Аннотация:
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.
