Gödel and Turing: we cannot use classical mathematics to build an interpreter that runs classical mathematics Church and Turing: we can use constructive mathematics to run constructive mathematics Minsky and Turing: we can use constructive mathematics to run classical mathematics
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there is good hope though; the program of constructive analysis in math seems to be successful, and there does not seem to be any obvious limit to the capacities of computer algebra systems
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so, basically, the solution is: while we cannot build math in math, and computation is much weaker than math, every bit of math that we will ever come across will have been built by computation
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This is indeed curiously interesting. Are there formulations of QM that are based on constructive mathematics? In fact, how often do proofs in physics assume an excluded middle?
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t‘Hooft has written up some of his work on that (in his “cellular automaton interpretation of qm”)
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In what way is quantum mechanics hypercomputational?
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the actually computable stuff is all functions from int to int. classical physics is real to real, and qm is real to complex and back to real
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But isn't mind and universe NP-complete? The whole mapping of classical to constructive math breaks up because of P!=NP
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what
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