# Luitzen Egbertus Jan Brouwer

Harald Bohr and Bertus Brouwer (1932)

Luitzen Egbertus Jan (Bertus) Brouwer (born February 27, 1881 in Overschie (today in Rotterdam ), † December 2, 1966 in Blaricum ) was a Dutch mathematician . He created basic topological methods and concepts and proved important topological theorems. After him is Brouwer fixed point theorem named. Through his founding of intuitionism , he became a protagonist in the so-called fundamental dispute of mathematics , which reached its peak in the 1920s and 1930s.

Brouwer's later work was groundbreaking for the development of constructive mathematics . Formalizations of his views on the nature of logic gave birth to the discipline of intuitionist logic . In his writings on the philosophy of mathematics , he dealt with the relationship between logic and mathematics, especially with the role of existence statements and the use of the principle of the excluded third in mathematical proofs .

## Life

Brouwer was the eldest of three sons of Egbertus Luitzens Brouwer and Henderika Poutsma. His father, like some relatives, was a teacher. His younger brother was the future geology professor Hendrik Albertus Brouwer . After moving a few times and attending school in Hoorn and Haarlem, sixteen-year-old Brouwer graduated from high school in 1897 and enrolled at the University of Amsterdam . In the course of a transfer to the Remonstrantse Kerk in the following year, Brouwer's idealistic and solipsistic religious creed was handed down.

Well-known people such as the physicist Johannes Diederik van der Waals and the biologist Hugo de Vries worked at the Faculty of Mathematics and Natural Sciences . The mathematical lectures were mainly given by Diederik Johannes Korteweg . Korteweg, who would later also accept Brouwer's dissertation , offered him fascination, but no inspiration. He worked in a wide range of applied mathematics , mainly physics .

Among Brouwer's student acquaintances, the poet Carel Adema van Scheltema (1877–1924) stands out, with whom Brouwer had a lifelong friendship. Brouwer himself wrote poetry and always had literary interests. After graduating in 1904, he carefully took note of the recently propagated philosophy of GJPJ Bolland , published several articles on cultural philosophical issues, and finally gave a series of lectures in Delft in 1905 . Moral and mystical themes, contemplation , the disappearance of innocence and language form its content; they were published under the title Leven, Kunst en Mystiek (Life, Art and Mysticism).

### dissertation

Above all, the philosopher and mathematician Gerrit Mannoury exerted influence on Brouwer . The private lecturer for the logical foundations of mathematics sensitized Brouwer to the new developments in set theory and logical notation by Giuseppe Peano and Bertrand Russell . Brouwer dealt with this in detail in his dissertation, which, in addition to a small part of mathematical results, is exclusively devoted to the difference between logic and mathematics ( Over de grondslagen der wiskunde , 1907; dt. On the basics of mathematics ).

In 1908 Brouwer published the article De onbetrouwbaarheid der logische principes (Eng. The unreliability of logical principles ), where he clearly formulated the rejection of the principium exclusii tertii for the first time . He also identified this principle with the problem of solving any mathematical problem, which was the aim of the program formulated by the German mathematician David Hilbert .

### topology

A visit to the International Congress of Mathematicians in Rome in 1908 marked the beginning of the actual topological creative period in Brouwer's life. He had already published works on geometry for a number of years. Now this occupation intensified; the basics of mathematics should only be considered again later.

The paper Zur Analysis Situs (1910) referred entirely to the developments in the set-theoretical topology of the time. Brouwer supplemented and improved the work of Arthur Moritz Schoenflies , for which he was able to give several counterexamples. He had previously published on Lie groups and vector fields on surfaces . This in turn led him to discover the degree of imaging . He proved the theorem of domain invariance and generalized the Jordanian curve theorem to dimensions ( Jordan-Brouwer decomposition theorem ). He also clarified the concept of dimension . In addition, he developed the method of simplicial approximation . His best-known result today is Brouwer's Fixed Point Theorem . ${\ displaystyle n}$

Many of these works were printed in the German journal Mathematische Annalen . David Hilbert , who is considered the leading mathematician of the era at the end of the 19th and beginning of the 20th century, was one of the three main editors in the editorial team of the Mathematische Annalen . Brouwer initially reached an amicable collaboration with Hilbert, which then came to an end in the context of the fundamental dispute in mathematics .

