Date: Fri Jan 18 2002 - 11:18:43 MST
> It has been commonly argued, on the basis of Godel's theorem and related
> mathematical results, that true artificial intelligence cannot exist. Penrose
> has further deduced from the existence of human intelligence that fundamental
> changes in physical theories are needed. I provide an elementary
> demonstration that these deductions are mistaken. Is real articial
> intelligence possible? Are present-day theories of physics suficient for a
> reductionist explanation of consciousness?
Penrose's argument has been refuted so many ways it is amazing that anyone
even bothers any more. However I don't think this particular counter-
argument works. It amounts to concern over Penrose's reasoning where
he proposes to stump the AI by giving it what amounts to its own Godel
sentence. One problem with Penrose's plan is that a real-world AI is
not self-contained but would interact with the world; Penrose proposes
to fix this by providing the AI with a simulated world, making it become
self-contained and non-interactive, and giving it a well defined Godel
sentence. The author challenges this fix, writing:
Penrose  gives a number of examples, that appear to show that it
is easy to construct the requisite non-interactive subroutine using
the interactive program as a component.
However, there is a big problem in figuring out how to present the
input to the program, to tell it what theorem is to be proved. Now
the program, which we can call an artificial mathematician, is in the
position of a research scientist whose employer specifies a problem
to be worked on. To be effective, such a researcher must be able
to question the employer's orders at any point in the project. The
researcher's questions will depend on the details of the progress of
the research. ("What you suggested didn't quite work out. Did you
intend me to look at the properties of XXYZ rather than XYZ?") As
every scientist knows, if the researcher does not have the freedom
to ask unanticipated questions, the whole research program may fail
to achieve its goals.
Therefore to construct the non-interactive program needed by Penrose
one must discover the questions the artificial mathematician will
ask and attach a device to present the answers in sequence. The
combination of the original computer and the answering machine is
the entity to which Turing's halting theorem is to be applied.
The author, John Collins, goes on to show that this strategy of providing
answers in advance won't work because each time you change the set of
answers you change the system, hence the Godel sentence, hence you have to
start over from the beginning with a different proof challenge to the AI.
The problem with this reasoning is that this claim that the AI must be
able to ask question seems false. It is given a fully self-contained
mathematical formula which it is asked to evaluate. There is no scope
for ambiguity or confusion in this problem. It is not provided in
some loose language like English; it is in hard mathematical symbols.
This is different from Collins' model where the AI must ask questions to
clarify his problem just like a scientist given some research problem by
I've worked on many mathematical problems, challenges and contests over
the years, and they are self-contained. You don't get to ask questions;
in fact in national contests where the problems are provided in writing,
there is typically no one around who would be remotely qualified to
answer any questions about the problems. And these problems are often
described much less formally than the Godel statement which Penrose
would propose to provide.
So Collins' claim that the AI would have to be able to ask questions
about its problem, and that this would require answers to be prepared,
which would change the system in a never-ending cycle, does not seem a
strong refutation of Penrose's program.
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