A Critique of a Formalist-Mechanist Version of the Justification of Arguments in Mathematicians' Proof Practices

In a recent article, Azzouni has argued in favor of a version of formalism according to which ordinary mathematical proofs indicate mechanically checkable derivations. This is taken to account for the quasi-universal agreement among mathematicians on the validity of their proofs. Here, the author subjects these claims to a critical examination, recalls the technical details about formalization and mechanical checking of proofs, and illustrates the main argument with aanalysis of examples. In the author's view, much of mathematical reasoning presents genuine meaning-dependent mathematical characteristics that cannot be captured by formal calculi. ‘…there is a conflict between mathematical practice and the formalist doctrine.’ [Kreisel, 1969, p. 39]