by **gib** » Tue Oct 03, 2017 6:15 am

I am now waffling. I'm trying to image the points that comprise the circumference of the circle--I'm trying to zoom in with my mind's eye as close as I can to see just a few points (maybe a dozen or so) that are consecutive to each other and try to make out the curvature. I'm having difficulty. I'm finding that the simplest curve I can imagine must consist of at least 3 points. You can't have just 2 points because that would comprise a line (which is problematic in itself given that a circle is not supposed to be composed of lines). But now if these three points make up the smallest curve the circle can have, then it seems curvature is reduced to angles (3 points also make an angle). And furthermore, this angle would have to be so obtuse as to be indistinguishable from a straight line (comprised of the 3 points). Otherwise, you'd be saying the angle is a finite amount (however small) and it would take only a finite number of these angles attached together to form the circle, effectively throwing out the notion that the circle is comprise of infinite points.

^ No doubt, all this is due to the same fact I brought up earlier--that when you're skipping an infinity (from the circle as a whole to its individual points), you're already doing something paradoxical--it shouldn't be a surprise when other paradoxes follow.

Maybe we've been looking at geometry wrong over the past 2,500 years. Maybe we shouldn't say that circles and squares and lines, etc., etc., etc., are made of points, but that points are the smallest geometric entity they can be decomposed into. Points don't "exist" per se on the circumference of the circle but you can mark a point on the circle and say "let that be point A." <-- IOW, we invent points as we need them. As for what the circle (and other geometric shapes) are made of, I'd say segments. Segments can be straight or curved, and they can be infinitely divided, and the result of any such division is just smaller segments. So if you take a quarter of the circle's circumference, that's a segment 1/4 of the circle's circumference in length and half a radian in curvature. But then you can divide that segment in two equal halves, each being an eighth the circle's circumference and a quarter radians, or two unequal halves, .1 radians and .4 radians, or any ratio you want. <-- That's what shapes, lines, and curves are made of. Points end up being, not something these objects divide into, but things used to mark a certain position on these objects. You can then imagine that at that position, you've invented a point (like inventing a border between countries--it's real because we say it's real), and therefore the circle is comprised of that point at that exact location, but that doesn't mean it's got a "neighbouring" point, or that you can count twelve points to the left and be a bit further along the circle's circumference. On either side of the point are segments whose length depends on where you want to mark the other points constituting their other ends. So it's still true that the point constitutes the smallest thing you can decompose the circle into, but unlike segments, they aren't just all "there" before you mark out specific ones.

^ How's that sound to everyone?

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