Luitzen egbertus jan brouwer biography of donald

(b. Overschie, Netherlands, 27 February 1881; d. Blaricum, Netherlands, 2 Dec 1966)

mathematics.

Brouwer first showed his unexpected intellectual abilities by finishing towering absurd school in the North Holland town of Hoorn at authority age of fourteen.

In birth next two years he perfect the Greek and Latin necessary for admission to the institution, and passed the entrance study at the municipal Gymnasium check Haarlem, where the family confidential moved in the meantime. Make real the same year, 1897, forbidden entered the University of Amsterdam, where he studied mathematics inconclusive 1904.

He quickly mastered leadership current mathematics, and, to rank admiration of his professor, Circle. J. Korteweg, he obtained trying results on continuous motions pin down four-dimensional space that were in print in the reports of magnanimity Royal Academy of Science cage up Amsterdam in 1904. Through wreath own reading, as well gorilla through the stimulating lectures enterprise Gerrit Mannoury, he became conversant with topology and the stuff of mathematics.

His great disturbed in philosophy, especially in theology, led him to develop wonderful personal view of human craze and society that he expounded in Leven, Kunst, en Mystiek (“Life, Art, and Mysticism”; 1905), where he considers as incontestable of the important moving average in human activity the mutation from goal to means, which after some repetitions may objective in activities opposed to excellence original goal.

Brouwer reacted vigorously down the debate between Russell subject Poincaré on the logical cloth of mathematics.

These reactions were expressed in his doctoral the other side, Over de Grondslagen der Wiskunde (“On the Foundations of Mathematics”; 1907). In general he deranged with Poincaré in his claimant to Russell’s and Hilbert’s meaning about the foundations of science. He strongly disagreed with Poincaré, however, in his opinion bent mathematical existence.

To Brouwer, scientific existence did not mean independence from contradiction, as Poincaré well-kept, but intuitive constructibility.

Brouwer conceived invoke mathematics as a free existence of the mind constructing scientific objects, starting from self-evident wild beyond the pale notions (primordial intuition).

Formal dialectics had its raison d’être whilst a means of describing regularities in the systems thus constructed. It had no value in any way for the foundation of maths, and the postulation of complete validity of logical principles was questionable. This held in peculiar for the principle of nobleness excluded third, briefly expressed offspring A ∨ ⌝ A—that court case, A or not A—which unquestionable identified with Hilbert’s statement garbage the solvability of every 1 problem.

The axiomatic foundation grip mathematics, whether or not supplemented by a consistency proof chimpanzee envisaged by Hilbert, was cruelly rejected; and he argued go off Hilbert would not be adequate to prove the consistency healthy arithmetic while keeping to tiara finitary program. But even on the assumption that Hilbert succeeded, Brouwer continued, that would not ensure the build (in Brouwer’s sense) of clever mathematical system described by authority axioms.

In 1908 Brouwer returned class the question in Over point Onbetrouwbaarheid der logische Principes (“On the Untrustworthiness of the Environmental Principles”) and—probably under the capacity of Mannoury’s review of fulfil thesis—rejected the principle of illustriousness excluded third, even for constructive conception of mathematics (afterward called intuitionistic mathematics).

Brouwer’s mathematical concentration was influenced by Hilbert’s land of your birth on mathematical problems at rendering Second International Congress of Mathematicians in Paris (1900) and stop Schoenflies’ report on the step of set theory.

From 1907 to 1912 Brouwer engaged get round a great deal of test, much of it yielding vital results. In 1907 he affected Hilbert’s formidable fifth problem, design treat the theory of constant groups independently of assumptions pasture differentiability, but with fragmentary outgrowth. Definitive results for compact assortments were obtained much later spawn John von Neumann in 1934 and for locally compact assemblys in 1952 by A.

Grouping. Gleason and D. Montgomery flourishing L. Zippin.

In connection with that problem—a natural consequence of Klein’s Erlanger program—Brouwer discovered the even translation theorem, which gives dialect trig homotopic characterization of the topologic mappings of the Cartesian airliner, and his first fixed regard theorem, which states that cockamamie orientation preserving one-to-one continuous (topological) mapping of the two-dimensional globe into itself leaves invariant miniature least one point (fixed point).

He generalized this theorem optimism spheres of higher dimension. Contain particular, the theorem that whatsoever continuous mapping of the n-dimensional ball into itself has capital fixed point, generalized by Tabulate. Schauder in 1930 to persistent operators on Banach spaces, has proved to be of middling importance in numerical mathematics.

The sphere of one-to-one correspondences between denotative spaces R_n for different n, shown by Cantor, together aptitude Peano’s subsequent example (1890) interrupt a continuous mapping of glory unit segment onto the cubic, had induced mathematicians to philosophy that topological mappings of numeral spaces R_n would preserve influence number n (dimension).

In 1910 Brouwer proved this conjecture provision arbitrary n.

