Aristotle's wheel paradox

[Brant Gaede]

1 hour ago, merjet said:

The nitpicking ankle biter, obnoxious ignoranus, reality-faking (link), lying, incompetent, hysterical jackass wrote: “I haven't used the word "molesting" on this thread.”

So you are saying I did not molest Wikipedia and you are calling Jon a liar. 

Beep-beep. Vroom.

Nitpicking idiot, you have arbitrarily and dishonestly accused me of dishonestly editing, messing with, and polluting the Wikipedia article, and “fucking with it” (link). You might fool yourself into believing your accusations don’t amount to molesting, but you don’t fool me and your dishonesty is in plain sight.

 

 

Nice segue.

--Brant

[Max]

2 hours ago, merjet said:

Duh! Your analogy fails, since you misunderstood the argument.

P1. The smaller circle rolls further horizontally 2*pi*R.

P2. It would roll horizontally 2*pi*r by pure rolling separately.

P3. 2*pi*R > 2*pi*r

P1, P2, and P3 are all about straight lines, i.e. translation vectors. P1 is not about a cycloid the way you botched it. That is your fallacy.

. . . .

You got that backwards. You’re helpless with your crutch snatched away by whitewater.  

. . . .

“But he doesn’t find that, as his cycloid argument is fallacious (see above). It is really worse than you think" (link).

It is not fallacious. The fallacy is your analogy (see above). The alleged “new paradox” is your fabrication.

 

 

This is your original post I reacted to:

 

On 11/23/2018 at 2:06 PM, merjet said:

Wrong. It's obvious that the smaller circle's cycloid is shorter than the outer circle's cycloid, and longer than the center's path. I conclude that the length of the cycloid path of every point on the smaller circle is identical. Since the center's path is 2*pi*R, every cycloid of the smaller circle is greater than 2*pi*r, i.e. its circumference. Therefore, the smaller circle rolls farther than it would by pure rolling. 

That's made with no mention of any slipping or skidding on some fantasized horizontal line tangent to the smaller circle.  

Max: "There is nothing to understand" (link).

You missed the boat.

 

 

Observe the strange logic: 

1. Since the center's path is 2*pi*R, every cycloid of the smaller circle is greater than 2*pi*r, i.e. its circumference. 

Every cycloid of a circle with radius r is greater than 2*pi*r (unless in the degenerate case when you force the circle to remain in the same place, then the cycloid is identical with the circle itself), this has nothing to do with the fact that the center's path is 2*pi*R.

2. Therefore, the smaller circle rolls farther than it would by pure rolling. 

So the fact that every cycloid of the smaller circle is greater than 2*pi*r implies that it rolls farther than it would by pure rolling? That is what you write. Now you claim that you meant that the fact that the circumference of the small circle is smaller than that of the large circle implies that the small circle rolls farther than it would by pure rolling. But that is not what you wrote! Ironically you write now that it is "not about a cycloid", but you are yourself continuously talking about cycloids in your "argument". So you admit now that these cycloids have nothing to do with solving Aristotle's paradox! 

[Max]

1 hour ago, merjet said:

Asinine.

P.S. https://en.wikipedia.org/wiki/Chariot#Greece

Ever heard of the concept "irony"?

Didn't you realize that I was just paraphrasing a post of your comrade Anthony? 

Tsk tsk!

[Max]

4 hours ago, merjet said:

 

Also, here and here.

Heh. A dogmatist – and like a propaganda agent from Nineteen Eighty-Four’s Ministry of Truth -- chants his mantra again, while blanking out reality as follows.

- It is Aristotle’s wheel paradox, not Aristotle’s wheels paradox.

Sez who? Aristotle himself?

 

4 hours ago, merjet said:

- The ‘Wrong problem entirely’ section of the Wikipedia Talk page

And who wrote that section?

 

4 hours ago, merjet said:

- The problem as posed for the first time in Mechanica is about two circles, not two wheels.

But talking about Aristotle's wheel paradox is no problem? It is allowed translate one circle as a wheel, but not the second circle? Sez who?

