Shape

306 II Seeing How It Works
Table 12
Alternative Ways to Describe a Shape A
Points Parts Closed parts
Topology Euclidean space Stone space Closure operation c
Closure of x a A Point set þ boundary Parts are closed and
open
Smallest closed part
containing x
Boundary of x Point set boundary Empty shape cðxÞðA xÞ
Connectivity Multiply connected Totally disconnected Partially connected
Algebra Boolean Boolean Heyting
Most of the mathematics I use is pretty elementary. In a few places, however, I
probably exceed the limits of everyday familiarity. There are (1) M. H. Stone’s generalized
Boolean algebras (Boolean rings), (2) his extension of these systems to standard
Boolean algebras, and (3) his theory of representations. 33 Earl presents a nice review
of the techniques in the latter—Stone spaces, etc.—with respect to shapes and their
topologies. 34 For example, closure for shapes and Stone’s filters describe parts by the
parts containing them. But, of greater interest, Earl summarizes alternative ways of
describing shapes in terms of sets and parts. These are in table 12, and are familiar
from what I’ve covered, using point sets, parts made up of points as well as lines, etc.,
and closed parts. Still, table 12 is only a snapshot, and there’s more to see and do. It’s
crucial to what I’ve been trying to show that the closure operation c in the rightmost
column is defined as rules are applied—for example, using the mapping h 1 on page 286
and either the first or both of the formulas that follow there. This allows for anything
to happen as meaning unfolds in an Aristotelian process that’s finite and continuous.
Nothing is fixed before I begin to calculate, and everything is free to change as I go on.
A Heyting algebra in which the complement of a closed part is the closure of its
Boolean complement is defined via c. This is especially telling for h 1 and the first formula
alone. (Heyting algebras also suggest a link with logic that I haven’t followed up.)
The Stone space topology in the middle column is really neat once you see how it
works—even as an artifact of mathematical abstraction—and confirms what I’ve been
saying about shapes with all of their concrete properties. And the mathematics for
parts and closed parts—this goes together in closure algebras (þ, , , and c)—doesn’t
intrude in what’s going on. It shows how shapes and rules work as if it weren’t there.
With parts and closure, you’re free to think with your eyes—to see and do whatever
you want. That’s the real mathematics. And it doesn’t need to be formalized to be
used, even if that provides an impressive pedigree. Elsewhere, I show how generalized
Boolean algebras and Boolean algebras are related, using figure-ground-like properties
of shapes. 35 Monoids (page 194) are used in the theory of formal languages and automata
instead of algebras with more Boolean-like properties. 36 Concatenation is the sole
operation on strings of symbols from a given vocabulary. But, in general, monoids
won’t do for rule application. 37 Concatenation isn’t enough to recognize and replace