Split (2)

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57280c03ff5b5019007d9bad	What was Von Neumann's given name as member of Austro-Hungarian nobility?	Margittai Neumann János	[ "John_von_Neumann\n\nIn 1913, his father was elevated to the nobility for his service to the Austro-Hungarian Empire by Emperor Franz Joseph. The Neumann family thus acquired the hereditary appellation Margittai, meaning of Marghita. The family had no connection with the town; the appellation was chosen in referenc...
57280cffff5b5019007d9bcc	At what age did schooling begin in Hungary?	ten	[ "John_von_Neumann\n\nFormal schooling did not start in Hungary until the age of ten. Instead, governesses taught von Neumann, his brothers and his cousins. Max believed that knowledge of languages other than Hungarian was essential, so the children were tutored in English, French, German and Italian. By the age of ...
57280cffff5b5019007d9bcd	Who taught children before they began school in Hungary?	governesses	[ "John_von_Neumann\n\nFormal schooling did not start in Hungary until the age of ten. Instead, governesses taught von Neumann, his brothers and his cousins. Max believed that knowledge of languages other than Hungarian was essential, so the children were tutored in English, French, German and Italian. By the age of ...
57280cffff5b5019007d9bce	What languages were Von Neumann and his siblings taught?	English, French, German and Italian	[ "John_von_Neumann\n\nFormal schooling did not start in Hungary until the age of ten. Instead, governesses taught von Neumann, his brothers and his cousins. Max believed that knowledge of languages other than Hungarian was essential, so the children were tutored in English, French, German and Italian. By the age of ...
57280cffff5b5019007d9bcf	At what age was Von Neumann familiar with calculus?	8	[ "John_von_Neumann\n\nFormal schooling did not start in Hungary until the age of ten. Instead, governesses taught von Neumann, his brothers and his cousins. Max believed that knowledge of languages other than Hungarian was essential, so the children were tutored in English, French, German and Italian. By the age of ...
57280e013acd2414000df341	What did Von Neumann enter in 1911?	Lutheran Fasori Evangelikus Gimnázium	[ "John_von_Neumann\n\nVon Neumann entered the Lutheran Fasori Evangelikus Gimnázium in 1911. This was one of the best schools in Budapest, part of a brilliant education system designed for the elite. Under the Hungarian system, children received all their education at the one gymnasium. Despite being run by the Luth...
57280e013acd2414000df342	In what room did children get education in Hungary?	gymnasium	[ "John_von_Neumann\n\nVon Neumann entered the Lutheran Fasori Evangelikus Gimnázium in 1911. This was one of the best schools in Budapest, part of a brilliant education system designed for the elite. Under the Hungarian system, children received all their education at the one gymnasium. Despite being run by the Luth...
57280e013acd2414000df343	What collective name was given to the generation of scholars produced by the Hungarian school system?	Martians	[ "John_von_Neumann\n\nVon Neumann entered the Lutheran Fasori Evangelikus Gimnázium in 1911. This was one of the best schools in Budapest, part of a brilliant education system designed for the elite. Under the Hungarian system, children received all their education at the one gymnasium. Despite being run by the Luth...
57280e013acd2414000df344	In what year did Wigner win Nobel Prize?	1963	[ "John_von_Neumann\n\nVon Neumann entered the Lutheran Fasori Evangelikus Gimnázium in 1911. This was one of the best schools in Budapest, part of a brilliant education system designed for the elite. Under the Hungarian system, children received all their education at the one gymnasium. Despite being run by the Luth...
57280f6cff5b5019007d9c1c	At what age did Von Neumann begin to study advance calculus?	15	[ "John_von_Neumann\n\nAlthough Max insisted von Neumann attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gáb...
57280f6cff5b5019007d9c1d	Who did Von Neumann begin his studies of advanced calculus under?	Gábor Szegő	[ "John_von_Neumann\n\nAlthough Max insisted von Neumann attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gáb...
57280f6cff5b5019007d9c1e	By 19 how many papers had Von Neumann published?	two major mathematical papers	[ "John_von_Neumann\n\nAlthough Max insisted von Neumann attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gáb...
57280f6cff5b5019007d9c1f	Von Neumann's modern definition of ordinal numbers superseded whose definition?	Georg Cantor	[ "John_von_Neumann\n\nAlthough Max insisted von Neumann attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gáb...
57280f6cff5b5019007d9c20	What prize did Von Neuman win at the conclusion of his gymansium formal education?	Eötvös Prize, a national prize for mathematics	[ "John_von_Neumann\n\nAlthough Max insisted von Neumann attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gáb...
