[FOM] Disproving Godel's explanation of incompleteness

[paolo mancosu]

People interested in Steiner's model and a 
criticism of it might want to look at:
J. Hafner, P, Mancosu, The varieties of 
mathematical explanation, in P. Mancosu et al., 
eds., Visualization, Explanation and Reasoning 
Styles in Mathematics, Springer, 2005, pp. 
215-250.
While it is true that there is no agreement on 
what counts as explanation in mathematics, recent 
work in this area shows that the problem is 
attracting more and more attention and should be 
considered an open problem in the discipline. I 
do agree with Richard Heck that we are not yet at 
the stage where we can map a conceptual chart of 
all the possible positions. In order to do that 
one needs more case studies from mathematical 
practice where explanatory concerns play an 
essential role. The paper mentioned above uses a 
case study from real analysis (Kummer's 
convergence criterion). In a forthcoming paper 
with Hafner we also test Kitcher's unification 
theory of mathematical explanation against a case 
from real algebraic geometry.
Apologies for the self-promotion.
Paolo


>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