*WDEshleman@aol.com*

*Sun, 5 Sep 1999 00:13:28 EDT*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**WDEshleman@aol.com: "[time 674] Re: [time 664] Reply to NOW/PAST question"**Previous message:**Stephen P. King: "[time 672] Re: [time 667] Stephen's duality theory, Plus Infinite Products"**In reply to:**WDEshleman@aol.com: "[time 667] Stephens duality theory, Plus"**Next in thread:**Stephen P. King: "[time 677] Re: [time 673]Stephen's duality theory, Plus Infinite Products = Infinite Worlds"

Stephen,

[WDE]

Stephen, restate your duality theory, in 100 words or less,

then I will comment. :-]

[SPK]

*> I am still in the formative stage of my thinking of the duality theory.
*

*> I use a strange combination of graph theory, category theory and other
*

*> formalisms that I have picked up here and there...
*

*> This is a very bad sketch of what comes to mind right now. It is not
*

*> even wrong as it is here presented! I intend it to be fixed as we
*

*> discuss the ideas further. :-)
*

[WDE]

100 words, not 1029 words (yes I counted them)!!!

[SPK]

*> Simply put, the Universe U is the totality of Existence and as such is
*

*> infinite (with a "undecidable" cardinality). It is everything that
*

*> exists and this existence is tenseless. The particular subsets u_i of
*

*> the Universe form a powerset U^U that admits any possible decomposition,
*

*> e.g. any possible combination of u_i is contained in U^U. The u_i are
*

*> singletons that may be {0} under certain circumstances that I still need
*

*> to work out. :-).
*

[WDE]

I suspect that the entropy (S) of U^U is k*U or { U * log(U) } ???

And that the probability of being found in u_i is:

P_i = u_i / (U^U) ??? So that,

S = U * log(U) = sum{ -P_i * log(P_i) : i = 1, infinity} ???

[SPK]

