ЯКІСНА ТЕОРІЯ ДИФЕРЕНЦІАЛЬНИХ РІВНЯНЬ: ФІЛОСОФСЬКІ ПРОБЛЕМИ ТА ЕВОЛЮЦІЙНІ ПРОЦЕСИ

Abstract

There are discussed the problems of philosophy in general position and evolutionary processes on the example of the formation and development of a qualitative theory of differential equations and its methods. A qualitative theory of differential equations is a science answering the question «How this world really works?» – «The world (evolutionary processes) is described by differential equations». It substantiates the fact that the theory created by astronomers Tycho Brahe, Johannes Kepler on only one empirical material, has been confirmed and applied in mathematics. Isaac Newton proved Kepler's laws analytically, using differential equations. From the point of view of philosophy, the effectiveness of mathematics, including differential equations, is another confirmation of the principle of material unity of the world and the principle of determinism. The qualitative theory of differential equations is considered as a living knowledge that is constantly evolving. There are reviewed main achievements in the theory of differential equations of a group of Odessa mathematicians under the leadership of Grabovskaya R. G. It is concluded that the history of qualitative theory of differential equations is a history from insight to proof. It is a history that is about changing misconception and understandings. Every misconception stimulates a new understanding, and every understanding is a leap after which great development follows. Then there are new misconceptions and new understandings in trying to go beyond what has been achieved in this development.