Shape

287 Erasing and Identity
and every part of C is left alone. Calculating is continuous if the topology of C is the
same from step to step—from my previous formulas, I get x ¼ C y ¼ y. But then I
need to update the topology every time an identity A fi A is applied, so that tðAÞ is a
closed part. Conditions 1 and 2 are independent. It’s no surprise, either, that there’s a
comparable account of any rule with a left side that’s part of its right side—in particular,
any rule that answers to the degenerate schema
fi x
with an empty left side, or to the familiar schema for spatial relations
x fi A þ B
Mappings show how rules with widely different uses are alike.
Examples for both of my mappings show that what I can say about a shape that’s
definite—at least for parts—may have to wait until I stop calculating. Descriptions
are retrospective—they record what I’ve done without limiting what I can do. In
many ways, the past is reconfigured in the present, while the future is always open.
My description of what happened before can change radically as I continue to
calculate. What happens now makes a difference for the past because something else
is going on whenever another rule is tried. History varies as I see and do more in new
ways.
Before I turn to specific examples, it’s worth emphasizing that describing what
rules do is really the same as describing shapes in terms of their parts. There are alternative
ways to do it—lots of them. I can be a purist and treat all my rules in the same
way, or I can let my descriptions wander all over the place—even describing the same
rule differently at different times. I’ve scarcely touched the surface with what I’ve said
here about mappings and topologies and how they might be used. The foregoing presentation
is pretty impressionistic—it’s still about seeing. But I can fix things up a little
with closure operations and closure algebras. And for those who enjoy mathematical
depth and rigor, there’s M. H. Stone’s theory of representations. Only this means seeing
in a complementary way that’s abstract rather than concrete. Whatever you like—
I’ll stick with my finite Aristotelian perspective and what I’m able (need) to see—it
seems to me that continuity is a good place to start. It develops the idea that things
that coalesce can change at any time. In retrospect, unbroken community and licentious
freedom are compatible. Playing around with descriptions of rules is another
way to inform seeing, even impressionistically. It leads to a host of new insights about
how calculating with shapes works and what it implies. There’s plenty to do, but then
calculating with shapes means going on.
Erasing and Identity
Some easy illustrations show how my three scenarios work. Suppose I want to use the
first scenario to define decompositions for the shape