Aristotle's wheel paradox

[merjet]

32 minutes ago, Ellen Stuttle said:

Exactly.  Merlin cheated.  I think that he has to have known, in his edit getting rid of the assumption that the smaller wheel was rolling freely, that he was eliminating the basis for the original so-called "paradox."

Ellen

Hogwash.

25 minutes ago, Jon Letendre said:

Maybe he knows he cheated.

Maybe he doesn’t know and simply cannot comprehend what’s going on.

I lean to the latter.

If he ever actually engaged and discussed with one of us, we could see whether he gets what’s going on, but he doesn’t engage.

Earlier in the thread he did, and almost all of it established that he didn’t understand what is going on.

So I lean to the latter.

More hogwash.

[Jonathan]

Hargwarsh!

 

[Jonathan]

Thems is jes cairtoorns! Cretches! Hargawarish 'lusions.

 

[Peter]

hogwash. pig sty. makin' bacon. We have a lot of metaphors and derogatory meanings for pig stuff. I mentioned before how I would torment my Aunt Myrtle's male hog with a squirt gun. The 500 plus pound guy was scared to death of it.

[Jon Letendre]

16 minutes ago, merjet said:

Hogwash.

More hogwash.

Tell us what is the paradox in the new version?

Two differently-colored dotted lines - which are simply equivalent statements of road traversed by the wheel in one rotation - have equivalent length.

Where’s the “paradox”?

[Jonathan]

Muh, muh paradox!

Muh loozhuns.

Muh hargwarsh.

[anthony]

People, please. May we stay in touch with reality?

Fact: The Aristotle circle diagram is an accurate representation of wheels rotating. Since it ain't broke, don't fix it. Any time you want to review this in action, go check a car going past slowly, watch a wheel, pay attention to the inner metal wheel rim turning within the outer rubber tyre. Watch the whole wheel revolve once, and every inner part turn with it.

See any slippage at the cusp of rim and tyre? See any track for the wheel rim to roll on? No - and yet the wheel works.

Transfer this image back to the diagram on your computer. Exactly as a car wheel, the inner circle *must always*, in wheel-reality, move further transversely than its length of circumference  - by definition. It is an "inner wheel"! It has a smaller diameter! The big wheel has a bigger diameter!

All established without recourse to 'tracks'. 

I.e. No matter its dimension, the concentric inner circle traces a path from the point measured by the dotted pink lines from start to finish point, so the distance moved by any small circle = the distance travelled by the large one. A wheel /circle is "a unit", and moves as one. Second, the distance moved laterally by any inner wheel has to always exceed its circumference. If these facts weren't true, then we *would* definitely have a contradiction.

Instead, the diagram is true to reality. Only the entity of the outer wheel counts (for the purpose of the 'paradox').

The interventions, adjustments and compensations attempted here, endlessly, because of one little visual suggestion that the inner wheel's path must be ~corrected~ (with 'slippage' etc.) to match its circumference, amount to trying to force facts to fit into a theory, rationalistically. If this could be physically accomplished, you won't "solve" a paradox, the nature of the wheel will no longer exist, qua wheel.

If Aristotle's (intellectual, and maybe, mischievous) purpose was to highlight the methodological flaws of both rationalism and empiricism, which he anticipated will come out in a debate around his innocuous-looking little diagram, I'd say he succeeded brilliantly.

 

[Ellen Stuttle]

37 minutes ago, anthony said:

[....]

Tony: Next post above:

"Exactly as a car wheel, the inner circle 'must always' move transversely further than its length of circumference "

The supposed "paradox" in the "Aristotle's Wheel Paradox" resulted from the assumption that the inner circle should not move transversely further than its length of circumference, that it should only move the length of its circumference, but instead it moved the length of the large circle's circumference. Hence the apparent contradiction, according to whoever posed the "paradox."  (It might be incorrectly attributed to Aristotle.)

Ellen

[anthony]

Ellen, Would "will always" or " does always" (...move further...) read better?

Sure, I think the whole 'paradox' hangs on the "assumption" that the small wheel circumference dictates its travel distance (as the big one's certainly does). What goes for x, should go for y. Except, it doesn't.

[Ellen Stuttle]

19 minutes ago, anthony said:

Ellen, Would "will always" or " does always" (...move further...) read better?

