Shape

77 What Makes Calculating Visual?
with an ancient history and a superb record of success, and it works like magic. Yes,
it’s simply an illusion. And it’s easy to expose when calculating and sensible experience
are asked to agree. But this is nothing new. James says it squarely—there
are ‘‘many ways in which our conceptual transformation of perceptual experience
makes it less comprehensible than ever.’’ What’s gone wrong? Maybe there’s no
such thing as visual calculating. Do all of these negative examples mean that I should
give up, or are they reason to try another way? What kind of evidence would make a
difference?
I’m going to have to show that analysis—segmenting or dividing shapes into
lowest-level constituents—isn’t something necessary in order to calculate, but rather
something contingent that changes or evolves—even discontinuously—as a byproduct
of what I see and do. Analysis is one way to show what happens as I go on
calculating. What I do as I calculate—the rules I have and the way I use them—
determines how constituents are defined. There’s nothing to see or do before I start.
There are still rules to try.
What Makes Calculating Visual?
Sometimes I try an informal rule of thumb to decide when calculating is visual. Both
the dimension (dim) of the elements (el) and the dimension of the embedding relation
(em) I use to calculate are the same. I can state this in a nifty little formula that’s a good
mnemonic—
dimðelÞ ¼dimðemÞ
I’m pretty sure the rule is sufficient—whenever my formula is satisfied, it’s visual calculating.
I’m just not as sure the rule is necessary. There may be examples of visual calculating
that don’t meet this standard. Perhaps it would be better to try and make sense
of the relationship dimðelÞ a dimðemÞ. But whatever the answer—and I’m ready to bet
on equality, especially because the formula is so elegant—I want the equivalence
(biconditional)
dimðelÞ ¼0 1 dimðemÞ ¼0
to be satisfied. I want to ensure that zero-dimensional embedding relations are only
used with zero-dimensional elements, that is to say, with things that behave like
points. This rubs against canonical practice when it comes to calculating. But I want
to try something else.
As formulas go, mine is pretty vague. I don’t say how to evaluate either side except
in a few ad hoc cases. Still, the formula is enough for the time being. And there’s
no reason to avoid ambiguous or vague ideas when they stimulate calculating. In fact,
they may be indispensable to what I’m trying to show. I can’t imagine anything more
ambiguous and vague than a shape that isn’t divided (analyzed) into meaningful constituents,
so that it’s without definite parts and evident purpose. I’m going to use my