On 15 Jul 2017, at 20:54, Brent Meeker wrote:
On 7/15/2017 1:26 AM, Bruno Marchal wrote:
We, the universal machines, are distributed in an infinity of
(arithmetical) computations below our substitution level on which
our First Person experience is undetermined, and the laws to
predict the observable must be given by a statistic on all
computations, or it needs to invoke some metaphysical entity, be it
Matter or God, doing a magical (non Turing emulable nor FPI
recoverable) selection. That entails a "many computations" view of
reality, and the quantum logical aspect of Nature is intuitively
explained already.
Now, why negative amplitude of probability? This remains counter-
intuitive, but is explained by the fact that the statistic are not
on the 3p representational bodies in arithmetic, but on the 1p
views, which necessitate to be defined by machine-self-reference.
In that case, the non intuitive matter features are recovered by
the nonintuitive logic of self-reference and its 1p variants, and
indeed, we get a quantum logic which makes us expect a unique
measure. It obeys, up to now at least, everything needed to have
those negative probability amplitudes, as required.
That's interesting; but I don't see that it follows.
That is technical. It is based on the fact that the "material
hypostases", given by the three logics S4Grz1, Z1*, X1* are enough
close to the logic, known as B (in Modal logic) which provides an
epistemic reading of a minimal quantum logic (a result obtained by
Goldblatt(*)). That gives non commuting observable, having linear and
symmetrical aspect, from which we expect to derive the unique measure
à la Gleason. We should get the unitary representation of the
evolution. More difficult: to get the tensor products. To progress we
must solve mathematical open problem. My goal was , using digital
mechanism, to transform a problem in philosophy (the mind-body
problem) into a problem in mathematics (and let other solve it).
How do you get negative probabilities in your theory and what is
this "statistic on all computations"?
The negative probabilities should come from the uniqueness of the
measure itself coming from the mathematics of the material hypostases.
The statistic on all computations is explained, intuitively, by UDA
(step 0-7): it is the global FPI, that is the FPI on the whole
universal dovetailing, where we have an infinity of 3p extensions (but
the proba are on the 1p extensions). To get the logic of the
observable, we need a notion of "probability one". UDA suggests to
define it by []p & <>t. []p, with p sigma_1, entails that p remains
true in all extensions, and <>t makes sure that there is at least one
extension (elimination of cul-de-sac worlds). We have to add it
because the machine, by incompleteness, cannot prove <>t.
I illustrated this with the cup of coffee promised to all
reconstitutions. P(coffee) = 1, despite the subject cannot know in
advance if it will be american or russian coffee!
Note that "sigma_1" entails computable, and vice versa. Sigma_1 is the
arithmetical definition of "computable" (this is related to a well
known normal form theorem by S.C. Kleene). The sigma_1 p verifies p ->
[]p, which is used to get the symmetries needed for the quantization
(p -> []<>p).
Bruno
(*)
Goldblatt, R. I. (1974). Semantic Analysis of Orthologic. Journal of
Philosophical Logic, 3:19-35. Also in Goldblatt 1993, page 81-97.
Goldblatt, R. I. (1993). Mathematics of Modality. CSLI Lectures Notes,
Stanford California.
Brent
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