 

4 hours ago, merjet said:

Also, the journal article ‘Aristotle's Wheel: Notes on the History of a Paradox’ by Israel E. Drabkin, which is referenced 6 times (!) in the Wikipedia Article includes the following:  “For though the smaller circle traverses a distance equal to that traversed by the larger, it does not keep pace with the larger by sliding over the tangent, if by ‘sliding’ we mean that a point on the circumference is at any time in contact with a finite segment of the tangent” [my bold].

That is what happens with a real wheel. In the idealized continuum case we should take the limit → 0 of a segment of the circle touching a larger segment of the tangent. But why should we accept the authority of Drabkin? I think you should add this link to the Wikipedia page: https://www.humanities.mcmaster.ca/~rarthur/articles/aristotles-wheelfinal.pdf for a good discussion of the paradox and Drabkin's ideas.

[anthony]

17 hours ago, Jonathan said:

Heh. So, now your theory is that the smaller wheel starts our faster than the larger wheel,  but then slows down somewhere during the middle of the trip, and then speeds up again right at the end? And therefore the relative accelerations and declarations of both wheels have to be averaged in order to arrive at the truth? The wheels know where the start and finish lines are, and choose to behave differently when near them? Your little theory is that the image that I posted, of the smaller wheel's point covering more territory faster than the larger wheel's point, would not be true of any paired points on the large and small wheels during any and every moment of the cycle, but it is only true at the start and finish lines?

Buffoon.

J

The beginning and the end indicate conflicting actions, not clear so not relevant.

If you had visuo-spatial recognition

1. you'd see what is quite clear, the inner wheel turning relatively slower as they rotate, in mid-cycle particularly.

2. you'd have inferred from experience that something moving a further distance than something else ~ in the same time ~ MUST have moved faster.*

3. finally, as the last resort: measure and calculate d/t = v for each circle.

Two boys A and B have a race. B's starting position is ahead of A's. They start at the same instant and they reach the finish together in a dead heat. Who ran faster?

 

* corollary of that: something that moves faster than something else will cover a distance sooner.

[Jon Letendre]

“not clear so not relevant.”

So, when Tony can’t make the what why how clear in his mind, his next following thought is that it is therefore not relevant.

[Peter]

In the meantime, the Mars rover's wheels are working. A close encounter of the third kind is expected soon. If you look at some of the images from the current and past rovers you can see things. Strange things. Bones. Faces staring back at you. 

[Jonathan]

1 hour ago, anthony said:

1. you'd see what is quite clear, the inner wheel turning relatively slower as they rotate, in mid-cycle particularly.

 

The wheels don't change their speed relative to each other. They are attached to one another. Locked. Fixed.

You're so easily fooled.

 

1 hour ago, anthony said:

2. you'd have inferred from experience that something moving a further distance than something else ~ in the same time ~ MUST have moved faster.*

Hmmm. What would that be called? What's the word for what happens when a wheel rolls farther than it's circumference in one rotation? It's on the tip of my tongue. I think that maybe it starts with an "s." Hmmm.

 

1 hour ago, anthony said:

Two boys A and B have a race. B's starting position is ahead of A's. They start at the same instant and they reach the finish together in a dead heat. Who ran faster?

That's not what's happening here.

The paired points on the two wheels start at the same starting line exactly at the same time, and end, exactly at the same time, at the same finish line. Every point on the small wheel contacts the surface beneath the small wheel precisely at the same moment that the corresponding point on the large wheel contacts the surface beneath the large wheel.

There's no change of rotational speed of either of the wheels in relation to each other.

You're still not properly envisioning what's happening. That's because your mind can't wrap itself around all of the entities involved while they're in motion. Too complex for you.

It's kind of like "crow epistemology," only Tony & Merlin Epistemology.

J

[Jonathan]

7 minutes ago, Peter said:

In the meantime, the Mars rover's wheels are working. A close encounter of the third kind is expected soon. If you look at some of the images from the current and past rovers you can see things. Strange things. Bones. Faces staring back at you. 

