Xref: utzoo comp.ai:2603 talk.philosophy.misc:1568
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From: ok@quintus.uucp (Richard A. O'Keefe)
Newsgroups: comp.ai,talk.philosophy.misc
Subject: Re: Artificial Intelligence and Intelligence
Message-ID: <686@quintus.UUCP>
Date: 16 Nov 88 10:11:39 GMT
References: <490@soleil.UUCP>
Sender: news@quintus.UUCP
Reply-To: ok@quintus.UUCP (Richard A. O'Keefe)
Organization: Quintus Computer Systems, Inc.
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In article <490@soleil.UUCP> peru@soleil.UUCP (Dave Peru) writes:
>In article <484@soleil.UUCP> I write:
>>>Definition of Intelligence:
>>>1. Know how to solve problems.
>>>2. Know which problems are unsolvable.
>>>3. Know #1, #2, and #3 defines intelligence.

>There is a misunderstanding what I meant by this statement, especially #2.
>Human beings KNOW the "halting problem for Turing machines", my point is
>that machines can NOT know the "halting problem for Turing machines".
>Please describe how you would give this knowledge to a computer.

I was afraid for a minute there that you were going to say "know how to
solve _practical_ problems, have a _practical_ grasp of which problems
are _feasible_", but no, it's the halting-problem type of problem after all.

What does it mean to say 'Human beings KNOW the Halting Problem'?
As a plain matter of fact, most of them _don't_.  I think I do, but what
I mean by that is that I have enough formal knowledge about mathematical
logic to follow the books, to relate some of the key concepts to each
other, and to deploy this information in further formal reasoning.  I do
_not_ know, for any given program, whether or not it halts until I have
examined that particular program, and even then the Law I _really_ rely
on is Murphy's 1st law of bugs:  "There is a mistake in this program."

Boyer & Moore have a paper on a machine-generated proof of Goedel's Theorem.
Read that paper to see how to "give this knowledge to a computer".
Getting a computer to "know" mathematical things is a Small Matter of
Programming.

>All uncomputability problems come from dealing with infinity.

In a strict sense, yes.  But you can find oodles of NP-hard problems
without thinking once about infinities, and NP-hard is just as good as
uncomputable for most practical purposes.

>Like naive set theory, naive Artificial Intelligence does not deal with
>paradoxes and the concept of "infinity".

There have been several conferences and workshops on knowledge
representation.  There is no shortage of papers discussing paradoxes
in these areas.  Smullyan has a fine paper on some paradoxes of belief.
Not all AI work is naive.  (There is an infinite regress in "I think 
that he thinks that I think ..." which has to be headed off; this _has_
occurred to people.)

Surely set theory has taught us by now that there is no such thing as
THE concept of infinity:  omega is not Aleph-null is not the-point-at-
infinity is not ... is not the common-sense notion of infinity.

>Human beings understand the concept of "infinity", most of mathematics would
>be meaningless if you took out the concept of "infinity".

As for the second clause, clearly you are not an Intuitionist.
As for the first, this is simply false:  the vast majority of human beings
have not had the technical training to distinguish between omega, Aleph-null,
and the reciprocal of an infinitesimal in non-standard arithmetic, and those
who _have_ had such training would probably be humble enough to admit that
we are still nibbling at the edges of the concepts.

>Finite machines cannot understand "infinity".

Why not?  The whole human race has so far encountered only a finite number
of facts about the various infinities.  By starting this joust in a FORMAL
field you've lost the game in advance.

>For the concept of "infinity" to have any meaning at all you MUST have the
>computational strength of reality.

For the concept of infinity to have any meaning at all to WHOM?
If you mean "in order to understand infinity correctly, the understander
must have the computational strength of infinity", maybe, but it is not
clear to me that any human being understands infinity that well,
particulary not one who talks about THE concept of infinity.
What _is_ "the computational strength of reality", and how is it
possible for finite beings to possess it?

[Someone who believes that we are god could consistently believe that
 humans are not finite bounded creatures and so can "know" infinity.
 New AIge?  On the other hand, a god so easily deceived could be wrong...]