X-Message-Number: 23889 References: <> From: Peter Merel <> Subject: Constructivists and Formalists Date: Thu, 15 Apr 2004 07:43:11 +1000 Thomas Donaldson writes, > I will point out (for both Bob Ettinger and Peter Merel) that the > underlying idea of constructive mathematics consists exactly of using > a system in which proofs by contradiction aren't allowed Thanks Thomas. Indeed constructivists reject the law of the excluded middle because it is necessarily ambiguous. They hold you cannot legitimately assert the truth of (P Or Not P) unless you can specifically affirm the truth of P, or you can specifically affirm the truth of Not P. Constructivists generally use a 3 valued logic (true, false, indeterminate) to include the possibility of ambiguity. But this doesn't go far enough to beat Godel because it is possible to enumerate determinacy in a 3 valued logic through exactly the same procedure used to enumerate truth in a 2 valued one. One winds up with a statement that the system cannot consistently justify as determinate or indeterminate, and the nut is busted again. And you can keep adding values to your logic to account for the trail of busted nuts ... but Godel keeps busting 'em until you either get tired or run out. There is an intuitionistic, or at least quasi-humanistic, path through Godel, but it doesn't come this way. Instead of dealing in predicates with values, and hence a machine with states blah blah blah, chuck that whole theoretical frame and pick one closer to empiricism. We don't find any "true" or "false" in life. We make distinctions concerning patterns of behaviors of our sensing processes when they interact with signals of other processes. Formalize the correspondence of process, behaviors, signals, and distinctions, and you can make a calculus in which there is no terminal encoding truth or falsity. Hence no nuts to bust. Cryonet isn't the right forum for this, and anyway it's already covered reasonably well in Barnsley and in G. Spencer Brown. Naturally one hesitates to recommend the latter to a mathematician ... Peter Merel. Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=23889