You can see those things in clouds, too. It's proof that Martians are making our clouds into familiar shapes.

J

[anthony]

52 minutes ago, Jon Letendre said:

“not clear so not relevant.”

So, when Tony can’t make the what why how clear in his mind, his next following thought is that it is therefore not relevant.

Scraping the barrel for arguments, looks like.  I conceded a minor point about clarity and called it even-stevens - since while it is visible that the end of the cycle obviously shows one point on one circle in front -- at the start it looks behind. This last, the only one that J referred to, tellingly. A bad idea to concede anything with all this dishonest nit-picking going on.

 

[Jonathan]

5 minutes ago, anthony said:

Scraping the barrel for arguments, looks like.  I conceded a minor point about clarity and called it even-stevens - since while it is visible that the end of the cycle obviously shows one point on one circle in front -- at the start it looks behind. This last, the only one that J referred to, tellingly. A bad idea to concede anything with all this dishonest nit-picking going on.

 

No, you didn't successfully concede anything, but actually bungled everything further than you already had.

You're lost and confused.

J

[anthony]

38 minutes ago, Jonathan said:

The wheels don't change their speed relative to each other. They are attached to one another. Locked. Fixed.

You're so easily fooled.

 

Hmmm. What would that be called? What's the word for what happens when a wheel rolls farther than it's circumference in one rotation? It's on the tip of my tongue. I think that maybe it starts with an "s." Hmmm.

 

That's not what's happening here.

The paired points on the two wheels start at the same starting line exactly at the same time, and end, exactly at the same time, at the same finish line. Every point on the small wheel contacts the surface beneath the small wheel precisely at the same moment that the corresponding point on the large wheel contacts the surface beneath the large wheel.

There's no change of rotational speed of either of the wheels in relation to each other.

You're still not properly envisioning what's happening. That's because your mind can't wrap itself around all of the entities involved while they're in motion. Too complex for you.

It's kind of like "crow epistemology," only Tony & Merlin Epistemology.

J

"Locked. Fixed". And that's the end of the story? Geez, if you can't make an inference, at least apply some math.

d/t = v.

Think - *rotational* velocity.

I'll simplify. Two boys enter a race. The one, A, will run on an outer, circular track - the other, B, runs on an inner circular track. A start/finish line bisects both tracks. They start together, and after a lap, finish exactly together.

Q 1. Did they run at an identical speed?  Q2. If not, who was faster?

(Q3. Did either 'skid and slip''?) Haha.

[Jonathan]

26 minutes ago, anthony said:

"Locked. Fixed". And that's the end of the story? Geez, if you can't make an inference, at least apply some math.

d/t = v.

Think - *rotational* velocity.

I'll simplify. Two boys enter a race. The one, A, will run on an outer, circular track - the other, B, runs on an inner circular track. A start/finish line bisects both tracks. They start together, and after a lap, finish exactly together.

Q 1. Did they run at an identical speed?  Q2. If not, who was faster?

(Q3. Did either 'skid and slip''?) Haha.

You're having conversations with straw men again.

Read and comprehend what I write before attempting to reply.

Slow down, Tony. Pay attention. Try to focus harder.

J

[Max]

9 hours ago, merjet said:

[The cycloids are “omitted measurements.” Said translations hold for many possible transitions between start and end. Which particular one doesn’t matter.

The first solution considers the before and after positions and the transition. The transitions of points are cycloid shaped. The cycloids are not “omitted measurements.”

Those transition-cycloids may be interesting in themselves, but they don't contribute anything to solving the paradox. Perhaps you're inspired by the cycloids in Mr. Drabkins's book, but these don't give a solution either.

 

9 hours ago, merjet said:

[2] Duh! They say nothing about inner circles slipping either. When they use slipping, it’s about the whole wheel! They don’t use crutches to fake reality either. Unlike you and the two idiotic, hysterical jackasses do!

