axioms

Socrates generally applied his method of examination to concepts that seem to lack any concrete definition; e.g., the key moral concepts at the time, the virtues of piety, wisdom, temperance, courage, and justice. Such an examination challenged the implicit moral beliefs of the interlocutors, bringing out inadequacies and inconsistencies in their beliefs, and usually resulting in puzzlement known as aporia. In view of such inadequacies, Socrates himself professed his ignorance, but others still claim ...

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Socratic method, Socratic method - Method, Socratic method - Practice, Socratic method - Application, Socratic method - Typical Application in Legal Education, Socratic method - Socratic Method in Psychotherapy

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Several properties of Euclidean geometry are logically equivalent to Euclid's parallel postulate, meaning that they can be proven in a system where the parallel postulate is true, and that if they are assumed as axioms, then the parallel postulate can be proven. Strictly speaking, some of these are actually equivalent to the conjunction of Euclid's parallel postulate and its converse, and thus can be used to distinguish Euclidean geometry from both elliptic geometry and hyperbolic geometry simultaneously. One of the most important of these p ...

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Parallel postulate, Parallel postulate - Converse of Euclid's parallel postulate, Parallel postulate - Logically equivalent properties, Parallel postulate - History

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A topological space is a set X together with a collection T of subsets of X satisfying the following axioms: The empty set and X are in T. The union of any collection of sets in T is also in T. The intersection of any pair of sets in T is also in T. The set T is a topology on X. The sets in T are the open sets, and their complements in X are the closed sets. Th ...

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Topological space, Topological space - Definition, Topological space - Comparison of topologies, Topological space - Continuous functions, Topological space - Alternative definitions, Topological space - Examples of topological spaces, Topological space - Topological constructions, Topological space - Classification of topological spaces, Topological space - Topological spaces with algebraic structure, Topological space - Topological spaces with order structure

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Bertrand Russell - Analytic philosophy. Russell is generally recognised as one of the founders of analytic philosophy, indeed, even of its several branches. At the beginning of the 20th century, alongside G. E. Moore, Russell was largely responsible for the British "revolt against Idealism", a philosophy greatly influenced by Georg Hegel and his British apostle, F. H. Bradley. This revolt was echoed 30 years later in Vienna by the logical positivists' "revolt against metaphysics". Russell was particularly appalle ...

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Bertrand Russell, Bertrand Russell - Biography, Bertrand Russell - Russell's philosophical work, Bertrand Russell - Analytic philosophy, Bertrand Russell - Epistemology, Bertrand Russell - Ethics, Bertrand Russell - Logical atomism, Bertrand Russell - Logic and mathematics, Bertrand Russell - Philosophy of language, Bertrand Russell - Philosophy of science, Bertrand Russell - Religion and theology, Bertrand Russell - Influence on philosophy, Bertrand Russell - Russell's activism, Bertrand Russell - Pacifism war and nuclear weapons, Bertrand Russell - Communism and socialism, Bertrand Russell - Women's suffrage, Bertrand Russell - Sexuality, Bertrand Russell - Eugenics and race, Bertrand Russell - Russell summing up his life, Bertrand Russell - Comments about Russell, Bertrand Russell - As a man, Bertrand Russell - As a philosopher, Bertrand Russell - As a writer and his place in history, Bertrand Russell - As a mathematician and logician, Bertrand Russell - As an activist, Bertrand Russell - As a recipient of the Nobel Prize for Literature, Bertrand Russell - From a daughter, Bertrand Russell - Quotes, Bertrand Russell - Asides, Bertrand Russell - Succession

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Many problems in continuous mathematics do not possess a closed-form solution. Examples are finding the integral of exp(−x2) (see error function) and solving a general polynomial equation of degree five or higher (see Abel-Ruffini theorem). In these situations, one has two options left: either one tries to find an approximate solution using asymptotic analysis or one seeks a numerical solution. The latter choice describes the field of numerical analysis. See also:

