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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: <1991Jan29.044227.6344@sics.se>
Date: 29 Jan 91 04:42:27 GMT
References: <17928.9101281842@s4.sys.uea.ac.uk>
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Organization: Swedish Institute of Computer Science, Kista
Lines: 61
In-Reply-To: jrk@information-systems.east-anglia.ac.uk's message of 28 Jan 91 18:42:50 GMT

In article <17928.9101281842@s4.sys.uea.ac.uk> jrk@information-systems.
east-anglia.ac.uk (Richard Kennaway CMP RA) writes:

   >>My question is, what proofs in elementary
   >>analysis, if any, do you regard as proving anything?

   >All of them - provided elementary analysis is consistent.

   First let me note that this only makes any obvious sense if what
you mean is that you accept a proof of a statement A in analysis as
proving "if analysis is consistent then A."

   Now in the case of FLT, it is indeed correct that "if analysis is
consistent then FLT" follows from the existence of a proof in analysis
of FLT. However, since we have only very poor statistical evidence for
the consistency of analysis, on your view, we then have only very poor
statistical evidence for any theorem of analysis, as you present the
matter.  This is a very radical doctrine and goes far beyond ordinary
misgivings about consistency. For example, would you say that we have
only very poor statistical evidence for our belief that every natural
number is the sum of four squares? Or for the validity of Sturm's
algorithm for finding the number of real zeros of a polynomial?

   Furthermore, consistent theories may well have false consequences,
viz. consequences not of pi-zero-one form. For example, it is perfectly
compatible with FLT being true that it is provable in elementary analysis
that there is a counterexample to FLT, even if elementary analysis is
consistent. Or, to take a common class of statements: theorems of the
form "the algorithm R terminates for every input". The fact that such a
theorem is provable in a consistent theory does not imply that it is
true. And, in case you wonder about this point, to say that the theorem
is true is to say that the algorithm R terminates for every input.

   >Quite so.  Consistency is no more worth remarking on than that the earth
   >turns.  One does not usually bother to preface a mathematical proof with
   >"provided arithmetic (or analysis, ZF, etc.) is consistent".

  I'm afraid these remarks of yours further illustrate the ritual
character of much talk of consistency, since you apparently silently
invoke consistency as a supposed justification of mathematical
conclusions, without having taken the trouble to notice that it is
insufficient to justify them, except in the case of pi-zero-one statements.

   >It begins
   >to matter, when people start claiming (as in the discussion that this
   >subthread arose from) that arithmetic "really" is true, rather than
   >merely exhibiting a certain syntactic object (viz. a proof of consistency
   >of arithmetic in elementary analysis).  There is no problem with the
   >latter.  The former is - to me - unclear and unnecessary.

  For some reason you are intent on trying to make something
mysterious out of a perfectly ordinary mathematical use of the word
"true". There's no need, you know, to put a "really" before it. If I
say that the induction principle is true, what I am saying is simply
this, that any set of natural numbers which contains 0 and is closed
under the successor operation contains every natural number. If you
then ask me, in portentous tones, whether I am really claiming that
the induction principle is really true, I don't know what you're
talking about. On the other hand I'd be interested to hear if you have
any real justification for the idea that conclusions obtained by means of the
induction principle have merely very poor, statistical support.