Infinitesimals

Abstract

Infinitesimals, i.e., numbers which are in some respect like zero and in some other respect unlike zero, have been used in mathematics since the beginning of the calculus. Robinson [Rob96] gives an overview of the use of infinitesimals in the history of the calculus: Leibniz uses infinitely small numbers in the development of the calculus without admitting their existence; he considers them to be useful fictions. De l’Hospital seemed to believe in the existence of infinitesimals, and he formulated Leibniz’ principles in a way which made the main inconsistency stand out – that infinitesimal quantities are sometimes treated as being equal to zero, and sometimes as being not equal to zero.