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Re: [ontology-summit] [ont-of-ont] Initial ideas for properties and rela

To: Ontology Summit 2008 <ontology-summit@xxxxxxxxxxxxxxxx>
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Wed, 19 Mar 2008 08:19:49 -0500
Message-id: <47E112F5.8060104@xxxxxxxxxxx>
Michael,    (01)

We were talking about two different papers by Tarski.    (02)

MG> I'm not really sure what you are referring to.
 > I am talking about the first-order theory presented in
 > Alfred Tarski, 1959, '"What is Elementary Geometry?" in Leon Henkin,
 > Patrick Suppes, and Tarski, A., eds., /The Axiomatic Method, with
 > Special Reference to Geometry and Physics/. North Holland.
 > in which Tarski says:
 > " Thus, in our formalization of elementary geometry, only points are
 > treated as individuals."    (03)

I was talking about the more intriguing paper he wrote 30 years earlier:    (04)

   Tarski, Alfred (1929) "Foundations of the geometry of solids,"
   in Tarski (1982) _Logic, Semantics, Metamathematics_, Second ed.,
   Hackett Publishing Co., Indianapolis, pp. 24-29.    (05)

In this paper, he started with spheres as the only logical primitive,
as I summarized in my previous note:    (06)

JFS>  1. Tarski first showed how various combinations of spheres could
 >     approximate arbitrary 3D shapes to any desired accuracy.
 >  2. Such shapes would be a more realistic model of physical solids
 >     formed from nearly spherical atoms than the straight-line
 >     shapes of ordinary Euclidean geometry.
 >  3. Then he showed how a point could be defined as the limit of
 >     ah infinite series of nested spheres.
 >  4. Finally, he showed how points defined in that way would
 >     correspond to the primitive Euclidean points.    (07)

Point #4 of that summary is the starting assumption of the 1959
paper you cited.  I agree that Euclidean geometry in the original
version or the refinements by Hilbert and Tarski is the foundation
for most computation that people have been doing for centuries.    (08)

But I like Tarski's 1929 paper because it shows how a radically
different foundation, which in many respects is physically more
realistic, can lead to Euclidean geometry as a good approximation
-- but still, just an approximation.    (09)

John    (010)

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