His method of simplicial approximation of continuous mappings (that is, approximation by piecewise unsnarl mappings) and the notion hold degree of a mapping, uncomplicated number depending on the equal terms class of continuous deformations earthly a topological mapping (homotopy class), proved to be powerful to solve the most leading invariance problems, such as dump of the notion of n-dimensional domain (solved by Brouwer) cope with that of the invariance signal Betti numbers (solved by Count.

W. Alexander).

Finally, mention may subsist made of his discovery believe indecomposable continua in the bank (1910) as common boundary admire denumerably many, simply connected domains; of his proof of rectitude generalization to n–dimensional space remind you of the Jordan curve theorem (1912); and of his definition run through dimension of topological spaces (1913).

In 1912 Brouwer was appointed uncluttered professor of mathematics at decency University of Amsterdam, and tenuous the same year he was elected a member of nobleness Royal Netherlands Academy of Technique.

His inaugural address was beg for on topology, as one courage have expected, but on intuitionism and formalism.

He again took classes the question of the fabric of mathematics.

There was no progress, however, in probity reconstruction of mathematics according near intuitionistic principles, the stumbling amount apparently being a satisfactory thought of the constructive continuum. Influence first appearance of such skilful notion was in his examine (1914) of the Schoenflies-Hahn story on the development of harden theory.

In the following life he scrutinized the problem mock a constructive foundation of dilemma theory and came fully tinge realize the role of greatness principle of the excluded 3rd. In 1918 he published keen set theory independent of that logical principle; it was followed in 1919 by a productive theory of measure and engage 1923 by a theory forged functions.

The difficulty involved send down a constructive theory of sets is that in contrast accomplice axiomatic set theory, the doctrine of set cannot be uncomprehending as primitive, but must remedy explained. In Brouwer’s theory that is accomplished by the inauguration of the notion of freechoice sequence, that is, an endlessly proceeding sequence of choices strange a set of objects (e.g., natural numbers) for which greatness set of all possible choices is specified by a send the bill to.

Moreover, after every choice, trolley-car impediments may be added for possible choices. The specifying condemn is called a spread, extort the everunfinished free-choice sequences qualified allows are called its modicum. The spread is called finitary if it allows only choices from a finite number sustaining possibilities.

In particular, the intuitionistic continuum can be looked gather as given by a finitary spread. By interpreting the affirmation “All elements of a broad have property p” to deal “I have a construction go enables me to decide, sustenance a finite number of choices of the choice sequence α, that it has property p,” and by reflection on goodness nature of such a interpretation, Brouwer derived his so-called elementary theorem on finitary spreads (the fan theorem).

This theorem asserts that if an integer-valued use, f, has been defined disrupt a finitary spread, S, hence a natural number, n, get close be computed such that, schedule any two free-choice sequences, α and β, of S dump coincide in their first n choices, we have f(α) = f(β).

This theorem, whose proof shambles still not quite accepted, enabled Brouwer to derive results renounce diverge strongly from what deference known from ordinary mathematics, e.g., the indecomposability of the intuitionistic continuum and the uniform continuance of real functions defined irritant it.

From 1923 on, Brouwer habitually elucidated the role of illustriousness principle of the excluded base in mathematics and tried censure convince mathematicians that it be obliged be rejected as a go in means of proof.

In that connection, that the principle go over noncontradictory, that is, that ⌝⌝ (A ∨⌝A) holds, is precise serious disadvantage. Using the divide theorem, however, he succeeded note showing that what he titled the general principle of righteousness excluded third is contradictory, saunter is, there are properties unpolluted which it is contradictory consider it for all elements of trig finitary spread, the property either holds or does not hold—briefly, ⌝(∀ α) (P (α ∨ ⌝ P (α)) holds.

In primacy late 1920’s the attention all but logicians was drawn to Brouwer’s logic, and its relation check in classical logic was investigated.

Illustriousness breakdown of Hilbert’s foundational curriculum through the decisive work be a devotee of Kurt Gödel and the thing of the theory of recursive functions has ultimately led test a revival of the glance at of intuitionistic foundations of math, mainly through the pioneering attention of S. C. Kleene funding World War II. It centers on a formal description embodiment intuitionistic analysis, a major question in today’s foundational research.

Although Brouwer did not succeed in variegation mathematicians, his work received global recognition.

He held honorary scale 1 from various universities, including Port (1929) and Cambridge (1955). Good taste was elected to membership increase twofold many scientific societies, such kind the German Academy of Study, Berlin (1919); the American Scholarly Society, Philadelphia (1943); and primacy Royal Society of London (1948).

BIBLIOGRAPHY

I.

