There are quite a few studies on late Bolzano’s notion of a collection (Inbegriff). We try to broaden the perspective by introducing the forerunner of collections in Bolzano’s early writings, namely the entities referred to by expressions with the technical term ‘et’. Special emphasis is laid on the question whether these entities are set-theoretical or mereological plenties. Moreover, similarities and differences to Bolzano’s mature conception are pointed out.

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 We also discuss the second, unfinished and unpublished issue, where Bolzano develops his views on universal mathematics.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. (shrink)

The aim of the paper is to present and evaluate Bolzano's theory of grounding, that is, his theory of the concept expressed and the relation brought into play by 'because'. In the first part of the paper (Sections 1-4) the concept of grounding is distinguished from and related to three other concepts: the concept of an epistemic reason}, the concept of causality, and the concept of deducibility (i.e., logical consequence). In its second part (Sections 5-7) Bolzano's positive account of grounding (...) is reconstructed in axiomatic form and critically discussed. (shrink)

In his Foundations of a General Theory of Manifolds, Georg Cantor praised Bernard Bolzano as a clear defender of actual infinity who had the courage to work with infinite numbers. At the same time, he sharply criticized the way Bolzano dealt with them. Cantor’s concept was based on the existence of a one-to-one correspondence, while Bolzano insisted on Euclid’s Axiom of the whole being greater than a part. Cantor’s set theory has eventually prevailed, and became a formal basis of contemporary (...) mathematics, while Bolzano’s approach is generally considered a step in the wrong direction. In the present paper, we demonstrate that a fragment of Bolzano’s theory of infinite quantities retaining the part-whole principle can be extended to a consistent mathematical structure. It can be interpreted in several possible ways. We obtain either a linearly ordered ring of finite and infinitely great quantities, or a partially ordered ring containing infinitely small, finite and infinitely great quantities. These structures can be used as a basis of the infinitesimal calculus similarly as in non-standard analysis, whether in its full version employing ultrafilters due to Abraham Robinson, or in the recent “cheap version” avoiding ultrafilters due to Terence Tao. (shrink)

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.

Le but de cet article est de présenter et d’évaluer la théorie de la fondation de Bernard Bolzano, c’est-à-dire sa théorie du concept exprimé et de la relation mise en jeu par « parce que ». Dans la première partie , le concept de fondation est distingué et mis en relation avec trois autres concepts : le concept de raison épistémique, le concept de causalité et le concept de déductibilité . Dans la seconde partie , je reconstruis la théorie bolzanienne (...) de la fondation sous forme axiomatique et j’en offre une discussion critique.The aim of the paper is to present and evaluate Bernard Bolzano’s theory of grounding, i.e. his theory of the concept expressed and the relation brought into play by ‘because’. In the first part the concept of grounding is distinguished from and related to three other concepts : the concept of an epistemic reason, the concept of causality and the concept of deducibility . In its second part Bolzano’s positive account of grounding is reconstructed in axiomatic form and critically discussed. (shrink)