Artificial Intelligence and Foundations of Mathematics

Abstract

The authors consider the problem: what is the essence of real mathematics? This is not axiomatization, because the axiomatization of any new branch of mathematics is performed later, since its essence has appeared. Axiomatization is carried out in such a way that already "existing" notions and statements become theorems or are declared as primary notions and axioms. The authors propose the following framework definition: the concepts of the foundations of real mathematics are abstract objects being self-evident to a thinking being, including artificial intelligence. To identify such objects, the authors suggest using the Socratic method – that is, posing questions as 2D-pictures to a thinking being, but without the use of any conventional signs. Such objects are formed: As quotient objects from large quantities of real objects based on an "evident" equivalence relation (closeness, similarity); by means of the operation of idealization (passing to the limit). The authors conducted experiments with children, with people whose language is unknown, and with artificial intelligence. The existence of a notion of "continuity" without the notion of a real number is proven.