Peirce's place in mathematics

This article compares treatments of the infinite, of continuity and definitions of real numbers produced by the German mathematician Georg Cantor and Richard Dedekind in the late 19th century with similar interests developed at virtually the same time by the American mathematician/philosopher C. S. Peirce. Peirce was led, not by the internal concerns of mathematics which had motivated Cantor and Dedekind, but by research he undertook in logic, to investigate orders of infinite sets (multitudes, in his terminology), and to introduce the related concept of infinitesimals. His arguments in support of the mathematical and logical validity of infinitesimals (which were rejected by such eminent mathematicians as Cantor, Peano, and Russell at the turn of the century) are considered. Attention is also given to the connections between Peirce's mathematics, his philosophy, and especially his interest in continuity as it was related to his Pragmatism.