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Gödel believed the human brain used non-mathematical reasoning (i.e. inexpressible with a Turing machine) to derive the axioms and thus could "see outside" of any particular axiom schema.

The Church-Turing thesis throws cold water on this: since the human body (including brain) is describable by a finite system of Schrödinger equations, and these equations can be solved numerically by a Turing machine, the human process of creating an axiom scheme should be Turing-computable. But some recent results on very large finite numbers (busy beaver) suggest there may be a subtlety here, e.g. complexity blows up to the point that it takes far more energy than the sun to simulate one human.

The more interesting subtlety: for a physically meaningful result you would need to define configuration space very carefully, e.g. not screwing up the boundary consitions or causal order of subsystems. Perhaps defining this is actually not computable, and after every delta(t) in a computer simulation, a human has to check the physics and redefine certain parameters of the system. Solving the Schrodinger equation numerically is certainly Turing-computable, but the process of ensuring that solution is physically meaningful isn't even slightly formalized. It may be unformalizable.

by hardbass11 hours ago|

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Some problem being too chaotic or computationally expensive to perform is a completely different thing that claiming something supernatural. And there is hardly any reason for computers to be confined to silicon digital logic, using coprocessors is a standard procedure, so whatever element if any which exhibits hypercomputation (which is highly doubtful as the universe barely even reaches even a small fraction of the possibilities of turing machines let alone beyond them) or otherwise more efficient classical computation, we can isolate the element and use it as a chip.

by Diogenesian9 hours ago|

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To be clear the limitation here isn't silicon logic, it's theoretical logic (specifically general recursive functions). It sure seems like any possible computation can be expressed as a general recursive function, but that's a scientific thesis, not a mathematical theorem. As we have yet to formalize "define a physical system" it is possible that this task isn't actually expressible in 21st century mathematical logic. I suppose some custom hardware which doesn't use logic at all might help, but then Gödel's theorem wouldn't apply at all. (Likewise with modal logic.)