From Osher Doctorow
The fact that the golden mean and Fibonacci numbers involve sqrt(5)
and the Pell Numbers involve sqrt(2), and the relationships of both to
other prime (and non-prime) square roots, raises the question of
whether Probable Causation/Influence (PI) can clarify the Causation
involved in square roots, which is important for example in a deeper
analysis of SR's gamma or beta or inverse gamma or beta factor sqrt(1
- v^2/c^2).
We'll examine quantities of the form:
1) 1 - sqrt(n), n nonnegative integer
Let's let y (the "effect" variable in PI) be sqrt(n) normalized or a
generalization of it, and let x (the "cause" variable in PI) be
defined as:
2) x = (1 - y)^2, y = sqrt(n) normalized or generalization of it
So for example if y = sqrt(n), then x is:
3) x = (1 - sqrt(n))^2 if y = sqrt(n)
Now consider version 2 of PI:
4) P ' (A-->B) = 1 + y - x, y = P(B), x = P(A), y < = x
Consider the equation:
5) P ' (A-->B) = 1 + y - x = y
which implies that x = 1 ("deterministic" Cause up to sets/events of
probability 0) and is equivalent to x = 1.
We know that in conditional probability, the "corresponding" equation
is:
6) P(B|A) = y/x = y, x not 0, y = P(AB), x = P(A)
which coincides with Statistical Independence of A and B or, when
generalized to random Variables, of X and Y.
However, P ' (A-->B) = y is just equivalent to x = 1, so it does not
define Statistical Independence but rather "deterministic up to
probability 0" Causation.
Let's expand (2):
7) x = (1 - y)^2 = 1 - 2y + y^2
which is equivalent to:
8) 2y = 1 + y^2 - x
or in other words:
9) P(x --> y^2) = 2y = 1 + y^2 - x
In order to fit the pattern of (5), we require that:
10) 2y = (approximately) y^2
which implies that:
11) y = approximately 0 or 2, x = 1
When y = sqrt(n), we get from (11) n = 0 or 4, x = 1.
Ordinarily, we'd throw out y = 2 as a solution since y < = x, but in
previous threads I found that [0, 1], [1, 2], [2, 3], etc. can be
regarded as different phase domains except for their endpoints (which
are in 2 phases), so if we remove the restriction that y < = x and
also generalize beyond usual probabilities, then y has jumped into a
different phase domain from [0, 1], namely [1, 2] or [2, 3]. Notice
that in the second phase domain, [1, 2], both y ( = 2) and x ( = 1)
are in the same domain.
I'll try to come back to this next time.
Osher Doctorow
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