Aristotle's wheel paradox

[Jonathan]

22 minutes ago, anthony said:

Michael,  My repetitious "You're wrong" has been nothing like what I've been getting, personal slurs in with that.

You haven't been getting "slurs." You've been getting accurate, objective appraisals of your cognitive limitations.

 

23 minutes ago, anthony said:

Then, not myself only, but the arguers against Objectivism (misunderstood, misrepresented, or evaded, as always)...

No one has argued against Objectivism. That's a lie.

 

25 minutes ago, anthony said:

I put a lot of effort into understanding this topic..

Yeah, you're not going to get it no matter how much effort that you put in. You're like a blind man putting effort into seeing.

 

29 minutes ago, anthony said:

I have looked at the topic inside out and objectively.

You've failed. You've erred. And you're not being honest. You haven't looked at the other side's arguments and evidence. You've dodged it all, avoided questions, sidestepped challenges.

 

30 minutes ago, anthony said:

I have not rejected experimentation, math, nor anything, I simply stress, repeatedly, that they can't replace identity and reason.

No one has suggested replacing identity and reason.

You have failed to properly identify reality. In this case, it is too complex for your brain to grasp.

 

28 minutes ago, anthony said:

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

Bullshit. You're lying. You're making up straw men again, and making them say things that no one has said.

 

32 minutes ago, anthony said:

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.

False. We're not forcing preconceptions, but explaining what we see/know to a simpleton who can't see/know despite our slowing everything way down and spoon-feeding it to him bit by tiny bit.

 

34 minutes ago, anthony said:

This, in a time when identity, reason and the mind are under heavy attacks all over the place, needs a strong response.

I agree. So stop attacking reality with your stupidity.

 

35 minutes ago, anthony said:

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.

No, mankind isn't going to dive into epistemological relativism just because you're cognitively incompetent in the realm of visuospatial/mechanical reasoning. I know that you want us to all be doomed, but that's just not reality.

 

37 minutes ago, anthony said:

"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.

 

You're babbling again. Nonsense freak tangent. Imagining all sorts of disconnected consequences to positions that no one has taken.

J

[anthony]

1 hour ago, Max said:

 

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.

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...

 

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!

 I don't know about you, but the archery targets I've seen, are circular. Yes, I've seen one rolled like a wheel. No matter. This is not a mechanical challenge, involving a second track, friction, drag, slippage, etcetera - to make any sense and to teach anything, it must pose a -conceptual- challenge.   

Extract the characteristics of "the wheel", and they may be transposed elsewhere onto many round entities. Get hung up on one wheel, and one gets mired in concretism.

Your "Experiment and mathematical analysis" come into play after.

 

[Jonathan]

1 hour ago, anthony said:

This, in a time when identity, reason and the mind are under heavy attacks all over the place, needs a strong response.

Tony, apply identity and reason to the geometry of this:

You've claimed that it's an illusion, and that anyone can do anything with a video, as if making such declarations is sufficient.

Objectively demonstrate which aspects of it do not conform to reality and to proper geometry. Back up your assertions with objective measurements.

You've been claiming to be objective, and that you practice objectivity here against your opponents who want to destroy it. Well, demonstrate objectivity. It should be very simple. The circles and lines are right there on the screen in reality, easily measured. So do it.

J

P.S. Please note that the animated diagram is not a real wheel. It is circles and lines. It is geometry rather than a physical incarnation such as wheels and surfaces.

[anthony]

51 minutes ago, Brant Gaede said:

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

Don't let yourself be talked into a 'second track' and 'slippage'. That is obfuscating nonsense.

[Jonathan]

3 minutes ago, anthony said:

Don't let yourself be talked into a 'second track'...

Why are you still opposed to the second track? It was included in the original, ancient formulation of the so-called paradox. You're rejecting that reality. Why? Why do you need to alter it?

J

[anthony]

14 minutes ago, Jonathan said:

Why are you still opposed to the second track? It was included in the original, ancient formulation of the so-called paradox. You're rejecting that reality. Why? Why do you need to alter it?

J

You should quote it then, should you not? There was no "track" mentioned in any version of the "paradox", earlier or now.

[And none mentioned in Aristotle's "In Antiquity" passage you quoted on p.48] 

[Jon Letendre]

Jonathan posted it, Tony. Go see, you’ll find it. Very recent.

