Shape

222 II Seeing How It Works
In particular, notice how singleton shapes are used to produce new basic elements. Let
e and f be basic elements in A and B. Then for the twin differences A B and
B A , new basic elements are defined by feg B and ff g A , and for the product
A B ,byfegff g. And once this is done for all basic elements, resultant ones with
equal weights are added together to make sure that everything is maximal. Basic elements
defined in differences and products with singleton shapes are always discrete,
but they may still have overlapping boundary elements.
In like fashion, the difference of two weighted shapes A and B is formed by dividing
their basic elements into pieces that contribute to the shapes A B and A B .
The weights assigned to the basic elements that contribute to A B are defined in relative
complements. However, weights can’t be empty for basic elements to be included
in the difference of the weighted shapes A and B. And once again, the process is easy to
illustrate for lines
The Boolean algebra for sets of labels is nothing special. Nonetheless, it shows
how a familiar abstract device is enough to change any algebra of labeled shapes into
an isomorphic algebra of weighted shapes. Putting independent labels in sets lets me
combine the labels. (I can do the same using individuals, only with atoms—one for
each label—instead of members. But I like sets better. Labels are pretty abstract already,
so there’s no harm in sets and no reason to be like shapes. The contrast is worth keeping
to stress the difference between classifying parts and finding them.) This trick
makes labels unnecessary—weights will do. Of greater interest, though, there are more
exotic algebras of weights that aren’t Boolean. One is worth seeing in a little detail, because
of its relationship to drawing and because of what it shows about cooking up