Matthew E. Moore

Peirce’s topical theory of continuity

In the last decade of his life C.S. Peirce began to formulate a purely geometrical theory of continuity to supersede the collection-theoretic theory he began to elaborate around the middle of the 1890s. I argue that Peirce never succeeded in fully formulating the later theory, and that while that there are powerful motivations to adopt that theory within Peirce’s system, it has little to recommend it from an external perspective.

A Cantorian Argument Against Infinitesimals

In 1887 Georg Cantor gave an influential but cryptic proof of theimpossibility of infinitesimals. I first give a reconstruction ofCantors argument which relies mainly on traditional assumptions fromEuclidean geometry, together with elementary results of Cantors ownset theory. I then apply the reconstructed argument to theinfinitesimals of Abraham Robinsons 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.