Shape

88 I What Makes It Visual?
But this doesn’t mean—as it did in Evans’s example—that triangles have to be
described (represented) in three different ways for the identity to work in the way it’s
supposed to. For the identity
the embedding relation is given for lines, so that they can fuse and divide in any way
whatsoever, but for Evans’s rule
Three lines fi Triangle
embedding requires that constituents—they neither fuse nor divide—match in an
exact correspondence, as elements do in sets. Perhaps this is another way of seeing it.
A shape—at least with lines, or basic elements that are planes or solids—is not a set.
For points, shapes and sets coincide. Whether or not embedding is identity is key in
saying how shapes work.
My identity applies to shapes, not to their descriptions. As far as the identity is
concerned, a triangle is whatever I draw. It’s simply this
with no divisions of any kind—no sides, no angles, and no anything else. And it’s
always there to see if it can be embedded. I only have to trace over it. Evans gets into
trouble because of the way he calculates. He confuses a triangle—something sensible—
with a solitary description—an abstract definition—that’s only one of many conceptual
transformations. This is the kind of thing that can happen whenever my formula
dimðelÞ ¼dimðemÞ isn’t satisfied, and in particular, when the embedding
relation is zero dimensional and elements aren’t. Shapes and descriptions are different
sorts of things. But more important, I can calculate with shapes without referring
to their descriptions. This idea provides another useful way to think about visual
calculating.
Only before doing so, let’s look at Evans’s approach in a slightly different
way. Let’s ask how to make it work, so that lowest-level constituents—points, lines, or
whatever—correspond with what I see. They’re numerically distinct, and I can pick
them out whenever I look. This shows more of what’s at stake—why lowest-level constituents
are just another way of thinking about points.