Bolzano’s early quest for a priori synthetic principles: Mereological aspects of the analytic-synthetic distinction in Kant and the early Bolzano

The aim of the present study is to gain insight into the development of the ideas of Bolzano, often described as the grandfather of analytic philosophy, out of the German philosophy of the eighteenth century. It provides an analysis of Wolff's influential mathematical method and argues that the notion of construction plays an important role in Wolff's account of definitions and geometric demonstrations. Investigation of Wolff's work, however, reveals that he does not provide a philosophical account for the role of construction in the mathematical method. As a result, Kant's philosophy of mathematics can be understood as filling the gap in Wolff's mathematical method rather than as a replacement.
Contrary to existing studies, the present study substantiates the following claim: the Kantian idea that mathematical knowledge extends (synthetic) rather than clarifies our knowledge (analytic) plays a crucial role in the work of the early Bolzano. An unexpected outcome of this study is that so called mereological distinctions (part-whole relations) are fundamental to the philosophy of mathematics of both Kant and the early Bolzano. Detailed investigation of Bolzano's manuscripts and notes reveals that part-whole relations play a central role in his conception of general mathematics. According to a reconstruction of his theory of numbers, Bolzano regards arithmetic as synthetic because it relies on the laws of associativity and commutativity as synthetic principles.