Luitzen Egbertus Jan Brouwer

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Brouwer, Luitzen Egbertus Jan

 

Born Feb. 27, 1881, in Overschie; died Dec. 2, 1966, in Amsterdam. Dutch mathematician. Member of the Netherlands Academy of Sciences in Amsterdam (1912), and corresponding member of the Paris and Göttingen academies of sciences. Professor at the University of Amsterdam (1912-51).

From 1908, Brouwer consistently criticized the so-called pure mathematical existence proofs based on the logical principle of the excluded middle, which ultimately laid the basis for a whole trend in the foundations of mathematics, mathematical intuitionism. But Brouwer’s analysis of the mathematical existence proofs from the point of view of the constructive synthesis of those objects whose existence is being proved has value independent of the philosophy of intuitionism. In particular, A. N. Kolmogorov has shown that the rules of so-called intuitionist logic find their true realization in the logic of the constructive solution of mathematical problems. Between 1911 and 1913, Brouwer established a number of important concepts and results in the field of topology. Among them are the concepts of simplicial approximation and the degrees of continuous mapping, the concept of homotopic classification of mappings, the theorem on the homotopic equivalence of two mappings (of a sphere into itself) of identical degree, the theorem of the invariance of the number of dimensions and the invariance of internal points (in topological mapping of a set lying in an n-dimensional space into that space), the fixed-point theorem, and Jordan’s n-dimensional theorem. These results and the methods found for their proof were the basis of Brouwer’s significant influence on the development of topology in the period between World War I and World War II.

REFERENCES

Aleksandrov, P. S. Kombinatornaia topologiia. Moscow-Leningrad, 1947.
Weyl, H. O filosofii matematiki, sb. rabot. Moscow-Leningrad, 1934. (Translated from German; see section “O novom krizise osnov matematiki.”)