In 1897 Brouwer he entered the University of Amsterdam, where he studied mathematics until 1904. He obtained his doctorate in 1907 for his thesis, Over de Grondslagen der Wiskunde.
Career
Gallery of Luitzen Brouwer
Luitzen Egbertus Jan Brouwer
Gallery of Luitzen Brouwer
Luitzen Egbertus Jan Brouwer
Achievements
Brouwer crater, on the far side of the moon.
Membership
Royal Netherlands Academy of Arts and Sciences
1912 - 1966
The Royal Society of London for Improving Natural Knowledge
In 1897 Brouwer he entered the University of Amsterdam, where he studied mathematics until 1904. He obtained his doctorate in 1907 for his thesis, Over de Grondslagen der Wiskunde.
(Luitzen Egburtus Jan Brouwer founded a school of thought ...)
Luitzen Egburtus Jan Brouwer founded a school of thought whose aim was to include mathematics within the framework of intuitionistic philosophy; mathematics was to be regarded as an essentially free development of the human mind. What emerged diverged considerably at some points from tradition, but intuitionism has survived well the struggle between contending schools in the foundations of mathematics and exact philosophy.
Luitzen Egbertus Jan Brouwer was a Dutch mathematician and philosopher. He laid the foundation of intuitionism, an approach that views the nature of mathematics as mental constructions governed by self-evident laws.
Background
Luitzen Egbertus Jan Brouwer was born on February 27, 1881, in Overschie, Netherlands, a son of Hendrika Poutsma and Fgbertus Luitzens Brouwer, schoolmaster at a primary school at Overschie. Hendrika was of Friesian stock; her earliest registered forebear Tammerus Gerhardi (1579-1644) was a minister of the Dutch Reformed Church at Joure and IJIst in Friesland. The Poutsma family tree is adorned with a good number of parsons and schoolmasters, and towards the end of the nineteenth century there is a definite tendency to rise to the higher strata of the teaching profession.
Education
Brouwer first showed his unusual intellectual abilities by finishing high school in the North Holland town of Hoorn at the age of fourteen. In the next two years, he mastered the Greek and Latin required for admission to the university, and passed the entrance examination at the municipal Gymnasium in Haarlem, where the family had moved in the meantime. In the same year, 1897, he entered the University of Amsterdam, where he studied mathematics until 1904. He quickly mastered the current mathematics, and, to the admiration of his professor, D. J. Korteweg, he obtained some results on continuous motions in four-dimensional space that were published in the reports of the Royal Academy of Science in Amsterdam in 1904. Through his own reading, as well as through the stimulating lectures of Gerrit Mannoury, he became acquainted with topology and the foundations of mathematics.
He obtained his doctorate in 1907 for his thesis, Over de Grondslagen der Wiskunde. Brouwer also received honorary doctorates from the universities of Oslo (1929) and Cambridge (1954).
Even during his university years, Luitzen Brouwer was interested in philosophical matters, as evidenced by his Leven, Kunst, en Mystiek, published in 1905. In his doctoral thesis, 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, 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.
In 1909 Luitzen Egbertus Jan Brouwer was appointed as a privatdocent at the University of Amsterdam and served as a professor there from 1912 until his retirement in 1951. He gave his inaugural lecture on 12 October 1909 on 'The nature of geometry' in which he outlined his research program. A couple of months later he made an important visit to Paris, around Christmas 1909, and there met Poincaré, Hadamard, and Borel. Prompted by discussions in Paris, he began working on the problem of the invariance of dimension.
In a number of publications beginning in 1918 and extending through the 1920s, Brouwer developed intuitionist mathematics and worked out in detail his critique of classical mathematics, determining for different branches of mathematics which of their theorems are intuitionistically true. In "Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten," Brouwer undertook to develop an intuitionist set theory, on which a theory of the continuum could be based. In this work he introduced his concept of set (Menge ; later, in "Points and Spaces," 1954, he called it "spread") and therefore the idea of an arbitrary infinite sequence as generated by successive free choices. He also introduced the notion of species, which led to his own version of a predicative hierarchy of classes. The principle that the value of a function everywhere defined on a spread must, for a given sequence as argument, be determined by a sufficiently large finite number of its terms is already present in "Begründung der Mengenlehre." This "continuity axiom" is the first of the two distinctive principles of intuitionist analysis.
In "Beweis, dass jede volle Funktion gleichmässig stetig ist" (1924), Brouwer announced a proof that a function everywhere defined on the closed unit interval is uniformly continuous. In this proof Brouwer used two fundamental assertions about spreads, later called the bar and fan theorems. The bar theorem, or an equivalent assertion, constitutes the other distinctive principle of intuitionist analysis. Brouwer's proof was presented in full in "Über Definitionsbereiche von Funktionen" (1927) and reworked, in a more general setting, in "Points and Spaces."
After World War II Brouwer published a long series of short papers in which he developed a new type of counterexample to classical theorems, based on another new principle.
After retiring in 1951, Brouwer lectured in South Africa in 1952, and the United States and Canada in 1953. His wife died in 1959 at the age of 89 and Brouwer, who himself was 78, was offered a one year post in the University of British Columbia in Vancouver; he declined. In 1962, despite being well into his 80s, he was offered a post in Montana.
