[FOM] The Lucas-Penrose Thesis

[praatika@mappi.helsinki.fi]

Lainaus Robbie Lindauer <robblin at thetip.org>:

> 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.

If this refers to me, I should point out that what I actually said 
was: "And as far as I can see, Lucas and Penrose have never really replied 
this basic objection ... They have only added various epicycles" 
 
> In particular, he's got well-reasoned responses to the questions 
> brought up in this thread thus far 

Just how well-reasoned they are can be disputed.

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

I think they are. 

> >> 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.

A consistent machine can. The claim was that no machine can do what a 
human mind can, not that an inconsistent machine cannot. 
 
> >> 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 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.  

Elsewhere, Lindauer wrote:   

> An inconsistent machine will DEFINITELY prove 0=1.  A human, even if 
> they're inconsistent on some matters, may not prove 0=1.  They could do 
> this, for instance, by stubbornly refusing to recognize proofs that 
> they ought to believe that given the other things they believe.

A machine using e.g. Rosser's provability predicate can do the same thing 
(i.e., refrain from deriving 0=1). No difference here between a human and 
a possible machine.  

> So we can eliminate the 
> possibility that the machine proposed is inconsistent.

I don't think we can. 
 
> 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.

No, given what I've said above. 

> > 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. 

It is interesting as Lucas, Peronse and other have denied this possibility.

> > 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.

OK, whether there could be any *consistent* formal system which would be 
able to prove everything that a human mind can.
 
> No, the question is whether or not any particular turing machine could 
> be a model of a particular human mind.

I repeat what Lucas originally claimed: “given any machine which is 
consistent and capable of doing simple arithmetic, there is a formula it 
is incapable of producing as being true ...but which we can see to be 
true”.

Sounds like rather general (and strong) claim to me.


Best, Panu


Panu Raatikainen
Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
Department of Philosophy 
University of Helsinki
Finland


E-mail: panu.raatikainen at helsinki.fi
 
http://www.helsinki.fi/collegium/eng/Raatikainen/raatikainen.htm