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In mathematics, non-standard calculus is the modern application of infinitesimals, in the sense of non-standard analysis, to differential and integral calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic. Calculations with infinitesimals were widely used before Karl Weierstrass sought to replace them with the (ε, δ)-definition of limit starting in the 1870s. (See history of calculus.) For almost one hundred years thereafter, mathematicians like Richard Courant viewed infinitesimals as being naive and vague or meaningless.〔Courant described infinitesimals on page 81 of ''Differential and Integral Calculus, Vol I'', as "devoid of any clear meaning" and "naive befogging". Similarly on page 101, Courant described them as "incompatible with the clarity of ideas demanded in mathematics", "entirely meaningless", "fog which hung round the foundations", and a "hazy idea".〕 Contrary to such views, Abraham Robinson showed in 1960 that infinitesimals are precise, clear, and meaningful, building upon work by Edwin Hewitt and Jerzy Łoś. According to Jerome Keisler, "Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century."〔Elementary Calculus: An Infinitesimal Approach〕 == History == The history of non-standard calculus began with the use of infinitely small quantities, called infinitesimals in calculus. The use of infinitesimals can be found the foundations of calculus independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s. John Wallis refined earlier techniques of indivisibles of Cavalieri and others by exploiting an infinitesimal quantity he denoted in area calculations, preparing the ground for integral calculus.〔Scott, J.F. 1981. "The Mathematical Work of John Wallis, D.D., F.R.S. (1616–1703)". Chelsea Publishing Co. New York, NY. p. 18.〕 They drew on the work of such mathematicians as Pierre de Fermat, Isaac Barrow and René Descartes. In early calculus the use of infinitesimal quantities was criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley in his book ''The Analyst''. Several mathematicians, including Maclaurin and d'Alembert, advocated the use of limits. Augustin Louis Cauchy developed a versatile spectrum of foundational approaches, including a definition of continuity in terms of infinitesimals and a (somewhat imprecise) prototype of an ε, δ argument in working with differentiation. Karl Weierstrass formalized the concept of limit in the context of a (real) number system without infinitesimals. Following the work of Weierstrass, it eventually became common to base calculus on ε, δ arguments instead of infinitesimals. This approach formalized by Weierstrass came to be known as the ''standard'' calculus. After many years of the infinitesimal approach to calculus having fallen into disuse other than as an introductory pedagogical tool, use of infinitesimal quantities was finally given a rigorous foundation by Abraham Robinson in the 1960s. Robinson's approach is called non-standard analysis to distinguish it from the standard use of limits. This approach used technical machinery from mathematical logic to create a theory of hyperreal numbers that interpret infinitesimals in a manner that allows a Leibniz-like development of the usual rules of calculus. An alternative approach, developed by Edward Nelson, finds infinitesimals on the ordinary real line itself, and involves a modification of the foundational setting by extending ZFC through the introduction of a new unary predicate "standard". 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Non-standard calculus」の詳細全文を読む スポンサード リンク
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