Shape

63 Does Reasoning Include Calculating?
But when I look at mine, I worry about which way the inclusions should go—what’s
embedded in what? What makes me think I have the right setup? There are a couple
dozen ways to permute the names of the areas, with novel consequences that might
be useful. Maybe the names should be reversed like this
so that reasoning and calculating are special cases of visual reasoning and visual calculating.
Is this an example of how visual reasoning is supposed to work? I don’t know.
There’s a nifty change in perspective, with the kind of switch that’s common in design
in the interplay between results and goals. Many things alter freely the way drawings
(shapes) do when I look at them a second time. Ambiguity is something to use. And no
matter which diagram I try, visual reasoning and calculating need to be explained. My
three questions still hold with undiminished force. What’s more, I’m sure that each of
my diagrams is correct on its own, for calculating by counting (this is the standard
method of calculating that’s taught in school) and then by seeing (this is my alternative
that relies on shapes). And there are telling equivalencies, as well. It’s uncanny
how everything converges so quickly, and without any fuss. I’m right back to the
scheme I outlined at the beginning of the introduction to relate verbal and visual expression.
This depends on the contrast between identity and embedding—between
points (symbols) of dimension i ¼ 0 and basic elements of other kinds—lines, planes,
and solids—of dimension i > 0. What this contrast means for reasoning and calculating,
visual and not—why the value of i matters—is mostly what I’m going to explore.
Let’s see how all of this works out in some detail. What about question 3?
Does Reasoning Include Calculating?
I want to show that reasoning includes calculating—or at least that there’s a reason to
think so. Most of the people I’ve asked agree that it does. Some see calculating as what
the best in reasoning is all about, while others see it as a narrow kind of process among
many of greater scope and use. Nonetheless, I only need the inclusion to go on, so that
I can explain visual reasoning in terms of visual calculating. This isn’t new territory.
The relationship between reasoning and calculating has been described many times
before—Thomas Hobbes was apparently the first to try—in a variety of ways from
which I can select. Sifting through the possibilities may take some reasoning—at least
a little judgment if not actual calculating—but the stakes aren’t high. There’s enough
agreement to decide on pragmatic grounds alone. And in fact, this points the way.