Numerical analysis, Numerical analysis - General introduction, Numerical analysis - Direct and iterative methods, Numerical analysis - Discretization, Numerical analysis - The generation and propagation of errors, Numerical analysis - Applications, Numerical analysis - Areas of study, Numerical analysis - Computing values of functions, Numerical analysis - Interpolation extrapolation and regression, Numerical analysis - Solving equations and systems of equations, Numerical analysis - Optimization, Numerical analysis - Evaluating integrals, Numerical analysis - Differential equations, Numerical analysis - History, Numerical analysis - Software

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Many of these names have alternative meanings in some of mathematical literature, as explained on History of the separation axioms; for example, the meanings of "normal" and "T4" are sometimes interchanged, similarly "regular" and "T3", etc. Many of the concepts also have several names. Most of these axioms have alternative definitions with the same meaning; the definitions given here are those which fall into a consistent pattern relating the various notions of separation defined in the previous section. Other ...

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Separation axiom, Separation axiom - Separated sets and topologically distinguishable points, Separation axiom - Definitions of the axioms, Separation axiom - Relationships between the axioms, Separation axiom - Other separation axioms, Separation axiom - Sources

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While Euclidean geometry (named for the Greek mathematician Euclid) includes some of the oldest known mathematics, non-Euclidean geometries were not widely accepted as legitimate until the 19th century. The debate that eventually led to the discovery of non-Euclidean geometries began almost as soon as Euclid's work Elements was written. In the Elements, Euclid began with a limited number of assumptions (five axioms and five postulates) and sought to prove all the other results (propositions) in the work. The most notorious of t ...

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Non-Euclidean geometry, Non-Euclidean geometry - History, Non-Euclidean geometry - Reference

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The Peano axioms may be interpreted in the general context of category theory. Let US1 be the category of pointed unary systems; i.e. US1 is the following category: The objects of US1 are all ordered triples (X, x, f), where X is a set, x is an element of X, and f is a set map from X to itself. For each (X, x, f), (Y, y, g) in US1, HomUS1((X, x, f), ...

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Peano axioms, Peano axioms - The axioms, Peano axioms - Peano arithmetic, Peano axioms - Existence and uniqueness, Peano axioms - Binary operations and ordering, Peano axioms - Categorical interpretation, Peano axioms - Metamathematical discussion

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Theoretical physics is the study of matter and energy through the development of theory and was begun 2300 years ago by the ancient Greek natural philosophers, most notably, Aristotle. The development of theoretical physics was the development of science itself. As theories of matter and energy progressed through the ages, other sciences began to specialize and break off from natural philosophy t ...

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Theoretical physics, Theoretical physics - Overview, Theoretical physics - Mainstream theories, Theoretical physics - Examples, Theoretical physics - Proposed theories, Theoretical physics - Examples, Theoretical physics - Fringe theories, Theoretical physics - Examples, Theoretical physics - Notes

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A vector space over a field F (such as the field of real or of complex numbers) is a set V together with two operations: vector addition: defined on the Cartesian product V × V with values in V and denoted v + w, where v, w ∈ V, and scalar multiplication: defined on the Cartesian product F × V with values in V and denoted a v, where a< ...

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Vector space, Vector space - Formal definition, Vector space - Elementary properties, Vector space - Examples, Vector space - Subspaces and bases, Vector space - Linear transformations, Vector space - Generalizations and additional structures

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The axioms of ZFC are: Axiom of extensionality: Two sets are the same if and only if they have the same elements. Axiom of empty set: There is a set with no elements. We will also use {} to denote this empty set. Axiom of pairing: If x, y are sets, then there exists a set containing x and y as its only elements, which we denote by {x,y} or {x} ∪ {y}. < ...

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Zermelo-Fraenkel set theory, Zermelo-Fraenkel set theory - The Axioms

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Although the formalisation of logic was much advanced by the work of such figures as Gottlob Frege, Peano, Russell and Dedekind, conventionally the story of modern proof theory is seen as being established by David Hilbert, who initiated what is called Hilbert's program in the Foundations of mathematics. Kurt Gödel's seminal work on proof theory first advanced, then refuted this program: his completeness theorem seemed to bring Hilbert's problem of reducing all mathematics to a finitist formal system, then his incompleteness theorems showed that was unattainable. All of this work was carried out with the pr ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

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NLP is a divergent subject, and so different individuals will have different formulations for what they consider "principles of NLP". However there is common agreement that some principles, which date from very early on and in some cases were borrowed from other fields can be identified as "principles of NLP" with annotation describing their universality if need be. According to Jane Revell, a British NLP trainer, the presuppositions of NLP "are not a philosophy or a credo or a set of rules and regulations. Rather, they are assumptions upon which individuals ba ...

