[FOM] Lucas, Penrose, and the Church-Kleene ordinal
praatika@mappi.helsinki.fi
praatika at mappi.helsinki.fi
Mon May 31 05:32:48 EDT 2004
"Timothy Y. Chow" <tchow at alum.mit.edu>:
> In other words, I would say that Penrose's argument is best described
> not as trying to use Goedel's theorem to show that human thought is
> non-recursive, but as a philosophical argument that RP is true and
> hence that "humanly known mathematical truths" are best modeled as
> CK sets.
Certainly there is - and I think there is no dispute on this - a version of
the Godelian argument against (a quite naive form of) mechanism that is
sound; roughly, assume someone claims that the mathematical truths he can
possibly know are captured by certain formal system F he knows with
mathematical certainty to be sound, or at least consistent. Quite obviously
this must be false, by Godel's theorems.
However, the anti-mechanist conclusion of Lucas, Penrose etc. just does not
follow from this. First, all truths knowable by us may still in fact follow
from some formal system, but such that we do not know that system, or don't
know certainly that it is consistent. Or, we may even have some less
conclusive evidence (short of mathematical certainty) for the consistency
of the system. etc.
I say more on these issues in my paper:
"On the Philosophical Relevance of Godel's Incompleteness Theorems",
http://www.helsinki.fi/collegium/eng/Raatikainen/godelfinal.pdf
Those interested in the relation of these issues to transfinite
progressions of theories etc. might also check:
Shapiro, Stewart (1998) “Incompleteness, mechanism, and optimism”, Bulletin
of Symbolic Logic 4, 273–302.
Best
Panu Raatikainen
PhD., Docent in Theoretical Philosophy
Fellow, Helsinki Collegium for Advanced Studies
University of Helsinki
Address:
Helsinki Collegium for Advanced Studies
P.O. Box 4
FIN-00014 University of Helsinki
Finland
E-mail: panu.raatikainen at helsinki.fi
http://www.helsinki.fi/collegium/eng/Raatikainen/raatikainen.htm
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