Sure, I think the whole 'paradox' hangs on the "assumption" that the small wheel circumference dictates its travel distance (as the big one's certainly does). What goes for x, should go for y. Except, it doesn't.

If you understand what the supposed problem in the "paradox" is, then why are you persistently objecting when people give the answer?  (The answer is that the smaller wheel slips relative to its track, with the amount of slippage being the difference between the larger wheel's circumference and the smaller wheel's.)

Ellen

PS: "must always," "will always," "does always" - whichever.  What I was trying to explain to you is what in the situation was taken to be paradoxical.  I haven't seen pervious indications that you understood this.

[Jon Letendre]

You have a long way to go, Ellen.

Tony and Merlin still think that our talk about slippage indicates that we believe, for instance of automobiles, that the circular metal hub fails to hold tight to the rubber tire mounted on it, allowing the rubber tire to rotate at a rate different than the hub rotates.

They also seem to believe that we sometimes mean by slippage that the rubber tires come off of the hubs they are mounted to and go down the road separate from each other.

There is a profound reading comprehension or reading effort thing going on, on top of the problem Jonathan has indicated.

[Michael Stuart Kelly]

2 hours ago, anthony said:

... the concentric inner circle traces a path from the point measured by the dotted pink lines from start to finish point, so the distance moved by any small circle = the distance travelled by the large one.

Tony,

This is the problem you are not seeing.

To measure something, you have to establish a point for where to start the measurement. 

A point on the diameter of either circle does not follow a path represented by a straight line. Yet this is essentially what you claim is happening. 

It doesn't matter whether a line is dotted and pink or thick and puce. A point on either circle will move along a circular path. The only part of the wheel where you can put a point that moves in a straight line--if the straight line is all you want to measure--is the center of the wheel. All other points on either wheel run in circles, not straight lines. So if you want to measure their paths, you need to measure circles, not straight lines.

Hell, notice how much of the circular path is "wrong" at the very start even before anything moves. Half of the diameter is already outside the boundary (to the left) of the starting point before anything happens--and it ends up that way, too. (If I were to be vulgar, I would say it's ass is hanging out, but who wants to be vulgar?  )

Once you start measuring the right thing, you will see the problem and why the slippage happens if both circles are in contact with identical stretches of surface. The points to be measure on them are moving in circles, and the straight line of the surface you are measuring is not a circle. Since the points are running in circles, there is more area each point covers than represented by the straight line (except for the center point)--and this extra area is what you want to measure by the straight line only.

But the circle paths exist. No need to blank them out. They exist. They want to be measured, too. They deserve to be measured. Yet you would deny them their due measurement. Why are you so mean to them?

 

Michael

[Jon Letendre]

2 hours ago, anthony said:

... the concentric inner circle traces a path from the point measured by the dotted pink lines from start to finish point, so the distance moved by any small circle = the distance travelled by the large one.

Tony,

The above is 100% correct.

[Jon Letendre]

25 minutes ago, Michael Stuart Kelly said:

Tony,

This is the problem you are not seeing.

1. To measure something, you have to establish a point for where to start the measurement. 

2. A point on the diameter of either circle does not follow a path represented by a straight line. Yet this is essentially what you claim is happening. 

3. It doesn't matter whether a line is dotted and pink or thick and puce. A point on either circle will move along a circular path. The only part of the wheel where you can put a point that moves in a straight line--if the straight line is all you want to measure--is the center of the wheel. All other points on either wheel run in circles, not straight lines. So if you want to measure their paths, you need to measure circles, not straight lines.

Hell, notice how much of the circular path is "wrong" at the very start even before anything moves. Half of the diameter is already outside the boundary (to the left) of the starting point before anything happens--and it ends up that way, too. (If I were to be vulgar, I would say it's ass is hanging out, but who wants to be vulgar?  )

Once you start measuring the right thing, you will see the problem and why the slippage happens if both circles are in contact with identical stretches of surface. The points to be measure on them are moving in circles, and the straight line of the surface you are measuring is not a circle. Since the points are running in circles, there is more area each point covers than represented by the straight line (except for the center point)--and this extra area is what you want to measure by the straight line only.

But the circle paths exist. No need to blank them out. They exist. They want to be measured, too. They deserve to be measured. Yet you would deny them their due measurement. Why are you so mean to them?

