In the 18th and 19th centuries two transitions took place in the development of mathematical analysis: a shift from the geometric approach to the formula-centered approach, followed by a shift from the formula-centered approach to the concept-centered approach. We identify, on the basis of Bolzano's Purely Analytic Proof [Bolzano 1817], the ways in which Bolzano's approach can be said to be concept-centered. Moreover, we conclude that Bolzano's attitude towards the geometric approach on the one hand and the formula-centered approach on (...) the other were of a different nature; the for­mer being one of rejection, the latter of non-participation. Bolzano supports his concept-centered methodology by philosophical views, which were partially shared by mathematicians with a formula-centered approach to analysis. (shrink)

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, …. (shrink)