ending universe history h is computed by a finite >nonhalting program p. To simulate randomness and noise etc, p invokes a >short pseudorandom generator subroutine q which also never halts. The >n-th pseudorandom event of history h is based on q's n-th output bit >q(n) which is initialized by 0 and set to 1 as soon as the n-th element >of an ordered list of all possible program prefixes halts. Whenever q >modifies some q(n) that was already used in the previous computation of >h, p appropriately recomputes h since the n-th pseudorandom event. > >Such a virtual reality or universe is perfectly well-defined. At some >point each history prefix will remain stable forever. Even if we know p >and q, however, in general we will never know for sure whether some q(n) >that is still zero won't flip to 1 at some point, because of Goedel etc. >So this universe features lots of unprovable aspects. > >But why should this lack of provability matter? It does not do any harm. > >Note also that observers evolving within the universe may write >books about all kinds of unprovable things; they may also write down >inconsistent axioms; etc. All of this is computable though, since the >entire universe history is. So again, why should provability matter? > >Juergen Schmidhuber http://www.idsia.ch/~juergen/toesv2/
Re: Provable vs Computable
Dear Juergen: I am not so much interested in provability as I am in whether or not the "noise" in a universes history is pseudorandom or random and forging an . At 5/4/01, you wrote: >Which are the logically possible universes? Max Tegmark mentioned >a somewhat vaguely defined set of ``self-consistent mathematical >structures,'' implying provability of some sort. The postings of Bruno >Marchal and George Levy and Hal Ruhl also focus on what's provable and >what's not. > >Is provability really relevant? Philosophers and physicists find >it sexy for its Goedelian limits. But what does this have to do with >the set of possible universes? > >I believe the provability discussion distracts a bit from the >real issue. If we limit ourselves to universes corresponding to >traditionally provable theorems then we will miss out on many formally >and constructively describable universes that are computable in the >limit yet in a certain sense soaked with unprovable aspects. > >Example: a never ending universe history h is computed by a finite >nonhalting program p. To simulate randomness and noise etc, p invokes a >short pseudorandom generator subroutine q which also never halts. The >n-th pseudorandom event of history h is based on q's n-th output bit >q(n) which is initialized by 0 and set to 1 as soon as the n-th element >of an ordered list of all possible program prefixes halts. Whenever q >modifies some q(n) that was already used in the previous computation of >h, p appropriately recomputes h since the n-th pseudorandom event. > >Such a virtual reality or universe is perfectly well-defined. At some >point each history prefix will remain stable forever. Even if we know p >and q, however, in general we will never know for sure whether some q(n) >that is still zero won't flip to 1 at some point, because of Goedel etc. >So this universe features lots of unprovable aspects. > >But why should this lack of provability matter? It does not do any harm. > >Note also that observers evolving within the universe may write >books about all kinds of unprovable things; they may also write down >inconsistent axioms; etc. All of this is computable though, since the >entire universe history is. So again, why should provability matter? > >Juergen Schmidhuber http://www.idsia.ch/~juergen/toesv2/
Re: Provable vs Computable
> Juergen Schmidhuber wrote: > > Which are the logically possible universes? Max Tegmark mentioned > > a somewhat vaguely defined set of "self-consistent mathematical > > structures'' implying provability of some sort. The postings of Bruno > > Marchal and George Levy and Hal Ruhl also focus on what's provable > > and what's not. > > Is provability really relevant? Philosophers and physicists find > > it sexy for its Goedelian limits. But what does this have to do with > > the set of possible universes? scerir writes an enjoyable version on the last part of the quote. Let me address the first part about "possible universes". Of course Juergen was cautious and included "logically" in his phrase. "Logically" most likely refers to human (on this list: even mathematical) logic. Do we really think that human (math) logic is the restrictive principle for nature? What we see (what we want to see?) seems to point to that, but do we see'em all? Isn't "possible" what we don't see or understgand or realize? Didn't our horizon (logic, math) increase over some time? Are we at the end? In considering plenitude/multiverse, does it make sense to select part of it (maybe a small, unimportant segment only)? Even if we cannot develop "knowledge" about the rest, we should not deny its "possibility" of existence. The farthest from this list would be a closed mind! John Mikes
Re: Provable vs Computable
Juergen Schmidhuber wrote: > Which are the logically possible universes? Max Tegmark mentioned > a somewhat vaguely defined set of "self-consistent mathematical > structures'' implying provability of some sort. The postings of Bruno > Marchal and George Levy and Hal Ruhl also focus on what's provable > and what's not. > Is provability really relevant? Philosophers and physicists find > it sexy for its Goedelian limits. But what does this have to do with > the set of possible universes? Many people think that if a formal statement is neither provable nor refutable, then it should be considered neither true, nor false. But it is not that way that we - normally - use the term "true". Somebody wrote: "Suppose that I have a steel safe that nobody knows the combination to. If I tell you that the safe contains 100 dollars - and it really does contain 100 dollars - then I'm telling the truth, whether or not anyone can prove it. And if it doesn't contain 100 dollars, then I'm telling a falsehood, whether or not anyone can prove it." (A multi-valued logics can deal with statements that are either definitely true or definitely false, but whose actual truth value may, or may not, be known, or even be knowable.). - scerir
