Editing Nonstandard calculus (section)

{{Short description|Modern application of infinitesimals}}
In [[mathematics]], '''nonstandard calculus''' is the modern application of [[infinitesimal]]s, in the sense of [[nonstandard analysis]], to infinitesimal [[calculus]]. It provides a rigorous justification for some arguments in calculus that were previously considered merely [[heuristic]].

Non-rigorous calculations with infinitesimals were widely used before [[Karl Weierstrass]] sought to replace them with the [[(ε, δ)-definition of limit]] starting in the 1870s. For almost one hundred years thereafter, mathematicians such as [[Richard Courant]] viewed infinitesimals as being naive and vague or meaningless.<ref>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".</ref> <!--On this paragraph, see Talk, three sections that mention Courant and/or quotations-->

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 [[Howard Jerome Keisler|Howard 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."<ref>[[Elementary Calculus: An Infinitesimal Approach]], p. iv.</ref>