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)
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