On 4/24/2017 7:49 PM, Russell Standish wrote:
On Mon, Apr 24, 2017 at 07:12:38PM -0700, Brent Meeker wrote:
On 4/24/2017 2:15 AM, Bruno Marchal wrote:
This world is 'objective' in the sense that there is
intersubjective agreement about it.
That happens in multi-user video games, and all the multi-user
games are implemented by all universal numbers, with all players
in arithmetic. The only problem is the relative measure, but we
have already that the measure one obeys a quantum logic.
How do we "have" that? Can you derive, from computationalism, that
the description of the world must be in terms of vectors in a
complex Hilbert space?
I looked into that claim, so maybe I can offer a different
perspective. Quantum logics are the logic of events in a complex
Hilbert space that have probability 1, ie the logic of Hilbert
subspaces. For example, if x is the statement that the system is in
subspace X and y the statement that the system is in subspace Y, we
can speak of x∧y being the statement that the system is in the
subspace X∩Y, and x∨y being the statement that the system is in X⊕Y
(X∪Y is not a subspace). It turns out that these logics (apparentally
a family of them, all quite distinct from classical logic) satisfy the
same axioms as Z and X, modal logics describing two of Bruno's hypostases
(that of the believer and the observer IIRC).
If you can explain why the state of systems should be described by
vectors in a complex Hilbert space, the derivation of Born's rule might
follow from Gleason's theorem.
Brent
The significance of all of this? Bit hard to say - it would be nice to
handle the more usual QM statements where probability is less than
1. Also, it is open whether Z describes exactly Birkhoff and Neumann's
quantum logic, or merely something like it.
Nevertheless an intriguing result.
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