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Article 2883 of comp.ai.philosophy:
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>From: dirish@math.utah.edu (Dudley Irish)
Subject: Is understanding algorithmic?
Sender: news@math.utah.edu
Date: Sat, 18 Jan 1992 22:58:27 GMT
Organization: Department of Mathematics, University of Utah
Message-ID: <DIRISH.92Jan18155827@jeeves.math.utah.edu>

Recently one of the discussions in this group worked its way around to
the question, "Is understanding algorithmic?"  Let's look at this
question a bit.

Now as any inveterate reader of this list knows, we have no good
definition of understanding, so let's put that aside for a little while
and see if we can come up with a definition for algorithmic.  (You can
guess were I am going.)

In attempting to define algorithmic we might be tempted to follow the
reasoning of either Alan Turing or Alonzo Church (or for the well
initiated, Emil Post) and suggest a definition in terms of Turing
Machines or Recursive functions.  Unfortunately, we would find that we
can't really prove that algorithmic means that which can be done by a
Turing machine or recursive function.  	In order to get somewhere we
might suggest the following thesis:

	Algorithmically computable functions are those which are
	computable by a Turing Machine.

Does this sound familar?  Yes that's right; it's the Church-Turing
thesis.  Now, if we accept this thesis, we can ask the question, "Is
understanding algorithmic?" and know that what we mean is, "Is
understanding Turing computable?"  So we have made a little progress.

But let's turn to a question that we skimmed over: Why is it that we
can't prove the Church-Turing thesis?  Well, obviously, since we don't
have a rigorous definition of algorithmic we can't prove that
algorithmic is anything.  Before we can prove anything rigorously we
must start with rigorous definitions; and we don't have these.

But, given that we can't prove the Church-Turing thesis, can we list
some reasons why we might want to accept it?

Well, certainly all algorithmically computable functions in
mathematics are Turing computable.  I don't think that anybody will
object to this assertion.  Recursive functions or Turing machines do
seem to capture the mathematical notion of algorithmic.  This isn't
too suprising since that is exactly what the mathematicians were
trying to capture.  However, being interested in philosophy, we would
like to ask if some things that seem algorithmic to us are Turing
computable: for instance, understanding.

Now we are stuck.  We can't prove that understanding is Turing
computable because we don't have a rigorous definition of
understanding.  And there are lots of people who are not going to
accept the claim that understanding is Turing computable unless we can
prove it or demonstrate it.  Afterall, it is this claim of
computability that's at issue.

Well, what about demonstrating it?  No, that isn't going to work
because lots of people argue that no behavioral test is ever going to
demonstrate understanding.  So what would convince one of these rabid
skeptics that our Turing machine can understand?

Well, unless the denial is a religious belief on their part, we should
be able to convince them by showing how the machine implements the
basic actions of understanding.  But first, we must decide what the
basic actions of understanding are.

Hopefully, up to now I haven't lost anybody and I haven't sent anybody
running down the hall screaming "What an idiot!"  But at this point, I
am going to make a sugestion that will probably get me into trouble.
Let me suggest that one of the basic actions required for
understanding is reference, the operation or action of refering to
something.  Hardly any one denies that we can refer to objects in the
real world (if it exists), things like hamburgers, cars, people, etc;
lots of us believe we can refer to thoughts, ideas and plans, things
that don't have space/time loci.  Some us of think we can refer to
universals, wetness, hardness, etc.  Some think we can refer to
events, WWII, the Spanish American War, etc.  For my purposes, it
doesn't matter where you draw the line; all that matters is that you
have some idea of how 'reference'/'refers' is used and what it means.

So let us turn to our Turing machine and see if we can explain to our
skeptical philosopher how such a machine could refer to something.  We
have three parts to the Turing machine, the tape, the read/write head,
and the state machine which controls the read/write head and the
movement of the tape.  (I repeat all this just to make sure
that we are all talking about the same notion of a Turing machine.)
The state machine has some state which is identified as the starting
state.  The tape is arbitrarily long and can have symbols from some
language written on it.  The read/write head can read these symbols
and write these symbols.  The next state function of the state machine
is defined as an array with two indices, the first is the current
state and the second is the symbol currently being read by the
read/write head.  The action function of the state machine is
similarly defined with the allowed actions being to write any symbol
from the language, to advance the tape, to backspace the tape, or to
halt.  The machine is operated by putting symbols on the tape,
positioning the tape so that the first symbol is being read by the
read/write head, then starting execution of the state machine in its
starting state.  Following Searle's example we now ask, "Where is the
reference?"

Does the tape refer to anything?  Does the read/write head refer to
anything?  Does the state machine refer to anything?  I have to answer
no.  And if the answer is no, then: Where is the understanding?

At this point I will assume that at least one or two people have stuck
with this rather long posting and might be interested in what we can
conclude from this discusion of "understand" and "algorithmic".

We can picture the situation as follows.  We have a four assumptions:

	0) Understanding is algorithmic
	1) Church-Turing thesis is true
	2) Reference is required for understanding
	3) A Turing machine cannot refer

(We have a few more assumptions, but I don't believe that anybody is
going to question them so I won't list them.)  Taken together these
four assumptions are incompatible; at least one of them must be false
to avoid a contradiction.  Which should we reject?

0) Clearly understanding is algorithmic because we all do it all the
time and I believe that anything that we all do all the time is
algorithmic by definition.  Therefore, (0) must stand.

2) For me, this is basic: anyone who rejects it should stop reading
here; though a look at Roger Shank's work on scripts (or any recent
work on natural language understanding (damn, there's that word
again)) might convince to come back to this later.

This leaves (1) and (3).  Either we reject the Church-Turing thesis
(1) or we assert that a Turing machine can refer (3).  It seems to me
(and this is supported by recent conversations on the Net) that
anybody who is likely to assert that Turing machines can refer is
going to be very unlikely to reject the Church-Turing thesis and vice
versa.  But to assert that Turing machines can refer needs some
argumentation and support; bald assertion won't do.  Further, it
stands in the face of both the reference question and the fact that
there is no proof of the Church-Turing thesis.  Since the evidence
favors the truth of the claim that Turing machines can't refer and
since the Church-Turing thesis is unproven, this seems the most
reasonable course of action.  Therefore, my solution is to reject the
Church-Turing thesis, but if anybody can suggest a reference mechanism
for Turing machines I would be happy to entertain it.

There are many other reasons one can propose for rejecting the
Church-Turing thesis, but I will not go into them here.  It seems to
me much more important to understand why we are unwilling to assert
that a Turing machine can refer.  I therefore invite all and sundry to
attempt to convince us (I have admitted it now, I am one of those
rabid skeptics) that a Turing machine can refer.

One last thing before I go.  All this argument attempts to show is
that understanding is not Turing computable.  It does not prove that
AI is impossible.  I do not mean it to prove that.  What I mean it to
show is that the notion of algorithmic captured by Turing machines
is not identical with the common notion of algorithmic.  It seems
clear to me, and hopefully this discussion has made this clear for some
of the readers, that whereas all mathematical functions may be Turing
computable, the transformation of raw ingredients into cookies as
specified by a recipe is not Turing computable.

Dudley Irish
--
________________________________________________________________________
dirish@ced.utah.edu, Center for Engineering Design, University of Utah

The views expressed in this message do not reflect the views of the
Center for Engineering Design, the University of Utah, or the State of Utah.



