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Ultraintuitionism
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, 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 in practice because of physical restrictions in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality. Truth and proof The fundamental distinguishing characteristic of intuitionism is its interpretation of what it means for a mathematical statement to be true. In Brouwer's original intuitionism, the truth of a mathematical statement is a subjective claim: a mathematical statement corresponds to a mental construction, and a mathematician can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophy Of Mathematics
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's lives. The logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts. The philosophy of mathematics has two major themes: mathematical realism and mathematical anti-realism. History The origin of mathematics is subject to arguments and disagreements. Whether the birth of mathematics was a random happening or induced by necessity during the development of other subjects, like physics, is still a matter of prolific debates. Many thinkers have contributed their ideas concerning the nature of mathematics. Today, some philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that ... [...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 arithmetic. In philosophy of mathematics he advocated the view of formalism rather than 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, stayed in St. Petersburg in connection with issues related to prisoners of war. After the war, his family returned to Italy a ... [...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 space, if , , are the lines of shortest distance respectively for the pairs , and , and if , and are the lines of shortest distance respectively for .... 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 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 Husserlian phenomenology). Vopěnka served as the Minister of Education of the Czech Republic (then part of Czechoslovakia) from 1990 to 1992 within the government of Prime Minister Petr Pithart. Biography Petr Vopěnka grew up in small town of Dolní Kralovice. After finishing gymnasium in Ledeč nad Sázavou in 1953 he went to study mathematics at the Mathematics and Physics Faculty of Charles University in Prague, graduating in 1958. In 1962 he was made Candidate of Sciences (CSc) and in 1967 Doctor of Science (D ... [...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. 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. He was a great-great-grandson of the architect and artist Joseph Gandy (1771–1843). 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 speech encipherment project, and he became one of Turing's lifelong friends and associates. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ludwig Wittgenstein
Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He is considered by some to be the greatest philosopher of the 20th century. From 1929 to 1947, Wittgenstein taught at the University of Cambridge. In spite of his position, during his entire life only one book of his philosophy was published, 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. His voluminous manuscripts were edited and published posthumously. The first and best-known of this posthumous series is the 1953 book ''Philosophical Investigations''. A su ... [...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 Pseudosc ... [...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, epistemic logic, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doron Zeilberger
Doron Zeilberger (דורון ציילברגר, born 2 July 1950 in Haifa, Israel) is an Israeli 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. Contributions Zeilberger has made contributions to combinatorics, hypergeometric identities, and q-series. Zeilberger 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 ... [...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 from containing u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