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Principles of NLP, Principles of NLP - Stated purpose of NLP, Principles of NLP - NLP as described by its major developers and promoters, Principles of NLP - Philosophical stance of NLP, Principles of NLP - Self-declared scope of NLP, Principles of NLP - Specific principles within NLP, Principles of NLP - The map is not the territory, Principles of NLP - Life and 'Mind' are Systemic Processes, Principles of NLP - Behind every behavior is a positive intention, Principles of NLP - Rapport, Principles of NLP - There is no failure only feedback, Principles of NLP - Choice is better than no choice and flexibility is the way one gets choice, Principles of NLP - The meaning of your communication is the response you get, Principles of NLP - People already have all the resources they need to succeed, Principles of NLP - Multiple descriptions are better than one, Principles of NLP - Other beliefs, Principles of NLP - NLP's approach to clinical conditions, Principles of NLP - Criticisms of the principles of NLP

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Universal algebra - Groups. To see how this is supposed to work, let's consider the definition of a group. Normally a group is defined in terms of a single binary operation *, subject to these axioms: Associativity (as in the previous paragraph): x * (y * z)  =  (x * y) * z. Identity element: There exists an element e such that e * x  =  x  ...

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Universal algebra, Universal algebra - Basic idea, Universal algebra - Examples, Universal algebra - Groups, Universal algebra - Modules, Universal algebra - Further issues

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Presuppositional apologetics - Van Tillian presuppositionalism. Apologists who follow Van Til earned the label "presuppositional" because of their central tenet that the Christian must at all times presuppose (that is, assume from the beginning) the supernatural revelation of the Bible as the ultimate arbiter of truth and error in order to know anything. Christians, they say, can assume nothing less because all human thought presupposes (that is, requires by prerequisite) the existence of the God of the Bible. Th ...

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Presuppositional apologetics, Presuppositional apologetics - History of presuppositional apologetics, Presuppositional apologetics - Varieties of presuppositionalism, Presuppositional apologetics - Van Tillian presuppositionalism, Presuppositional apologetics - Clarkian presuppositionalism, Presuppositional apologetics - Circularity, Presuppositional apologetics - Notes, Presuppositional apologetics - Resources, Presuppositional apologetics - Books, Presuppositional apologetics - Websites, Presuppositional apologetics - Debates utilizing a presuppositional approach, Presuppositional apologetics - Debates and discussions on apologetic method

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Given two sets A and B, we may construct their union. This is the set consisting of all objects which are elements of A or of B or of both (see axiom of union). It is denoted by A ∪ B. The intersection of A and B is the set of all objects which are both in A and in B. It is denoted by A ∩ B. Finally, the relative complement of B relative to A, also known as the set theoretic differenc ...

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Naive set theory, Naive set theory - Introduction, Naive set theory - Sets membership and equality, Naive set theory - Specifying sets, Naive set theory - Subsets, Naive set theory - Universal sets and absolute complements, Naive set theory - Unions intersections and relative complements, Naive set theory - Ordered pairs and Cartesian products, Naive set theory - Some important sets, Naive set theory - Paradoxes, Naive set theory - Footnote

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Quantum field theory - Canonical quantization. Quantum field theory solves these problems by consistently quantizing a field. By interpreting the physical observables of the field appropriately, one can create a (rather successful) theory of many particles. Here is how it is: 1. Each normal mode oscillation of the field is interpreted as a particle with frequency f. 2. The quantum number n of each normal mode (which can be tho ...