In 1935 Brouwer entered local politics when he was elected as Neutral Party candidate for the municipal council of Blaricum. He continued to serve on the council until 1941.
Views
Although he was primarily a mathematician, Brouwer was always preoccupied with general philosophy and had elaborated a highly individual philosophical vision. Indeed, the most remarkable feature of Brouwer's work in the foundations of mathematics was the boldness and consistency with which, starting from his own philosophical position, he questioned the principles on which the mathematics he inherited was based, down to so elementary a principle as the law of excluded middle, and then proceeded to criticize these principles in detail and to begin to reconstruct mathematics on a basis he regarded as sound.
Although he later presented them more systematically, the essentials of Brouwer's philosophy were already present in his thesis of 1907 and, in certain respects, in Life, Art, and Mysticism (1905). These works antedate the decisive steps in the development of mathematical intuitionism. In effect, Brouwer argued in his thesis that logic is derivative from mathematics and dependent for its evidence on an essentially mathematical intuition that rests on a basis close to Immanuel Kant's notion of time as the "form of inner sense." Intellectual life begins with "temporal perception," in which the self separates experiences from each other and distinguishes itself from them. Brouwer describes this temporal perception as "the falling apart of a life moment into two qualitatively different things, of which the one withdraws before the other and nonetheless is held onto by memory". This perception, however, belongs to an attitude (which Brouwer earlier termed "mathematical consideration") that the self adopts to preserve itself; the adoption of this attitude is an act of free will, in a broad sense that Brouwer probably derived from Arthur Schopenhauer. The fundamental intuition of mathematics is this structure of temporal perception "divested of all content"; in mathematics one sees that the process of division and synthesis can be iterated indefinitely, giving rise to the series of natural numbers. In the temporal order thus revealed, one can always imagine new elements inserted between the given ones, so that Brouwer could say that the theories of the natural numbers and of the continuum come from one intuition, an idea that, from his point of view, was made fuller and more precise by his theory of free choice sequences, although one might argue that it was made superfluous by that theory.
Brouwer's constructivism was developed in this context. His constructivism was probably motivated less by an insistence on absolute evidence and a rejection of hypotheses (which might have led to "finitism" in David Hilbert's sense of the term or even to a still narrower thesis) than by Brouwer's subjectivism and his insistence on the primacy of will over intellect. On these grounds, mathematics should consist in a constructive mental activity, and a mathematical statement should be an indication or report of such activity. Brouwer credited this way of looking at mathematics to the inspiration of his teacher, Gerrit Mannoury.
In his thesis Brouwer limited himself to criticizing alternative theories of the foundations mathematics and to criticizing Cantorian set theory, but in "De Onbetrouwbaarheid der Logische Principes" (1908), perhaps urged on by Mannoury, Brouwer raised doubts about the validity of the law of excluded middle, although he still regarded the question as open. In Intuitionisme en Formalisme (1912) Brouwer did not say flatly that the law of excluded middle is false, but he gave an instance of his standard argument, an example like that presented in the section on intuitionism in the entry titled "Mathematics, Foundations of," which also gives a fuller exposition of constructivism.
Brouwer's philosophy is not limited to what is relevant to the foundations of mathematics. Mathematical consideration has a second phase, which he called causal attention. In this phase "one identifies in imagination certain series of phenomena with one another," an operation by which one can pick out objects and postulate causal rules. (The relation between temporal perception and causal attention is analogous to that between Kant's mathematical and dynamical categories.) The whole point of mathematical consideration lies in the fact that it makes possible the use of means: One produces a phenomenon that will be followed in a certain repeatable series by a desired phenomenon that cannot be directly reproduced. This makes the pursuit of instinctual satisfaction more efficient.
Membership
Brouwer was a member of the Royal Dutch Academy of Sciences, the Royal Society in London, the Berlin Academy of Sciences, and the Göttingen Academy of Sciences.
Royal Netherlands Academy of Arts and Sciences
1912 - 1966
The Royal Society of London for Improving Natural Knowledge
Berlin-Brandenburg Academy of Sciences
Göttingen Academy of Sciences
Personality
Brouwer seems to have been an independent and brilliant man of high moral standards, but with an exaggerated sense of justice, making him at times pugnacious. As a consequence, in his life, he energetically fought many battles.
Quotes from others about the person
"Brouwer is most famous ... for his contribution to the philosophy of mathematics and his attempt to build up mathematics anew on an Intuitionist foundation, in order to meet his own searching criticism of hitherto unquestioned assumptions. Brouwer was somewhat like Nietzsche in his ability to step outside the established cultural tradition in order to subject its most hallowed presuppositions to cool and objective scrutiny; and his questioning of principles of thought led him to a Nietzschean revolution in the domain of logic. He in fact rejected the universally accepted logic of deductive reasoning which had been codified initially by Aristotle, handed down with very little change into modern times, and very recently extended and generalised out of all recognition with the aid of mathematical symbolism." - G. T. Kneebone
Connections
In 1904 Luitzen Brouwer married Lize de Holl who was eleven years older than Brouwer and had a daughter from a previous marriage. The marriage produced no children.