nungslehre (1844, 4th ed. 1969).
..... Click the link for more information. , who showed that several different consistent algebras may be derived by choosing different sets of axioms governing the operations on the elements of the algebra.
These developments continued with the group theory of M. S. LieLie, Marius Sophus
, 1842–99, Norwegian mathematician. He is noted for his contributions to the theories of differential equations and continuous transformation groups.
..... Click the link for more information. in the late 19th cent. and reached full expression in the wide scope of modern abstract algebra. Number theory received significant contributions in the latter half of the 19th cent. through the work of Georg CantorCantor, Georg
, 1845–1918, German mathematician, b. St. Petersburg. He studied under Karl Weierstrass and taught (1869–1913) at the Univ. of Halle. He is known for his work on transfinite numbers and on the development of set theory, which is the basis of modern
..... Click the link for more information. , J. W. R. DedekindDedekind, Julius Wilhelm Richard
, 1831–1916, German mathematician. Dedekind studied at Göttingen under the German mathematician Carl Gauss and in 1852 received his doctorate there for a thesis on Eulerian integrals.
..... Click the link for more information. , and K. W. WeierstrassWeierstrass, Karl Wilhelm Theodor
, 1815–97, German mathematician. From 1864 he was professor of mathematics at the Univ. of Berlin. His development of the modern theory of functions is described in his Abhandlungen aus der Funktionenlehre
..... Click the link for more information. . Still another influence of Gauss was his insistence on rigorous proof in all areas of mathematics. In analysis this close examination of the foundations of the calculus resulted in A. L. CauchyCauchy, Augustin Louis, Baron
, 1789–1857, French mathematician. He was professor simultaneously (1816–30) at the École polytechnique, the Sorbonne, and the Collège de France in Paris. While a political exile (1830–38) he taught at 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 GödelGöd