On The Epistemological Justification of Hilbert's Metamathematics

LEGRIS, JAVIER

Philosophia Scientiae

Editions Kimé

Lugar: Paris; Año: 2005 vol. 9 p. 225 - 238

1281-2463

ON THE EPISTEMOLOGICAL JUSTIFICATION OF HILBERT´S METAMATHEMATICS Javier Legris University of Buenos Aires and CONICET (Argentina) Abstract This paper deals with the interplay between logic and philosophy in Hilbert´s metamathematical program. It examines the idea of metamathematical deduction in Hilbert´s program showing its dependence of epistemological notions, specially the notion of intuitive knowledge. The distinction will be made between two foundational levels or layers which can in fact be observed in the last stages (in the 20s) 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. This metamathematical deduction led to a characterization of logic as a theory of formal deductions that was fundamental for the historical development of the proof-theoretical understanding of logic. 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 notion and with the idea of symbolic construction due to Herrmann Weyl.