 the Univ.
..... Click the link for more information. 's theory of limits (1821), which in turn yielded new and clearer definitions of continuity, the derivative, and the definite integral. A further important step toward rigor was taken by Weierstrass, who raised new questions about these concepts and showed that ultimately the foundations of analysis rest on the properties of the real number system.
In the Twentieth Century
In the 20th cent. the trend was toward increasing generalization and abstraction, with the elements and operations of systems being defined so broadly that their interpretations connect such areas as algebra, geometry, and topology. The key to this approach was the use of formal axiomatics, in which the notion of axioms as "self-evident truths" was discarded. Instead the emphasis was placed on such logical concepts as consistency and completeness. The roots of formal axiomatics lie in the discoveries of alternative systems of geometry and algebra in the 19th cent.; the approach was first systematically undertaken by David Hilbert in his work on the foundations of geometry (1899).
The emphasis on deductive logic inherent in this view of mathematics and the discovery of the interconnections between the various branches of mathematics and their ultimate basis in number theory led to intense activity in the field of mathematical logic after the turn of the century. Rival schools of thought grew up under the leadership of Hilbert, Bertrand RussellRussell, Bertrand Arthur William Russell, 3d Earl,
1872–1970, British philosopher, mathematician, and social reformer, b. Trelleck, Wales.
..... Click the link for more information. and A. N. WhiteheadWhitehead, Alfred North,
1861–1947, English mathematician and philosopher, grad. Trinity College, Cambridge, 1884. There he was a lecturer in mathematics until 1911. At the Univ.
..... Click the link for more information. , and L. E. J. Brouwer. Important contributions in the investigation of the logical foundations of mathematics were made by Kurt GödelGödel, 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 proc