Nonstandard calculus

In mathematics, nonstandard calculus is the modern application of infinitesimals, 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. (See history of calculus.) For almost one hundred years thereafter, mathematicians such as Richard Courant viewed infinitesimals as being naive and vague or meaningless. 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 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." The history of nonstandard calculus began with the use of infinitely small quantities, called infinitesimals 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 Cavalieri and others by exploiting an infinitesimal quantity he denoted in area calculations, preparing the ground for integral calculus. 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.