[FOM] The Lucas-Penrose Thesis
Hartley Slater
slaterbh at cyllene.uwa.edu.au
Thu Sep 28 21:06:41 EDT 2006
First, there is a subscriber to FOM who thinks the Lucas-Penrose
Thesis (or at least one version of it) is true, namely me.
Remarkably this very day I am reading a paper on the subject here at
UWA, and next week at Monash University (copies are available on
request). 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, and specifically the point needs to be
remembered when people say Goedel's result means (see Panu
Raatikainen's contribution): '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.'
But if it is 'true' it can only be true on this or that
interpretation of the symbols. 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? All it can do is generate formulae of the
form 'Fn' where 'n' runs through the numerals, and not produce the
corresponding formula '(x)Fx'. But that is OK, since (given the
soundness of the system on the standard interpretation) it only
follows from the individual cases that every natural number is F, not
that every element in every model is F.
And is it so difficult to assess the soundness of the system on the
standard interpretation? Panu Raatikainen says here (following Putnam
and others): '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.' 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.
--
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, M207 School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 6488 1246 (W), 9386 4812 (H)
Fax: (08) 6488 1057
Url: http://www.philosophy.uwa.edu.au/staff/slater
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