chapter  7
44 Pages

The Recovery of Certainty

Wittgenstein did not suggest or assume that the foregoing resolution of the consistency problem would dispose of the 'foundations crisis'. On Hilbert's approach such would indeed be the case, but Wittgenstein proceeded on the firm understanding that the two issues are distinct albeit related areas of philosophical confusion, which thus demand independent investigation. In the former we are concerned with the reliability of the calculi we employ; it is not the concept of mathematical truth per se that is under consideration, but rather, the problem is one of guaranteeing that some future mathematical discovery will not undermine our pre-existing trust in the theorems yielded by a calculus. In other words, the problem is one of elucidating what is meant by regarding a calculus as sound. But the heart of the foundations problem lies, as Frege indicated in Foundations of Arithmetic, at a deeper philosophical level (FA §3). The challenge here is ultimately that of clarifying the logical status of mathematical truth in order to establish the basis for our conviction that mathematics is, as Hilbert described it, 'a paragon of certitude'. Thus, the 'foundations crisis' does not simply amount to the anxiety that mathematical systems might unbeknownst to us contain 'hidden' contradictions. More importantly, it derives from the fundamental philosophical dilemma that it is not at all clear what it means to describe a mathematical proposition as a 'necessary truth'.