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Quantum field theory, Quantum field theory - Why quantum field theory, Quantum field theory - What QFT is, Quantum field theory - Technical statement, Quantum field theory - Quantizing a classical field theory, Quantum field theory - Canonical quantization, Quantum field theory - Path integral methods, Quantum field theory - The axiomatic approach, Quantum field theory - Renormalization, Quantum field theory - Gauge theories, Quantum field theory - Supersymmetry, Quantum field theory - Beyond local field theory, Quantum field theory - History, Quantum field theory - Suggested reading

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Albert Einstein - Youth and college. Einstein was born on March 14, 1879 at Ulm in Baden-Württemberg, Germany, about 100 km east of Stuttgart. His parents were Hermann Einstein, a featherbed salesman who later ran an electrochemical works, and Pauline, whose maiden name was Koch. They were married in Stuttgart-Bad Cannstatt. The family was Jewish (non-observant); Albert attended a Catholic elementary school and, at the insist ...

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Albert Einstein, Albert Einstein - Biography, Albert Einstein - Youth and college, Albert Einstein - Work and doctorate, Albert Einstein - Middle years, Albert Einstein - Final years, Albert Einstein - Personality, Albert Einstein - Religious views, Albert Einstein - Political views, Albert Einstein - Nationality: German Swiss or American?, Albert Einstein - Popularity and cultural impact, Albert Einstein - Entertainment, Albert Einstein - Licensing, Albert Einstein - Honors, Albert Einstein - Works by Albert Einstein, Albert Einstein - Notes

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Lobachevsky was born in Nizhny Novgorod, Russia. His parents were Ivan Maksimovich Lobachevsky, a clerk in a landsurveying office, and Praskovia Alexandrovna Lobachevskaya. In 1800, his father died and his mother moved to Kazan. In Kazan, Nikolai Ivanovich Lobachevsky attended Kazan Gymnasium, graduating in 1807 and then Kazan University which was founded just three years earlier, in 1804. At Kazan University, Lobachevsky was influenced by professor Martin Bartels (1769–1833), a friend of Carl Friedrich Gauss. Lobachevsky received a ...

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Nikolai Ivanovich Lobachevsky, Nikolai Ivanovich Lobachevsky - Biography, Nikolai Ivanovich Lobachevsky - Mathematical results, Nikolai Ivanovich Lobachevsky - In popular culture, Nikolai Ivanovich Lobachevsky - External link

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An associative algebra A over a field K is defined to be a vector space over K together with a K-bilinear multiplication A x A → A (where the image of (x,y) is written as xy) such that the associative law holds: (x y) z = x (y z) for all x, y and z in A. The bilinearity of the multiplication can be expressed as (x + y) z = x z + y z  
...

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Associative algebra, Associative algebra - Definition, Associative algebra - Examples, Associative algebra - Algebra homomorphisms, Associative algebra - Index-free notation, Associative algebra - Generalizations, Associative algebra - Coalgebras, Associative algebra - Representations, Associative algebra - Motivation for a Hopf algebra, Associative algebra - Motivation for a Lie algebra

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The axiom schema of separation can almost be derived from the axiom schema of replacement. First, recall this axiom schema: for any functional predicate F in one variable that doesn't use the symbols A, B, C or D. Given a suitable predicate P for the axiom of specification, define the mapping F by F(D) = D if P(D) is true and F(D) = E if P(D) is false, where E is any member of ASee also:

Axiom schema of specification, Axiom schema of specification - Relation to the axiom schema of replacement, Axiom schema of specification - Unrestricted comprehension, Axiom schema of specification - In NBG class theory, Axiom schema of specification - In second order logic, Axiom schema of specification - In Quine's New Foundations

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Suppose P is any predicate in two variables that doesn't use the symbol B. Then in the formal language of the Zermelo-Fraenkel axioms, the axiom schema reads: or in words: If, given any set X, there is a unique set Y such that P holds for X and Y, then, given any set A, there is a set B such that, given any set C, C is a member of B if and only if there is a set D such that D is a member of A ...

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Axiom schema of replacement, Axiom schema of replacement - Statement, Axiom schema of replacement - Example applications, Axiom schema of replacement - History and philosophy, Axiom schema of replacement - Relation to the axiom schema of specification

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