The paper deals with the significance of mathematics in examining of various forms of existence and evolution of material systems. It shows heuristic application of mathematical methods in biological and social sciences.
The author concentrates on those mathematical methods which are capable, at the contemporary stage of scientific cognition, to reflect the evolutionary dynamics of live systems, viz. from the macroscopic level down to the level of the structure of cells and of genetic material — differential equation, difference equations, mathematical structures, the theory of singularities, of catastrophies, of bifurcations, the theory of dynamic systems, ate.
Simultaneously the author grasps the integrative function of mathematics in the examination of principles of the rise, existence and evolution of complex real systems.