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Preintuitionism
In the philosophy of mathematics, the pre-intuitionists is the name given by L. E. J. Brouwer to several influential mathematicians who shared similar opinions on the nature of mathematics. The term was introduced by Brouwer in his 1951 lectures at Cambridge where he described the differences between his philosophy of intuitionism and its predecessors:Luitzen Egbertus Jan Brouwer (edited by Arend Heyting, ''Collected Works'', North-Holland, 1975, p. 509. Of a totally different orientation Dedekind, Cantor, Peano, Zermelo, and Couturat, etc.">Louis Couturat">Couturat, etc./nowiki> was the Pre-Intuitionist School, mainly led by Poincaré, Borel and Lebesgue. These thinkers seem to have maintained a modified observational standpoint for the introduction of natural numbers, for the principle of complete induction ...">/nowiki>.../nowiki> For these, even for such theorems as were deduced by means of classical logic, they postulated an existence and exactness independent of language ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophy Of Mathematics
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathematical objects are purely abstract entities or are in some way concrete, and in what the relationship such objects have with physical reality consists. Major themes that are dealt with in philosophy of mathematics include: *''Reality'': The question is whether mathematics is a pure product of human mind or whether it has some reality by itself. *''Logic and rigor'' *''Relationship with physical reality'' *''Relationship with science'' *''Relationship with applications'' *''Mathematical truth'' *''Nature as human activity'' (science, the arts, art, game, or all together) Major themes Reality Logic and rigor Mathematical reasoning requires Mathematical rigor, rigor. This means that the definitions must be absolutely unambiguous and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Poincaré
Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, 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. He has further been called "the Carl Friedrich Gauss, Gauss of History of mathematics, modern mathematics". Due to his success in science, along with his influence and philosophy, he has been called "the philosopher par excellence of modern science". As a mathematician and physicist, he made many original fundamental contributions to Pure mathematics, 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. Poincaré is regarded as the cr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Induction
Mathematical induction is a method for mathematical proof, 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), \dots all hold. This is done by first proving a simple case, then also showing that if we assume the claim is true for a given case, then the next case is also true. 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 trut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rigour
Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as mathematical proofs which must maintain consistent answers; or socially imposed, such as the process of defining ethics and law. Etymology "Rigour" comes to English through old French (13th c., Modern French '' rigueur'') meaning "stiffness", which itself is based on the Latin">Wiktionary:fr:rigueur">rigueur'') meaning "stiffness", which itself is based on the Latin ''rigorem'' (nominative ''rigor'') "numbness, stiffness, hardness, firmness; roughness, rudeness", from the verb ''rigere'' "to be stiff". The noun was frequently used to describe a condition of strictness or stiffness, which arises from a situation or constraint either chosen or experienced passively. For example, the title of the book ''Theologia Moralis Inter Rigorem et Lax ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Temporal Logic
In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time (for example, "I am ''always'' hungry", "I will ''eventually'' be hungry", or "I will be hungry ''until'' I eat something"). It is sometimes also used to refer to tense logic, a modal logic-based system of temporal logic introduced by Arthur Prior in the late 1950s, with important contributions by Hans Kamp. It has been further developed by computer scientists, notably Amir Pnueli, and logicians. Temporal logic has found an important application in formal verification, where it is used to state requirements of hardware or software systems. For instance, one may wish to say that ''whenever'' a request is made, access to a resource is ''eventually'' granted, but it is ''never'' granted to two requestors simultaneously. Such a statement can conveniently be expressed in a temporal logic. Motivation Consider the statement "I am hungry". Though it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Excluded Middle
In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction and the law of identity; however, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law/principle of the excluded third, in Latin ''principium tertii exclusi''. Another Latin designation for this law is ''tertium non datur'' or "no third ossibilityis given". In classical logic, the law is a tautology. In contemporary logic the principle is distinguished from the semantical principle of bivalence, which states that every proposition is either true or false. The principle of bivalence always implies the law of excluded middle, while the converse is not always true. A commonly cited counterexample uses statements unprovable n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luitzen Egbertus Jan Brouwer
