[FOM] The Lucas-Penrose Thesis

[praatika@mappi.helsinki.fi]

Lainaus Hartley Slater <slaterbh at cyllene.uwa.edu.au>:

> Panu Raatikainen comments on two of my previous remarks:
> 
> >  >But how can the machine determine what interpretation
> >>  is given to its symbols? 
> >
> >And how exactly can a human mind do that?

> The answer to the first question is that a human can choose the 
> interpretation given to some symbols - in the given case, reading 
> '(x)Fx' as about the natural numbers, or some larger domain.

A word of explanation is in order here. I don't really want to 
straightforwardly deny a human can do that, in some sense, maybe he can. I 
only wanted to point out that the issue is not trivial. There are 
apparently three possible cases: 

1. Neither a human nor a machine can nail down the intended intepretation.

2. A human can, but a machine cannot.

3. A human can, but so can a machine.   

(I ignore the case that a machine can but a human cannot)

I think Putnam, at some point, suggested something like 1. I don't like 
it, but it is a position one needs arguments to refute. Slater and others 
adhere 2. Maybe they're right, but much more needs to be said in its 
defense. And if we can tell a more specified story on how exactly a human 
can do that, it may turn out that there is nothing there that a more 
sophisticated machine could not imitate. 

For example, one way to view the issue is to note that we can fix the 
intended interpretation from the perspective of model theory, which 
assumes some amount of set theory. But so can a machine which is not 
restricted to the language of arithmetic but can use the language of set 
theory. 
Of course, FO set theory has its own non-standard models... So the option 
1. threatens after all. 

I don't really have any clear view about these matter; as I noted, I just 
wanted to point out that the issue is not trivial. 

> >  >All a person needs to do, of course, is check that
> >  > each axiom of T is true, on the given interpretation, and that each
> >>  rule of inference of T preserves truth on that interpretation.
> >
> >And how on earth can one check that the axioms of any given formal
> system are true?

> The answer to the second question starts with the same choice process, 
> but it also requires acknowledgement that our knowledge of facts 
> about elements in any model is not available axiomatically - which 
> might be what Panu's hang-up is about.  Are'n't Peano's (Dedekind's) 
> Axioms true when read as about the natural numbers?  To check that 
> requires a prior knowledge of facts about the natural numbers, and so 
> any axiomatisation claimed to be of Arithmetic can only *presuppose* 
> a knowledge of numbers, and not be the origin of it.

I am not sure if I understood that, but as much as I did, I am inclined to 
agree. However, my point was about formal systems in general. If we are 
given some very complex and unfamiliar formal system, it is simply 
impossible to check whether its axioms are true in the standard model or 
not. And that is the point relevant for the Lucas-Penrose argument. 

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