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Constructive Nonstandard Analysis
In mathematics, constructive nonstandard analysis is a version of Abraham Robinson's nonstandard analysis, developed by Moerdijk (1995), Palmgren (1998), Ruokolainen (2004). Ruokolainen wrote: :The possibility of constructivization of nonstandard analysis was studied by Palmgren (1997, 1998, 2001). The model of constructive nonstandard analysis studied there is an extension of Moerdijk’s (1995) model for constructive nonstandard arithmetic. See also *Smooth infinitesimal analysis *John Lane Bell References *Ieke Moerdijk, ''A model for intuitionistic nonstandard arithmetic'', Annals of Pure and Applied Logic, vol. 73 (1995), pp. 37–51. : "Abstract: This paper provides an explicit description of a model for intuitionistic nonstandard arithmetic, which can be formalized in a constructive metatheory without the axiom of choice* Erik Palmgren, ''Developments in Constructive Nonstandard Analysis'', Bulletin of Symbolic Logic Volume 4, Number 3 (1998), 233–272. : "Abst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abraham Robinson
Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorporated into modern mathematics. Nearly half of Robinson's papers were in applied mathematics rather than in pure mathematics. Biography He was born to a Jewish family with strong Zionist beliefs, in Waldenburg, Germany, which is now Wałbrzych, in Poland. In 1933, he emigrated to British Mandate of Palestine, where he earned a first degree from the Hebrew University. Robinson was in France when the Nazis invaded during World War II, and escaped by train and on foot, being alternately questioned by French soldiers suspicious of his German passport and asked by them to share his map, which was more detailed than theirs. While in London, he joined the Free French Air Force and contributed to the war effort by teaching himself aerodynamics an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonstandard Analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Nonstandard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers. Nonstandard analysis originated in the early 1960s by the mathematician Abraham Robinson. He wrote: ... the idea of infinitely small or ''infinitesimal'' quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection ... that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ieke Moerdijk
Izak (Ieke) Moerdijk (; born 23 January 1958) is a Dutch mathematician, currently working at Utrecht University, who in 2012 won the Spinoza prize. Education and career Moerdijk studied mathematics, philosophy and general linguistics at the University of Amsterdam. He obtained his PhD ''cum laude'' in 1985 at the same institution. His thesis was entitled ''Topics in intuitionism and topos theory'' and was written under the supervision of Anne Sjerp Troelstra. After that, he worked as postdoctoral researcher at the University of Chicago and Cambridge. From 1988 to 2011 he was professor at Utrecht University. After working at the Mathematical Institute of the Radboud University Nijmegen for a few years, he returned to Utrecht University in 2016. In 2000 Moerdijk was an invited speaker to the 3rd European Congress of Mathematics. He was elected member of the Royal Netherlands Academy of Arts and Sciences in 2006 and of the Academia Europaea in 2014. Moerdijk received the 2011 De ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructive Mathematics
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation. There are many forms of constructivism. These include the program of intuitionism founded by Brouwer, the finitism of Hilbert and Bernays, the constructive recursive mathematics of Shanin and Markov, and Bishop's program of constructive analysis. Constructivism also includes the study of constructive set theories such as CZF ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Infinitesimal Analysis
Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of being expressed in terms of discrete entities. As a theory, it is a subset of synthetic differential geometry. The ''nilsquare'' or ''nilpotent'' infinitesimals are numbers ''ε'' where ''ε''² = 0 is true, but ''ε'' = 0 need not be true at the same time. Overview This approach departs from the classical logic used in conventional mathematics by denying the law of the excluded middle, e.g., ''NOT'' (''a'' ≠''b'') does not imply ''a'' = ''b''. In particular, in a theory of smooth infinitesimal analysis one can prove for all infinitesimals ''ε'', ''NOT'' (''ε'' ≠0); yet it is provably false that all infinitesimals are equal to zero. One can see that the law of excluded middle cannot hold from the following basic theorem (again, unde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Lane Bell
