[FOM] The Lucas-Penrose Thesis
Robbie Lindauer
robblin at thetip.org
Thu Sep 28 12:12:27 EDT 2006
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
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