Aristotle's wheel paradox

[Jon Letendre]

On 12/29/2018 at 5:03 AM, merjet said:

LOL. Slippage is not a solution; it is a mere restatement of the paradox.

You have it reversed, Merlin.

Your cycloid exposition merely restates or reconfirms the paradox - you merely demonstrate that the small wheel indeed accomplishes a traversing of road length in excess of its circumference, a fact already understood, indeed the very fact that the paradox is all about. Your cycloids confirm the problem, they don’t resolve  it.

The small wheel’s skidding on its road and the precise calculation of that skidding resolves and explains the paradox.

[merjet]

9 minutes ago, Jon Letendre said:

You have it reversed, Merlin.

Your cycloid exposition merely restates or reconfirms the paradox - you merely demonstrate that the small wheel indeed accomplishes a traversing of road length in excess of its circumference, a fact already understood, indeed the very fact that the paradox is all about. Your cycloids confirm the problem, they don’t resolve  it.

The small wheel’s skidding on its road and the precise calculation of that skidding resolves and explains the paradox.

Hogwash.

[Jon Letendre]

What’s really funny is the centuries long record of thinkers dealing with the small wheel at its road and trying to grasp their interaction while you and Tony deny that record and babble on and on, for years, gleefully denying historical fact that that surface has been the topic of thousands of years of discussion of this paradox.

That surface, on which the small wheel appears to perform the impossible, IS the paradox.

[Jon Letendre]

Just now, merjet said:

Hogwash.

I marvel at the depth of your intellectuality.

[Max]

 

7 hours ago, anthony said:

Max: "...an inner track *is* given".

But I'm quite happy to accept an inner track, and such.  

The problems it brings in become greater than the original 'problem'. As my last post, this wheel isn't going anywhere, let alone solve anything.

But the inner track is part of the original problem, as you easily can verify by reading the original text. After all this is Aristotle's paradox and not a "What I Find Intersting About Wheels" discussion. Aristotle wasn't trying to design some new wheel or commenting on the quality of the Greek wheels of his time, this is about kinematics, about circles rolling over tracks, not about dynamics. These may become relevant when constructing functioning wheels, but they are completely irrelevant for solving the paradox. That can be done by purely geometrical/mathematical methods. If you remove that second track,  you throw the paradox-baby out with the track-bathwater. That is not the same as solving it.

[anthony]

On 12/31/2018 at 1:06 AM, Max said:

 

But the inner track is part of the original problem, as you easily can verify by reading the original text. After all this is Aristotle's paradox and not a "What I Find Intersting About Wheels" discussion. Aristotle wasn't trying to design some new wheel or commenting on the quality of the Greek wheels of his time, this is about kinematics, about circles rolling over tracks, not about dynamics. These may become relevant when constructing functioning wheels, but they are completely irrelevant for solving the paradox. That can be done by purely geometrical/mathematical methods. If you remove that second track,  you throw the paradox-baby out with the track-bathwater. That is not the same as solving it.

First, I doubt this was the point. Let's define.

Paradox:

"Statement contrary to received opinion; statement that, whether true or not, seems absurd at first hearing...etc."

Also, in philosophical terms, "an apparent contradiction" (which turns out to not be). So Aristotle's "strange" comment, leads one to think these definitions and the use of "paradox" is correct. ("Strange" - but true - he might have added).

And as it is a "paradox", by definition, the solution *always* is epistemological -- not metaphysical. You have to comprehend the facts you're seeing. One has to induce and integrate the behavior of the wheel into one's knowledge. NOT - force the wheel motion to fit into one's notions of how it should act, or subjectively, how you would will it to act, NOR - 'fix' the wheel.

Second, and briefly entertaining these premises of slippage,

If you've understood, as anyone will, after a brief observation and past experience, that an inner wheel ~IS~ acting precisely according to every wheel, i.e. you see and know its circumference cannot possibly roll a distance equal to the main wheel, but must in fact roll much LESS, since it is (duh!) a smaller wheel -  

and you still have to compulsively 'fix' the problem, via "slippage", thereby hopefully, in theory, extending the inner's circumference to be identical to the outer's -

-what you'll have achieved is gone from a normal wheel to an subnormal wheel. You'll have distorted the causality of wheel motion;

IOW, induced "slippage" will cause the whole wheel/track apparatus to 1. also slide, or 2. halt.

Simply, it doesn't work.

A prime case of "the operation was successful, but the patient died". 

We (I think I speak for Merlin) are realists who start from reality and need to see a wheel still functioning when the paradox is resolved, you guys seemingly are theoretical fantasists who, after all efforts, end up with a non-operating wheel, and what then of "kinematics"?

