X-Message-Number: 10239
Date: Sat, 15 Aug 1998 09:36:26 -0400
From: Thomas Donaldson <>
Subject: CryoNet #10229 - #10236

Hi everyone!

Yet more on Goedel's Theorem etc. There is another approach entirely
to the results Goedel found. The question it might raise in some minds
is whether or not we want to use a formal system at all. We are after
all living in the world without benefit of a formal system which 
explains the world to us. The whole idea of formal systems began with
the Greeks, and has now been developed far enough that some people
such as Goedel have come upon one of their central problems. People
were able to invent and think things before the Greeks, and the 
Chinese (with no well developed notion of theory) went quite far --
for some time farther than any other civilization on Earth.

I am not doubting Goedel's Theorem. I am raising a question about
what it means.

It's even likely that we can do mathematics, though it would not be
the kind done with formal systems, axiomatically. Even the Pythagorean
theorem, as a statement about how lengths and triangles behaved, was
known to the Assyrians if not before them. And we might even avoid
the problems of such systems by refusing to admit proofs by contradiction
and insisting that the entities we discuss must always be constructible.

Symbols are NOT the world. We do not live in symbols, we live in the
world, and ultimately our language and words are defined not by 
other symbols and language but by their use in the world, sometimes
by just pointing, sometimes other ways. And the entire idea of formal
systems has a history, and may someday be abandoned for other means of
doing math, or physics, or in any way dealing with the world. After all,
it was experimental science, not mathematics, which brought the West
first even and then far ahead of the Chinese. 

As someone who taught math at college level and beyond, I will add that
axioms do provide a useful form of exposition. But for some time now
that idea has begun to look shaky --- Goedel's Theorem providing an 
example. I doubt that axiomatic math and formal systems will disappear
entirely, but they may well end up as useful tools for exposition, to
which we do not and should not attach any special metaphysical 

Or does anyone on Cryonet seriously believe that we've come to the
acme of understanding, even of mathematics and how to do it?

			Best and long long life,

				Thomas Donaldson

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