At 8:19 AM -0500 3/19/08, John F. Sowa wrote:
> Tarski, Alfred (1929) "Foundations of the geometry of solids,"
> in Tarski (1982) _Logic, Semantics, Metamathematics_, Second ed.,
> Hackett Publishing Co., Indianapolis, pp. 24-29.
>In this paper, he started with spheres as the only logical primitive,
>as I summarized in my previous note:
>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.
>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.
>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. (01)
The construction Tarski used to get points from limits of nested
spheres is a special case of a general construction (now) called an
ultrafilter, the complement of a prime ideal. This can be used to
create timepoints from time-intervals and other similar tricks. I
used it in a recent paper to show that any notion of context which
satisfies some fairly weak conditions can be reduced to 'pointlike'
contexts which are transparent to the Boolean connectives. (02)
I learned this idea in an undergraduate topology course and have
always liked it. I believe it was invented by A.N.Whitehead, and that
he regarded this construction of points from non-points as one of his
most original and important ideas. (03)
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