*> I believe that the u_i can be considered in two ways, as "independent
*

*> sets" or "complete graphs". These are complements in that the complement
*

*> of a graph G which has all nodes connected to each other is a graph with
*

*> no edges connecting them. I am identifying clusters of material
*

*> "particles" with the independent sets and the information "content" of
*

*> them with the complete graph. I am identifying the complete graph with
*

*> Complete Atomic Boolean Algebras (CABAs). These denote the n-ary
*

*> relations that exist between the u_i. Note that any u_i by itself is
*

*> isomorphic with U.
*

*> I am considering all subsets are dynamical systems when we allow for
*

*> the identification of the elements in one u_i (the 'independent set')
*

*> with the relations (the 'complete graph') among the elements of another
*

*> u_i. This identification is at least symmetrical iff they share an
*

*> element in common. The subsets can evolve to become identical to each
*

*> other and thus U by stepwise changing their relations, this collapses
*

*> the CABAs into singletons as Pratt describes in ratmech.ps. The key is
*

*> "how many steps does it take to collapse all possible CABAs into
*

*> singletons, given an infinite number of them?" (Remember that singletons
*

*> are identified with the subsets of U.) Tentative answer: Forever!
*

*> ...
*

*> Now, the english version of this: The Universe is all that could
*

*> possibly exist. So we get an infinity of "existents" or "possibilities".
*

*> At this level we have no time or motion or change of any type, thus no
*

*> mass, charge, or any other property other than mere existence.
*

*> The Universes is identical to the powerset of its existents and is an
*

*> element there of (as the empty set {0}, I think). The possible subsets
*

*> contained in the Universe can have elements in common. These constitute
*

*> the subsets of the Universe. The allowance that the subsets of the
*

*> Universe can have elements in common allows for the definition of n-ary
*

*> relations between the subsets. I identify the n-ary relations with that
*

*> is called information and the subsets themselves with material
*

*> particles.
*

*> The "evolution" of the subsets of the Universe is given by the
*

*> possibility that the relations can connect subsets, converting them into
*

*> singletons, such that they become identical to the Universe itself. This
*

*> evolution is seem most clearly in thermodynamic entropy, where material
*

*> events evolve such that they become identical to each other. This
*

*> "evolution" has a directionality to it that is identified with the
*

*> "directionality" of time. One key implication of the duality theory is
*

*> that for every change there is a dual one such that the two add to zero
*

*> change, thus the evolution of material particles is dual to the
*

*> evolution of the information "content". This evolution is called logic
*

*> and it defines the chaining of inference of the bits of information.
*

*> The subsets take forever to accomplish the task of becoming identical
*

*> to each other, and thus this gives us an Eternity of time to experience
*

*> "what it is like to experience some sequence of particular
*

*> observations".
*

*> I will quit here before I cause even more confusion!
*

*>
*

*> references:
*

*> http://one.ececs.uc.edu/cs543/4-22.html
*

*> http://www.askdrmath.com/problems/randazzo3.19.96.html
*

*> ***
*

*> http://www.cs.utwente.nl/amast/links/v02/i03/AL0203.html
*

*>
*

*> A First Course in Category Theory
*

*>
*

*> by Jaap van Ooosten
*

*>
*

*> Jaap van Oosten has written a first course in category theory which is
*

*> intended to contain what's presumed knowledge in not too specialized
*

*> papers and theses (in computer science). It's 75 pages long. The
*

*> synopsis is:
*

*>
*

*> 1.Categories and functors. Definitions and examples. Duality.
*

*> 2.Natural transformations. Exponents in Cat. Yoneda lemma. Equivalent
*

*> categories; Set^op equivalent to Complete Atomic Boolean
*

*> Algebras.
*

*> 3.Limits and Colimits. Functors preserving (reflecting) them.
*

*> (Finitely) complete categories. Limits by products and
*

*> equalizers.
*

*> 4.A little categorical logic. Regular categories, regular epi-mono
*

*> factorization, subobjects. Interpretation of coherent logic in
*

*> regular categories. Expressing categorical facts in the logic. Example
*

*> of \Omega -valued sets for a frame \Omega.
*

*> 5.Adjunctions. Examples. (Co)limits as adjoints. Adjoints preserve
*

*> (co)limits. Adjoint functor theorem.
*

*> 6.Monads and Algebras. Examples. Eilenberg Moore and Kleisli as
*

*> terminal and initial adjunctions inducing a monad. Groups monadic
*

*> over Set. Lift and Powerset monads and their algebras. Forgetful functor
*

*> from T-Alg creates limits.
*

*> 7.Cartesian closed categories and the \lambda-calculus. Examples of
*

*> ccc's. Parameter theorem. Typed \lambda calculus and its
*

*> interpretation in ccc's. Ccc's with natural numbers object: all
*

*> primitive recursive functions are representable.
*

*>
*

*> the paper: ftp://ftp.daimi.aau.dk/pub/BRICS/LS/95/1/BRICS-LS-95-1.ps.gz
*

*> ***
*

*>
*

*> B. Roy Frieden's work appear to me as a confirmation of this thinking.
*

*> See Frieden, B. R. & Soffer, B. H., Physics Review E, 52, 2274- (1995))
*

*>
*

*> Echoing Frieden's quote of d'Espagnat's interpretation of E. P.
*

*> Wigner's idea: "...The observer 'consciously' measures, obtaining data
*

*> at the information level I. Corresponding to I is the 'matter' form J.
*

*> These are distinct 'realities in themselves' which 'mutually interact'
*

*> during the information transfer game."
*

*> I am going further that either Pratt or Frieden in that I consider that
*

*> the "world" of any given observer (object) is given by those objects
*

*> that it can bisimulate. Thus is is not the Universe, but some
*

*> approximation thereof! Hitoshi's discussion of the time uncertainty
*

*> principle gets into details of the nature of this asymptotic
*

*> approximation. The key notion is that Fisher information decreases
*

*> ("decreasing ability to estimate") as thermodynamic entropy increases.
*

*> There is much to be worked out, and I must admit, I could be in error!
*

*> I need to understand Matti's "issue" with Frieden's notion!
*

*>
*

[WDE]

When you can say this in 100 words or less, then you will know whether

you are right (consistent) or wrong (inconsistent).

