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Point-pair Separation
In a cyclic order, such as the real projective line, two pairs of points separate each other when they occur alternately in the order. Thus the ordering ''a b c d'' of four points has (''a,c'') and (''b,d'') as separating pairs. This point-pair separation is an invariant of projectivities of the line. Concept The concept was described by G. B. Halsted at the outset of his ''Synthetic Projective Geometry'': Given any pair of points on a projective line, they separate a third point from its harmonic conjugate. A pair of lines in a pencil separates another pair when a transversal crosses the pairs in separated points. The point-pair separation of points was written AC//BD by H. S. M. Coxeter in his textbook ''The Real Projective Plane''. Application The relation may be used in showing the real projective plane is a complete space. The axiom of continuity used is "Every monotonic sequence of points has a limit." The point-pair separation is used to provide definitions: * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cyclic Order
In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "". One does not say that east is "more clockwise" than west. Instead, a cyclic order is defined as a ternary relation , meaning "after , one reaches before ". For example, [June, October, February], but not [June, February, October], cf. picture. A ternary relation is called a cyclic order if it is #The ternary relation, cyclic, asymmetric, transitive, and connected. Dropping the "connected" requirement results in a partial cyclic order. A set (mathematics), set with a cyclic order is called a cyclically ordered set or simply a cycle. Some familiar cycles are discrete, having only a Finite set, finite number of element (mathematics), elements: there are seven days of the week, four cardinal directions, twelve notes in the chromatic scale, and three plays in rock-paper-scissors. In a finite cycle, each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Betweenness Relation
Betweenness is a noun derived from the proposition between. It may refer to: * The ternary relation of intermediacy or betweenness, a feature of ordered geometry. * Betweenness problem - an algorithmic problem. The input is a collection of ordered triples of items; the task is to decide whether there is a single total order such that such that, for each of the given triples, the middle item in the triple appears in the output somewhere between the other two items. * Betweenness centrality - a measure of centrality in a graph, based on shortest paths. The betweenness centrality of a vertex is the number of shortest paths that pass through the vertex. * Metric betweenness - given a metric ''d'', a point ''y'' is said to be ''between'' ''x'' and ''z'' if all three points are distinct, and d(x,y)+d(y,z)=d(x,z). See convex metric space. See also * Between (other) * In Between (other) {{Disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
The Principles Of Mathematics
''The Principles of Mathematics'' (''PoM'') is a 1903 book by Bertrand Russell, in which the author presented Russell's paradox, his famous paradox and argued his thesis that mathematics and logic are identical. The book presents a view of the foundations of mathematics and Alexius Meinong, Meinongianism and has become a classic reference. It reported on developments by Giuseppe Peano, Mario Pieri, Richard Dedekind, Georg Cantor, and others. In 1905 Louis Couturat published a partial French translation that expanded the book's readership. In 1937 Russell prepared a new introduction saying, "Such interest as the book now possesses is historical, and consists in the fact that it represents a certain stage in the development of its subject." Further editions were published in 1938, 1951, 1996, and 2009. Contents ''The Principles of Mathematics'' consists of 59 chapters divided into seven parts: indefinables in mathematics, number, quantity, order, infinity and continuity, spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic philosophy.Stanford Encyclopedia of Philosophy"Bertrand Russell", 1 May 2003. He was one of the early 20th century's prominent logicians and a founder of analytic philosophy, along with his predecessor Gottlob Frege, his friend and colleague G. E. Moore, and his student and protégé Ludwig Wittgenstein. Russell with Moore led the British "revolt against British idealism, idealism". Together with his former teacher Alfred North Whitehead, A. N. Whitehead, Russell wrote ''Principia Mathematica'', a milestone in the development of classical logic and a major attempt to reduce the whole of mathematics to logic (see logicism). Russell's article "On Denoting" has been considered a "paradigm of philosophy". Russell was a Pacifism, pacifist who ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Giovanni Vailati
Giovanni Vailati (24 April 1863 – 14 May 1909) was an Italian proto-analytic philosopher, historian of science, and mathematician. Life Vailati was born in Crema, Lombardy, and studied engineering at the University of Turin. He went on to lecture in the history of mechanics there from 1896 to 1899, after working as assistant to Giuseppe Peano and Vito Volterra. He resigned his university post in 1899 so that he could pursue his independent studies, making a living from high-school mathematics teaching. During his lifetime he became internationally known, his writings having been translated into English, French, and Polish, though he was largely forgotten after his death in Rome. He was rediscovered in the late 1950s. He did not publish any complete books, but left about 200 essays and reviews across a range of academic disciplines. Philosophy Vailati's view of philosophy was that it provided a preparation and the tools for scientific work. For that reason, and because ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Proceedings Of The American Academy Of Arts And Sciences
