Implicit definitions and the development of modern axiomatics

GIOVANNINI, EDUARDO N.; SCHIEMER, GEORG

Sevilla

Workshop; International Workshop: Mathematics and Mechanics in the Newtonian Age; 2017

Universidad de Sevilla

Among the many epistemological and methodological issues triggered by the radical transformation that mathematics underwent in the nineteenth century, the problem of understanding and explaining what is exactly the subject?matter of a pure mathematical theory was perhaps one of the most urgent and pressing ones. More specifically, in the context of the emergence of abstract or modern axiomatics, this problem was translated into the question of what exactly an axiom system characterizes or defines, that is, into the inquire about the exact nature of the so?called method of implicit definitions. Accordingly, the notion ? or better, notions ? of ?implicit definition? is nowadays identified as one of the most fundamental methodological innovations of early modern axiomatics. However, even though this fundamental role of implicit definitions in the development of modern axiomatics is often stressed, it is fair to saythat we still lack a clear historical and conceptual understanding of this notion.The main goal of this presentation is to offer a historically sensitive account of the development of the notion of implicit definition in nineteenth and early twentieth century axiomatics. The talk will survey different contributions to the understanding of this notion in the works of Dedekind, Pasch, Frege, Hilbert, Schlick and Carnap. Firstly, we will claim that in this period one can distinguish two main approaches or positions regarding the question of what an axiom system defines. On the one hand, the view that axiomatic systems define higher?order entities, such as higher level concepts, relations or ?structures?; on the other hand, the view that what is defined by means of a set of axioms are the primitive terms or concepts of the system. Secondly, we will argue that while the first approach had a clear mathematical motivation, the second position was mainly suggested by philosophical reasons. More precisely,while the first approach was intimately bounded with the emergence of the structural understanding of mathematical theories, the second approach was rather prompted by the philosophical problem of explaining the nature of the primitive terms of a mathematical theory. In other words, the view that a system of axioms define implicitly its primitive terms was motived by the philosophical problem of explaining how the primitive terms of a pure mathematical theory acquire its meaning and how its reference is determined. Finally, we will argue that a more refined account of the method of implicit definitions turns out to be highly relevant for the examination of the problem of the applicability of mathematics in the nineteenth century.