5728105e3acd2414000df37b	What career path was decided upon for Von Neuman?	chemical engineer	[ "John_von_Neumann\n\nSince there were few posts in Hungary for mathematicians, and those were not well-paid, his father wanted von Neumann to follow him into industry and therefore invest his time in a more financially useful endeavor than mathematics. So it was decided that the best career path was to become a che...
5728105e3acd2414000df37c	At what university was it arranged for Von Neumann to enter a chemistry program?	University of Berlin	[ "John_von_Neumann\n\nSince there were few posts in Hungary for mathematicians, and those were not well-paid, his father wanted von Neumann to follow him into industry and therefore invest his time in a more financially useful endeavor than mathematics. So it was decided that the best career path was to become a che...
5728105e3acd2414000df37d	Where did Von Neumann study as a PHD candidate?	Pázmány Péter University in Budapest	[ "John_von_Neumann\n\nSince there were few posts in Hungary for mathematicians, and those were not well-paid, his father wanted von Neumann to follow him into industry and therefore invest his time in a more financially useful endeavor than mathematics. So it was decided that the best career path was to become a che...
5728105e3acd2414000df37e	What was the topic of Von Neuman's thesis?	axiomatization of Cantor's set theory	[ "John_von_Neumann\n\nSince there were few posts in Hungary for mathematicians, and those were not well-paid, his father wanted von Neumann to follow him into industry and therefore invest his time in a more financially useful endeavor than mathematics. So it was decided that the best career path was to become a che...
5728115c3acd2414000df39b	Where did Von Neumann begin to lecture in 1928?	University of Berlin	[ "John_von_Neumann\n\nVon Neumann's habilitation was completed on December 13, 1927, and he started his lectures as a privatdozent at the University of Berlin in 1928. By the end of 1927, von Neumann had published twelve major papers in mathematics, and by the end of 1929, thirty-two papers, at a rate of nearly one ...
5728115c3acd2414000df39c	By the end of 1927 how many papers had Von Neuman written?	twelve major papers in mathematics	[ "John_von_Neumann\n\nVon Neumann's habilitation was completed on December 13, 1927, and he started his lectures as a privatdozent at the University of Berlin in 1928. By the end of 1927, von Neumann had published twelve major papers in mathematics, and by the end of 1929, thirty-two papers, at a rate of nearly one ...
5728115c3acd2414000df39d	How many papers had Von Neumann written by 1929?	thirty-two papers	[ "John_von_Neumann\n\nVon Neumann's habilitation was completed on December 13, 1927, and he started his lectures as a privatdozent at the University of Berlin in 1928. By the end of 1927, von Neumann had published twelve major papers in mathematics, and by the end of 1929, thirty-two papers, at a rate of nearly one ...
5728115c3acd2414000df39e	What better offer came for Von Neumann in 1930?	Princeton University	[ "John_von_Neumann\n\nVon Neumann's habilitation was completed on December 13, 1927, and he started his lectures as a privatdozent at the University of Berlin in 1928. By the end of 1927, von Neumann had published twelve major papers in mathematics, and by the end of 1929, thirty-two papers, at a rate of nearly one ...
5728121c2ca10214002d9d34	When did Von Neumann get married?	New Year's Day in 1930	[ "John_von_Neumann\n\nOn New Year's Day in 1930, von Neumann married Mariette Kövesi, who had studied economics at the Budapest University. Before his marriage he was baptized a Catholic. Max had died in 1929. None of the family had converted to Christianity while he was alive, but afterwards they all did. They had ...
5728121c2ca10214002d9d35	What was Von Neumann's wife's name?	Mariette Kövesi	[ "John_von_Neumann\n\nOn New Year's Day in 1930, von Neumann married Mariette Kövesi, who had studied economics at the Budapest University. Before his marriage he was baptized a Catholic. Max had died in 1929. None of the family had converted to Christianity while he was alive, but afterwards they all did. They had ...
5728121c2ca10214002d9d36	Before marriage what faith did Von Neumann join?	he was baptized a Catholic.	[ "John_von_Neumann\n\nOn New Year's Day in 1930, von Neumann married Mariette Kövesi, who had studied economics at the Budapest University. Before his marriage he was baptized a Catholic. Max had died in 1929. None of the family had converted to Christianity while he was alive, but afterwards they all did. They had ...
5728121c2ca10214002d9d37	What was Von Neumann's child's name?	Marina	[ "John_von_Neumann\n\nOn New Year's Day in 1930, von Neumann married Mariette Kövesi, who had studied economics at the Budapest University. Before his marriage he was baptized a Catholic. Max had died in 1929. None of the family had converted to Christianity while he was alive, but afterwards they all did. They had ...
