[FOM] Query: references on intuitionistic Euclidean geometry

Kreinovich, Vladik vladik at utep.edu
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

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

Dehn has proved that some are not, and moreover, he proved the existence
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
sian); English translation: Siberian Mathematical Journal, 1980, Vol.
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

 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

 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
> books, papers, or  any other source on this topic.
> Thank you very much in advance for collaboration
> Giovanni Sambin

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