Shape

147 What’s That or How Many?
it’s something different. Then Putnam’s second point—at least my version of it—is
also clear. It’s not up to me how many triangles there are in the shape; the number is
the result of calculating with the rule
to see how triangles are embedded. The only problem is that the rule can be used in
different ways to get different results—‘‘the figures alter erratically.’’ Of course, I can
always insist that the rule be applied everywhere there’s a triangle. And I even have
an algorithm for this—in fact, my algorithm is good for any rule. So maybe there’s a
definite answer after all.
But I have another way to look at counting that gives inconsistent results that
aren’t so easy to bypass. Suppose I start with the rule
that erases equilateral triangles—that’s simple enough—and then apply the rule to
the shape
In the series of shapes
there are clearly four independent triangles. That’s precisely how many I erase in order
to make the shape
disappear. What better test for counting could I ever have? Every time I count a
triangle, I take it away to define a one-to-one correspondence between triangles and
numbers. Each triangle is counted once—that’s how counting works—and all triangles
are accounted for when the shape is gone, or my rule doesn’t apply anymore. But I
count five triangles in each series in this trio