Shape

268 II Seeing How It Works
and its inverse that I used earlier, to show that calculating with shapes needn’t be
reversible. The rule has the symmetry properties of the fifth rule in table 10.
Classifying Rules with Parts
So far, I’ve been using transformations to classify rules. This relies on the Euclidean
properties of shapes, but it’s not the only way to describe rules. I can take a Boolean
approach and think about rules in terms of how many parts there are in the shapes in
their left and right sides. The classification of rules in this way by counting their parts
is the crux of the Chomsky hierarchy for generative grammars. The hierarchy gives
compelling evidence for the idea that calculating is counting.
Generative grammars produce strings of symbols that are words or well-formed
sentences in a language. A rule
AqR fi AAqAL
links a pair of strings—in this case, three symbols A, q, and R, and three others AA, qA,
and L—by an arrow. If the strings were shapes, then the rule would be like one of my
rules. The rule is context free if there’s just one symbol in its left side, context sensitive
if the number of symbols in its left side isn’t more than the number of symbols in its
right side—my rule is context sensitive—and belongs to a general rewriting system
otherwise. Languages vary in complexity according to the kind of rules that are needed
to define them. Moreover, the class of languages generated by context-free rules is
included in the larger class of languages generated by context-sensitive rules, and so
on. How well does this idea work for rules defined in my algebras? What difference
does it make whether or not i is zero?
There’s no problem counting for shapes and other things when they’re zero dimensional.
Points—or whatever corresponds to them, say, the parts in a decomposition
of a shape—are uniquely distinguished in exactly the same way symbols are. But
what happens in an algebra when i isn’t zero? What can I find in the algebra U 12 , for
example, that corresponds to points? Is there anything I can count on?
The rule
looks as if it should be context free, at least with respect to the unremarkable
description