On the Procedural Character of Hilbert’s Axiomatic Method

Abstract

Hilbert’s methodological reflection has certainly shaped a new image of the axiomatic method. However, the discussion on the procedural character of the method is still open, with commentators subscribing to three differing points of view: (1) some have seen it as a synthetic method, i.e. a method to derive theorems from axioms already and arbitrarily established; (2) others have counter-argued in favour of its analytic nature, i.e. given a particular scientific field, the method is useful to reach the conditions (axioms) for the known results of the field (theorems) and to rightly place both in a well-structured theory; (3) still others have underlined the meta-theoretical character of the axiomatic reflection, i.e. the axiomatic method is the method to verify whether axioms already identified satisfy properties such as completeness, independence and consistency. Each of these views has highlighted aspects of the way Hilbert conceived and practiced the axiomatic method and, therefore, they can be harmonized into an image better suited to the function the method was called to fulfil: deepening the foundations of given scientific fields, to recall one of Hilbert’s well-known expressions. Here, I argue that the axiomatic method is, in Hilbert’s hands, a very flexible tool of inquiry, and that for the method to lead analytically to an axiomatic well-structured and reasonably grounded theory it needs to include both synthetic procedures and meta-theoretical reflections in a dynamic interplay. Therefore, in Hilbert’s thought, the expression “deepening the foundations” denotes the whole set of considerations, permitted by the axiomatic method, that allow the theoretician first to identify and then to present systems of axioms for given scientific fields.