Aristotle's wheel paradox

[Peter]

11 minutes ago, Jon Letendre said:

State the paradox, Tony. Include at least one illustration.

Can a person "start a paradox?" 

[Jonathan]

26 minutes ago, Jon Letendre said:

State the paradox, Tony. Include at least one illustration.

How many times have we asked Tony and Merlin such questions?

There's definitely been a lot of dodging of substance on their parts.

The question that Jon asks above really is something that I'd like to hear answered by the deficient duo, especially since Merlin messed with the Wikipedia page.

J

[Peter]

"Cuz they want to believe that they do." ? That's no excuse, by golly. Leah Remini has said Tom Cruise "personally punished" wavering Scientologists. Well at least those Schitzoids keep it "in house" unlike that frightful species Muhammodinis Triceratops.  

[Jon Letendre]

21 minutes ago, Jon Letendre said:

State the paradox, Tony. Include at least one illustration.

Don’t go quiet, Tony. State the paradox.

We have a wheel rolling and two dotted lines that indicate road length traversed.

Whats paradoxical about it?

[Jonathan]

18 minutes ago, Jon Letendre said:

Don’t go quiet, Tony. State the paradox.

We have a wheel rolling and two dotted lines that indicate road length traversed.

Whats paradoxical about it?

...And...he's gone.

Question successfully dodged.

J

[Ellen Stuttle]

1 hour ago, Jon Letendre said:

Seriously. Just read today’s contributions. That’s what Tony still thinks we mean.

Proceed with him, go ahead and try, [...].

No thanks.  Much else to do besides spinning cognitive wheels.

I did think of a really simple visual illustration of slippage on a track.

Place a string of equally-spaced vari-colored dots along both the outer wheel's and the inner wheel's tracks and wrap an identical string around each of the wheels.  Then roll.  It would soon be visually obvious that, although the dots continue to line up color-by-color between the larger wheel and its track, they start diverging between the smaller wheel and its track.

But since Tony doesn't even understand the relevance of the tracks, what use?

Ellen

[Jonathan]

Here, once again, is what the Wikipedia page had on the "paradox" prior to Merlin's fucking with it:

Aristotle's wheel paradox is a paradox appearing in the Greek work Mechanica traditionally attributed to Aristotle.[1] There are two wheels, one within the other, whose rims take the shape of two circles with different diameters. The wheels roll without slipping for a full revolution. The paths traced by the bottoms of the wheels are straight lines, which are apparently the wheels' circumferences. But the two lines have the same length, so the wheels must have the same circumference, contradicting the assumption that they have different sizes: a paradox.

 

[Jon Letendre]

He’ll be back tomorrow confidently spewing his doctrinaire 

22 minutes ago, Jonathan said:

...And...he's gone.

Question successfully dodged.

J

He’ll be back tomorrow confidently spewing the same nonsense he spewed today, having read nothing and  considered nothing we’ve said.

He’s getting nowhere and doesn’t want to try. Our efforts are totally wasted. He’s not interested.

[Jon Letendre]

I don’t know Wiki.

When someone sees how Merlin molested history and geometry there and tries to repair it, will we get to see the back and forth as Merlin defends keeping it as is? Will we get to see? That’s going to be a must-see.

[Max]

5 hours ago, anthony said:

Math formulations (of wheel rotation, here) were *derived from* the identity and causal actions of circle-cum-wheel, they can hardly be used *to prove* any controversial aspects - assuming one thinks circular reasoning is invalid. 

In the 17th century he would have told Newton that Math formulations (of planetary orbits around the Sun) were *derived from* the identity and causal actions of sun-cum-planets, they can hardly be used *to prove* any controversial aspects - assuming one thinks circular reasoning is invalid, Newton's calculations like the derivation of Keplers laws was therefore invalid. Circular reasoning! Or perhaps elliptical reasoning?

How does he think that the "identity" of something is determined? By divine relevation? Or by Peikoff speaking ex cathedra?

It's totally hopeless, a rational argument with him is impossible.

