Brouwer adhered to an intuitionist philosophy of mathematics, which is sometimes characterized by saying that its adherents refuse to use the law of excluded middle in mathematical reasoning, and wrote books on the subjects mentioned above in which he proceeded accordingly.
His ideas were initially exposed in Beweis des Jordanschen Satzes für N Dimensionen (1912) ("Proof of Jordan's theorem for N dimensions").
He was involved in an eventually demeaning controversy with David Hilbert.
He was member of the Significs group, containing others with a generally neo-Kantian philosophy. It formed part of the early history of semiotic study, around Victoria, Lady Welby in particular.
LEJBrouwer founded the doctrine of mathematical intuitionism, which views mathematics as the formulation of mental constructions that are governed by self-evident laws.
Brouwer's doctrine differed substantially from the formalism of David Hilbert and the logicism of Bertrand Russell.
LEJBrouwer was elected to the Royal Society of London in 1948.
Early in his career, Brouwer proved a number of theorems that were breakthroughs in the emerging field of topology.
Brouwer also proved the simplicial approximation theorem in the foundations of algebraic topology, which justifies the reduction to combinatorial terms, after sufficient subdivision of simplicial complexes, of the treatment of general continuous mappings.
Brouwer in effect founded the mathematical philosophy of intuitionism as an opponent to the then-prevailing formalism of David Hilbert and his collaborators Paul Bernays, Wilhelm Ackermann, John von Neumann and others.
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