From Osher Doctorow
The Birkhoff Causation of x on y is given by dy/dx. Although often x
is taken to be time t or some function of time, the notion is also
extended to spatial functions in simultaneous scenarios, etc.
We have:
1) Dx(log(x)) = d(log(x))/dx = 1/x, x > 0
2) Dxx(log(x)) = -1/x^2
3) Dxxx(log(x)) = 2/x^3
and so on. Since each (n + 1)st derivative is the derivative of the
nth derivative, we have a Causal sequence from "bottom up" which stays
paradoxical at x = 0, precisely where the MacLaurin Series is supposed
to be "centered". In fact, we have infinity, -infinity, infinity,
-infinity, etc. as alternating extended-real-values (the extended real
valued line, R# or R^#, is R and/or infinity or the point at infinity
and arguably -infinity). This extends to the Taylor Series expanded
about x = k for real k, again paradoxical in the analogues of (1) to
(3) above at x = k.
As if this isn't bad enough, looking at Dxxxx(log(x)) gives us
-3*2*1*x^(-4) where * is real multiplication, and we get the general
term (+/-)n!/x^(n+1) which is the multiplicative inverse of the Causal
expression x^n/n! up to x and the sign where x^n/n! is the general term
of exp(x) as an infinite series. So we're going in the opposite
direction to Probable Influence/Causation (PI) as well as wildly
oscillating from + to -.
Shouldn't we be happy with an alternating series or sequence in view of
PI and determinants and so on? Not if the "Cause" itself alternates
from + to - infinitely. Alternation has the effect of closing PI in
"additive" expressions like 1 + p1 - p2 + p3 - ... It is a
mathematical requirement. Alternation of Cause is a bizarre physical
and mathematical notion. And logarithms don't survive this scrutiny.
Osher Doctorow
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