Aristotle's wheel paradox

[Jon Letendre]

4 hours ago, merjet said:

The top one, nitwit.

Wheels roll on ceilings? In which direction do they rotate to pull that off?

[Max]

 

4 hours ago, merjet said:

Yes, it will "slip", that term being a confused metaphor. Do you know what scare quotes are? Why have you never used them around slips? You even get the metaphor wrong. The correct metaphor is "skid", meaning that the smaller circle rotates less than it would by pure rolling as an independent entity for the same horizontal distances.  But even that is not literal skidding. Literal slipping or skidding requires a point of contact. See here.

In my post of Januari 26 I said that I don‘t make a distinction between slipping and skidding. And I‘m not the only one. According to your view, everyone who writes “rolling without slipping” is wrong, as it then should be “rolling without slipping or skidding”. Better even, look at the previous link you gave yourself: click on “summary” and scroll down to “rolling and slipping”: those guys don’t make a distinction between slipping and skidding either, they call it all “slipping” positive or negative. And they don't put "slipping" between scare quotes! In a link that you recommend!

 

5 hours ago, merjet said:

Do you see a point of contact tangent to the bottom of the hub in the picture there? It is reasonable to assume a surface tangent to the bottom of the wheel that the wheel rolls on, but the authors omitted it. In reality a wheel on a car or truck rolls on a surface. It is NOT reasonable to assume a surface tangent to the bottom of the hub/rim that the hub/rim rolls on. 

Ever seen a train wheel? Anyway the point is moot, as Aristotle’s paradox is not about the practicability of wheel designs, old Greek or modern, it is a thought experiment. The only important point is that the system can in principle be built in reality (which will show that the condition that both wheels can roll without slipping cannot be met). In the original Wikipedia article there was also a line/surface drawn under the small wheel.

 

5 hours ago, merjet said:

In reality a wheel on a car or truck rolls on a surface. It is NOT reasonable to assume a surface tangent to the bottom of the hub/rim that the hub/rim rolls on. In reality no car or truck wheel does such a thing. In reality, is there a horizontal surface tangent to the inner round surface of a roll of duct tape? In reality, is there a horizontal surface tangent to the neck or mouth of a rolling round wine bottle? Like Jonathan, you arbitrarily add a tangent to the bottom of an inner circle that is totally unnecessary. Like Jonathan, it is your "crutch" to fake reality, and you are helpless without it. 

It is not a crutch, it is the crux of the paradox, as the paradox description states that both wheels roll without slipping. Whether such a surface exists in reality (it does for train wheels) is irrelevant. If you like you can avoid Aristotle's mechanical language and translate into mathematical terms. Then you can substitute “traces out the circumference of the circle” for “rolls without slipping”.

 

5 hours ago, merjet said:

Show me where I made an error in my proofs, if you can, and with arguments firmly tied to reality, not faking reality with a confused metaphor and a "crutch." 

You wrote: Summarizing, the smaller circle moves horizontally 2πR because any point on the smaller circle travels a shorter, more direct path than any point on the larger circle.

That “because” here is nonsense. The small circle moves horizontally 2πR because the large circle moves 2πR, and those circles have a common center, which implies that the small circle must slip, oh sorry, I mean “does not trace out its circumference”, as was implied in the description of the paradox. Further, without the latter point, you have not solved the paradox, you've only told us that both circles moved the same distance, what we knew all along, as part of the paradox description.

That those cycloids are completely superfluous, you show yourself in your second “solution”, in which no cycloid at all is mentioned.

5 hours ago, merjet said:

All four of you epitomize closed-minded dogmatism. You FEEL you have the only possibly correct answer, and any other answer must be wrong. Are you agents from the Ministry of Truth? 

No, we are the four horsemen of the apocalypse. 

Well, three horsemen and one horselady.

[Jon Letendre]

5 hours ago, merjet said:

Everything that Jonathan and Jon have said about my "molesting" the Wikipedia article or similar is hogwash. I vastly improved it. I have received thanks and compliments for doing so. J and Jon insist the paradox is about two wheels each on their own road. That contradicts both history and the 'Wrong problem entirely' section on the Wikipedia Talk page. Note how Jonathan conveniently omits mentioning that section when he accuses me of "molesting" Wikipedia and similar such rot.

It will be fixed soon, idiot, don’t worry.

We insist no such thing.

The paradox setup is about one rolling wheel.

The Paradox is in the appearance that the small wheel rolls like a normal wheel and yet goes farther than it should.

