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Ultrafinitism
In the philosophy of mathematics, ultrafinitism (also known as ultraintuitionism,International Workshop on Logic and Computational Complexity, ''Logic and Computational Complexity'', Springer, 1995, p. 31. strict formalism,St. Iwan (2000),On the Untenability of Nelson's Predicativism, ''Erkenntnis'' 53(1–2), pp. 147–154. strict finitism, actualism, predicativism, and strong finitism) is a form of finitism and intuitionism. There are various philosophies of mathematics that are called ultrafinitism. A major identifying property common among most of these philosophies is their objections to totality of number theoretic functions like exponentiation over natural numbers. Main ideas Like other finitists, ultrafinitists deny the existence of the infinite set \N of natural numbers, on the basis that it can never be completed (i.e., there is a largest natural number). In addition, some ultrafinitists are concerned with acceptance of objects in mathematics that no one can construct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Esenin-Volpin
Alexander Sergeyevich Esenin-Volpin (also written Ésénine-Volpine and Yessenin-Volpin in his French and English publications; rus, Алекса́ндр Серге́евич Есе́нин-Во́льпин, p=ɐlʲɪˈksandr sʲɪrˈɡʲejɪvʲɪtɕ jɪˈsʲenʲɪn ˈvolʲpʲɪn, a=Alyeksandr Syergyeyevich Yesyenin-Vol'pin.ru.vorb.oga; May 12, 1924March 16, 2016) was a Russian-American poet and mathematician. A dissident, political prisoner and a leader of the Soviet human rights movement, he spent a total of six years incarcerated and repressed by the Soviet authorities in psikhushkas and exile. In mathematics, he is known for his foundational role in ultrafinitism. Life Alexander Volpin was born on May 12, 1924, in the Soviet Union. His mother, Nadezhda Volpin, was a poet and translator from French and English. His father was Sergei Yesenin, a celebrated Russian poet, who never knew his son. Alexander and his mother moved from Leningrad to Moscow in 1933. His first psy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rohit Jivanlal Parikh
Rohit Jivanlal Parikh (born November 20, 1936) is an Indian-American mathematician, logician, and philosopher who has worked in many areas in traditional logic, including recursion theory and proof theory. He is a Distinguished Professor at Brooklyn College at the City University of New York (CUNY). Research Parikh worked on topics like vagueness, ultrafinitism, belief revision, logic of knowledge, game theory and social software (social procedure). This last area seeks to combine techniques from logic, computer science (especially logic of programs) and game theory to understand the structure of social algorithms. Personal life and politics Rohit Parikh was married from 1968 to 1994 to Carol Parikh (née Geris), who is best known for her stories and biography of Oscar Zariski, ''The Unreal Life of Oscar Zariski''. Parikh is a nontheist opposing abortions. To fight abortions he joined the Atheist and Agnostic Pro-Life League. In 2018, a Facebook post by Parikh, called for de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edward Nelson
Edward Nelson (May 4, 1932 – September 10, 2014) was an American mathematician. He was professor in the Mathematics Department at Princeton University. He was known for his work on mathematical physics and mathematical logic. In mathematical logic, he was noted especially for his internal set theory, and views on ultrafinitism and the consistency of Peano arithmetic, arithmetic. In philosophy of mathematics he advocated the view of Formalism (mathematics), formalism rather than Platonism (mathematics), platonism or intuitionism. He also wrote on the relationship between religion and mathematics. Biography Edward Nelson was born in Decatur, Georgia, in 1932. He spent his early childhood in Rome where his father worked for the Italian YMCA. At the advent of World War II, Nelson moved with his mother to New York City, where he attended high school at the Bronx High School of Science. His father, who spoke fluent Russian language, Russian, stayed in St. Petersburg in connection wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophy Of Mathematics
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathematical objects are purely abstract entities or are in some way concrete, and in what the relationship such objects have with physical reality consists. Major themes that are dealt with in philosophy of mathematics include: *''Reality'': The question is whether mathematics is a pure product of human mind or whether it has some reality by itself. *''Logic and rigor'' *''Relationship with physical reality'' *''Relationship with science'' *''Relationship with applications'' *''Mathematical truth'' *''Nature as human activity'' (science, the arts, art, game, or all together) Major themes Reality Logic and rigor Mathematical reasoning requires Mathematical rigor, rigor. This means that the definitions must be absolutely unambiguous and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johannes Hjelmslev
Johannes Trolle Hjelmslev (; 7 April 1873 – 16 February 1950) was a mathematician from Hørning, Denmark. Hjelmslev worked in geometry and history of geometry. He was the discoverer and eponym of the Hjelmslev transformation, a method for mapping an entire hyperbolic plane into a circle with a finite radius. He was the father of Louis Hjelmslev. Originally named Johannes Trolle Petersen, he changed his patronymic to the surname Hjelmslev to avoid confusion with Julius Petersen. Some of his results are known under his original name, including the Petersen–Morley theorem In geometry, the Petersen–Morley theorem states that, if , , are three general skew lines In three-dimensional geometry, skew lines are two Line (geometry), lines that do not Line-line intersection, intersect and are not Parallel (geometry), p .... Publications *Johannes Hjelmslev, ''Grundprinciper for den infinitesimale Descriptivgeometri med Anvendelse paa Læren om variable Figurer. Afhandling for d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Petr Vopěnka
