Shape

65 A First Look at Calculating
he puts greater emphasis on the former. Perhaps this is surprising, but I think he’s right
about ‘‘the novel step in thought.’’ Embedding—‘‘the ability to seize fresh aspects in
concrete things’’—is the key. Of course, James also relies on learning. This isn’t about
the kind of facts that are taught in school where the emphasis is on counting things,
but about finding rules and remembering them independent of things and their parts.
I can decide what to see and do without ‘‘translat[ing] everything into results.’’ There is
no final analysis, only lots of temporary analyses that can change erratically as rules are
tried. This is a dynamic process with plenty of ambiguity. Nothing ever has to stay the
same. Novelty is everywhere. Once you see how this works—and it does depend on
seeing—it’s evident that the standard version of rules in terms of symbols—what
Susanne Langer has in mind when she considers vocabulary and syntax—is neither
James’s nor mine.
So question 3 is easy enough, and the answer I have helps with question 2. First,
I can use embedding to distinguish visual calculating and calculating in the ordinary
way with symbols. And then, James’s double take on creativity works for shapes and
rules. There’s always the chance for something new. Rules are defined when shapes
are combined in pairs—there’s learning what to try—and rules apply via embedding—
there’s sagacity seeing in alternative ways. Is this all that’s needed? What does embedding
allow that ‘‘results’’ miss? How does it work?
Reasoning is often surprising when it succeeds—at least for James—and may be
equally so when it fails. I want to begin with an instance of the latter to illustrate a few
details of embedding, and what these mean for shapes and rules. My example deals
with line drawings, but this in itself doesn’t make it visual. If drawings are handled
logically—that is to say, rationally in terms of given parts—shapes aren’t really
defined. Neither reasoning nor calculating is automatically aligned with seeing. Things
can break down. In fact, my example shows how easy it is for calculating and seeing to
disagree. Of course, there’s a way to reconcile them that I’m going to show, too. Visual
calculating and hence visual reasoning are neatly practicable with shapes and rules.
A First Look at Calculating
T. G. Evans applies the rules of a ‘‘grammar’’—it’s syntax in Langer’s sense—to define
shapes in terms of their ‘‘lowest-level constituents’’—or alternatively atoms, bits, cells,
components, features, primitives, simples, symbols, units, etc. This illustrates some
notions that have been used widely in computer applications for a long time. They’re
ideas that are as fresh now as they ever were. (Rules like this were applied early on in
‘‘picture languages’’ to combine picture atoms and larger fragments, and just afterward
in Christopher Alexander’s better known yet formally derivative ‘‘pattern language’’
for building design. The story today in AI, computer graphics, design, etc. is
the one I told on page 51 about units, Lindenmayer systems, and cellular automata.
My set grammars are also alike—but maybe more inclusive, being the same as Turing
machines. Of even greater interest, though, set grammars work the way shape grammars
do for points. For lines, things diverge with important consequences for visual