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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: <1991Feb20.215853.22149@sics.se>
Date: 20 Feb 91 21:58:53 GMT
References: <6270.9102172336@s4.sys.uea.ac.uk>
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Organization: Swedish Institute of Computer Science, Kista
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In-Reply-To: jrk@information-systems.east-anglia.ac.uk's message of 17 Feb 91 23:36:10 GMT

In article <6270.9102172336@s4.sys.uea.ac.uk> jrk@information-systems.
east-anglia.ac.uk (Richard Kennaway CMP RA) writes:

   >I can't give any satisfactory justification for my choice, any more than
   >you can for the non-empirical evidence.

  Apparently you're saying that the fact that no inconsistency has yet
been derived from the axioms actually used so far in mathematics is
good evidence that no inconsistency can be derived from the full set
of axioms. Since you give no explanation of how this differs
from the step from "hypothesis P has been verified for all natural
numbers checked so far" to "hypothesis P is true of all numbers" - an
argument which we have agreed is even "empirically" poor - and, moreover,
say that you have no justification for your claim, there's little I
can say about this.

  You do make some general remarks about generalizations and support,
but they're completely pointless in view of the fact that they apply
equally to any formal theories. Thus, there is nothing in your remarks
on this point that would not equally support the idea that any
arbitrary extension T of ZFC is consistent, given that our
mathematical theorems are theorems of T. "You see the observations as
supporting at most the consistency of A: I take them as supporting the
consistency of T."

  You do also say that

   >What corresponds in mathematics, and
   >specifically the hierarchy of induction axioms, to that background of
   >physical theory that makes prediction of the sunrise more than just "what
   >has been, will be"?  The fact that those induction axioms are not an
   >arbitrary collection, but seem to form a natural whole.

but of course make no attempt to make this out to have anything to do
with "empirical evidence". Anybody can claim that the axioms of any
theory T seem to form a natural whole. Nothing in what you say is any
less arbitrary than the most romantic blathering about intuition, to
which you so sternly objected in your first messages on this topic.
Your observation that successive sunrises are logically independent of
each other again simply ignores the fact that we have a theoretical
explanation of sunrises, while you propose no theoretical explanation
of the fact that no contradiction has appeared in arithmetic.

  In short, the vacuous character of your present remarks makes it
impossible for me to criticize them in any worthwhile way, and I don't
think it's a very meaningful exercise to argue in detail that your
remarks are vacuous.  Essentially, you rely on the idea that any
alternative to your views is equally arbitrary. But this is mere
obscurantism. You have preferred to ignore my questions about what, if
anything, is proved in mathematical proofs. If we reject the notion of
mathematical evidence in favor of your supposed "empirical" evidence,
we are left with the idea that the truth of ordinary theorems of
mathematics (in so far as they are at all meaningful?) is a matter of
their being generalizations that have stood the test of time - a
particularly feeble form of mathematical "empiricism". To be sure,
J.S.Mill resolutely and bravely argued that the truth of "11+6=17" etc
is a matter of empirical experience; and perhaps you wish to follow in
his footsteps.  If so, there is a considerable body of thinking on
this topic that you should take into account.