Carefully analysing how it really rolls is thereby necessary for resolution of said paradox.

[Jon Letendre]

3 hours ago, Max said:

 

Merlin 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.

Max said:

Applying the same argument to the large circle: every cycloid of the large circle (cycloid length = 8*R) is greater than 2*pi*R . Therefore, the large circle rolls farther than it would by pure rolling? You've created a new paradox!

 

 

Max, I see the error, too. Merlin meant to write “ ... 2*pi*R, i.e., the circumference of the main wheel.”

He surely meant that. Then his statement is fine.

I have gone down the cycloid path with Merlin, I follow him just fine. All of it is fine, from what I can see, (aside from some minor textual errors, such as you found above.) The cycloids are an abstract geometric concept and examining them here is perfectly appropriate, why not have a look, just to see what you see?

Merlin examined them. And what did he find? In his words :

”the smaller circle rolls farther than it would by pure rolling.” Which is just another telling of the paradox. The small wheel appears to do the impossible.

 

The paradox, as stated, I assume  by Merlin, at Wiki

” The distances moved by both circles are the same length, as depicted by the blue and red ... 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.”

 

So, he helped confirm, with cycloids, that there really is a paradox here.

I can’t wait to see what resolution he might offer.

 

[Max]

43 minutes ago, Jon Letendre said:

Merlin 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.

Max said:

Applying the same argument to the large circle: every cycloid of the large circle (cycloid length = 8*R) is greater than 2*pi*R . Therefore, the large circle rolls farther than it would by pure rolling? You've created a new paradox!

 

 

Max, I see the error, too. Merlin meant to write “ ... 2*pi*R, i.e., the circumference of the main wheel.”

He surely meant that. Then his statement is fine.

I don’t think he meant that, after all he also writes “...greater than 2*pi*r… with small r, so he’s referring to the small wheel. But even if he meant 2*pi*R, his conclusion doesn’t follow: the cycloid of the large wheel is also greater than its circumference, but the large wheel is rolling without slipping (tracing out its circumference), so this condition is no guarantee for rolling farther than its circumference.

 

45 minutes ago, Jon Letendre said:

Merlin examined them. And what did he find? In his words :

”the smaller circle rolls farther than it would by pure rolling.”

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

[Jon Letendre]

36 minutes ago, Max said:

I don’t think he meant that, after all he also writes “...greater than 2*pi*r… with small r, so he’s referring to the small wheel. But even if he meant 2*pi*R, his conclusion doesn’t follow: the cycloid of the large wheel is also greater than its circumference, but the large wheel is rolling without slipping (tracing out its circumference), so this condition is no guarantee for rolling farther than its circumference.

 

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

I see it now. You are correct, the conclusion does not follow, even after helping the text. I missed that, thanks.

It IS worse than I thought.

In any case, we already knew the big conclusion we wants and asserts. We already knew that “the small wheel rolls farther than it would by pure rolling.”

We already knew that it rolls farther than its circumference. That is supposed to be impossible and that IS the paradox.

Finding news ways to show that ““the small wheel rolls farther than it would by pure rolling” gets us no closer to a resolution, but Merlin is strutting around as though it IS a resolution.

[Brant Gaede]

15 hours ago, 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.

 

Since "rolls" and "pure rolling" are by you two different things then by you there is no paradox. However, if only one rolls physically the other can only roll metaphorically. Thus the paradox can only exist if the conflict between the real and metaphorical is resolved with both still standing without contradiction as an integrated whole.

Good luck with that.

--Brant

are you for or against the existence of this - or any - paradox? 

[anthony]

Slippage - bleh. As I can see, everyone is dependent on the suggestion of a physical "track", by a dotted line at the base of the inner circle. As soon as the line is moved anywhere else that notion collapses, to all round confusion.

Seeing that no one chose to properly consider my simple transition from a car wheel to the diagram and back, for another instance, consider an archery target with its concentric coloured circles. The paradox diagram could quite as well represent this. "Fixed inner circle" now, more of them.

Vertically rotate the archery target once, and it could be easily measured that all the inner, painted circles and bullseye equally shift x meters (laterally). Identical to the outer perimeter. It can also be calculated that this distance - in each case for each circle - exceeds the circumference of each inner circle.

Of far greater measured difference for the innermost circle, proportionately reducing towards the outer ones. BUT, constantly a distinct discrepancy between the two lengths.

For you visuo-spatial experts, who don't seem very visual to me, there should be an easy transposition back to the diagram.

Where's the problem? Where the need for 'tracks'? Where's slippage? 

