[FOM] Intermediate value theorem and Euclid

[Vaughan Pratt]

On 5/20/2009 11:07 AM, Andre.Rodin at ens.fr wrote:
> From my part I don't think that such notions as
> *the right* notion of Euclidean plane  (= what is the Euclidean plane
> "essentially") makes sense - althought I assume that there are more and less
> viable modernisations of this old mathematical notion.

Is there *the right* notion of abelian group?  If not, why not?  If so, 
why that and not *the right* notion of Euclidean plane?  And would your 
answer change at all if "Euclidean plane" were replaced by "affine 
space" (the abstraction obtained by Euler in the 18th century by 
omitting the concepts of length and angle from Euclidean space)?  That 
is, is the Euclidean plane elusive in your view on account of having the 
difficult notions of length and angle, or for some other reason?

Vaughan Pratt