5728121c2ca10214002d9d38	When did Von Neumann get married for a second time?	October 1938	[ "John_von_Neumann\n\nOn New Year's Day in 1930, von Neumann married Mariette Kövesi, who had studied economics at the Budapest University. Before his marriage he was baptized a Catholic. Max had died in 1929. None of the family had converted to Christianity while he was alive, but afterwards they all did. They had ...
572813202ca10214002d9d4e	In what year was Von Neumann offered a lifetime professorship?	1933	[ "John_von_Neumann\n\nIn 1933, von Neumann was offered a lifetime professorship on the faculty of the Institute for Advanced Study when the institute's plan to appoint Hermann Weyl fell through. He remained a mathematics professor there until his death, although he announced that shortly before his intention to resi...
572813202ca10214002d9d4f	In what year did Von Neumann's mother and siblings join him in U.S?	1939	[ "John_von_Neumann\n\nIn 1933, von Neumann was offered a lifetime professorship on the faculty of the Institute for Advanced Study when the institute's plan to appoint Hermann Weyl fell through. He remained a mathematics professor there until his death, although he announced that shortly before his intention to resi...
572813202ca10214002d9d50	In what year did Von Neumann become a naturalized citizen of US?	1937	[ "John_von_Neumann\n\nIn 1933, von Neumann was offered a lifetime professorship on the faculty of the Institute for Advanced Study when the institute's plan to appoint Hermann Weyl fell through. He remained a mathematics professor there until his death, although he announced that shortly before his intention to resi...
572813202ca10214002d9d51	Why was Neuman rejected from joining US Army?	age	[ "John_von_Neumann\n\nIn 1933, von Neumann was offered a lifetime professorship on the faculty of the Institute for Advanced Study when the institute's plan to appoint Hermann Weyl fell through. He remained a mathematics professor there until his death, although he announced that shortly before his intention to resi...
572814ba3acd2414000df401	What type of "off color" humor did Von Neumann especially enjoy?	limericks	[ "John_von_Neumann\n\nVon Neumann liked to eat and drink; his wife, Klara, said that he could count everything except calories. He enjoyed Yiddish and \"off-color\" humor (especially limericks). He was a non-smoker. At Princeton he received complaints for regularly playing extremely loud German march music on his gr...
572814ba3acd2414000df402	What complaints did Von Neumann's neighbor's have?	regularly playing extremely loud German march music	[ "John_von_Neumann\n\nVon Neumann liked to eat and drink; his wife, Klara, said that he could count everything except calories. He enjoyed Yiddish and \"off-color\" humor (especially limericks). He was a non-smoker. At Princeton he received complaints for regularly playing extremely loud German march music on his gr...
572814ba3acd2414000df403	What famous neighbor did Von Neumann have that may have been distracted by the music Von Neumann played?	Albert Einstein,	[ "John_von_Neumann\n\nVon Neumann liked to eat and drink; his wife, Klara, said that he could count everything except calories. He enjoyed Yiddish and \"off-color\" humor (especially limericks). He was a non-smoker. At Princeton he received complaints for regularly playing extremely loud German march music on his gr...
572815ebff5b5019007d9cc6	What mathematician was Von Nemann's closest friend?	Stanislaw Ulam	[ "John_von_Neumann\n\nVon Neumann's closest friend in the United States was mathematician Stanislaw Ulam. A later friend of Ulam's, Gian-Carlo Rota writes: \"They would spend hours on end gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk.\" When von Neumann was dying in hosp...
572815ebff5b5019007d9cc7	What topics of conversation would Ulam and Von Neumann typically share?	gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk	[ "John_von_Neumann\n\nVon Neumann's closest friend in the United States was mathematician Stanislaw Ulam. A later friend of Ulam's, Gian-Carlo Rota writes: \"They would spend hours on end gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk.\" When von Neumann was dying in hosp...
572815ebff5b5019007d9cc8	What would Ulam use to cheer up Von Neumann in the hospital?	a new collection of jokes	[ "John_von_Neumann\n\nVon Neumann's closest friend in the United States was mathematician Stanislaw Ulam. A later friend of Ulam's, Gian-Carlo Rota writes: \"They would spend hours on end gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk.\" When von Neumann was dying in hosp...
57281714ff5b5019007d9cf6	What caused a setback in naive set theory at the beginning of 20th century?	Russell's paradox	[ "John_von_Neumann\n\nThe axiomatization of mathematics, on the model of Euclid's Elements, had reached new levels of rigour and breadth at the end of the 19th century, particularly in arithmetic, thanks to the axiom schema of Richard Dedekind and Charles Sanders Peirce, and geometry, thanks to David Hilbert. At the...
