Shape

53 Background
and count as I please. Moreover, they show how the contrasting things that people do
hang together. Of course, this was evident from the start with Turing, photography,
and machines.
Langer’s book is Philosophy in a New Key. Her discussion of language and drawing,
etc.—some of which I quote on page 18—is used to distinguish presentational and
discursive forms of symbolism. 28 The distinction corresponds more or less to the informal
one I make between visual and verbal expression. These are so obviously separate
that it’s easy to miss how they’re related. Certainly, for me, there’s a decisive link, and
elaborating it to equivalence is what a lot of this book is about. I enjoy reading the initial
few pages of Langer’s final chapter, ‘‘The Fabric of Meaning.’’ I like the opening
line—‘‘All thinking begins with seeing . . .’’—and more. First there’s a strand of the
idea that structure is a temporary (evanescent) record of our own activity. Topologies
(vocabulary and syntax) are one way to show the structure of shapes. And I show that
topologies are by-products of using rules—afterthoughts that evolve in fits and starts as
calculating goes on, in the way a designer might explain a developing body of work.
Then there’s an apparent behaviorist twist—seeing is ‘‘sheer’’ response. Chomsky’s
old review of B. F. Skinner’s Verbal Behavior puts this in doubt. But behaviorist ideas
are of some use when it comes to saying what it means to calculate with shapes. One
way to describe rules minimally is as stimulus-response pairs—I see this and do that.
It’s a stark reminder that shapes aren’t represented either in rules or as the result
of what rules do. Everything fuses and divides anew as I go on. Every outcome is a
surprise.
I was pretty clear about the relationship between grammar and syntax in Langer’s
sense and calculating with shapes more than twenty years ago. I framed the contrast in
terms of set grammars on the one hand and shape grammars on the other. Set grammars
parody combinatorial systems, exaggerating their use of identity via set membership.
Still, the technical details are unassailable—set grammars are equivalent to Turing
machines.
Set grammars treat [shapes] as symbolic objects; they require that [shapes] always be parsed into
the elements of the sets from which they are formed. The integrity of the compositional units in
[shapes] is thus preserved, as these parts cannot be recombined and decomposed in different ways.
In contrast, shape grammars treat [shapes] as spatial objects; they require no special parsing of
[shapes] into fixed [parts]. Spatial ambiguities are thus allowed, as given compositional units in
[shapes] can be recombined and decomposed in different ways. 29
There’s creativity in combining shapes and in dividing them. But the one without
the other is just reciting by rote, merely counting out. It’s all memory when shapes
are divided in advance, but otherwise, everything is always new. No one took any notice
of this. Maybe the difference between sets and shapes in calculating—between
identity and embedding—is too subtle. Or perhaps rigor and formality don’t work.
I’m less technical now, and as informal as I can be. The message is the same, and I
don’t want it to be missed. It’s all about seeing—there are no units; shapes fuse and divide
when I calculate.