 

Michael

1. Ok

2. Correct, Michael, such a point actually curves through space during rotation, not straight movement.

However, What Tony means is that the straight line connecting the start to the end point is the length of road the wheel travels.

He means simply: Every inner circle travels the same distance the wheel travels, and that is correct. Take a point anywhere on the wheel. Draw a line from where it started to where it ended. That line length is again, the length of road the wheel travels.

You are almost correct about any such point (except the center point) traveling a circular path. It travels a path known as a cycloid, have a quick look: https://en.wikipedia.org/wiki/Cycloid

3. He doesn’t want to measure their paths. He wants to measure how far the wheel went, and he is doing it the right way.

Might be enough for now, ok so far?

[Peter]

Ellen wrote: Every inner circle travels the same distance the wheel travels, and that is correct. end quote

My parents bought me and my older brother Schwinn’s one Christmas and mine was too big for me. I needed a wooden block attached to each pedal to raise its height but I finally mastered the big bike. I found that I could out travel kids with smaller bike tires and I pedaled less hard. I remember switching bikes with another kid and his little wheeled bike required that I work harder and harder to go the same distance as the bigger Schwinn. He wanted to trade.

Now if the inner and outer portions of the tire were not attached and instead made up two separate bikes one with big wheels and one with small wheels the smaller wheeled bike would initially out sprint me but after X amount of revolutions my bigger Schwinn would win the race.

The other point I want to make is that “girls’ bikes” make more sense for a boy’s anatomy because the boy’s genitals are in less danger of being squished. Peter

[Jonathan]

My favorite rule of geometry is the one that says that no reasoning or proof is required to dismiss any geometrical drawing or animation. If you don't like a geometrical diagram, or animated sequence of diagrams, because you have feelings and don't want it to be true, you don't have to show any errors in it, or identify any specific false measurements or angles, etc. All that you have to do is to say that it's a mere silly "cartoon," like Donald Duck or Scooby Doo, and that it is therefore automatically invalid. Or shout "Hogwash!" That's enough to officially refute it. 

No work, no geometrical principles or theorems cited or applied. Easy peasy, just call it a con job, and then everyone else has to go along with you and blank it out of existence. It's a rule. A rule of real geometry.

 

https://goo.gl/images/w4wnP3

 

Hargwarsh!

 

https://goo.gl/images/1krXuA

 

Cartoon! Crutch!

 

https://goo.gl/images/taMZk1

 

Scam! Con!

 

https://goo.gl/images/cDgMuY

 

Diseased mind illusions! 

 

J

[Michael Stuart Kelly]

2 hours ago, Jon Letendre said:

Might be enough for now, ok so far?

Jon,

That's OK with the way I think.

In the cycloid illustration...

... it is clear to me that people who only consider the whole wheel are treating the entire small line in the middle of the circle as the middle point only. They ignore the rest of the small line. That makes them leave out the extra distance of the arcs, which is drawn in the animation with the end point of the line, not the center point. The end point is both part of the line, the end of it, and it sits on the wheel's circumference as the circle rotates along a surface.

I get why Tony misses this because I was doing the same thing at first. The fact is, though, based on the way he has described what he sees when watching a whole wheel with two or more rims attached rotate (like a wheel on a car), the outer rim is in contact with a hard surface, but the inner rim(s) is in contact with air. Nothing more. So he simply doesn't see any slipping by the inner rim(s) because who the hell slips against air? Hell, I don't see it, either.

I suppose, technically, a kind of slippage is there with air, but who can see it?

The standard diagram for the paradox sure as hell doesn't show that one line is a hard surface and the other line (or lines) is air. To a casual observer with a "whole wheel" perspective, it looks like all lines are hard surfaces in contact with the hard rims of their respective circles. Thus it looks like a paradox, but it's an illusion due to diagramming an abstract surface for the inner circle that is, in reality, nothing but air along an imaginary line.

That, to me, is especially the part where I agree with you--the diagram does not represent the reality of rotating circles, at least not the reality of hard rims rotating against hard surfaces.

This is an interesting problem, both in figuring it out and in trying to figure out how to teach people to see it.

This brings up a tangent in my mind. In storytelling (including the neuroscience of story), there is an equivalent for the "whole wheel" perspective that I, also, used to hold as the only valid one. It's called a frame--or core story, or core narrative model, etc.