[Jon Letendre]

1 hour ago, Max said:

In the 17th century he would have told Newton that Math formulations (of planetary orbits around the Sun) were *derived from* the identity and causal actions of sun-cum-planets, they can hardly be used *to prove* any controversial aspects - assuming one thinks circular reasoning is invalid, Newton's calculations like the derivation of Keplers laws was therefore invalid. Circular reasoning! Or perhaps elliptical reasoning?

How does he think that the "identity" of something is determined? By divine relevation? Or by Peikoff speaking ex cathedra?

It's totally hopeless, a rational argument with him is impossible.

You appear to be correct. A discussion with him is impossible.

Its like playing chess with a pigeon. You can’t. The pigeon is going to knock over all the pieces and strut around like it won.

[Peter]

2 hours ago, Jon Letendre said:

I don’t know Wiki.

When someone sees how Merlin molested history and geometry there and tries to repair it, will we get to see the back and forth as Merlin defends keeping it as is? Will we get to see? That’s going to be a must-see.

Molesting history was one of the early criticisms of Wikipedia. And incontinence. Let's lighten up for the holidays. Who's doing well? The Food Lion was packed at 3:30 today, Wednesday.

 

 

[Jon Letendre]

2 minutes ago, Peter said:

Molesting history was one of the early criticisms of Wikipedia. And incontinence. Let's lighten up for the holidays. Who's doing well? The Food Lion was packed at 3:30 today, Wednesday.

 

 

Do you see the paradox yet?

[Brant Gaede]

On 11/20/2018 at 9:38 AM, Michael Stuart Kelly said:

Brant,

Tell that to quantum physics.

 

Michael

I don't think quantum physics understands itself.

Not that I know what I'm talking about.

--Brant

[Michael Stuart Kelly]

I found a great solution to the paradox yesterday--at least from an optical illusion perspective--and have not had time to write about it. I found it in my sleep right before waking up.

I'll try to present it graphically a little later. Ah, hell. Let me present the same GIF as before.

Where I got this GIF from is an egghead site called Wolfram Mathworld (see here).

They said:

Quote

A paradox mentioned in the Greek work Mechanica, dubiously attributed to Aristotle. Consider the above diagram depicting a wheel consisting of two concentric circles of different diameters (a wheel within a wheel). There is a 1:1 correspondence of points on the large circle with points on the small circle, so the wheel should travel the same distance regardless of whether it is rolled from left to right on the top straight line or on the bottom one. this seems to imply that the two circumferences of different sized circles are equal, which is impossible.

The fallacy lies in the assumption that a one-to-one correspondence of points means that two curves must have the same length. In fact, the cardinalities of points in a line segment of any length (or even an infinite line, a plane, a three-dimensional space, or an infinite dimensional Euclidean space) are all the same:  (the cardinality of the continuum), so the points of any of these can be put in a one-to-one correspondence with those of any other.

And of course I didn't understand jack when I first read it.  But then something in the second paragraph stuck in the back of my mind: "one-to-one correspondence of points."

Later my eureka moment happened while coming out of dozing.

The line going through the small circle and ending on the rim of the large circle has nothing to do with both circumferences. Nothing at all. It exists only to represent the circumference of only one of the circles. And, it doesn't matter which. That will depend on which circumference corresponds to the straight line in reality (the stretch of road, so to speak). The line intersecting the rim of the other circle is a projection inward or outward, sort of like a perspective view in a painting.

If that "perspective view" aspect is eliminated, and the same stretch of road remains constant, one wheel has to slip if the other does not.  

I'm not a graphic artist, so I am loathe to draw a perspective view of a person near the viewer and a same size person farther away, but to get the effect of distance, the nearer person will have to be drawn larger than the one farther away. However, if we put a drawing of a tree next to the near person (the larger-drawn person) and a same size tree next to the farther away person (the smaller drawn person), we either destroy the effect of distance, or destroy the effect of the two people being the same size. If we keep one, we destroy the other, it doesn't matter which.

This corresponds to what happens with the slippage of one wheel when the other doesn't slip.. 

