[FOM] The Lucas-Penrose Thesis

[Hartley Slater]

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