[FOM] The Lucas-Penrose Thesis

[Robbie Lindauer]

On Sep 28, 2006, at 12:21 AM, praatika at mappi.helsinki.fi wrote:

> The basic error of such an argument is actually rather simply pointed
> out.  The argument assumes that for any formalized system, or a finite
> machine, there exists the Gödel sentence (saying that it is not 
> provable
> in that system) which is unprovable in that system, but which the human
> mind can see to be true. Yet Gödel’s theorem has in reality the
> conditional form, and the alleged truth of the Gödel sentence of a 
> system
> depends on the assumption of the consistency of the system.
>
> The anti-mechanists argument thus also requires that the human mind can
> always see whether or not the formalized theory in question is 
> consistent.
> However, this is highly implausible. After all, one should keep in mind
> that even such distinguished logicians as Frege, Curry, Church, Quine,
> Rosser and Martin-Löf have seriously proposed mathematical theories 
> that
> have later turned out to be inconsistent. As Martin Davis has put
> it: “Insight didn’t help”.


Actually, Lucas replied to this at length in the Freedom of the Will.

In particular the (short version) reply is this:

If the machine proposed by a mechanist as a model of the mind is NOT 
consistent, it will produce ANY statement as true, and hence not be a 
model of a human mind.

In particular, a machine which is inconsistent will produce "1 + 1 = 3" 
as a  theorem.  A human (sane one) will be able to see that that is 
obviously false.

The argument is structured thus:

1)  IF the machine proposed as a model of the mind is consistent then 
there exists a godel sentence G for the formalism represented by the 
machine which a Human can recognize true and which that machine can not 
produce as true.  (Godel's Theorem)

2) IF the machine proposed as a model of the human mind is INCONSISTENT 
then it will produce nonsense that a human will recognize as such.  In 
particular, if it is an arithmetic machine, the machine has as a 
theorem '1 = 0'.

(for now we're imagining the excluded third applies)

His smaller expositions of this point are available on his website.

Best Regards,

Robbie Lindauer