The two circles have the same line intersecting their rims and rotating with them, not two different lines. That's because they are connected. If they were separate with each running on their own, they could have two center-to-rim lines. But when there is only one center-to-rim line, that will mean there is only one end-point that counts in relation to the road, not two end-points like it seems (one end point for each circle). 

Note that the same size road line was used for both circles in the diagram, sort of like the same-size tree was for both people. That's why, when there are two roads for real against two wheels for real, and the wheels are connected, there is slippage in one wheel. 

Michael

[Ellen Stuttle]

Guys, re Tony:

Tony is not actually denying the fact that the smaller wheel moves farther laterally in one revolution than the length of its circumference.  He states that fact himself.  It's the label - slippage - for this fact which he goes into outer space conniptions about.

Ellen

[Brant Gaede]

6 hours ago, Michael Stuart Kelly said:

I found a great solution to the paradox yesterday--at least from an optical illusion perspective--and have not had time to write about it. I found it in my sleep right before waking up.

I'll try to present it graphically a little later. Ah, hell. Let me present the same GIF as before.

Where I got this GIF from is an egghead site called Wolfram Mathworld (see here).

They said:

And of course I didn't understand jack when I first read it.  But then something in the second paragraph stuck in the back of my mind: "one-to-one correspondence of points."

Later my eureka moment happened while coming out of dozing.

The line going through the small circle and ending on the rim of the large circle has nothing to do with both circumferences. Nothing at all. It exists only to represent the circumference of only one of the circles. And, it doesn't matter which. That will depend on which circumference corresponds to the straight line in reality (the stretch of road, so to speak). The line intersecting the rim of the other circle is a projection inward or outward, sort of like a perspective view in a painting.

If that "perspective view" aspect is eliminated, and the same stretch of road remains constant, one wheel has to slip if the other does not.  

I'm not a graphic artist, so I am loathe to draw a perspective view of a person near the viewer and a same size person farther away, but to get the effect of distance, the nearer person will have to be drawn larger than the one farther away. However, if we put a drawing of a tree next to the near person (the larger-drawn person) and a same size tree next to the farther away person (the smaller drawn person), we either destroy the effect of distance, or destroy the effect of the two people being the same size. If we keep one, we destroy the other, it doesn't matter which.

This corresponds to what happens with the slippage of one wheel when the other doesn't slip.. 

The two circles have the same line intersecting their rims and rotating with them, not two different lines. That's because they are connected. If they were separate with each running on their own, they could have two center-to-rim lines. But when there is only one center-to-rim line, that will mean there is only one end-point that counts in relation to the road, not two end-points like it seems (one end point for each circle). 

Note that the same size road line was used for both circles in the diagram, sort of like the same-size tree was for both people. That's why, when there are two roads for real against two wheels for real, and the wheels are connected, there is slippage in one wheel. 

Michael

There is slippage because there is slippage. That's the real starting point. The reason is simple and obvious. That's not the real starting point for the paradox which is 100 percent in the head. All the pages on this thread is climbing out of nonsense versus staying  in the nonsense. And that means the nonsense wins for it's invincible ignorance.

--Brant

[Jon Letendre]

4 hours ago, Ellen Stuttle said:

Guys, re Tony:

Tony is not actually denying the fact that the smaller wheel moves farther laterally in one revolution than the length of its circumference.  He states that fact himself.  It's the label - slippage - for this fact which he goes into outer space conniptions about.

Ellen

 

Because he cannot mentally isolate the inner circle’s motions. He can’t keep all the elements in play at once: The rotation, the translation forcing travel in excess of circumference being applied to road, leading directly to slippage. He can’t “make it go” correctly in his head.

He thinks the inner circle’s motions are too difficult to model at home, let alone in the mind. He doesn’t see that the lid and straw accurately model the motions of the wheel and inner circle on the road and drawn road.

 

 

If he was capable of mentally isolating the relevant aspects of motion, he would see the models do indeed depict the motions correctly. Instead he nitpicks aspects that do not matter and throws it all in the trash. Here is one response to the models from Tony:

 

Full marks for your effort, Jon. Of course, this is not rigorous enough for science. Too much flexibility and too little weight in a plastic straw; and any slight imbalance of levels or in the manually applied downward force - is naturally going to cause drag.  A controlled scientific experiment will do the utmost to obviate these variables.