Sidestepping my point. You were berating me for not making a distinction between slipping and skidding. I pointed out that the authors of the link you recommended don't distinguish between slipping and skidding either, that they used slipping, positive or negative, for both cases. And the point about using scare quotes: the small wheel/circle is also slipping against its support/tangent in the original article, so there is no reason to use scare quotes. That you insist on removing that tangent from your "solution" doesn't make it disappear.

 

[anthony]

1 hour ago, anthony said:

 

I'll simplify. Two boys enter a race. The one, A, will run on an outer, circular track - the other, B, runs on an inner circular track. A start/finish line bisects both tracks. They start together, and after a lap, finish exactly together.

Q 1. Did they run at an identical speed?  Q2. If not, who was faster?

(Q3. Did either 'skid and slip''?) Haha.

 

Then, answer Q1, Q2. When you're ready.

[Max]

3 hours ago, anthony said:

"Locked. Fixed". And that's the end of the story? Geez, if you can't make an inference, at least apply some math.

d/t = v.

Think - *rotational* velocity.

First, you should define your terms accurately. Velocity is a vector, speed a scalar. Rotational speed ω for a point on a circle with radius r is the number of revolutions/time unit. Linear speed is the distance traveled/time unit. For a circular motion, linear speed = tangential speed = rω, proportional to r.

 

Quote

I'll simplify. Two boys enter a race. The one, A, will run on an outer, circular track - the other, B, runs on an inner circular track. A start/finish line bisects both tracks. They start together, and after a lap, finish exactly together.

Q 1. Did they run at an identical speed?  Q2. If not, who was faster?

This is trivial. The rotational speed is for both boys the same, but boy A has a larger tangential speed. But this is not relevant to the problem of Aristotle's rolling wheels. In this case the rotational speed is for both wheels the same, the tangential speed is for the small circle r/R smaller than for the large circle. But the translational speed, defined as the distance traveled by the center of the wheels after one revolution, is the same for both wheels, and equals 2*pi*R if the large wheel rolls without slipping (essential condition). The smaller wheel would travel 2*pi*r if it also rolled without slipping, but that is in contradiction to the fact that it travels a distance of 2*pi*R. The difference must be made up by the small wheel, and that implies that it must be slipping against its tangent. It cannot roll without slipping, like the large wheel. Clear?

 

3 hours ago, anthony said:

(Q3. Did either 'skid and slip''?) Haha.

Silly question. Dit they roll?

[anthony]

13 hours ago, Max said:

First, you should define your terms accurately. Velocity is a vector, speed a scalar. Rotational speed ω for a point on a circle with radius r is the number of revolutions/time unit. Linear speed is the distance traveled/time unit. For a circular motion, linear speed = tangential speed = rω, proportional to r.

 

This is trivial. The rotational speed is for both boys the same, but boy A has a larger tangential speed. But this is not relevant to the problem of Aristotle's rolling wheels. In this case the rotational speed is for both wheels the same, the tangential speed is for the small circle r/R smaller than for the large circle. But the translational speed, defined as the distance traveled by the center of the wheels after one revolution, is the same for both wheels, and equals 2*pi*R if the large wheel rolls without slipping (essential condition). The smaller wheel would travel 2*pi*r if it also rolled without slipping, but that is in contradiction to the fact that it travels a distance of 2*pi*R. The difference must be made up by the small wheel, and that implies that it must be slipping against its tangent. It cannot roll without slipping, like the large wheel. Clear?

 

Silly question. Dit they roll?

"This is trivial". Untrivial, I'd think,, when I have had to argue the self-evident with somebody - that the "tangential" (thanks for the reminder)speeds are dissimilar for different points in a circle. This could be no more than a sidebar to the main 'paradox', but not unimportant.

(If it is not recognized, for only one example, that in a race around a circular track, runners have to start in "staggered" lane positions, one needs to restate the obvious). 

Your mathematics do not convince. They strike me as reverse justification. For I have seen, or can envisage, an archery target rotate-roll, and like anybody, many more kinds of round objects, and I have "more faith" in observation than seeing facts forced to conform to math i.e. to prove "slippage".

