Shape

46 Introduction: Tell Me All About It
Then, I’ve also tried to make the presentation visual in the way all of the ideas fit
together. I like to think of this book as if it were a shape itself. There’s a lot of redundancy
in shapes—pieces recur all over the place—and I’ve tried to do this with what I
show and what I say about it. This doesn’t mean you’ll notice anything twice, but only
that you’ll find the same thing in many different guises. There’s plenty of overlap and
interpenetration of this kind. I’ve found it’s a useful way to describe how things work.
Once you see that shapes and calculating with them aren’t about what you think,
you’ll see what’s going on.
There’s another way in which this book is like a shape. The different parts of a
shape are shapes themselves. And the shape and its parts can stand alone. In the same
way, the three parts of this book can be read separately, and for the most part are selfcontained.
Each was written with the others in mind, and tried on different audiences.
My guess is that if the parts work separately, then they’ll also work together. Whenever
shapes combine, they fuse.
Background
The following background notes mostly follow the order of presentation in the preceding
text. When coordinates seem necessary, I have indicated earlier page numbers. I
thought of this when I saw ten-bars. You can find them on page 42.
My use of references in these notes and elsewhere in this book deserves brief notice.
To start with, my references aren’t comprehensive. Nor are they useful for their
technical content, although it’s largely current. And they aren’t select points in the
locus of a new debate. That visual and verbal expression are different doesn’t surprise
anyone. No, the surprises are found in the relationships between seeing and calculating,
and how they tie visual and verbal expression together. By and large, my references
are about what it means to calculate—about what calculating is and isn’t
supposed to involve. Some of the things I cite are so well known that they may seem
gratuitous. This is a convenient way to highlight ideas that have become too obvious
to discuss. In particular, nearly everyone I talk to naively assumes that generative processes
are framed in combinatorial terms, so that calculating and seeing are both
diminished. There’s a given vocabulary of symbols and rules to combine them. And
some apply this to thinking (cognition)—‘‘[humans] have combinatorial minds.’’ 1 I
try to show how there’s more. I also use references without regard to their original purpose.
This is easy—it’s dealing with shapes. And it’s just as easy to trust anecdotal evidence
before facts and results. This also has spatial aspects. It’s no good predicting
what people will see and do next unless it shows how they’re free to go on in another
way. Anything that can happen can be useful. In the end, what you get is how things
appeared to me at the time. This is all you can ask for calculating with shapes. And
what this shows is what I really want you to see. I have one thing foremost in mind—
to explain in whatever way I can what it’s like to calculate by seeing and why this is
different than counting. Design and calculating with shapes are much the same. The
trick is to be creative. This is a licentious process in which nothing is fixed. There’s