```X-Message-Number: 23917
Date: Sat, 17 Apr 2004 10:52:36 -0400
From: Thomas Donaldson < var s1 = "73647.1215"; var s2 = "compuserve.com"; var s3 = s1 + "@" + s2; document.write("<a href='mailto:" + s3 + "'>" + s3 + "</a>"); >
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

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