Actual Infinity and Newton’s Calculus

Abstract

In developing and presenting the algorithms of his calculus Newton used two systems of concepts — the fluxional and the infinitesimal systems. He declared that these were merely presentational means, and when he goes over to the justification of the algorithms they are not mentioned anymore and in their place he uses the conceptual system of limits or “first and last ratios”. It would seem that what constitutes the system of first and last ratios as a justificatory system, is the fact that infinitesimals are not needed in it. In fact, however, his specific manner of justification by means of limits is logically equivalent to that of using infinitesimals, since both are based on the assumption of actual infinity in his conceptual system. I’ll show this by analysing some texts in his Principia, and compare them with the Aristotelian conception which grounds the Greek theory of exhaustion.