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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 recent 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'' [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history and philology. 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. In addition, he contributed to the field of library science: while serving as overseer of the Wolfenbüttel library in Germany, he devised a cataloging system that would have served as a guide for many of Europe's largest libraries. Leibniz's contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters and in unpublished manuscripts. He wrote in several languages, primarily in Latin, ... [...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". People named "Henk" include: 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 (born 1939), Dutch Sanskrit scholar *Henk J. M. Bos (born 1940), Dutch historian of mathematics * Henk Braakhuis (born 1939), Dutch historian of philosophy *Henk Buck (born 1930), 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 *Hen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. 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. Infinitesimals regained popularit ... [...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 scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... of 0.594. References External links * History of science journals Springer Science+Business Media academic journals Bimonthly journals English-language journals Publications ... [...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,Karin Usadi Katz and Mikhail G. Katz (2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. Foundations of Science.Searxiv The authors refer to the Fermat-Robinson standard part. 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 derivati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperreal Number
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form :1 + 1 + \cdots + 1 (for any finite number of terms). Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948. The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic law of continuity. The transfer principle states that true first-order statements about R are also valid in *R. For example, the commutative law of addition, , holds for the hyperreals just as it does for the reals; since R is a real closed field, so is *R. Since \sin()=0 for all integers ''n'', one also has \sin()=0 for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Łoś' theorem of 1955. ... [...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 need ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Continuity
The law of continuity is a heuristic principle introduced by Gottfried Leibniz based on earlier work by Nicholas of Cusa and Johannes Kepler. It is the principle that "whatever succeeds for the finite, also succeeds for the infinite". Kepler used the law of continuity to calculate the area of the circle by representing it as an infinite-sided polygon with infinitesimal sides, and adding the areas of infinitely many triangles with infinitesimal bases. Leibniz used the principle to extend concepts such as arithmetic operations from ordinary numbers to infinitesimals, laying the groundwork for infinitesimal calculus. The transfer principle provides a mathematical implementation of the law of continuity in the context of the hyperreal numbers. A related law of continuity concerning intersection numbers in geometry was promoted by Jean-Victor Poncelet in his "Traité des propriétés projectives des figures". Leibniz's formulation Leibniz expressed the law in the following terms in 17 ... [...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''''METHOD FOR THE STUDY OF MAXIMA AND MINIMA'' English translation of Fermat's treatise ''Methodus ad disquirendam maximam et minimam''. (a treatise circulated in France c. 1636) to calculate of functions, s to curves, |
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 day. 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] |
