**Stephen P. King** (*stephenk1@home.com*)

*Sat, 04 Sep 1999 15:14:22 -0400*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**WDEshleman@aol.com: "[time 673] Re: [time 672] Re: [time 667] Stephen's duality theory, Plus Infinite Products"**Previous message:**Matti Pitkanen: "[time 671] Color constancy, sensory organs as primary experiencers and hologramic brain"**Next in thread:**WDEshleman@aol.com: "[time 673] Re: [time 672] Re: [time 667] Stephen's duality theory, Plus Infinite Products"

Dear Bill,

WDEshleman@aol.com wrote:

*>
*

*> In a message dated 9/3/99 12:34:35 PM Eastern Daylight Time,
*

*> stephenk1@home.com writes:
*

*>
*

*> > Hi Bill,
*

*> >
*

*> > Thanks for the reference. I really appreciate Kevin Brown's work and
*

*> > thinking! BTW, what do you make of my duality idea? (I just found out
*

*> > that Frieden supports duality! See: Frieden, B. R. & Soffer, B. H.
*

*> > Physics Review E, 52, 2274- (1995))
*

*> > Is there any connection between the infinite products and the "logistic
*

*> > map"? (See:
*

*> > http://www.wiwi.uni-bielefeld.de/~boehm/members/klaus/logistic/) The
*

*> > "1/(1-x)" term is key component.
*

*> >
*

*> > Later,
*

*> >
*

*> > Stephen
*

*> >
*

*>
*

*> 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. :-)

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 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!

*> 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?

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

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

*> all objects are observers.
*

I also consider this as fundamental! I an 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").

Onward,

Stephen

**Next message:**WDEshleman@aol.com: "[time 673] Re: [time 672] Re: [time 667] Stephen's duality theory, Plus Infinite Products"**Previous message:**Matti Pitkanen: "[time 671] Color constancy, sensory organs as primary experiencers and hologramic brain"**Next in thread:**WDEshleman@aol.com: "[time 673] Re: [time 672] Re: [time 667] Stephen's duality theory, Plus Infinite Products"

*
This archive was generated by hypermail 2.0b3
on Sat Oct 16 1999 - 00:36:39 JST
*