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
I don’t think that Minsky and Turing have proven that minds are computers, but I think each of them made a very good case, which lead to a whole academic field.
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None of what you just wrote is about "we can use constructive mathematics to run classical mathematics". I guess your argument is that computers are based on constructive math and can be intelligent and thus think about classical mathematics. Okay, but...
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... there are purely mathematical ways to study the extent to which constructive math can "run" classical math. These avoid a detour through more speculative territory and let one prove a bunch of theorems, both positive and negative, about this question.
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What did Turing say about human minds? In 1948, he wrote "Intelligent Machinery" http://www.alanturing.net/turing_archive/archive/l/l32/L32-001.html … which was never published. He writes "I propose to investigate the question as to whether it is possible for machinery to show intelligent behavior"
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Turing's summary is not unlike present day Deep Learning: "the investigation centres round an analogous teaching process applied to machines .. the idea of an unorganized machine is define and it is suggested that the infant human cortex is of this nature."
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