On 27 Oct 2014, at 21:08, meekerdb wrote:
On 10/27/2014 9:53 AM, Bruno Marchal wrote:
What remains amazing is the negative amplitude of probability, but
then that is what I show being still possible thanks to the
presence of an arithmetical quantization in arithmetic, at the
place we need the probabilities.
I don't recall you having shown that. Can you repeat it.
By a result of Goldblatt, you have that QL proves A iff the modal
logic B proves some transformation T(A), defined by
T(p) = []<>p (Which Rawling and Selesnick called the quantization
of p), p atomical sentence (that is arithmetical sentence without
quantifier and variable, like s(s(0) + s(0) = s(s(s(0))).
T(A & B) = T(A) & T(B)
T(~A) = [] T(~A)
This makes the modal logic B a classical modal rendering of quantum
logic, a bit llike Gödel and others saw that S4 was a classical
epistemological rendering of intuitionist logic.
The Kripke semantics of B is symmetry and reflexivity: B's main axiom
are A -> []<>A, and []A -> A, the accessibility relation is
symmetrical and reflexive. Note that the complement relation with
alpha R beta iff NOT (alpha R beta) gives a proximity relation, and an
abstract orthogonality condition. If the arithmetic material
hypostases, defining "the probability one" for the FPI on the sigma_1
sentences (roughly, the UD*) dis not have such an abstract
orthogonality conditions, then classical comp as I defined it (in
AUDA) would be refuted.
Now, I showed that the arithmetical hypostases (S4Grz1, X1*, Z1*) does
verify that orthogonality conditions, on the sigma_1 sentences,
despite the modal logic is bot a weakening of B (we loss the closure
for the necessitation rule), and a strengthening we get new axioms
(like we get the new Grz for the internal solipsist, the first person,
or Pltinus' universal soul).
Those logic verifies the two main axiom of B, and suggest that the
bottom physics, which sums on all sigma_1 sentences, is indeed
symmetrical, linear, reflexive, ... let us say that we can hope for
some "Gleason theorem" there, which would determine entirely the
measure on the directly accessible sigma_1 state.
Do you show that the Hilbert space of QM must be over the complex
numbers? Or does your proof allow quaternion or octonion QM?
You know I share with Ramanujan (and thanks to him) some love for the
number 24, so I would be happy if the Octonion, a famous divisor of 24
could play some rôle, but that, I would say, has to wait for the
"Gleason theorem" of the introspective physics of the universal, and
Löbian, machine.
I do have argument for octonions playing some key role, but I keep
them for myself, because if the number theorist find physics before
the theologians, theology could sleep again for one millennium or
more. I can imagine a number theoreticalism capable of eliminating
consciousness too (thats' why I am reassured that David Nyman avoided
that trap consciously or explicitly so).
Come on, Brent, the greeks discovered the Automobile of "Science", to
explore deep questions, they use it from
-500 to +500. After that it was declared illegal in Occident, so to
speak. We get half of it back at the enlightenment period, and I am
just pointing that computer science + the computationalism offers the
second half, or at least a second half. I am only the guy who tries to
restart the Automobile, that is science including theology, the mother
of math and physics (before the political pseudo-religious
recuperation).
You ask me if QM is octonionic ? There are now two questions: what
does nature say? and what does the universal machine see in arithmetic
from inside. And we can compare, even if today it requires "hard math"
like the modal logic of arithmetical (and non arithmetical) self-
references. Good textbooks exists, as I have given references.
Keep in mind we try to figure what happens, not how to make bombs and
rockets. We want just a coherent picture of the possible whole, and
this without eliminating persons, consciousness, etc.
Bruno
Brent
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