### Intuitionism

In 1912 Brouwer became a full professor at the University of Amsterdam. His inaugural lecture again took up thoughts from his dissertation. He lectured on intuitionism and formalism and turned against the growing trend towards formalization. In particular, he attacked the axiomatization of set theory by Ernst Zermelo . In 1914 Brouwer was appointed co-editor of the Mathematische Annalen; therefore, and also because of his teaching activity, Brouwer's research stagnated. He turned to a philosophical project, the Significant , founded by Victoria Lady Welby . The spiritus rector of the company was Mannoury , Brouwer's friend and teacher. The Significant aimed for a comprehensive language reform, which however did not materialize.

During the First World War , Brouwer designed a set theory based on intuitionist principles. His establishment of set theory independent of the logical proposition of the excluded third party (1918) is a technical piece of work, free from the polemics of his dissertation, and attempts to establish analysis on a constructive basis ; further such work will follow and build on this study.

Hermann Weyl had made similar attempts to justify the continuum differently than with the cuts introduced by Richard Dedekind . Weyl received Brouwer's writings enthusiastically and defended Brouwer's constructive basis. It was mainly Weyl's operations that sparked the fundamental dispute in mathematics . In an extremely provocative and influential article ( On the new basic crisis in mathematics , 1921) he made Brouwer's ideas known to a wide audience.

### Fundamental dispute

Hilbert was alarmed by this development, but he still felt motivated to clarify the logical foundations of mathematics. He developed his proof theory and confirmed his views on axiomatization and foundation in logic , where the principle of the excluded third party was naturally used and constructability was not decisive for an assumption of existence (provided only the consistency of the axioms was required ).

Brouwer, on the other hand, was mainly occupied in the 1920s with proving classical mathematical results anew and reformulating them intuitionistically up to the design of a new theory of functions. The international science policy after the war, the founding of the Conseil International de Recherches and the Union Mathématique Internationale : Brouwer had tried early and unsuccessfully to lift its boycott against German scientists. When these societies held an international congress in Bologna years later (1928) , Brouwer called on the Germans who had now been invited to boycott. Hilbert, who attended the conference, viewed this as undue interference in German affairs and as harm to science.

Hilbert excluded Brouwer shortly afterwards from the editorship of the Mathematische Annalen , which led to a dispute with the other editors, especially Einstein and Carathéodory. These also no longer belonged to the group of editors. This surprising blow finally broke the friendly relationship between the two mathematicians, and put a great strain on Brouwer. Brouwer himself attributed it to the fact that he had not followed an earlier call (1919) to Göttingen , the seat of the Hilbertkreis. In the vicinity of Hilbert it was suspected that he feared to die soon and that Brouwer could become too influential after his death.

The discussion about the basics of mathematics was intensively pursued by others. Hesseling speaks of over 250 works that reacted to the dispute in the twenties and thirties.

### 1930 to 1966

Public lectures in Berlin and Vienna in 1927 and 1928 were the last two major public appearances for the time being. After the scandal over the mathematical annals, Brouwer was not present in the mathematical public and hardly published. He got involved in local politics and dealt with the failure of a private investment.

The years after World War II were marked by differences Brouwer's in Amsterdam . The journal Compositio Mathematica , which he founded , was withdrawn from his influence, and a research center was founded independently of him. Arend Heyting finally became his mathematical successor. Brouwer had retired in 1951.

Brouwer went on lecture tours to the USA, Canada and South Africa. He gave various lectures in Europe, notably the longer series in Cambridge . The later publications did not produce any significant new results, but revolved around the concept of the creative subject and had a solipsistic impression.

Brouwer died in a traffic accident in Blaricum in 1966, seven years after the death of his wife Lize Brouwer-de Holl . They had no children together. However, Lize Brouwer-de Holl had a daughter from her first marriage, in whose upbringing Brouwer participated.

