Shape

210 II Seeing How It Works
First, there are the famous individuals of logic. Shapes are like them in important
ways. In particular, both shapes and individuals can be divided into parts, now in one
way and again in some other way, independent of any agreed formula. As a result,
shapes, like individuals, aren’t like sets. Henry Leonard and Nelson Goodman clarify
this difference. The following quotation strikes at the heart of the matter.
The concept of an individual and that of a [set] may be regarded as different devices for distinguishing
one segment of the total universe from all that remains. In both cases, the differentiated
segment is potentially divisible, and may even be physically discontinuous. The difference in the
concepts lies in this: that to conceive a segment as a whole or individual offers no suggestion as to
what these subdivisions, if any, must be, whereas to conceive a segment as a [set] imposes a definite
scheme of subdivision—into [subsets] and members.
Embedding and the part relation are crucial. But the likeness between shapes and individuals
fades as additional details are checked.
The way shapes and individuals are used distinguishes them most of the time.
Rules are applied to change shapes—in fact, I defined shapes and rules in concert to
make sure that this was so—while individuals aren’t handled in this way. There’s no
distinction between individuals and individuals in rules. This is difference enough
when there’s calculating to do, but it may not seem decisive otherwise. And perhaps
the contrast is simply a question of emphasis, with a common purpose. Individuals
distinguish things in the total universe just as rules do in shapes. Only how? In the
latter, it’s calculating with parts that vary as rules are tried. Though here, too, there
are worthwhile shadings. Goodman and W. V. Quine set up the machinery—‘‘shape
predicates’’—to calculate with ink marks (individuals) in a ‘‘nominalistic syntax.’’
Marks and lines, etc., are much the same in terms of embedding and parts, yet neither
relation is exploited. Instead, shape predicates are used to recognize symbols
and texts largely in accordance with the rudimentary conventions of printing and
reading—pretty much as described in part I. Predicates are framed to block ambiguity
and to discourage my way of calculating with shapes. But this is no surprise—symbols
and texts are supposed to stay the same in a syntax.
Differences in definition are easier to compare. Individuals form a complete Boolean
algebra that has the zero excised. The algebra may have atoms or not—it seems to,
more often than not—and may be finite or infinite. Yet worries about zero—having
something for nothing—pale against the possibility of taking infinite sums and products.
I’m happy with zero (the empty shape) as an unremarkable technical device,
because it makes a nice algebra and sticks to standard mathematical usage. And I’m
unhappy with infinite operations, because I can’t figure out how to do them in a practical
way I can understand, or to see the extent of the results in every case. Still, this
isn’t all that’s different. Unlike shapes, there are no added operators for individuals—
transformations aren’t defined. To be completely honest, though, the nonempty parts
of any nonempty shape in an algebra U 0 j are possible individuals. But if j isn’t zero,
then this isn’t so for all of the nonempty shapes in the algebra taken at once. Shapes
only go together finitely. Maybe the real difference is that logicians and philosophers