Shape

145 What’s That or How Many?
The idea that I can change how I describe things with rules is at the heart of visual
calculating, but it seems to be an idea that’s not easy to accept or use. And in fact
it would be easy to dismiss if it only had roots in sensible experience. Art and such are
important, but it’s science that really counts. It’s funny how things turn out. The idea
that things have incompatible descriptions is strongly rooted in science, as well. Hilary
Putnam tells the story.
Since the end of the nineteenth century science itself has begun to take on a ‘‘non-classical’’—
that is, a non-seventeenth-century—appearance. [Earlier] I described the phenomenon of conceptual
relativity—one which has simple illustrations, like [mine for a few individuals], but which has
become pervasive in contemporary science. That there are ways of describing what are (in some
way) the ‘‘same facts’’ which are (in some way) ‘‘equivalent’’ but also (in some way) ‘‘incompatible’’
is a strikingly non-classical phenomenon. Yet contemporary logicians and meaning theorists
generally philosophize as if it did not exist.
This contains a good description of the shapes in the series
that are produced when I calculate with the rule
and its inverse. The five-pointed star is produced from the initial shape in one way, and
then divided to produce the final shape in another way. This doesn’t vex the eye,
because it embodies my rules. All the same, the triangle and chevron at the start are
incompatible with their reflection at the end. The star is made up of a triangle and a
chevron in alternative ways. The facts are equivalent—‘‘a rose is a rose.’’ But the polygons
consistent with the initial shape are inconsistent with the ones consistent with
the final shape—‘‘a rose is a rose.’’ No one expects things to vanish if they’re moved.
Conservation is a solid law of physics. The outcome is never in doubt. The triangle in
the initial shape must be in the star. Only how? Separate things can’t occupy the same
place at the same time. Impenetrability is another solid law of physics that has the ring
of logic. It’s impossible to jam two distinct triangles and their conjugate chevrons