[FOM] The Lucas-Penrose Thesis
Hartley Slater
slaterbh at cyllene.uwa.edu.au
Fri Sep 29 21:09:53 EDT 2006
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?
> ...
> >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 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 (hence
the connection with Lucas' ideas about the Freedom of the Will). 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.
The problem with the modern 'axiomatic' understanding of mathematics
goes back to Hilbert (as I have explained more fully in the paper
previously offered). Hilbert established the plausibility of his
line of meta-mathematical research with his axiomatisation of
Geometry, which dispensed with Euclidean figures, and proceeded
entirely by means of logic from completely explicit geometrical
postulates. The removal of diagrams, a philosopher might say, took
foundational studies away from 'intuition', and thus 'the synthetic a
priori' in the traditional, Kantian sense. More plainly, it simply
takes one away from what the language in the axioms is about. As a
result, despite wanting to say he had provided a foundation for
'Geometry', Hilbert had nothing to say about the lines and points in
Euclid. Certainly the words 'line' and 'point' appear in Hilbert's
axioms, but they were taken to apply merely to anything which
satisfied the axioms. So the fact that those axioms did apply to
Euclid's elements was something Hilbert did not attempt to provide
any foundation for.
The basic error in Hilbert's programme was that it gave no account at
all of what is true in a model of some formulae, being deliberately
concerned entirely with the formulae themselves.
--
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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