Yvon Gauthier

My discussion centers around Ulrich Kohlenbach's Applied Proof Theory: Proof Interpretations and their Use in Mathematics [21 ] which appears as a major work in the "proof mining project" of recent proof theory. The emphasis is on the tradition of proof theory originating with Hilbert and his motivations. I examine also Hilbert's Kroneckerian inspiration and I maintain that Hilbert's programme is the continuation of Kronecker's programme by logical and metamathematical means. I conclude that despite Kreisel's proposal of a shift of emphasis in traditional proof theory, Hilbert's original proof-theoretical ideal is still alive and proof mining is in the purview of Hilbert programme. © 2011 Elsevier B.V., All rights reserved.

The theme « Truth and Certainty » is reminiscent of Hegel’s dialectic of prominent in the Phänomenologie des Geistes, but I want to treat it from a different angle in the perspective of the constructivist stance in the foundations of logic and mathematics. Although constructivism stands in opposition to mathematical realism, it is not to be considered as an idealist alternative in the philosophy of mathematics. It is true that Brouwer’s intuitionism, as a variety of constructivism, has idealistic overtones, but my main concern in this paper is located in the mathematical tradition of constructive mathematics from the Greeks to Fermat, Gauss and Kronecker, and from the logical side, in the finitist doctrine of Hilbert and his followers.