The priority of arithmetical truth over arithmetical provability (Q1607593)

As is well known, intuitionism regards the notion of intuitive proof as the fundamental semantic notion on which the essential meaning of a theorem is founded: the proof-conditions are primordial. Following \textit{J. Hintikka} [The principles of mathematics revisited. Cambridge: Cambridge Univ. Press (1996; Zbl 0869.03003)], the author aims to criticize this standpoint as resting on a confusion between dertermining and recognizing the truth of a mathematical proposition and then he wants to defend the priority of truth over provability. Sometimes assimilating ``psychological'' intuitionism with rigorous logical constructivism [Heyting, Bishop, Dummet, Troelstra, Van Dalen], the author considers that Heyting's attempt to clarify what are concrete and logical conditions for a complete, receivable proof destroys the very nature of mathematical proof, oriented towards truth. He also emphasizes that the intuitionistic conception of integers entails a sort of ``computational realism'', which rests on the computational activity of an ideal agent in a ideal spatio-temporal world. For him, such a purely imaginary criterion of receivability of a proof should be rejected. As often happens, the interest in logics and in philosophy of mathematics is not supported by a deep mathematical culture, which makes the debate rather narrow in comparison to the explosion of mathematical knowledge.