*> > The paper is over 1 mB zipped; thanks
*

*> > for figuring out what I'll be doing for the future. And that is
*

*> > exactly the point I'm trying to make about 1/(1 - x). You may
*

*> > think it is contrary to common sense when I propose that
*

*> > NOW is NOT "pushed" from the PAST by a PAST operator,
*

*> > but that the PAST was attracted to all possible NOW's by
*

*> > an operator that only becomes evaluated in the NOW.
*

*>
*

[SPK]

*> I see these NOW's as the related observations of other observers (the
*

*> simultaneity frames).
*

*>
*

[WDE]

*> > Another way of saying this is that NOW is attracted to
*

*> > all FUTUREs by an operator to be measured in the FUTURE.
*

*>
*

*>
*

*> Oh, I agree completely with this thought! We are "pulled" into the
*

*> future ( a common future)! It is as if we are being pulled toward a
*

*> singularity, all time arrows of those observers that we can communicate
*

*> effectively with are pointing in its direction. In a black hole, all
*

*> motions are restrained to point to the singularity, but this is a
*

*> space-like restriction. In the former case we appear to have a time-like
*

*> restriction. I am curious about how it is that the particular observers
*

*> are given, or in other words, why these observers? I think that is is
*

*> because they have a minimum amount of overlap in their respective sets
*

*> of observables and thus can communicate with each other (via
*

*> bisimulation). BTW, does the bisimulation concept make sense to you?
*

*>
*

[WDE]

I think of your bisimulation as being more analogous to interference

than to interaction.

*>
*

*> > My disclaimer is that this state of affairs is due to a subjective
*

*> > limitation of the observer and by "psychophysical parallelism",
*

*> > all objects are observers.
*

*>
*

[SPK]

*> I also consider this as fundamental! I am a bit more specific in
*

*> thinking that all objects are definable as quantum mechanical Local
*

*> Systems, and as such are observers, if only of nothing at all!
*

*>
*

[WDE]

*> > And, that the underlying objective
*

*> > structure has been programmed to subjectively mimic an
*

*> > attraction to the FUTURE by objectively requiring every
*

*> > augmentation of state in a given world to be accompanied by
*

*> > related augmentations in a majority of other worlds. That is,
*

*> > ( 1 + x ) objectively in multiplicity leads to a subjective
*

*> > reality where the FUTURE seems to attract the PRESENT.
*

*>
*

[SPK]

*> Yes, this follows, for me, from a consideration that the act of
*

*> bisimulation itself, is given in terms of the changes that occur within
*

*> an LS, by the propagator, is "accompanied by related augmentations in a
*

*> majority of other worlds" which are the posets of observations of LSs
*

*> that have at least one state in common. (I think that this relates to
*

*> the formal concept of a fixed point!)
*

*> This corresponds to the idea that the LSs are evolving toward
*

*> equilibrium with each other. Thus, if two LSs are at equilibrium, they
*

*> are identical in information content. Metaphorically put: If two persons
*

*> are exactly the same, their minds are exactly the same.
*

*>
*

[WDE]

*> > My infinite products are simply candidates for role of the
*

*> > objective multiplicity that subjectively offers the seemingly
*

*> > non-intuitive conclusions drawn above.
*

[SPK]

*> I think that the infinite product offer a way to construct coordinate
*

*> systems that are "subjective" yet can be "shared". It is as if each
*

*> framing of observations by any observer (object) is constructed from the
*

*> observations of all of the other objects that it can bisimulate (read
*

*> "interact with").
*

[WDE]

Sounds good to me...except that I prefer "bisimulate (read "INTERFERE with")"

Sincerely,

Bill

**Next message:**WDEshleman@aol.com: "[time 674] Re: [time 664] Reply to NOW/PAST question"**Previous message:**Stephen P. King: "[time 672] Re: [time 667] Stephen's duality theory, Plus Infinite Products"**In reply to:**WDEshleman@aol.com: "[time 667] Stephens duality theory, Plus"**Next in thread:**Stephen P. King: "[time 677] Re: [time 673]Stephen's duality theory, Plus Infinite Products = Infinite Worlds"

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