The Ultra-Intuitionistic Criticism and the Antitraditional Program for Foundations of Mathematics

Publisher Summary This chapter discusses the ultra-intuitionistic criticism and the antitraditional program for foundations of mathematics. The program aims to banish faith from the foundations of mathematics, faith being defined as any violation of the law of sufficient reason (for sentences). This law is defined as the identification (by definition) of truth with the result of a (present or feasible) proof, in spite of the traditional incompleteness theorem, which deals with very narrow kinds of proofs. The methods of traditional mathematical logic are not sufficient for this program: and the domains of means that are explicitly studied in logic have to be enlarged. The rejection of the axiomatic method does not mean the expulsion of the axioms and rules of inference.