Make a drawing of what’s described in that ancient text. Use pencil. With a good eraser.

[anthony]

1 hour ago, Jon Letendre said:

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

Filthy mystics. Fucking Kantians. 

The type of skepticism I notice has more the effects of David Hume in it, I reckon. He said empiricism - considering only the 'facts', without one's concept-forming (reason) - must lead one to skepticism. He would know, and that's one insight he was spot-on about. Of course, the Left today are generally the greatest skeptics, not that you seemingly care to know. 

[anthony]

24 minutes ago, anthony said:

You should quote it then, should you not? There was no "track" mentioned in any version of the "paradox", earlier or now.

[And none mentioned in Aristotle's "In Antiquity" passage you quoted on p.48] 

 

[Jonathan]

1 hour ago, anthony said:

You should quote it then, should you not? There was no "track" mentioned in any version of the "paradox", earlier or now.

[And none mentioned in Aristotle's "In Antiquity" passage you quoted on p.48] 

Look again. I've bolded it for you:

 

In antiquity, the wheel problem was described in the Aristotelian Mechanica, as well as in the Mechanica of Hero of Alexandria.^[1] In the former it appears as "Problem 24", where the description of the wheel is given as follows.

For let there be a larger circle ΔZΓ a smaller EHB, and A at the centre of both; let ZI be the line which the greater unrolls on its own, and HK that which the smaller unrolls on its own, equal to ZΛ. When I move the smaller circle, I move the same centre, that is A; let the larger be attached to it. When AB becomes perpendicular to HK, at the same time AΓ becomes perpendicular to ZΛ, so that it will always have completed an equal distance, namely HK for the circumference HB, and ZΛ for ZΓ. If the quarter unrolls an equal distance, it is clear that the whole circle will unroll an equal distance to the whole circle, so that when the line BH comes to K, the circumference ZΓ will be ZΛ, and the whole circle will be unrolled. In the same way, when I move the large circle, fitting the small one to it, their centre being the same, AB will be perpendicular and at right angles simultaneously with AΓ, the latter to ZI, the former to HΘ. So that, when the one will have completed a line equal to HΘ, and the other to ZI, and ZA becomes again perpendicular to ZΛ, and HA to HK, so that they will be as in the beginning at Θ and I.^[2]

The problem is then stated:

Now since there is no stopping of the greater for the smaller so that it [the greater] remains for an interval of time at the same point, and since the smaller does not leap over any point, it is strange that the greater traverses a path equal to that of the smaller, and again that the smaller traverses a path equal to that of the larger. Furthermore, it is remarkable that, though in each case there is only one movement, the center that is moved in one case rolls a great distance and in the other a smaller distance.^[1]

-----

J

[Jonathan]

1 hour ago, Jon Letendre said:

Jonathan posted it, Tony. Go see, you’ll find it. Very recent.

Make a drawing of what’s described in that ancient text. Use pencil. With a good eraser.

He doesn't possess the skills to accurately create such a simple diagram.

J

[Max]

7 minutes ago, Jonathan said:

Look again. I've bolded it for you:

 

In antiquity, the wheel problem was described in the Aristotelian Mechanica, as well as in the Mechanica of Hero of Alexandria.^[1] In the former it appears as "Problem 24", where the description of the wheel is given as follows.

For let there be a larger circle ΔZΓ a smaller EHB, and A at the centre of both; let ZI be the line which the greater unrolls on its own, and HK that which the smaller unrolls on its own, equal to ZΛ. When I move the smaller circle, I move the same centre, that is A; let the larger be attached to it. When AB becomes perpendicular to HK, at the same time AΓ becomes perpendicular to ZΛ, so that it will always have completed an equal distance, namely HK for the circumference HB, and ZΛ for ZΓ. If the quarter unrolls an equal distance, it is clear that the whole circle will unroll an equal distance to the whole circle, so that when the line BH comes to K, the circumference ZΓ will be ZΛ, and the whole circle will be unrolled. In the same way, when I move the large circle, fitting the small one to it, their centre being the same, AB will be perpendicular and at right angles simultaneously with AΓ, the latter to ZI, the former to HΘ. So that, when the one will have completed a line equal to HΘ, and the other to ZI, and ZA becomes again perpendicular to ZΛ, and HA to HK, so that they will be as in the beginning at Θ and I.^[2]

The problem is then stated:

Now since there is no stopping of the greater for the smaller so that it [the greater] remains for an interval of time at the same point, and since the smaller does not leap over any point, it is strange that the greater traverses a path equal to that of the smaller, and again that the smaller traverses a path equal to that of the larger. Furthermore, it is remarkable that, though in each case there is only one movement, the center that is moved in one case rolls a great distance and in the other a smaller distance.^[1]

-----

J

You beat me to it...

