Magnus Anderson wrote:Gib wrote:I think I'm partial to the infinite sides faction of the debate. A circle has infinite sides. Yet at the same time, I see how this is equivalent to the notion of a circle having no sides.

Circles are first and foremost objects of our experience. Yes. They are shapes that we can SEE in our everyday life. That's what they are. And we know what they are thanks to our intuition. We need no dictionary definitions. That's true.

When you say something like "circles are a set of points equidistant from some fixed point" you are providing an alterantive way of judging whether any given shape is a circle or not.

This method of judgment involves a ruler that is anchored in the center of the shape. It also involves choosing a number of points on the boundary of the shape you are measuring. And it involves rotating the ruler so that one of its ends passes through one of the selected points on the boundary. What you have to do is to measure the distance between the center and each one of the selected points. If all of the distances are equal you declare that the shape is a circle. Correct!

The other approach is polygon approximation but this approach has a weakness in that it involves the redundant concept of side. So Occam's Razor might have something to say about this. Sides cannot help us to determine any given shape is a circle or not. Sure, they could be ovals. They are quite simply useless in this regard. Where they are useful is in measuring the circumference of a circle. That is when describing circles in terms of polygons is useful.

The polygon approximation approach gives us a more complete view of circles.

So it sounds like you're saying there are at least 3 approaches to defining circles: intuition (we know a circle when we see it), point geometry (all points on the circumference are equidistant to the center), and polygon geometry (a polygon with infinite sides all at equal angles to each other).

You also say that the polygon geometry approach has the drawback of holding onto the redundant concept of side, yet you also say that it gives us a more complete view of circles. Would you say that this redundancy is a good thing then? That it allows for a more complete view of circles?