Shape

35 Trying to Be Clear
and twenty-four of these L’s altogether. Yet indefinitely many bigger and smaller L’s
with equal and unequal arms remain. This kind of arithmetic is disconcerting. Counting
symbols and seeing shapes aren’t the same.
Of course, there are still rules. And they show in another way how counting and
seeing differ just as symbols and shapes do. Rules to calculate with symbols are easy to
classify in terms of how many symbols they contain. There are two symbols in each of
the left sides of the two rules that I described earlier—the rule h i fi hh ii and the rule
h i fi h i h i—and four symbols in each of their right sides. This is the basis for the
so-called Chomsky hierarchy that ranks alternative ways of calculating by their power
and complexity. But the rules I use to calculate with shapes can’t be classified in this
fashion. There’s no way to describe them by counting because there’s no vocabulary.
And even if there were, it might not help. Shapes don’t have definite constituents.
Their parts aren’t numerically distinct. There were different descriptions of the triangle
in the left side of the rule
when I tried to count parts. The triangle was just a single constituent, then three constituents
in two ways
or six
And the left side of the rule
had twenty constituents and also forty when I used it to rotate the trio of squares in
the shape