Re: Dynamical Systems and Expansion-Contraction

>>From Osher Doctorow

Suppose that we do a Thought Experiment with an observer at a point
that's about to expand based on a Decision mechanism of some sort which
the observer can determine. We'll suppose that the point can
distinguish directions of expansion (infinitely many), amounts of
expansion, and speeds of expansion in each direction. If Decision
mechanisms seem disturbing, think of them as just input-output
connected by a program or black box or electrical network or whatever.

Although there might not be an "outside" in some sense as with the Big
Bang at its beginning instant, the point might have a "picture" of an
outside in its program as well as a "picture" of an inside even though
supposedly it is an irreducible point.

The decision of how to expand would seem to depend on what the point
via its program regards itself as, what it regards its environment as,
and what it regards as the relationship between/among them.

If it thought of itself as a sphere and all of its outside as a sphere
(whether these are really accurate or not), then there is some way in
which it "makes sense" for the point to expand via:

1) rho = kt

where rho is the distance of expansion in any direction and k is a
constant and t is time. Here it is assumed that there is no "urgency"
to expand differently now vs later, no "memory" of having done that
before, etc. A spherical coordinate system would fit (1) very nicely.

If, however, there is deviation from (1) in the Decision, then we would
be tempted to attribute it to something additional if we were the
observer. The point might have a picture of a hyperboloid, ellipsoid,
paraboloid, hyperbolic paraboloid, etc., and we could attribute the
expansion to "simplicity of the equation and picture" since these all
have equations that are rather simple, and so on. For each of these
and cones, we could use particular coordinate systems and attribute
them to the point's "picture", e.g., conic coordinates, etc. That's
all rather symmetric or symmetrical, and we could isolate "symmety" as
a principle of the program, although we still would be lacking a basis
for choosing one versus another symmetric object.

In the case of asymmetric expansion, especially one not following any
known simple equation, and worse and worse if the expansions in
different directions differ more and more and the rates in different
directions differ more and more, the questions of expansion may look
unrelated, but they're all about expansion, and they're all about what
determines how something changes in time from some past state to some
future state. To make a system of equations (even infinitely many,
maybe uncountably many) that can handle either one type of asymmetric
expansion or many (all?) types, it seems that we would have to go
beyond merely restricting things to symmetry or to particular geometric
or topological "pictures" or closed form equations.

The one thing that fits these requirements is the operation of
(probable) influence or (probable) causation which takes an influencing
set A, an influenced set B, an "influence connection" written -->, and
a probability P( ) or P', and relates them via:

2) P(A-->B) = P(A' U B) = 1 + P(AB) - P(A)
3) P' (A-->B) = 1 + P(B) - P(A) where P(B) < = P(A)
4) P(A<-->B) = P{(A-->B)(B-->A)} = P(AB U A' B' ) = P(AB) + P(A' B' )

We can regard these as changing in time with or without the subscript t
for time on P, P_t or Pt or more simply just P or P' if one keeps track
of the fact that they're changing in time and calculates the derivative
dP(A-->B)/dt, etc.

Osher Doctorow

.

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