57281714ff5b5019007d9cf7	Who resolved the problem of adequate axiomatization of set theory?	Ernst Zermelo and Abraham Fraenkel	[ "John_von_Neumann\n\nThe axiomatization of mathematics, on the model of Euclid's Elements, had reached new levels of rigour and breadth at the end of the 19th century, particularly in arithmetic, thanks to the axiom schema of Richard Dedekind and Charles Sanders Peirce, and geometry, thanks to David Hilbert. At the...
57281714ff5b5019007d9cf8	What 2 techniques did Von Neumann use to exclude sets in his doctoral thesis in 1925?	the axiom of foundation and the notion of class.	[ "John_von_Neumann\n\nThe axiomatization of mathematics, on the model of Euclid's Elements, had reached new levels of rigour and breadth at the end of the 19th century, particularly in arithmetic, thanks to the axiom schema of Richard Dedekind and Charles Sanders Peirce, and geometry, thanks to David Hilbert. At the...
572818a43acd2414000df465	What is the method to demonstrate that no contradictions were created by the addition of axiom of foundation?	method of inner models	[ "John_von_Neumann\n\nThe axiom of foundation established that every set can be constructed from the bottom up in an ordered succession of steps by way of the principles of Zermelo and Fraenkel, in such a manner that if one set belongs to another then the first must necessarily come before the second in the successi...
572818a43acd2414000df466	Zermelo and Fraenkel's axiom of foundation established that every set be constructed how?	from the bottom up in an ordered succession of steps	[ "John_von_Neumann\n\nThe axiom of foundation established that every set can be constructed from the bottom up in an ordered succession of steps by way of the principles of Zermelo and Fraenkel, in such a manner that if one set belongs to another then the first must necessarily come before the second in the successi...
572818a43acd2414000df467	What excluded a set from belonging to itself in set theory?	if one set belongs to another then the first must necessarily come before the second in the succession	[ "John_von_Neumann\n\nThe axiom of foundation established that every set can be constructed from the bottom up in an ordered succession of steps by way of the principles of Zermelo and Fraenkel, in such a manner that if one set belongs to another then the first must necessarily come before the second in the successi...
57281ad82ca10214002d9df4	What is a proper class?	a class which does not belong to other classes	[ "John_von_Neumann\n\nThe second approach to the problem took as its base the notion of class, and defines a set as a class which belongs to other classes, while a proper class is defined as a class which does not belong to other classes. Under the Zermelo–Fraenkel approach, the axioms impede the construction of a s...
57281ad82ca10214002d9df5	Under what approach does an axiom impede a set of all sets which do not belong to themselves?	Zermelo–Fraenkel	[ "John_von_Neumann\n\nThe second approach to the problem took as its base the notion of class, and defines a set as a class which belongs to other classes, while a proper class is defined as a class which does not belong to other classes. Under the Zermelo–Fraenkel approach, the axioms impede the construction of a s...
57281ad82ca10214002d9df6	Under Von Neumann's approach can a class of all sets that do not belong to themselves be constructed?	class of all sets which do not belong to themselves can be constructed, but it is a proper class and not a set.	[ "John_von_Neumann\n\nThe second approach to the problem took as its base the notion of class, and defines a set as a class which belongs to other classes, while a proper class is defined as a class which does not belong to other classes. Under the Zermelo–Fraenkel approach, the axioms impede the construction of a s...
57281c093acd2414000df4bd	In 1930 who gave a strongly negative answer to Von Neuman's approach to the axiomatic system of theory of sets?	Kurt Gödel	[ "John_von_Neumann\n\nWith this contribution of von Neumann, the axiomatic system of the theory of sets became fully satisfactory, and the next question was whether or not it was also definitive, and not subject to improvement. A strongly negative answer arrived in September 1930 at the historic mathematical Congres...
57281c093acd2414000df4be	What was the central theme of Godel's announcement that the axiomatic system was not complete?	they cannot prove every truth which is expressible in their language	[ "John_von_Neumann\n\nWith this contribution of von Neumann, the axiomatic system of the theory of sets became fully satisfactory, and the next question was whether or not it was also definitive, and not subject to improvement. A strongly negative answer arrived in September 1930 at the historic mathematical Congres...
57281c093acd2414000df4bf	Where was Godel's statement about axiomatic system made in 1930?	Congress of Königsberg	[ "John_von_Neumann\n\nWith this contribution of von Neumann, the axiomatic system of the theory of sets became fully satisfactory, and the next question was whether or not it was also definitive, and not subject to improvement. A strongly negative answer arrived in September 1930 at the historic mathematical Congres...
57281d523acd2414000df4cb	How long did it take von Neumann to figure a response to theory of incompleteness?	less than a month	[ "John_von_Neumann\n\nBut von Neumann, who had participated at the Congress, confirmed his fame as an instantaneous thinker, and in less than a month was able to communicate to Gödel himself an interesting consequence of his theorem: namely that the usual axiomatic systems are unable to demonstrate their own consist...