It can be (and is) engineered on purpose in propaganda to get people to not see things they otherwise would, and to be resistant to seeing them even when explained correctly. (Think "muh Russians" for instance. Merely debunking it with facts doesn't work to persuade the victims of the kind of story indoctrination they have been subjected to.)

If they see something is in front of them that contradicts their core story, and they are dimly aware that their view is flawed, in order for them to see the thing correctly, they have to see it through a different frame, a different core story, a different core model of how the world works. That kind of meta-thinking is a skill that has to be learned. It's a damn hard skill to learn, too. It doesn't come naturally. In fact, in my experience, most people who can't do it get scared shitless when they contemplate (for real) abandoning their core frames, even for a thought experiment.

I've got a whole theory about anchor neurons (my term) that control neural pathways. (Actually they are neural networks, but I like the term "pathway" for ease of imagining.) Many of these anchor neurons are situated in the hippocampus and when overly excited through stress, they actually cause people to snap and commit violence or other deeds without remembering a thing.

(There are many experiments where scientists have artificially triggered this snapping process in lab rats by drilling holes in a rat's skull, ramming fiber optic cables down through the brain until each end point sits on a specific neuron. When they run light through a fiber optic cable and stop for, say, an aggression neuron, the rat immediately goes ballistic and stops like with an on-off switch. Ditto for sex, basking, and other actions. btw - The image of a rat with what looks like whiskers coming out of the top of its head is really weird to think about.  

Back to the indoctrinated. This core story model--or at least a part of it--gets engraved on these anchor neurons as the entire neural pathways they control get myelinated from emotional and/or focused repetition. This structure is physical, not just abstract, and that's what makes it such a bitch to change certain ideas (in others or even in ourselves--think the success rate of New Years Resolutions for an easy example) or communicate around them.

Anyway, I see parallels here with the different perceptions of the wheel paradox. But enough, already!  

End of tangent. 

Michael

[Jon Letendre]

13 hours ago, Michael Stuart Kelly said:

Jon,

A real life example might make what's in my head clearer. Take this cool-ass bike of yours:

Now imagine the wheels get arranged and connected as in the so-called paradox, side by side..

Then remove the tires, cut them (each at one point only) and lay them out on the ground as straight lines. There will be one really long line and one really short one.

They will show completely different lengths for a stretch traveled when measuring the distance from a beginning point to an end point referencing only the axle where your pedals and feet are.

Now with the tires back on, run the bike from one point to another. To me, at least, I see the distance traveled by the axle between the points is not the same as the distance covered by a single point on the circumference of each wheel. Why? Because the axle travels in a line (one dimension), not in a circle. The point on the circumference of each wheel covers the same length of line as the axle, but they also cover the rest of each respective circumference as it goes up and down around each circle (two dimensions).

I probably was not clear in my previous post because I talked only about the animation I presented and said both circumferences were each longer than the straight line if they were made into straight lings. That would obviously not be the case in the bike example due to the really small size of the smaller wheel. (Sorry... I got excited by actually figuring the damn thing out. So I didn't run through any other mental permutations even though you had already presented a straw and jar lid with large size differences.  )

At any rate, I fully get what you say when you say that the animation "is not a valid or accurate depiction of a rolling wheel." 

It really isn't.

The point on the circumference is not traveling the same distance as the straight line. It's covering a lot more distance. People who think it is the same distance are leaving out the up and down distance it travels around the circle. In other words, in my own way of saying it, they leave out the second dimension.

But reality doesn't leave out the second dimension. The full circle (or wheel) exists in two-dimensional space on a two-dimensional diagram, not just on a one dimensional line. (A wheel actually exists in reality in three dimensions, but who's talking real reality except for people who live and work in it?  )

So it's evident--to me at least--that the only way both circles can be in contact with surfaces of the same length, travel that same length in a line (one dimension), and one of the circles doesn't slip on the surface, is for the other one to slip. The extra distance traveled by the points on the circumferences as the circles go around need to be accounted for or "metaphysically compensated" so to speak. The only way for that to happen is to periodically disconnect one of the wheels from the surface or drag it, i.e., slip.

The diagrams in Aristotle's paradox, even those I've seen that are not animated, are very misleading as to what actually happens in reality. That's because they induce you blank out the second dimension in calculating the distance a point actually travels through space.

