[FOM] The Lucas-Penrose Thesis

[Robbie Lindauer]

On Sep 29, 2006, at 1:44 AM, praatika at mappi.helsinki.fi wrote:

> Robbie Lindauer <robblin at thetip.org>:
>
>> Actually, Lucas replied to this at length in the Freedom of the Will.
>
> Certainly, but whether the reply is any good is a different matter.

Someone had said, incorrectly, that he hadn't responded to the various 
objections, when in fact Lucas has at length both in journals and in 
two useful books on the subject.

In particular, he's got well-reasoned responses to the questions 
brought up in this thread thus far and a thorough investigation of just 
his website would have revealed that.  In particular the contention:

1)  That the human need not be told which machine it is that is 
representing him.
2)  That the human might be inconsistent (and so correctly represented 
by the machine)
3)  That the human need not understand the machine being proposed as 
him.

None of these objections are damning to the argument in anyway.

>
>> 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.
>
> So can a machine, say, one which lists the theorems of Robinson
> arithmetic.

Not an inconsistent machine.  It will prove that 1 + 1 = 2 and 1 + 1 = 
3.

>> 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)
>
> No, he/she cant. Only if he/she could "see" that the formal system is
> consistent. But that is not in general possible.

This is irrelevant to the point at hand since the human has a handy 
proof that they (themselves) are not consistent (e.g. that they won't 
claim that 1=0) and we know that IF the machine being proposed as him 
is inconsistent, then it will prove that 1=0.  So we can eliminate the 
possibility that the machine proposed is inconsistent.

The point being we can rule out the possibility that the human is 
inconsistent (since we don't prove that 1=0) and therefore that IF the 
machine is a representation of us, then (supposedly) it is also 
consistent.  It follows that there is a Godel sentence for it.


> Anyway, this reply demands that a mechanist must provide a particular
> machine as a model of the human mind.

The theory "There may be a machine that is a model of your mind but we 
can never say which one it is" is uninteresting, at best obtuse and 
certainly wouldn't qualify as a scientific hypothesis.  For instance, 
try entering this hypothetical machine into a competition with the 
AliceBot!

I liken this objection to "There is an apple in the sky, but we can't 
detect it."


Once it is specified, e.g. the candidate for being the model of a 
particular human's mind, then the Godel sentence can be generated (and 
it need not be generated by the Human), and it will be a sentence the 
human (logically) can decide which the machine (logically) can not.

> But this amounts to changing the
> subject. Orginally, the claim at stake was whether there could be a 
> Turing
> machine which would be able to prove everything that a human mind can.

Obviously THAT's not the question.  If it were, then an inconsistent 
turing machine could clearly do that and that would be the end of the 
discussion.

No, the question is whether or not any particular turing machine could 
be a model of a particular human mind.

Since that machine must be consistent (to repeat) because humans are 
minimally consistent (because we don't prove that 0=1), it follows that 
there is a godel sentence G which is provable mechanically on the 
machine but which the machine itself can not prove.  While there is no 
absolute criteria of identity for machine specifications, certainly one 
will have to be that they can (logically) prove all and only the same 
theorems.

Robbie Lindauer