Shape

129 What Schemas Show
For example, I can use maximal lines or their halves, thirds, fourths, etc. The principle
is also true for shapes that contain points, because I can specify finitely many in countless
ways with alternative expressions in logic. But of special interest right now, I can
redo the predicate I have, so that y is n triangles rather than twin polygons with n sides
apiece. If my rule is originally this
with two quadrilaterals, then it’s also this
with four triangles. There are some wonderful twists and turns when I reason in terms
of these contrasting descriptions.
This series of shapes
is defined using my new schema. It begins with a quadrilateral that’s replaced with four
triangles. And this is repeated in the same way to put four triangles in a quadrilateral.
But this doesn’t add up. The second quadrilateral comes from nowhere. And what’s
up with the four triangles? If I add four more, there are eight. I’ve said it before, and
it bears repeating. The parts I see aren’t fixed. They may alter at any time in number
or by kind. Are there four triangles or two quadrilaterals? How I describe a rule and
the shape to which it’s applied don’t have to agree. I can fool around with descriptions
as long as I like to produce the results I want in a sensible way. Rules apply to shapes,
not to descriptions. Everything follows, at least when I calculate with shapes. There’s
no such thing as a non sequitur or an inconsistency—only the elaboration that
comes from looking again and then going on independent of anything that I may have
seen or done before. Nothing is ever meaningless or contradictory. It’s impossible to be
irrational. I don’t have to look back to look forward. There’s nothing to remember to
go on.
This sounds wrong. How can calculating be so confused? Definite descriptions
define the shapes in a rule gðxÞ fi gðyÞ. But my description of gðxÞ may be incompatible
with the description I have for the shape to which I apply the rule. It just isn’t calculating
unless both of these descriptions agree. But what law says that shapes must