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Hello everyone.

Something for the Physicist, Mathematician, and Philosopher in each of us.

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The difference is in the attempt to apply a linear dimension to both circumferences and equate them with the distance traveled. Only the outer diameter rotates where the relationship between the angular progression and the circumferential engagement are 1:1. The inner diameter goes along at 1:1 with regard to the angular progression.

There may be some quibble with the terminology, but it appears that a paradox is contrived by trying to combine unlike terms.

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22 hours ago, dream_weaver said:

The difference is in the attempt to apply a linear dimension to both circumferences and equate them with the distance traveled. Only the outer diameter rotates where the relationship between the angular progression and the circumferential engagement are 1:1. The inner diameter goes along at 1:1 with regard to the angular progression.

There may be some quibble with the terminology, but it appears that a paradox is contrived by trying to combine unlike terms.

Is it possible to "dissect" the paradox to identify exactly where and how (perhaps even why) one is "potentially" led astray?  Clearly, some people were and possibly are of the kind who see this as being some problem or riddle... they perceive a paradox...  is it of any use for us to try to understand why?

Should we merely chalk it up to a fictional problem in a bad thinker's mind, and dismiss it completely from our thought and effort?

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Posted (edited)

It looks to me that the second resolution of the paradox resolves it completely. A point on any of the concentric circles will travel through a curvilinear path (a cycloid) in the plane while the single revolution of the wheel takes place. The translational advance of a point on every circle is the same. The outer circle gets to determine how much that translational advance over the road is in one revolution, but it too, like a point on any other circle, does not trace a path of that length in its course through the plane. The center point is a circle of zero radius, and the curve it traces in space is the straight line limit that the other curves approach as we look at smaller and smaller radii.

Gregg's remark seems headed this way also. And there is a visualization error here too, in that one might neglect to imagine the path traced in the plane by a point on the biggest circle and just stay fixed on the fact that that point went around in merely a circle, neglecting the translational element in the path of that point. It is mysterious to me why any of the thinkers mentioned in the article who lived after Galileo were unable to diagnose this paradox. (I'd bet that impression of inabilities reported is false.) It is mysterious that the author gives this diagnoses (their second one) without any suggestion of who was earliest in seeing this resolution.

If we consider the rotation of a water wheel, the translational distance in one revolution is zero. A point on any of the concentric circles of the wheel translate that same amount: none. The the length of the curvilinear path (a circle) traced by a point on any of those circles varies with radius of the circle, as was shown also for the non-zero translation case (lengths of cycloids).

Edited by Boydstun
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1 hour ago, Boydstun said:

It looks to me that the second resolution of the paradox resolves it completely. A point on any of the concentric circles will travel through a curvilinear path (a cycloid) in the plane while the single revolution of the wheel takes place. The translational advance of a point on every circle is the same. The outer circle gets to determine how much that translational advance over the road is in one revolution, but it too, like a point on any other circle, does not trace a path of that length in its course through the plane. The center point is a circle of zero radius, and the curve it traces in space is the straight line limit that the other curves approach as we look at smaller and smaller radii.

Gregg's remark seems headed this way also. And there is a visualization error here too, in that one might neglect to imagine the path traced in the plane by a point on the biggest circle and just stay fixed on the fact that that point went around in merely a circle, neglecting the translational element in the path of that point. It is mysterious to me why any of the thinkers mentioned in the article who lived after Galileo were unable to diagnose this paradox. (I'd bet that impression of inabilities reported is false.) It is mysterious that the author gives this diagnoses (their second one) without any suggestion of who was earliest in seeing this resolution.

If we consider the rotation of a water wheel, the translational distance in one revolution is zero. A point on any of the concentric circles of the wheel translate that same amount: none. The the length of the curvilinear path (a circle) traced by a point on any of those circles varies with radius of the circle, as was shown also for the non-zero translation case (lengths of cycloids).

You are at an intellectual advantage, whether by nature, nurture, or self made and you’ve shown that once more.

I do begin to wonder whether as a compliment or adjunct to philosophy (perhaps it already forms part of it) the study of what leads the mind astray, causes confusion, gives rise to the appearance of paradox etc should not have a more prominent place, perhaps even outranking epistemology (maybe it’s a small subset of epistemology) in importance … it seems to me that

insufficient knowledge was never the cause of all the woes of a man or mankind but the over abundance of false impressions and ideas masquerading as knowledge.

