# [FOM] Query: references on intuitionistic Euclidean geometry

Sun Nov 22 13:05:11 EST 2009

```One more related reference:

O. M. Kosheleva, "Hilbert Problems (Almost) 100 Years Later
(From the Viewpoint of Interval Computations)",
Reliable Computing, 1998, Vol. 4, No. 4, pp. 399-403.

mentions that

"Let us
give an example of a problem where an algorithm was not expected: 3rd,
the axiomatization of volume in elementary geometry. Traditional
description of
a volume requires not only additivity but also the so-called method of
exhaustion.

* In the plane, every two polygons of equal area are equi-decomposable,
and
therefore, any additive function on polygons is either an area or a
function
of an area.
* In 3D case, in 1900, it was not known whether any two polytopes of
equal
volume are equi-decomposable.

Dehn has proved that some are not, and moreover, he proved the existence
of
an additive function that is neither a volume nor a function of a
volume. This
function was strongly non-constructive (used axiom of choice), and it
was widely
believed that no constructive function of this type is actually
possible. This was
proven in [14]."

[14] O. M. Kosheleva, "Axiomatization of volume in elementary geometry",
Siberian Mathematical Journal, 1980, Vol. 21, No. 1, pp. 106-114 (in
Rus-
sian); English translation: Siberian Mathematical Journal, 1980, Vol.
21,
pp. 78-85.

-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf
Of Arnon Avron
Sent: Sunday, November 22, 2009 5:34 AM
To: fom at cs.nyu.edu
Subject: Re: [FOM] Query: references on intuitionistic Euclidean
geometry

Another work that should perhaps be mentioned is a series
of papers by Victor Pambuccian about constructive geometry.

The last published one is:

Constructive Axiomatizations of Plane Absolute, Euclidean and
Hyperbolic Geometry. Math. Log. Q. 47(1): 129-136 (2001)

(see there for references to other papers on the subject by the same
author).

Arnon Avron

> Dear FOMers,
> a student of mine plans to write a thesis on Euclidean geometry
> developed over intuitionistic logic. I would appreciate any reference
to
> books, papers, or  any other source on this topic.
> Thank you very much in advance for collaboration
> Giovanni Sambin

```