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Mathematical Platonism
Mathematical Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the view most people have of numbers. Overview The term ''Platonism'' is used because such a view is seen to parallel Plato's Theory of Forms and a "World of Ideas" (Greek: ''eidos'' (εἶδος)) described in Plato's allegory of the cave: the everyday world can only imperfectly approximate an unchanging, ultimate reality. Both Plato's cave and Platonism have meaningful, not just superficial connections, because Plato's ideas were preceded and probably influenced by the hugely popular ''Pythagoreans'' of ancient Greece, who believed that the world was, quite literally, generated by numbers. A major question considered in mathematical Platonism is: Precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Realism (philosophy)
Philosophical realismusually not treated as a position of its own but as a stance towards other subject mattersis the view that a certain kind of thing (ranging widely from abstract objects like Mathematical realism, numbers to Moral realism, moral statements to the physical world itself) has ''mind-independent existence'', i.e. that it exists even in the absence of any mind perceiving it or that its existence is not just a mere Illusion, appearance in the eye of the beholder. This includes a number of positions within epistemology and metaphysics which express that a given thing instead exists independently of knowledge, thought, or understanding. This can apply to items such as the physical world, the past and future, The problem of other minds, other minds, and the self, though may also apply less directly to things such as Universal (metaphysics), universals, mathematical truths, moral, moral truths, and thought itself. However, realism may also include various positions which i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Mathematical Experience
''The Mathematical Experience'' (1981) is a book by Philip J. Davis and Reuben Hersh that discusses the practice of modern mathematics from a historical and philosophical perspective. The book discusses the psychology of mathematicians, and gives examples of famous proofs and outstanding problems. It goes on to speculate about what a proof really means, in relationship to actual truth. Other topics include mathematics in education and some of the math that occurs in computer science. The first paperback edition won a U.S. National Book Award in Science."National Book Awards – 1983" . Retrieved 2012-03-07. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modern Platonism
Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary Platonists do not necessarily accept all doctrines of Plato. Platonism has had a profound effect on Western thought. At the most fundamental level, Platonism affirms the existence of abstract objects, which are asserted to exist in a third realm distinct from both the sensible external world and from the internal world of consciousness, and is the opposite of nominalism." Philosophers who affirm the existence of abstract objects are sometimes called platonists; those who deny their existence are sometimes called nominalists. The terms "platonism" and "nominalism" have established senses in the history of philosophy, where they denote positions that have little to do with the modern notion of an abstract object. In this connection, it is essential to bear in mind that modern platonists (with a small 'p') need not accept any of the doctrines of Plato, just as modern nominalis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Object Theory
Abstract object theory (AOT) is a branch of metaphysics regarding abstract objects. Originally devised by metaphysician Edward Zalta in 1981, the theory was an expansion of mathematical Platonism. Overview ''Abstract Objects: An Introduction to Axiomatic Metaphysics'' (1983) is the title of a publication by Edward Zalta that outlines abstract object theory. AOT is a dual predication approach (also known as "dual copula strategy") to abstract objectsDale Jacquette, ''Meinongian Logic: The Semantics of Existence and Nonexistence'', Walter de Gruyter, 1996, p. 17. influenced by the contributions of Alexius Meinong and his student Ernst Mally. On Zalta's account, there are two modes of Predicate (mathematical logic), predication: some objects (the ordinary Abstract and concrete, concrete ones around us, like tables and chairs) ''exemplify'' properties, while others (abstract objects like numbers, and what others would call "nonexistent objects", like the Round square copula, round ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Naturalism (philosophy)
In philosophy, naturalism is the idea that only Scientific law, natural laws and forces (as opposed to supernatural ones) operate in the universe. In its primary sense, it is also known as ontological naturalism, metaphysical naturalism, pure naturalism, philosophical naturalism and antisupernaturalism. "Ontological" refers to ontology, the philosophical study of what exists. Philosophers often treat naturalism as equivalent to materialism, but there are important distinctions between the philosophies. For example, philosopher Paul Kurtz argued that nature is best accounted for by reference to Matter, material principles. These principles include mass, energy, and other Physical property, physical and Chemical property, chemical properties accepted by the scientific community. Further, this sense of naturalism holds that spirits, Deity, deities, and ghosts are not real and that there is no "Teleology, purpose" in nature. This stronger formulation of naturalism is commonly ref ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stanford–Edmonton School
