el, Kurt
, 1906–78, American mathematician and logician, b. Brünn (now Brno, Czech Republic), grad. Univ. of Vienna (Ph.D., 1930). He came to the United States in 1940 and was naturalized in 1948.
..... Click the link for more information. and A. Church.
See R. Courant and H. Robbins, What Is Mathematics? (1941); E. T. Bell, The Development of Mathematics (2d ed. 1945) and Men of Mathematics (1937, repr. 1961); J. R. Newman, ed., The World of Mathematics (4 vol., 1956); E. E. Kramer, The Nature and Growth of Mathematics (1970); M. Kline, Mathematical Thought from Ancient to Modern Times (1973); D. J. Albers and G. L. Alexanderson, ed., Mathematical People (1985).
Mathematics is the science of the quantitative relations and spatial forms of the real world.
“Pure mathematics studies the spatial forms and quantitative relations of the real world and thus its subject matter is rooted in reality. The highly abstract form of this subject matter hardly conceals its derivation from the outside world. However, in order to investigate these forms and relations as such, one must separate them entirely from their underlying content, which must be set aside as something irrelevant” (F. Engels in K. Marx and F. Engels, Soch., 2nd ed., vol. 20, p. 37). The abstractness of mathematics does not, however, imply the separation of mathematics from material reality. The number of quantitative relations and spatial forms studied by mathematics is inseparably connected with the demands of technology and the natural sciences and is growing continually; therefore the definition of mathematics given above is endowed with ever richer content.
Mathematics and other sciences. The applications of mathematics are extremely varied. In principle, the domain of application of the mathematical method is unlimited: all types of motion of matter can be studied mathematically. However, the role and significance of the mathematical method vary with each case. No single mathematical scheme can cover all aspects of real phenomena, and therefore the process 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 ac