Bolzano’s Theory of meßbare Zahlen: Insights and Uncertainties Regarding the Number Continuum

During the first half of the 1830s, and as part of his project for a Größenlehre, Bernard Bolzano worked on a manuscript entitled Reine Zahlenlehre in which he introduced the notion of what he called “meßbare Zahlen.” The various additions and corrections to its three extant versions are evidence of an unfinished work, the definitive edition of which was not published until 1976. The present chapter casts light upon the links between, on the one hand, his theory of “measurable numbers” and its conceptual framework, and, on the other hand, his insights and uncertainties with regard to the notions of number and quantity prior to the writing of that work. While Bolzano’s proposal has usually been considered as an attempt at a theory of what nowadays is called the real-number continuum, this chapter shows that a more faithful reading must consider it as a pioneering and transitional theory of the number continuum which provided relevant insights into this latter but which remained, nonetheless, still bound to a not-yet-modern conception of mathematics and numbers.