On 12/16/2013 5:23 PM, LizR wrote:
On 17 December 2013 14:03, meekerdb <[email protected] <mailto:[email protected]>> wrote:On 12/16/2013 4:41 PM, LizR wrote:On 17 December 2013 13:07, meekerdb <[email protected] <mailto:[email protected]>> 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/0521540259Surely 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.
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