[Brant Gaede]

The big wheel operates and dominates by sheer torque force. Thus the little wheel gets dragged the difference in their circumferences. If somehow the little wheel had greater torque it would pull the big wheel. But this is physically impossible.

--Brant

[Jonathan]

33 minutes ago, anthony said:

We (I think I speak for Merlin) are the realists who start from reality and need to see a wheel still functioning when the paradox is resolved, you guys seemingly are theoretical fantasists who, after all efforts, end up with a non-operating wheel.

No, you're not realists, you're retards.

J

[anthony]

2 hours ago, Brant Gaede said:

The big wheel operates and dominates by sheer torque force. Thus the little wheel gets dragged the difference in their circumferences. If somehow the little wheel had greater torque it would pull the big wheel. But this is physically impossible.

--Brant

But the inner wheel's track and slippage are supposed to ~extend~ its circumference-distance, not be "dragged".

Put this another way, without the track, both wheels roll together "in harmony".

Those elements of mass, torque and friction are all concentrated onto the large wheel alone, rolling on a single surface.

Now we have an inner track, which compounds these elements upon the second wheel too. There must certainly be a certain amount of additional friction as also there is its individual weight to consider, and therefore, extra torque distributed onto this track. 

And (by some means) this wheel also must skid (lose all frictional drag!) an exact amount, one necessary to equalize their circumferences.

You see what I meant that the track only worsens the minor, original problem? I am sure beyond anything intended by the Paradox.

And without getting anywhere, since, over all, I maintain this wheel can't function. The inner 'balance' by the inner wheel's rotation-velocity has been interfered with. The big wheel will be affected.

[anthony]

On 1/3/2019 at 12:33 AM, Brant Gaede said:

The big wheel operates and dominates by sheer torque force. Thus the little wheel gets dragged the difference in their circumferences. If somehow the little wheel had greater torque it would pull the big wheel. But this is physically impossible.

--Brant

Once a track is put in, the inner wheel's motion isn't any longer just the 'effect' of the big wheel's 'cause'. The two achieve causal parity. The total weight, drag, etc. and velocity of both (fixed) wheels has been transferred onto two surfaces, equalising them, so skidding the smaller one will cause the big wheel to skid too, and vice-versa. Ignoring causality is partly where the slipists have slipped up. When the front wheel on your bicycle skids, doesn't the bike skid? 

[Jonathan]

5 minutes ago, anthony said:

Once a track is put in, the little wheel isn't any longer just the 'effect' of the big wheel's 'cause'. The two achieve causal parity. The total weight etc. of both (fixed) wheels has been transferred and equalized onto two surfaces, so skidding the small one will cause the big wheel to skid too, and vice-versa. Ignoring causality is partly where the slipists have slipped up. When the front wheel on your bicycle skids, doesn't the bike skid? 

You're still an idiot, and you're not using your volition to choose to overcome your idiocy.

J

[anthony]

Slipists prefer slippage/track without the inconvenient consequences - or, having the cake after eating it.

[Ellen Stuttle]

On January 2, 2019 at 6:12 PM, anthony said:

But the inner wheel's track and slippage are supposed to ~extend~ its circumference-distance, not be "dragged".

[....]

And (by some means) this wheel also must skid (lose all frictional drag!) an exact amount, one necessary to equalize their circumferences.

[....]

Now you're making up something else (the idea that the small wheel's circumference is extended) that no one has said - just as you've kept talking as if people were saying that the small circle/wheel slips in relation to the surrounding configuration, when you've been told and hold that no one is saying that.

Ellen

[Jon Letendre]

6 minutes ago, Ellen Stuttle said:

Now you're making up something else (the idea that the small wheel's circumference is extended) that no one has said - just as you've kept talking as if people were saying that the small circle/wheel slips in relation to the surrounding configuration, when you've been told and hold that no one is saying that.

Ellen

It is not fair to get mad.

He cannot understand what we are saying, nor keep it clear in his mind. He can not grasp the paradox setup, it has too many parts, motions and conditions.

[Ellen Stuttle]

9 minutes ago, Jon Letendre said:

It is not fair to get mad.

He cannot understand what we are saying, nor keep it clear in his mind. He can not grasp the paradox setup, it has too many parts, motions and conditions.

Who's mad?

Ellen

[anthony]

26 minutes ago, Ellen Stuttle said:

Now you're making up something else (the idea that the small wheel's circumference is extended) that no one has said - just as you've kept talking as if people were saying that the small circle/wheel slips in relation to the surrounding configuration, when you've been told and hold that no one is saying that.

Ellen

I'd like to hear definitively what the intention is. To 'solve' the paradox? and by what method? 

No one argued with the excerpt by an unknown author, concluding: "...the small wheel will slide to catch up with the pace of the big circle". 

