Understanding in mathematics: From classics to non-classics and post-non-classics. Part One

Abstract

The article deals with an issue of understanding in the classical, non-classical and post-non-classical tradition by the example of understanding mathematics. The prime part of the article looks at the matter of classical and post-non-classical science and poses an issue of understanding. Mathematics itself can be classical or non-classical, as well as logic. Classical logic and mathematics were characterized by intuitive clarity and connection with the world, whether it be the surrounding world or the world of thought. Non-classical logic and mathematics are “sciences-in-themselves”, the only requirement for which is consistency. A connection is made between the idea of J. Gray about modernist mathematics and non-classical mathematics. The understanding of mathematics is considered on the example of the works of E. Husserl. Husserl describes logical experiences and the constitution of mathematical meaning in the acts of intuition, realization and reactivation of sense in passing it on in tradition. An important question is the relationship between intuition and logic in the new mathematics. Poincaré contrasts intuition and logic, while Husserl speaks of a specific logical discernment, which can be called logical intuition. The idea of two cognitive abilities is introduced: intuitive-logical and formal-logical.