[FOM] The Lucas-Penrose Thesis vs The Turing Thesis

[Robbie Lindauer]

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