Things, Collections and Numbers

Abstract

While Bolzano’s theory of collections has been compared to set-theory and Lesniewskian-type mereology, it can be reduced to neither.¹ His analyses present a vast historical interest and arguably a philosophical one as well — the latter as long as they are considered for what they are: an investigation in the nature of collective entities whose results are put to work in Bolzano’s semantics as well as in his philosophy of mathematics. Bolzano’s collections (Inbegriffe) are neither sets, nor mereological sums, nor classes. What they are follows from the following tentative definition:

One very important genus of complex ideas that we encounter everywhere are those in which the idea of collection (Inbegriff) appears. There are many types of the latter […] I must first determine with more precision the concept I associate with the word collection. I use this word in the same sense as it is used in the common usage and thus understand by a collection of certain things exactly the same as what one would express by the words: a combination (Verbindung) or association (Vereinigung) of these things, a gathering (Zusammensein) of the latter, a whole (Ganzes) in which they occur as parts (Teile). Hence the mere idea of a collection does not allow us to determine in which order and sequence the things that are put together appear or, indeed, whether there is or can be such an order. […] A collection, it seems to me, is nothing other than something complex (das Zusammengesetztheit hat). (1837, §82, 393)