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From: Osher Doctorow on 13 Jul 2010 04:42
From Osher Doctorow

Causation occurs through time, whether explicitly as in the Cause A
operating on the Effect B from a prior time to a later time, or
implicitly when the Cause A and Effect B operate simultaneously (the
latter often referred to as (non-spurious) Correlation)).

Yet we might nevertheless ask: Why time? Why not space or something
else? Here we come to a rather curious insight: we need something
like "time" to compare Causation to, in which a way that "time" is the
OPPOSITE of Causation in a sense, because "time" INEVITABLY UNWINDS
WITHOUT DEVIATION! So in the Newtonian Derivative dy/dt for some
variable y, reflecting the change in y in time, we are really doing
something like approximately taking the dimensional or dimensionless
ratio of a "Caused y" to an "Uncaused t" at least in the most
physically meaningful contexts. Then y plays the role of an Effect,
and the ratio approximately plays the role of a Cause although that
could in a sense be attributed to t also even though it is in a sense
itself "Uncaused".

To better see what I am saying, let us consider the Logistic
Differential Equation, which is a special case of the Riccati
Differential Equation, where the former is:

1) dy/dt = ky(1 - y), where y is in [0, 1].

Let us consider the possibility that the actual nature of y itself may
be such as to clarify Causation, or roughly speaking that there is a
type of y which is more deeply an Effect than other types of y.

In particular, let y be the "radius" or "principal radius" or "scale
factor" of an expanding Universe:

2) y = R (radius, principal radius, scale factor of expanding
Universe).

Then we rewrite (1) as:

3) dR/dt = kR(1 - R), R in [0, 1], scaling R in such a way that for an
unbounded Universe 1 corresponds to infinity.

In such a case, we get the remarkable result that R itself could serve
as "time" because it "unfolds" in the same "direction" at least up to
a certain point. We could also say that the PROJECTION of R on time
t roughly looks like 1, especially in homogeneous and isotropic
expansion - R looks like a "distinguished axis" that grows with time,
at least up to a point, and that distinguishes it from the innumerable
other spatial configurations or curves or surfaces or solids or
whatever.

There is actually both an advantage and a disadvantage in regarding R
as roughly "time", and so let us consider the possibility that R in
the expanding Universe is a SECOND type of time. Formally:

4) PRINCIPLE of Expanding Universe: the Scale Factor R is a second
type of time, while "ordinary" time t is the first type of time.

Now we obtain a remarkable result:

5) dR/dt in (3), that is in the Logistic Differential Equation, is
(locally) maximized with respect to "R-time" at R = 1/2.

This is easy to prove. dR/dt = kR(1 - R) = kR - kR^2. Assume that k
> 0. Then d(dR/dt)/dR = k - 2kR = 0 iff R = 1/2, and the second
derivative is -2k < 0 so we have a relative maximum. There dR/dt =
(1/4)k. It also yields an absolute maximum since at 0 and 1, dR/dt =
0 < (1/4)k, so at the boundaries 0 and 1 of [0, 1], dR/dt is less than
at R = 1/2.

It turns out, however, that R(1 - R) involves two opposite tendencies,
growth or expansion R and contraction or anti-growth 1 - R, just as
Gravitation and Repulsion are opposite tendencies and have a
transition at P(A) = 1/2 from earlier posts here. This link between
them has to be added to the picture.

Osher Doctorow
From: Osher Doctorow on 13 Jul 2010 04:50
From Osher Doctorow

I meant to type: "The second derivative is -2kR < 0," rather than "The
second derivative is -2k < 0."

Osher Doctorow
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