The Continuum: History, Mathematics, and Philosophy

Abstract

The main aim of this dissertation is to depict a wide variety of the conceptions of the continuum by tracing the history of the continuum from the ancient Greece to the modern times, and in so doing, to find a new way to look at the continuum.

In the first part, I trace the history of the continuum with a special emphasis on unorthodox views at each period. Basically, the history of the continuum is the history of the rivalry between two views, namely, between the punctiform and the non-punctiform views of the continuum. According to the punctiform view, the continuum is composed of indivisibles; on the other hand, according to the non-punctiform view, the continuum cannot be composed of indivisibles.

In the second part, I present Richard Dedekind’s and Georg Cantor’s standard mathematical theories of the continuum as the modern representative of the punctiform view of the continuum, and then examine Charles Saunders Peirce’s non-punctiform view of the continuum.

In the last chapter, I give some mathematical interpretations to Aristotle’s and Peirce’s theories of the continuum according to both of which the continuum cannot be composed of points. In interpreting Aristotle’s view, I use modern topology and show that Aristotle’s view can be nicely captured by topology. On the other hand, in interpreting Peirce’s view, I appeal to the theory of category and show that in the category-theoretic framework the continuum appears quite differently from the standard one conceived in the Dedekindian and Cantorian ways.

In conclusion, I try to defend a sort of pluralistic view concerning the conceptions of the continuum.