From Osher Doctorow
So far it looks as though the gap is widening between the Quaternion
Causal representation of the Universe and Quantum Theory which has a
rather "loose" Causation if any. But if we examine quaternions
carefully, they really have 4 components: 3 parts analogous to vectors
(i, j, k and their coefficients), and one scalar part with no
vector.
We have only accounted for 3 of these: y = 1 (or y = 0), y = exp(-t),
and y = exp(t). Where does the 4th component come in?
Let's go back to Fuzzy Multivalued Logics (FMLs) and Pavel Hajek of
the Czech Republic in Metamathematics of Fuzzy Logics, Kluwer:
Dordrecht 1998. There are three Fundamental FMLs which generate
"generalized Boolean Logic", namely Lukaciewicz/Rational Pavelka
(corresponding to PI), Product/Goguen (corresponding to conditional
probability-statistics), and Godel (corresponding to independent
probability-statistics). Any two of these generate generalized
Boolean Logic, essentially FML.
The characteristic of PI (Probable Causation/Influence) that most
concerns us is high dependence in a Causal sense, while conditional
probability is usually much lower and independent probability-
statistics is lowest of all. Since our quaternions already
incorporate PI via y = 1 (or y = 0), let's insert for our 4th
component independent probability-statistics, in which the nearest
analog to PI is:
1) P(B|A) = P(B)
which is to say the probability of B "given" (or "fixing") A is P(B),
the probability of B. Another way of saying this is that B is not
influenced by A when A is fixed. This may seem trivially but
universally true, but it is far from that. For example, you might
think that an engine doesn't influence a car's motion when the engine
is "given", or "fixed", but since an engine can be "fixed" in any
usual type of motion of its parts and its output and input, this is
obviously false.
In the expression:
2) 1 + y - x
for Probable Influence P(A-->B), x is P(A), y is P(AB). In the
alternative expression designated P ' (A-->B), we have:
3) 1 + P(B) - P(A), P(B) < = P(A)
y can be taken as P(B), or to avoid confusion z can be used for
P(B).
So P(B) is put into the quaternion equation:
4) y = P(B) + 1i + exp(-t)j and exp(t)k
or we could interchange P(B) and 1 in their positions:
5) y = 1 + P(B)i + exp(-t)j + exp(t)k
Here on the left hand side y represents just the quaternion as a
composite or mixture of its influences, and so let's replace y by z
and P(B) by y as in the P ' (A-->B) scenario:
6) z = 1 + yi + exp(-t)j + exp(t)k
which therefore reserves x = exp(-t) or exp(t) (contraction or
expansion) by analog with P ' (A-->B) = 1 + y - x. What "drives" the
Universe is exponential expansion or contraction (x) or both, and its
effect is represented by y, the probability of the influenced variable
or set/event or process, but there is an implicit understanding that
the 1 of (6) represents the optimum of P ' (A-->B) and in fact of P(A--
B) both of which have maximum value 1. So the scenario of (6) is
the optimum scenario of the Universe.
Readers may have begun to fidget because quaternions are usually
multiplied in ordinary physics and mathematics. Indeed, many people
assume that the purpose of quaternions, loosely speaking, is to
provide products. But from earlier discussions in this and previous
threads, multiplication and division are not optimal in PI. In
short, quaternions in addition and subtraction are optimal, but not
their products or quotients.
Osher Doctorow
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