Newsgroups: sci.physics,sci.skeptic,alt.consciousness,sci.psychology,comp.ai.philosophy,sci.bio,sci.philosophy.meta,rec.arts.books
Path: cantaloupe.srv.cs.cmu.edu!das-news2.harvard.edu!news2.near.net!howland.reston.ans.net!EU.net!sun4nl!cwi.nl!olaf
From: olaf@cwi.nl (Olaf Weber)
Subject: Re: Roger Penrose's New Book (in HTML) 1.0
In-Reply-To: baez@guitar.ucr.edu's message of 11 Nov 1994 18:12:38 GMT
Message-ID: <Cz95IJ.CDz@cwi.nl>
Followup-To: alt.consciousness,comp.ai.philosophy,sci.philosophy.meta
Sender: news@cwi.nl (The Daily Dross)
Nntp-Posting-Host: havik.cwi.nl
Organization: CWI, Amsterdam
References: <39drsi$7nl@crl10.crl.com> <39vac3$ba6@news.halcyon.com> <Cz3t22.E3s@cwi.nl>
	<3a0c6m$5u9@galaxy.ucr.edu>
Distribution: inet
Date: Mon, 14 Nov 1994 10:08:45 GMT
Lines: 67
Xref: glinda.oz.cs.cmu.edu sci.physics:100147 sci.skeptic:95155 sci.psychology:29670 comp.ai.philosophy:22022 sci.bio:23077 sci.philosophy.meta:14739

I've restricted followups, although I'm not completely sure about
which groups to keep or drop.  I'm reading this in comp.ai.philosophy,
so please keep that one in any answers.

In article <3a0c6m$5u9@galaxy.ucr.edu>, baez@guitar.ucr.edu (john baez) writes:
> In article <Cz3t22.E3s@cwi.nl> olaf@cwi.nl (Olaf Weber) writes:

>> More seriously, I've reached more or less the same conclusion, if
>> by another route: Goedel's theorem states that any "sufficiently
>> complex" formal system is either incomplete or inconsistent.  It
>> seems to me that a formal system describing how people think would
>> be inconsistent rather than incomplete.

> I agree that people are quite inconsistent, but it's also important
> to realize that describing people via a formal system is a gross
> oversimplification, if by that you mean that somebody believes all,
> and only, the consequences of some particular axiom system.  On the
> one hand, we don't proceed by doing mathematics within a fixed axiom
> system.

Well, I'd argue that we do this, in the sense that you could describe
the way we think according to "mathematical" rules and axioms.
However, when interpreted at the level of ideas and thoughts, I don't
expect the system to even resemble anything mathematically sound.

> Perhaps the only sense in which we are truly "axiomatic" is that as
> physical beings we satisfy the laws of physics.

I hold that to be "trivially" true, hence my assertion above.

> We do not, however, get around to working out *all* the consequences
> of any set of axioms, even the laws of physics.

At the levels of beliefs there would seem to be four problems: (1) the
axioms may contradictory, (2) the inference rules may be logically
invalid, (3) people only believe a subset of all that can be proven in
the system, because the implications of the rules and axioms are
worked out "on demand", and (4) the system isn't fixed: when an
obvious contradiction arises, people will often change the axioms or
the rules in a way that is believed to prevent future contradictions
of the same kind.

The above is (in my opinion) a consequence of the fact that our
intelligence evolved as a solution to the problem of making "correct"
decisions based on incomplete data under severe real-time constraints.
A consistent system proceeding only by logically valid steps probably
isn't up to that task.

> A fortiori we will never work out all "mathematical truths".

Indeed not.  If it were easy to know _all_ the consequences of a set
of axioms and inference rules, then things like the last theorem of
Fermat should have been shelved years ago as either true, false, or
undecidable.

> This is why it is completely silly to worry about whether or not
> people can "sneak around Goedel's theorem" --- we are not even
> remotely the sort of things to which the theorem applies.

As pointed out above, I think you _can_ apply the theorem to us, and
the result (IMHO) is that we are inconsistent.

So when people claim to have "sneaked around Goedel's theorem" - in
that they can see a "truth" that is unprovable within the system -
have they actually done so or are they just sloppy thinkers?

-- Olaf Weber
