Reflections on Concrete Incompleteness

How do we prove true but unprovable propositions? Gödel produced a statement whose undecidability derives from its ad hoc construction. Concrete or mathematical incompleteness results are interesting unprovable statements of formal arithmetic. We point out where exactly the unprovability lies in the ordinary ‘mathematical’ proofs of two interesting formally unprovable propositions, the Kruskal-Friedman theorem on trees and Girard's normalization theorem in type theory. Their validity is based on robust cognitive performances, which ground mathematics in our relation to space and time, such as symmetries and order, or on the generality of Herbrand's notion of ‘prototype proof’.