Hermann Cohen’s rationalist philosophy of mathematics

Scott Edgar

Saint Mary’s University, Canada

Abstract: 
In his 1883 Principle of the Infinitesimal Method and its History, the Marburg School neo-Kantian Hermann Cohen gives a philosophically critical history of calculus’ development in the early modern period. Despite being an interpreter of Kant, his account is deeply rationalist. He is centrally concerned with establishing that the foundational concepts of calculus are valid -- that is, that we are justified in using them in our mathematical natural scientific descriptions of reality. However, he explicitly and repeatedly emphasizes the significance of a Leibnizian rationalist principle -- the principle of continuity: the principle that “there are no jumps” in natural scientific representations of reality -- in establishing the validity of those concepts. Cohen offers the Kantian-sounding argument that calculus’ foundational concepts are necessary conditions of natural science’s representation of physical reality. But it is the principle of continuity that, for Cohen, establishes that argument’s premise -- namely, that the foundational concepts of calculus, and in particular the concept of infinitesimals, are necessary for natural science’s representation of physical reality. This paper seeks to identify the commitments that justify the principle of continuity for Cohen, and to locate them in his text. In particular, I identify two further principles that Cohen is committed to, and I argue that they entail the principle of continuity. Cohen never appeals to these two principles by name, which explains why they have gone unnoticed in the existing literature on Cohen’s Infinitesimal Method. Nevertheless, I argue, he appeals to them repeatedly throughout his various discussions of ancient, renaissance, and early modern mathematics of the infinite and continuous. The first is the principle that our concepts of the infinite and continuous are conceptually prior to our concepts of the finite and discrete: that is, that our concepts of the finite and discrete can never define our concepts of the infinite and continuous, but must always be defined by them. The second principle, which entails both the principle of continuity and Cohen’s other, unnamed principle, is Leibniz’s principle of sufficient reason. Consequently, I argue, for Cohen the principle of continuity -- and so too his argument for the validity of calculus’ foundational concepts -- depends essentially on the principle of sufficient reason. I thus defend a Leibnizian rationalist interpretation of Cohen’s philosophy of mathematics. This interpretation is of interest for two reasons: first, for how it illustrates a broader Marburg School project of synthesizing Kantian and rationalist doctrines in their philosophy of science; but second, for how it foregrounds the significance of Leibnizian themes in philosophy of mathematics in the immediate background to early analytic philosophy.