57281d523acd2414000df4cc	What was the consequence of the theory of incompleteness?	usual axiomatic systems are unable to demonstrate their own consistency	[ "John_von_Neumann\n\nBut von Neumann, who had participated at the Congress, confirmed his fame as an instantaneous thinker, and in less than a month was able to communicate to Gödel himself an interesting consequence of his theorem: namely that the usual axiomatic systems are unable to demonstrate their own consist...
57281d523acd2414000df4cd	Upon revising the theory of incompleteness, what was the name of the new Godel theory?	second incompleteness theorem	[ "John_von_Neumann\n\nBut von Neumann, who had participated at the Congress, confirmed his fame as an instantaneous thinker, and in less than a month was able to communicate to Gödel himself an interesting consequence of his theorem: namely that the usual axiomatic systems are unable to demonstrate their own consist...
57281f30ff5b5019007d9d86	What field did Von Neuman establish?	continuous geometry.	[ "John_von_Neumann\n\nVon Neumann founded the field of continuous geometry. It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is an analogue of complex projective geometry, where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an elem...
57281f30ff5b5019007d9d87	What is the distinction of continuous geometry?	instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval	[ "John_von_Neumann\n\nVon Neumann founded the field of continuous geometry. It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is an analogue of complex projective geometry, where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an elem...
57281f30ff5b5019007d9d88	What was the first example of continuous geometry?	projections of the hyperfinite type II factor	[ "John_von_Neumann\n\nVon Neumann founded the field of continuous geometry. It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is an analogue of complex projective geometry, where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an elem...
572821fc3acd2414000df54f	In what way did von Neumann make spectacular contributions to measure theory?	In a series of famous papers	[ "John_von_Neumann\n\nIn a series of famous papers, von Neumann made spectacular contributions to measure theory. The work of Banach had implied that the problem of measure has a positive solution if n = 1 or n = 2 and a negative solution in all other cases. Von Neumann's work argued that the \"problem is essentiall...
572821fc3acd2414000df550	What concept was relevant to the solvability of the problem of measure?	algebraic concept of solvability of a group is relevant	[ "John_von_Neumann\n\nIn a series of famous papers, von Neumann made spectacular contributions to measure theory. The work of Banach had implied that the problem of measure has a positive solution if n = 1 or n = 2 and a negative solution in all other cases. Von Neumann's work argued that the \"problem is essentiall...
572821fc3acd2414000df551	According to von Neumann what is the most important part of the problem of measure?	change of group that makes a difference, not the change of space	[ "John_von_Neumann\n\nIn a series of famous papers, von Neumann made spectacular contributions to measure theory. The work of Banach had implied that the problem of measure has a positive solution if n = 1 or n = 2 and a negative solution in all other cases. Von Neumann's work argued that the \"problem is essentiall...
5728230b4b864d1900164540	What is often considered more important than the results of von Neumann's papers?	methods of argument he employed	[ "John_von_Neumann\n\nIn a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the ...
5728230b4b864d1900164541	In 1938 what was von Neumann awarded?	Bôcher Memorial Prize	[ "John_von_Neumann\n\nIn a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the ...
5728230b4b864d1900164542	What did von Neumann use the Haar theorem to solve in a 1936 paper?	Hilbert's fifth problem in the case of compact groups	[ "John_von_Neumann\n\nIn a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the ...
5728241e3acd2414000df587	What is von Neumann algebra?	algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator	[ "John_von_Neumann\n\nVon Neumann introduced the study of rings of operators, through the von Neumann algebras. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. The von Neumann bicommutant theorem shows that ...
5728241e3acd2414000df588	What does the von Neumann bicommutant theorem show?	the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries.	[ "John_von_Neumann\n\nVon Neumann introduced the study of rings of operators, through the von Neumann algebras. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. The von Neumann bicommutant theorem shows that ...
5728241e3acd2414000df589	When was direct integral introduced by von Neumann?	1949	[ "John_von_Neumann\n\nVon Neumann introduced the study of rings of operators, through the von Neumann algebras. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. The von Neumann bicommutant theorem shows that ...
5728241e3acd2414000df58a	What was introduced in von Neumann algebras?	study of rings of operators	[ "John_von_Neumann\n\nVon Neumann introduced the study of rings of operators, through the von Neumann algebras. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. The von Neumann bicommutant theorem shows that ...
