Re: Penrose vs the Robot


"Rupert" <rupertmccallum@xxxxxxxxx> wrote in message
news:1132884982.182337.101260@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>
>> So, yes, it is like the liar paradox, except that it isn't a paradox.
>>
>
> The contradiction only appears at the level of Penrose's belief system,
> so it's not as startling. However, when Penrose reasons about this
> sentence, he almost certainly will get himself into contradictions. I
> think that's a good enough reason to call the sentence paradoxical.
>> --
>> Daryl McCullough
>> Ithaca, NY
>

Recall that Penrose is making a claim about AI, which must use
a formal language (program) which is restricted from recognizing
all truths in a given formal system.

Penrose is claiming that he as a representative human, can divine
unassailable truths, which is of course equivalent to the claim that
he can perceive all mathematical truths.

The Penrose argument is true if you grant this premise because
the argument is circular. The problem is that it's not at all plausible
that any human belief system encompasses the truth or scope of
reality. However, because humans can't do it, and computers can't
do it, does not make them equivalent. If Penrose humans actually
thinks that some human mathematicians can discriminate all of any
category, unboundedly finite+, then that thinking is best not described
as paradoxical. This idea again returns to the inability to translate
abstractions into physical reality with a one-to-one correspondence.
The situation seems closer to the Sorites Paradox, in any event.
Distilling natural languages into a formal system produces a <subset.

Humans intuit truth/belief for a mathematical idea using the same
brain mechanism they use to play chess which is not the same way
computers do it (brute force) in complex positions; although both
methods may output the same move. How to generalize from the
finite to the infinite does not seem to be a listable or computable
property. This quote from "The Scientist" refers to John Myhill.

"Most logical systems have the property of being listable but not
computable: all their theorems can be listed but there is no automatic
procedure for inspecting a statement and deciding whether or not it is
a theorem.

If the mathematical world had no Godel theorem, then every property
of any system that contained arithmetic would be listable. We could
write a definite program to carry out every activity. Without the
restrictions of Turing and Church on computability, every property
of the world would be computable.

Not every feature of the world is either listable or computable. For
example, the property of being a true statement is neither listable nor
computable. One can approximate the truth to greater and greater
accuracy by introducing more and more rules of reasoning and adding
further axiomatic assumptions, but it can never be captured by any finite
set of rules. These attributes that have neither the property of listability
nor that of computability--the "prospective" features of the world--are
those that we cannot recognize or generate by a series or sequence of
logical steps. They witness to the need for ingenuity and novelty; for
they cannot be encompassed by any finite collection of rules or laws.
Beauty, simplicity, truth; these are all properties that are prospective."

SH: Also see:
http://groups.google.com/group/sci.logic/msg/e47fa27c85ecc2a0?hl=en&;

www.math.wisc.edu/~alfeld/computnotes.pdf

"So a productive set is one which allows us to, given a c.e. subset,
find an element in the set but outside the c.e. subset. A creative
set is "effectively noncomputable". Namely if we think that C is
computable then C^_ should W\/_x for some x, but as C^_ is productive
and Wx is a proper subset of C^_ we can find an element in C -- Wx so
Wx can not be C. The "effectively" comes from that we know a
counterexample, namely \|/(x), for any candidate x.

Any productive set has a one to one total computable productive
function p. A set is creative if it's complement is productive."

>From MathWorld/Wolfram
"Productive sets are not recursively enumerable.
A recursively enumerable set A is creative if its complement is productive.
Creative sets are not recursive."

"A creative set is "effectively noncomputable".

"These attributes that have neither the property of listability
nor that of computability--the "prospective" features of the world--are
those that we cannot recognize or generate by a series or sequence of
logical steps. They witness to the need for ingenuity and novelty; for
they cannot be encompassed by any finite collection of rules or laws."


"The property of creativeness coincides with completeness.
Namely, set A is creative iff if it is many-one complete."


SH: Penrose is claiming/stipulating that a human mathematician
can recognize a member of an "effectively noncomputable" set,
that a computer obviously cannot, so therefore an AI which is a
program written in a formal language which must be computable,
cannot be the equivalent of a human mathematician reasoner. He
is also claiming that there is a complete mapping of a mathematical
result (Godel Inc.) unto physical reality, so that Godel Inc. can be
used to justify a consequence impacting physical reality, existence
or non-existence of a program using a formal system to completely
emulate human thought potential. What is paradoxical in a way is
that Godel Inc. used to propel his argument does not support such
a complete mapping or description of the membership of its reality.

Therefore if the rules of physical reality are actually formalistic
no such mapping as Penrose attempts is consistent/complete.

So Penrose is assuming when using his premise that there is no
in principle mathematical truth unavailable to the mathematician.
If physical reality works that way, then it is not isomorphic to
a physical reality which in principle proscribes some human
mathematical discovery or recognition of truth. Which is another
way of saying his argument is circular.

If one assumes that the human mathematician does not have
complete mastery of his/her province, then observing that a
formalized program also does not, does not permit one to
make conclusions about their properties in other areas.

Like snow is white and paper is white doesn't make them
both wet or both foldable. A black dog and black car
don't both bark just because you can put them into a
larger category: capable of making noise.

Penrose's argument boils down to one about using
an adjective, a description of a property, and attempting
to generalize that to what a brain is (a non-computable
mechanism he says) and what a computer is (a computable
mechanism) = nouns, as if his arguments encompasses
how two things behave decide what they are. A human
can sing well but not play basketball well, does not mean
that both aren't human underneath. As I see it, refuting
Penroses's argument refutes Computationalism, that
the observed output determines whether two things are
equivalent, not what is under the hood that produces
comparable (or not) behavior is what matters. This comes
up a good deal more in Searle's Chinese Room Argument.

http://www-formal.stanford.edu/jmc/towards/node14.html

"A recursive ordinal is a recursive ordering of the integers
that is a well-ordering in the sense that any subset of the
integers has a least member in the sense of the ordering.
Thus, we have a contest in trying to name the largest
recursive ordinal. Here we seem to be stuck, because the
limit of the recursive ordinals is not recursive."

If wishes were horses then beggars would ride,
Stephen


.

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