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

To: Ontology Summit 2008 <ontology-summit@xxxxxxxxxxxxxxxx>
From: Michael Gruninger <gruninger@xxxxxxxxxxxxxxx>
Date: Tue, 18 Mar 2008 10:18:00 -0400
Message-id: <47DFCF18.3040906@xxxxxxxxxxxxxxx>

Hi John,    (01)

I'm not really sure what you are referring to.
I am talking about the first-order theory presented in    (02)

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.    (03)

in which Tarski says:
" Thus, in our formalization of elementary geometry, only points are 
treated as individuals."    (04)

The word "sphere" does not appear anywhere in that paper.    (05)

Wikipedia also contains a presentation of the axioms (see Tarski's Axioms).    (06)

- michael    (07)

John F. Sowa wrote:    (08)

>I just wanted to comment on that point:
> > Hilbert's Geometry has points,lines, and planes as primitives.
> > Tarski's Geometry only has points as primitives.  Each of these
> > ontologies are definable interpretations of each other.
>Actually, Tarski's basic geometry had only one kind of primitive:
>spheres of arbitrary (but finite) size.  Summary:
>  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.
>But note that if you assume a finite limit on the size of the
>spheres, you could never get an exact correspondence between
>Tarski's shapes and Euclid's shapes.
>This result has similarities to Heisenberg's uncertainty principle:
>a precise measurement in one ontology or one system of coordinates
>might not have a precise mapping into another ontology of system
>of coordinates.
>When we get down to the details of various ontologies, I suspect
>we may find that such discrepancies between systems are common.
>    (09)

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