One, two, three . . . continuity: C.S. Peirce and the nature of the continuum

The nature of the one and the many is an immemorial problem. This paper begins with Parmenides and the paradoxes presented by his disciple Zeno, then presents at some depth the mathematical concepts of limit and the continuum developed in the second half of the 19th-century by Karl Weierstrass, Richard Dedekind, and Georg Cantor. These interpretations are explicitly contrasted with C. S. Peirce's view of the nature of the continuum, and how this implies the actual existence of infinitesimals. A brief description of Peirce's concept of ‘one, two, three’ is presented, showing how this new view on continuity completes this model of all reality, which he now termed synechism. Finally, several modern scientific examples of a similar view of continuity are presented.