On Mar 11, 2010, at 7:09 AM, John F. Sowa wrote:
> 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
> 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. (01)
John, I don't disagree with anything you say, but I don't understand what you
are proposing, since an ontology of 3D individuals is not just an isomorphic
slice of an ontology of 4D individuals. As you note, a 3D ontology is in fact
fully 4D -- "3+1 D" as you put it -- in the sense that time is not ignored (the
way the "z-axis" of R^3 is ignored in R^2, say). Rather, the difference
between the two ontologies concerns how they conceive the relation between
individuals and time. AFAICS, the endurantist ontology doesn't "embed"
isomorphically in any obvious way into the perdurantist ontology (unless,
perhaps, you also introduce temporal parts into the endurantist ontology, which
seems sort of self-defeating). So what, exactly, are you proposing? (02)
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