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)
>Michael,
>
>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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