Path: utzoo!attcan!uunet!tut.cis.ohio-state.edu!ucbvax!information-systems.east-anglia.ac.uk!jrk
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: <4361.9102051326@s4.sys.uea.ac.uk>
Date: 5 Feb 91 13:26:08 GMT
Sender: daemon@ucbvax.BERKELEY.EDU
Lines: 49

In message <1991Feb3.205621.10828@sics.se> torkel@sics.se (Torkel
Franzen) writes:

>Previously you explicitly agreed that the
>"statistical" argument for the consistency of elementary arithmetic is
>a very poor one, and claimed we have no other.  Are you now saying
>that it is an excellent argument?

Yes.  I was earlier swayed by the force of your rhetoric into an
unnecessary retreat.  I claim (now) that the empirical argument is an
excellent one, and that there is no other.  (Of course, I might change
my mind again in future.  What I am claiming, or as you put it -
portentously? - "in fact" claiming, is a matter of empirical judgement
(for me as well as for you) of which the statements I just made are but
the latest evidence available to you.  Such is life.  If you feel that
the changeableness of my views makes them impossible to discuss, that's
up to you.)

In regard to the first part of my claim, you believe that the empirical
evidence (please stop calling it "statistical") is poor.  In regard to
the second part, that whatever one thinks of the empirical evidence,
there is no other, you never quite divulge your attitude, let alone
justify it, as I remarked in my last message.

You rightly objected to the changeableness of my own views.  I am
getting rather frustrated at the absence of yours on this matter.  You
have claimed that my views ascribe a "radical" uncertainty to
mathematics which presumably you do not regard it as having.  This claim
is also in two parts.  The second - that mathematics is not so uncertain
as you argue that I am arguing it to be - you have only hinted at.  So I
ask you once more: are you claiming that there is better evidence than
the empirical for the consistency of arithmetic?  And if so, what is it? 
If it is "intuition", why do you regard it as reliable?  If something
else, what?  You have asked me why I think there is no other evidence; I
can only answer that I see none.  Your turn.  Lighten my darkness.

>   But what does a proof in a consistent system prove? I'm afraid this is
>entirely unclear from your presentation. Are you saying that a proof in
>a consistent theory T of a statement A proves "if T is consistent, then A"?
>It doesn't, you know, unless A is a pi-zero-one statement.

Can you clarify your question?  If A is a theorem of T, then A is
satisfied by every model of T.  This is so even (trivially) for
inconsistent T.  If you can be clearer about what you mean by "if T is
consistent, then A", then perhaps I can answer your question more clearly.

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