On the epistemological justification of Hilbert's metamathematics

Javier Legris

pp. 225-238

in: Gerhard Heinzmann, Manuel Rebuschi (eds), Aperçus philosophiques en logique et en mathématiques, Philosophia Scientiae 9 (2), 2005.

Abstract

The aim of this paper is to examine the idea of metamathematical deduction in Hilbert’s program showing its dependence of epistemological notions, specially the notion of intuitive knowledge. It will be argued that two levels of foundations of deduction can be found in the last stages (in the 1920s) of Hilbert’s Program. The first level is related to the reduction – in a particular sense – of mathematics to formal systems, which are ‘metamathematically’ justified in terms of symbolic manipulation. The second level of foundation consists in warranting epistemologically the validity of the combinatory processes underlying the symbolic manipulation in metamathematics. In this level the justification was carried out with the aid of notions from modern epistemology, particularly the notion of intuition. Finally, some problems concerning Hilbert’s use of this notion will be shown and it will be compared with Brouwer’s

Cited authors

Hilbert David