57282557ff5b5019007d9dfa	What did von Neumann work on in 1937 - 39?	lattice theory	[ "John_von_Neumann\n\nVon Neumann worked on lattice theory between 1937 and 1939. Von Neumann provided an abstract exploration of dimension in completed complemented modular topological lattices: \"Dimension is determined, up to a positive linear transformation, by the following two properties. It is conserved by pe...
57282557ff5b5019007d9dfb	What properties determine dimension in lattice theory?	conserved by perspective mappings ("perspectivities") and ordered by inclusion	[ "John_von_Neumann\n\nVon Neumann worked on lattice theory between 1937 and 1939. Von Neumann provided an abstract exploration of dimension in completed complemented modular topological lattices: \"Dimension is determined, up to a positive linear transformation, by the following two properties. It is conserved by pe...
57282557ff5b5019007d9dfc	Who wrote that "John von Neumann's brilliant mind blazed over lattice theory like a meteor?"	Garrett Birkhoff	[ "John_von_Neumann\n\nVon Neumann worked on lattice theory between 1937 and 1939. Von Neumann provided an abstract exploration of dimension in completed complemented modular topological lattices: \"Dimension is determined, up to a positive linear transformation, by the following two properties. It is conserved by pe...
572827144b864d19001645da	What type of axioms were used by von Neumann in his lattice work?	novel axioms	[ "John_von_Neumann\n\nAdditionally, \"[I]n the general case, von Neumann proved the following basic representation theorem. Any complemented modular lattice L having a \"basis\" of n≥4 pairwise perspective elements, is isomorphic with the lattice ℛ(R) of all principal right-ideals of a suitable regular ring R. This ...
572827144b864d19001645db	How long was von Neumann's lattice theory paper?	140 pages	[ "John_von_Neumann\n\nAdditionally, \"[I]n the general case, von Neumann proved the following basic representation theorem. Any complemented modular lattice L having a \"basis\" of n≥4 pairwise perspective elements, is isomorphic with the lattice ℛ(R) of all principal right-ideals of a suitable regular ring R. This ...
572827144b864d19001645dc	In what room did von Neumann often write at home?	living room writing-table	[ "John_von_Neumann\n\nAdditionally, \"[I]n the general case, von Neumann proved the following basic representation theorem. Any complemented modular lattice L having a \"basis\" of n≥4 pairwise perspective elements, is isomorphic with the lattice ℛ(R) of all principal right-ideals of a suitable regular ring R. This ...
572828782ca10214002d9f78	Who established the framework for quantum mechanics?	Von Neumann	[ "John_von_Neumann\n\nVon Neumann was the first to establish a rigorous mathematical framework for quantum mechanics, known as the Dirac–von Neumann axioms, with his 1932 work Mathematical Foundations of Quantum Mechanics. After having completed the axiomatization of set theory, he began to confront the axiomatizati...
572828782ca10214002d9f79	What paper did von Neumann produce in 1932?	Mathematical Foundations of Quantum Mechanics	[ "John_von_Neumann\n\nVon Neumann was the first to establish a rigorous mathematical framework for quantum mechanics, known as the Dirac–von Neumann axioms, with his 1932 work Mathematical Foundations of Quantum Mechanics. After having completed the axiomatization of set theory, he began to confront the axiomatizati...
572828782ca10214002d9f7a	What von Neumann work preceded the axiomatiztion of quatum mechanics?	axiomatization of set theory	[ "John_von_Neumann\n\nVon Neumann was the first to establish a rigorous mathematical framework for quantum mechanics, known as the Dirac–von Neumann axioms, with his 1932 work Mathematical Foundations of Quantum Mechanics. After having completed the axiomatization of set theory, he began to confront the axiomatizati...
572829bd4b864d190016461e	What are the physics of quantum mechanics?	mathematics of Hilbert spaces and linear operators acting on them	[ "John_von_Neumann\n\nThe physics of quantum mechanics was thereby reduced to the mathematics of Hilbert spaces and linear operators acting on them. For example, the uncertainty principle, according to which the determination of the position of a particle prevents the determination of its momentum and vice versa, is...
572829bd4b864d190016461f	What is the uncertainty principle?	determination of the position of a particle prevents the determination of its momentum and vice versa	[ "John_von_Neumann\n\nThe physics of quantum mechanics was thereby reduced to the mathematics of Hilbert spaces and linear operators acting on them. For example, the uncertainty principle, according to which the determination of the position of a particle prevents the determination of its momentum and vice versa, is...
572829bd4b864d1900164620	The physics of quantum mechanics included special cases for what work?	formulations of both Heisenberg and Schrödinger	[ "John_von_Neumann\n\nThe physics of quantum mechanics was thereby reduced to the mathematics of Hilbert spaces and linear operators acting on them. For example, the uncertainty principle, according to which the determination of the position of a particle prevents the determination of its momentum and vice versa, is...
