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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this is where Penrose disagrees: Penrose believes that the mind and by extension the universe are implemented in non constructive mathematics. (that should imply that he thinks that QM is the wrong foundation for physics, since it is a [slightly hyper]computational model)
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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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The issue is not about whether one can do math using a brain or on a computer, it is that no matter how powerful your formal system is, there will be axioms it can not prove. Whether we decide to use them can be cultural. A computer deciding to invent an axiom isn't a good idea
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