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From: jrk@information-systems.east-anglia.ac.uk (Richard Kennaway CMP RA)
Newsgroups: comp.ai.philosophy
Subject: Intuition and doubt (was Re: Minds, machines, and Godel)
Message-ID: <5794.9101222235@s4.sys.uea.ac.uk>
Date: 22 Jan 91 22:35:29 GMT
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In article <1991Jan21.130344.19720@sics.se> torkel@sics.se (Torkel Franzen)
writes:

>Anyway, the
>"statistical" argument for the consistency of elementary arithmetic or
>ZFC is a very poor one.

Agreed.  But (IMHO) it's the only one we've got.

>  On the other hand, there is nothing mysterious about the claim that
>e.g. first order arithmetic is obviously consistent. There is no need
>to speak poetically about intuition: the theory is obviously consistent
>because its axioms are obviously true. The mere occurrence of the word
>"true" in an observation is often, it seems, taken to imply that
>the observation must be metaphysical or otherwise deplorable. To say
>that the axioms are true, however, is to say no more than what is said
>in asserting the recursion equations for addition and multiplication,
>and the principle of mathematical induction. Now if somebody does in fact
>find these assertions obscure or doubtful or false, he will have no reason to
>believe that first order arithmetic is consistent. It is mere ritual
>however, to speak of the consistency of arithmetic as dubious without
>attempting to establish just what is obscure, false, or doubtful about
>the axioms. If one does in fact have serious doubts concerning the
>consistency of arithmetic, one must have serious doubts concerning the
>validity of practically every theorem of mathematics.

Dealing with this is wandering away from the original subject, and out of
comp.ai.philosophy for that matter.  But I'm not sure if I can cross-post
(broken news software), so I won't attempt to move the thread to another
group.

To speak of the consistency of arithmetic as obvious is just as much a
ritual (i.e. non-productive activity making no practical difference to
anything).

The axioms seem intuitive; no inconsistency has been found; they underpin
"the validity of practically every theorem of mathematics"; so we might
as well use them, pro tem.  No-one needs to believe in them to use them.

If you like, I have non-serious doubts.  But they're serious enough to
torpedo, for me, at its outset, the argument that we are more powerful
than machines because we can intuit the truth of various statements,
including the consistency of arithmetic.  I do not regard myself as able
to intuit the consistency of arithmetic any better than a machine fed
with knowledge about the history of mathematics.

As for finding the axioms obscure, doubtful, or false...well, I don't
suppose I do.  But here is an analogy.  I am sitting on a chair, at a
desk, on which is a computer.  These facts seem to me as clear, obvious,
and true as the axioms of arithmetic or of the various flavours of ZF
(all of which can be true simultaneously, of different concepts of
"set").  Yet when I ask myself how I create from my senses the notions of
"this chair", "this desk", etc., I see that I have not the faintest idea. 
And neither does anyone else, it would seem, or it wouldn't be so
difficult to make a robot that can pick parts out of a jumbled bucket. 
(Ah, comp.ai relevance!)  More than once, I have had the experience -
surely not unique to me - of seeing something, and then realising that it
is something else entirely; thereafter seeing it as such.  My fundamental
uncertainty regarding the physical senses applies just as much to my
awareness of my own thoughts.  The axioms of arithmetic are only obvious
to me as long as I don't think too closely about them.  When I do, I
cannot find any reason to assume that someone will not eventually prove
from those axioms that 0=1, only reason enough not to be concerned about
the possibility in practice.  It only makes a difference in philosophical
arguments like these - I continue to sit on "this chair", typing on "this
keyboard", expounding "these thoughts".  I just don't really believe in
any of it.

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

"As I was sitting in my chair
I knew the seat just wasn't there
Nor legs, nor back, but I just sat
Ignoring little things like that."