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From: torkel@sics.se (Torkel Franzen)
Newsgroups: comp.ai.philosophy
Subject: Re: Intuition and doubt (was Re: Minds, machines, and Godel)
Message-ID: <1991Jan23.045926.17528@sics.se>
Date: 23 Jan 91 04:59:26 GMT
References: <5794.9101222235@s4.sys.uea.ac.uk>
Sender: news@sics.se
Organization: Swedish Institute of Computer Science, Kista
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In-Reply-To: jrk@information-systems.east-anglia.ac.uk's message of 22 Jan 91 22:35:29 GMT

In article <5794.9101222235@s4.sys.uea.ac.uk> jrk@information-systems.
east-anglia.ac.uk (Richard Kennaway CMP RA) writes:

  >If you like, I have non-serious doubts.  But they're serious enough to
  >torpedo, for me, at its outset, the argument that we are more powerful
  >than machines because we can intuit the truth of various statements,
  >including the consistency of arithmetic.

  What is this talk about 'intuiting truth'? It means nothing to me. My
claim is that arithmetic is obviusly consistent because there is a
trivial consistency proof for it: the axioms are true, the rules
of inference are truth-preserving, hence all theorems are true, hence no
contradiction is a theorem.

  Now I am not claiming that it is impossible to reject this proof.
Intuitionistist will reject it because of the use of classical logic in
elementary arithmetic; finitists will reject it because of its use of
quantification over an infinite totality. So these people do in fact
take the view that the axioms are obscure, doubtful, or false (at least
coupled with the use of classical logic).

  What is mere ritual is to reject this particular proof while not raising
any comparable hullabaloo about mathematical proofs in general. As I tried
to emphasize: if the proof appears insufficiently mathematical, consider
the fact that it is easily formalizable in elementary analysis. And if we
say of one particular arithmetical theorem of elementary analysis that
the only reason we have for believing it to be true is that no counterexample
has been found (and thus that its proof proves nothing at all), we are
indulging in a very peculiar ritual unless we go on to consider just which
proofs of elementary analysis, if any, prove anything at all.