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

To: Ontology Summit 2008 <ontology-summit@xxxxxxxxxxxxxxxx>
From: Pat Hayes <phayes@xxxxxxx>
Date: Wed, 19 Mar 2008 10:55:31 -0500
Message-id: <p06230903c406e5cf8ce8@[]>
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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