It seems strange to say that Godel didn't know it at the time -- Godel's work was part of the effort to mechanize logic, Hilbert's program, and the axiomatization effort started in response to contradictory calculus theorems, and followed Frege, Russel, Whitehead, et al.
Or rather, Godel's work showed that Russel's efforts to create a consistent and complete foundation for mathematics was fundamentally insufficient. Turing extended this work, by providing an explicit model of a calculating machine, to show that no algorithm could effectively determine the truth -- even without providing a proof -- locking the door on Hilbert's program that Godel had slammed shut.
Regardless, the entire point of Godel's work was exploring the relationship between our ability to "reason" and our ability "calculate" or "perform rote tasks".
(Missing details of the narrative aside -- it was a really good read.)