And as I keep repeating, friction, drag, rolling resistance or gravity are not explicitly or implicitly imparted by the paradox diagram.

It is not a mechanical puzzle. It's a thinking exercise about circles/wheels.  

Even so. You've got what, 20-30 times differences of circumferences of straw to that of the lid? That's approx 25: 1 ratio. For them to travel the same distance, by your theory, the straw must drag almost continuously. I don't see that in the video, I see it revolve at least half way, and I had no 'drag and skid' in my practical tests.

[Jon Letendre]

Merlin, Tony:

State the Paradox.

You have a wheel rolling.

There are two dotted lines indicating road length traversed by that wheel in one rotation.

What’s paradoxical about this?

[Michael Stuart Kelly]

3 hours ago, Brant Gaede said:

All the pages on this thread is climbing out of nonsense versus staying  in the nonsense.

Brant,

You're calling it nonsense. I'm calling it an optical illusion in the drawing.

(As a Randian, how dare you say Aristotle seriously entertained nonsense and was befuddled by it! The ghost of Ayn Rand will now haunt you to the end of your days...  )

I'm going back to my original idea of dimensions, except with a change.

Two dimensional drawings can have elements in them to represent three dimensions (like the perspective trick of making one object larger than another). In the diagram of the paradox, there's a third dimension element not represented well. Some people attribute the diagram as a correct representation of two-dimensions of reality, thus arrive at a paradox. And it sure as hell looks like a paradox from that premise.

Other people look at actual reality, they look at real stuff the two-dimensional diagram implies (movement, circumference, stretch of road and all) and correctly notice there has to be slippage in one of the circles for it to work since the circles are attached to each other and cannot move separately.

Notice that I just said they "notice there has to be slippage," not "imagine there has to be slippage." They notice the slippage because the slippage happens and they see it. I mean, everyone does.

So why do these these two different views occur? It's because the diagram is flawed and misleading for what it is supposed to represent, although it looks like it is correct. 

It kinda reminds me of a magic trick where you trick the eye to focus on one hand with a large movement and do the reality move with as little movement as possible in the other. When people focus in on the large movement, then look back to the whole, it looks like a paradox happened. It's magic!  

Humans, like almost all animals, innately track movement with their eyes. In the paradox diagram, when they see the line that intersects the rims of both circles moving in a circle like hands in two different clocks, and moving in a manner that looks like two distinct lines instead of one (one line for each circle), they automatically assume there are two different situations. Only there are not.

One of the lines represents movement around the circumference of one of the circles and that is in perfect alignment with reality. The other line does not represent the circumference of its respective circle, although it looks like it does. It actually represents the circumference of the other circle from a different angle--in other words, the angle is part way through the entire whole line if the larger circle is the anchor circle (so to speak), or the angle is from the end of the whole line and projecting outward just because, or maybe to look pretty or whatever, if the smaller circle is the anchor circle. There are two line components being mixed up--one section of line that represents reality and a section of line with a false attribution. 

Lets look at the "one true surface" perspective. When there is only one real surface, we have one circle with line--metaphorically, one clock with a hand, representing reality while the other clock has a false hand. But they look the same on the diagram. To use Objectivist jargon (Peikoff's in fact), one clock is integrated and the other is misintegrated like a Frankenstein monster where parts are put together and look like a whole person, but they don't make a living being.

Another way of putting it is like this. If the true road is for the larger circle, the whole line covering both circles moving like a clock hand is the correct representation and the smaller line in the smaller circle is merely part of the larger circle's clock-hand line. If the true road is for the smaller circle, the line extending to the larger circle is a projection of the whole line, extra made-up stuff, and does not represent the circumference of the smaller circle, i.e., it does not represent reality. 

Now the "two separate but equal surfaces" perspective. If there are identical true roads (roads in reality, not just depictions of roads) for both wheels, which is what the reality-oriented people build (and who can blame them since that is exactly what the diagram looks like?), one of the circles has to slip since the wheels are attached to each other.

