X-Message-Number: 23917 Date: Sat, 17 Apr 2004 10:52:36 -0400 From: Thomas Donaldson <> Subject: response to Peter Merel Hi everyone! This message is to Peter Merel. As a mathematician I work and worked on partial differential equations and problems related to them. I cannot claim deep familiarity with Goedel's theorem or those who have written about it. When Iread your message which gave two sources for what you said (Barnsley and Spencer- Brown) I looked them up. There is a Barnsley, involved mostly in work with fractals. Spence-Brown, in 1968, published a book on logic that doesn't seem to have gotten lots of attention. When I read parts of it it wasn't clear at all that he had any special answer either for constructivism or for Goedel's theorem. Perhaps you're thinking of some other references which I could not find? If so, please quote them with adequate detail so that I can recover them. The main thing that happens with constructivism is that you haven't proved anything unless you produce either an example/counterexample showing that some claim is true or false, or have a method which provably (note what I just said about examples/counterexamples) works in a given, constructably specifiable domain. This means, among other results, that we're not interested in developing a FORMALISM which doesn't allow any use of the excluded middle or proofs by contradiction. We're trying to see just how much math can be developed if we don't use such methods at all. As for formalisms, anyone who can give me a reference to some way to develop something like Goedel's theorem without using an argument or statement which relies simply on the excluded middle (if it's not false it must be true; if it's not true it must be false) or proofs by contradiction will get my thanks. Incidentally, even though constructivism so far cannot get to lots of nice math (transfinite numbers, etc, for instance) it can get to lots of the math which is constantly applied by others, like vector spaces or metric spaces. Best wishes and long long life for all, Thomas Donaldson Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=23917