Aristotle's wheel paradox


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Of course the usual noisy reply to drown out dissent. Evading the rest of a sentence, conveniently.

"Slippage" has been advanced as the 'solution' to the paradox. Yes? If slippage does not go for the cause, it does not go for the solution. That was my full context. "If one cannot entertain the notion that [...] how can the same notion be presented as the solution..."

There's been a lot of dishonesty and bad faith here. Any rational answer from anyone, which responds to my full question? 

 

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Thanks for noticing, Max. It’s easy for me, to be honest. I don’t mind stupid, it doesn’t rub me the wrong way at all. It’s only when snippy gets added to stupid that I have to explode or walk aw

These are not solutions. "Solution" 1 is nothing else than a short recapitulation of the paradox: Aristotle: If I move the smaller circle I am moving the same centre, namely Α; now let the larger

No, it isn't. Artistotle wrote: "nowhere does the greater stop and wait for the less in such a way as to remain stationary for a time at the same point" and "the smaller does not skip any point", so h

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31 minutes ago, anthony said:

Can anyone assert that the wheels as diagramed and 'wheels' in an archery target are essentially different?

No. No one can or would assert that they are different. No one has asserted that, or even remotely suggested it. You are chasing a straw man. You are not understanding what people are saying. When they tell you that there is skidding/slippage, you are not grasping to what they are referring, even though they've diagramed it, illustrated it, animated it, video taped it, and otherwise shoved it into in your face and spelled it out in the simplest terms possible. Alas, you are incapable of understanding it. You will never understand it. You mind is not capable of it. Your cognitive limititations prevent it.

J

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14 hours ago, anthony said:

Correct! As are the painted circles on the 'target'. ("...along for the ride"). You have affirmed my entire argument.

(Btw, How, have lines appeared crossing out part of my post?! Weird).

No matter how many wheels are attached to each other they only get to roll on one track. Thus slippage is impossible not considering torque over coming friction. If you have two wheels and two tracks but the wheels roll independent of each other for the same distance the smaller wheel will turn over more than the big wheel. No slippage. If two wheels of different sizes are firmly attached to each other and they roll on two different tracks the smaller wheel will slip or skid for it can't turn over more than the big wheel does. NOW throw these facts at the paradox and integration is impossible. As reality beats abstraction the paradox is destroyed.

Considering any paradox as a total abstraction unto itself, that's for fun. So is getting drunk.

--Brant

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7 hours ago, anthony said:

The observable wheel-within-a wheel, which could equally be delineated by drawn circles or by painted rings - or by solid, attached wheels - must follow the same law of identity. The fundaments of the wheel don't change identity, although 'wheels' may come in many forms.

Very secondarily and less important, the brief itself, reminds us and stipulates "The wheels roll without slipping..." May one over-rule that rule, at whim? 

Still, this wasn't meant to be resolved by a simple mechanical or a complex mathematical solution. And those applications come after identification.

A good example of the nonsense you get with all that talk about "identity". You apparently think that painted archery rings are essentially the same as car wheels, just while both can approximately described as concentric circles, and that the fact that archery rings usually don't roll or slip "proves" that car wheels can't roll or slip (let's hope you don't drive a car). If you want to use the abstract property that archery rings and car wheels share, then you should concentrate on circles. Circles can roll and slip, as I've several times demonstrated, even if painted archery rings can't. Of course Aristotle chose a mechanical implementation with wheels, as these are a natural implementation of rolling circles, that are at the core of his paradox. It is a mechanical/mathematical puzzle, that of course can and should be solved by mechanical/mathematical reasoning. And the mathematical solution isn't complex at all, it is in fact very simple. Well, except for people who insist on "identification". Where is that "identification" now? Does it come soon?

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10 hours ago, Michael Stuart Kelly said:

Brant,

That is one way to interpret the diagram. And that interpretation is correct. There is nothing wrong with it.

Another way to interpret the diagram is to have both wheels functioning as wheels, albeit suck together. That interpretation is also correct. There is nothing wrong with that interpretation, either.

The diagram works both ways. Just not at the same time.

One interpretation is incorrect only when the other is considered as correct. When people who hold opposite sides on that point meet and claim their interpretation is the only correct one for a premise, all hell breaks out. People talk past each other. And so on.

From what I can see, all the disagreements boil down to taking sides on which interpretation is correct for the premise. This gets bolstered by the fact that the technical stuff for one interpretation doesn't work for the other and vice-versa. Discussing this technical stuff while using different premises is generating all the cognitive parts of the disagreements. (The passions that generate the normative parts of the disagreements... well... you know... :) )

Once I became aware of all this, I can now bounce back and forth at will in my mind between the two interpretations. It's kinda cool, actually. And it is not making any false equivalencies. When I see the problem through one interpretation, the diagram works and the other interpretation is obviously false. Ditto for vice-versa.

Like I said above, the diagram is misleading. Any diagram that can induce such opposite interpretations and still work according to the interpretation adopted has an internal shortcoming. Something seriously got left out or did double duty when it should not have or whatever.

Michael

Michael!

It’s great to see you have it now. I struggled to parse your earlier writings about this paradox. I could not tell if you really understood or not, It is plain to me now that you get it. That’s great.

Regarding your last paragraph - is there really anything wrong with the illustration? I don’t think the illustration is faulty. What’s funny is the appearance we are supposed to get sucked in by - namely the appearance the small wheel is a magic one that can “go farther than it should.” That’s a goofy thought, but I cannot think of a better way to illustrate it. We need the reader to be sucked in. It would be a mistake to design the illustration such that it immunizes the reader from falling for the goofy thought.

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31 minutes ago, Jonathan said:

No. No one can or would assert that they are different. No one has asserted that, or even remotely suggested it. You are chasing a straw man. You are not understanding what people are saying. When they tell you that there is skidding/slippage, you are not grasping to what they are referring, even though they've diagramed it, illustrated it, animated it, video taped it, and otherwise shoved it into in your face and spelled it out in the simplest terms possible. Alas, you are incapable of understanding it. You will never understand it. You mind is not capable of it. Your cognitive limititations prevent it.

J

You've got nothing, not without a 'track".

With the 'track' you also have nothing. If a wheel in a wheel turns unaided, it will also turn evenly with a second track. Assuming all is equal. Assuming it is not one of those crude experiments and is balanced accurately, it will reproduce what it does without a track. Or else there is a deliberate bias. Can't have it both ways.

Try to visualize and think.

 

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19 minutes ago, anthony said:

You've got nothing, not without a 'track".

With the 'track' you also have nothing. If a wheel in a wheel turns unaided, it will also turn evenly with a second track. Assuming all is equal. Assuming it is not one of those crude experiments and is balanced accurately, it will reproduce what it does without a track. Or else there is a deliberate bias. Can't have it both ways.

Try to visualize and think.

 

 

 If a wheel in a wheel turns unaided, it will also turn evenly with a second track.”

 

It does not. 

We’ve shown you it does not a million ways.

We’ve explained that the circumferences are different, so it does not. Because it cannot.

 

Here is a Wheel in a wheel.

It is your contention that they can both simultaneously roll without slip or skid on their respective tracks.

 

 

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3 hours ago, Max said:

A good example of the nonsense you get with all that talk about "identity". You apparently think that painted archery rings are essentially the same as car wheels, just while both can approximately described as concentric circles, and that the fact that archery rings usually don't roll or slip "proves" that car wheels can't roll or slip (let's hope you don't drive a car). If you want to use the abstract property that archery rings and car wheels share, then you should concentrate on circles. Circles can roll and slip, as I've several times demonstrated, even if painted archery rings can't. Of course Aristotle chose a mechanical implementation with wheels, as these are a natural implementation of rolling circles, that are at the core of his paradox. It is a mechanical/mathematical puzzle, that of course can and should be solved by mechanical/mathematical reasoning. And the mathematical solution isn't complex at all, it is in fact very simple. Well, except for people who insist on "identification". Where is that "identification" now? Does it come soon?

"The wheels roll without slipping for a full revolution";

"The wheels roll without slipping for a full revolution". Read carefully. See? "Wheels"; without slipping?

You have missed the premise of the exercise, you are not alone. If "slipping" were admissible, this would be elementary to 'fix', mechanically. But it goes deeper than that.

Here is the premise of the 'paradox': Solve, explain, resolve, justify, or whatever - the phenomenon that the travel of an inner wheel, without slipping, will be extended beyond its circumference.

That's all.

On the face of it, the inner wheel's extension appears counter-intuitive. And too, we have the dimensions/circumference/distance of the outer wheel, to support this theory for the inner wheel. But it's a wrong conflation.

I show that this behavior is normal and inherent to a wheel inside a wheel. You can visibly see it in many ways. The large wheel is the determining factor. End of paradox.

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4 minutes ago, anthony said:

"The wheels roll without slipping for a full revolution";

"The wheels roll without slipping for a full revolution". Read carefully. See? Wheels; without slipping.

You have missed the premise of the exercise, you are not alone. If "slipping" were admissible, this would be elementary to 'fix' ,mechanically. But it goes deeper than that.

Here is the premise of the 'paradox': Solve, explain, resolve, justify, or whatever - WHY the travel of an inner wheel, without slipping, will be extended beyond its circumference.

That's all.

I show that this behavior is normal and inherent in a wheel. You can see it. The large wheel is the determining factor. End of paradox.

But Tony, both wheels roll their road without slipping their road was a lie in the setup,of the paradox.

They can’t do that.

Watch the above two videos again.

Its just tape, a straw and a lid and a book and table. Try it.

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Solutions ... disputes ... solutions ... dispute. It's like a wave function.

Tony, this 3D animation below is really good at instantiating an example of a wheel with two 'rails' or roads.  The inner wheel drags, slips, skids, what have you.  The difference between this and the 2D conceptualizations show the apparent paradox as (in my mind, at least) an illusion.  The 3D is a reality, the 2D is an impossibility in the real 3D world.

2 hours ago, anthony said:

"Slippage" has been advanced as the 'solution' to the paradox.

Slip, skid, drag, rub, scrape ...

Tony, what do you see in this three dimensional illustration -- versus what you see in the earlier animation posted from Wolfram MathWorld?

Quote

Aristotle's Wheel Paradox

DOWNLOAD Mathematica Notebook Asristotle's wheel paradox animation

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.

Spoiler

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: c (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.

 

As I think Jon Letendre has pointed out several times, along with Jonathan and Max -- the inner-wheel circumference cannot 'unroll' as in the illustration. It is (again, in my mind) more a kind of mental-visual illusion.

In my imperfect comment a few pages back I pointed to a real-word example of 'slip,  drag, skip, skid ...' what have you as an example of the differential angle and circumference between the flanges and the rails on a railway's curve.  If you have ever ridden a train, the screeching on sharp corners is the skip, drag, skid.  The angle of a fixed-axle bogie negotiating a curve is the problem. The axle rotates, the wheels rotate, but the inside-of-the-curve flanged-wheel and the outside-of-the-curve flanged-wheel on the bogie are negotiating different distances.

I probably have botched this attempt at helping you out, but I am trying to suggest to you that besides the slurs and slings and arrows, the folks here are trying to help you understand their points of view.  Not everyone is a great teacher, but the lessons still have objective value.

Edited by william.scherk
Screeching noises, spelling
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1 hour ago, Jon Letendre said:

 

 If a wheel in a wheel turns unaided, it will also turn evenly with a second track.”

 

It does not. 

We’ve shown you it does not a million ways.

We’ve explained that the circumferences are different, so it does not. Because it cannot.

 

Here is a Wheel in a wheel.

It is your contention that they can both simultaneously roll without slip or skid on their respective tracks.

 

 

 

 

56 minutes ago, Jon Letendre said:

But Tony, both wheels roll their road without slipping their road was a lie in the setup,of the paradox.

They can’t do that.

Watch the above two videos again.

Its just tape, a straw and a lid and a book and table. Try it.

Indeed, "they" can do that. Else, this whole thing becomes childs-play. Simple to solve mechanically and little thought required.

This way, is much better and I'm certain as was intended. One has to review one's premises.

I don't need to try it again, I've done my own tests. Obviously, friction is introduced with another track. So you get drag. One millimeter and 0.1 grams pressure variability will skew the outcome, so the experiment has to be rigorous to be accurate.

An experiment reproduces what you observe: i.e. a bottle rolling "unaided". It rolls straight. Therefore, the experiment must begin with that and make it consistent. 

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37 minutes ago, william.scherk said:

Solutions ... disputes ... solutions ... dispute. It's like a wave function.

Tony, this 3D animation below is really good at instantiating an example of a wheel with two 'rails' or roads.  The inner wheel drags, slips, skids, what have you.  The difference between this and the 2D conceptualizations show the apparent paradox as (in my mind, at least) an illusion.  The 3D is a reality, the 2D is an impossibility in the real 3D world.

Slip, skid, drag, rub, scrape ...

Tony, what do you see in this three dimensional illustration -- versus what you see in the earlier animation posted from Wolfram MathWorld?

As I think Jon Letendre has pointed out several times, along with Jonathan and Max -- the inner-wheel circumference cannot 'unroll' as in the illustration. It is (again, in my mind) more a kind of mental-visual illusion.

I was saying - one millimeter of track height difference - anywhere - in that demonstration, and it becomes arbitrary. You've got rolling resistance and the weight-disparity of a massive main wheel and a tiny inner wheel, like a hub. Even so, you can see the 'hub' gripping and turning for a while. A dubious experiment that proves little..

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1 hour ago, anthony said:

If a wheel in a wheel turns unaided, it will also turn evenly with a second track.

Reality says otherwise. Your limited brain is incapable of grasping reality in regard to this issue. Plus, you don't want to grasp it.

1 hour ago, anthony said:

Assuming it is not one of those crude experiments and is balanced accurately, it will reproduce what it does without a track.

False. We've reproduced it here several times, and the results in reality do not conform to your opinion of what happens. You are rejecting reality.

1 hour ago, anthony said:

 

Try to visualize and think.

 

You're a moron. The funniest thing to me is that you believe that you've tested all of this with a bottle, and that you've confirmed your belief that there is no skidding/slipping of the bottle's mouth on its surface. Just trust Tony, he tested it, and he did it scientifically with the right balance and everything, and not like the crude experiments that lesser thinkers have performed, and he found conclusively and irrefutably that there is no slippage. And he doesn't need to do it again while paying closer attention and while considering that he might have missed something. Nope, he dun it already, and it's a proven fact of reality that there's no slippage.

Heh.

J

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9 minutes ago, anthony said:

I was saying - one millimeter of track height difference - anywhere - in that demonstration, and it becomes arbitrary. You've got rolling resistance and the weight-disparity of a massive main wheel and a tiny inner wheel, like a hub. Even so, you can see the 'hub' gripping and turning for a while. A dubious experiment that proves little..

Hahahahaha!!!!

HAHAHAHA HAHAHAHA HAHAHAHAH!!!!!!!!

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22 minutes ago, Jonathan said:

Reality says otherwise. Your limited brain is incapable of grasping reality in regard to this issue. Plus, you don't want to grasp it.

False. We've reproduced it here several times, and the results in reality do not conform to your opinion of what happens. You are rejecting reality.

You're a moron. The funniest thing to me is that you believe that you've tested all of this with a bottle, and that you've confirmed your belief that there is no skidding/slipping of the bottle's mouth on its surface. Just trust Tony, he tested it, and he did it scientifically with the right balance and everything, and not like the crude experiments that lesser thinkers have performed, and he found conclusively and irrefutably that there is no slippage. And he doesn't need to do it again while paying closer attention and while considering that he might have missed something. Nope, he dun it already, and it's a proven fact of reality that there's no slippage.

Heh.

J

One can do whatever one wants with a videoed experiment. You can reproduce what you want it to show.

Do you mean I should "trust" your word, and not my own experience? Heh. With adjustments, I can make a bottle run straight on two tracks, no slip. Does that prove anything? No. 

 

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So I don't misunderstand, what 'you' can  see is a solid wheel, is that right, Tony?

47 minutes ago, anthony said:

You've got rolling resistance and the weight-disparity of a massive main wheel and a tiny inner wheel, like a hub. Even so, you can see the 'hub' gripping and turning for a while.

That is not what I am seeing. I am seeing a single, solid object-wheel, with an extrusion.  When I watch that object-wheel rotate, the extrusion 'spins' at the exact same rate as the larger (one revolution of the extrusion equals one revolution of the larger edge) -- it must be so as the wheel is depicted as if a real-life single unified object. 

It might be easier for you to see that the extrusion scrapes against the upper 'rail' surface if there were a fixed dot on the granite.

The illustration does nothing to trick the eye with the wheel, Tony, to my eyes at least.  The wheel is depicted as a solid; it 'spins' or rotates as one thing, one "A" ...

Quote

A dubious experiment that proves little.. 

I don't understand. Can you describe exactly what is 'dubious' about that animated illustration from Jonathan? There is no trickery, no added frames or illusory elements. I've just slowed it down to 30% of the original speed in the loop above.

If you are telling us that you see 'grip,' would it help to zoom in on the extruded part of the 3D illustration, that you may point to the 'grip' before 'slip'? I mean, would you find a zoomed-in animation helpful to your argument?

Edited by william.scherk
Slow roll zoom on the solid 'granite wheel's extrusion.
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6 minutes ago, william.scherk said:

So I don't misunderstand, what 'you' can  see is a solid wheel, is that right, Tony?

That is not what I am seeing. I am seeing a single, solid object-wheel, with an extrusion.  When I watch that object-wheel rotate, the extrusion 'spins' at the exact same rate as the larger (one revolution of the extrusion equals one revolution of the larger edge) -- it must be so as the wheel is depicted as if a real-life single unified object. 

It might be easier for you to see that the extrusion scrapes against the upper 'rail' surface if there were a fixed dot on the granite.

The illustration does nothing to trick the eye with the wheel, Tony, to my eyes at least.  The wheel is depicted as a solid; it 'spins' or rotates as one thing, one "A" ...

I don't understand. Can you describe exactly what is 'dubious' about that animated illustration from Jonathan? There is no trickery, no added frames or illusory elements. I've just slowed it down to 30% of the original speed in the loop above.

If you are telling us that you see 'grip,' would it help to zoom in on the extruded part of the 3D illustration, that you may point to the 'grip' before 'slip'? I mean, would you find a zoomed-in animation helpful to your argument?

 

You'd know that 'inner wheel" and "outer wheel" is used representatively, by now. That is obviously a solid wheel.

I don't buy your premises. "Without slippage" was specified by the original Wiki piece, and more interesting than these selective practical tests in mechanics.

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1 hour ago, anthony said:

"The wheels roll without slipping for a full revolution";

"The wheels roll without slipping for a full revolution". Read carefully. See? Wheels; without slipping?

You have missed the premise of the exercise, you are not alone. If "slipping" were admissible, this would be elementary to 'fix' ,mechanically. But it goes deeper than that.

Here is the premise of the 'paradox': Solve, explain, resolve, justify, or whatever - the phenomenon that the travel of an inner wheel, without slipping, will be extended beyond its circumference.

That's all.

On the face of it, the action appears counter-intuitive.And we have the dimensions/circumference/distance of the outer wheel to support this theory for the inner wheel, too. But it's wrong.

I show that this behavior is normal and inherent to a wheel inside a wheel. You can visibly see it in many ways. The large wheel is the determining factor. End of paradox.

The joke is of course that the fact that this premise is false (both wheels cannot roll without slipping) is the cause of a paradox emerging. You can solve the paradox by showing that it is impossible that both wheels roll without slipping and why that is so. It is like a magic trick: the magician tells you something or suggests you something by his actions that is not true, but that the naive onlooker accepts for true, which makes the next actions of the magician seem to be impossible. A general recipe for creating a paradox is to tell a plausible story, with a plausible argument, but with a hidden error in the premises or in the argument, leading to an apparent contradiction. So the fact that the premise of two wheels rolling without slipping is false is essential for the existence of the paradox, otherwise there wouldn't be a paradox at all!.

Further, if you can't be convinced by the (excellent) videos and animations in this thread, you can always check the mathematical derivation I gave, it is quite simple. Avoiding it is avoiding reality.

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Jonathan seems to me correct in saying that Tony does not posses the visualization skills to cognitively isolate the small wheel and deduce its necessary motions over its road. After all, there are several motions and conditions to manage at once.

He can’t do it. As a result, he cannot judge correctly any depiction of the small wheel interacting with its road.

That’s not a put-down. Tony has the superior ability in other areas.

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1 hour ago, anthony said:

Do you mean I should "trust" your word, and not my own experience?

 

No, I mean that you are incapable of grasping reality in this instance (and some other instances as well, but that's a discussion for another day). You don't have the cognitive ability to understand, nor do you have the cognitive ability to recognize that you don't understand. You don't even understand the alleged problem, let alone the solution, and you don't understand how to go about testing and measuring without being fooled.

J

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2 hours ago, Max said:

The joke is of course that the fact that this premise is false (both wheels cannot roll without slipping) is the cause of a paradox emerging. You can solve the paradox by showing that it is impossible that both wheels roll without slipping and why that is so. It is like a magic trick: the magician tells you something or suggests you something by his actions that is not true, but that the naive onlooker accepts for true, which makes the next actions of the magician seem to be impossible. A general recipe for creating a paradox is to tell a plausible story, with a plausible argument, but with a hidden error in the premises or in the argument, leading to an apparent contradiction. So the fact that the premise of two wheels rolling without slipping is false is essential for the existence of the paradox, otherwise there wouldn't be a paradox at all!.

Further, if you can't be convinced by the (excellent) videos and animations in this thread, you can always check the mathematical derivation I gave, it is quite simple. Avoiding it is avoiding reality.

Well,  math "derivation"s and "videos and animation" - actually are deductions from, and representations of "reality". They are not "reality" per se. Eh, but objectivity and Objectivism are of little interest... 

On the other hand, I agree, the circumvention of reality in this thread is rife. 

You want to explain mechanically, or by math, what is a perceptual-conceptual exercise in "reality", how we see and how we understand. It interests me and I'm not surprised that the methodologies emerging here, switch between concretism and rationalism. 

Anyhow, you have the Paradox premise completely wrong. Do you make this up as you go? You have accepted 2 tracks - since I guess this fits your agenda and preferred methods, also catered to by all the others. Where is this extra track stated?

I have repeated this too often, in several ways. Explicitly asserted, and evident also from the lines and circles of the diagram -- this puzzle concerns the distance travelled by the smaller wheel exceeding its circumference. That's it.

No one has challenged me on that, but nobody states their own version. 

"Otherwise there wouldn't be a paradox at all!". Yup. 

There is no slippage, as there are not two tracks. "Two tracks" and "slippage" reduces the puzzle to boring problems of friction at work. Why think?

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14 minutes ago, anthony said:

Well, a math "derivation" and "videos and animation" - actually are deductions from and representations of "reality". They are not "reality". Eh, but objectivity and Objectivism are of little interest... 

On the other hand, I agree, the circumvention of reality in this thread is rife. 

You want to explain mechanically, or by math, what is a perceptual-conceptual exercise in "reality", how we see and how we understand. It interests me and I'm not surprised that the methodological versions emerging here, switch between concretism and rationalism. 

Anyhow, you have the Paradox premise completely wrong. Do you make this up as you go? You have accepted 2 tracks - since I guess this fits your agenda and preferred methods, also catered to by all the others.

I have repeated this too often, in several ways. Explicitly stated, and evident from the lines and circles of the diagram -- this puzzle concerns the distance travelled by the smaller wheel exceeding its circumference. That's it.

No one has challenged me on that. 

"Otherwise there wouldn't be a paradox at all!". 

There is no slippage, as there are not two tracks. "Two tracks" and "slippage" reduces the puzzle to boring problems of friction at work. 

See?

He thinks he understands the paradox just fine.

And his resolution is that it travels in excess of its circumference because it is just a drawing on a wheel, so it has to go where the wheel goes. The End.

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