Taking formalism seriously

Abstract

Publisher Summary This chapter discusses the concept of formalism. Formalism is that view of the foundations of mathematics that maintains that the natural numbers and other mathematical entities do not exist, that mathematics is the manipulation of marks according to specified rules. There is a gap between the professions of formalists and mathematical practice. Few mathematical works contain full proofs that obey, in detail, explicitly formulated rules. Formalism is a tactical device designed to enable mathematicians to continue to dwell in Cantor's paradise. The chapter outlines the consequences of taking formalism seriously. Robinson's theory is the simplest theory in which one can do nontrivial mathematics. It is essentially Peano arithmetic without induction. There are several consistency proofs for Q. The most familiar is the infinitary argument that the natural numbers are a model for Q. But, in a world from which numbers have vanished, this carries no conviction. There will be a central data bank of theorems, arranged hierarchically to facilitate search by mathematicians attempting to prove new theorems. Interactive programs will be developed to help construct fully formalized proofs, and when these are submitted, they will be verified and entered into the data bank with the name of the inventor and date of construction.