Luitzen Egbertus Jan Brouwer
Encyclopedia
Luitzen Egbertus Jan Brouwer ˈlœyt.sən ɛx.ˈbɛʁ.təs jɑn ˈbʁʌu.əʁ FRS (February 27, 1881 – December 2, 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch
mathematician
and philosopher, a graduate of the University of Amsterdam, who worked in topology
, set theory
, measure theory and complex analysis
.
of dimension
. Among his further results, the Brouwer fixed point theorem
is also well known. 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 complex
es, 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 (cf. Kleene (1952), p. 46–59). As a variety of constructive mathematics
, intuitionism is essentially a philosophy of the foundations of mathematics
. It is sometimes and rather simplistically characterized by saying that its adherents refuse to use the law of excluded middle
in mathematical reasoning.
Brouwer was member of the Significs group, containing others with a generally neo-Kantian philosophy. It formed part of the early history of semiotics
—the study of symbols—around Victoria, Lady Welby in particular. The original meaning of his intuitionism probably can not be completely disentangled from the intellectual milieu of that group.
In 1905, at the age of 24, Brouwer expressed his philosophy of life in a short tract Life, Art and Mysticism described by Davis as "drenched in romantic pessimism" (Davis (2002), p. 94). Schopenhauer
had a formative influence on Brouwer, not least because he insisted that all concepts be fundamentally based on sense intuitions. Brouwer then "embarked on a self-righteous campaign to reconstruct mathematical practice from the ground up so as to satisfy his philosophical convictions"; indeed his thesis advisor refused to accept his Chapter II " 'as it stands, ... all interwoven with some kind of pessimism and mystical attitude to life which is not mathematics, nor has anything to do with the foundations of mathematics' " (Davis, p. 94 quoting van Stigt, p. 41). Nevertheless, in 1908:
"After completing his dissertation (1907 - see Van Dalen), Brouwer made a conscious decision to temporarily keep his contentious ideas under wraps and to concentrate on demonstrating his mathematical prowess" (Davis (2000), p. 95); by 1910 he had published a number of important papers, in particular the Fixed Point Theorem. Hilbert—the formalist with whom the intuitionist Brouwer would ultimately spend years in conflict—admired the young man and helped him receive a regular academic appointment (1912) at the University of Amsterdam (Davis, p. 96). It was then that "Brouwer felt free to return to his revolutionary project which he was now calling intuitionism " (ibid).
He was combative for a young man. He was involved in a very public and eventually demeaning controversy in the later 1920s with Hilbert over editorial policy at Mathematische Annalen
, at that time a leading learned journal. He became relatively isolated; the development of intuitionism at its source was taken up by his student Arend Heyting
.
About his last years, Davis (2002) remarks:
Netherlands
The Netherlands is a constituent country of the Kingdom of the Netherlands, located mainly in North-West Europe and with several islands in the Caribbean. Mainland Netherlands borders the North Sea to the north and west, Belgium to the south, and Germany to the east, and shares maritime borders...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and philosopher, a graduate of the University of Amsterdam, who worked in topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, measure theory and complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
.
Biography
Early in his career, Brouwer proved a number of theorems that were breakthroughs in the emerging field of topology. The most celebrated result was his proof of the topological invarianceTopological property
In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic...
of dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
. Among his further results, the Brouwer fixed point theorem
Brouwer fixed point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to...
is also well known. Brouwer also proved the simplicial approximation theorem
Simplicial approximation theorem
In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be approximated by ones that are piecewise of the simplest kind. It applies to mappings between spaces that are built up from simplices — that is,...
in the foundations of algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, which justifies the reduction to combinatorial terms, after sufficient subdivision of simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
es, of the treatment of general continuous mappings.
Brouwer in effect founded the mathematical philosophy of intuitionism
Intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism , is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied...
as an opponent to the then-prevailing formalism
Formalism (mathematics)
In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules....
of David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
and his collaborators Paul Bernays
Paul Bernays
Paul Isaac Bernays was a Swiss mathematician, who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant to, and close collaborator of, David Hilbert.-Biography:Bernays spent his childhood in Berlin. Bernays attended the...
, Wilhelm Ackermann
Wilhelm Ackermann
Wilhelm Friedrich Ackermann was a German mathematician best known for the Ackermann function, an important example in the theory of computation....
, John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
and others (cf. Kleene (1952), p. 46–59). As a variety of constructive mathematics
Constructivism (mathematics)
In the philosophy of mathematics, constructivism asserts that it is necessary to find a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its...
