VI.75 Luitzen Egbertus Jan Brouwer
b. Overschie, the Netherlands, 1881;
d. Blaricum, the Netherlands, 1966
Lie groups; topology; geometry; intuitionistic mathematics; philosophy of mathematics
Brouwer entered the University of Amsterdam at the age of sixteen, where his teacher was D. J. Korteweg. The young Brouwer taught himself modern mathematics, as well as a fair amount of philosophy. As a graduate student he published some original papers on the decomposition of rotations in four-dimensional space. He also published a brief monograph on mysticism that contained a number of ideas that became prominent in his later philosophy. In his dissertation of 1907 he solved a special case of HILBERT’S [VI.63] fifth problem (the elimination of differentiability conditions from the axioms of LIE GROUPS [III.48 §1]), and he presented his first program for “constructive mathematics.”
The basis of his mathematics was the ur-intuition of mathematics: the continuum and the natural numbers are simultaneously created from intuition. Mathematical objects (including proofs) are mental creations. After sketching the development of the basic parts of mathematics, Brouwer went on to criticize contemporary mathematics for transcending the bounds of the human mind. In particular, he criticized CANTOR [VI.54] for introducing sets beyond human recognition, and Hilbert for the axiomatic method and for formalism. He criticized the latter’s consistency program and denied that “consistency implies existence.”
In his 1908 paper “The unreliability of the logical principles,” Brouwer explicitly rejected the principle of the excluded middle as unreliable (and also rejected Hilbert’s dogma that “all mathematical problems can be solved in one way or another”). Between 1909 and 1913 Brouwer worked in topology. He continued his work on Lie groups, and noted that topology (in the style of Cantor-Schoenflies) was in need of a sound basis. In his paper “Zur Analysis Situs” (1910), he spelled out a number of notions and examples (curves, indecomposable continua, three domains with one common boundary). This was the beginning of his revision of set-theoretic topology. At the same time he started two lines of research: one on homeomorphisms from surfaces to themselves, establishing FIXED POINT THEOREMS [V.11] on the sphere and the plane translation theorem (a characterization of homeomorphisms of the Euclidean plane that have no fixed points); and one on vector distributions on the sphere, yielding existence theorems for singular points, and a characterization of these points. The best-known theorem in this area is Brouwer’s “hairy ball theorem” (no matter how one combs a hairy ball, there is always a crown). In 1910 Brouwer published a direct topological proof of the Jordan curve theorem, which remains one of the most elegant proofs. The so-called new topology opened with Brouwer’s “invariance of dimension” theorem (1910). He then laid the basis for topology of MANIFOLDS [I.3 §6.9], where his basic tool was the Brouwer degree of continuous mappings. The basic paper is his “Über Abbildungen von Mannigfaltigkeiten” (“On mappings of manifolds,” 1911), which contained most of the tools for the new topology, e.g., simplicial approximation, mapping degree, HOMOTOPY [IV.6 §§2, 3], singularity index (in his own terminology), and also the fundamental properties of the new notions.
Brouwer’s new topological insights and techniques led him to a wealth of spectacular results: the Brouwer fixed point theorem, the invariance-of-domain theorem, the higher-dimensional Jordan theorem, and the definition of dimension, including the soundness proof (that 
n has dimension n). He also applied his invariance-of-domain theorem to the theory of automorphic functions and uniformization, thus proving the correctness of the Klein-Poincaré continuity method (1912).
During World War I Brouwer returned to the foundations of mathematics; he conceived his mature INTUITIONISTIC MATHEMATICS [II.7 §3.1], which fully exploited the potential of constructive mathematics, based on mentally created objects and notions. The key notions were (infinite) choice sequences (i.e., sequences determined by more or less free choices (by the mathematician) of mathematical objects, say natural numbers), well-orderings, and intuitionistic logic. In “Brouwer’s universe” strong results can be obtained: the “continuity principle,” for example, which says that a function that assigns natural numbers to choice sequences is continuous (i.e., the output is determined by a finite piece of the (infinite) input); and certain transfinite induction principles, in particular the novel principle of “bar-induction. With the help of these principles he showed that (i) all real functions on a closed segment are uniformly continuous and (ii) the continuum is indecomposable (cannot be split). This enabled him to refute the principle of the excluded middle in a strong sense: it is not the case that each real number is zero or nonzero. In Brouwer’s universe many classical theorems, such as the intermediate value theorem and the Bolzano-Weierstrass theorem, fail.
Brouwer’s mathematical universe lacked the logical “principle of the excluded middle,” but instead it had certain strong constructive principles at its disposal, which turned it into an alternative to the traditional universe, with a comparable strength.
His foundational program brought him into conflict with Hilbert (“intuitionism versus formalism”). In 1928 matters came to a head and, in an incident famously described by Einstein as “the war of frogs and mice,” Hilbert succeeded in getting Brouwer removed (after fourteen years’ service) from the editorial board of Mathematische Annalen.
Brouwer was unconventional and had wide-ranging interests: art, literature, politics, philosophy, mysticism. He was a staunch internationalist.
He was a professor at the University of Amsterdam from 1912 until 1951.
Further Reading
Brouwer, L. E. J. 1975-76. Collected Works, two volumes. Amsterdam: North-Holland.
van Dalen, D. 1999-2005. Mystic, Geometer and Intuitionist. The Life of L. E. J. Brouwer, two volumes. Oxford: Oxford University Press.