Shape

146 I What Makes It Visual?
together all at once in the star—even visually. Try and see both alternatives at the same
time—
But there they are—at least until I move one of the triangles. This is in fact a striking
way to calculate. That it’s left out of logic is no real surprise. But Putnam isn’t worried
about what logicians ignore. He wants to explain this nonclassical phenomenon. And
what he finds meshes with some of the things I’ve been trying to say about visual
calculating.
There are two important points. One works with visual calculating, and the other
does—when it goes against convention—and doesn’t—when it embraces counting.
Counting is a reliable test of how things are, but it doesn’t bother me if numbers
change without rhyme or reason as I calculate. Let’s see how this plays out. The first
point is this—
[The] phenomenon [of conceptual relativity] turns on the fact that the logical primitives themselves,
and in particular the notions of object and existence, have a multitude of different uses rather than one
absolute ‘‘meaning.’’
And second,
Once we make clear how we are using ‘‘object’’ (or ‘‘exist’’), the question ‘‘How many objects
exist?’’ has an answer that is not at all a matter of ‘‘convention.’’
Putnam’s first point is pretty obvious whenever I calculate with shapes. What I
see before me depends on the rules I try. Evans’s shape
isn’t anything in particular until I apply the rule (identity)
to pick out triangles. And if I try the rule