Some Mathematicians Are Not Turing Machines

Abstract

A certain mathematician M, considering some hypothesis H, conclusion C and text P, can arrive at one of the following judgments: (1) P does not convince M of the fact that since H, it follows that C; (2) P is the proof that since H, it follows that C (judgment of the type "Proved"). Is it possible to replace such a mathematician with an arbitrary Turing machine? The paper provides a proof that the answer to the question is negative under the two following conditions: (1) M is faultless, namely his judgment "Proved" always implies that since H, it actually follows that C; (2) M recognizes a certain P' as the correct proof of the fact that for certain H' and C', if H', then C' (where P', H', and C' are stated in the paper).