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

 To: Ontology Summit 2008 Pat Hayes Wed, 19 Mar 2008 10:55:31 -0500
 ```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) Pat -- --------------------------------------------------------------------- IHMC (850)434 8903 or (650)494 3973 home 40 South Alcaniz St. (850)202 4416 office Pensacola (850)202 4440 fax FL 32502 (850)291 0667 cell http://www.ihmc.us/users/phayes phayesAT-SIGNihmc.us http://www.flickr.com/pathayes/collections    (04) _________________________________________________________________ Msg Archives: http://ontolog.cim3.net/forum/ontology-summit/ Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontology-summit/ Unsubscribe: mailto:ontology-summit-leave@xxxxxxxxxxxxxxxx Community Files: http://ontolog.cim3.net/file/work/OntologySummit2008/ Community Wiki: http://ontolog.cim3.net/cgi-bin/wiki.pl?OntologySummit2008 Community Portal: http://ontolog.cim3.net/    (05) ```
 Current Thread [ontology-summit] [ont-of-ont] Initial ideas for properties and relationships between ontologies, Michael Gruninger Re: [ontology-summit] [ont-of-ont] Initial ideas for properties and relationships between ontologies, Natasha Noy Re: [ontology-summit] [ont-of-ont] Initial ideas for properties and relationships between ontologies, John F. Sowa Re: [ontology-summit] [ont-of-ont] Initial ideas for properties and relationships between ontologies, Michael Gruninger Re: [ontology-summit] [ont-of-ont] Initial ideas for properties and relationships between ontologies, John F. Sowa Re: [ontology-summit] [ont-of-ont] Initial ideas for properties and relationships between ontologies, Pat Hayes <=