Editing Nonstandard calculus (section)

== History ==
The history of nonstandard calculus began with the use of infinitely small quantities, called [[infinitesimal]]s in [[calculus]].  The use of infinitesimals can be found in the foundations of calculus independently developed by [[Gottfried Leibniz]] and [[Isaac Newton]] starting in the 1660s. [[John Wallis]] refined earlier techniques of [[indivisibles]] of [[Bonaventura Cavalieri|Cavalieri]] and others by exploiting an [[infinitesimal]] quantity he denoted <math>\tfrac{1}{\infty}</math> in area calculations, preparing the ground for integral [[calculus]].<ref>Scott, J.F. 1981. "The Mathematical Work of John Wallis, D.D., F.R.S. (1616&ndash;1703)". Chelsea Publishing Co. New York, NY. p. 18.</ref> 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 [[George Berkeley|Bishop Berkeley]] in his book ''[[The Analyst]]''.

Several mathematicians, including [[Colin Maclaurin|Maclaurin]] and [[Jean le Rond d'Alembert|d'Alembert]], advocated the use of limits. [[Augustin Louis Cauchy]] developed a versatile spectrum of foundational approaches, including a definition of [[continuous function|continuity]] in terms of infinitesimals and a (somewhat imprecise) prototype of an [[(ε, δ)-definition of limit|ε, δ argument]] in working with differentiation. [[Karl Weierstrass]] formalized the concept of [[Limit of a function|limit]]<!--correct link?--> 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 [[nonstandard analysis]] to distinguish it from the standard use of limits.  This approach used technical machinery from [[mathematical logic]] to create a theory of [[hyperreal number]]s 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".