On 12/16/2013 5:23 PM, LizR wrote:
On 17 December 2013 14:03, meekerdb <meeke...@verizon.net <mailto:meeke...@verizon.net>>
wrote:
On 12/16/2013 4:41 PM, LizR wrote:
On 17 December 2013 13:07, meekerdb <meeke...@verizon.net
<mailto:meeke...@verizon.net>> wrote:
In a sense, one can be more certain about arithmetical reality than the
physical reality. An evil demon could be responsible for our belief in
atoms,
and stars, and photons, etc., but it is may be impossible for that same
demon
to give us the experience of factoring 7 in to two integers besides 1
and 7.
But that's because we made up 1 and 7 and the defintion of factoring.
They're
our language and that's why we have control of them.
If it's just something we made up, where does the "unreasonable
effectiveness" come
from? (Bearing in mind that most of the non-elementary maths that has been
found to
apply to physics was "made up" with no idea that it mighe turn out to have
physical
applications.)
I'm not sure your premise is true. Calculus was certainly invented to
apply to
physics. Turing's machine was invented with the physical process of
computation in
mind. Non-euclidean geometry of curved spaces was invented before Einstein
needed
it, but it was motivated by considering coordinates on curved surfaces like
the
Earth. Fourier invented his transforms to solve heat transfer problems.
Hilbert
space was an extension of vector space in countably infinite dimensions.
So the
'unreasonable effectiveness' may be an illusion based on a selection
effect. I'm on
the math-fun mailing list too and I see an awful lot of math that has no
reasonable
effectiveness.
Well, maybe my sources are misinformed (Max Tegmark for example). I imagine the
"selection effect" comes about because it's hard to think of completely abstract topics,
so a lot of maths problems will originate from something in the "real world". My point
was that they weren't invented (or discovered) with the relevant physics application in
mind (with exceptions where the physics drove the maths, like calculus).
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.
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.
(The lack of application in some cases would I suppose fit with Max Tegmark's suggestion
that maths is "out there" and different parts of it are implemented as different universes.)
Another answer is that we're physical beings who evolved in a physical
world and
that's why we think the way we do. That not only explains why we have
developed
logic and mathematics to deal with the world, but also why quantum
mechanics seems
so weird compared to Newtonian mechanics (we didn't evolve to deal with
electrons).
There's a very nice, stimulating and short book by William S. Cooper "The
Evolution
of Reason" which takes this idea and develops it and even projects it into
the
future.
http://www.amazon.com/The-Evolution-Reason-Cambridge-Philosophy/dp/0521540259
Surely the maths we "made up" to deal with the "classical" world applies to quantum
mechanics, too? Or are you saying that we had to make up a new load of maths to deal
with QM, and that "quantum maths" is incommensurate with "Relativistic maths" and
"Newtonian maths" ?
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.
Brent
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