[FOM] The Lucas-Penrose Thesis
praatika at mappi.helsinki.fi
Fri Sep 29 05:04:18 EDT 2006
Hartley Slater <slaterbh at cyllene.uwa.edu.au>:
> First, there is a subscriber to FOM who thinks the Lucas-Penrose
> Thesis (or at least one version of it) is true, namely me.
> The reason we are different from Turing machines is simply
> that we, unlike them, can give interpretations to the symbol strings
> produced by such machines,
Whether or not this is true, one should note that this is very different
from the the Lucas-Penrose argument.
> On the standard interpretation (where
> the variables range over just the natural numbers) the Goedel
> sentence is true, while on a non-standard interpretation (where the
> variables range over a domain other than (just) the natural numbers)
> it is false. But how can the machine determine what interpretation
> is given to its symbols?
And how exactly can a human mind do that?
> But a
> recent paper by Hartry Field 'Truth and the Unprovability of
> Consistency' (MIND 115.459 (2006), 567-605, see specifically p568),
> shows how easy the provability of consistency can be, while at the
> same time analysing in very close detail why a machine cannot follow
> the same path. 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
Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
Department of Philosophy
University of Helsinki
E-mail: panu.raatikainen at helsinki.fi
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