On Mar 11, 2010, at 7:09 AM, John F. Sowa wrote: (01)
> Folks,
>
> 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.
>
> 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.
>
> 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.
>
> 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.
>
> 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. (02)
John, the continuant/occurrent model is NOT isomorphic to the view
that 3D entities are temporal slices of 4D entities. The issues do not
arise from questions about relating 3D temporal slices to 4D
entities. Even posing the issue in those terms begs the question:
temporal slices are *logically incoherent* in the continuant/occurrent
view of things, and continuants are *necessarily impossible* in the 4D
view. Your historical exegesis, while all correct, is therefore
irrelevant to the topic. (03)
Pat (04)
>
> John
>
>
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> (05)
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