Bolzano and the Foundation of the Real Continuum

This paper offers a historical and conceptual examination of Bernard Bolzano’s 1817 Rein analytischer Beweis, in light of modern perspectives on the foundations of real analysis. While it is well known that Bolzano did not fully succeed in establishing the Intermediate Value Theorem, we argue that his work nonetheless demonstrates a remarkably advanced understanding of the necessary conditions for such a proof. In particular, we show that Bolzano effectively formulated the property now known as the Supremum Axiom and, under assumptions weaker than those adopted later by Dedekind and Cantor, gathered the essential properties that today define the real number continuum. Thus, what we present is an interpretation of what Bolzano in fact accomplished through his proof. Our reading highlights Bolzano’s consistent aim, which is expressed across his works, to provide mathematical analysis with a purely analytical foundation, independent of geometric intuition. By analysing his use of convergent geometric series, his implicit reliance on the Archimedean property, and his philosophical reflections on mathematical concepts, we argue that Bolzano’s contributions deserve a central place in the prehistory of the modern theory of the real numbers.