On a New Dialectical Approach to Mathematical Creativity

The article sets the task to show how the dialectical form of thinking is con­cretized in the form of a new dialectical approach to mathematical creativity. Its philosophical basis is the fundamental ontology of Martin Heidegger, from which the operation of dialectical negation, called privation, is borrowed. Pri­vation in its ultimate expression finds a place in the paraconsistent (“Imagi­nary”) logic of N.A. Vasiliev, operating with logical antinomies and revealing their difference from logical contradictions (from logical absurdity). In logical and mathematical research, a clear example of the application of privation are the well-known Gödel incompleteness theorems, which establish the existence of positive integers with properties opposite to the properties of inductive numbers (N.N. Luzin). Within the framework of the task, the descriptive set theory of Luzin and the non-Euclidean (hyperbolic) geometry of Lobachevsky are analyzed. They are united by the problem of ordering the arithmetic conti­nuum. In this context, the question is posed as to why the set of real points of the continuum is not countably ordered, and an answer is given. The answer is provided by the dialectical approach to mathematical creativity described here as part of the logical discipline of thinking called complementary-dialecti­cal logic. The dialectical approach to mathematical creativity includes the trans­formation of mathematical logic developed on the basis of classical formal logic. This transformation is presented in the article as a synthesis of the exten­sional and intensional aspects of logical thinking, which are often called (for brevity) extensional and intensional logic.