[FOM] The Lucas-Penrose Thesis

[praatika@mappi.helsinki.fi]

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