ess of cognition of the concrete always involves two conflicting tendencies: the isolation of the form of the phenomena under study and the logical analysis of this form, on the one hand, and the disclosure of aspects that do not fit into established forms and the transition to the consideration of new forms that are more flexible and more completely encompass the phenomena, on the other. If the difficulties in studying some range of phenomena bear largely on the second tendency in the sense that each new step in the investigation entails the consideration of qualitatively new aspects of the phenomena, then the mathematical method recedes into the background; in this case, dialectical analysis of all aspects of a phenomenon can only be obscured by mathematical schematization. On the other hand, if comparatively simple and stable basic forms of the phenomena under study encompass these phenomena with great accuracy and completeness and if within the limits of these forms there arise quite difficult and complex problems requiring special mathematical investigation, in particular, the creation of special symbolic notation and a special algorithm for their solution, then we enter the sphere of dominance of the mathematical method.
Celestial mechanics, particularly the study of planetary motion, is a typical example of the complete dominance of the mathematical method. The law of universal gravitation, which has a very simple mathematical expression, almost completely determines the range of phenomena studied under this heading. With the exception of the theory of lunar motion, we may, within the limits of observational accuracy, disregard the shape and dimensions of celestial bodies and think of them as “material points.” The solution of the resulting problem of the motion of n material points subject to gravitational forces presents great difficulties as early as the case of n = 3. However, every result obtained by mathematical analysis of the adopted model of the phenomenon describes a real event with remarkable accuracy: a logically simple model reflects well the selected range of phenomena, and all difficulties consist in extracting the mathematical consequences from the adopted model.
No appreciable reduction of the role of the mathematical method occurs upon transition from mechanics to physics, but the difficulties of application increase significantly. There is almost no field of physics that does not require the use of a sophisticated mathematical apparatus, but the main difficulty involved in the investigations often lies in the selection of the assumptions for mathematical treatment and in the interpretation of the results obtained by mathematical means rather than in the development of the mathematical theory.
The ability of the mathematical method to encompass the very process of transition from one level of insight into reality to the next higher, and qualitatively new, level, can be observed by using a number of physical theories as examples. A classical example is the relation between the macroscopic theory of diffusion, which presupposes a continuously distributed diffusing substance, and the statistical theory of diffusion, which proceeds from consideration of the motion of individual particles of the diffusing substance. In the first theory the density of the diffusing substance satisfies a specific partial differential equation and the study of various problems pertaining to diffusion reduces to finding solutions of this differential equation for appropriate boundary and initial conditions. The continuous theory of diffusion describes with very high precision the real course of events insofar as it deals with ordinary (macroscopic) spatial and time scales. However, for small regions of space (containing only a small number of particles of the diffusing substance), the very concept of density loses meaning. The statistical theory of diffusion proceeds from consideration of the microscopic random motions of the diffusing particles induced by the molecules of the solvent. We do not know precise quantitative 