🚀 go-pugleaf

In article <90Mar3.152728est.6160@neat.cs.toronto.edu> radford@ai.toronto.edu
(Radford Neal) writes:
>
>Let's say that one day Penrose announces that he is able to solve, say,
>the word problem for semi-groups - a well-known non-computable problem.
>People give him instances of this problem. After a period of time
>that goes up only reasonably with the size of the instance he announces
>the answer: YES or NO. In those cases where the true answer is 
>subsequently determined, he always turns out to be right. This holds
>even for very difficult cases that require increasingly subtle 
>arguments to establish that the answer is NO, as well as cases where
>extremely complex reductions are needed to demonstrate that the answer
>is YES.
>
I think this gets to the heart of the matter behind Penrose's argument very
nicely.  It also raises some rather troubling issues which probably need
further discussion.  I realize that, to some extent, these issues conflict
with a personal tendency to try to abstract the world into deductive
operations;  but, in spite of the fact that I know this may only be
a personal bias, there are aspects of this example which worry me.

I think the problem is one of the epistemological foundations of issues of
BELIEF.  Penrose announces that he can solve the word problem.  For any case
we give him, he responds with the correct answer.  Has he convinced us?

Let me illustrate the difficulty by constructing what I hope is a valid
analogy:  Penrose announces that he can solve the general medical diagnosis
problem.  For any case from the medical literature we give him, he responds
with the correct answer.  If you were running a hospital, would you want him
going on rounds through your wards?

The point I am trying to get at here is that there is more to reasoning than
getting the right answer.  If Penrose says (using whatever reasoning powers
"work" for him), "This patient has a malignant brain tumor which will be fatal
if not corrected by surgery within one hour," would YOU rush the patient into
the operating room WITHOUT ASKING ANY QUESTIONS?  Whether you are an
administrator or a fellow physician, chances are, you would want some
kind of JUSTIFICATION for Penrose's decision before taking any hasty
action;  and if, when confronted with the question of justification,
Penrose were to respond, "I just know it," you would be in quite a quandary.
(Believe me, I have no idea how I would respond in such a situation,
particularly if I knew that Penrose's track record had been flawless
prior to this incident.)

I think the point we have to confront here is that CONVINCING reasoning lies at
neither extreme.  If we are talking about "the process by which real
mathematicians [or any other thinkers, for that matter] establish new
truths," then we have no reason to believe that deductive systems are
the only mechanisms which may kick in.  Instead, as Marvin Minsky pointed
out in THE SOCIETY OF MIND, such systems for formal reasoning are better
qualified to SUMMARIZE and JUSTIFY those "new truths," once they have been
encountered.  The question then reduces to whether or not there are MECHANISMS,
deductive or otherwise, for establishing (or even hypothesizing) them.  AI
argues that such mechanisms do, indeed, exist;  but, because we have given
so much of our attention to deductive systems, we are probably still a far
cry from a better understanding of their nature.

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