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Transcendental Law Of Homogeneity
In mathematics, the transcendental law of homogeneity (TLH) is a heuristic principle enunciated by Gottfried Wilhelm Leibniz most clearly in a 1710 text entitled ''Symbolismus memorabilis calculi algebraici et infinitesimalis in comparatione potentiarum et differentiarum, et de lege homogeneorum transcendentali''. Henk J. M. Bos describes it as the principle to the effect that in a sum involving infinitesimals of different orders, only the lowest-order term must be retained, and the remainder discarded. Thus, if a is finite and dx is infinitesimal, then one sets :a+dx=a. Similarly, :u\,dv+v\,du+du\,dv=u\,dv+v\,du, where the higher-order term ''du'' ''dv'' is discarded in accordance with the TLH. A 2012 study argues that Leibniz's TLH was a precursor of the standard part function over the hyperreals. See also * Law of continuity *Adequality Adequality is a technique developed by Pierre de Fermat in his treatise ''Methodus ad disquirendam maximam et minimam'' (a Latin t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gottfried Wilhelm Leibniz
Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many other branches of mathematics, such as binary arithmetic and statistics. Leibniz has been called the "last universal genius" due to his vast expertise across fields, which became a rarity after his lifetime with the coming of the Industrial Revolution and the spread of specialized labor. He is a prominent figure in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history, philology, games, music, and other studies. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in probability theory, biology, medicine, geology, psychology, linguistics and computer science. Leibniz contributed to the field ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henk J
Henk is a Dutch male given name, originally a short form of Hendrik. It influenced " Hank" which is used in English-speaking countries (mainly in the US) as a form of " Henry". Academics * Henk Aertsen (born 1943), Dutch Anglo-Saxon linguist * Henk Barendregt (born 1947), Dutch logician * Henk Jaap Beentje (born 1951), Dutch botanist * Henk Blezer (born 1961), Dutch Tibetologist, Indologist, and scholar of Buddhist studies * Henk Bodewitz (1939–2022), Dutch Sanskrit scholar * Henk J. M. Bos (born 1940), Dutch historian of mathematics * Henk Braakhuis (born 1939), Dutch historian of philosophy *Henk Buck (1930–2023), Dutch organic chemist * Henk van Dongen (1936–2011), Dutch organizational theorist and policy advisor * Henk Dorgelo (1894–1961), Dutch physicist and academic *Henk van der Flier (born 1945), Dutch psychologist * Henk A. M. J. ten Have (born 1951), Dutch medical ethicist * Henk van de Hulst (1918–2000), Dutch astronomer and mathematician * Henk Lombaers (19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinitesimal
In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the "infinity- th" item in a sequence. Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities; the augmentations are the reciprocals of one another. Infinitesimal numbers were introduced in the development of calculus, in which the derivative was first conceived as a ratio of two infinitesimal quantities. This definition was not rigorously formalized. As calculus developed further, infinitesimals were replaced by limits, which can be calculated using the standard real numbers. In the 3rd century BC Archimedes used what ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archive For History Of Exact Sciences
''Archive for History of Exact Sciences'' is a peer-reviewed academic journal currently published bimonthly by Springer Science+Business Media, covering the history of mathematics and of astronomy observations and techniques, epistemology of science, and philosophy of science from Antiquity until now. It was established in 1960 and the current editors-in-chief are Jed Z. Buchwald and Jeremy Gray. Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... of 0.594. References External links * History of science journals Springer Science+Business Media academic journals Bimonthly journals English-language journals Academic j ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Part Function
In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such hyperreal x, the unique real x_0 infinitely close to it, i.e. x-x_0 is infinitesimal. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de Fermat, as well as Leibniz's Transcendental law of homogeneity. The standard part function was first defined by Abraham Robinson who used the notation ^x for the standard part of a hyperreal x (see Robinson 1974). This concept plays a key role in defining the concepts of the calculus, such as continuity, the derivative, and the integral, in nonstandard analysis. The latter theory is a rigorous formalization of calculations with infinitesimals. The standard part of ''x'' is sometimes referred to as its shadow. Definition Nonstandard analysis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperreal Number
In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number x is said to be finite if, and only if, , x, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erkenntnis
''Erkenntnis'' is a journal of philosophy that publishes papers in analytic philosophy. Its name is derived from the German word " Erkenntnis", meaning "knowledge, recognition". The journal was also linked to organisation of conferences, such as the Second Conference on the Epistemology of the Exact Sciences, of which it published the papers and accounts of the discussions. First series (1930–1940) When Hans Reichenbach and Rudolf Carnap took charge of ''Annalen der Philosophie und philosophischen Kritik'' in 1930 they renamed it ''Erkenntnis'', under which name it was published 1930–1938. The journal was published by the ''Gesellschaft für Empirische Philosophie'', or the Berlin Circle and the Verein Ernst Mach, Vienna. In the first issue Reichenbach noted that the editors hoped to gain a better understanding of the nature of all human knowledge through consideration of the procedures and results of a variety of scientific disciplines, whilst also hoping that philosophy nee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Continuity
Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior, with its precise definition a matter of longstanding debate. It has been variously described as a science and as the art of justice. State-enforced laws can be made by a legislature, resulting in statutes; by the executive through decrees and regulations; or by judges' decisions, which form precedent in common law jurisdictions. An autocrat may exercise those functions within their realm. The creation of laws themselves may be influenced by a constitution, written or tacit, and the rights encoded therein. The law shapes politics, economics, history and society in various ways and also serves as a mediator of relations between people. Legal systems vary between jurisdictions, with their differences analysed in comparative law. In civil law jurisdictions, a legislature or other central body codifies and consolidates the law. In common law systems, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adequality
Adequality is a technique developed by Pierre de Fermat in his treatise ''Methodus ad disquirendam maximam et minimam'' (a Latin treatise circulated in France c. 1636 ) to calculate maxima and minima of functions, tangents to curves, area, center of mass, least action, and other problems in calculus. According to André Weil, Fermat "introduces the technical term adaequalitas, adaequare, etc., which he says he has borrowed from Diophantus. As Diophantus V.11 shows, it means an approximate equality, and this is indeed how Fermat explains the word in one of his later writings." (Weil 1973). Diophantus coined the word παρισότης (''parisotēs'') to refer to an approximate equality. Claude Gaspard Bachet de Méziriac translated Diophantus's Greek word into Latin as ''adaequalitas''. Paul Tannery's French translation of Fermat's Latin treatises on maxima and minima used the words ''adéquation'' and ''adégaler''. Fermat's method Fermat used ''adequality'' first to find max ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
History Of Calculus
Calculus, originally called infinitesimal calculus, is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series. Many elements of calculus appeared in ancient Greece, then in China and the Middle East, and still later again in medieval Europe and in India. Infinitesimal calculus was developed in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz independently of each other. An argument over priority led to the Leibniz–Newton calculus controversy which continued until the death of Leibniz in 1716. The development of calculus and its uses within the sciences have continued to the present. Etymology In mathematics education, ''calculus'' denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. The word ''calculus'' is Latin for "small pebble" (the diminutive of '' calx,'' meaning "stone"), a meaning which still persists in medicine. Because such pebbles were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
