[FOM] Fwd: invitation to comment
Andre.Rodin at ens.fr
Andre.Rodin at ens.fr
Sun May 22 15:40:10 EDT 2011
>Thank you... I think that explanation helps clarify for me how your
>point of view differs from that of some others on this list.
thank you, Kevin, for the interesting discussion
>One might imagine a (superficially) similar argument made by a
>geometer in regard to the trisection of the angle. Namely, that the
>trisection of the angle should be considered an open problem, because
>the purported "proof" that the trisection is impossible is not
>geometric: the impossibility proof hinges on replacing geometrical
>constructions with certain formal objects representing geometrical
>constructions (polynomials over a field), and according to the
>geometer's point of view, this replacement (of a geometrical
>construction by its formal counterpart in terms of polynomials over a
>field) has done such violence to the original question of whether the
>angle can be trisected that the problem, from the geometrical point of
>view, should be considered open.
>It could be that you sympathize with the geometer in this story. Or
>it could be that I am overlooking a crucial difference between your
>point of view and the geometer's.
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. The more recent foundational change that
brought about what Harvey Friedman calls "classical fom" also produced some
important impossibility results, most prominently the two Godel's theorems. I
assume we both agree that impossibility results play a crucial role in the
history of maths.
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. In particular, I don't
consider as a mathematical proof of Con(PA) the argument according to which one
who endorses the proof of Poincare-Perelman theorem must endorse Con(PA) too.
Thanks again for the fruitful discussion!
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