Re: Dynamical Systems and Expansion-Contraction

>>From Osher Doctorow

Let's stop here and ask: is it true that if geometry or topology gives
you no clues, then the next clue comes from probable influence or
probable causation?

In my opinion, the answer is yes, and in rather a curious way.
Birkhoff causation is embodied in differential equations, in particular
the derivative operator which takes a particular point at a particular
time and relates it to an "infinitesimally later or earlier" point in
time. There are also partial derivatives and derivatives not with
regard to time but space or even other variables, but notice a curious
Hint:

Hint 1. If geometry and topology give no knowledge/information in
spatial terms, then only time is left as either a geometry or
topological knowledge/information.

But Probable Influence (PI) is the "quintessence" of time-related
expressions, as we can see for example by comparing the set (A-->B) and
the proposition (a-->b) where the latter says either "if a then b" or
"a implies b" for a, b propositions. In logic, we know that:

1) (a-->b) = ~(a ^ ~b) = ~a V b (read: "either not a or b")

In set theory, we have correspondingly with the first = sign as a
definition:

2) (A-->B) = (AB' )' = A' U B (read: either "outside A" or "in B" )

with A, B sets/events. In the logical form, you have a proposition a,
and regard it as a premise in mathematical and physics and engineering
contexts, and some theorem has a conclusion b, and theorems tell you
that when you see or have a, then b holds or you have b. For example,
"When you have a right triangle with longest side h and other two sides
a, b, then h^2 = a^2 + b^2." Although it seems to just mean that
under the conditions of the premise the conclusion holds, it actually
has a Knowledge-Information sense that the Knowledge of the premise is
enough to obtain Knowledge of the conclusion or that the key causal
factors of the conclusion are in the premise. In fact, under the truth
or Material Implication interpretation, the truth of the premise is
enough to establish the truth of the conclusion.

In a computer program, you would have to have h, a, b, and the
Knowledge or Information that you have a right triangle, before being
able to output the conclusion h^2 = a^2 + b^2.

Now in set/events, propositions can generalize to anything. The
proposition a which says that "object A is a triangle" corresponds to a
set A (or we could label it A* ) such that A is a triangle. The
proposition "John runs" corresponds to the set of all spacetime and/or
additional points such that "John runs" applies to them, including
interestingly the set of all random or sample points in a probability
space for which John runs with a probability of 1.0 on a probability
scale of 0.0 to 1.0.

When we in addition put a probability on these things, like P(A-->B),
then we're talking about "Probable Influence," "Probable Causation".

There isn't any other clue - not in arithmetic and not in algebra. The
"mother of all Lie Algebras" and Fermat's Last Theorem give us not on
iota of clues, for example.

Osher Doctorow

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