Internal logic

from Kronecker to Hilbert and beyond

Yvon Gauthier

pp. 186-213

in: Yvon Gauthier, Internal logic, Berlin, Springer, 2002

Abstract

Internal logic is the logic internal to mathematical discourse, primarily the arithmetical discourse. The logic in question tends to vanish as a component of arithmetic and is readily identified to the inferential structure of arithmetic. Internal logic becomes arithmetical or polynomial logic — or modular logic as we shall say later on. The internal structure can be exhibited with the help of ordinary logic (Hilbert says Aristotelian logic) or intuitionistic (constructive) logic. The internalization of logic in the case of arithmetic means the arithmetization of logic, that is the polynomial interpretation of logic which I have achieved on the model of Kronecker's general arithmetic. I have claimed that Hilbert's programme was conceived originally along the same line of thought. The idea that consistency (and decidability) were internal properties of mathematical theories was Hilbert's own motive in his first attempts at defining the consistency problem — which had to be solved, as we have seen, in terms of polynomial equations.