[FOM] Intermediate value theorem and Euclid

[Andre.Rodin@ens.fr]

Vaughan Pratt wrote:

> Oh, ok, you're using "notion" synonymously with "presentation" or
> "definition."  Yes, I have no difficulty with multiple presentations or
> definitions of "the same thing."
>
> However for me the "notion" of a concept is the common element of its
> different presentations when there is a consensus that they are
> equivalent up to a mutually agreed-on level of detail.
>
> So now that we've established that we have different notions of
> "notion," you have me worried that maybe I'm using the term
> nonstandardly.  What do other people mean by "the notion of integer" or
> "the notion of a topological space?"


I agree that the notion of notion is at stake here. The talk of concept as "the
common element of ... different presentations" can be understood in different
ways. I do NOT think about a concept as an invariant core of its
different presentations. It seems that you do so, or correct me if this is not
the case. I think of it rather as a trajectory in a conceptual space-time, not
just a point in a conceptual space (quotient by the assumed equivalence of
presentations). The analogy with Classical 3D and Relativistic 4D ontology can
be helpful here. I'm on the 4D side.



>Would you consider glaringly
> different yet demonstrably equivalent definitions of "topological space"
> to be different notions thereof or different definitions of the same notion?


When I spoke about "different notions of topological space" I meant different
pieces of the "conceptual trajectory" (i.e. of concept-as-trajectory).
Concerning equivalent definitions. This is a tricky point. In order to prove
the equivalence of two definitions one needs to assume a certain framework for
doing this proof. But the issue I'm stressing rather concerns such frameworks
themselves, i.e. different foundations of mathematics. Here the situation is
different. As a general rule one can reasonably translate older maths into new
maths. This provides a kind of historical continuity allowing for my notion of
concept-as- -trajectory. But again as a general rule one cannot translate the
new maths into older maths. For example, one can reasonably translate Euclid's
planimetry in terms of R^2 but one cannot reasonably translate the modern
theory about R^2 into Euclid's language. And this means that the two things are
not equivalent. Translations and interpretations are, generally,
non-reversible.


> If I'm simply off base here (as often the case) then I'll start using
> "notion" more standardly.  If however there is no consensus then perhaps
> the word "notion" should be avoided when talking about the foundations
> of mathematics to avoid talking at cross purposes.


Actually I think you're quite right stressing the issue of notion of notion in
this context. Hegel famously tried - perhaps not very successfully - to develop
a dynamic and historically-laden notion of notion. I'm trying this too but
differently.


> > 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
>
> Indeed, or for that matter as a *space* with structure---why should a
> structured object need an underlying set when equational logic
> (organized as a monad) can furnish the objects of other categories than
> Set with structure?

>
> If the constructs assumed by the proofs of Book I of Euclid, when
> expressed as operations, turned out to be just another basis for the
> linear and bilinear operations of the standard two-dimensional real
> inner product space, we could then say that Euclid had the modern
> "notion" (or whatever the correct term is) of the Euclidean plane in the
> same sense that Boolean lattices and Boolean rings are just different
> bases for the same notion of Boolean algebra.
>

I agree that there are *different* interesting translations of Euler's maths and
Euclid's maths into today's maths.  In particular, one may translate Euler's
stuff into a category-theoretic framework as well as into the (already
old-fashioned) set-theoretic framework. I don't know exactly how to compare
such different translation and this, in my view, is an interesting question.
But I disagree that the mere existence of such translations is a sufficient
ground for claiming that "Euclid had the modern notion of the Euclidean plane"
and the like. To interpret Euclid authentically and elaborate on his ideas in
modern terms are two very different tasks.

best,
Andrei