Luitzen Egbertus Jan "Bertus" Brouwer (27 February 1881 – 2 December 1966) was a Dutch mathematician and philosopher who worked in topology, set theory, measure theory and complex analysis. Regarded as one of the greatest mathematicians of the 20th century, he is known as one of the founders of modern topology, particularly for establishing his fixed-point theorem and the topological invariance of dimension. Brouwer also became a major figure in the philosophy of intuitionism, a constructivist school of mathematics which argues that math is a cognitive construct rather than a type of objective truth. This position led to the Brouwer–Hilbert controversy, in which Brouwer sparred with his formalist colleague David Hilbert. Brouwer's ideas were subsequently taken up by his student Arend Heyting and Hilbert's former student Hermann Weyl. In addition to his mathematical work, Brouwer also published the short philosophical tract ''Life, Art, and Mysticism'' (1905). Biography B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knowledge
Knowledge is an Declarative knowledge, awareness of facts, a Knowledge by acquaintance, familiarity with individuals and situations, or a Procedural knowledge, practical skill. Knowledge of facts, also called propositional knowledge, is often characterized as Truth, true belief that is distinct from opinion or guesswork by virtue of Justification (epistemology), justification. While there is wide agreement among philosophers that propositional knowledge is a form of true belief, many controversies focus on justification. This includes questions like how to understand justification, whether it is needed at all, and whether something else besides it is needed. These controversies intensified in the latter half of the 20th century due to a series of thought experiments called ''Gettier cases'' that provoked alternative definitions. Knowledge can be produced in many ways. The main source of empirical knowledge is perception, which involves the usage of the senses to learn about ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Language
Language is a structured system of communication that consists of grammar and vocabulary. It is the primary means by which humans convey meaning, both in spoken and signed language, signed forms, and may also be conveyed through writing system, writing. Human language is characterized by its cultural and historical diversity, with significant variations observed between cultures and across time. Human languages possess the properties of Productivity (linguistics), productivity and Displacement (linguistics), displacement, which enable the creation of an infinite number of sentences, and the ability to refer to objects, events, and ideas that are not immediately present in the discourse. The use of human language relies on social convention and is acquired through learning. Estimates of the number of human languages in the world vary between and . Precise estimates depend on an arbitrary distinction (dichotomy) established between languages and dialects. Natural languages are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logicism
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Synthetic Proposition
Synthetic may refer to: Science * Synthetic biology * Synthetic chemical or compound, produced by the process of chemical synthesis * Synthetic elements, chemical elements that are not naturally found on Earth and therefore have to be created in experiments * Synthetic organic compounds synthetic chemical compounds based on carbon (organic compounds). * Synthetic peptide * Synthetic population * Synthetic population (biology) Industry * Synthetic fuel * Synthetic oil * Synthetic marijuana * Synthetic diamond * Synthetic fibers, cloth or other material made from other substances than natural (animal, plant) materials Other * Synthetic position, a concept in finance * Synthetic-aperture radar, a type or radar * Analytic–synthetic distinction, in philosophy * Synthetic language in linguistics, inflected or agglutinative languages * Synthetic intelligence a term emphasizing that true intelligence expressed by computing machines is not an imitation or "artificial." * Sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Proposition
Analytic or analytical may refer to: Chemistry * Analytical chemistry, the analysis of material samples to learn their chemical composition and structure * Analytical technique, a method that is used to determine the concentration of a chemical compound or chemical element * Analytical concentration Mathematics * Abstract analytic number theory, the application of ideas and techniques from analytic number theory to other mathematical fields * Analytic combinatorics, a branch of combinatorics that describes combinatorial classes using generating functions * Analytic element method, a numerical method used to solve partial differential equations * Analytic expression or analytic solution, a mathematical expression using well-known operations that lend themselves readily to calculation * Analytic geometry, the study of geometry based on numerical coordinates rather than axioms * Analytic number theory, a branch of number theory that uses methods from mathematical analysis M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