Brouwer was a member of numerous scientific societies (including the Royal Society of London and the Royal Society of Edinburgh ), and he received honorary doctorates from the Universities of Oslo (1936) and Cambridge (1955). In 1924 he was elected a member of the Leopoldina in Halle (Saale) .

## Brouwers intuitionism

### philosophy

Brouwer rejected academic philosophy. In many cases he expresses himself against philosophical reasoning; he was skeptical of professional philosophers like GJPJ Bolland and tried to prevent the integration of philosophy in the science curriculum. In Leven, Kunst en Mystiek, he mocked alleged clarifications of epistemology . Nonetheless, his attempts to base mathematics on intuition and distrust of the foundations in logic were preceded by extensive philosophical reflection. Brouwer's philosophy is subjectivistic and begins with a consideration of the mental constitution of man.

Brouwer's philosophy deals with the mental functions of the subject. The perspective thus gained is applied not only to the foundation of mathematics, but also to life. In earlier writings this creates moral undertones.

Experiences of transcendental truth, the reunification of the world with the self, the striving for a free life, turning away from economic categories, freedom within, man's falling away from the natural order and Brouwer's views on the linguistic expression of mystical experiences, for example in art form the thematic block of Leven, Kunst en Mystiek . The book was hardly noticed as a philosophical argument. Nevertheless, traces of later differentiations can be seen in it.

In self-reflection, in mysticism, you experience freedom. The external reality, on the other hand, is weakened as a sad world . Brouwer is critical of language, which is difficult to use as a means of expressing inner reality. At the same time as language is the intellect. He also causes human apostasy. The original condition of the human being had been damaged by civilization (based on the intellect); culture appears as a special case of human sin . - Brouwer consistently raises the critical voice against the assumption of a universal and independent reality that binds people and their intellect to one another. The meaning of language does not come from such a reality. The language can only be understood in consideration of the respective will and is an expression of an inner reality. The work is in part a reaction to the Hegelian GJPJ Bolland. It should be a reply to his rhetorical flaming appearances.

Some of Brouwer's writings, including those on intuitionist mathematics, have a moralizing or pessimistic appeal; he also speaks of sin or sinfulness. Brouwer's term “sin” can be described as a state of consciousness of centralizing and externalizing: Sin indicates the transition from free, undirected contemplation in the self to concentration on very specific aspects as well as the shifting of the experienced concepts into an independent exterior. In a brief private note, he called mathematics, its application, and the intuition of time (see below) sinful.

In later writings Brouwer distinguished three phases of consciousness:

1. the naive phase that arises with the creation of the world of sensations
2. the isolated causal phase of scientific activity
3. the social phase of social action and language

The consciousness of the naive phase receives sensations spontaneously in the silence. It doesn't connect them, there is silence in between. Responses to these sensations are direct, spontaneous. There is no activity of the will.

In the wake of the change in sensations, consciousness begins to hold back a sensation as past and to distinguish the past from the present. The consciousness rises above the alternation of the two sensations and becomes spirit . (In Dutch, Brouwer writes the English at least ).

The consciousness now identifies different sensations and their complexes in order to create a sequence. Special cases of such successive mental perception are things and causal consequences .

In the second phase, things are already recognized. A transition from mind to will happens when objects of sensation are seen as causally following one another. This is the act of intellect and characterizes the scientific approach, Brouwer also calls it the mathematical view.

The transition to free will, to the acting human being, takes place through the process with which a change of impressions is consciously achieved through action: the goal-oriented action. The third and social phase now includes all phenomena in which the will itself is changed in its direction: for example through command or suggestion. Laws derive their effect from this. For Brouwer, language originally represented nothing other than the transfer of will to others. Starting with simple gestures and primitive sounds, the development of human society brought with it a more sophisticated language that is also used as a memory aid.

### Philosophy of mathematics

Brouwer's dissertation Over de grondslagen der wiskunde (1907) sets out the basic element that he would use as the basis for all other writings on the philosophy of mathematics. It is the primordial intuition of time .

Through the primordial intuition of the time Brouwer tries to arrive at a genetic understanding of mathematics in experience. Ultimately, for Brouwer, mathematics means nothing but an exact activity of the mind, above all language, which consists of mental constructions. The possibility of spiritual constructions is guaranteed by the primal intuition of time.

