Conversation

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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basic train of thoughts: Gödel's incompleteness theorem (and Turing's subsequent adaptation to computational machinery) shows that mathematics itself is incomplete. that was a shock: mathematics is the domain of all formal languages, but mathematics cannot be generated using them
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there is a small branch of mathematics that only accepts statements as true that have actually been constructed. constructive mathematics turns out to be identical to computation. Church and Turing demonstrated that computation contains itself, i.e. we can compute all computers
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constructive mathematics is not time-less like classical math. a function that has not been computed yet does not have a defined value. pi is a function, not a number. the axiom of choice does not hold. true infinities and continua cannot be constructed
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construction means that there has to be a machine that can perform the actual computations. in the strong, intuitionist form, we must actually run the proof, i.e. we need to compress it into a form that actually runs on our machine (which leads to a small foundational crack)
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