Where, in fact, does the abstract theory of a circle (and inner circles) differ from a tangible wheel (and inner wheels)?!  This cognitive split is the cause and effect of rationalism.

Everyone has been taken in by the idea that the Wheel Paradox poses "an anomaly*, which necessitates *correction*. 

There is NO anomaly. The diagram is true. This is the reality, the nature of, of a circle/wheel - *by necessity* - which one needs to get one's head around, visually, perceptually, conceptually.

[The wheel] "to be commanded, must be obeyed". (I am messing with the famous dictum). By "obeyed" I think can be taken to mean, "comprehended and accepted".

[Brant Gaede]

Tony, there are two wheels only one functioning as a wheel. The second wheel is along for the ride.

---Brant

[anthony]

3 hours ago, Brant Gaede said:

Tony, there are two wheels only one functioning as a wheel. The second wheel is along for the ride.

---Brant

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).

[anthony]

On 11/22/2018 at 9:31 PM, Jon Letendre said:

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

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

"You seem to think..." false. What I've repeated all along, there's no paradox. No aberrancy, no anomaly, no problem. The "paradox" is your own.

Is everyone so obsessed with the discrepancy (circumference to distance) of the INNER wheel, they can't see what is real?

The outer wheel, and its circumference alone, determines distance travelled. The inner wheel complies. its distance travelled has no relation to its circumference. Period. Full stop. 

What I've also maintained several times, the inner wheel turns slower. Despite its smaller circumference and a single, identical rotation, therefore, it 'ends up in the same place'  (so to speak) as the entire wheel, at the finish point. "By necessity".

[anthony]

On 11/23/2018 at 1:12 AM, Jon Letendre said:

I think you will be proved correct.

Tony has a chance, if he would go back to the beginning and state what is the paradox.

Up to now he does not understand what is the paradox.

His attempt, “the assumption that the small wheel dictates distance” is absurd and incoherent. Many small wheels of various diameters can be drawn onto the wheel - which one does he imagine someone believes dictates distance? No one would think that and that’s not Aristotle’s Wheel Paradox.

He can’t state the paradox, because up to now, he doesn’t grasp that there is a paradox.

Once he sees there is a paradox and he truly understands the paradox, then he will soon thereafter understand the resolution easily. He has well more than the cognitive capacity required.

But the horse has to want to drink.

Total misrepresentation. I don't think you are worth replying to if that's what you do. I've stated the 'suggested' premise of the  'paradox' several times.

 

[anthony]

On 11/21/2018 at 10:04 PM, Max said:

 

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

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

 

I see you don't understand reasoning: "identity" is what a mind perceives and integrates of existence without dependence on what any authority informs one: Peikoff, Rand, or in your case, I am sure, Hume. But you have made known your disdain for metaphysics, as you apparently do for epistemology and all things Objectivist. Skepticism is your thing, I notice. But that was amusing, "elliptical" or circular reasoning. 

Fitting to this debate, in which wheel-reality has been obfuscated, Magellan could clearly see - and identify - in spite of opposing Authority.

"The Church says the Earth is flat, but I know it is round. For I have seen the Shadow of the earth on the Moon and I have more Faith in the Shadow than in the Church".

 

[Michael Stuart Kelly]

3 hours ago, Brant Gaede said:

Tony, there are two wheels only one functioning as a wheel. The second wheel is along for the ride.

---Brant

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

[Jules Troy]

[anthony]

A question.

If one cannot entertain, seriously, the notion that there is any ostensible skid/slippage among the painted rings of the archery target, and improbable between an auto tyre and rim, how can the same notion be presented as the solution to the 'Paradox'? If the first is ridiculous, so must be the second.

Is there a difference one should know about, between the diagram of circles necessitating 'a track', and real-life wheels which do not? Are theory and practice in contradiction? The answer to that, dump the theory.

[Max]

2 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.

I don't understand that. I see no difference between "the second wheel is along for the ride" and "two wheels stuck together". In both representations there is only one rigid object, that consists of two wheels with a common center. The only difference in interpretation would be either to treat those wheels as a mechanical system or to treat the paradox as a mathematical problem, but as I've shown in earlier posts, the two descriptions are equivalent. 

[Max]

22 minutes ago, anthony said:

A question.

If one cannot entertain, seriously, the notion that there is any ostensible skid/slippage among the painted rings of the archery target, or in an auto wheel, etc., how can the same notion be presented as the solution to the 'Paradox'? If the first is ridiculous, so must be the second.