I think this is the crux of the matter I ask of everyone: To comply with the "wheel within a wheel" - if one superimposed on any ring in the archery target, a fixed, protruding, (inner)wheel, would there be any change? 

How could there be? A target may be revolved like a circle or wheel, and one can *see* all its components rotate correspondingly. No slip. Why should "circles" and "wheels" act at all differently? What is there so remarkable about a 'track' and 'slippage'  which we don't need with circles -- but have to have to 'correct' wheels? Why do many here separate the practice from the theory?

[Jonathan]

3 hours ago, anthony said:

"This is trivial". Untrivial, I'd think,, when I have had to argue the self-evident with somebody - that the "tangential" (thanks for the reminder)speeds are dissimilar for different points in a circle. This could be no more than a sidebar to the main 'paradox', but not unimportant.

(If it is not recognized, for only one example, that in a race around a circular track, runners have to start in "staggered" lane positions, one needs to restate the obvious). 

Your mathematics do not convince. They strike me as reverse justification. For I have seen, or can envisage, an archery target rotate-roll, and like anybody, many more kinds of round objects, and I have "more faith" in observation than seeing facts forced to conform to math i.e. to prove "slippage".

I think this is the crux of the matter I ask of everyone: To comply with the "wheel within a wheel" - if one superimposed on any ring in the archery target, a fixed, protruding, (inner)wheel, would there be any change? 

How could there be? A target may be revolved like a circle or wheel, and one can *see* all its components rotate correspondingly. No slip. Why should "circles" and "wheels" act at all differently? What is there so remarkable about a 'track' and 'slippage'  which we don't need with circles -- but have to have to 'correct' wheels? Why do many here separate the practice from the theory?

Still doesn't get any of it. But still yapping. Still pretending, posing, preening and preaching.

Heh.

J

[Peter]

Jonathan wrote: You can see those things in clouds, too. It's proof that Martians are making our clouds into familiar shapes. end quote

Whoa. We are having sustained winds of 10 to 30 MPH here.

The Martians must be laughing their tentacles off at our “supposed but dumb” representations of what they look like, Jonathan. Every time I see a Martian landscape, my mind interprets it as an arid region on earth. And I still think any landings on Mars should be near the polar cap where fairly abundant water exists under the surface. Of course that is where there may be remnants of Martian life which is a problem for cross contamination.     

Remember “Galaxy Quest” with Tim Allen, Sigourney Weaver, Alan Rickman, and Tony Shalhoub scifi movie where aliens thought our “fiction” was real? Peter

From IMDb: Cmdr. Peter Quincy Taggart is based on Captain James T. Kirk, Dr. Lazarus on Spock, Tawny Madison on both Lieutenant Uhura and Deanna Troi, Tech Sergeant Chen on Chief Engineer Scott, Lieutenant Laredo on both Ensign Chekov and Wesley Crusher, and Guy, aka Crewman Number 6, is of the infamous "red shirts", one-off characters who would go down to a planet with the officers and be quickly killed by aliens

Pure Imagination sung by Gene Wilder

Come with me and we'll be
In a world of pure imagination
Take a look and you'll see
Into your imagination
We'll begin with a spin
Traveling in a world of my creation
Look and see
We'll defy explanation

[Chorus]
If you want to view paradise
Simply look around and view it
Anything you want to, do it
Want to change the world?
There's nothing to it . . . . Songwriters: ANTHONY NEWLEY, LESLIE BRICUSSE

[Jonathan]

8 minutes ago, Peter said:

Remember “Galaxy Quest” with Tim Allen, Sigourney Weaver, Alan Rickman...

I love Galaxy Quest.

"By Grabthar's Hammer, by the Sons of Warvan, you shall be avenged."

J

[anthony]

On 11/27/2018 at 1:58 AM, Michael Stuart Kelly said:

  

That means, according to my own standards, there's nothing to discuss. I'll discuss this issue with people who want to discuss it. I actually learned something I didn't know before by doing precisely that on this very thread. I worked through an idea with a little help from those who knew more than I did and now I have more knowledge than before. (Yay me.  And thank you to them.  )

You have communicated--and communicated well--that you have no need for this knowledge and no wish for it. What's more, you don't even believe it's knowledge and think those who do are dummies. OK. I am perfectly fine with that decision.