Petr Vopěnka (16 May 1935 – 20 March 2015) was a Czech people, Czech mathematician. In the early seventies, he developed alternative set theory (i.e. alternative to the classical Cantor theory), which he subsequently developed in a series of articles and monographs. Vopěnka’s name is associated with many mathematical achievements, including Vopěnka's principle. Since the mid-eighties he concerned himself with philosophical questions of mathematics (particularly vis-à-vis Edmund Husserl, Husserlian Phenomenology (philosophy), phenomenology). Vopěnka served as the Minister of Education of the Czech Republic (then part of Czech and Slovak Federative Republic, Czechoslovakia) from 1990 to 1992 within the government of List of Prime Ministers of the Czech Republic, Prime Minister Petr Pithart. Biography Petr Vopěnka grew up in small town of Dolní Kralovice. After finishing Gymnasium (school), gymnasium in Ledeč nad Sázavou in 1953 he went to study mathematics at the Mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robin Gandy
Robin Oliver Gandy (22 September 1919 – 20 November 1995) was a British mathematician and logician. He was a friend, student, and associate of Alan Turing, having been supervised by Turing during his PhD at the University of Cambridge, where they worked together. Education and early life Robin Gandy was born in the village of Rotherfield Peppard, Oxfordshire, England. A great-great-grandson of the architect and artist Joseph Gandy (1771–1843), he was the son of Thomas Hall Gandy (1876–1948), a general practitioner, and Ida Caroline née Hony (1885–1977), a social worker and later an author. His brother was the diplomat Christopher Gandy and his sister was the physician Gillian Gandy. Educated at Abbotsholme School in Derbyshire, Gandy took two years of the Mathematical Tripos, at King's College, Cambridge, before enlisting for military service in 1940. During World War II he worked on radio intercept equipment at Hanslope Park, where Alan Turing was working on a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ludwig Wittgenstein
Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. From 1929 to 1947, Wittgenstein taught at the University of Cambridge. Despite his position, only one book of his philosophy was published during his entire life: the 75-page ''Logisch-Philosophische Abhandlung'' (''Logical-Philosophical Treatise'', 1921), which appeared, together with an English translation, in 1922 under the Latin title ''Tractatus Logico-Philosophicus''. His only other published works were an article, "Some Remarks on Logical Form" (1929); a book review; and a children's dictionary. #Works, His voluminous manuscripts were edited and published posthumously. The first and best-known of this posthumous series is the 1953 book ''Philosophical Investigations''. A 1999 survey among American university and college teachers ranked the ''Investigations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Paul Van Bendegem
Jean Paul Van Bendegem (born 28 March 1953 in Ghent) is a mathematician, a philosopher of science, and a professor at the Vrije Universiteit Brussel in Brussels. Career Van Bendegem received his master's degree in mathematics in 1976. Afterwards, he went to study philosophy. He attended lectures on the philosophy of mathematics from Leo Apostel. He received his master's degree in philosophy in 1979. Van Bendegem wrote his PhD thesis in philosophy on the subject of finitism under the supervision of Diderik Batens while at Ghent University. He defended his thesis in 1983. The content of the thesis was on notation systems, number theory, analysis, physics and logic in a finite empirical framework. Van Bendegem was the dean of the faculty of Arts and philosophy, and was until his retirement in September 2018 head of the CLPS (Centre for Logic and Philosophy of Science) at the same university. He is an honorary chairman of SKEPP (Research Society for Critical Evaluation of Pseudos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doron Zeilberger
Doron Zeilberger (; born 2 July 1950) is an Israeli-American mathematician, known for his work in combinatorics. Education and career He received his doctorate from the Weizmann Institute of Science in 1976, under the direction of Harry Dym, with the thesis "New Approaches and Results in the Theory of Discrete Analytic Functions." He is a Board of Governors Professor of Mathematics at Rutgers University. Mathematical work Zeilberger has made contributions to combinatorics, hypergeometric identities, and q-series. He gave the first proof of the alternating sign matrix conjecture, noteworthy not only for its mathematical content, but also for the fact that Zeilberger recruited nearly a hundred volunteer checkers to "pre-referee" the paper. In 2011, together with Manuel Kauers and Christoph Koutschan, Zeilberger proved the ''q''-TSPP conjecture, which was independently stated in 1983 by George Andrews and David P. Robbins. Zeilberger is an ultrafinitist. He is also known ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zermelo–Fraenkel Set Theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
