[FOM] Disproving Godel's explanation of incompleteness

[Richard Zach]

On Thu, 2005-10-20 at 14:06 -0400, Richard Heck wrote:
> A.P. Hazen wrote:
> 
> >    What counts as an EXPLANATION is one of the great open problems in the philosophy of science, and what counts as an explanation in MATHEMATICS is.... 
> >
> so hard as not even to count as an open problem yet? That's how I feel
> about it. Jamie Tappenden's recent paper "Proof Style and
> Understanding", available on his web page, starts to make some strides
> towards an understanding of what the problem is, though, and how it
> might be addressed.

Not recognized as a problem in "mainstream" philosophy of mathematics,
perhaps, but there's a growing number of people working on explanation
in mathematics and related questions (the role of visualization, e.g.),
Jamie among them.   The following are just the four papers I assigned on
the topic in my Philosophy of Math seminar last term; you'll find plenty
of references in particular in Mancosu's paper (as well as some more
recent ones, unfortunately I don't have the references handy).

Mark Steiner. 1978. Mathematical explanation. Philosophical Studies 34,
135--151. 
 
Michael D. Resnik and David Kushner. 1987. Explanation, independence,
and realism in mathematics. British Journal for the Philosophy of
Science 38, 141--158.

David Sandborg. 1998. Mathematical explanation and the theory of
why-questions. British Journal for the Philosophy of Science 49,
603--624.

Paolo Mancosu. 2001. Mathematical explanation: problems and prospects.
Topoi 20, 97--117.

Also relevant is Jeremy Avigad's recent work on the relevance of
automated theorem proving to models of explanation and understanding in
mathematics, see, e.g., the forthcoming paper "Mathematical method and
proof" (available on his website
http://www.andrew.cmu.edu/user/avigad/).

The book in which Jamie's article appeared is in the new collection

"Visualization, Explanation and Reasoning Styles in Mathematics"
Synthese Library, Vol. 327 (2005)
Mancosu, Jørgensen, Pedersen (Eds.) 
http://www.springer.com/sgw/cda/frontpage/0,11855,5-40385-72-44241889-0,00.html


Regards,
Richard