Shape

228 II Seeing How It Works
see what I want to in an ongoing process than to remember or learn what to do. My
immediate perception takes precedence over anything I’ve decided or know. But
there’s an option. I don’t have to divide the shape into meaningful parts. I can cut it
into anonymous units, small enough to model (approximate) as many new parts as I
want. This is a standard practice, but the parts of the shape and their models are too
far apart. Shapes containing lines, for example, aren’t like shapes that combine points.
This is what the algebras U 0 j and U 1 j show. Moreover, useful divisions are likely to be
infrequent and not very fine. Getting the right ones at the right time is what matters.
Yet even if units do combine to model the parts I want, they may also multiply unnecessary
parts, beyond interest and use, to cause unexpected problems. Point sets show
this perfectly. I have to deal with the parts I want, and also with the parts I don’t.
Dividing the shape into units exceeds my intuitive reach. I can no longer engage it
directly in terms of what I see. There’s something definite to know beforehand, so I
have to think twice about how the shape works. It’s an arrangement of arbitrary units
that go together in an arbitrary way that has to be remembered. What happens to
novelty and new experience? Memory blocks it all. Any way you cut it, there’s got to
be a better way to handle the shape. Decompositions are preposterous before I calculate.
They provide a record of what I’ve done, not a map of what to do. They keep me
from going on—at least in new ways.
But decompositions are fine if I remember they’re only descriptions, and that
descriptions don’t count. They’re what I get whenever I talk about shapes—to say
what they’re for as I show how they’re used. Without decompositions, I could only
point at shapes in a vague sort of way. Still, words are no substitute for shapes when I
calculate. Parts aren’t permanent but alter freely every time I try a rule. Shapes and
words aren’t the same.
How Rules Work When I Calculate
Most of the things in the algebras I’ve been describing—I’ll call all of these things
shapes from now on unless it makes a serious difference—are useless jumbles. This
should come as no surprise. Most of the numbers in arithmetic are totally meaningless,
too. The shapes that count—like the numbers I get when I balance my checkbook—are
the ones there are when I calculate. I have to see something and do something for
things to have any meaning. And seeing and doing can change things freely, even as
calculating goes on.
Shapes are defined in different algebras. But how I calculate in each of these algebras
has a common mechanism. Rules are defined and applied recursively to shapes in
terms of the part relation and Boolean sum and difference—or whatever corresponds to
the part relation and these operations—and the Euclidean transformations. This uses
the full power of the algebras. You can’t get away with anything less.
Rules are given by ostension. Any two shapes whatsoever—empty or not and the
same or not—shown one after the other determine a rule. Suppose that these shapes
are first A and next B. Then the rule they define is