57282bb1ff5b5019007d9e72	Von Neumann's abstract treatment allowed him to work on what foundational issue?	determinism versus non-determinism	[ "John_von_Neumann\n\nVon Neumann's abstract treatment permitted him also to confront the foundational issue of determinism versus non-determinism, and in the book he presented a proof that the statistical results of quantum mechanics could not possibly be averages of an underlying set of determined \"hidden variabl...
57282bb1ff5b5019007d9e73	What was John Bell's argument?	proof contained a conceptual error and was therefore invalid	[ "John_von_Neumann\n\nVon Neumann's abstract treatment permitted him also to confront the foundational issue of determinism versus non-determinism, and in the book he presented a proof that the statistical results of quantum mechanics could not possibly be averages of an underlying set of determined \"hidden variabl...
57282bb1ff5b5019007d9e74	What did Jeffrey Bub argue in 2010?	Bell had misconstrued von Neumann's proof	[ "John_von_Neumann\n\nVon Neumann's abstract treatment permitted him also to confront the foundational issue of determinism versus non-determinism, and in the book he presented a proof that the statistical results of quantum mechanics could not possibly be averages of an underlying set of determined \"hidden variabl...
57282bb1ff5b5019007d9e75	Did von Neumann rule hidden variable theories?	von Neumann did not claim that his proof completely ruled out hidden variable theories	[ "John_von_Neumann\n\nVon Neumann's abstract treatment permitted him also to confront the foundational issue of determinism versus non-determinism, and in the book he presented a proof that the statistical results of quantum mechanics could not possibly be averages of an underlying set of determined \"hidden variabl...
57282d9cff5b5019007d9e9e	What results came from von Neumann's deep analysis of the measurement problem?	physical universe could be made subject to the universal wave function	[ "John_von_Neumann\n\nIn a chapter of The Mathematical Foundations of Quantum Mechanics, von Neumann deeply analyzed the so-called measurement problem. He concluded that the entire physical universe could be made subject to the universal wave function. Since something \"outside the calculation\" was needed to collap...
57282d9cff5b5019007d9e9f	What "outside of calculation" variable could initiate collapse in von Neumann's conclusion?	consciousness of the experimenter	[ "John_von_Neumann\n\nIn a chapter of The Mathematical Foundations of Quantum Mechanics, von Neumann deeply analyzed the so-called measurement problem. He concluded that the entire physical universe could be made subject to the universal wave function. Since something \"outside the calculation\" was needed to collap...
57282d9cff5b5019007d9ea0	Were von Neumann's conclusions of collapse of entire universe cause accepted widely?	Von Neumann–Wigner interpretation never gained acceptance amongst the majority of physicists	[ "John_von_Neumann\n\nIn a chapter of The Mathematical Foundations of Quantum Mechanics, von Neumann deeply analyzed the so-called measurement problem. He concluded that the entire physical universe could be made subject to the universal wave function. Since something \"outside the calculation\" was needed to collap...
57282ec4ff5b5019007d9ea4	With whom did von Neuman work on a paper in 1936 that introduce quantum logic?	Garrett Birkhoff	[ "John_von_Neumann\n\nIn a famous paper of 1936 with Garrett Birkhoff, the first work ever to introduce quantum logics, von Neumann and Birkhoff first proved that quantum mechanics requires a propositional calculus substantially different from all classical logics and rigorously isolated a new algebraic structure fo...
57282ec4ff5b5019007d9ea5	What concept was created for quatum logic?	propositional calculus	[ "John_von_Neumann\n\nIn a famous paper of 1936 with Garrett Birkhoff, the first work ever to introduce quantum logics, von Neumann and Birkhoff first proved that quantum mechanics requires a propositional calculus substantially different from all classical logics and rigorously isolated a new algebraic structure fo...
57282ec4ff5b5019007d9ea6	What is the difference of logic in quantum theory?	laws of distribution of classical logic, and , are not valid for quantum theory	[ "John_von_Neumann\n\nIn a famous paper of 1936 with Garrett Birkhoff, the first work ever to introduce quantum logics, von Neumann and Birkhoff first proved that quantum mechanics requires a propositional calculus substantially different from all classical logics and rigorously isolated a new algebraic structure fo...
572831002ca10214002da042	What year was game theory established?	1928	[ "John_von_Neumann\n\nVon Neumann founded the field of game theory as a mathematical discipline. Von Neumann proved his minimax theorem in 1928. This theorem establishes that in zero-sum games with perfect information (i.e. in which players know at each time all moves that have taken place so far), there exists a pa...