The primordial phenomenon is no more than the intuition of time, in which repetition of 'thing-in-time and again thing' is possible, but in which (and this is a phenomenon outside mathematics) a sensation can fall apart in component qualities, so that a single moment can be lived-through as a sequence of qualitatively different things.

The process is nothing other than the above-described linking of two sensations in consciousness. A duality arises in consciousness, which includes two entities and the connection between them. Through this human capacity , things, causal consequences, relationships in nature can be seen. Sensory stimuli become perceptions through the actually mathematical primal intuition of time . Every scientific experiment is based on this intuition of duality.

Unlike Immanuel Kant , Brouwer emphasizes that the intuition of time is not a permanent property of the human way of thinking, but is only conveyed through an event from which the consciousness can act freely. In the naive phase before, neither things nor causality are recognized.

Furthermore, in the primal intuition, the properties do not diverge discretely and continuously : they are integrated into one another and cannot be mutually distinguished. This is what sets Brouwer apart from Henri Bergson , who strives to differentiate the discrete (as individual points in time) from the continuous.

For Brouwer, scientifically measurable time is a derived phenomenon. For him, number and measure are initially isolated. The primordial intuition of time is only concerned with the duality that can be drawn from a sequence of times.

construction

Brouwer's original intuition describes the basis of the intellectual faculty. Consciousness can create things through the content of sensory stimuli and mathematical intuition and thus construct the external world as it were (externalization). Second, however, the mind can create new, artificial entities by simply combining elements that exist exclusively in the primordial intuition. This is pure math and independent of experience. Constructive elements that come from the original intuition are for example: unity, continuum , repetition.

For Brouwer, mathematical thinking consists in this construction (Dutch gebouw , building), which is limited to elements of the original intuition. The objects produced in this way exist mathematically . The process of construction is, however, bound to the individual consciousness; Recordings of this process in a symbolic medium cannot replace it. They are suitable for example for exposure. Brouwer was extremely skeptical about the use of special symbols even in his writings.

Brouwer distinguishes between three things:

• Knowledge that is gained firsthand and recorded individually
• its recording in a symbolic , physical medium, as a memory aid
• the interpersonal communication of these symbols and the recording of collective knowledge

The intuitive construction itself is not linguistic, but remains a mental reality based on the primordial intuition of the time. Any analysis of knowledge, according to Brouwer, should focus on the first point.

This is where Brouwers' sharp criticism of the then common philosophies of mathematics begins. Nowhere was language clearly separated from mathematics. Even the intuitionism of the French mathematicians Henri Poincaré , Émile Borel and Henri Lebesgue , who appeared in opposition to logicism and formalism , did not differentiate so sharply. Compared to Brouwer, they use the term intuition vaguely and did not build a systematic theory on it. In particular, the intuition seemed to suffice only for the postulate of the natural series of whole numbers , but not for the real numbers , the Dedekindian introduction of which Brouwer considered to be a matter that was merely linguistically fixed. Brouwer later called his separation of mathematics and mathematical language "the first act of intuitionism".

Application of math

In the beginning, mathematics jumped from the means to the end, i.e. in the third phase of consciousness. A conscious action is based on the previous discovery of regularity. If you intervene in what is happening yourself, you do not get exactly the set purpose through a certain means. In the wake of the refinement of the means that is now beginning, one discovers even more regularities in a concentrated area. Finally, one area of ​​phenomena can be singled out that can be treated intellectually independently of others: mathematics. These regularities (or causal consequences) can be used wherever such regularity is naturally seen. In an attempt to refine the steps and isolate regularity, one can also use virtual causal sequences, which may ultimately be easier to rearrange and also fit again in specific cases. An example is Euclidean geometry , which consists of such virtual causal sequences.

Natural sciences, in turn, have their origin exclusively in the application of mathematics. Discussing the Kantian views of the a priori of time and space , Brouwer notes that - as independent of experience - one must understand all of mathematics (including Euclidean and non-Euclidean geometry) as a priori. On the other hand, there is only one thing from which mathematics is constructed and what connects it with the natural sciences, namely the primordial intuition of time. That is why one could nevertheless assert that ultimately the only a priori element in science is time. Brouwer then rejects the Kantian spatial arguments in his dissertation.

