Implicit definition and modern axiomatics

GIOVANNINI, EDUARDO N.; SCHIEMER, GEORG

Groningen

Congreso; 12th Congress of the International Society for History of Philosophy of Science (HOPOS); 2018

University of Groningen. Faculty of Philosophy

Among the many epistemological and methodological issues triggered by theradical transformation that mathematics underwent in the nineteenth century, theproblem of understanding and explaining what is exactly the subject?matter of apure 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 say that we still lack a clear historical and conceptual understanding of this notion.The main goal of this presentation is to offer a historically sensitive accountof the development of the notion of implicit definition in nineteenth and earlytwentieth-century axiomatics. We will survey different contributions to the understanding of this notion both in some key representative cases in the history of modern axiomatics ? especially in the works of Dedekind, Pasch and Hilbert ? as well as in early philosophical reflections, in particular in the works of Frege, Schlick and Carnap. Firstly, we will claim that in this period it is possible to 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 concepts or relations ? or in modern model?theoretic terms, a class of models or structures. On the other hand, the view that axioms can be regarded as definitions of the meaning of the primitive terms of a mathematical theory. 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. Finally, we will suggest that it is instructive to relate these two conceptions of implicit definitions to two important traditions in nineteenth and early twentieth century axiomatics, i.e., one which takes axiom systems primarily in a semantic way, as means to define the subject-matter of a theory, and the other which gives more importance to the proof?theoretical role of axioms, that is, axiom systems as tools or theoretical devices to prove theorems.