That the word "track" isn't mentioned there, does of course not mean that the concept "track" is absent. Unrolling the large circle to the line ZI  means that ZI is the track over which the large circle rolls, and unrolling the smaller circle to the line HK means that HK is the track over which the small circle rolls. So those two tracks are an essential part of the original paradox. Taking those away is destroying the paradox, not solving it. Child and bathwater.

 

[Jonathan]

15 hours ago, Max said:

You beat me to it...

That the word "track" isn't mentioned there, does of course not mean that the concept "track" is absent. Unrolling the large circle to the line ZI  means that ZI is the track over which the large circle rolls, and unrolling the smaller circle to the line HK means that HK is the track over which the small circle rolls. So those two tracks are an essential part of the original paradox. Taking those away is destroying the paradox, not solving it. Child and bathwater.

 

And the word "unrolls" is important in that it means that the circles become the lines as they roll, as depicted in the Wolfram animated diagram that people have been posting.

its an idea that Ellen addressed a few pages back asking people to imagine strings wrapped around the wheels and unfurling when contacting the surfaces on which they roll.

Hmmm. Let's take that a step further. Referring to the animated diagram above, the two wheels are cable spool cylinders, rigidly affixed to one another. Red represents cables, which are unbreakable and do not stretch. The beginning end of each cable is affixed, at the starting point, to the surface on which its wheel will roll. Each cable is wrapped around its wheel exactly once, and the other end of each cable is then attached to its wheel aligned with the starting point. Thus, we have two different lengths of cable -- each is the circumference of the wheel around which it wrapped.

Now, we apply an overwhelming force to push the wheels forward and unfurl the cables as the wheels roll. Given that all items in the scenario are unbreakable, cannot be stretched or otherwise distorted in form, and that the wheels must move due to the overwhelming force, what will happen? What MUST happen, and why?

If marks were to be placed on both wheels, and their movements traced, what paths must they trace?

J

[PDS]

On 9/17/2017 at 3:28 PM, Jonathan said:

Is it even possible for this thread to get any more entertaining?

Perhaps!

What if another [hypothetical regular poster] happens to share Merlin's degree of visual/spatial/mechanical ineptitude, agrees with his nonsensical position on this thread, and would be willing to step up and help to argue Merlin's case?

That would be just freaking amazingly awesome!

J

I just had to come off my lurking bench to note that Jonathan’s rapier/rapist wit is only exceeded by his prescience.  

This is, by far, the most entertaining OL thread I have lurked at in some time.  

Do carry on!

[anthony]

18 hours ago, Jonathan said:

Tony, apply identity and reason to the geometry of this:

You've claimed that it's an illusion, and that anyone can do anything with a video, as if making such declarations is sufficient.

Objectively demonstrate which aspects of it do not conform to reality and to proper geometry. Back up your assertions with objective measurements.

You've been claiming to be objective, and that you practice objectivity here against your opponents who want to destroy it. Well, demonstrate objectivity. It should be very simple. The circles and lines are right there on the screen in reality, easily measured. So do it.

J

P.S. Please note that the animated diagram is not a real wheel. It is circles and lines. It is geometry rather than a physical incarnation such as wheels and surfaces.

I see. It is a straightforward animation of two attached circles, revolving one to one. The question is, do you see?

Are you seeing slippage? In which case, the two will not remain in synch, evidently.

It may be that everyone believes they see "slip" on the inner - when this is fully explained by distinct,"tangential" speed differences: i.e., a point on the outer rim moves further than a corresponding point on the inner rim - in the same time - therefore, the outer circle is moving faster. And the inner 'seems' to drag, relatively.

And then "tangential velocity" comes back into play.

[Jonathan]

14 minutes ago, anthony said:

Are you seeing slippage? In which case, the two will not remain in synch, evidently.

