mathematics: In the Twentieth Century

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 Russell and A. N. Whitehead, and L. E. J. Brouwer. Important contributions in the investigation of the logical foundations of mathematics were made by Kurt Gödel and A. Church.

Sections in this article:

Introduction
In the Twentieth Century
In the Nineteenth Century
Western Developments from the Twelfth to Eighteenth Centuries
Chinese and Middle Eastern Advances
Greek Contributions
Development of Mathematics
Applied Mathematics
Geometry
Analysis
Algebra
Foundations
Bibliography

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