Editing Nonstandard analysis (section)

{{short description|Calculus using a logically rigorous notion of infinitesimal numbers}}
{{Use dmy dates|date=June 2019}}
[[File:Gottfried Wilhelm Leibniz, Bernhard Christoph Francke.jpg|thumb|right| [[Gottfried Wilhelm Leibniz]] argued that idealized numbers containing [[infinitesimal]]s be introduced.]]

The [[history of calculus]] is fraught with philosophical debates about the meaning and logical validity of [[fluxion]]s or [[infinitesimal]] numbers. The standard way to resolve these debates is to define the operations of calculus using [[(ε, δ)-definition of limit|limits]] rather than infinitesimals. '''Nonstandard analysis'''<ref>Nonstandard Analysis in Practice. Edited by [[Francine Diener]], [[Marc Diener]]. Springer, 1995.</ref><ref>{{cite book|title=Nonstandard Analysis, Axiomatically|last1=Kanovei|first1=V. Vladimir Grigorevich|author1-link=V. Vladimir Grigorevich Kanovei|last2=Reeken|first2=Michael|author2-link=Michael Reeken|publisher=Springer|year=2004}}</ref><ref>Nonstandard Analysis for the Working Mathematician. Edited by [[Peter A. Loeb]], [[Manfred P. H. Wolff]]. Springer, 2000.</ref> instead reformulates the calculus using a logically rigorous notion of [[infinitesimal]] numbers.

Nonstandard analysis originated in the early 1960s by the mathematician [[Abraham Robinson]].<ref>Non-standard Analysis. By [[Abraham Robinson]]. Princeton University Press, 1974.</ref><ref>[http://www.mcps.umn.edu/philosophy/11_7dauben.pdf Abraham Robinson and Nonstandard Analysis] {{Webarchive|url=https://web.archive.org/web/20140415224619/http://mcps.umn.edu/philosophy/11_7Dauben.pdf |date=15 April 2014 }}: History, Philosophy, and Foundations of Mathematics. By [[Joseph W. Dauben]]. www.mcps.umn.edu.</ref> He wrote:

<blockquote>... 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.''</blockquote>

Robinson argued that this [[law of continuity]] of Leibniz's is a precursor of the [[transfer principle]]. Robinson continued:

<blockquote>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.<ref name="NSA">[[Abraham Robinson|Robinson, A.]]: Non-standard analysis. North-Holland Publishing Co., Amsterdam 1966.</ref></blockquote>

Robinson continues:

<blockquote>... Leibniz's ideas can be fully vindicated and ... 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]].</blockquote>

In 1973, [[Intuitionism|intuitionist]] [[Arend Heyting]] praised nonstandard analysis as "a standard model of important mathematical research".<ref>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.</ref>