From Osher Doctorow

The one-direction-at-a-time or no-direction-at-a-time tangential
physicists and mathematicians who run mainstream QCD and QG tend to
center coordinates (when they use them) of all types at "zero" which is
usually some axiomatic "physical center" that ranges from a point
particle or little string to some ill-defined quantum "volume"
field-particle region that's replaced point particles, and then their
equations if any tend to dampen with distance from those "objects".

Probable Influence/Causation (PI)

uses the function 1 + y - x and its
vector generalizations instead of the Euclidean, Euclidean-like, or the
Mainstream Non-Euclidean metrics/distance functions, and 1 + y - x is a
partial one-sided inverse of most of those metrics, so it involves an
"opposite" of "farness" which is "nearness" or "proximity". It is, in
simple language, boundary-oriented rather than "central-oriented",
although it can also formulate centers.

On boundaries of finite-volume objects for example in Euclidean
n-dimensional space R^n or E^n, probabilities are zero if a continuous
random variable is associated to them, and this yields x = 0 and since
y < = x we get y = 0 for 0 < = y < = x < = 1, so the proximity function
1 + y - x = 1 which is maximal on its [0, 1] interval.

It is true that 1 + y - x is maximal whenever y = x.

Since y = P(AB)
and x = P(A), it follows that 1 + y - x is maximal whenever P(AB) =
P(A) which except for a set of probability 0 occurs iff AB = A which
occurs iff A is a subset of B or else P(A) = P(B) = 0. Since A is
exerting probable influence/causation on B, it could be argued that A
can be taken as the "center" of B or any singleton ("point") or set
inside B. However, if B is a continuous/connected set in R^n, there
is nothing distinguishing about the center of B unless B is symmetric
(which it usually isn't), and it would require specifying uncountably
many points in B to get a geometric picture of B even if B has a finite
volume if B is not symmetric. For finite volume B, however, although
the boundary of B is lower dimensional than B and has uncountably many
points if the dimension of B is 2 or more, the boundary of B is a
proper subset of B which itself specifies what we usually think of as
the geometric shape of B. Every point on that boundary is relevant to
the description of the shape of B, unlike every point of B itself.

If the volume of B is infinite, then it also makes more sense to base
coordinates at "infinity" for asymmetric B, and we can "append" the
center of B for symmetric B to the set of relevant coordinates just as
in complex analysis the reverse "point at infinity" is appended to
usually finite sets. The argument is longer, but even intuitively it
makes sense.

So both central and boundary "perspectives" and central and boundary
forces make sense in PI, and this is where the center-based gravitation
and the infinity-based opposite-directed forces for example make sense
in my recent QCD and QG threads.

Osher Doctorow

From Osher Doctorow