From Osher Doctorow
Under rather plausible assumptions on probabilities, the Macroscopic
domain appears to exert Causation on the Microscopic domain more than
vice versa, in the following sense.
Theorem. Let the probability of a set/event/process A, represented as
P(A), increase with the Lebesgue measure m(A) of the set A. Then:
1) If B is a subset of A or A is a subset of B up to sets of
probability 0, then P(A-->B) = 1 (that is to say, is optimized) only
when the "smaller" subset equals the larger one up to sets of
probability 0, which means that the set with larger Lebesgue measure
(and hence larger volume) directs the Causation. The same with P
' (A-->B), where as usual P(A-->B) = 1 + P(AB) - P(A) and P ' (A-->B)
= 1 + P(B) - P(A) with P(B) < = P(A).
2) If B and A are not subsets of each other (that is, neither is a
subset of the other), but their intersection is not null (so they are
not disjoint) and the probability of the intersection is not 0, then
P(A-->B) cannot be optimized at 1 (because P(AB) = P(A) iff A is a
subset of B up to sets of probability 0, but this contradicts the
assumption that neither A nor B is a subset of the other). There is
no restriction apparently on P ' (A-->B).
3) In the modification of (2) above for which P(AB) = 0, then we have
P(A-->B) = 1 iff P(A) = 0 which is equivalent to the "extreme"
Holographic Principle in the sense that only sets/events/processes of
0 Lebesgue measure/volume are Causal and neither positive volume
microscopic nor positive volume macroscopic sets are Causal. This
would invalidate both GR and Quantum theory, although technically
things like planar waves (of A. Bohm et al's Rigged Hilbert Space and
Nested/Lattices of Hilbert and Banach Spaces) would "survive" as
Causal, as would strings and points and branes. There is no
restriction apparently on P ' (A-->B).
It should be noted that since sets/events/processes which are
"infinitely thin" in some Euclidean dimension, such as points, lines,
curves, planes, surfaces in 3 dimensions, have Lebesgue measure 0,
then they must have probability 0 by the above assumptions or even by
more general assumptions, so it would not make sense to postulate that
probability decreases as Lebesgue measure increases for example.
Osher Doctorow
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