Some developments in the philosophy of mathematics, 1790–1870

Abstract

Introductions to the philosophy of mathematics rarely focus on the period 1790–1870, if they mention it at all. Aside from Mill’s empiricist program, what was happening in the philosophy of mathematics between Kant and Frege? Certainly there were crucial mathematical developments. For example, non-Euclidean geometry was discovered in the 1830s, and this paved the way for later conventionalist and formalist philosophies of mathematics such as those offered by Poincare and Hilbert. In addition, the rigorization of analysis during our period was an important precedent to logicism, a philosophical program espoused by Frege, Dedekind, and Russell just after 1870. But where is the influential philosophy? Of course, there were philosophers, so there was also philosophy of mathematics. One approach to this period would be to survey what philosophers such as Hegel, Herbart, Fries, Mill, and Comte said about mathematics. However, few of these authors are known for their contributions to philosophy of mathematics, with the notable exception of Mill. Moreover, most of this philosophical work seems far removed from the revolutionary developments in mathematics. On the other hand, entwined in the mathematical developments are crucial and fascinating philosophical arguments. In fact, it is precisely in its close connection to mathematical developments that the relevant philosophy is so exciting, for it shows that the border between mathematics and its philosophy is both permeable and dynamic. Philosophical issues arise in mathematics, and mathematicians deal with them, explicitly and implicitly, in their creative work and in their teaching. Furthermore, how the philosophical issues are treated affects mathematics.