[time 141] Re: [time 130] On Pratt's Duality

Stephen P. King (stephenk1@home.com)
Sun, 04 Apr 1999 12:35:42 -0400

Hitoshi Kitada wrote:
> Dear Stephen,
> The parts you quoted from Prugovecki are all well-known in math, and do not
> reflect the special features of the dualism of Chu spaces.
> Best,
> Hitoshi

Dear Hitoshi,

        The fact that they are "all well-known in math" is what makes my task
so difficult. It is only because something is so close and "obvious"
that blinds us to the implications. As a philosopher, it is the
implications of these mathematical formalisms that matters. When we
think of causality, what is it that we mean? We say A causes B; but if
we look past the "fact" and look to the "how" of this fact, we are
forced to face the difficulty of explaining them in a way that is
logically consistent with all other facts available, thus I say: A
causes B iff Y implies X, A encodes information X and B encodes
information Y.
        When we recognize that the Universe is not fixed ab initio by some
divine act, but is constructed by the interactions of finite subsets of
it, we cannot just say "it is so because we observe it to be so", even
though this statement is undeniably true.
        When we come to the problem of consciousness within the Universe, we
have many ideas available with which to construct models. Pratt has
found that the original Cartesian proposal of Mind-body dualism can be
corrected so that it does provide a viable paradigm with which to make
sense of how it is that information and matter can influence each other.
        Perhaps a little toy model would help. I have been working on a
gedankenexperiment for a long time now and am lacking making it
mathematically rigorous. :( We start with the now infamous Maxwell Demon
who attempts to bypass the laws of thermodynamics. We substitute a
computer for the imaginary demon. The computer's "power" is supplied by
using the difference in temperature/pressure/gravitational
curvature/charge/etc. between the two closed and isolated chambers, and
it opens and closes the valve/wormhole/switch/etc. between the two using
a string of bits. For a binary sting reading computer, we think of the
1s as open and 0s as closed. We can easily generalize this to a [0, 1]
n-ary stream, where the binary is a mere special case.
        What makes this idea concrete is that we can think of the two Chambers
as closed manifolds that differ in their clock readings. Thus we can
think of the Demon as "connecting" together two separate instants "in
time". Alternatively, we can think of the pair of spaces that are dual
to each other in the manner "that is well know in mathematics". There is
also a duality existing between the information content of the pair!
        If the two manifolds/spaces are identical, or that there is a complete
bijective mapping between the two (or totally orthogonal : A x X = 0 <->
B x Y = 0), then the computer is unable to derive any power to operate.
The Totality Universe, being a bound state has this property between any
of its proper subsets, thus we arrive to the conclusion that no time
exist at that level. What is interesting, is that if there is no time
there is also no consciousness possible and vise versa!
        Pratt's residuations between Chu spaces and Wegner et al's
infomorphisms between classifications are connections of the same type
but they explicitly take information content into account. We can say
that information entropy *is* thermodynamic entropy, but we are very
wrong! They are only identical (bisimilar/bisimulational equivalence
becomes isomorphism) at the Universe level, were we have infinite
completeness and Eternity to compute the bijection, the problem is that
such a computation is IMPOSSIBLE. In Eternity, there is no free energy,
no differences on matter to encode bits on/in, no shifts, no nothing!

other sources

        In Maxwell's original model the "survival" of the demon was not
dependent on the existence of a difference of some kind between the two
chambers/space/manifolds/etc.. But, here it is obvious that any demon,
being a "real" computer operating over some n-ary stream or a
supernatural entity, has to exist "within" the Universe and thus we see
that its survival does depend on the fact that there exist subsets of
the Universe that are
scattering/evolving/non-static/far-from-equilibrium from each other.
        If there is no way at all allowed to "connect" subsets (LSs) to each
other, then we are back to were Newton started: A Universe at
equilibrium between itself and all of its subsets. When we postulate
that there is no uniquely a priori defined connection, language,
wormhole, mapping, parallel transport, etc. We are allowing for
consciousness and time to be concrete and real. "All is allowed that is
not Forbidden" (Peter Wegner)
        Wegner and Pratt's ideas, for me, speak to this Connection. The
Universe is Both Information and Matter, it is their interaction that is
us. :)

Onward to the Unknown,


PS, Robert's discovery of Phi is a direct manifestation of a fundamental
principle involved here!
> ----- Original Message -----
> From: Stephen P. King <stephenk1@home.com>
> To: Hitoshi Kitada <hitoshi@kitada.com>; Time List <time@kitada.com>; Vaughan
> Pratt <pratt@CS.Stanford.EDU>
> Sent: Sunday, April 04, 1999 1:58 PM
> Subject: [time 129] Re: [time 128] On Pratt's Duality
> > Dear Hitoshi,
> >
> > I don't think that he knows Pratt's work, but...
> >
> > page 33.
> >
> > "An M-coordinate-independent definition of 'covariant vectors'... can be
> > obtained by introducing the *cotangent space* T_x^*M above x as the
> > algebraic dual of T_xM. -i.e.as consisting of real-valued linear
> > functionals w over T_xM. An equivalent definition of cotangent space can
> > also be given in terms of the family of all smooth real-valued functions
> > defined on some neighborhood N_x of x, which forms the basis of the
> > definition (1.5), by introducing for each element f in that family the
> > following linear maps:
> >
> > df: X |-> Xf \element R^1, X \element T_xM. (1.8)"
> >
> > ...
> >
> > Page 63. Note 11
> > "Note that, in the case that g is a matrix that acts by matrix
> > multiplication on the elements of R^n, for its action from the
> > rightthose elements have to be viewed as one-row matrices, whereas for
> > its action on the left they have to be viewed as one-column matrices, so
> > that one mode of such action can be related to the other by taking the
> > transposes of the matrices in question."
> >
> > These properties are consistent with a Chu space. I may have gotten a
> > bit exited and missed something... :) There is much to cover and I am a
> > bit tired. :)
> >
> > Later,
> >
> > Stephen
> >
> > Hitoshi Kitada wrote:
> > >
> > > Dear Stephen,
> > >
> > > ----- Original Message -----
> > > From: Stephen P. King <stephenk1@home.com>
> > > To: Hitoshi Kitada <hitoshi@kitada.com>
> > > Cc: Time List <time@kitada.com>
> > > Sent: Sunday, April 04, 1999 11:19 AM
> > > Subject: [time 127] Re: [time 121] RE: [time 115] On Pratt's Duality
> > >
> > > > Dear Hitoshi,
> > > >
> > > > I apologize for the length of this... :) BTW, I think that Prugovecki's
> > > > formalism already has Chu_2 spaces built in, he just does not understand
> > > > the implications! More on this later... ;)
> > >
> > > At which points or where in the book does Prugovecki include Chu spaces?
> > >
> > > Best,
> > > Hitoshi
> >

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