Michael, (01)
I just wanted to comment on that point: (02)
> 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. (03)
Actually, Tarski's basic geometry had only one kind of primitive:
spheres of arbitrary (but finite) size. Summary: (04)
1. Tarski first showed how various combinations of spheres could
approximate arbitrary 3D shapes to any desired accuracy. (05)
2. Such shapes would be a more realistic model of physical solids
formed from nearly spherical atoms than the straightline
shapes of ordinary Euclidean geometry. (06)
3. Then he showed how a point could be defined as the limit of
ah infinite series of nested spheres. (07)
4. Finally, he showed how points defined in that way would
correspond to the primitive Euclidean points. (08)
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. (09)
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. (010)
When we get down to the details of various ontologies, I suspect
we may find that such discrepancies between systems are common. (011)
John (012)
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