Pat, (01)
I agree with the technical points you've made, but the following
claim is still controversial: (02)
PH> But set theory is not particularly discrete. Sets are the
> basis for all mathematics, including continuous mathematics. (03)
For example, what happens to the midpoint when you break a line
segment in two equal parts? (04)
1. In classical Euclidean geometry, the two halves are isomorphic. (05)
2. But if the line is considered isomorphic to a set of points
that are in a one-to-one correspondence with the real numbers,
the former midpoint must belong to one side or the other. (06)
Assumption #2 implies that the two "halves" cannot be isomorphic
because the one that contains the former midpoint will be closed
at that end. But the one from which the former midpoint was
removed will be open at that end. (07)
This puzzle has troubled many brilliant mathematicians, including
Kurt Gödel. (08)
Tarski's solution to that puzzle (and related issues) was to use
a version of mereology combined with an ontology whose basic
constructs were not points, but 3-dimensional spheres of arbitrary
finite size. (09)
Tarski then showed that it was possible to define a Euclidean point
as the limit of an infinite sequence of decreasing concentric
spheres. This solution implies that points are not truly "parts"
of a line (or any other geometric structure). Instead, they are
limits and each half of the line segment can have its own limit
point. (For that matter, even lines and planes are limits of
constructions made of spheres.) (010)
For the fourth volume on geometry of the _Principia Mathematica_.
Whitehead independently developed a version of mereology in which
the basic construct was a 4-D blob and points were decreasing
sequences of nested blobs. He never completed that volume, but
he published some of the material in other writings. (011)
It's significant that Gödel couldn't find a solution that used
only set theory and that Tarski and Whitehead adopted mereology
rather than set theory to address such issues. (012)
And by the way, the Whitehead-Tarski approach can be adapted
to relate a time ontology based on intervals to one that is
based on instants. In effect, an instant is the limit of a
decreasing nest of intervals. With this mapping, it is
possible to map any observation statement made in terms of
one ontology to an equivalent statement in terms of the other. (013)
John (014)
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