Shape

62 I What Makes It Visual?
(2) What makes calculating visual?
(3) Does reasoning include calculating?
I’ll take a meandering route through their answers that digresses now and then to explore
other questions like the unnumbered ones in the previous paragraph. This isn’t
exactly what Plato recommends in the quote that heads the introduction, but wandering
around aimlessly to see what’s what—changing your mind freely—is a useful way
to get new results. It’s a nice way to see what calculating with shapes and rules is all
about. It feels right, and there’s some logic to it. If my answer to question 3 is yes,
then I can argue by analogy that visual reasoning includes visual calculating. And if I
can answer question 2, then I can use visual calculating to understand visual reasoning.
It’s reasoning by inference—from the properties of a part to the properties of the
whole. Of course, this assumes that parts and wholes are alike. I’ve already said so for
shapes and embedding. This makes calculating with shapes and reasoning (inference)
the same in at least one respect. Plus, I’m more confident about the mechanics of calculating
than I ever will be about reasoning generally.
I’ve always liked engineering, with its sensible outlook and palpable results. It’s
the same when I calculate. I can point to examples—some of my own invention—
and go through them step by step. But it’s hard to know about reasoning. My own reasoning
when it goes beyond calculating is as suspect as any. Whenever I think I have a
good argument, someone soon comes along and proves the opposite. And it’s just the
same if I try to follow the reasoning of others. I go from thinking I’m thinking to
thinking I’m not. I’d rather stick with engineering and show how things work out sensibly.
It’s easy to do applying rules to calculate with shapes. I can’t be sure until I show
you, but I’m almost positive—you’re going to be surprised at how much more there is
to visual reasoning than you imagine, if it’s anything at all like visual calculating. The
kinds of things I have in mind don’t come from complicated ways of counting or
clever coding tricks that take real brainpower—from what’s customarily valued and
encouraged in calculating. They come straight from seeing. It’s all there whenever you
look.
I’ve framed my questions with respect to the following diagram
that shows how everything is supposed to fit together. This extends counting to calculating
in what I said in the final pages of the introduction about reasoning at its best
and why its visual counterpart is something to explain. Diagrams are meant to see.