, intuitionism is essentially a philosophy of the foundations of mathematics
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
. It is sometimes and rather simplistically characterized by saying that its adherents refuse to use the law of excluded middle
Law of excluded middle
In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
in mathematical reasoning.
Brouwer was member of the Significs group, containing others with a generally neo-Kantian philosophy. It formed part of the early history of semiotics
Semiotics
Semiotics, also called semiotic studies or semiology, is the study of signs and sign processes , indication, designation, likeness, analogy, metaphor, symbolism, signification, and communication...
—the study of symbols—around Victoria, Lady Welby in particular. The original meaning of his intuitionism probably can not be completely disentangled from the intellectual milieu of that group.
In 1905, at the age of 24, Brouwer expressed his philosophy of life in a short tract Life, Art and Mysticism described by Davis as "drenched in romantic pessimism" (Davis (2002), p. 94). Schopenhauer
Arthur Schopenhauer
Arthur Schopenhauer was a German philosopher known for his pessimism and philosophical clarity. At age 25, he published his doctoral dissertation, On the Fourfold Root of the Principle of Sufficient Reason, which examined the four separate manifestations of reason in the phenomenal...
had a formative influence on Brouwer, not least because he insisted that all concepts be fundamentally based on sense intuitions. Brouwer then "embarked on a self-righteous campaign to reconstruct mathematical practice from the ground up so as to satisfy his philosophical convictions"; indeed his thesis advisor refused to accept his Chapter II " 'as it stands, ... all interwoven with some kind of pessimism and mystical attitude to life which is not mathematics, nor has anything to do with the foundations of mathematics' " (Davis, p. 94 quoting van Stigt, p. 41). Nevertheless, in 1908:
- "... Brouwer, in a paper entitled "The untrustworthiness of the principles of logic", challenged the belief that the rules of the classical logic, which have come down to us essentially from Aristotle (384--322 B.C.) have an absolute validity, independent of the subject matter to which they are applied" (Kleene (1952), p. 46).
"After completing his dissertation (1907 - see Van Dalen), Brouwer made a conscious decision to temporarily keep his contentious ideas under wraps and to concentrate on demonstrating his mathematical prowess" (Davis (2000), p. 95); by 1910 he had published a number of important papers, in particular the Fixed Point Theorem. Hilbert—the formalist with whom the intuitionist Brouwer would ultimately spend years in conflict—admired the young man and helped him receive a regular academic appointment (1912) at the University of Amsterdam (Davis, p. 96). It was then that "Brouwer felt free to return to his revolutionary project which he was now calling intuitionism " (ibid).
He was combative for a young man. He was involved in a very public and eventually demeaning controversy in the later 1920s with Hilbert over editorial policy at Mathematische Annalen
Mathematische Annalen
Mathematische Annalen is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann...
, at that time a leading learned journal. He became relatively isolated; the development of intuitionism at its source was taken up by his student Arend Heyting
Arend Heyting
Arend Heyting was a Dutch mathematician and logician. He was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a footing where it could become part of mathematical logic...
.
About his last years, Davis (2002) remarks:
- "...he felt more and more isolated, and spent his last years under the spell of 'totally unfounded financial worries and a paranoid fear of bankruptcy, persecution and illness.' He was killed in 1966 at the age of 85, struck by a vehicle while crossing the street in front of his house." (Davis, p. 100 quoting van Stigt. p. 110.)
See also
- Gerrit MannouryGerrit MannouryGerrit Mannoury was a Dutch philosopher and mathematician, professor at the University of Amsterdam and communist, known as the central figure in the signific circle, a Dutch counterpart of the Vienna circle.- Biography :...
- Arend HeytingArend HeytingArend Heyting was a Dutch mathematician and logician. He was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a footing where it could become part of mathematical logic...
- George F C GrissGeorge F C GrissGeorge François Cornelis Griss , usually cited as G. F. C. Griss, was a Dutch mathematician and philosopher, who was occupied with hegelian idealism and Brouwers intuitionism and stated a negationless mathematics....
- Philosophy of mindPhilosophy of mindPhilosophy of mind is a branch of philosophy that studies the nature of the mind, mental events, mental functions, mental properties, consciousness and their relationship to the physical body, particularly the brain. The mind-body problem, i.e...
- Philosophy of mathematicsPhilosophy of mathematicsThe philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
- Brouwer-Hilbert controversy