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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: <1991Jan25.075046.4464@sics.se>
Date: 25 Jan 91 07:50:46 GMT
References: <11656.9101241836@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 24 Jan 91 18:36:29 GMT

In article <11656.9101241836@s4.sys.uea.ac.uk> jrk@information-systems.
east-anglia.ac.uk (Richard Kennaway CMP RA) writes:

   >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.

  Your remarks are unrelated to what I wrote. I didn't say that you
should ascribe equal 'statistical' weight to FLT and the consistency
of arithmetic. I was assuming that FLT turned out to be provable in
elementary analysis. My question is, what proofs in elementary
analysis, if any, do you regard as proving anything? If FLT turned out
to be provable in analysis, would you still say that we have no reason
for believing it to be true (i.e.  that the equation x^n+y^n=z^n has
no non-trivial solutions for n>2) except that no counterexample has
been found - as you do in the case of the theorem "elementary
arithmetic is consistent"?

  Perhaps I had better say this one more time? The 'hullabaloo' you raised
in the case of one particular theorem of elementary analysis consisted
in claiming that its proof proves nothing; that we still have no reason for
believing its conclusion to be true (i.e. arithmetic to be consistent)
except that no counterexample has been found. The question is, would you
raise a corresponding hullabaloo in the case of any and all proofs of
elementary analysis? And if not, what are the grounds for this distinction?

  Again, I am not saying that good answers to such questions can't be given.
But no good answer - so I claim - will give any special status to those
theorems of classical mathematics which have the form "T is consistent".
Hence my initial remarks on the merely ritual character of much talk about
consistency.