The concept of continuity recurs in many different philosophical contexts. Aristotle and Kant believed it to be an essential feature of space and time. Medieval scholars believed it to be the key to unlock the mysteries of motion and change. Bertrand Russell believed that, while everyone talked about continuity, no one quite knew what it was they were talking about.

The subject of this dissertation is mathematical continuity in particular. By mathematical continuity, I mean continuity as it applies to or is found in mathematical systems such as sets of numbers. Mathematical continuity is a relatively recent concern. The need to address whether numerical systems are continuous came about with the creation of calculus, specifically, of limit theory.

The dissertation focuses on four mathematicians/philosophers from the late nineteenth and early twentieth centuries who were concerned with mathematical continuity. Richard Dedekind and Georg Cantor, in the 1870s and 1880s, developed the concept of a 'point-continuum;' i.e. a continuum composed of discrete entities, such as a collection of numbers arranged on a straight line. Paul du Bois-Reymond, in 1882, and Charles S. Peirce, especially in his post-1906 essays, criticized this compositional point-continuum. Du Bois-Reymond believed infinitesimals were necessary for continuity; Peirce believed no compositional continuum could ever satisfy our intuitions.

My ultimate conclusions are that (1) the concept of the mathematical point-continuum does suffer from philosophical difficulties, (2) the concept of the infinitesimal is neither as philosophically problematic nor as mathematically useless as is often charged, but that (3) infinitesimals by themselves cannot solve the problems raised by a compositional view of continuity.