Quantum Gravity Via Expansion-Contraction 41.0: Probable Causation of Tensors, Matrices, Determinants

From Osher Doctorow mdoctorow@xxxxxxxxxxx

In the last few Sections of this thread, I showed that alternating sign
series derive from Probable Influence/Causation's additivity, in the
sense that:

1) 1 + (1 + y - x) - (1 + u - v) = 1 + (y + v) - (x + u)

which generalizes to any number of terms. Let's look more closely at
this equation using the usual Probable Influence/Causation language, in
which:

2) P(A --> B) = 1 + y - x, y = P(AB), x = P(A)
3) P(C --> D) = 1 + v - u, v = P(CD), u = P(C)

Since the left hand side of (1) is also of a similar form to a Probable
Influence/Causation, it is rather easy to see that it is in fact:

4) P{(P(A-->B) --> P(C-->D)}

But until now, we have been taking P(A-->B) to have A and B (random)
set/events, while here we have generalized this to probabilities
replacing random set/events. This is fine, since generalizing is a
quite useful method in physics and mathematics.

What (4) represents is in some ways like composition of functions in
Category Theory, but much more specific to + and -. In words, (4)
expresses "the probability that the Probable Influence of A on B
influences the Probable Influence of C on D". There is indeed
something "composite" about this, but at the same time very explicitly
additive-subtractive.

Determinants of square matrices are (finite) alternating series and
with traces more or less characterize the matrix (they are invariants
of the matrix). They are quintessentially "informational" or as I
prefer the term "Knowledge" to refer to Semantic rather than just
Syntactic Information from first principles. So it is arguable that
the REASON WHY determinants alternate in sign is that Probable
Influence/Causation in its "compositional additive-subtractive" form
alternates in sign.

In other words, the Causation inherent in matrices via determinants is
essentially (generalized) Probable Influence/Causation.

All this works only if determinants and matrices have entries in [0, 1]
since Probability is restricted to [0, 1]. Since normalization of
variables is so common, there should be no objection to reformulating
matrix equations in terms of [0, 1]-variables, that is to say variables
whose values are between 0 and 1 inclusive.

More mathematically, the alternating sign knocks out the double 1 of
the two terms and adding 1 restores a single 1, making the result a
generalized Probable Influence/Causation in (1) and in (4). The sign
has to alternate to achieve "closure" of Probable Influence/Causation
under addition/subtraction, that is to say to make the result a
Probable Influence/Causation.

To put it simply, alternation of sign is a principle of closure of
Probable Influence/Causation.

We see then that tensors, which have matrix representations which are
different in different coordinate systems, are missing the boat with
regard to determinant Causation properties in each particular matrix
representation. It could be, of course, that all these different
determinant Causation properties (Probable Influence/Causations) fall
into larger families which tensor equations summarize. But now it
begins to be arguable that larger families of matrices in turn reflect
Probable Causation which leads the way and needs to be addressed rather
than "following behind" the tensor "intuition".

Osher Doctorow

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