Gödel, Nagel, minds and machines

One could no more disprove this modified version of the idealized mechanist’s thesis than the version considered by Gödel, et al., simply by applying the mechanist’s empiricist argument. Nevertheless, it is difficult to conceive of any formal system of the sort with which we are familiar, from Peano Arithmetic (PA) up to Zermelo-Fraenkel Set Theory (ZF) and beyond, actually underlying mathematical thought as it is experienced.

And mathematical practice certainly supports the conclusion drawn by Nagel and Newman that “mathematical proof does not coincide with the exploitation of a formalized axiomatic method,” even if that can’t be demonstrated unassailably as a consequence of Gödel’s incompleteness theorems.

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