[FOM] Fwd: invitation to comment
Monroe Eskew
meskew at math.uci.edu
Mon May 23 03:25:55 EDT 2011
On Sun, May 22, 2011 at 12:40 PM, <Andre.Rodin at ens.fr> wrote:
>
> You are making a very good point bringing this historical analogy into the
> discussion. To support this analogy I would like also to stress the fact that
> the algebraic proof of the impossibility to trisect an angle by ruler and
> compass required a dramatic revision of the ancient foundations of geometry and
> bringing new foundations instead.
What do you mean by "revision"? One plausible view is that the
changes brought to geometry by Descartes did not require an alteration
of any of the ancient foundations or concepts. Only new machinery and
techniques were introduced, which could all be reduced to the original
language and techniques. Further, the eventual axiomatization by
Hilbert did no violence to Euclid's geometry. It completely respected
the concepts and approach; it merely clarified, brought out implicit
assumptions, and made things fully rigorous. Therefore, the fact that
the appropriate correspondences between primitive geometrical spaces,
objects, and procedures on the one hand, and numbers, equations, and
field extensions on the other hand, can be rigorously established
using modern axiomatizations and techniques, really DOES solve the
original question of trisection of an angle, in the same sense that it
was originally posed. No revision took place, only progress. If such
a view is plausible regarding geometry, then similar ones about
formalization should be plausible.
> However the hypothetical argument supporting the view that trisection remains
> an open problem is not really similar to mine. I don't suggest that Con(PA)
> *can* be proved in a violation of Godel's theorem (this would be the precise
> analogy) - and I also don't consider as proper mathematical proofs of Con(PA)
> certain arguments that some people do qualify as such.
Analogously, the geometer who is reluctant to accept analytic
techniques will not therefore assert that the angle *can* be
trisected. She will just not accept analytic proofs of the fact that
it cannot, and consider the problem unsolved, just as you will not
accept perfectly ordinary proofs that PA is consistent.
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