Rethinking Leibniz\u27 Infinitesimals

Newton and Gottfried Leibniz both used infinitesimals—numbers which are nonzero, yet smaller in magnitude than any real number—in the early 1700s to describe instantaneous rates of change in their developments of calculus. However, they were unable to provide a rigorous foundation for the existence of these quantities, and mathematicians instead began to embrace the now-ubiquitous epsilon-delta approach to the foundations of calculus, avoiding the notion of infinitesimal numbers altogether. However, in the 1960s, Abraham Robinson finally provided these foundations through what he called “non-standard analysis,” and his work provides an extension of the real numbers to include both infinitesimal and infinite quantities: the so-called system of hyperreal numbers. My project is to reconsider the standard construction of hyperreal numbers using a relatively new approach to set theory, which is based on W. F. Lawvere\u27s “Elementary Theory of the Category of Sets.” This approach places emphasis on sets and functions between them, rather than on sets and the notion of elementhood as in a traditional set theory, such as that of Zermelo-Fraenkel. This shift in perspective is prompted by the fact that the axioms of a function-based set theory are thought to be more intuitive and accessible to an undergraduate who is encountering foundational mathematics for the first time. During my presentation, we will consider constructions of familiar number systems, e.g., the integers, and conclude with a construction of hyperreal numbers, beginning from the axioms of this function-based set theory. We will see that the elements of these number systems may be regarded as equivalence classes of functions, and we will consider the role of the Axiom of Choice in the construction of these hyperreals