Re: Kress's Probability Trilogy Q's

Michael Ash <mike@xxxxxxxxxxx> wrote:

James A. Donald <jamesd@xxxxxxxxxxx> wrote:
Wayne Throop
His justification for why algorithmic computation
isn't adequate to explain the human mind was related
to Goedel's incompleteness theorem, was it not? Thus
implicitly, he's claiming that humans can determine
the truth of *any* Goedel sentence. This seems wildly
unlikely.

A human can see that the rules of simple arithmetic must
be consistent. An algorithm cannot.

Could you please provide more support for this statement? There appears to
be a proof that first-order arithmetic is consistent, and I see no reason
why an algorithm can not "see" this proof. Remember that the
Incompleteness Theorem only prohibits using a logical system to prove *its
own* consistency, there is nothing prohibiting consistency proofs in
general.

I really don't understand why anybody uses Goedel to even attempt to
disprove AI. These fundamental truths which we can ascertain without
having any proof could simply be wrong, or could be arrived at through
simple means such as trying a bunch of examples and not finding any
counterexamples.

Something else often overlooked in these kinds of argument:

If a theorem prover is complete (in some domain, e.g. first order
predicate calculus) that means it is guaranteed to find a proof of any
true statement (eventually).

So what "incomplete" means is that it is *not* guaranteed to find a
proof. That's not the same as "can't find a proof" -- it just means that
you can't guarantee it.

Which means that if a mathematician comes up with a proof (of something)
in a domain where a *complete* theorem prover is impossible, it just
means he lucked out (maybe indigestion kept him awake all night). Yeah,
human beings are particularly good at coming up with hunches - but
that's nothing to do with completeness theorems.

It would be extremely easy to program a computer to "guess" proofs,
check them for correctness, and reject those that are incorrect. Set it
running on some "impossible" problem and leave it -- and you are
absolutely not guaranteed it will come up with anything. But it might.

That's not AI, either, of course :-)

Jonathan

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Relevant Pages

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