I only made a long post before to you because I thought you were interested for real in understanding this issue and, having recently worked my own way through it, thought I could shed a little light on where the blind spots were. After all, I experienced them myself right here on this thread before checking my premises and you have the same ones I had. But so far, you only show interest in repeating what you do know of the wheel paradox issue and only doing that. Result: waste of time for everyone. My mistake.

We don't have to have the same wants and needs and we certainly don't have to agree on everything to be friends.

 

 

Michael

3

Michael,  My repetitious "You're wrong" has been nothing like what I've been getting, personal slurs in with that. At this childish level, I'm only giving some of it back to the loudmouths. I'd find it preferable not to have either. Then, not myself only, but the arguers against Objectivism (misunderstood, misrepresented, or evaded, as always) deserve a little of their medicine in return -- by pointing out *their* thought methodologies.

I put a lot of effort into understanding this topic ("you have no need for this knowledge and no wish for it..."- so that's flat wrong). The "knowledge" which is important here is the approach, the way of thinking, not new factoids about circles and wheels, for god's sake. I have accumulated a lifetime of those as referents. I have looked at the topic inside out and objectively. I have not rejected experimentation, math, nor anything, I simply stress, repeatedly, that they can't precede and replace identity and reason.

This 'paradox' isn't exactly rocket science. A wheel behaves the way it does, it's perfectly unparadoxical. As with everything, understanding grows more complex, hierarchically, as it delves further. Where this subject is absorbing is in the basic errors of non-objective thinking and that has lengthened the thread with confusions. Much of what's heard here, is forcing one's preconceptions onto what a wheel is *supposed to do*, not what one sees/ knows it is. Some deep-rooted premises have needed reviewing and as we all know, that can be painful.

I've heard a few times that identity has no place here.  

This, in a time when identity, reason and the mind are under heavy attacks all over the place, needs a strong response. Starting with something as simple as the wheel. If there is no 'one reality' which each can work to understand and conceptualise, mankind will dive deeper into an epistemological relativism and skepticism. "What you think is cool with me, bro, everybody is right. Who can know anything for sure, anyhow?" That 'thinking' by those people makes them ripe for total mind control by the lusters for power (No worries, we'll tell you what to think and do...") and please don't anyone believe the good, thinking people will not go down with them.

 

[Peter]

The WWWeb makes the truth and the lies available to everyone. Charitable organizations gave away a million phones to people around the world and the next give away may include more advanced amalgamations of phones and computers. The younger people I know already have them. Surprisingly the bigger phones seem more usable than the tiny ones, which reverses a trend. 

[Jon Letendre]

Wow, Tony has really had it with those arguers against Objectivism.

Filthy mystics. Fucking Kantians. 

[Brant Gaede]

21 minutes ago, anthony said:

Michael,  My repetitious "You're wrong" has been nothing like what I've been getting, personal slurs in with that. At this childish level, I'm only giving some of it back to the loudmouths. I'd find it preferable not to have either. Then, not myself only, but the arguers against Objectivism (misunderstood, misrepresented, or evaded, as always) deserve a little of their medicine in return -- by pointing out *their* thought methodologies.

I put a lot of effort into understanding this topic ("you have no need for this knowledge and no wish for it..."- so that's flat wrong). The "knowledge" is the approach, the way of thinking, not new factoids about circles and wheels, for god's sake. I have accumulated a lifetime of those as referents. I have looked at the topic inside out and objectively. I have not rejected experimentation, math, nor anything, I simply stress, repeatedly, that they can't replace identity and reason.

This 'paradox' isn't exactly rocket science. A wheel behaves the way it does, it's perfectly unparadoxical. Where this subject is absorbing is in the basic error of non-objective thinking and that has lengthened the thread. Much of what's heard here, is forcing one's preconceptions onto what a wheel is *supposed to do*, not what one sees/ knows it is. Some deep-rooted premises have needed reviewing and as we all know, that can be painful.

