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From: jrk@information-systems.east-anglia.ac.uk (Richard Kennaway CMP RA)
Newsgroups: comp.ai.philosophy
Subject: Re: Intuition and doubt (was Re: Minds, machines, and Godel)
Message-ID: <11656.9101241836@s4.sys.uea.ac.uk>
Date: 24 Jan 91 18:36:29 GMT
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In article <1991Jan23.180914.8116@sics.se> torkel@sics.se (Torkel Franzen)
writes:

>  This doesn't even begin to touch on the question I raised. You were
>saying, of a particular theorem of elementary analysis of the form
>"every natural number has property P", with P a primitive recursive
>predicate, that we have no reason whatever to believe that every
>natural number has property P, except that no counterexample has been
>found. So my question is, would you extend this to every theorem of
>analysis? If it were proved in elementary analysis that the equation
>x^n+y^n=z^n has no non-trivial solution for n>2, would you proclaim to
>the net that we still have no reason whatever to believe that this is
>so, except that no counterexample has been found? I rather suspect
>not.

You suspect rightly.  I fail to see the relevance of the fact that both
FLT and the consistency of arithmetic can be expressed as statements of
the given form.  You might equally well have imputed to me the claim that
every long-undecided conjecture, of whatever syntactic form, may be
accepted as true.  The difference between "arithmetic is consistent" and
FLT is that there has been vastly greater opportunity for an inconsistency
to show up in arithmetic than a counterexample to FLT.  The effort
expended on the latter is a tiny fraction of that expended, not
necessarily on the metamathematical study of arithmetic itself, but on
ordinary, everyday mathematics that uses it and can be formalised in it. 
On the whole, I suspect that arithmetic is consistent; I have no opinion
regarding FLT.

But looking back on this thread, I believe you have misinterpreted the
thrust of my comments on consistency.  I was primarily rejecting the
argument to consistency from intuition: "the axioms are obviously true and
the inference rules obviously truth-preserving".  I meant only to remark
in passing that to justify using arithmetic, the fact that it has worked
so far is enough reason for not worrying.  Weak as it might be, I see is
no other.

I also notice that you have generally refrained from expressing personal
opinions about the consistency of various systems.  You may feel that
personal tastes are not relevant to the discussion; but I would be
interested in knowing what they are anyway.  Do you have opinions on the
consistency of arithmetic, ZFC, and ZFC+UM?  If so, what are they, and on
what grounds do you hold them?

--
Richard Kennaway          SYS, University of East Anglia, Norwich, U.K.
Internet:  jrk@sys.uea.ac.uk		uucp:  ...mcsun!ukc!uea-sys!jrk

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