In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that – for some coherent meaning of 'logic' – mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. Bertrand Russell and Alfred North Whitehead championed this programme, initiated by Gottlob Frege and subsequently developed by Richard Dedekind and Giuseppe Peano. Overview Dedekind's path to logicism had a turning point when he was able to construct a model satisfying the axioms characterizing the real numbers using certain sets of rational numbers. This and related ideas convinced him that arithmetic, algebra and analysis were reducible to the natural numbers plus a "logic" of classes. Furthermore by 1872 he had concluded that the naturals themselves were reducible to sets and mappings. It is likely that other logicists, most importantly Frege, were also guided by the new theories of the rea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Benacerraf's Epistemological Problem
Paul Joseph Salomon Benacerraf (; 26 March 1930 – 13 January 2025) was a French-born American philosopher working in the field of the philosophy of mathematics who taught at Princeton University his entire career, from 1960 until his retirement in 2007. Benacerraf was appointed Stuart Professor of Philosophy in 1974, and retired as the James S. McDonnell Distinguished University Professor of Philosophy. Life and career Benacerraf was born in Paris on 26 March 1930, to a Moroccan-Venezuelan Sephardic Jewish father, Abraham Benacerraf, and Algerian Jewish mother, Henrietta Lasry. In 1939 the family moved to Caracas and then to New York City. When the family returned to Caracas, Benacerraf remained in the United States, boarding at the Peddie School in Hightstown, New Jersey. He attended Princeton University for both his undergraduate and graduate studies. He was elected a fellow of the American Academy of Arts and Sciences in 1998. Benacerraf died on 13 January 2025, at the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metaphysical Naturalism
Metaphysical naturalism (also called ontological naturalism, philosophical naturalism and antisupernaturalism) is a philosophical worldview which holds that there is nothing but natural elements, principles, and relations of the kind studied by the natural sciences. Methodological naturalism is a philosophical basis for science, for which metaphysical naturalism provides only one possible ontological foundation. Broadly, the corresponding theological perspective is religious naturalism or spiritual naturalism. More specifically, metaphysical naturalism rejects the supernatural concepts and explanations that are part of many religions. Definition In Carl Sagan’s words: "The Cosmos is all that is or ever was or ever will be." According to Arthur C. Danto, naturalism, in recent usage, is a species of philosophical monism according to which whatever exists or happens is ''natural'' in the sense of being susceptible to explanation through methods which, although paradigmati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of ''naive set theory''. After the discovery of Paradoxes of set theory, paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Penelope Maddy
Penelope Maddy (born 4 July 1950) is an American philosopher. Maddy is Distinguished Professor Emerita of Logic and Philosophy of Science and of Mathematics at the University of California, Irvine. She is well known for her influential work in the philosophy of mathematics, where she has worked on mathematical realism (especially set-theoretic realism) and mathematical naturalism. Education and career Maddy first became interested in mathematics in her first algebra class in middle school. After being given a book on abstract algebra by her teacher, she entered the 1968 Westinghouse Science Talent Search, becoming a finalist and placing seventh overall. She went on to study mathematics at University of California, Berkeley and received her bachelor's degree in 1972. Maddy's interest in the continuum hypothesis—which she had initially learned of during high school—and the fact that it could not be proved without introducing a new axiom, led her to question what could count a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ivor Grattan-Guinness
Ivor Owen Grattan-Guinness (23 June 1941 – 12 December 2014) was a historian of mathematics and logic. Life Grattan-Guinness was born in Bakewell, England; his father was a mathematics teacher and educational administrator. He gained his bachelor degree as a Mathematics Scholar at Wadham College, Oxford, and an MSc (Econ) in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966. He gained both the doctorate (PhD) in 1969, and higher doctorate ( D.Sc.) in 1978, in the History of Science at the University of London. He was Emeritus Professor of the History of Mathematics and Logic at Middlesex University, and a Visiting Research Associate at the London School of Economics. He was awarded the Kenneth O. May Medal for services to the History of Mathematics by the International Commission on the History of Mathematics (ICHM) on 31 July 2009, at Budapest, on the occasion of the 23rd International Congress for the History of Science. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
