From Osher Doctorow

The unit sphere in spherical coordinates is:

1) rho = 1

Let's convert this to Probable Influence/Causation by subtracting rho
from both sides:

2) 0 = 1 - rho

Now comes the strangest part. Add 0 to both sides, which although it
doesn't change anything numerically, has the following familiar form:

3) 0 = 1 + 0 - rho = P(rho --> 0) = 1 + y - x (y = P(AB) = 0, x = P(A)
= rho)

Equation (3) says that the probability of the radius rho of the sphere
influencing the center 0 of the sphere is 0, which by (1) occurs when
the radius of the sphere equals 1.

A Probable Influence/Causation of 0 means that 1 and 0 have "lost
contact causally". This is what we expect at infinity, but not at
radius 1 if radius 1 just is another numerical value.

Readers who followed my postings at geometry.research in previous
years, and to my recollection some postings here as well, may recall
that Probable Influence/Causation (PI) is a one-sided partial inverse
of the Euclidean distance-function/metric.

Define the "Proximity"
p(x, y) between x and y from the viewpoint of x as:

4) p(x, y) = 1 + y - x, 0 < = y < = x < = 1

The smallest distance between x and y occurs when x = y, in which case
the Euclidean distance is 0, but the quantity p(x, y) = 1. The
biggest distance between x and y for y < = x occurs when y = 0 and x =
1, precisely the case of (3) above, but equally important, for these
values p(x, y) = 1 + 0 - 1 = 0 whereas the Euclidean distance is 1.
Therefore, for the one-sided case y < = x specified by (4), the
proximity function and the Euclidean distance function are "inverses"
at 0 and 1 (one is 0 when the other is 1, and vice versa). It turns
out that p(x, y) can be generalized to vectors x = (x1, x2, ..., xn), y
= (y1, y2, ..., yn) with x and y in (4) replaced by the arithmetic mean
of the coordinates of x and of y respectively, and everything still
goes through the same with the new expression pn(x, y) or p_n(x, y) = 1
+ y-bar - x-bar where y-bar is the arithmetic mean of y coordinates and
x-bar is the arithmetic mean of x coordinates.

Although the geometry.research readers reacted almost entirely with
silence since they didn't recognize novelty any more than physics
people,

the unit sphere/ball

in Euclidean distance equals the
"infinite" proximity/Probable Influence/Causation sphere/ball (coded as
1) in proximity representation.

There may well be "something" beyond the unit sphere/ball in Euclidean
distance, but whatever it is, it's

arguably
in a different phase. 1
represents "infinity" in our phase.

Osher Doctorow

From Osher Doctorow