Overview

For centuries prior to 1800, infinitesimals—infinitely small numbers—were an indispensable tool in the calculus practiced by the great mathematicians of the age. Between the mid-1800s and the mid-1900s, however, infinitesimals were excluded from calculus because they could not be rigorously established. This changed in 1960, when Abraham Robinson resurrected their use with his creation of nonstandard analysis. Since that time, nonstandard analysis has had an important effect on several areas of mathematics as well as on mathematical physics and economics.

Background

An infinitesimal is an infinitely small number. More precisely, it is a nonzero number smaller in absolute value than any positive real number. But merely defining a mathematical entity does not guarantee its existence. For example, we can define an obtuse-angled triangle as a triangle all of whose...