Due to the strict sense in which Brouwer understands intuition, it is also clear that it is by no means denoting a “vague feeling”. From his explanations of space it becomes clear that, in contrast to the etymological connotation, Brouwer's intuition does not conceal a visual or spatial metaphor. After all, he does not understand it to be an obvious truth, but rather the mere ability, based on a duality, to maintain regularity.

### language

Brouwer's intuitionism originally considers language and thoughts to be separate. The subjective thought precedes the language. This, in turn, is initially a purely social phenomenon, used to influence the actions of others. The words that are attached to things do not refer to an external reality, but to the experience of the subject. They are therefore not independent of the “causal” attention. Understanding a word is in this respect a reflex, which, however, has its origin in the primal intuition of the time.

Even if language is an originally social phenomenon to influence the will of others, it can also be found out of habit in the individual subject himself: language plays a role in reflective thinking or as a mnemonic aid. Language is also the means of communicating conceptual constructions; in this respect the language is defective and unstable. The reproduction of a thought, its verification in another subject can lead to different results. However, rational considerations (e.g. mathematical) are, at least hypothetically, structured in the same way, and this interface enables mutual understanding. Incidentally, accuracy would only be possible in solitude and with an unlimited memory.

Brouwer's early work Leven, Kunst en Mystiek polemicizes against the excessive trust in language in philosophical treatises; The use of language is also ridiculous where there is no agreement of the will.

Brouwer later joined the significant circle around Gerrit Mannoury . The aim of the members was to improve people's understanding of the language by making them better understood. The development of the language over time, from primitive sounds to demanding levels, should be taken into account. Brouwer himself wanted on the one hand to create words that conveyed spiritual values ​​to Western societies, on the other hand to show where these values ​​only appear to appear in words that stand for other ideals. These plans did not come about.

Just as language does not refer to a world of objects independent of personal experience, so truth does not refer to an external reality. Rather, truth is also experienced by the subject and means nothing other than the presence of meaning. The truth of an utterance consists in nothing other than the fact that its content has appeared to the consciousness of the subject. Therefore, expectations of future experience or statements about the experience of others are only true insofar as they are anticipations or hypotheses. A sentence only conveys truth if the truth is also experienced.

### logic

Since working on his dissertation, Brouwer tried to make an original contribution to the fundamentals of mathematics. His teacher, Gerrit Mannoury , made him aware of the tendency towards axiomatization and formalization . Following Gottlob Frege , logic was further developed as a discipline. Giuseppe Peano and Bertrand Russell created a new symbolic notation, Georg Cantor created set theory , Ernst Zermelo axiomatized it and proved the well-ordered theorem .

It was believed that the newly discovered logic was the foundation of mathematics. Hilbert axiomatized geometry and based it on certain propositions in which its basic concepts were in certain relationships. He no longer defined it explicitly and left the underlying interpretation open. A few years later, after axiomatization could also be successfully applied elsewhere, he called for all mathematics to be axiomatically founded. In order that the resulting theories should be secure, the consistency of the important axiom systems should be demonstrated in a comprehensive program in a special way.

The ideas for this were already known at the time Brouwer wrote Over de grondlsagen der wiskunde . Brouwer subjected Hilbert's corresponding work to an analysis and came to the conclusion that most of it was an unmathematical unconscious act. Brouwer's analysis results in eight levels, he recognizes three systems of mathematics in it, which occur once with and then without language. The following scheme illuminates his basic idea and describes the transition from first-order mathematics to second-order mathematics:

1. Recording of mathematical constructions (language of mathematics)
2. Perception of a structure in it, conscious use of this structure (classical logic)
3. Isolation of symbols and structure, abstraction from mathematical content, formal constructions (formal logic)

This is the level Peano has reached. Hilbert, who by means of his proof theory with finite methods the consistency wanted to establish, would have as Brouwer analyzed, found on the third level. Hilbert's program was abandoned as implausible due to the results of Kurt Gödel .

