Brouwer, Luitzen Egbertus Jan

Luitzen Egbertus Jan Brouwer, (born February 27, 1881, Overschie, Netherlands—died December 2, 1966, Blaricum), Dutch mathematician who founded mathematical intuitionism (a doctrine that views the nature of mathematics as mental constructions governed by self-evident laws) and whose work completely transformed topology, the study of the most basic properties of geometric surfaces and configurations.

Brouwer studied mathematics at the University of Amsterdam from 1897 to 1904. Even then he was interested in philosophical matters, as evidenced by his Leven, Kunst, en Mystiek (1905; “Life, Art, and Mysticism”). In his doctoral thesis, “Over de grondslagen der wiskunde” (1907; “On the Foundations of Mathematics”), Brouwer attacked the logical foundations of mathematics, as represented by the efforts of the German mathematician David Hilbert and the English philosopher Bertrand Russell, and shaped the beginnings of the intuitionist school. The following year, in “Over de onbetrouwbaarheid der logische principes” (“On the Untrustworthiness of the Logical Principles”), he rejected as invalid the use in mathematical proofs of the principle of the excluded middle (or excluded third). According to this principle, every mathematical statement is either true or false; no other possibility is allowed. Brouwer denied that this dichotomy applied to infinite sets.

Brouwer taught at the University of Amsterdam from 1909 to 1951. He did most of his important work in topology between 1909 and 1913. In connection with his studies of the work of Hilbert, he discovered the plane translation theorem, which characterizes topological mappings of the Cartesian plane, and the first of his fixed-point theorems, which later became important in the establishment of some fundamental theorems in branches of mathematics such as differential equations and game theory. In 1911 he established his theorems on the invariance of the dimension of a manifold under continuous invertible transformations. In addition, he merged the methods developed by the German mathematician Georg Cantor with the methods of analysis situs, an early stage of topology. In view of his remarkable contributions, many mathematicians consider Brouwer the founder of topology.

In 1918 he published a set theory, the following year a theory of measure, and by 1923 a theory of functions, all developed without using the principle of the excluded middle. He continued his studies until 1954, and, although he did not gain widespread acceptance for his precepts, intuitionism enjoyed a resurgence of interest after World War II, primarily because of contributions by the American mathematician Stephen Cole Kleene.

His Collected Works, in two volumes, was published in 1975–76.