The Truth of Proof: A Hegelian Perspective on Constructivism

Abstract

In the constructivist movement, which started with Brouwer’s attack on the principles of classical logic and mathematics, the history of mathematics seems to repeat itself in a specific, self-conscious way. The main objective of this paper is to describe this approach by adopting the phenomenological method of Hegel. Its starting point consists in looking at knowledge as a continuous, yet painful, process—the Calvary of the Spirit—with its stations corresponding to the naive, direct concepts of experience as based on sense certainty or belief in the independent realms of objects and their transformation into more sophisticated, socially charged theories. The signs of this advance are the patterns of self-consciousness, such as Cantor’s diagonal results, adopted by constructivism in a different but still powerful way. The key concept against which the progress of this development will be measured within the constructivist movement is the concept of proof, particularly with respect to Gödel theorems and the resulting split of knowledge into proof and truth. I will read this split, with the help of Lorenzen and Brandom, as a relative differentiation between two co-dependent aspects of self-consciousness that are to be prospectively conceived as two idealized dialogue partners.