Paul Rusnock

Bernard Bolzano of Prague was one of the few thinkers of his time who combined real talent in mathematics and philosophy. He was especially drawn to the common ground between these fields, interested in questions of method and what would today be called foundations . Interestingly, he was neither a professional mathematician nor a professional philosopher. As a young man, he had decided that his first priority must be to work for the reform and improvement of society. This led him, after much reflection, to become a priest and to occupy a newly-created chair in religious science at the Charles University in 1805. In this post, he was widely known as what would later be called a dissident, denouncing abuses of power and the ills that were the consequence of the still largely feudal institutions and attitudes of the Austrian Empire, and inspiring a generation of students to transform these institutions from within. While his activities brought him a large and committed following, they also gained him powerful enemies in Vienna and Rome, who eventually convinced the Emperor to remove him from his post in December of 1819. Amazingly, during these years of energetic public life, in which he suffered horribly from lung disease and a variety of other ailments, Bolzano remained quite active as a mathematician. As his notebooks attest, he read the latest mathematical literature voraciously and systematically.1 He also made a number of important discoveries, especially in the foundations of real analysis, and published four mathematical works between 1810 and 1817 [Bolzano, 1810; 1816; 1817a; 1817b].2 After his dismissal, he continued his work, pursuing his earlier inquiries with greater depth and precision, …

Alongside his groundbreaking work in logic, Bernard Bolzano (1781?1848) made important contributions to ontology, notably with his theory of collections. Recent work has done much to elucidate Bolzano's conceptions, but his notion of a sum has proved stubbornly resistant to complete understanding. This paper offers a new interpretation of Bolzano's concept of a sum. I argue that, although Bolzano's presentation is defective, his conception is unexceptionable, and has important applications, notably in his work on the foundations of arithmetic