If the point is the center (or the axle), it travels one distance through space. It travels in one dimension. If the point is on the circumference, it travels a different distance through space. It travels in two dimensions. Yet the diagram implies that only one dimension counts for both points.

Is that clearer? (My problem is not conceptual right now. It's the damn words that are getting in the way.  )

Michael

Getting clearer. I think I follow most of it and it most of it is sensible.

[Jon Letendre]

Michael,

I really think we need to start at the beginning.

The Paradox setup requires an abstraction.

The paradox is: A wheel rolls a road and it appears lesser diameters within the wheel roll the same road length, which is impossible due to their reduced circumference.

The Paradox doesn’t exist until we abstract the lesser diameters as wheels in themselves, rolling on a drawn road. Until we do that, you are right, an inner circle is not a wheel and it is moving through air, not in contact with any road. There is just a rolling wheel, and there is no paradox.

There is only a paradox after we abstract the inner circle as a wheel in itself rolling down its drawn road. NOW we have a paradox: Inner diameters which, when thought of as wheels rolling on their road, can and do roll distances in excess of their circumferences, which is impossible. That’s a paradox. That’s Aristotle’s Wheel Paradox.

[Jon Letendre]

8 hours ago, anthony said:

People, please. May we stay in touch with reality?

Fact: The Aristotle circle diagram is an accurate representation of wheels rotating. Since it ain't broke, don't fix it. Any time you want to review this in action, go check a car going past slowly, watch a wheel, pay attention to the inner metal wheel rim turning within the outer rubber tyre. Watch the whole wheel revolve once, and every inner part turn with it.

See any slippage at the cusp of rim and tyre? See any track for the wheel rim to roll on? No - and yet the wheel works.

Transfer this image back to the diagram on your computer. Exactly as a car wheel, the inner circle *must always*, in wheel-reality, move further transversely than its length of circumference  - by definition. It is an "inner wheel"! It has a smaller diameter! The big wheel has a bigger diameter!

All established without recourse to 'tracks'. 

I.e. No matter its dimension, the concentric inner circle traces a path from the point measured by the dotted pink lines from start to finish point, so the distance moved by any small circle = the distance travelled by the large one. A wheel /circle is "a unit", and moves as one. Second, the distance moved laterally by any inner wheel has to always exceed its circumference. If these facts weren't true, then we *would* definitely have a contradiction.

Instead, the diagram is true to reality. Only the entity of the outer wheel counts (for the purpose of the 'paradox').

The interventions, adjustments and compensations attempted here, endlessly, because of one little visual suggestion that the inner wheel's path must be ~corrected~ (with 'slippage' etc.) to match its circumference, amount to trying to force facts to fit into a theory, rationalistically. If this could be physically accomplished, you won't "solve" a paradox, the nature of the wheel will no longer exist, qua wheel.

If Aristotle's (intellectual, and maybe, mischievous) purpose was to highlight the methodological flaws of both rationalism and empiricism, which he anticipated will come out in a debate around his innocuous-looking little diagram, I'd say he succeeded brilliantly.

 

Then tell us - what is the paradox?

We know a wheel is rolling. Anything else? Anything paradoxical?

[Jon Letendre]

Tony likely will answer “the assumption that the small wheel circumference dictates its travel distance.”

But that is not paradoxical, it is merely a trivially false assumption. Wheels stay together. Drawings on them stay where you drew them. A wheel cannot outpace the drawings on it. Only the total diameter dictates distance, it is quick business, simple false assumption.

Indeed it is a comically incoherent assumption. There are an infinitude of small wheels, so which one exactly did you imagine anyone might believe dictates distance?

Besides. Wait. Before I say this, Merlin, Tony are you there? Are you listening? Do you have your reading glasses on? That is NOT what has been passed to us through the millennia, you fucking retards. Look it up. (And not at Wikipedia where Merlin molested history, geometry, mechanics, etc.)

What history gives us is this: “the appearance of the small wheel traversing road length in excess of its circumference, which is impossible.”

That is Aristotle’s Wheel Paradox.

[anthony]

12 hours ago, Michael Stuart Kelly said:

Tony,

This is the problem you are not seeing.

To measure something, you have to establish a point for where to start the measurement. 

A point on the diameter of either circle does not follow a path represented by a straight line. Yet this is essentially what you claim is happening. 

It doesn't matter whether a line is dotted and pink or thick and puce. A point on either circle will move along a circular path. The only part of the wheel where you can put a point that moves in a straight line--if the straight line is all you want to measure--is the center of the wheel. All other points on either wheel run in circles, not straight lines. So if you want to measure their paths, you need to measure circles, not straight lines.

Hell, notice how much of the circular path is "wrong" at the very start even before anything moves. Half of the diameter is already outside the boundary (to the left) of the starting point before anything happens--and it ends up that way, too. (If I were to be vulgar, I would say it's ass is hanging out, but who wants to be vulgar?  )

Once you start measuring the right thing, you will see the problem and why the slippage happens if both circles are in contact with identical stretches of surface. The points to be measure on them are moving in circles, and the straight line of the surface you are measuring is not a circle. Since the points are running in circles, there is more area each point covers than represented by the straight line (except for the center point)--and this extra area is what you want to measure by the straight line only.

But the circle paths exist. No need to blank them out. They exist. They want to be measured, too. They deserve to be measured. Yet you would deny them their due measurement. Why are you so mean to them?

 

Michael

Michael,

I think for certain the cyclical movement of a point in a circle is superfluous to this 'paradox'. But of course, we know a point in a wheel, and any point in a wheel and circle, revolves. All that matters here, though, is where it begins and where it ends - linearly. Point a. to point b. can be physically marked on a wheel and observed. Or, imagined.

To reiterate.

The diagram shows that the distance traversed (linearly) after a single rotation, is more than the inner circle's circumference. A). Circumference (ic) is less than total distance. And increasingly less as the smaller inner circle is reduced. Until one can easily arrive at a 25: 1 ratio (distance: circumference) and more, as the case in Jon's experiment. That's a huge amount of 'slippage' necessary to accomodate the hypothesis.

We can see visually by its line paths, and are informed by the explanation, that the outer circle covers exactly its own circumference. B). Circumference (oc) = total distance.

A vs. B creates a (false) contradiction. But linear distance travelled comparative to circumference, is the ~only~ premise of this 'paradox' (which isn't a paradox, but it is true to the reality of a wheel).

So it's not blanking out the "circle paths", here they are not relevant - and a confusing red herring.

This is all and only about *distance*. After identification of a wheel's nature, all the math and geometry and mechanics then will play an important part.

[Jonathan]

5 hours ago, Michael Stuart Kelly said:

Jon,

That's OK with the way I think.

In the cycloid illustration...

Cartoon?!!!

Hawrgwarsh!!!

Loozhuns!!!

[anthony]

13 hours ago, Ellen Stuttle said:

If you understand what the supposed problem in the "paradox" is, then why are you persistently objecting when people give the answer?  (The answer is that the smaller wheel slips relative to its track, with the amount of slippage being the difference between the larger wheel's circumference and the smaller wheel's.)

Ellen

 

 

Clearly I disagree. ;) No slippage. If a track is "the answer" and has to be added for the smaller wheel to run on, done in an accurate experiment it won't slip on that either. The problem is created I believe by 'taking out' (reductionism) a single part of the entire wheel and treating it in isolation. "Tracks" , slippage and stuff, however specially "defined" (as some have insisted) are in self-contradiction to the identity of a wheel.

The inner wheel does (has to) rotate at a proportionately lesser velocity (rotational, not transitional) than the outer. Therefore, it linearly covers the identical distance in the identical time, in a single revolution.., as is obvious if you roll a wine bottle on the floor. A secondary track not required.

[Max]

1 hour ago, anthony said:

Clearly I disagree. No slippage. If a track is "the answer" and has to be added for the smaller wheel to run on, done in an accurate experiment it won't slip on that either. The problem is created I believe by 'taking out' (reductionism) a single part of the entire wheel and treating it in isolation. "Tracks" , slippage and stuff, however specially "defined" (as some have insisted) are in self-contradiction to the identity of a wheel.

 

Yes, it will slip, as I've proved mathematically (see my post of February 4) and as Jon and Jonathan have very clearly visualized in their videos and animations (do you insinuate that these are optical illusions?). Show me were I made an error in my proof, if you can, and with real hard arguments, not with some confused metaphysical nonsense like "self-contradiction to the identity of the wheel".