On Friday, February 27, 2015 at 9:15:40 AM UTC-5, Osher wrote:
> On Thursday, February 26, 2015 at 10:11:06 PM UTC-5, noTthaTguY wrote:
> > quaternions are not commutative
> > ;that is what makes them the aboriginal vector
>
Readers will find papers on gauge groups, GUTs, etc. typically using symmetry groups and other groups typically derived from geometry, which raises the question of how PCI (Probable Causation-Influence) relates to such groups. Recall that "group" in mathematics is not an arbitrary collection of objects but a collection having an identity, an inverse, associative law, etc., using written multiplicatively like xy (x "times" y), although they can be written additively (x "+" y).
A new perspective can be seen from an example that I have given before, but which I did not develop thoroughly. In one dimension, e.g., the real line or the segment [0, 1] of the real line, we have:
1) P'(A-->B) = 1 + P(B) - P(A) = 1 - [P(A) - P(B)] for P(B) < = P(A).
The bracketed expression, however, is precisely one-dimensional Euclidean distance d(P(A), P(B)) for P(A) > = P(B), which is |P(A) - P(B)|.
From (1), therefore, we can generalize:
2) GEOMETRY + NON-GEOMETRY = UNIVERSE, where PCI is included in non-geometry even though it does have one "inverse geometry" indication, namely "nearness" as an opposite of geometric "farness".
But we know from my previous and concurrent threads here that PCI is the (random) set-event-measure analog of Lukaciewicz infinite multivalued logic, so we can go even further:
3) LOGIC (at least, Lukaciewicz) is a subset or subcategory of Non-Geometry.
Osher Doctorow