|
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Non-standard analysis〔Nonstandard Analysis in Practice. Edited by Francine Diener, Marc Diener. Springer, 1995.〕〔Nonstandard Analysis, Axiomatically. By V. Vladimir Grigorevich Kanovei, Michael Reeken. Springer, 2004.〕〔Nonstandard Analysis for the Working Mathematician. Edited by Peter A. Loeb, Manfred P. H. Wolff. Springer, 2000.〕 instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers. Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson.〔Non-standard Analysis. By Abraham Robinson. Princeton University Press, 1974.〕〔(Abraham Robinson and Nonstandard Analysis ): History, Philosophy, and Foundations of Mathematics. By Joseph W. Dauben. www.mcps.umn.edu.〕 He wrote: () the idea of infinitely small or ''infinitesimal'' quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection () that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were ''to possess the same properties as the latter'' Robinson argued that this law of continuity of Leibniz's is a precursor of the transfer principle. Robinson continued: However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits.〔Robinson, A.: Non-standard analysis. North-Holland Publishing Co., Amsterdam 1966.〕 Robinson continues: It is shown in this book that Leibniz's ideas can be fully vindicated and that they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory. In 1973, intuitionist Arend Heyting praised non-standard analysis as "a standard model of important mathematical research".〔Heijting, A. (1973) Address to Professor A. Robinson. At the occasion of the Brouwer memorial lecture given by Prof. A.Robinson on the 26th April 1973. Nieuw Arch. Wisk. (3) 21, pp. 134—137.〕 ==Introduction== A non-zero element of an ordered field is infinitesimal if and only if its absolute value is smaller than any element of of the form , for , a standard natural number. Ordered fields that have infinitesimal elements are also called non-Archimedean. More generally, non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle. A field which satisfies the transfer principle for real numbers is a hyperreal field, and non-standard real analysis uses these fields as ''non-standard models'' of the real numbers. Robinson's original approach was based on these non-standard models of the field of real numbers. His classic foundational book on the subject ''Non-standard Analysis'' was published in 1966 and is still in print. On page 88, Robinson writes: The existence of non-standard models of arithmetic was discovered by Thoralf Skolem (1934). Skolem's method foreshadows the ultrapower construction () Several technical issues must be addressed to develop a calculus of infinitesimals. For example, it is not enough to construct an ordered field with infinitesimals. See the article on hyperreal numbers for a discussion of some of the relevant ideas. 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「nonstandard analysis」の詳細全文を読む スポンサード リンク
|