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Article 1949 of comp.ai.philosophy:
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>From: jmc@SAIL.Stanford.EDU (John McCarthy)
Subject: Re: Existence
In-Reply-To: egnilges@phoenix.Princeton.EDU's message of 7 Dec 91 19:03:38 GMT
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Unaccustomed as I am to agreeing with Mikhail Zeleny, I even agree
with his use of the word "presumptuous" applied to Bill Skaggs's
posting.  Namely, Skaggs has taken as obvious a position on an
issue which is still highly controversial among mathematicians
and philosophers - the reality of certain mathematical objects,
real numbers, sets, etc.

I think most mathematicians are anti-Platonist when they come
to philosophy, although they are often effectively Platonist
in their work.  However, Goedel was explicitly Platonist,
carefully distinguishing what was true from what might be
proved and even from what might be known in any sense.

Not only did Goedel hold this view, which was rather unpopular
throughout his life, but this view motivated his three greatest
pieces of work, his completeness theorem for first order logic,
his even more famous incompleteness theorem and his work on
the continuum hypothesis.

To oversimplify only slightly, the first two pieces of work
related truth and provability.  In one case he showed they
agreed and in the second case he showed they were different.
When he showed the consistency of the continuum hypothesis,
he distinguished between the real numbers, which are not
denumerable and certain models of the axioms of the real
numbers in set theory, which could be denumerable because
of the inability of the axioms of set theory to cover all
the facts about sets.  Cohen's proof of the independence
of the continuum hypothesis used the same basic idea -
only the technical details were different.

Of course, many people, perhaps most mathematical philosophers, still
disagree with Goedel's Platonism, but they don't dismiss it so
cavalierly as Bill Skaggs does.  I happen to agree with Goedel for
reasons of artificial intelligence.  Concisely, put I think success in
AI theory will require a theory of how a mind embedded in a world can
find out about the world.  Such a theory has got to assume the world
exists and there are facts about it that the mind may or may not
discover.  Even if the facts are conjectured, the mind may never be
completely sure of them.  It is presumptuous and narcissistic to
declare a question meaningless just because you have no way of
answering it conclusively.  This holds true both for questions
about the material world and for purely mathematical questions.

The need for such a theory should put AI on the side of Platonism
rather than on the side of positivism.  Of course, there are
positions not well classified along this spectrum.
--
John McCarthy, Computer Science Department, Stanford, CA 94305
*
He who refuses to do arithmetic is doomed to talk nonsense.



