[FOM] The Lucas-Penrose Thesis
praatika@mappi.helsinki.fi
praatika at mappi.helsinki.fi
Thu Sep 28 03:21:38 EDT 2006
F.A. Muller wrote:
> 1. Is there any subscriber on the FOM-list
> who thinks the Lucas-Penrose Thesis is
> true or plausible or can be made true or
> plausible?
>
> I have the impression that most informed
> people think the thesis is false or
> implausible or at least that its argument
> is no good. Yet there a quite a few papers
> published on the subject and they provide
> different analyses.
This is what Lucas originally claimed: “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”. Penrose has made similar claims.
The basic error of such an argument is actually rather simply pointed
out. The argument assumes that for any formalized system, or a finite
machine, there exists the Gödel sentence (saying that it is not provable
in that system) which is unprovable in that system, but which the human
mind can see to be true. Yet Gödel’s theorem has in reality the
conditional form, and the alleged truth of the Gödel sentence of a system
depends on the assumption of the consistency of the system.
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. After all, one should keep in mind
that even such distinguished logicians as Frege, Curry, Church, Quine,
Rosser and Martin-Löf have seriously proposed mathematical theories that
have later turned out to be inconsistent. As Martin Davis has put
it: “Insight didn’t help”.
[The above is taken from my paper "On the Philosophical Relevance of
Gödel’s Incompleteness Theorems", Revue Internationale de Philosophie 59
(2005):
http://www.helsinki.fi/collegium/eng/Raatikainen/godelfinal.pdf
I think all the informed critics agree about this standard objection.
And as far as I can see, Lucas and Penrose have never really replied this
basic objection (how could they?).They have only added various epicycles,
and at this point things become convoluted. It is simply hard to make
sense of these further developments, so unclear and confused they are. So,
if the critics interpret Lucas or Penrose differently here, the blame may
not be only on the critics.
For some good criticism of Penrose by people who are undeniably among the
leading experts in the field (note that Penrose is a dilettante in these
issues), see:
http://psyche.csse.monash.edu.au/v2/psyche-2-07-feferman.html
Martin Davis: "Is Mathematical Insight Algorithmic?" and "How Subtle is
Gödel's Theorem", in his homepage:
http://cs.nyu.edu/cs/faculty/davism/
Penrose's further arguments are discussed critically in the following
papers:
Per Lindström: "Penrose's New Argument", Journal of Philosophical Logic
Volume 30, Number 3 / June, 2001.
Per Lindström: "Remarks on Penrose’s 'New Argument'", Journal of
Philosophical Logic Volume 35, Number 3 / June, 2006.
Stewart Shapiro: "Mechanism, Truth, and Penrose's New Argument", Journal
of Philosophical Logic Volume 32, Number 1 / February, 2003
All the 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
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