Hermann Cohen’s Principle of the Infinitesimal Method: A Defense

Abstract

In Bertrand Russell’s 1903 The Principles of Mathematics, he offers an apparently devastating criticism of The Principle of the Infinitesimal Method and Its History (PIM) by the neo-Kantian Hermann Cohen. Russell’s criticism is motivated by a concern that Cohen’s account of the foundations of calculus saddles mathematics with the paradoxes of the infinitesimal and continuum and thus threatens the idea of mathematical truth. This article defends Cohen against Russell’s objection and argues that, properly understood, Cohen’s views of limits and infinitesimals do not entail the paradoxes of the infinitesimal and continuum. Essential to that defense is an interpretation, developed in the article, of Cohen’s positions in the PIM as deeply rationalist. The interest in developing this interpretation is not just that it reveals how Cohen’s views in the PIM avoid the paradoxes of the infinitesimal and continuum. It also reveals elements of what is at stake, both historically and philosophically, in Russell’s criticism of Cohen.