[time 673] Re: [time 672] Re: [time 667] Stephen's duality theory, Plus Infinite Products

Sun, 5 Sep 1999 00:13:28 EDT


Stephen, restate your duality theory, in 100 words or less,
then I will comment. :-]

> 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. :-)

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

> 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. :-).

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} ???

> 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!

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.

> I see these NOW's as the related observations of other observers (the
> simultaneity frames).

> > 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?
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.

> 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!

> > 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.

> 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.

> > My infinite products are simply candidates for role of the
> > objective multiplicity that subjectively offers the seemingly
> > non-intuitive conclusions drawn above.

> 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").

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



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