On 22 Nov 2012, at 18:38, Stephen P. King wrote:
How exactly does the comparison occur?
By comparing the logic of the observable inferred from observation
(the quantum logic based on the algebra of the observable/linear
positive operators) and the logic obtained from the arithmetical
quantization, which exists already.
How does the comparison occur? I will not ask what or who is
involved, only how. What means exists to compare and contrast a pair
of logics?
The logic exists, because, by UDA, when translated in arithmetic,
makes a relative physical certainty into a true Sigma_1 sentence,
which has to be provable, and consistent. So the observability with
measure one is given by []p = Bp & Dt & p, with p arithmetical sigma_1
(this is coherent with the way the physical reality has to be
redefined through UDA). Then the quantum logic is given by the
quantization []<>p, thanks to the law p -> []<>p, and this makes
possible to reverse the Goldblatt modal translation of quantum logic
into arithmetic.
Comparison is used in the everyday sense. Just look if we get the
quantum propositions, new one, different one, etc.
Comp seems to necessitate all possible physical worlds in an
equiprobable way.
?
Does not comp require all possible 1p to exist?
Comp makes all possible 1p existing in arithmetic, from the possible
arithmetical pov.
There is a deep problem with notions of priors as it seems that we
cannot escape from the problem of subjectivity as we see in the
(so-called) anthropic principle: each observer will necessarily
find itself in a world what has laws compatible with its
existence. It seems to me that the observational act itself is a
breaking of the perfect symmetry of equiprobability of possible
worlds.
?
But this claim implies violence to the idea of a 3p.
I found at http://higgo.com/qti/Mallah.htm an exchange between
Mallah and Standish that seems to illustrate this problem:
"Russell Standish: The predictions can easily depend of the
'picture' but must be consistent with each other. Let me give a
simple example: In one picture, observer A decides to measure the
spin of an electron in the x direction. In the other, observer B
decides to measure the spin of the electron in the y direction.
Observer A will see the spin of the electron aligned with x axis,
and Observer B will see it aligned with the y axis. Both
observations are correct in the first person picture of that
observer. A "person" with the third person perspective, sees
observers A and B as inhabiting separate `worlds' of a multiverse,
each with appropriate measure that can be computed from Quantum
Mechanics.
Jacques Mallah: On the contrary, this is a textbook example of the
way I said it works. The theory predicts some measure distribution
of observers; an individual observer sees an observation drawn
from that distribution. There are no different sets of predictions
for different pictures, just the measure distribution and the
sample from it.
Russell Standish: It sounds to me like you don't think the
prediction changes according to what the observer chooses to
observe? An electron cannot have its spin aligned with the x axis
and the y axis at the same time. Once the experimenter has chosen
which direction to measure the spin, the history of that
particular is observer is constrained by that fact, and the
predictions of QM altered accordingly. This is true both in MWI
and the Copenhagen interpretation, and is the "spooky" nature of
QM. I used to think that QM gave predictions in terms of
distributions, and that because one didn't see isolated particles,
rather ensembles of such particles, I didn't see a problem. The
properties of an ensemble are well defined. However, the ability
of experimenters to isolate a single particle, such as a photon,
or an atom, means we have to take this "spookiness" seriously."
The idea of a 3p cannot be applied consistently to the notion
of a 'person' or observer if one is considering the 1p of
observers in separate 'worlds' of a multiverse unless, for
example, A and B have observables that mutually commute and thus
have some chance of being mutually consistent and capable of being
integrated into a single narrative. I think that this problem is
being overlooked because the problem of Satisfiability is being
ignored.
?
I hope that we can agree that there is at least an illusion of a
physical world that 'we' - you, me, Russell, .... can
consider... Is it necessarily inconsistent with comp?
? ? ?
Not at all. The whole point of UDA is in explaining why the
physical reality is unavoidable for the dreaming numbers, and how
it emerges from + and * (in the "number base"). It is indeed a
first person plural product, with the persons being all Löbian
machines, etc.
