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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