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>From: boroson@spot.Colorado.EDU (BOROSON BRAM S)
Subject: Re: Searle (was Re: Daniel Dennett (was Re: Comme
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Date: Tue, 26 Nov 1991 02:26:32 GMT
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What are the implications of G\"odel's theorem for the Platonic
and formalist philosophies of mathematics?

First, I'm not a mathematician or a philosopher, just a grad student
studying astrophysics.  My knowledge of these topics comes from reading
_G\"odel, Escher, Bach_, _The Emperor's New Mind_, and a one-semester
course on Zermelo-Frankel set theory I took as a freshman (it went way
over my head).  This is my first post, and though it may be lacking 
in erudition, here goes:
______________________________________________________________________

I've read that G\"odel's theorem was a major setback for the formalist
philosophy and a major victory for Platonic philosophy.  But I'm not
convinced.  To work with a concrete example, assume that Fermat's Last 
Theorem is a G\"odel-undecidable proposition:

FLT:  There exist no Fermat Triples (x,y,z), for any n>2.

Then a G\"odel-numbering scheme can turn propositions into numbers and
"provability" into a numerical relation.  The numerical relation
corresponding to "FLT is undemonstrable" is simply FLT.  Therefore,
by metamathematical reasoning, if FLT is provable, it is false (since
by translation it asserts that FLT is unprovable).  Thus if number theory
is to be consistent, FLT must be unprovable.  But since FLT asserts the
unprovability of FLT, it must be true.  **Therefore mathematical truth,
though it exists, can never be formalized.**

My first problem with the above argument is that ~FLT can be added to
the axioms of number theory just as consistently as FLT can.  The
situation is analagous to Euclidean vs. non-Euclidean geometry:
neither FLT nor ~FLT can be considered "true" because neither leads
to a contradiction.

Turning ~FLT into an axiom leads to a formal system that is not
"omega-consistent."  That is, it asserts the existence of a Fermat
Triple, but nobody will ever be able to exhibit one.  But I see no
real drawback to a system that is omega-inconsistent (maybe there is
one I don't know?)--a system with a true inconsistency collapses because 
all propositions are demonstrable.  From a formalist point of view,
the "upside-down A" and "backwards E" do not have to correspond to our
intuitive ideas of universal and existential quantifiers any more than
a line must have only one parallel through a given point.

It seems that the G\"odelesque argument above stacks the deck against
formalism by assuming FLT must be either true or false, that there
even is such a thing as mathematical truth.  But a formalist must,
in light of G\"odel's theorem, not only argue that FLT is neither
true nor false, but also that it is neither true nor false that FLT is
a theorem of number theory.  So G\"odel's theorem forces a formalist
to become more radical.

So here is a formalist conception of mathematics:

Some abstract entities exist.  Only I would say that combinations of
formal symbols exist, and not the kind of informal ideas we have in
our heads.  A subset of these combinations of formal symbols obey a
regular syntax; a subset of those, in turn, can be considered deductive
systems--but the "deduction" is simply human intuition attempting to
find order in the set of formulae.

Now what is the argument for a Platonic conception and why does it fail?

Assume there is a "real" geometry.  What is a point and what is a line,
if they are not formal elements?  Maybe points and lines are represented
as formal elements, but they have meaning only when compared with "real
geometry."  That is, the axioms are true or false, based on whether 
they describe geometry or not.  Certainly, non-Euclidean geometries
have proved beautiful and useful, which just means that a more general
"truth" (differential geometry) had to be "discovered" to accomodate both.

Anyone should see that this choice of "truth" is as subjective and arbitrary
as the choice of "beauty" in the arts.

Everything must be a pattern of formal symbols, because nothing else is 
well-defined.  Thus our own universe is such a pattern, and our ideas and 
minds must be formed by such a pattern (if your introspection does not let
you imagine this, your introspection is stronger than mine, or your 
imagination weaker.)  Platonic mind/matter duality is therefore not a
valid objection to Artificial Intelligence.

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| BRAM      | I have nothing to say, and I'm Saying it. | BramBo roson  |
| Recursive |                                           |               |
| Acronym   |                --John Cage                | boroson.jila. |
| Man       |                                           | colorado.edu  |
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