John Lane Bell (born March 25, 1945) is an Anglo-Canadian philosopher, mathematician and logician. He is Professor Emeritus of Philosophy at the University of Western Ontario in Canada. His research includes such topics as set theory, model theory, lattice theory, modal logic, quantum logic, constructive mathematics, type theory, topos theory, infinitesimal analysis, spacetime theory, and the philosophy of mathematics. He is the author of more than 70 articles and of 13 books. In 2009, he was elected a Fellow of the Royal Society of Canada. Biography John Bell was awarded a scholarship to Oxford University at the age of 15, and graduated with a D.Phil. in Mathematics: his dissertation supervisor was John Crossley. During 1968–89 he was Lecturer in Mathematics and Reader in Mathematical Logic at the London School of Economics. Bell's students include Graham Priest Graham Priest (born 1948) is Distinguished Professor of Philosophy at the CUNY Graduate Center, as well as a re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of Symbolic Logic
Bulletin or The Bulletin may refer to: Periodicals (newspapers, magazines, journals) * Bulletin (online newspaper), a Swedish online newspaper * ''The Bulletin'' (Australian periodical), an Australian magazine (1880–2008) ** Bulletin Debate, a famous dispute from 1892 to 1893 between Henry Lawson and Banjo Paterson * ''The Bulletin'' (alternative weekly), an alternative weekly published in Montgomery County, Texas, U.S. * ''The Bulletin'' (Bend), a daily newspaper in Bend, Oregon, U.S. * ''The Bulletin'' (Belgian magazine), a weekly English-language magazine published in Brussels, Belgium * ''The Bulletin'' (Philadelphia newspaper), a newspaper in Philadelphia, Pennsylvania, U.S. (2004–2009) * ''The Bulletin'' (Norwich) * ''The Bulletin'' (Pittsburgh), a monthly community newspaper in Pittsburgh, Pennsylvania, U.S. * ''London Bulletin'', surrealist monthly magazine (1938–1940) * ''The Morning Bulletin'', a daily newspaper published in Rockhampton, Queensland, Austral ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Errett Bishop
Errett Albert Bishop (July 14, 1928 – April 14, 1983) was an Americans, American mathematician known for his work on analysis. He expanded constructive analysis in his 1967 ''Foundations of Constructive Analysis'', where he Mathematical proof, proved most of the important theorems in real analysis by Constructivism (mathematics), constructive methods. Life Errett Bishop's father, Albert T. Bishop, graduated from the United States Military Academy at West Point, ending his career as professor of mathematics at Wichita State University in Kansas. Although he died when Errett was less than 4 years old, he influenced Errett's eventual career by the math texts he left behind, which is how Errett discovered mathematics. Errett grew up in Newton, Kansas. Errett and his sister were apparent math prodigies. Bishop entered the University of Chicago in 1944, obtaining both the BS and MS in 1947. The doctoral studies he began in that year were interrupted by two years in the US Army, 1950â ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Juha Ruokolainen
Juha is a masculine given name of Finnish origin derived from Johannes (or John in English language contexts). Notable people with the name include: * Juha Alén * Juha Gustafsson * Juha Hakola * Juha Harju * Juha Haukkala * Juha Hautamäki * Juha Helppi * Juha Hernesniemi * Juha Hirvi * Juha Hurme * Juha Ikonen * Juha Isolehto * Juha Janhunen * Juha Jokela * Juha Järvenpää * Juha Kankkunen * Juha Kaunismäki * Juha Kilpiä * Juha Kivi * Juha Kylmänen * Juha Lallukka * Juha Laukkanen * Juha Leimu * Juha Leiviskä * Juha Leskinen * Juha Lind * Juha Malinen * Juha Mannerkorpi * Juha Metsola * Juha Metsäperä * Juha Mieto * Juha Pasoja * Juha Pekka Alanen * Juha Peltola * Juha Pentikäinen * Juha Pirinen * Juha Pitkämäki * Juha Plosila * Juha Rantasila * Juha Rehula * Juha Reini * Juha Riihijärvi * Juha Riippa * Juha Ruusuvuori * Juha Salminen * Juha Salo * Juha Sihvola * Juha Sipilä * Juha Soukiala * Juha Suoranta * Juha Tapio * Juha K. Tapio * Juha Tiaine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructivism (mathematics)
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation. There are many forms of constructivism. These include the program of intuitionism founded by Brouwer, the finitism of Hilbert and Bernays, the constructive recursive mathematics of Shanin and Markov, and Bishop's program of constructive analysis. Constructivism also includes the study of constructive set theories such as CZ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
