In order to test whether any given shape is a circle (or not) we need to decide how many points there are on the boundary. This is not an objective parameter. The boundary of the shape under the test does not have a finite number of points. Instead, it is us who has to decide, entirely subjectively, how many points there are. What this means is that testing whether any given shape is a circle or not is in actuality a test of how circular that shape is. No shape is inherently a circle. Instead, shapes are MORE OR LESS circular. When we choose a smaller number of points, we are testing for a lower degree of circularity. When we choose a larger number of points, we are testing for a higher degree of circularity. This is perfectly in line with the holistic stance that there are no absolutes, only degrees.
Sure, we can run binary true/false tests. Any given shape either passes the test or it fails the test. But what we are testing for can never be “the perfect circle”. Why? Because there is no such a thing. Can anyone here define what a perfect circle is? But without taking things out of context. Don’t just say “a perfect circle is every point in the plane that is equidistant from some fixed point”. Such a definition does not specify the number of points that have to be tested. If you say that the number of points does not matter, then the test becomes too lax . . . nearly every shape can pass it. The 12-sided polygon? It’s a perfect circle and James is wrong! On the other hand, if you say that the number of points is, say, quadrillion, then it becomes too strict . . . no shape can pass it. And the problem is not only strictness but the arbitrariness of choosing a number of points that defines the perfect circle. It’s entirely subjective. Why quadrillion and not, say, quintillion? The question cannot be answered without taking context into account. The problem is that analytical thinkers, such as James, don’t want to take context into consideration. They are absolutists.
When someone says that a circle is “an infinite sided polygon” what is meant is that the greater the number of sides a circular polygon (i.e. a shape with its key points being equidistant from the center) has, the more circular it is. That’s all it means. Pentagon is less circular than hexagon which is less circular than octagon which is less circular than 12-sided polygon which is less circular than 24-sided polygon and so on and so forth.