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 mathematical method often recedes into the background.
Mathematics and technology. The principles of arithmetic and elementary geometry arose, as will be seen from the history section below, in response to the needs of everyday life; the subsequent creation of new mathematical methods and ideas took place under the influence of the mathematical natural sciences, such as astronomy, mechanics, and physics, whose development responds to practical needs. But the direct ties between mathematics and technology have frequently taken the form of application of existing mathematical theories to technological problems. Let us point out, however, examples of the rise of new general mathematical theories in response to direct technological requirements: the development of the method of least squares was connected with geodetic work; the study of many new types of partial differential equations first began with the solution of technical problems; and operational methods of solving differential equations were developed in connection with electrical engineering. The requirements of communications led to the formation of a new branch of probability theory—information theory. The problems of synthesizing control systems led to the development of new branches of mathematical logic. Along with the needs of astronomy, technical problems also played a decisive role in the development of the methods of approximate solution of differential equations. Many methods of approximate solution of partial differential equations and of integral equations were developed entirely to meet technological needs. The problem of rapidly obtaining numerical solutions became quite acute with the growing complexity of technical problems. Numerical methods are acquiring ever greater importance in connection with the possibilities opened up by computers for the solution of practical problems. The high level of theoretical mathematics has made it possible to develop rapidly the methods of computer science, which has played a major role in the solution of a nu