No, that does not logically follow. Not even close. You're still not getting it. You're never going to grasp it.

 

21 minutes ago, anthony said:

It may be that everyone believes they see "slip" on the inner...

So, you're saying that we don't see what we see? We just believe that we see it? And your proof of this is the fact that you don't see it? If you don't see or understand something, then, therefore, no one else does either? No one can see or understand anything that Tony can't, because Tony is the universal limit of human cognition?

 

16 minutes ago, anthony said:

...when this is fully explained by distinct,"tangential" speed differences: i.e., a point on the outer rim moves further than a corresponding point on the inner rim - in the same time - therefore, the outer circle is moving faster. And the inner 'seems' to drag, relatively.

You've gotten it backwards. As usual. The opposite is true. Any point on the smaller wheel travels faster than its corresponding point on the larger wheel.

Look again at the animated diagram. Slow down, try to focus, and pay attention.

Do you notice anything in addition to the circles and lines?

Can you see the yellow and orange segments? How about the letters identifying point on the circles and lines? Can you see them?

See them now? Okay, now watch point E in comparison to point A (actually, first spit out your gum -- we don't want you multitasking while trying to do this). Okay. Which point, E or A, is traveling faster? Which is covering more ground/space in the same amount of time?

See? It's pretty easy if you look and keep your attention on it.

This is where your little theory goes to hell. Your theory is about wheels rotating on a point, like a ferris wheel. That doesn't work here because these wheels are ROLLING!!!! Different stuff happens when something is rolling versus when it is only rotating on a stationary point. See?

No, of course you don't see.

J

[anthony]

18 hours ago, Max said:

You beat me to it...

That the word "track" isn't mentioned there, does of course not mean that the concept "track" is absent. Unrolling the large circle to the line ZI  means that ZI is the track over which the large circle rolls, and unrolling the smaller circle to the line HK means that HK is the track over which the small circle rolls. So those two tracks are an essential part of the original paradox. Taking those away is destroying the paradox, not solving it. Child and bathwater.

 

"...does not mean that the *concept* track is absent". (Logically, it doesn't mean that the concept is present, either).

But, good that you didn't pretend that "track" was explicitly mentioned.

All that we read here is of a "line" - i.e. a possible representation of a track, more like an imaginary "path".

But you all need to have a tangible, physical "track" to fulfil the "slippage solution" - so, track it must be...You are "destroying" reality, not solving the paradox.

"So those two tracks are an essential part of the original paradox".

NO. You are including your conclusion in validating your conclusion. Two tracks it must be, so that's the only thing that makes sense, which is illogical.

Simply, again:

A wheel of circumference x rotates once, moving distance x. An fixed inner wheel of circumference y rotates once -- but moves also distance x.

How can it be!! For reasons I've repeated.

Slippage must be introduced!!

Aristotle: "The problem is then stated"-

"...and since the smaller does not leap over any point, it is strange [...] that the smaller traverses a path equal to the larger".

"Strange", and perhaps counter-intuitive, but that is indeed what happens. Aristotle merely reported his observation, presumedly;  he explained what he saw and didn't attempt to solve it.

 

[Jon Letendre]

One cannot play chess with a pigeon.

The pigeon will knock over the pieces and strut around like it won.

[Jonathan]

5 minutes ago, anthony said:

Simply, again:

A wheel of circumference x rotates once, moving distance x. An fixed inner wheel of circumference y rotates once -- but moves also distance x.

 

That's a lie. That's NOT what the setup states. Rather, it states that the circles UNROLL. It does not state that they "rotate" while "moving" a "distance." Your false formulation separates the rotation from the translation, as if their relationship is not united and defined by ROLLING.

Like Merlin, you're lying about the wording and meaning of the original problem, and substituting your own stupid interpretation because you can't grasp the original problem.

J

[anthony]

56 minutes ago, Jonathan said:

No, that does not logically follow. Not even close. You're still not getting it. You're never going to grasp it.

 

So, you're saying that we don't see what we see? We just believe that we see it? And your proof of this is the fact that you don't see it? If you don't see or understand something, then, therefore, no one else does either? No one can see or understand anything that Tony can't, because Tony is the universal limit of human cognition?

 

You've gotten it backwards. As usual. The opposite is true. Any point on the smaller wheel travels faster than its corresponding point on the larger wheel.

