Shape

52 Introduction: Tell Me All About It
units or symbols. The whole reach of calculating as a creative way of reasoning depends
on the answer. Isn’t it high time for this ‘‘novel’’ issue to come to the fore once and for
all?
I read Noam Chomsky’s book Syntactic Structures when I was an undergraduate,
and have barely glanced at linguistics since. 23 I checked some current sources, and the
little I’ve said still holds up—from creativity to how sentences are described. 24 (In addition,
there’s the independent Cartesian hypothesis that rules are innate. Evidently,
language and embedding are equally automatic—‘‘Healthy adults [and children, for
that matter] make very few errors on this test.’’ 25 If our knack for language tells us
about ourselves, then surely our knack for embedding does, too. Both are parts of
what it means to be us.) A generative grammar is a model combinatorial system. Words
and syntactic categories—sentences, noun phrases, verb phrases, nouns, verbs, etc.—
combine in trees like this one
for the sentence ‘‘A rose is a rose.’’ 26 (The identity is logically necessary, but Gertrude
Stein repeats its predicate to be sure. 27 This makes another sound—Eros—so ‘‘a rose’’ is
a shape. And the tree is twin structures in perspective—two sheds back-to-side. Or is
this another trick, a clever facade to hide the truth? No doubt, the tree is a single edifice
in elevation. Only ‘‘a house is a house’’ may be a better foundation. There’s sound
evidence indeed—seeing and hearing are calculating. Now it’s eyes for shapes, hands
for the vertices, edges, and faces of polyhedra, and ears for words to apply rules sensibly.
The mind ties all of this together with symbols and counting to settle Plato’s question
in a surprising way. I said this before on page 47. Many things bear repeating—
some never enough.) Chomsky is as well known for his anarchic politics as he is for
his linguistics. And calculating with shapes seems to be as much like the former as the
latter. In fact, deciding when things are alike is a key part of using rules. But it’s more
important now that shapes and rules go with many things they’re not supposed to. I
show this again in part I for Gian-Carlo Rota’s phenomenology and combinatorics.
On the one hand, there’s how things look, for example, trees and hierarchically organized
list structures—strings like hhh i h ii hh i hh i h iiii, ((( ) ( )) (( ) (( ) ( )))), and
000101100100101111 in which symbols are combined in a certain way—and then on
the other hand, there’s counting distinct arrangements—the tree I’ve shown and my
three strings are all identical. This is perfectly clear. But shape grammars let me see