Stephen P. King (stephenk1@home.com)
Sat, 04 Sep 1999 15:14:22 -0400
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
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