If that's not clear, well, I'm doing the best I can.  It's clear to me, finally.

Simply put, the diagram is misleading. It seems like it represents two whole and separate things (two different circles) when it actually and only represents one whole thing (two connected circles that make up a whole wheel).

I still like the way Jon put it. The diagram does not correctly represent the way the two wheels actually roll.

Michael

[Jon Letendre]

30 minutes ago, Jon Letendre said:

Merlin, Tony:

State the Paradox.

You have a wheel rolling.

There are two dotted lines indicating road length traversed by that wheel in one rotation.

What’s paradoxical about this?

Here is the drawing.

[Jon Letendre]

47 minutes ago, Jon Letendre said:

Merlin, Tony:

State the Paradox.

You have a wheel rolling.

There are two dotted lines indicating road length traversed by that wheel in one rotation.

What’s paradoxical about this?

There are LOTS of dotted lines that show length of road traversed by the wheel in one rotation.

You seem to think there is a paradox here, so tell us what is the paradox?

[Michael Stuart Kelly]

Jon,

A recommendation. If you get SnagIt (not an affiliate link), I have a feeling you are going to go apeshit with it. I mean that in a good manner.  

It's extremely easy to learn and was practically made for taking screenshots and quickly changing them, mashing them up with each other, etc. It also does screencasting video and GIFs. At fifty bucks, I think it's a steal and I'm a cheapskate. (It's the damn Scottish in me.)  

Nothing open-source or paid even comes close unless you are a nerdy geek who dreams of sex in highly complex math formulas or Python computer code or something. God knows, I've tried a lot of image programs for doing screenshots, memes, etc. Of course, you can do images from scratch, too. There's a full set of image tools. You can also do technical drawings easily so long as you don't need the higher capabilities of AutoCad or something like that.

TechSmith, the parent company, has an image hosting site called "screencast.com." You can load images directly to Screencast from within Snagit to get URLs for sharing them online, but the images also automatically stay in the SnagIt library on your hard disk until you delete them for good. So you can later go back and rework them if you want without having to redo everything, or download them, etc.

I find this a lot more useful and reliable than Photobucket, which I can't stand ever since they deleted a bunch of images embedded here on OL. Nowadays, every time I have tried to use Photobucket, I run up against their marketing. They are pure spam in the aggravating way in trying to get your money. 

Anywho, just an idea. 

(How's that for a sales pitch? It just came off the top of my head. Dayaamm... I should have included an affiliate link.  )

Michael

[Max]

1 hour ago, Michael Stuart Kelly said:

So why do these these two different views occur? It's because the diagram is flawed and misleading for what it is supposed to represent, although it looks like it is correct.

The diagram is in so far incorrect, that it doesn't represent a wheel that is rolling without slipping (which was the supposition in the description of the paradox): the distance traveled after one revolution is smaller than the circumference of the large circle. But apart from that, it is a completely valid diagram, it's perfectly possible that both wheels are slipping.

1 hour ago, Michael Stuart Kelly said:

One of the lines represents movement around the circumference of one of the circles and that is in perfect alignment with reality. The other line does not represent the circumference of its respective circle, although it looks like it does. It actually represents the circumference of the other circle from a different angle--in other words, the angle is part way through the entire whole line if the larger circle is the anchor circle (so to speak), or the angle is from the end of the whole line and projecting outward just because, or maybe to look pretty or whatever, if the smaller circle is the anchor circle. There are two line components being mixed up--one section of line that represents reality and a section of line with a false attribution. 

No, that line doesn't represent the movement around the circumference of just one of the circles, it just marks two points on those circumferences, thereby forming a mark for the amount of rotation. You could very well paint such a line on a real wheel, and it would rotate exactly the same way. Those two intersection points rotate completely synchronously. However, a different thing is that at least one of those points is also slipping along its tangent line. You can see that also in slow motion.

 

[Jon Letendre]

Here is a wheel with a small wheel firmly attached...