[FOM] Intermediate value theorem and Euclid

Andre.Rodin@ens.fr Andre.Rodin at ens.fr
Wed May 20 14:07:09 EDT 2009

Arnon Avron wrote:

> Does any of the constructions you mention of R provide an isomorphism
> of R^2 with the Euclidean Plane *as was grasped by Euclid*
> (and I believe by all other people, living and dead)?

*As was grasped by Euclid* is not the same as grasped "by all other people,
living and dead". Euclid and his Greek fellows had quite a specific view on
geometry in which circle and straight line played a distinguished role. I don't
think that "all people, leaving and dead" conceive of the Euclidean plane in
the same way: this notion develops and today people usually think of it very
unlike the ancients did this. I also don't belive that behind these historical
variations there is - or there should be - any kind of unchangeable intuition
wired in human brains.

Paul Taylor wrote:

> Euclidean geometry is a more or less clearly defined system
> that, in modern terms, has a model in the field extension of
> the rationals that is obtained by adding all real square roots.

There are many ways to translate Euclid's notion of planes into different
conceptual frameworks. Even if Paul Taylor is right that as far as "modern
terms" are concerned a reasonable translation is essentially unique there are
other examples: in 17th century did this not quite in the same way people do
this today. But actually this very discussion on FOM shows that people often
disagree about "modern terms". 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.

In my understanding Foundations of Mathematics is and must be a
historically-laden issue. I believe that the idea of timeless a-historical
foundations of mathimatics is a mistake (albeit I think that the idea of
timeless mathematical truth is not).

I tried to spell out this view in the introductory part of a paper I recently
posted on PhilSci archive


the main body of the text is actually about Euclid

Andrei Rodin

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