Shape

81 What Makes Calculating Visual?
Suppose I keep Evans’s shape the same
I can draw it with eight lines—its maximal elements—the way an experienced draftsman
would. It might go like this—in sequence, the three horizontals
the three verticals
and the two diagonals
Only I’m not obliged to treat these lines as constituents. In particular, the parts of
the shape may be combinations of maximal lines—just four of Evans’s triangles are
defined in this way—or, allowing for everything I can including the other twelve triangles,
combinations of maximal lines and any of their segments. Now there are indefinitely
many parts, and none trumps another. But segments aren’t given all at once like
points or the members of a set. They’re not set out explicitly one by one; only maximal
lines are. This is something new that looks almost Aristotelian. Parts are potentially
infinite—I’m free to cut Evans’s shape wherever I like—and not actually so. The parts
I see—the limited number I resolve when I divide the shape or draw it—depend on my
rules and how I use them. I need an embedding relation for this that’s one dimensional
and that works for lines. (Linguists like to say that languages are potentially infinite
sets of sentences, and explain this with rules that combine words recursively. This is
calculating by counting. It’s why Chomsky’s grammars are generative and it’s what it
means to be creative. Sets of shapes are the same—my rules are recursive, too. Only my
rules do more when they divide shapes into parts. Seeing isn’t counting, nor does creativity
end with words and recursion, or searching through the myriad alternatives