For Brouwer, mathematics consists only of the first stage: mental constructions before any language. He dismissed the consistency, which was supposed to be established by the Hilbert program, as a mere linguistic phenomenon and therefore has no mathematical relevance. Brouwer made the real problem in the fact that a purely linguistic argument does not provide a mental construction. He counted the “pathological geometries of Hilbert”, the “logical constructions, certainly those of Bolyai, possibly also Lobatcheffsky”, Cantor's transfinite numbers and Dedekind cuts among these phenomena .

The logical language itself does not always refer directly to an identically structured mental construction. It can happen, for example, that even where the relation of the part to the whole is not included in the mathematical construction (which occurs, for example, in Brouwer's intuitionistic set theory as the basic phenomenon), in the literal expression the real relation is exchanged for the relation part-whole. (Brouwer has the syllogism in mind here.) Such phenomena may arise from the long tradition of logical expressions; nevertheless another language of communication would be possible with the same organization of the intellect and a question of culture .

Sentence of the excluded third party

Regularities of the language that accompanies mathematics , as they were picked up and classified by Aristotle , are for Brouwer mere patterns; they do not necessarily indicate an original construction. Conversely, however, the principle of the excluded third party can be applied to any mathematical construction and never leads to a contradiction . In the work De onbetrouwbaarheid der logische principes (1908) Brouwer explained why there was no reason to believe the principle to be true.

Brouwer used existential statements such as: “There is a sequence 012… 9 in the decimal expansion of Π.” According to Brouwer, there is no reason to consider the principle of the excluded third party to be true here, since one could not envisage any possibility of checking this . Brouwer considered the excluded third principle to be equivalent to claiming that every mathematical problem was solvable. Further "weak counterexamples" based on then unsolved problems can be found in the Brouwer entry in the Stanford Encyclopedia of Philosophy . Later Brouwer actually replaced the dichotomy of true and false with the four possibilities: that the statement has been proven true or false, further if there is no evidence that an algorithm is known for deciding whether to be true or falsehood, and fourthly, that too no such algorithm is known.

After Brouwer had set up an intuitionistic set theory, he was also able to "give strong counterexamples" (see below).

negation

The most fruitful application of Brouwer's views goes back to a few lines of his work Intuitionist Decomposition of Basic Mathematical Concepts (1925). There Brouwer tries, among other things, to provide intuitionist corrections for the negation and in doing so outlines the foundations of a new discipline, intuitionist logic . Brouwer speaks of absurdity and correctness instead of true and false and sets up some principles, whereby he interprets the double negation intuitionistically:

• Brouwer rejects the principle of the excluded third party ( )${\ displaystyle A \ lor \ neg A}$
• In particular, he rejects a special case of this, namely the principle of reciprocity of complementary sets (see the equation: in the article Complement (set theory) )${\ displaystyle A \ cup A ^ {\ rm {c}} = U}$
• So it is rejected: ${\ displaystyle A \ leftrightarrow \ neg \ neg A}$
• The following is retained: ${\ displaystyle A \ to \ neg \ neg A}$
• What is proven, however, is that absurdity-of-absurdity-of-absurdity is equivalent to absurdity. With a triple negation, two negations can therefore be shortened. ( )${\ displaystyle \ neg \ neg \ neg A \ leftrightarrow \ neg A}$

Arend Heyting was the first to formalize such a logic. The attempt was supported by Brouwer himself, but he regarded the task as sterile. Brouwer's intuitionist consideration is based in the corresponding mental construction on the relationship between part and whole, for example to understand the classical modus ponens . More complicated statements can also be obtained by interpreting the species (see below).

Towards the end of his life, Brouwer spoke out more and more favorably against formalization. For example, he praised George Boole's algebra and expressed his aesthetic appreciation for it.

### Intuitionistic math

The whole and rational numbers could be constructed from the mathematical primal intuition . The continuum is given for Brouwer by the experience of "intermediate" the duality of the primal intuition. Brouwer, on the other hand, rejects Cantor's transfinite ordinal numbers because they could not be put into a construction.

