ABSTRACT
Nineteenth-century mathematics was witness to the increasing importance of the concept of a set. In fact this ground-breaking theory of Georg Cantor's to which referred, as a theory of the infinite, was the first systematic mathematical theory of sets. Bernard Bolzano's stance, however, was even more radical, because he argued that it was possible to calculate with infinite quantities. He made some rather fumbling attempts to show how, later to be much refined and improved upon by Cantor. In the course of pursuing the programme, Gottlob Frege combined some of his own insights with insights that he shared with the likes of Bolzano, Richard Dedekind, and Cantor to help to vindicate new heterodoxy that mathematical study of the infinite was as capable as any other branch of mathematics of being put on a firm footing. The French philosopher Louis Couturat argued that Cantor's work was indispensable to the study of continuity, and would eventually be regarded with the mathematical equanimity.
