chapter  3
45 Pages

The Nature of Proof

Wittgenstein had an extraordinary gift, not simply for looking at stale problems from some entirely fresh perspective, but perhaps even more significant, for recognising problems where none was suspected. For example, philosophers of mathematics continue to debate the implications of Godel's second incompleteness theorem for the 'foundations crisis' without ever having considered the soundness of the 'meta-mathematical' framework on which GOdel's proof rests. Only Wittgenstein sought to clarify the significance of the latter by scrutinising the cogency of the former. So too, Wittgenstein warned that before we struggle with the Decision Problem - the question whether it is possible to construct an algorithm which will enable one to arrive in a finite number of steps at an answer to any question belonging to a given mathematical system - we must consider that most basic of all problems in the philosophy of mathematics: the nature of mathematical proof.