Sums, numbers and infinity Collections in Bolzano's mathematics and philosophy

This dissertation is a collection of essays almost exclusively focussing on Bernard Bolzano's theory of collections, and its connections to his conception of mathematics. The first significant contribution of this work is that it develops a novel approach for comparing and appraising Bolzano’s collections against the alternatives offered by set theory and mereology. This approach consists in focusing not on the metaphysical comparison but on the role Bolzano's collections play in his mathematics vis à vis the role the sets of set theory (ZFC) play for modern-day mathematics. The second important contribution of this dissertation is a novel interpretation of the interaction between Bolzano's theory of concepts on the one hand, and his way of comparing the size of infinite collections of natural numbers on the other. This culminates in a completely new interpretation of Bolzano's famous Paradoxes of the Infinite as a work that is not an anticipation of Cantorian set theory, but an attempt at deploying philosophical insights on the infinite to develop a sound treatment of infinite series, both converging and diverging ones.