Mechanical Procedures and Mathematical Experience

Source abstract

Abstract Wittgenstein’s terse remark1 captures the feature of Alan Turing’s analysis of calculability that makes it epistemologically relevant. Focusing on the epistemology of mathematics, I will contrast this feature with two striking aspects of mathematical experience implicit in repeated remarks of Kurt Gödel. The first, the conceptional aspect, is connected to the notion of mechanical computability through his assertion that “with this concept one has for the first time succeeded in giving an absolute definition of an interesting epistemological notion”; the second, the quasi-constructive one, is related to axiomatic set theory through his claim that its axioms “can be supplemented without arbitrariness by new axioms which are only the natural continuation of the series of those set up so far.” Gödel speculated on how the second aspect might give rise to a humanly effective procedure that cannot be mechanically calculated and thus provide a reason for his belief that the class of mental procedures is not exhausted by mechanical ones.