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 firstorder 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. 2429. (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 straightline
> 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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