Shape

233 How Rules Work When I Calculate
In the first stage, the rule A fi B is used to see. It works as an observational device.
If A can be embedded in C in any way at all—no matter what’s gone on before—
then C has a part like A. In the second stage, the rule changes C in accordance with A
and B. Now it’s a constructive device. But once something is added, it fuses with what’s
left of C and may or may not be recognized again. The shapes (parts) in the rule are lost
once it’s applied. There’s no record (memory) of the part identified in stage 1 or of the
part added in stage 2. Calculating always starts over with an undifferentiated shape.
And notice also that the two stages are intentionally linked via the same transformation
t. This completes the visual analogy
tðAÞ fi tðBÞ < A fi B
The rule A fi B is a convenient example of the kind of thing that’s supposed to
happen. The rule tðAÞ fi tðBÞ is the same and works for the given shape C. This is
one way of thinking about context. It’s evident when I transpose the terms in the
analogy—the part of C that’s subtracted is to A as the part that’s added is to B. But the
analogy stops there. It doesn’t go on. It ends with A and B, and the transformation t.
There are neither finer divisions nor additional relationships in A or B. The parts of
shapes are no less indefinite because they’re in rules, and they aren’t differentiated
when rules are tried.
The formal details of rule application shouldn’t obscure the main idea behind
rules. Observation—the ability to divide shapes into definite parts—provides the impetus
for meaningful change. This relationship is nothing new, but it’s no less important
for that. As I’ve already said, it’s seeing and doing, the behaviorist’s stimulus and response,
and Peirce’s habitual when and how. And James has it at the center of reasoning.
This is worth repeating.
And the art of the reasoner will consist of two stages:
First, sagacity, or the ability to discover what part, M, lies embedded in the whole S which is before
him;
Second, learning, or the ability to recall promptly M’s consequences, concomitants, or implications.
The twin stages in the reasoner’s art are formalized in the stages given to apply rules. In
fact, it’s the same kind of analogy contained in rules themselves. I can show it in this way
tðAÞ : tðAÞ fi tðBÞ < sagacity : learning
or, alternatively, so
tðAÞ a C : ðC
tðAÞÞ þ tðBÞ < sagacity : learning
in terms of what the rule does as it’s used. Every time a rule is tried, sagacity and
learning—redescription and inference—work jointly to create a new outcome. And
it’s sagacity that distinguishes rules the most as useful devices for calculating. Rules divide
shapes anew as they change in an unfolding process in which parts are picked out,
combine, and fuse. But learning isn’t remembering how shapes are divided into parts—