On 12/16/2013 6:28 PM, LizR wrote:
On 17 December 2013 14:54, meekerdb <meeke...@verizon.net <mailto:meeke...@verizon.net>>
wrote:
You asked where does the unreasonable effectiveness come from. Maybe I
should have
asked what you thought Wigner was referring to. I don't think he was
referring to
'all possible mathematics' like Tegmark was. Or even all computable
functions as
Tegmark has more recently. Wigner was probably still assuming a continuum.
He obviously wasn't referring to all possible maths, as you pointed out most of it
doesn't have any obvious effectiveness.
Shannon's theory of channel capacity turns out to use a form of Boltzmann's
entropy. Is that 'unreasonable effectiveness' or a real relation between
transmitting information and randomness in statistical mechanics.
I suspect it shows up a deep connection between the two subjects, which isn't too
surprising in this case.
It's not all or nothing. There was mathematics, like Fourier transforms
and Hilbert
space, that had already been invented before von Neumann formulated QM in
terms of
them. But the subsequent interest in QM inspired Gleason's theorem and the
Kochen-Specker theorem and the concept of POVMs and rigged Hilbert space.
William
Thompson proposed a vortex theory of matter which could be seen as the
forerunner of
braid and knot theory which developed as 'pure' math and then came back to
physics
in string theory.
As to whether they are incommensurate I'm not sure what that means. They
may have
contradictory axioms so that if you tried to axiomatize Newtonian mechanics
and
quantum mechanics together you'd get contradictions. But if you just take
them as
pure math, real valued differential equations and Hamiltonian functions vs
complex
Hilbert space and Hamiltonian operators then there's no contradiction
because
they're about different domains. Riemannian geometry is a consistent
theory which
include Euclidean geometry as a special case. But in a physical theory
about the
geometry of spacetime the geometry is either Euclidean or it's not.
My point, such as it is, is that we can use the same maths for both the Newtonian domain
in which things behave "roughly according to common sense" and the quantum domain in
which they very much don't. The fact that the same maths applies to these domains, which
as you pointed out are wildly different, at least implies that maths has an independent
(or at least physics-domain-independent) existence. Hence it probably isn't just
something we made up to work in one domain (roughly the Newtonian).
I don't see that it follows. Just like Shannon's information and Boltzmann's entropy, the
domains are very much related so it's no surprise that we can carry over some math
developed for Newtonian physics and apply it to quantum physics. After all the former
should be a kind of statistical mechanics of the latter.
Brent
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