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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: <16462.9102272325@s4.sys.uea.ac.uk>
Date: 27 Feb 91 23:25:44 GMT
Sender: daemon@ucbvax.BERKELEY.EDU
Lines: 61

In a message which will eventually appear, torkel@sics.se (Torkel Franzen)
will write:
>But to declare (to return to the very
>special case of the consistency of arithmetic) that the ordinary mathematical
>evidence for the truth of a theorem - i.e. its proof - is no evidence
>at all, that is what I call obscurantism.

I'm not saying it's no evidence at all, but that it is only evidence
relative to the consistency of the system in which the proof is conducted.
If one accepts that consistency, the proof is perfectly good evidence; to
the extent that one doubts it, so is that evidence weakened.

What is special about consistency theorems is that they are consistency
theorems, and that they generally cannot be proved in systems weaker than
those whereof they speak.  As a result, a proof of the consistency of a
system X conducted in a system Y containing X does not give one any reason
to believe in that consistency that one did not have already.  If one is
more sceptical of Y than of X, it is worth even less than one's preexisting
convictions, and is more interesting as a fact about Y than about X.  The
proof is worthless as evidence to support one's belief in the consistency of
X.  In contrast, if one does not know that, say, the four-squares theorem is
a theorem, it will be unlikely to appear obvious, and its truth will be
entirely uncertain; on reading a proof, one becomes as certain of its truth
as one is of the truth of the system in which the proof is made.

The fact that consistency theorems - some of them at least - can be seen as
"just" statements of number theory does not alter this.  Once one has read
them as asserting consistency of system X, that reading vitiates the
usefulness of proving them in system Y.

>You admit to an "intuitive
>understanding of and confidence in a mathematical theory", but seem
>reluctant to say that this yields any kind of evidence for the truth
>of mathematical theorems. I find it hard to take this seriously.

I am indeed reluctant.  But I suspect it is largely a matter of personal
taste, with, as you say, little to argue over.  I am far more impressed by
the fact that arithmetic is still standing after all these years than by
seeing the truth of the axioms.


In message <1991Feb24.095043.18175@sics.se>, torkel@sics.se (Torkel Franzen)
writes:
[concerning Wette]
>  In its details, his proof is clearly something out of the ordinary. That
>is, it is not the kind of thing that is done in ordinary mathematics. If
>only for this reason, truly empirically-minded scientists should give it
>close attention.

I don't think I'll bite.  From this and other things I've heard about his
proofs, it sounds like something I would be fascinated to learn more about
- but only if I don't have to lift a finger to find out. I'd rate his
chances at about the same level as the people who claim to make antigravity
machines with gyroscopes.

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

"Gods these days, they want FTL travel, causality, and Lorentz invariance.
I tell 'em to pick two out of three and call me back."