p A distinguishing feature of modern science is the mounting role of mathematical methods in scientific research. Mathematics is widely applied in natural sciences and is penetrating social sciences.

p It is common knowledge that all phenomena and processes have measure, unity of quantitative and qualitative features. Since qualities are manifested in properties that have quantitative gradations such as magnitude, level of development, etc., it is possible to use mathematics in investigating nature and society.

p Mathematics has penetrated other sciences because it is highly abstract and has an extraordinary broad range of principles. And this is consistent with one of the main trends in the development of modem science—the growth of abstract knowledge. Today there are scientific trends which cannot be visually expressed and substantiated by laboratory experiments. But mathematics has a range of concepts (function, set, group, infinite set, etc.), which’in terms of their breadth and universality are close to philosophical and thus make it possible to reflect the general quantitative characteristics of qualitatively different phenomena.

p Moreover, mathematics has an exceptionally strict inner logic. If there are certain premises and if they are true, then owing to the inner logic of mathematics their corollaries are true.

p Another reason why mathematics is penetrating other sciences is that the indices with which it operates are not necessarily numerical. It has a formidable arsenal of means (models, matrices, functions, graphics, symbols, etc.), which alongside quantitative can also express certain qualitative features of objects and phenomena, and disclose their 178 intricate relations and interdependence.

Yet it would be wrong to overestimate the role that mathematics can play. Mathematical methods produce results only when the qualitative nature, the essence of objects and phenomena are known. Hence the need for a close alliance of mathematics with other sciences.