Penrose vs the Robot
- From: stevendaryl3016@xxxxxxxxx (Daryl McCullough)
- Date: 17 Nov 2005 07:24:15 -0800
Rupert Mccallum and Stephen Harris have been posting recently
about Penrose' old argument that he is not a robot (or rather,
he has mathematical abilities above and beyond those of any
Turing machine program). I actually think now that Penrose'
argument falls apart on the very first step. (This counterargument
didn't occur to me when I initially debated with Penrose about
it.)
Penrose' argument simplified is this: Assume that there
is some robot with a Turing machine brain that is equivalent to
Penrose' human brain. Then Penrose can come up with a mathematical
statement S such that Penrose can be unassailably certain of the
truth of S, but the robot cannot.
I claim that Penrose *cannot* produce such a statement. Penrose
suggests that S can be constructed as follows:
Let T be the set of all statements in the language of Peano
arithmetic that the robot considers unassailably true. Then
since the robot's brain is a Turing machine, T is an r.e.
set, so T is actually definable in PA. Then we can let S
be the sentence "T is consistent". If T is indeed a consistent
extention of PA, then by Godel's theorem, S is not provable
by T.
But I claim that there *is* no such theory T, and so there
is no such statement S. The existence of such a theory T
does not follow from the fact that the robot has a Turing
machine brain.
What does follow is that the robot's behavior at any time
is a computable function of its current state, and its
current state is a computable function of its past inputs.
So there are two computable functions:
next_state(s,i) : if the current state is s, and the current
input is i, then this returns the next state
after receiving the input
output(s) : the output produced in state s
But these functions don't say anything about what the
robot "unassailably believes", and there is no way to
extract an r.e. set of such beliefs from this pair of
functions.
We can certainly extract the set of statements such that
the robot will *assert* that he unassailably believes them,
but that's not the same thing. Suppose we let T be the
set of all sentences Phi in the language of PA such that
if we ask the robot "Do you unassailably believe Phi?" the
robot will answer "yes". That's an r.e. set. So we can come
up with a sentence con(T) of arithmetic such that if T is a
consistent extention of PA, then con(T) is not a theorem
of T. So we ask the robot "Do you unassailably believe con(T)?"
The robot could very well answer as follows:
"It would be inconsistent for me to answer 'yes', but
I can say that it is correct to assume that that is one
of my unassailable beliefs."
The problem is that what is r.e. is the set of possible
*behaviors* of the robot. We don't have a foolproof means
of determining what "unassailable beliefs" are implied
by that behavior. So we can't go from an r.e. set of
behaviors to an r.e. set of "unassailable beliefs".
Or, at least I don't see any way to extract the set of
unassailable beliefs from the robot's program. Without
such an extraction procedure, Penrose' argument can't
even get started.
--
Daryl McCullough
Ithaca, NY
.
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