Shape

209 Boolean Divisions
doesn’t have a sum. No finite set of maximal lines corresponds to this infinite series
of segments. This is similar to what I did earlier with boundaries of basic elements
that were defined in the limit. And the construction applies equally to planes and
solids if I divide rectangles or rectilinear rods. What’s more, I can use fact 7 in table 6
to get the corresponding result for products. Now the shapes in my infinite set are
these
and their product gives the gaps separating the segments in my sum
This is another kind of figure-ground reversal with formal implications.
Shapes made up of lines, planes, and solids have underlying topologies in the
same way shapes made up of points do. Only instead of finite topologies, they’re
infinite ones. In particular, every shape has a Stone space topology. Parts correspond
to closed and open sets. They have empty boundaries and are disconnected. In this,
all shapes—with points and without—are exactly the same. They’re all ambiguous.
There’s nothing about shapes by themselves, no matter what basic elements they
have, that recommends any division over another. Stone spaces confirm what’s easy
to see, but they violate the Aristotelian scruples I invoked earlier on. Still, everything
is grounded when I calculate. The parts I see—the ones that rules pick out—are the
ones that count, with meaningful interactions and the possibility of substantial boundaries.
And these parts, too, combine in topologies—finite ones—to describe shapes.
So far, my classification of shapes and their algebras shows that things change
extensively once i is bigger than zero. And the pun isn’t gratuitous. It applies spatially
to basic elements, and then numerically to indicate the many differences between algebras
when shapes contain points and when shapes are made up of lines, etc. Whether
or not embedding implies identity makes a world of difference. But my algebras aren’t
the only way to think about shapes. There are two alternatives that deserve brief notice.
They locate my account of shapes in a wider landscape of formal possibilities that
includes philosophy and engineering.