The Ghosts of Departed Quantities

Nonstandard analysis is the branch of mathematics concerned with constructing a number system that includes infinitesimals. These are numbers smaller than any real number yet still greater than zero. Although nonstandard analysis was developed in 1966 by Abraham Robinson, it is based off of Gottfried Wilhelm Leibniz’s construction of the calculus, created some 300 years earlier. Nonstandard analysis differs from the standard approach to analysis in its use of infinitesimals. Infinitesimals in nonstandard analysis replace epsilon-delta technique in standard analysis; a method many would argue is far less intuitive than the nonstandard use of infinitesimals. While Robinson may have been the first to arithmetically combine infinitesimals with the real numbers and rigorously define the rules that govern this new number system, he certainly was not the first to use infinitesimals. The basic idea of an infinitesimal dates back to antiquity to a period when the Greeks were afraid of the infinite. This was the period in Greek history shortly after Zeno presented his paradoxes which shook geometers until they banned the use of infinity in mathematics. It was during this time when infinitesimals gained firm ground in mathematics. Archimedes incorporated infinitesimals into his “method of compression” which would later be formalized by Leibniz into his infinitesimal calculus. However, since neither Leibniz nor his followers could rigorously define the