572831002ca10214002da043	What discipline is game theory derived from?	mathematical discipline.	[ "John_von_Neumann\n\nVon Neumann founded the field of game theory as a mathematical discipline. Von Neumann proved his minimax theorem in 1928. This theorem establishes that in zero-sum games with perfect information (i.e. in which players know at each time all moves that have taken place so far), there exists a pa...
572831002ca10214002da044	what are the possible strategies in minimax theory?	a pair of strategies for both players that allows each to minimize his maximum losses	[ "John_von_Neumann\n\nVon Neumann founded the field of game theory as a mathematical discipline. Von Neumann proved his minimax theorem in 1928. This theorem establishes that in zero-sum games with perfect information (i.e. in which players know at each time all moves that have taken place so far), there exists a pa...
572831002ca10214002da045	What must a player consider when determining every possible strategy?	all the possible responses of his adversary.	[ "John_von_Neumann\n\nVon Neumann founded the field of game theory as a mathematical discipline. Von Neumann proved his minimax theorem in 1928. This theorem establishes that in zero-sum games with perfect information (i.e. in which players know at each time all moves that have taken place so far), there exists a pa...
572832a4ff5b5019007d9ee0	What is the difference of quantum disjunction from classic?	can be true even when both of the disjuncts are false	[ "John_von_Neumann\n\nThe reason for this is that a quantum disjunction, unlike the case for classical disjunction, can be true even when both of the disjuncts are false and this is, in turn, attributable to the fact that it is frequently the case, in quantum mechanics, that a pair of alternatives are semantically d...
572832a4ff5b5019007d9ee1	What is frequently the case in quantum mechanics when a pair of alternatives are semantically determinate?	each of its members are necessarily indeterminate	[ "John_von_Neumann\n\nThe reason for this is that a quantum disjunction, unlike the case for classical disjunction, can be true even when both of the disjuncts are false and this is, in turn, attributable to the fact that it is frequently the case, in quantum mechanics, that a pair of alternatives are semantically d...
572832a4ff5b5019007d9ee2	How can quantum disjuction be illustrated?	illustrated by a simple example	[ "John_von_Neumann\n\nThe reason for this is that a quantum disjunction, unlike the case for classical disjunction, can be true even when both of the disjuncts are false and this is, in turn, attributable to the fact that it is frequently the case, in quantum mechanics, that a pair of alternatives are semantically d...
572833752ca10214002da090	What is optimal strategy?	minimize the maximum loss for each player	[ "John_von_Neumann\n\nSuch strategies, which minimize the maximum loss for each player, are called optimal. Von Neumann showed that their minimaxes are equal (in absolute value) and contrary (in sign). Von Neumann improved and extended the minimax theorem to include games involving imperfect information and games wi...
572833752ca10214002da091	What von Neumann work was published in 1944?	Theory of Games and Economic Behavior	[ "John_von_Neumann\n\nSuch strategies, which minimize the maximum loss for each player, are called optimal. Von Neumann showed that their minimaxes are equal (in absolute value) and contrary (in sign). Von Neumann improved and extended the minimax theorem to include games involving imperfect information and games wi...
572833752ca10214002da092	Who was co-author with von Neman on the Theory of Games and Economic Behavior?	Oskar Morgenstern	[ "John_von_Neumann\n\nSuch strategies, which minimize the maximum loss for each player, are called optimal. Von Neumann showed that their minimaxes are equal (in absolute value) and contrary (in sign). Von Neumann improved and extended the minimax theorem to include games involving imperfect information and games wi...
5728359e2ca10214002da0d0	How did von Neumann raise the level of economics?	several stunning publications	[ "John_von_Neumann\n\nVon Neumann raised the intellectual and mathematical level of economics in several stunning publications. For his model of an expanding economy, von Neumann proved the existence and uniqueness of an equilibrium using his generalization of the Brouwer fixed-point theorem. Von Neumann's model of ...
5728359e2ca10214002da0d1	What did von Neumann prove with his model of expanding economy?	uniqueness of an equilibrium using his generalization of the Brouwer fixed-point theorem	[ "John_von_Neumann\n\nVon Neumann raised the intellectual and mathematical level of economics in several stunning publications. For his model of an expanding economy, von Neumann proved the existence and uniqueness of an equilibrium using his generalization of the Brouwer fixed-point theorem. Von Neumann's model of ...
5728359e2ca10214002da0d2	What did von Neumann consider for his model of expanding economy?	matrix pencil A − λB with nonnegative matrices A and B	[ "John_von_Neumann\n\nVon Neumann raised the intellectual and mathematical level of economics in several stunning publications. For his model of an expanding economy, von Neumann proved the existence and uniqueness of an equilibrium using his generalization of the Brouwer fixed-point theorem. Von Neumann's model of ...

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