I've heard a few times that identity has no place here.  

This, in a time when identity, reason and the mind are under heavy attacks all over the place, needs a strong response. Starting with something as simple as the wheel. If there is no 'one reality' which each can work to understand and conceptualise, mankind will further dive into an epistemological relativism and skepticism. "What you think, is cool with me, bro, everybody is right. Who can know anything for sure, anyhow?" That 'thinking' by those people makes them ripe for total mind control by the lusters for power (No worry, we'll tell you what to think and do...") and please don't anyone believe the good, thinking people will not go down with them.

 

The paradox is easy to understand, Tony, but not by you, so your great effort is, well, understandable.

Unfortunately you are defensively stuck behind the redoubt of denatured Objectivist-speak. It's just not working. A legitimate complaint about name calling does not in turn address the paradox.

--Brant

[Max]

 

6 hours ago, anthony said:

"This is trivial". Untrivial, I'd think,, when I have had to argue the self-evident with somebody - that the "tangential" (thanks for the reminder)speeds are dissimilar for different points in a circle. This could be no more than a sidebar to the main 'paradox', but not unimportant.

But the tangential speed for different points on a circle is the same. It is the speed in the rest frame of the circle, no translation. Of course it is different for points on circles with different radius, perhaps that's what you mean.

6 hours ago, anthony said:

(If it is not recognized, for only one example, that in a race around a circular track, runners have to start in "staggered" lane positions, one needs to restate the obvious). 

Your mathematics do not convince. They strike me as reverse justification. For I have seen, or can envisage, an archery target rotate-roll, and like anybody, many more kinds of round objects, and I have "more faith" in observation than seeing facts forced to conform to math i.e. to prove "slippage".

The original paradox was stated in terms of rollig circles. Any problem with circles is a geometrical/mathematical problem, so I don't see why a mathematical treatment of the paradox would not be the ideal method to solve it. Those circles are tracing out their circumference, a corresponing physical object would be a wheel, rolling without slipping it is the equivalent of a circle tracing out its circumference. Any objections so far? Now a wheel is a very good object in this case, as wheels are meant to roll without slipping, and a wheel concentric in a wheel (just as a circle within a circle in the original description) is easily realised (flange, hub), so a practical test of Aristotle's paradox is fairly easy to realize.

Now about your archery target: perhaps it isn't difficult to rotate it, but that isn't yet rolling. For that you'd have to accurately cut out the target at the outer circle and roll it over the ground or some other support. But then you still haven't one of those smaller circles rolling. Rotating, yes. But rolling needs a support and that circle has to be raised from the rest of the target to allow contact with that support. Now I seriously doubt that you've done that. Probably you just imagined that doing, but that is not good, reality-based evidence! Especially as you apparently already have great difficulty in observing the slipping in the animations and videos that we've seen here, and where many people clearly see the slippage. Some objects are just much better to visualize some effect than other object. The iris and pupil of the eye for also two concentric circles, but they are not well suited for a demonstration of Aristotles paradox. How would you roll an iris and a pupil? Yes in you imagination, but then you'd better concentrate on the mathematical solution. Experiment and mathematical analysis show definitively that slippage occurs on the smaller wheel, if the large wheel rolls without slipping. It that is not basing it on observing reality...

 

6 hours ago, anthony said:

I think this is the crux of the matter I ask of everyone: To comply with the "wheel within a wheel" - if one superimposed on any ring in the archery target, a fixed, protruding, (inner)wheel, would there be any change? 

How could there be? A target may be revolved like a circle or wheel, and one can *see* all its components rotate correspondingly. No slip. Why should "circles" and "wheels" act at all differently? What is there so remarkable about a 'track' and 'slippage'  which we don't need with circles -- but have to have to 'correct' wheels? Why do many here separate the practice from the theory?

But you do need them with circles, those are the equivalents of the tangents that form an essential part of the original problem. You may take them away, but then you take the problem also away. Child and bathwater!