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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the main claim of AI and cognitive science however is that the brain is a computer, and because all the mathematics we need to account for run on our brains, we can build something that can satisfy all desires a classical mathematician can hope to fulfill with a finite automaton
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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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