I am coming at the idea of a 'physical reality' as an emergent
structure and not some pre-defined ordering.
Good.
Comp gives the complete algorithm to extract bodies and physical
laws, making comp testable, even if that is technically difficult,
I claim that it is not even technically difficult; it is
impossible for the simple reason that there does not exist
a unique Boolean algebra for all possible 1p.
? (I agree such BA does not exist, but this is exactly what we need
to find a measure theorem à-la Gleason). We need a sufficiently
good quantum logic, and up to now the comp quantum logic fits
rather well.
Gleason's theorem is interesting:
http://en.wikipedia.org/wiki/Gleason%27s_theorem
"For a Hilbert space of dimension 3 or greater, the only possible
measure of the probability of the state associated with a particular
linear subspace a of the Hilbert space will have the form Tr(μ(a)
W), the trace of the operator product of the projection operator
μ(a) and the density matrix W for the system."
We sidestep the problem of how we define the transition from
pure states to density matrices. Andrew's discussion might be seen
as addressing this...
OK.
Why? Because it cannot be proven to be satisfiable(aka globally
self-consistent) by any finite sequence of algorithms.
Completeness and consistency for such cannot be assumed a priori.
?
Do you ever address the question of satisfiability?
Which satisfiability? I use it all the time. p->p is satisfiable by
all interpretation, and this is used all the time. I do not use the
complexity of satisfiability, as if this needed to be used, it has to
be justified by the modal logic extracted from self-reference.
but up to now, it fits remarkably, and that would not have been
the case without QM. That would not have the case if "p->[]<>p"
was not a theorem of the Z1* logics (matter).
Your reasoning is correct only because you are assuming the
impossible to be true a priori: that there exists a solution to
the Satisfiability problem
It exists. "Satisfability" is non tractable, not insoluble. The
first persons don't care "waiting exponential time" by the
invariance of first person experience on delays.
Of course, but an infinite BA requires eternity (infinitely many
steps) to solve its satisfiability problem.
But no machine ever need to do that (and can't). The BA might be
infinite, but not the proposition, unless you are using infinitary
logic, which does not play a big role in comp up to now.
I am not claiming non-solubility; I am pointing out that the
computation of satisfiability must run to obtain a solution,
The 1p depends on truth, not on proof.
otherwise it is false to claim that the solution is accessible.
The UD does "prove", or arithemtic proves, all the true sigma_1
sentences, which is enough for the computations to be emulated. then
the 1p are distirubuted non constructively on that, independently of
the complexity of the proofs. Without this, no measure problem.
And with no measure problem, you lost the reduction of physics to
computer science.
It is a profound mistake to claim that the existence of the largest
prime number defines the exact sequence of numerals that would
enumerate that prime number.
You need to decide in which base you write it, and then it is defined.
But we don't need this.
Similarly, the mere possibility of satisfiability of a BA
Satisfiability concerns sentences, not BA.
cannot be used to argue about the particular distribution of
propositions of the BA.
You are considering first persons in the eternal and ideal case,
but that does not connect omniscient machines to finite human brains.
The connection is explained by the UDA.
This is the challenge to Plato and Parmenides, how do we bridge
between the Realm of Truth and the world of appearances?
By the realtion between machines' belief and reality. With comp,
today, we can use the work of Tarski and others.
We could make claims forever but showing a proof requires physical
effort.
And time, money, if not a sense of public relation. But that is
relevant at some meta-meta-level.
There are no shortcuts to knowledge. You seem to be OK with the idea
that knowledge can obtain 'for free'.
Free of physics, yes. Free of math? No. You need to postulate enough
to get Turing universality.
Perhaps I am mistaken, but it seems that you are assuming the
impossible to be real.
I don't. Unless you come back with the idea that 1+1=2 requires a
physical world, or thing like that.
*and* that it is accessible for any finitely expressible logical
structure.
It is accessible, but then I don't see at all the relevance of this.
Please explain how it is accessible.
You were using the term. I am the one asking the question here.
Bruno
http://iridia.ulb.ac.be/~marchal/
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