chapter  6
39 Pages

Consistency

The foregoing chapters provide us with the necessary background to approach one of the murkiest areas of Wittgenstein's remarks on the foundations of mathematics: his puzzling insistence that the consistency problem is completely vapid. Once again the cries of incompetence have been most shrill where the least effort has been made to clarify the design of Wittgenstein's attack. Perhaps the most perplexing aspect of Wittgenstein's argument is the absence of any clear statement of its purpose. Why should Wittgenstein have dwelt so heavily on a topic which was made, if not quite irrelevant, certainly of no pressing mathematical importance following the publication of Godel's second theorem? Surely by the time he came to deliver his 1939 lectures on the foundations of mathematics, Wittgenstein had had sufficient opportunity to digest the significance of GOdel's work? Unless, of course, he was seeking in some obscure way to enhance GOdel's results: without fully realising the extent of this shared purpose! Not surprisingly, philosophers of mathematics have reacted to Wittgenstein's strictures with a mixture of bewilderment and unbridled animosity, seeing them as a crude assault on mathematical logic with little conceivable motive other than to fit in with Wittgenstein's perverse opinion that the general influence of mathematical logic on philosophy has been pernicious. Such a heresy could only be born from the seeds of technical confusion which ergo propter hoc is demonstrated by Wittgenstein's obtuse failure to grasp the mathematical problems involved in the discovery of a contradiction in an axiomatic system.