ABSTRACT
This chapter reviews what mathematics has to say concerning the continuous, the infinite, and the infinitesimal. The differential appeared as a philosophically unimportant application of the doctrine of limits. The view that the Calculus requires infinitesimals is apparently not thought open to question; at any rate, no arguments whatever are brought up to support it. This view is certainly assumed as self-evident by most philosophers who discuss the Calculus. Cohen proceeds at once to reject the view that the infinitesimal calculus can be independently derived by mathematics from the method of limits. The conception of magnitude, Cohen says, which is presupposed in limits, in turn presupposes limiting magnitudes. By limiting magnitudes, as appears from the context, he means infinitesimals, the ultimate differences between the terms of a series and its limit. The metaphysical theory by which infinitesimals are to be rescued seems, both mathematically and philosophically, destitute of grounds in its favour.
