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## Re: [ontolog-forum] Re Foundation ontology, CYC, and Mapping

 To: "[ontolog-forum]" "John F. Sowa" Thu, 11 Mar 2010 08:09:14 -0500 <4B98EB7A.2000400@xxxxxxxxxxx>
 ```Folks,    (01) From the time of Euclid, mathematicians have extended the 2D axioms and terminology to 3D. And since the 19th century they have generalized Euclidean and non-Euclidean geometries to N dimensions. In all the usual Euclidean geometries and the great majority of the others, an N-1 dimensional geometry is isomorphic to a slice of the corresponding N dimensional geometry.    (02) Euclid talks about the faces of a tetrahedron as triangles, and he applies exactly the same 2D axioms, theorems, and terms to those triangles that he used in the earlier 2D chapter. For all but some "weird" geometries, modern mathematicians do the same.    (03) When we are talking about 4D vs. 3D ontologies, we have some issues that are created by treating one of the dimensions (called time) as a special case. Those issues arise from questions about how we relate a 4D volume to its 3D time slices.    (04) Ordinary language uses a 3+1 D coordinate system that talks about "individuals" that "persist" in time. A 4D ontology would talk about those "same" individuals as 4D volumes that have a multiplicity of time slices, each of which is isomorphic to a 3D volume.    (05) There are many complex issues involved in mapping 4D terms and axioms to the 3+1 D terms and axioms. But let's follow common mathematical practice: use the same terminology for isomorphic structures unless there is some pressing need to do otherwise.    (06) John    (07) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (08) ```
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