Shape

150 I What Makes It Visual?
and its inverse in a specified way. Now look at the atoms in table 2. They take care of
any inconsistencies in the way the five-pointed star is described, so that the series is a
continuous process. And I can always do this somehow after I’m done calculating—
with topologies, Boolean algebras, lattices of different kinds, or comparable descriptive
devices. But more than the variability that’s brought in as a result of this, there are alternative
ways of going from the initial shape to the final shape that imply different
parts for shapes as I calculate. Just what these parts are isn’t a matter of convention, at
least not in the way Putnam has in mind. I can’t say what the parts are until I try my
rules. There’s real work to do to find out what’s what. I have to interact with shapes in
the same sort of way I interact with other things like letters or numerals when I read or
do arithmetic. I can probably contrive some method to reconcile any differences that
arise from calculating in this way or that afterward, if it’s useful. Yet there may be
no end of it. Differences—numerical discrepancies are only symptomatic—are bound
to arise as I go along, because shapes are ambiguous. And there’s no reason to think
that these differences aren’t real. It isn’t necessary to define objects all at once in a coherent
way. It’s OK to go on calculating, so that objects (parts) are defined piecemeal
with the possibility of revision every time a rule is tried. Things change with lots of
surprises.
One way to check on experience is to require that parts be numerically distinct—
to ask the question how many and get the same answer with the same parts every time.
An accountant is trained for this, and a schoolteacher expects it when taking roll. My
local selectman is proud of it—‘‘I’m logical, I’m an accountant.’’ I know just what she
means. I count at the market to make certain that I have all of the items on my grocery
list. Counting is an everyday practice that’s useful in business and science. But perhaps
it isn’t that elucidating. Suppose I have a superstar with around fifteen points