Axioms and Meaning

From: Lee Corbin (lcorbin@ricochet.net)
Date: Mon May 28 2001 - 07:12:57 MDT

There are two kinds of meaning. (The chapter "The Location
of Meaning" was my favorite in Godel, Escher, Bach.) One
is *conventional* meaning. For example, the five letters
"zebra" by convention denote a large striped animal. Other
some meanings, however, are *isomorphic*. For example, some
words attempt to imitate the sound of some physical phenomenon,
such as "whoosh". More importantly, some maps are so
detailed that they explicitly denote some object or
relationship in the real world.

This last claim is very controversial, and it really requires
an excursion into evolutionary epistemology to properly
substantiate it. But it can be looked upon as very obvious,
with the help of an example.

Suppose some awful plague wiped out humankind and all
the higher animals. Aliens land eventually, and in a
certain ruined city they come across an immense scale
model of the Mississippi river valley. At first they
don't know what to make of the structure, but finally
one of them breaks the code. He finds that for each
tributary depicted on the map, there is an actual tributary
of a large river system in the eastern half of the North
American continent. At first, this student thinks it's
coincidence, but eventually can not but help to believe
that he's discovered something objectively true of the
universe: namely that this object and the Mississippi have
a real one-to-one relationship. Here is the clincher: any
other piece of intelligent apparatus in the universe will
make the same discovery (or it's not very intelligent).

Sometimes meaning is conventional, sometimes it's isomorphic.
Those are the only two cases.

My question: what about an axiom system? Are all the
meanings (as in symbolic logic) only conventional? How
does this relate to Godel's theorem?

(Godel's theorem states that when a formal axiom system
is thought to be refering to number theory, there exist
true statements in that logical calculus (axiom system)
that have no demonstration within the axiom system. You
can see from my phrasing of the theorem the relevance of
my question.)

Lee Corbin

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