Is that not what you all are trying and doing?

It is not enough for some to keep repeating : but we didn't say that.

[Ellen Stuttle]

8 minutes ago, anthony said:

I'd like to hear definitively what the intention is. To 'solve' the paradox? and by what method? 

No one argued with the excerpt by an unknown author: "...the small wheel will slide to catch up with the pace of the big circle". 

Is that not what you all are doing?

It is not enough for some to keep saying : but we didn't say that.

We did say that the circumference of the small circle slides or slips in relationship to the track.  The small circle is not doing a true roll on the track.

This is not to say that the small wheel slips in relationship to the surrounding figure.  Nor is it to say that the small wheel's circumference is extended (your latest incorrect attribution).

Ellen

[Jonathan]

Jon is right that Tony's mind can't grasp it all at once. That's the main problem that we're having with him. His limited mind is trying to find something that it can hang on to, so it keeps going back to thinking that we are advocating positions that we've already told him that we are not.

The really surprising thing is that we can show him reality -- precisely geometrically accurate videos, animations and diagrams, including ones that are marked for measurement, slowed down, etc. -- and he STILL can't get it! He can't mentally track it. He can't see and comprehend what is right in front of him, and which is the simplest thing in the world for almost all of the rest of humanity.

Added to that is the fact that he doesn't want to understand. He doesn't want to believe that he is deficient in this area.

J

 

[anthony]

25 minutes ago, Ellen Stuttle said:

We did say that the circumference of the small circle slides or slips in relationship to the track.  The small circle is not doing a true roll on the track.

This is not to say that the small wheel slips in relationship to the surrounding figure.  Nor is it to say that the small wheel's circumference is extended (your latest incorrect attribution).

Ellen

If I'd put it "the circumference-length" - or, "circumference distance" - is "lengthened" - would that be more suitable? I have often used these phrases, too. A quibble, I thought my abbreviations would be understood by now.

No, the small wheel certainly cannot slip "in relationship to the surrounding figure". The wheels are fixed. However, either wheel CAN slip in relation to either the surface or the track. 

And as I just explained, any slippage can and will cause the other to slip. One can't remove the inner wheel from its context. 

[Jonathan]

He's like a blind man denying that he's blind, and trying to explain to the sighted what they should see.

J

[Jonathan]

6 minutes ago, anthony said:

And as I just explained, any slippage can and will cause the other to slip. One can't remove the inner wheel from its context. 

You didn't "explain" anything, but merely gave a false opinion.

The physical REALITY of the scenario is that if the large wheel rolls truly and freely on the surface beneath it without slippage, then the small wheel which is attached to the large wheel MUST and DOES slip while rolling on its own surface. THAT IS REALITY! We have explained it and illustrated it for you in the simplest terms possible. You are not grasping reality. You are denying reality, and you're doing so because you are not capable of grasping it.

J-

[Max]

3 minutes ago, anthony said:

And as I just explained, any slippage can and will cause the other to slip. . 

Wrong.

10 minutes ago, anthony said:

One can't remove the inner wheel from its context. 

It's that context that makes the inner wheel slip when the outer wheel doesn't slip.

[Ellen Stuttle]

30 minutes ago, anthony said:

If I'd put it "the circumference-length" - or, "circumference distance" - is "lengthened" - would that be more suitable?

[....]

No, that would not be more suitable.  There's no lengthening of the smaller wheel's circumference.

See Jonathan's and Max's replies to the rest of your post.

Ellen

[anthony]

45 minutes ago, Ellen Stuttle said:

No, that would not be more suitable.  There's no lengthening of the smaller wheel's circumference.

See Jonathan's and Max's replies to the rest of your post.

Ellen

But the only point of the paradox is the different circumferences and their identical travel distance! You're being difficult, you know what I mean.

The notion all you are expressing is the artificial lengthening of the small wheel's travel -- by sliding-- in order to - effectively - equalize the two wheels' circumference-lengths.

Anyhow, an intervention clearly was not what Aristotle was after, he was puzzled that's all, and was looking for an explanation for the phenomenon.

[anthony]

2 hours ago, anthony said:

Once a track is put in, the inner wheel's motion isn't any longer just the 'effect' of the big wheel's 'cause'. The two achieve causal parity. The total weight, drag, etc. and velocity of both (fixed) wheels has been transferred onto two surfaces, equalising them, so skidding the smaller one will cause the big wheel to skid too, and vice-versa. Ignoring causality is partly where the slipists have slipped up. When the front wheel on your bicycle skids, doesn't the bike skid? 

It should be easily understood that placing the two attached wheels on two surfaces ~spreads~ their combined weight equally, and therefore each experiences the same friction and drag -- or lack of.  

Skid one, the other skids too.