[FOM] Intermediate value theorem and Euclid

[Andre.Rodin@ens.fr]

Vaughan Pratt wrote:

> Is there *the right* notion of abelian group?  If not, why not?  If so,
> why that and not *the right* notion of Euclidean plane?

I think that NOT. Let's just talk about *group* (if I am right about *group* the
same holds for *abelian group*). Of course the group concept can look very
different: its today's standard presentation as a set with a binary operation
subject to the axioms is a more recent invention that the notion of group
itself. And there is today an alternative definition of group around: as a
category with exactly one objects and all morphisms reversible. What I want to
say is that comparing these different rendering of the same concept we cannot
get to a "higher" viewpoint from which all of them will look equivalent. What
we can do - and what people usually do - is to assume that the current
rendering is the "right" one, translate the older ones into the current one and
then forget about the older ones. If there is no smooth translation available
one might always say that this is because the older presentations are
erroneous, unclear, inaccurate, etc.

Perhaps this attitude is fine for doing research in the pure maths in general
but certainly not for doing FOM. This is because when we look at the history we
see that FOM change quickly and moreover change *radically* which means that
they are repeatedly wholly replaced rather than just upgraded. The
history strongly suggests that such permanent revision and renewing of FOMs is
something healthy for mathematics: think about Descartes "Geometry" for example.
(In the paper announced in the previous
posting I also provide theoretical justification for the "renewing
foundations".) So doing FOM we should learn how to manage this kind of change
and how to translate older - and current - mathematics into the future
mathematics. Studying history is helpful for it. The differences between
different notions of Euclidean plane or different notions of group, which I
stress, concern FOM rather than anything else.



>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?


No, it wouldn't change. I didn't mean anything specific about the notions of
angle and length. Of course when the historical distance is larger the
differences I stress are easier to notice. But anyway Euler didn't think about
the affine space as we usually do this today, he didn't have in mind our notion
of affine space as a set with a structure

best,
Andrei Rodin