Look again at the animated diagram. Slow down, try to focus, and pay attention.

Do you notice anything in addition to the circles and lines?

Can you see the yellow and orange segments? How about the letters identifying point on the circles and lines? Can you see them?

See them now? Okay, now watch point E in comparison to point A (actually, first spit out your gum -- we don't want you multitasking while trying to do this). Okay. Which point, E or A, is traveling faster? Which is covering more ground/space in the same amount of time?

See? It's pretty easy if you look and keep your attention on it.

This is where your little theory goes to hell. Your theory is about wheels rotating on a point, like a ferris wheel. That doesn't work here because these wheels are ROLLING!!!! Different stuff happens when something is rolling versus when it is only rotating on a stationary point. See?

No, of course you don't see.

J

You don't seem to make the translocation from static wheels to rolling wheels. "Any point on the smaller wheel travels faster..."

Not in this universe. Smaller = slower. (in this context).

You can find no principle connecting athletic tracks, archery targets, wine bottles - and I am sure, orbiting planets - if you haven't conceptualized the common denominators of circle/wheel..

The tangential speed of two circles or wheels or planets, or sprinters, must always be greater on the outer circumference. IF - they stay in alignment.

Spinning or rolling, no diff. The outer rim has farther to travel in one revolution and the equivalent time. Get it?

 

 

[anthony]

13 minutes ago, Jonathan said:

That's a lie. That's NOT what the setup states. Rather, it states that the circles UNROLL. It does not state that they "rotate" while "moving" a "distance." Your false formulation separates the rotation from the translation, as if their relationship is not united and defined by ROLLING.

Like Merlin, you're lying about the wording and meaning of the original problem, and substituting your own stupid interpretation because you can't grasp the original problem.

J

Getting so desperate? I stated the paradox in MY words - very clearly. 

Next, I made clear Aristotle's quote - and basically, he said the same. Quote my whole post in future.

 I've warned you before about accusations of lying.

[Jon Letendre]

A pigeon cannot lie.

[Jonathan]

5 hours ago, Jonathan said:

And the word "unrolls" is important in that it means that the circles become the lines as they roll, as depicted in the Wolfram animated diagram that people have been posting.

its an idea that Ellen addressed a few pages back asking people to imagine strings wrapped around the wheels and unfurling when contacting the surfaces on which they roll.

Hmmm. Let's take that a step further. Referring to the animated diagram above, the two wheels are cable spool cylinders, rigidly affixed to one another. Red represents cables, which are unbreakable and do not stretch. The beginning end of each cable is affixed, at the starting point, to the surface on which its wheel will roll. Each cable is wrapped around its wheel exactly once, and the other end of each cable is then attached to its wheel aligned with the starting point. Thus, we have two different lengths of cable -- each is the circumference of the wheel around which it wrapped.

Now, we apply an overwhelming force to push the wheels forward and unfurl the cables as the wheels roll. Given that all items in the scenario are unbreakable, cannot be stretched or otherwise distorted in form, and that the wheels must move due to the overwhelming force, what will happen? What MUST happen, and why?

If marks were to be placed on both wheels, and their movements traced, what paths must they trace?

J

Here's a diagram of the above cable/spool concept:

What must happen when the wheels roll in the direction of the arrow?

J

[Jonathan]

27 minutes ago, anthony said:

I stated the paradox in MY words - very clearly. 

No, you didn't. You misstated the "paradox" in your own words.

 

28 minutes ago, anthony said:

Next, I made clear Aristotle's quote - and basically, he said the same.

No, he did not basically say the same thing. You don't grasp what he said. And you don't want to grasp it. You want to blank it out and alter it.

 

29 minutes ago, anthony said:

I've warned you before about accusations of lying.

If you stop lying, I won't accuse you of it.

J

[Jon Letendre]

On 9/15/2017 at 7:39 AM, Jon Letendre said:

I love this stuff.

I have not read any responses, but I did see someone wrote "it's a mental paradox, not a physical one."

The setup description is a lie, red and blue lines are not same length,  the red lines should be dashed to represent that they cannot be continuous like the blue lines.

The larger wheel pulls the smaller one along, so that the smaller skips across its "ground."

If the red line is string, it will snap.

The cable would snap, break.