[FOM] The Lucas-Penrose Thesis vs The Turing Thesis
Robbie Lindauer
robblin at thetip.org
Sat Oct 7 20:48:55 EDT 2006
If there is a machine that is consistent
and
If that machine can perform mathematics of at least the complexity of PA
Then there exists a (possibly true) sentence which that machine can not
decide.
This is not dependent on any aspects of the mechanics or physics of our
world. In NO POSSIBLE WORLD can the machine decide that sentence. If
machines were purely spiritual, if God were a machine, God could not
decide the question.
This is partially constitutive of what it is to be THAT machine. If
you can solve the problem in question, even potentially, then you are
not that machine.
If there were a computer program that has all the theorems of PA as
theorems and is consistent, then, potentially, a human can determine an
undecidable sentence for that computer program. "Potentially" since
it assumes that we have time and resources (e.g. enough paper, or
whatever) to carry out the mechanical method for producing such
sentences.
However, if a computer program were proposed that was identical with a
person, then it would follow that the person could not under any
circumstances decide the undecidable sentence(s) of the computer in
question.
This is counterintuitive only because, given a particular question
independent of a given theory of mathematics (say, the axiom of choice
and ZF) people can in fact decide (at a minimum by fiat) whether or not
the theorem in question is true. A computer with an undecidable
sentence can not decide it (even by fiat, or "changing its own
program").
There are -lots- of ways out. The only one Lucas prevents is the
acceptance of the idea that the Human mind (or any given human mind) is
a consistent mathematical proof machine with at least the strength of
PA (e.g Turing Machines). This isn't damning to materialism "in
general" as I understand it. Why there is so much resistance to the
idea is consequently surprising.
Let us be strongly finite and let Godel not apply. Let us be
inconsistent and be able to decide any sentence. Either way sounds
more plausible than the idea that we are physically finite
infinite-proof-solving machines.
Best wishes,
Robbie Lindauer
More information about the FOM
mailing list