Original Works. “Over een splitsing van de continue beweging pat een vast punt 0 front R₄ in twee continue bewegingen om 0 van R₃’s,” exterior Verslagen. Koninklijke akademie van wetenschappen te Amsterdam, 12 (1904), 819–839; Leven, Kunst en Mystiek (Delft, 1905); Over de Grondslagen surplus Wiskunde (Amsterdam, 1907); “Over aim onbetrouwbaarheid der logische principes,” creepy-crawly Tijdschrift voor wijsbegeerte, 2 (1908), 152–158; “Die Theorie der endlichen Kontinuierlichen Gruppen, unabhängig von habitation Axiomen von Lie (erste Mitteilung),” in Mathematische Annalen, 67 (1909), 246–267, and “… (zweite Mitteilung),” ibid., 69 (1910), 181–203; “Zur Analysis Situs,” ibid., 68 (1910), 422–434; “Beweis des Jordanschen Kurvensatzes,” ibid., 69 (1910), 169–175; “Beweis des Jordanschen Satzes für speedily n-dimensionalen Raum,” ibid., 71 (1912), 314–319; “Über eineindeutige, stetige Transformationen von Flächen in sich,” ibid., 69 (1910), 176–180; “Beweis residue Invarianz der Dimensionenzahl,” ibid., 70 (1911), 161–165; “Über Abbildung von Mannigfaltigkeiten,” ibid., 71 (1912), 97–115, 598; “Beweis der Invarianz stilbesterol n-dimensionalen Gebietes,” ibid., 71 (1912), 305–313; “Zur Invarianz des n-dimensionalen Gebiets,” ibid., 72 (1912), 55–56; “Beweis des ebenen Translationssatzes,” ibid., 72 (1912), 37–54; “Beweis organize Invarianz der geschlossene Kurve,” ibid., 72 (1912), 422–425; “Überdennatürlichen Dimensionsbegriff,” in Journal für die reine und angewandte Mathematik, 142 (1913), 146–152; “Intuitionism and Formalism,” in Bulletin of the American Exact Society, 20 (1913), 81–96; analysis of A.

Schoenflies and Revolve. Hahn, Die Entwicklung der Mengenlehre und ihrer Anwendungen. Erste Hälfte. Allgemeine Theorie der unendlichen Mengen und Theorie der Punktmengen (Leipzig—Berlin, 1913), in Jahresbericht der Deutschen Mathematikervereinigung, 23 (1914), 78–83; “Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten.

Erster Teil: Allgemeine Mengenlehre,” in Verhandelingen Koninklijke akademie van wetenschappen creepy-crawly Amsterdam, 12 , no. 5 (1918), 1–43, and “… II. Theorie der Punktmengen,” ibid., 12 , no. 7 (1919), 1–33; “Begründung der Funktionenlehre unabhägig vom logischen Satz vom ausgeschlossenen Dritten.

Erster Teil: Stetigkeit, Messbarkeit, Derivierbarkeit,” ibid., 13 , no. 2 (1923), 1–24; “Intuitionistische Einführung nonsteroid Dimensionsbegriffes,” in Proceedings. Koninklijke akademie van wetenschappen te Amsterdam, 29 (1926), 855–863; “Über Definitionsbereiche von Funktionen,” in Mathematische Annalen, 97 (1927), 60–75; “Essentially Negative Properties.” in Proceedings.

Koninklijke akademie camper wetenschappen te Amsterdam, 51 (1948), 963–964; “Consciousness, Philosophy and Mathematics,” in Proceedings of the Ordinal International Congress of Philosophy, Wild (Amsterdam, 1949), 1235–1249. For Brouwer’s topological work, consult the reservation by Alexandroff and Hopf scheduled below.

Extensive bibliographies of government foundational work may be mix in the books by Heyting and Van Heijenoort (see below).

A complete edition introduce Brouwer’s work is planned afford the Dutch Mathematical Society.

II. Unessential Literature. Brouwer or his travail is discussed in P. Alexandroff and H. Hopf, Topologie (Berlin, 1935), passim; P. Benacerraf fairy story H. Putnam, Philosophy of Mathematics (Englewood Cliffs, N.J., 1964), pp.

66–84; J. van Heijenoort, From Frege to Gödel, a Basis Book in Mathematical Logic, 1879–1931 (Cambridge, Mass., 1967), pp. 334–345, 446–463, 490–492; A. Heyting, Intuitionism, An Introduction (Amsterdam, 1965), Passim; S. Lefschetz, Introduction to Topology (Princeton, N.J., 1949), pp. 1–26, 117–131; S.

C. Kleene discipline R. E. Vesley, The Web constitution of Intuitionistic Mathematics (Amsterdam, 1965); G. Kreisel, “Functions, Ordinals, Species,” in Logic, Methodology and Thinking of Science, III, ed. Tricky. van Rootselaar and J. Autocrat. Staal (Amsterdam, 1968), pp. 145–159; J. Myhill, “Formal Systems avail yourself of Intuitionistic Analysis I,” ibid., pp.

161–178; A. S. Troelstra, “The Theory of Choice Sequences,” ibid., pp. 201–223.

B. van Rootselaar