The goal of Brouwer's mathematics was to develop a theory of real numbers , the continuum. Only after his topological successes did Brouwer return to set theory and in 1918 published the foundation of set theory independently of the logical proposition of excluded third parties . Brouwer later calls the step taken the “second act of intuitionism”. Unlike his previous views, he now allows for the construction of sets ( spreads in English) not only points that would have to be indicated by a finite number of statements or by a law of construction, but also so-called election sequences . Electoral sequences contain an element of arbitrariness and cannot be fully specified. The concept of election sequences goes into the definition of a point set ( spread ):

First, an unlimited sequence of characters is determined by means of a first character and a law, since the next one is derived from each of these character series. We choose e.g. B. the sequence ζ of "numbers" 1, 2, 3, ... Then a set is a law, on the basis of which, if an arbitrary number is chosen over and over again, each of these choices produces either a certain character with or without termination of the process , or brings about the inhibition of the process including the definitive destruction of its result, whereby for each n > 1 after each unfinished and uninhibited sequence of n - 1 elections at least one number can be given, which, if it is chosen as the nth number does not inhibit the process.

A real point is created when nested intervals are selected. Point sets are special types of point species. A point species is defined by Brouwer as a property that can only be assigned to a point; the definition can also be generalized to higher species, which are properties of species. Species also allow classical operations of set theory (e.g. intersection, union); As noted above (negation), there is a constructive limitation with the complementary species.

The structural theorems about these sets (spreads) are the fan theorem and the bar theorem . Together with the continuity principle, this results in the surprising theorem for full (that is, defined on the entire closed interval from 0 to 1) functions :

Every full function is uniformly continuous .

This sentence is classically invalid. Brouwer used it to give "strong counterexamples" to the principle of the excluded third party . The application of the principle leads to a contradiction .

The function that assigns the value 0 to a real number if it is rational, but the value 1 if it is not rational, must, according to the proposition, be constant in the intuitionistic sense. It is therefore not possible to divide the continuum intuitionistically into rational and irrational numbers. However, it is precisely this result that results from applying the principle of the excluded third party with the property “rationality”, a contradiction. A detailed explanation can be found in the entry on Strong Counterexamples in the Stanford Encyclopedia of Philosophy .

## effect

The results that Brouwer produced from 1909 to 1913 had a lasting influence on the topology. Brouwer combined the set theoretical topology of Georg Cantor and Arthur Moritz Schoenflies with the methods of Henri Poincaré . In particular, Hermann Weyl's work on Riemann surfaces was based on Brouwer's topology. His fixed point theorem found numerous applications outside of topology.

Weyl's provocative article gave Brouwer's intuitionism, especially his rejection of the principle of the excluded third party, a high level of awareness that could not be attained through his own writings and lectures. He himself had little didactic ability to make intuitionism better known or more popular. However, AA Fraenkel , who supplemented the axioms of set theory by Ernst Zermelo , paid constant attention to intuitionism in his numerous books on set theory.

Later reactions to Brouwer's intuitionism refer mainly to Brouwer's pupil Arend Heyting , who formalized intuitionist logic in 1930. Such an attempt by the Russian mathematician Andrei Nikolajewitsch Kolmogorow in 1925 had gone unnoticed. Kurt Gödel and Valeri Iwanowitsch Gliwenko made significant contributions to the development of intuitionist logic. Also Alonzo Church responded in 1928 with an article on the law of the excluded middle. In the 1960s, the basic researcher Stephen Cole Kleene rekindled interest in intuitionist logic.

Representatives of constructive mathematics , on which Brouwer had at least some effect, are Errett Bishop and Paul Lorenzen .

It is not clear to what extent Brouwer could have had an influence on Gödel, who probably - like Ludwig Wittgenstein - heard him at his Vienna lecture in 1928. But that Wittgenstein's lecture was of philosophical interest is recorded in anecdotes by Herbert Feigl and Rudolf Carnap . Wittgenstein is said to have received the impetus for his later philosophical work there.

A moon crater has been named after him and Dirk Brouwer since 1970 .

