A Cantorian Argument Against Infinitesimals

In 1887 Georg Cantor gave an influential but cryptic proof of theimpossibility of infinitesimals. I first give a reconstruction ofCantor's argument which relies mainly on traditional assumptions fromEuclidean geometry, together with elementary results of Cantor's ownset theory. I then apply the reconstructed argument to theinfinitesimals of Abraham Robinson's nonstandard analysis. Thisbrings out the importance for the argument of an assumption I call theChain Thesis. Doubts about the Chain Thesis are seen to render thereconstructed argument inconclusive as an attack on the infinitelysmall.     

Infinitesimals are too small for countably infinite fair lotteries

We show that infinitesimal probabilities are much too small for modeling the individual outcome of a countably infinite fair lottery.     

Smooth Infinitesimals in the Metaphysical Foundation of Spacetime Theories

Journal of Philosophical Logic - I propose a theory of space with infinitesimal regions called smooth infinitesimal geometry (SIG) based on certain algebraic objects (i.e., rings), which regiments...     

Natural Numbers and Infinitesimals: A Discussion between Benno Kerry and Georg Cantor

During the first months of 1887, while completing the drafts of his Mitteilungen zur Lehre vom Transfiniten, Georg Cantor maintained a continuous correspondence with Benno Kerry. Their exchange essentially concerned two main topics in the philosophy of mathematics, namely, (a) the concept of natural number and (b) the infinitesimals. Cantor's and Kerry's positions turned out to be irreconcilable, mostly because of Kerry's irremediably psychologistic outlook, according to Cantor at least. In this study, I will examine and reconstruct the main points in the discussion around (a) and (b) and stress some interesting aspects of the philosophical and mathematical thought of Benno Kerry.     

Modern Infinitesimals as a Tool to Match Intuitive and Formal Reasoning in Analysis

We discuss various ways, which have been plainly justified in the secondhalf of the twentieth century, to introduce infinitesimals, and we considerthe new style of reasoning in mathematical analysis that they allow.