On 23 Oct 2014, at 04:14, meekerdb wrote:
On 10/22/2014 5:35 PM, Russell Standish wrote:
On Wed, Oct 22, 2014 at 05:23:38PM +0200, Bruno Marchal wrote:
On 22 Oct 2014, at 11:37, Richard Ruquist wrote:
Brent,
That is certainly true for Schrodinger's equations,
but is it also true for matrix theory?
Re: real and complex numbers.
Why would it be different for the matrix. In non relativistic QM,
the position observable in a continuous matrix of complex (and thus
couple of real numbers), same for momentum.
In a quantized space-time, that might be different. But we don't
find good quantization for space-time, I think. Loop gravity seems
to be refuted on this point.
Note also that if Brent is right that QM assume real numbers, it
does not imply that nature (whatever that is) needs them. All what
we can measure are rational numbers. Is there a circle in nature. I
think plausible that circle exists only in the mind of machine in
arithmetic, or they exists as infinite collection of natural numbers
with some relations, etc. Well, it has to like that if we assume
computationalism, and don't eliminate consciousness to save a
primary matter that nobody has seen or even can defined ...
To reiterate on Bruno's point, observables corresponding to x or d/dx
do not exist in reality. Every measurement made is done to some
finite
precision - the number of digits of a numerical readout, or the
needle of
an analogue meter lying between one graduation and the next.
Consequently, the actual observables have eigenvalues and
eigenvectors
drawn from the rational complex numbers. Reals do not exist except as
an approximation that is convenient for doing calculations. And
even then,
countable models of the reals' axioms exist, by virtue of the
Löwenheim-Skolem theorem. These countable models exist in Bruno's
ontology, and suffice for any practical purpose QM is put to.
But by the same kind of positivist attitude there's no reason to
think that every integer has a successor. It's just a convenient
assumption for doing proofs and calculations.
All depends on what you assume.
The idea is: let us start from simple, and if we need something more,
we can add more.
Once we assume the brain is turing emulable, then it is emulated
infinitely often in arithmetic, and the question is more: does this
define a unique universe, a unique multiverse, ... or not?
Arithmetic needs its "Gleason theorem", and the intensional variant of
provability shows that the "machine dreams" might be enough linear and
symmetrical for that. Does all physical realities have to exploit
Unitary = e^i * (self-adjointness)?
Bruno
Brent
Cheers
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