The Beyträge at 200: Bolzano's quiet revolution in the philosophy of mathematics

Abstract

This paper surveys Bolzano's Beyträge zu einer begründeteren Darstellung der Mathematik (Contributions to a better-grounded presentation of mathematics) on the 200th anniversary of its publication.  The first and only published issue presents a definition of mathematics, a classification of its subdisciplines, and an essay on mathematical method, or logic.  Though underdeveloped in some areas (including,somewhat surprisingly, in logic), it is nonetheless a radically innovative work, where Bolzano presents a remarkably modern account of axiomatics and the epistemology of the formal sciences.  We also discuss the second, unfinished and unpublished issue, where Bolzano develops his views on universal mathematics. Here we find the beginnings of his theory of collections, for him the most fundamental of the mathematical disciplines.  Though not exactly the same as the later Cantorian set theory, Bolzano's theory of collections was used in very similar ways in mathematics, notably in analysis.  In retrospect, Bolzano's debut in philosophy was a remarkably successful one, though its fruits would only become generally known much later.