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John Penn Mayberry
John Penn Mayberry (18 November 1939 – 19 August 2016) was an American mathematical philosopher and creator of a distinctive Aristotelian philosophy of mathematics to which he gave expression in his book ''The Foundations of Mathematics in the Theory of Sets''. Following completion of a Ph.D. at Illinois under the supervision of Gaisi Takeuti, he took up, in 1966, a position in the mathematics department of the University of Bristol. He remained there until his retirement in 2004 as a Reader in Mathematics. Philosophical work Mayberry's philosophy rejects the Platonic tradition, which holds mathematics to be a transcendental science concerned with discovering truths about immaterial, but intelligible, objective entities, as metaphysically conceited. This stance sets him apart from what probably is the “silent majority” view among practising mathematicians. Roger Penrose eloquently expresses a typical Platonic position. :“The natural numbers were there before there were hu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aristotelian Realism
Aristotle's Theory of Universals is Aristotle's classical solution to the Problem of Universals, sometimes known as the hylomorphic theory of immanent realism. Universals are the characteristics or qualities that ordinary objects or things have in common. They can be identified in the types, properties, or relations observed in the world. For example, imagine there is a bowl of red apples resting on a table. Each apple in that bowl will have many similar qualities, such as their red coloring or "redness". They will share some degree of the quality of "ripeness" depending on their age. They may also be at varying degrees of age, which will affect their color, but they will all share a universal "appleness". These qualities are the universals that the apples hold in common. The Problem of Universals asks three questions. Do universals exist? If they exist, where do they exist? Also, if they exist, how do we obtain knowledge of them? In Aristotle's view, universals are incorporeal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacob Klein (philosopher)
Jacob Klein (March 3, 1899 – July 16, 1978) was a Russian-American philosopher and interpreter of Plato, who worked extensively on the nature and historical origin of modern symbolic mathematics. Biography Klein was born in Libava, Russian Empire. He studied at Berlin and Marburg, where he received his Ph.D. in 1922. A student of Nicolai Hartmann, Martin Heidegger, and Edmund Husserl, he later taught at St. John's College in Annapolis, Maryland from 1938 until his death. He served as dean from 1949 to 1958. Klein was affectionately known as Jasha (pronounced "Yasha"). He was one of the world's preeminent interpreters of Plato and the Platonic tradition. As one of many Jewish scholars who were no longer safe in Europe, he fled the Nazis. He was a friend of fellow émigré and German-American philosopher Leo Strauss. Of Klein's first book ''Greek Mathematical Thought and the Origin of Algebra'', Strauss said: The work is much more than a historical study. But even if we take ... [...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] |
Recursive Definition
In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set ( Aczel 1977:740ff). Some examples of recursively-definable objects include factorials, natural numbers, Fibonacci numbers, and the Cantor ternary set. A recursive definition of a function defines values of the function for some inputs in terms of the values of the same function for other (usually smaller) inputs. For example, the factorial function ''n''! is defined by the rules :0! = 1. :(''n'' + 1)! = (''n'' + 1)·''n''!. This definition is valid for each natural number ''n'', because the recursion eventually reaches the base case of 0. The definition may also be thought of as giving a procedure for computing the value of the function ''n''!, starting from ''n'' = 0 and proceeding onwards with ''n'' = 1, ''n'' = 2, ''n'' = 3 etc. The recursion theorem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for ''n'' = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case ''n'' = ''k'', ''then'' it must also hold for the next case ''n'' = ''k'' + 1. These two steps establish that the statement holds for every natural number ''n''. The base case does not necessarily begin with ''n'' = 0, but often with ''n'' = 1, and possibly with any fixed natural number ''n'' = ''N'', establishing the truth of the statement for all natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peacock's Principle
George Peacock FRS (9 April 1791 – 8 November 1858) was an English mathematician and Anglican cleric. He founded what has been called the British algebra of logic. Early life Peacock was born on 9 April 1791 at Thornton Hall, Denton, near Darlington, County Durham. His father, Thomas Peacock, was a priest of the Church of England, incumbent and for 50 years curate of the parish of Denton, where he also kept a school. In early life Peacock did not show any precocity of genius, and was more remarkable for daring feats of climbing than for any special attachment to study. Initially, he received his elementary education from his father and then at Sedbergh School, and at 17 years of age, he was sent to Richmond School under James Tate, a graduate of Cambridge University. At this school he distinguished himself greatly both in classics and in the rather elementary mathematics then required for entrance at Cambridge. In 1809 he became a student of Trinity College, Cambridge. 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] |
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The Last Universalist", since he excelled in all fields of the discipline as it existed during his lifetime. As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology. Poincaré made clear the importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré discove ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His best known contribution is the definition of real numbers through the notion of Dedekind cut. He is also considered a pioneer in the development of modern set theory and of the philosophy of mathematics known as ''Logicism''. Life Dedekind's father was Julius Levin Ulrich Dedekind, an administrator of Collegium Carolinum in Braunschweig. His mother was Caroline Henriette Dedekind (née Emperius), the daughter of a professor at the Collegium. Richard Dedekind had three older siblings. As an adult, he never used the names Julius Wilhelm. He was born in Braunschweig (often called "Brunswick" in English), which is where he lived most of his life and died. He first attended the Collegium Carolinum in 1848 before transferring to the University ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dover
Dover () is a town and major ferry port in Kent, South East England. It faces France across the Strait of Dover, the narrowest part of the English Channel at from Cap Gris Nez in France. It lies south-east of Canterbury and east of Maidstone. The town is the administrative centre of the Dover District and home of the Port of Dover. Archaeological finds have revealed that the area has always been a focus for peoples entering and leaving Great Britain, Britain. The name derives from the River Dour that flows through it. In recent times the town has undergone transformations with a high-speed rail link to London, new retail in town with St James' area opened in 2018, and a revamped promenade and beachfront. This followed in 2019, with a new 500m Pier to the west of the Harbour, and new Marina unveiled as part of a £330m investment in the area. It has also been a point of destination for many illegal migrant crossings during the English Channel migrant crossings (2018-present) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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