Lessons on Causality

I passed 1st grade, unlike you and a few others here.

It could also be interpreted as a square with extra sides, as a dog with a few pieces missing, as the King of Cleveland on a good day, as a circle to a blind man…

:laughing:

And people who don’t think Chiliagons are circles can be interpreted as wrong.

Notice that he’s accepted it would be an interpretation.

I already said it was a matter of approximation pages ago. Some individuals need very specific definition and accuracy, like mathematicians, engineers, and physicists.

To a mathematician, a circle can be defined as (x^2)+(y^2)=r^2

It’s okay, you can admit I was right all along, now…

A dog is an approximation of a dog??
A tree is an approximation of a tree?? :confused:

What is a circle an approximation of?

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?

I’m not a fan of points. Lines and sides are practical.

Dogs are approximations of canines, yes.

Trees are approximations of fora and other plantlife, yes.

There are scales of interpretation in terms of accuracy. Wolves are not dogs but huskies are. Same mode of argument.

But a Chiliagon is still a circle.

No matter how close one could zoom into a circle there would always be room to zoom in even more. One could however zoom in beyond any individual
point. One could make the point smaller but one could still zoom in beyond it. So for both the point and the zoom it is infinite regress all the way down

If you accept that a circle has points then it must have sides too because sides are made up of points given that they are longer than them. And if it has
infinite sides [ and it does ] then it logically follows that it must have infinite points as well. But there are more points than there are sides even though
both are infinite as Cantor proved that not all infinities are the same

Two or more points constitute a line although there is no minimum length that a point can be. And two lines
touching at their edges constitute a side. This proves that circles have both infinite points and infinite sides

Two lines touching at their edges can be manipulated so they become a curve. So when that occurs there
is no way of knowing where those edges were and this is another reason why circles have infinite sides as
there are an infinite number of possible places on a curve where that could be. No matter how small it is

This discourse is now approaching your limits.

Backup. A point is not a physical object or entity. A point is a location, not an entity. There is no such thing as “two consecutive points” in the physical universe. There is always an infinity of points between any and all points, between all locations.

And what that means is that it isn’t two points that make up a line, fore there are always an infinity of points between those two. Two points can be used to define a particular line segment by designating the segment’s beginning and end locations. Again, the “points” are not a part of the physical line. They are merely locations on the line.

Between any two points on any line, there is a center location/point on the line that is also the center of a 180 degree angle that includes the first two points. No such center points/locations exist on a circumference or curve. That is the difference between the line and the curve or circle.

There can be no lines (3 points on a 180 degree angle) on a geometric circle because EVERY point must be equidistant from a circle center.

These are points which means they are a physical entity because you can see them
So although a location in spacetime is the definition of a point it is not the only one
This line can be manipulated to make a circle although each point is already a circle

Those are dots, not geometric points. Geometric points have no width, thus could never be seen.

The smallest possible distance between any two geometric points would be no longer than the points are wide … which is zero.

A dot and a point are the same thing but I accept that a geometric point is not physical
So this means that a circle can only be composed of physical points not geometric ones

A point is a Location, not a dot. A dot is used to represent a point. Or sometimes the center of a dot is said to be the point.

I agree wholeheartedly, James. My initial thoughts which you quoted were the result of “zooming in,” which as I said earlier, will introduce a whole suite of problems since zooming in to the point where you can actually see individual points is to skip over an entire infinity (allowing for paradoxes allows for more paradoxes). I assume you read on past that point as I dismissed that whole thought experiment on the grounds of these difficulties, and offered the alternative idea that circles (and lines and curves in general) are made of infinitely divisible segments, not points. Points still play a role but you have to invent them as you need them–say, for example, when you mark a point on the circumference of the circle. There’s an infinite range of possible locations you could mark a point between any two other points–I agree with that–but I think it’s a mistake to say the circle’s circumference is made of points.

That is exactly true. The entire physical universe if made of “segments” or regions (“afflates”), not geometric points.

No matter how small, infinitely small, there is always an infinity of space between any two points. A point has zero width. The smallest space between two points has zero width. So when we add them together, 0+0+0, we get a total distance of zero. Even an infinity of such point pairs will still make only zero distance. A line cannot be built of points. Nor can a volume.

So a circle is made of an infinity of identical curve segments.

Enough of the squabbling and ego-seeking James, this thread is not about you, time to get it back on track: