Aristotle's wheel paradox

[Max]

9 hours ago, Brant Gaede said:

I have no problem seeing all the physicalities involved and only wonder at the need for any math and highly abstract reasoning. I guess because it all started with such in those ancient days. It's a mind trap.

What highly abstract reasoning? It's all rather simple and the math is also quite elementary. I think that a big problem for those old guys was that they didn't know calculus, they didn't for example have a notion of the concept "instantaneous speed" (Zeno's arrow problem!), while that is now a piece of cake for us. So they were puzzling about wheels jumping over gaps in their supports, trying to make sense of it all.

[Jonathan]

15 hours ago, BaalChatzaf said:

I am computing the transverse of the center of a circle of radius r. If it doesn't slip  and it turns through angle theta then it will traverse   r*theta  (theta measured in radians).   Now look at the outer wheel radius R  where R > r.  If the little wheel rigidly affixed to the outer wheel turns an angle theta so does the outer wheel.  But the outer will will bring the common center R * theta to the right  which exceeds  r*theta  hence the inner wheel must have been dragged for a distance of   (R - r)*theta.

False. The reality is that the outer wheel must spin-out/over-spin.

 

15 hours ago, BaalChatzaf said:

Attaching a wire or cable to the inner wheel does not change the geometry.

False. Attaching a cable as I've described and illustrated changes the geometry. It makes the entire rig roll on the smaller wheel. Nothing else is possible. The small wheel cannot be dragged as you suggest due to its cable limiting its motion.

J

[Jonathan]

14 hours ago, Max said:

 

The point is that the inner wheel cannot be dragged, as it is held back by its own cable (that is fixed to the support). It cannot roll further than r*theta,  its cable is taut. From its attachment point on surface 1 then a piece of string with length r*theta lies stretched on surface 1, the rest ist still wound around the wheel. Perhaps it helps to look at the figure:

Therefore it is now the small wheel that determines the movement, the large wheel must "follow", that is, it is held back, slipping, while it rotates together with the small wheel. The large wheel has unrolled R*theta of its own cable, while it has only traveled over a distance of r*theta, there is "too much" unrolled cable from wheel 2, therefore it is slack, and lies there like a dead snake when you roll far enough.

Max is correct again.

 

14 hours ago, Max said:

I suppose Jonathan is now making a new picture or animation to make it even clearer...

I hadn't planned on it. Perhaps I will. I don't know. Diagrams, animations and videos of real things seem to have no effect on those who aren't cognitively suited to visuals. Showing such people reality seems to be a waste of time. "Look, this is what happens in reality." "No, that's a trick. Anyone can do optical tricks. I'm right cuz I know I am."

The only thing that might succeed is showing certain people the math of the cables' forces. I wouldn't know where to begin with that. Besides, even that would come down to their first recognizing and accepting what the cables are doing, and how the smaller wheel's cable affects the movement of the entire rig.

I think that those who don't get it pretty quickly on their own are very unlikely to ever be convinced by others' explanations, diagrams, math, or direct reality.

J

[Max]

30 minutes ago, Jonathan said:

I hadn't planned on it. Perhaps I will. I don't know. Diagrams, animations and videos of real things seem to have no effect on those who aren't cognitively suited to visuals. Showing such people reality seems to be a waste of time. "Look, this is what happens in reality." "No, that's a trick. Anyone can do optical tricks. I'm right cuz I know I am."

I think you're right. We have now posted so many diagrams, animations, videos and mathematical derivations, that anyone who is seriously committed to finding the solution to this puzzle has material enough for studying the problem, to either accept the slippage solution, or to come up with a valid counterargument. showing  what would be wrong in our examples and derivations and to give an alternative solution, and not a "solution" that isn't. 

 

30 minutes ago, Jonathan said:

The only thing that might succeed is showing certain people the math of the cables' forces. I wouldn't know where to begin with that. Besides, even that would come down to their first recognizing and accepting what the cables are doing, and how the smaller wheel's cable affects the movement of the entire rig.

The people who don't get the original problem, won't get this version either. Bob had directly the solution of the original version. I think he's still more or less thinking of that one, not yet seeing that the new puzzle is different (though a variation on the old theme). So I wouldn't bother too much about this version, he'll get it sooner or later.

30 minutes ago, Jonathan said:

I think that those who don't get it pretty quickly on their own are very unlikely to ever be convinced by others' explanations, diagrams, math, or direct reality.

Indeed, this is where the men are separated from the boys.

[anthony]

On 12/2/2018 at 2:10 AM, Ellen Stuttle said:

Right, and thanks.  The "should" was that of the poser of the problem. 

I thought after I'd posted that Tony might mistakenly take the "should" as mine, even though I'd said "Here's how the problem goes" (i,e., here's how the person posing the problem was thinking), but I didn't have time to edit by adding a parenthetical stating that the "should" was the assumption of the problem's formulator.

Regarding Tony's continuing to call the track superfluous:  The track, like the "should," was put into the problem by the person who formulated it.

Quite agreed - and I've said before - that the problem isn't a genuine paradox - defined as "an apparent contradiction between two true premises" - but instead an actual contradiction between a false assumption and reality.

Ellen

Exactly. You're bearing out my position, no paradox. "Should' is a general assumption, I should have made clear, and is the only premise of the paradox (i.e., the action of one wheel compared/contrasted to the action of the other).

Like this "should", why can't one also assume a second track "was put into the problem by the person who formulated it" ?Again, the 'track' serves no purpose but to lead astray with more "shoulds" or woulds.

This track is superfluous since it either a). inhibits/skews the rotation of the main wheel, and the 'paradox' is rendered null and void or b). it does nothing - has no effect (the small wheel continues to traverse a distance beyond its circumference). It can't be had both ways.

[anthony]

5 hours ago, Max said:

What highly abstract reasoning? It's all rather simple and the math is also quite elementary. I think that a big problem for those old guys was that they didn't know calculus, they didn't for example have a notion of the concept "instantaneous speed" (Zeno's arrow problem!), while that is now a piece of cake for us. So they were puzzling about wheels jumping over gaps in their supports, trying to make sense of it all.

haha. A problem with the young guys is to not identify before they calculate. Floating math abstractions.

[Jon Letendre]

Hey Jonathan,

I watched your stone wheel video again and Tony is actually right about it being too heavy. That creates too much drag, which throws everything off. Can it be wood?

[Jonathan]

5 minutes ago, Jon Letendre said:

Hey Jonathan,

I watched your stone wheel video again and Tony is actually right about it being too heavy. That creates too much drag, which throws everything off. Can it be wood?

Sure, it could be wood, or plastic, or styrofoam. Better yet, it could be card stock paper (perhaps printed with a photo of a stone surface just to confuse Tony and Merlin even further). Or it could remain stone, and we could add lubricants to eliminate drag. Or we could impose some mechanical means of controlling for drag and ensuring proper roll of one of the wheels. That's what scienctists would do. Whatever it takes to comply with the requirement of the large wheel rolling without slipping/skidding. Or vice versa.

J

 

[anthony]

Can't help thinking this is like obsessive market interventionists versus laissez-faire capitalists.

"We must force all things to be equal, it is not fair that the market doesn't enrich everyone the same!" (the small wheel must be made equal to the larger, and/or the large to conform to the smaller). 

vs.

"Leave it alone, the market precisely behaves according to its nature, you'll screw it up with your coercive regulations!" (it is all one wheel, you idiots).

[Jon Letendre]

Slippage is starting to smell like a smuggled-in stolen concept.

[Jon Letendre]

10 minutes ago, Jonathan said:

Sure, it could be wood, or plastic, or styrofoam. Better yet, it could be card stock paper (perhaps printed with a photo of a stone surface just to confuse Tony and Merlin even further). Or it could remain stone, and we could add lubricants to eliminate drag. Or we could impose some mechanical means of controlling for drag and ensuring proper roll of one of the wheels. That's what scienctists would do. Whatever it takes to comply with the requirement of the large wheel rolling without slipping/skidding. Or vice versa.

J

 

It’s always going to weigh something, though. Which screws up the delicate roll. Yet if it weighed nothing, it wouldn’t be a wheel, because those weigh something. I rolled a bottle on my floor, though, and you know what? It. Just. Rolled. There is no paradox here.

[Jon Letendre]

Obsession with slippage betrays an arbitrary sense of life.

[Jonathan]

51 minutes ago, Max said:

The people who don't get the original problem, won't get this version either.

Imagine confronting them with something a bit more complex, like, say, comparing a standard bevel gear differential system with a worm gear differential system (Torsen), and explaining the whats, whys and hows of it all. If A happens to B, then C will cause D to rotate faster than E.

We'd be standing here thousands of times more tardfounded than we are right now.

J

 

[Jonathan]

23 minutes ago, anthony said:

Can't help thinking this is like obsessive market interventionists versus laissez-faire capitalists.

"We must force all things to be equal, it is not fair that the market doesn't enrich everyone the same!" (the small wheel must be made equal to the larger, and/or the large to conform to the smaller). 

vs.

"Leave it alone, the market precisely behaves according to its nature, you'll screw it up with your coercive regulations!" (it is all one wheel, you idiots).

Straw man comment. You're still not getting it.

J

[Jon Letendre]

Slippage is collectivistic and second-handed.

[Jonathan]

22 minutes ago, Jon Letendre said:

It’s always going to weigh something, though. Which screws up the delicate roll. Yet if it weighed nothing, it wouldn’t be a wheel, because those weigh something. I rolled a bottle on my floor, though, and you know what? It. Just. Rolled. There is no paradox here.

I've heard that if a person rolls a bottle body and neck on two different surfaces, there's no slippage, no difference between traction and lack of traction. And do you know how to objectively measure these things? You just kind of pay attention or not. I didn't see nuthin.

That's proof.

J

[Jonathan]

3 minutes ago, Jon Letendre said:

Slippage is collectivistic and second-handed.

Kant invented slippage.

J

[Jonathan]

Just now, Jonathan said:

Kant invented slippage.

J

For proof, see his writings.

J

[Jon Letendre]

2 minutes ago, Jonathan said:

I've heard that if a person rolls a bottle on two different surfaces, there's no slippage, no difference between traction and lack of traction. And do you know how to objectively measure these things? You just kind of pay attention or not. I didn't see nuthin.

That's proof.

J

Exactly.

And funnels roll straight if you elevate the road the small end is on.

But everything has to be exact!

When you get it right there is no slippage and the funnel rolls straight.

That’s the experimental result I am getting at home right now.

If you get a different result, it’s likely because you have little knowledge of or respect for laissez-faire. Funnel laissez-faire.

[Max]

2 hours ago, anthony said:

haha. A problem with the young guys is to not identify before they calculate. Floating math abstractions.

Peikoff parrot. He also started to talk about "floating abstractions" when things went way over his head.

[Jonathan]

1 hour ago, Jon Letendre said:

Exactly.

And funnels roll straight if you elevate the road the small end is on.

But everything has to be exact!

When you get it right there is no slippage and the funnel rolls straight.

That’s the experimental result I am getting at home right now.

If you get a different result, it’s likely because you have little knowledge of or respect for laissez-faire. Funnel laissez-faire.

Ya know, now that you mention it, I have heard that cones and cylinders, despite appearing to be very different physically, behave exactly the same way when you roll them, but everything does have to be exact. Not too heavy and not too light, not too much traction and not too little. If you use Objectively proper epistemology, and eliminate floating abstractions and concrete-bound thinking, a cone will roll straight, exactly the same as a cylinder.

Okay so I just ran a scientific experiment. I just made myself become fully dedicated to being completely objective, and then I imagined rolling a cone and a cylinder, and it worked! They rolled exactly the same! Confirmed! I t think that they key is that I didn't artificially induce any slippage into my imagination, or allow anyone else to, so none appeared! Cones ARE cylinders if you have real actual Objective knowledge and respect for the law of identity.

J

 

[BaalChatzaf]

3 hours ago, Jon Letendre said:

Obsession with slippage betrays an arbitrary sense of life.

That is why I put sand on my steps when the sleet falls on them.

 

[Darrell Hougen]

On 11/18/2018 at 9:24 AM, anthony said:

Aristotle's wheel paradox

From Wikipedia, the free encyclopedia

 

 

Jump to navigationJump to search

Aristotle's wheel paradox is a paradox or problem appearing in the Greekwork Mechanica traditionally attributed to Aristotle.^[1] A wheel can be depicted in two dimensions using two circles. The larger circle is tangent to a horizontal surface (e.g. a road) that it can roll on. The smaller circle has the same center and is rigidly affixed to the larger one. The smaller circle could depict the bead of a tire, a rim the tire is mounted on, an axle, etc. Assume the larger circle rolls without slipping for a full revolution. The distances moved by both circles are the same length, as depicted by the blue and red dashed lines. The distance for the larger circle equals its circumference, but the distance for the smaller circle is longer than its circumference: a paradox or problem.

The paradox is not limited to a wheel. Other things depicted in two dimensions show the same behavior. A roll of tape does. A typical round bottle rolled on its side does -- the smaller circle depicting the mouth or neck of the bottle.

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

SHOW me the "slippage". You guys would be lost without an inner track, there is no "track" in the diagram and no possible slippage.

Quote: "The distance for the larger circle equals its circumference, but the distance for the smaller circle is longer than its circumference: a paradox or problem".

The dotted lines are distances. Not 'tracks". The crux of the paradox is different wheel circumferences, but identical distance traveled.

Once more, I point out that every and any point marked within the large circle will show that same dotted line, same distance moved from its 1st to its 2nd position. 

Enter a smaller, miniscule circle instead, and the dotted line will be the same length. WHY? because the circumference of the small circle is irrelevant. The distance moved by the entire wheel is all that counts, dictating the motion of its contents.

With that taken care of,

HOW does the small circle rotate once, precisely equal to the large circle's single rotation? It rotates relatively slower. At least 4 times I've explained that, not once has anyone acknowledged or argued it.

Enough with the nonsense. You've been chasing a red herring.

Tony,

Merlin edited the Wikipedia page so that it no longer contains an accurate description of Aristotle's paradox. The figure has also been edited and is no longer illustrative of the paradox.

Darrell

[anthony]

1 hour ago, Darrell Hougen said:

Tony,

Merlin edited the Wikipedia page so that it no longer contains an accurate description of Aristotle's paradox. The figure has also been edited and is no longer illustrative of the paradox.

Darrell

Darrell, yeah. I later accepted for the sake of argument and to prove a point, that there is a track.

Yet, I maintain that in practice a wheel will rotate the same as it always does, as we experience it to act, and as it does in theory (a circle diagram). You can observe a car wheel turn normally - and, let us say, you delineate the metal wheel rim to be the 'inner wheel'. Now extend that rim outwards. Now place a track for the extended rim to roll on. 

Now, you roll the car wheel for a revolution. What is possibly going to occur which did not occur when it was simply a car wheel and tyre on the road? No difference - surely? The entire wheel rolls forwards, on two tracks, where there was before just one surface. Assuming the weight on each track is carefully and evenly distributed, the outcome will be what the diagram denotes: the large wheel (tyre) rolls its circumference; the inner wheel (rim) rolls one revolution--but far past its circumference. It does not 'slip', it doesn't need to. The wheel assembly acts now exactly as it did on a single surface. Nothing essential has changed, inner and outer wheels keep integrity. 'Slippage' would contradict and destroy that. For this reason, the 'track' was put in as a red herring, imo.

[Jon Letendre]

On 11/19/2018 at 10:42 PM, Michael Stuart Kelly said:

Jon,

Hell, I'll play. I don't mind being the stooge when I'm learning something.

So...

drum roll...

3?

Michael

Yes, 3.