curacy: 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 laws governing these microscopic displacements. However, the mathematical theory of probability enables us (on the basis of general assumptions about the smallness of displacements over short time intervals and the independence of the displacements of a particle in two successive time intervals) to obtain definite quantitative consequences, that is, to determine (approximately) the laws governing the probability distribution of the displacements of particles over long (macroscopic) time intervals. Since the number of individual particles of the diffusing substance is very great, the laws governing the probability distribution of the displacement of individual particles lead, on the assumption that the displacements of each particle are independent of the others, to specific and nonrandom laws governing the motion of the diffusing substance as a whole. These laws turn out to coincide with the differential equations underlying the continuous theory. This example is quite typical in the sense that a statistic of random events frequently leads from one set of laws (in this case, the laws governing the motions of the individual particles of the diffusing substance) to a qualitatively different set of laws (in this case, the differential equations of the continuous theory of diffusion).
In the biological sciences the mathematical method plays a more subordinate role. In the social sciences and the humanities, to an even greater degree than in biology, the mathematical method gives way to the direct analysis of phenomena in all their specific complexity. The application of the mathematical method in the biological and the social sciences and in the humanities is accomplished chiefly through cybernetics. The importance of mathematics for the social sciences (as for the biological sciences) in the form of an auxiliary science—mathematical statistics—remains significant. But in the final analysis of social phenomena, aspects of the qualitative distinctness of each historical stage assume such a dominant position that the m