X-Message-Number: 10217
Date: Wed, 12 Aug 1998 08:32:41 -0400
From: Thomas Donaldson <>
Subject: CryoNet #10215 - #10216

Hi guys (and any women that are out there too):

In one sense, Bob, you are right about Goedel. However in quite another
you are wrong. The problem comes from any symbolic system which is rich
enough to say useful things, and in that way Goedel proved a very 
interesting result. If we do logic as most people would consider it,
then we inevitably run into such problems.

The plain fact is, though, that logic or symbols are no more than our
constructions, and behind them there is the workings of our brain. Since
humans particularly are very language-oriented, this may be hard to see,
but it does make a big difference. (Read the later Wittgenstein, not
the earlier Wittgenstein, and you will see him struggling with this 
fact). Our brain works WITH symbols but does not use them when it does.
If you want the rationality you are talking about not to run into the
problems Goedel described, you will need a notion of rationality which
runs without language. Since we live in a real world which is NOT
symbolic, this is not hard to DO, but almost impossible to explain 
symbolically ie. with words.

Do not think of logic. Think about neural nets. Even though the existing
computer ones probably do not emulate the workings of our brain, they
give a hint about how brains probably work.

It is also just possible that with the right setup we can avoid such
paradoxes, but we'll both gain and lose when we do so. There is a branch
of the philosophy of mathematics which insists that everything we do
must be explicitly constructible. Proofs by contradiction are declared
invalid. Since those who follow that philosophy do so by act rather than
arguments, very slowly large parts (but hardly all) of mathematics have
been turned into constructible proofs --- though sometimes the proofs
need a bit of rewording. The latest I've seen of this is the development
(entirely constructively) of the idea of metric spaces. General topology
has not been done and would probably become quite transformed. And as
part of this constructibility, we don't speak of ALL of anything without
provable, constructive tests that an object is one of these anythings.
Sets of course also must be explicitly constructible.

However so far as I know, no one has investigated whether or not such
a system will really avoid the problems Goedel pointed out. As for the
barber, perhaps she is a woman, or a castrato. Otherwise no constructive
proof of the existence of such a barber can be made.

			Best and long long life,

				Thomas Donaldson

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