In his honor, the Dutch Mathematical Society has awarded the Brouwer Medal every three years since 1970 .

## Fonts

Appeared after death:

• Collected Works. North Holland, Amsterdam.
1. Arend Heyting (Ed.): Philosophy and foundations of mathematics. 1975, ISBN 0-7204-2805-X . ( English review )
2. Hans Freudenthal (Ed.): Geometry, analysis, topology and mechanics. 1976, ISBN 0-7204-2076-8 .
• Dirk van Dalen (ed.): Intuitionism. BI Wissenschaftsverlag, 1992, ISBN 3-411-15371-7 . (introduced and commented by Dirk van Dalen; table of contents , PDF file, 60 kB)
• Life, art, and mysticism . In: Notre Dame Journal of Formal Logic. 37, Sommer 1996, pp. 389-429. (English translation by Leven, kunst en mystiek , 1905, by Walter P. Van Stigt)
• Dirk van Dalen (Ed.): LEJ Brouwer en de grondslagen van de wiskunde. Epsilon Uitgaven, Utrecht 2001, ISBN 90-5041-061-8 . (Dutch; annotated new edition of the dissertation, fragments and essays from the following years such as Onbetrouwbaarheid der logische principes ; table of contents , PDF file, 22 kB; Zentralblatt review )

## literature

• Dirk van Dalen: Mystic, geometer, and intuitionist: The Life of LEJ Brouwer. Clarendon Press, Oxford et al. a.
• Walter P. van Stigt: Brouwer's intuitionism. North Holland, Amsterdam a. a. 1990, ISBN 0-444-88384-3 . (Also includes brief biography and full bibliography of Brouwer's published writings)
• Dennis E. Hesseling: Gnomes in the fog: the reception of Brouwer's intuitionism in the 1920s. Birkhäuser, Basel a. a. 2003, ISBN 3-7643-6536-6 . (Monograph on the fundamental dispute)

## Individual evidence

1. Mark van Atten:  L. E. J. Brouwer. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy . , In particular Chronology 1928-1929.
2. ^ Dennis E. Hesseling: Gnomes in the fog: the reception of Brouwer's intuitionism in the 1920s. 2003, p. 346.
3. ^ Walter P. van Stigt: Brouwer's Intuitionism. 1990, p. 115ff.
4. Dirk van Dalen : Mystic, Geometer, and Intuitionist: The Life of LEJ Brouwer. Vol 1, 1999, pp. 82f.
5. ^ Walter P. van Stigt: Brouwer's Intuitionism. 1990, p. 137.
6. Quotation from the English translation of the parts of the dissertation deleted by Korteweg, p. 2, published in Walter P. van Stigt: Brouwer's Intuitionism. 1990, pp. 405-415.
7. ^ Walter P. van Stigt: Brouwer's Intuitionism. 1990, p. 149.
8. ^ Walter P. van Stigt: Brouwer's Intuitionism. 1990, p. 159.
9. David Hilbert: About the basics of logic and arithmetic. In: Negotiations of the Third International Congress of Mathematicians in Heidelberg from August 8 to 13, 1904. pp. 174-185.
10. Dirk van Dalen: Mystic, geometer, and intuitionist: The life of LEJ Brouwer. Vol 1, 1999, pp. 110f.
11. ^ Walter P. van Stigt: Brouwer's Intuitionism. 1990, p. 215.
12. ^ Walter P. van Stigt: Brouwer's intuitionism. 1990, p. 233. Van Stigt quotes here from LEJ Brouwer: Over de grondslagen der wiskunde. 1907, pp. 140f.
13. ^ Walter P. van Stigt: Brouwer's intuitionism. 1990, p. 221. - LEJBrouwer: Over de grondslagen der wiskunde. 1907, p. 129.
14. ^ LEJ Brouwer: Intuitionism. 1992, p. 23.
15. ^ Evidence and rigorous formulations see LEJ Brouwer: Intuitionismus. 1992.
16. Hermann Weyl : About the new basic crisis in mathematics . (May 9, 1920). In: Mathematical Journal. 10, 1921, pp. 39-79.
17. ^ Gazetteer of Planetary Nomenclature