Probable Influence/Causation Versus Algebraic Geometry/Topology 2: PI vs Scheme

From Osher Doctorow mdoctorow@xxxxxxxxxxx

Probable Influence/Causation (PI) involves two notions: that of a
causal/influence set (A-->B) and that of a probability P( ) on this
set, P(A-->B). They are respectively defined by:

1) (A-->B) = (AB' )' = A' U B
2) P(A-->B) = P(A' U B) = 1 + P(AB) - P(A)

where the right-most equality of (2) comes from the ordinary laws of
probability.

Both (A-->B) and P(A-->B) indicate the (Probable) Causation/Influence
of set/event A on set/event B. They extend to the influence of any
random variable X on any random variable Y (both taken continuous here
for simplicity) via:

3) P(X-->Y) = P(A-->B) where A = {w: X(w) < = x}, B = {w: Y(w) < = y}


What is the "power" or "usefulness" of the very fundamental concept of
"scheme" in algebraic geometry? Whatever it is, it has nothing to do
with (Probable) Causation/Influence. This is not to deny that schemes
do something. But one must search beyond Cause/Effect to find what
they do. Before searching beyond Cause/Effect, physicists need to
think: do I want to search beyond Cause/Effect in understanding and
explaining and predicting physics? Assuming that readers have
followed my previous posts, they know that (Probable) Correlation is
non-spurious "mainstream correlation" and is a type of Cause/Effect.
So, do physicists need anything except (Probable) Cause/Effect and
(Probable) Correlation?

The algebraic topologists and algebraic geometers would like to have us
believe that we need their concepts. But in the case of schemes as a
particular concept, their arguably most "beneficial" use is their
reduction to other concepts none of which reduces to (Probable)
Cause/Effect or Correlation. This is what algebraists love to do.
They love to have "pointers" from supersets to subsets or subsets to
supersets that reduce to other subsets or supersets unexpectedly and in
some sense they "generalize" or "specialize", and the more subsets a
superset "points to", the happier algebraists are. To make things
even less tractable, they prefer to use the word "object" in place of
sets, subsets, supersets.

Surely the subsets or supersets have some "deep" significance? No.
They turn out to be sets of functions, sets of values, sets of roots,
sets of functors, objects, etc. Their only "significance" is that
they've appeared somewhere before in the history of either algebra or
algebraic topology or algebraic geometry. Some of them have admittedly
been "applied" to physics, mostly in incredibly complicated scenarios
that simplify incredibly by using Probable Influence/Causation (see my
various prior threads for this).

Osher Doctorow

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