The constantly accelerating progress of contemporary natural science is indissolubly associated with the development and use of mathematics and with the processes of mathematical modeling of the phenomena of nature. The essence of this diverse and highly fertile interaction of mathematics and natural science and the dialectics of this interaction can only be disclosed through analysis of the nature of theoretical notions in general. Today, above all in the ranks of materialistically minded researchers, it is generally accepted that theory possesses (...) a value of its own. Contrary to positivist concepts of the nature of our knowledge, the meaning and significance of theory do not consist merely of the recording of experimental data, their classification and notation in abbreviated form. The meaning of theory is considerably more significant and is revealed, above all, in its explanatory and predictive functions. Contemporary theoretical knowledge is quite advanced and comprises a highly complex system, relatively self-contained and capable of internal development. Mathematics is the most interesting and distinctive phenomenon in the system of theoretical knowledge, in the system of science in general. It is in its dialectical development that the internal force and dynamics of the development of theory, in the very broadest sense of that word, find expression. Paul Lafargue tells us that Karl Marx held that "science only attains perfection when it succeeds in making use of mathematics" . In many fields of research the formulation of new ideas and concepts rests upon mathematics, its concepts and notions, and "is suggested" by the latter. "Mathematics," writes F. Dyson, "is the main source of the notions and principles out of which new theories are created" . The basic significance of mathematics to theory has been noted by outstanding thinkers over the course of the entire history of development of science. Today the expressive phrase, "the mathematization of knowledge," has come into general currency as a description of the principal direction of growth of theoretical notions in natural science. However, with respect to the growth of modern theoretical physics in general, the statement that it develops primarily by the method of the mathematical hypothesis has general validity. For example, the principal findings and discoveries of quantum theory and particle physics, starting from the corpuscularwave dualism and terminating with the omega minus hyperon and hypothetical quarks, were obtained or made "at the tip of a mathematician's pen.". (shrink)