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Summary Article: Brouwer, Luitzen Egbertus Jan (1881-1966)
from The Hutchinson Dictionary of Scientific Biography

Place: Netherlands

Subject: biography, maths and statistics

Dutch mathematician who founded the school of mathematical thought known as intuitionism.

Brouwer was born in Overschie on 27 February 1881. He studied mathematics at the University of Amsterdam, and on receiving his BA was appointed an external lecturer to the university in 1902. He remained at that post until 1912, when he was appointed professor of mathematics, a chair that he held until his retirement in 1951. His singular contribution to mathematics earned him numerous honours, most notably the Knighthood of the Order of the Dutch Lion in 1932. He died in Blaricum on 2 December 1966.

Brouwer's first important paper, a discussion of continuous motion in four-dimensional space, was published by the Dutch Royal Academy of Science in 1904, but the greatest early influence on him was Gerritt Mannoury's work on topology and the foundations of mathematics. This led him to consider the quarrel between Henri Poincaré and Bertrand Russell on the logical foundations of mathematics, and his doctoral dissertation of 1907 came down on the side of Poincaré against Russell and David Hilbert. He took the position that, although formal logic was helpful to describe regularities in systems, it was incapable of providing the foundation of mathematics.

For the rest of his career Brouwer's chief concern remained the debate over the logical, or other, foundations of mathematics. His inaugural address as professor of mathematics at Amsterdam in 1912 opened new ground in this debate, which had begun with the work of Georg Cantor in the early 1880s. In particular Brouwer addressed himself to problems associated with the law of the excluded middle, one of the cardinal laws of logic. He consistently took issue with mathematical proofs (so-called proofs, as he saw them) that were based on the law. In 1918 he published his set theory, which was independent of the law, explaining the notion of a set by the introduction of the idea of a free-choice sequence.

Having rejected the principle of the excluded middle as a useful mathematical concept, Brouwer went on to establish the school of intuitional mathematics. Put simply, it is based on the premise that the only legitimate mathematical structures are those that can be introduced by a coherent system of construction, not those that depend upon the mere postulating of their existence. So, for example, the intuitionist principle denies that it makes sense to talk of an actual infinite totality of natural numbers; that infinite totality is something that requires to be constructed.

Brouwer's work did not create an overnight sensation. But when, in the late 1920s, Kurt Gödel broke down Hilbert's foundation theory, it gained great pertinence. The result of Gödel's work was the theory of recursive functions, and in that field of mathematics Brouwer's work was of such fundamental significance that his intuitional theories and analysis have continued to be at the very centre of research into the foundations of mathematics.

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