Newsgroups: comp.ai.philosophy,sci.logic
Path: cantaloupe.srv.cs.cmu.edu!das-news2.harvard.edu!news2.near.net!howland.reston.ans.net!pipex!demon!prim.demon.co.uk!dave
From: dave@prim.demon.co.uk (Dave Griffiths)
Subject: Re: Godel, Lucas, Penrose, and Putnam
Message-ID: <1994Dec28.215202.511@prim.demon.co.uk>
Organization: Primitive Software Ltd.
References: <3ddp99$tc@usenet.ucs.indiana.edu> <1994Dec25.162020.17458@Princeton.EDU> <3dnekv$34g@usenet.ucs.indiana.edu>
Date: Wed, 28 Dec 1994 21:52:02 GMT
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Xref: glinda.oz.cs.cmu.edu comp.ai.philosophy:24123 sci.logic:9223

In article <3dnekv$34g@usenet.ucs.indiana.edu> chalmers@bronze.ucs.indiana.edu (David Chalmers) writes:
>
>Penrose would argue that he can simply "see" that PA is sound, because he
>can see that the axioms are all true and that the rules are valid,
>independent of any ability to deductively "show" that.  Actually, I would
>sympathize with such a claim; I think we know the truth and validity of
>those axioms and rules as well as we know anything in mathematics.  (How do
>we know the truth of *anything* in mathematics?  Now there's a question).

I think the answer to that is, we don't. When we parse a statement like
2+2=4, how do we know it's "really" true? Suppose our mind starting playing
tricks on us so that we'd get the same feeling of truth, of certainty with
an untrue statement like 2+2=5? I think that the formality and hard-edged
logical of mathematics is actually a powerful illusion, one that we are
trapped inside to a certain extent. Mathematics is actually a fuzzy empirical
science. There have been a lot of wrong turns and invalid assertions on the
road to the body of knowledge we have today.

What happens to an organism that believes that 2+2=5? It loses out in the
battle for survival to an organism that can perform arithmetic correctly. Our
mind is an organism that has had to adapt to the real world. It's through this
evolutionary pressure that there is a good correlation between our feelings
of certainty about mathematical rules and the real world. It's not infallible
though. None of us is sound. We have ideas all the time that on reflection
turn out to be wrong. Richard Feynman once said "We have a habit in writing
articles in scientific journals to make the work as finished as possible, to
cover up all the tracks, to not worry about the blind alleys or to describe
how you had the wrong idea first".

And this is the way out of the box Penrose has constructed. We don't generate
new ideas by mechanically obeying the rules we already have. Our unconscious
minds generate new ideas almost at _random_. Sometimes the new idea gives us
that 2+2=4 type feeling of certainty. There is no objective way to ascertain
the truth of this new statement though, only a fuzzy, neural net, pattern
matching sort of comparison with the existing body of ideas (is it consistent)
and with external reality. Sometimes this feeling of certainty turns out to
be wrong. Maybe we're just bad at maths.

If a neural net learns to recognize the number 2, how does it "know" that
the number is 2? It doesn't, it just guesses right most of the time. I don't
see any reason why a sufficiently complicated neural net shouldn't also
learn to recognize the truth of symbolic statements in mathematics. It only
needs to be right most of the time, just like us. All you need then is a
second neural net inventing new statements at random (well not quite random,
it would try certain avenues of attack in the same way a chess program does),
and have the first one check them. What could be simpler. :-)

Dave