Is there a difference one should know about, between the diagram of circles necessitating 'a track', and real-life wheels which do not? Are theory and practice in contradiction? The answer to that, dump the theory.

From the original version of the Wikipedia article: "The wheels roll without slipping for a full revolution." Rolling wheels, you know. No doubt chosen by Aristotle while it is rather natural for wheels to roll and to slip or not to slip, in contrast to rings of an archery target. Does the fact that rolling and slipping of archery target rings is a rather silly notion imply that rolling and slipping of car wheels or train wheels is ridiculous?

[anthony]

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.

[Jonathan]

10 hours ago, anthony said:

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

Yeah, um, that's yet another symptom of your being unaware and unobservant.

[Jonathan]

On 11/23/2018 at 7:12 AM, merjet said:

It is NOT reasonable to assume a surface tangent to the bottom of the hub/rim that the hub/rim rolls on.

False. It's perfectly reasonable to assume a surface that is tangent to the bottom of the hub that the hub rolls on.

On 11/23/2018 at 7:12 AM, merjet said:

In reality no car or truck wheel does such a thing.

False. When people, like Jon or I, place a surface under the hub, it is there in reality. It's real. And the fact that you haven't placed a surface under a hub doesn't cancel out the fact that we have --it doesn't erase from reality our having done it. So, now do you understand how your statement about "no car or truck" is false? See now how reality works? Your attempting to wish it away doesn't actually work. The diagrams and models that Jon and I have presented are real. They exist. The surfaces that we placed beneath the smaller wheels are there in reality. They don't cease to be there just because you're trying to blank them out.

It's sounds as if you were unaware of this, Merlin, but there isn't a natural default state of surfaces and wheels and what people don't do with them. Just because you have never seen a surface contacting a wheel hub before, it does not logically follow that therefore if anyone puts a surface into contact with a hub it can't be real, that it must be denied and wished out of existence. Doesn't work that way, Gramps. 

Merlin, have you heard of the scientific method? Well, the way that it works is that people test stuff. Often times that means arranging items in a way which you, personally, might not imagine ever seeing them, and then testing what happens. The idea is to grasp why the experiment is being performed, not to immediately object, "You went and done put a surface in thar. I ain't never seen no such surfaces when I drive muh car!" Retard.

Remember way back at the beginning of this thread, prior to your having dishonestly messed with the Wikipedia page, when you posted the link to video of the wheel rolling? The one with circles drawn on it, and the two lines at the bases of the circles? Do you remember that you didn't object to the setup in that video? You didn't gripe that, when you drive a car down the street, there are no strings contacting your wheels' hubs?

Heh.

J

 

[Jonathan]

On 11/23/2018 at 7:06 AM, merjet said:

It's obvious that the smaller circle's cycloid is shorter than the outer circle's cycloid, and longer than the center's path.

Illusion. Scam. Con art.

Besides, even if we accept it as not being an illusion, the shorter cycloid doesnt tell us anything new. We already knew that the smaller wheel's circumference was shorter that the larger wheel's. Why wasn't that enough to solve the "paradox"?

J: 

[Jonathan]

On 11/23/2018 at 7:23 AM, merjet said:

Jonathan conveniently omits mentioning that section when he accuses me of "molesting" Wikipedia and similar such rot.

???

Dipshit, I haven't used the word "molesting" on this thread. Dig your senile head out of you ass and try to pay attention to who has actually said what.

[Jonathan]

5 hours ago, anthony said:

A question.

If one cannot entertain, seriously, the notion that there is any ostensible skid/slippage among the painted rings of the archery target, and improbable between an auto tyre and rim, how can the same notion be presented as the solution to the 'Paradox'? If the first is ridiculous, so must be the second.

For the seven hundred billionth time, no one -- NO FUCKING ONE!!!!!!!! -- has entertained the notion that there is skidding/slippage among the circles/rings/wheels!!!! For fuck's sake!!!

No one has presented it as the solution to the "paradox." No one has taken the position that you're refuting. You are attacking straw men while still not having the slightest clue. Repeat, NO ONE HAS TAKEN THE POSITION THAT YOU ARE OPPOSING!!! NO ONE IS SUGGESTING THAT THERE IS SLIPPAGE/SKIDDING AMONG THE CIRCLES/RINGS/WHEELS!!!!!!!!!!!!!!

J

 

 

[Jonathan]

I've witnessed some Dunning-Kruger Effect stupidity out there in everyday life, but it's not even close to what you get here (not Objectivist Living per se, but all of Objectivist-land). Its astounding.

J