''Dædalus'' is a quarterly peer-reviewed academic journal that was established in 1846 as the ''Proceedings of the American Academy of Arts and Sciences'', obtaining its current title in 1958. The journal is published by MIT Press on behalf of the American Academy of Arts and Sciences and only accepts submissions on invitation. In January 2021, the journal moved to an open access Open access (OA) is a set of principles and a range of practices through which nominally copyrightable publications are delivered to readers free of access charges or other barriers. With open access strictly defined (according to the 2001 de ... model. References External links * {{humanities-journal-stub Academic journals established in 1846 MIT Press academic journals Quarterly journals English-language journals Multidisciplinary humanities journals Academic journals associated with learned and professional societies 1955 establishments in Massachusetts ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Quaternary Relation
In mathematics, a finitary relation over a sequence of sets is a subset of the Cartesian product ; that is, it is a set of ''n''-tuples , each being a sequence of elements ''x''''i'' in the corresponding ''X''''i''. Typically, the relation describes a possible connection between the elements of an ''n''-tuple. For example, the relation "''x'' is divisible by ''y'' and ''z''" consists of the set of 3-tuples such that when substituted to ''x'', ''y'' and ''z'', respectively, make the sentence true. The non-negative integer ''n'' that gives the number of "places" in the relation is called the ''arity'', ''adicity'' or ''degree'' of the relation. A relation with ''n'' "places" is variously called an ''n''-ary relation, an ''n''-adic relation or a relation of degree ''n''. Relations with a finite number of places are called ''finitary relations'' (or simply ''relations'' if the context is clear). It is also possible to generalize the concept to ''infinitary relations'' with infinite se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all rational numbers is often referred to as "the rationals", and is closed under addition, subtraction, multiplication, and division by a nonzero rational number. It is a field under these operations and therefore also called the field of rationals or the field of rational numbers. It is usually denoted by boldface , or blackboard bold A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Reduct
In universal algebra and in model theory, a reduct of an algebraic structure is obtained by omitting some of the operations and relations of that structure. The opposite of "reduct" is "expansion". Definition Let ''A'' be an algebraic structure (in the sense of universal algebra) or a structure in the sense of model theory, organized as a set ''X'' together with an indexed family of operations and relations φ''i'' on that set, with index set ''I''. Then the reduct of ''A'' defined by a subset ''J'' of ''I'' is the structure consisting of the set ''X'' and ''J''-indexed family of operations and relations whose ''j''-th operation or relation for ''j'' ∈ ''J'' is the ''j''-th operation or relation of ''A''. That is, this reduct is the structure ''A'' with the omission of those operations and relations φ''i'' for which ''i'' is not in ''J''. A structure ''A'' is an expansion of ''B'' just when ''B'' is a reduct of ''A''. That is, reduct and expansion are mutual converses. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Linear Order
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a ( strongly connected, formerly called totality). Requirements 1. to 3. just make up the definition of a partial order. Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, toset and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but generally refers to a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Real Projective Line
In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not intersect but seem to intersect "at infinity". For solving this problem, points at infinity have been introduced, in such a way that in a real projective plane, two distinct projective lines meet in exactly one point. The set of these points at infinity, the "horizon" of the visual perspective in the plane, is a real projective line. It is the set of directions emanating from an observer situated at any point, with opposite directions identified. An example of a real projective line is the projectively extended real line, which is often called ''the'' projective line. Formally, a real projective line P(R) is defined as the set of all one-dimensional linear subspaces of a two-dimensional vector space over the reals. The automorphisms of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Complete Space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. \sqrt is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the ''completion'' of a given space, as explained below. Definition Cauchy sequence A sequence x_1, x_2, x_3, \ldots of elements from X of a metric space (X, d) is called Cauchy if for every positive real number r > 0 there is a positive integer N such that for all positive integers m, n > N, d(x_m, x_n) < r. Complete space A metric space is complete if any of the following equivalent conditions are satisfied: #Every Cauchy se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