Perhaps the study of fools is the path to wisdom?

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9 hours ago, StrictlyLogical said:

Clearly, some people were and possibly are of the kind who see this as being some problem or riddle... they perceive a paradox...  is it of any use for us to try to understand why?

I don't think it's the right approach here to think of it as if it's a problem only because of a contrived epistemological issue.

Whether or not this problem is really from Aristotle or one of his students, the Aristotelian approach to problems is more like "that's an interesting observation, and not intuitively obvious things would be this way, so what is the explanation?" The approach would not be something like "the paradox shows that the world is broken, we can't find mathematical principles, the truth is uncertain!"

I thought the problem was interesting, but not particularly difficult to figure out an explanation. To me, this looks like a problem to invite an Aristotelian answer: the parts together as a whole, operating as a unity. We probably shouldn't think of it as a problem, but a question.

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Posted (edited)

The author of Mechanics, or Mechanical Problems, does not regard #24, which is the problem discussed in this thread, as unresolvable, only wondrous, and he offers a solution. I gather his solution fails; he's just spinning his wheels. The author is not Aristotle, all agree. A good argument has been made that the author was the famed Archytas, who was a contemporary of Plato.

Edited by Boydstun
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Posted (edited)
17 hours ago, Boydstun said:

It looks to me that the second resolution of the paradox resolves it completely.

I agree completely. It would be strange if I didn't, since I wrote the Analysis and Solutions section of the Wikipedia article.

17 hours ago, Boydstun said:

It is mysterious that the author gives this diagnoses (their second one) without any suggestion of who was earliest in seeing this resolution.

To the best of my knowledge, I am the only person who has ever resolved the paradox  based on cycloids. Mathematical Fallacies and Paradoxes deals with the paradox and mentions cycloids, but says the solution is that the smaller circle slips. "Skids" is better than "slips", but even "skids" is based on a flawed analogy. The behavior of the smaller circle is partly similar to that of a real-world wheel skidding, such as caused by the driver of a car braking hard and the wheel losing traction due to ice or snow on the road.

One can peek at the relevant pages of Mathematical Fallacies and Paradoxes using Amazon's "look inside" feature and the search terms "dime" and "cycloid." The author uses a dime and half-dollar glued together to illustrate the paradox. The analogy fails because the dime does not lose traction and is not even on its own surface.

Edited by merjet
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Posted (edited)
23 hours ago, Boydstun said:

It looks to me that the second resolution of the paradox resolves it completely. A point on any of the concentric circles will travel through a curvilinear path (a cycloid) in the plane while the single revolution of the wheel takes place. The translational advance of a point on every circle is the same. The outer circle gets to determine how much that translational advance over the road is in one revolution, but it too, like a point on any other circle, does not trace a path of that length in its course through the plane. The center point is a circle of zero radius, and the curve it traces in space is the straight line limit that the other curves approach as we look at smaller and smaller radii.

Gregg's remark seems headed this way also. And there is a visualization error here too, in that one might neglect to imagine the path traced in the plane by a point on the biggest circle and just stay fixed on the fact that that point went around in merely a circle, neglecting the translational element in the path of that point. It is mysterious to me why any of the thinkers mentioned in the article who lived after Galileo were unable to diagnose this paradox. (I'd bet that impression of inabilities reported is false.) It is mysterious that the author gives this diagnoses (their second one) without any suggestion of who was earliest in seeing this resolution.

If we consider the rotation of a water wheel, the translational distance in one revolution is zero. A point on any of the concentric circles of the wheel translate that same amount: none. The the length of the curvilinear path (a circle) traced by a point on any of those circles varies with radius of the circle, as was shown also for the non-zero translation case (lengths of cycloids).

Boydstun:

How do you feel about engaging one-on-one with me, in a Socratic, dispassionate, impersonal, and objective discussion, aimed at discovering exactly what is going on when one falsely perceives, at first blush or on an intuitive level, an "issue" or "problem" when presented with the kind of "description" or set-up presented for this rolling disc?

Edited by StrictlyLogical
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54 minutes ago, StrictlyLogical said:

aimed at discovering exactly what is going on when one falsely perceives, at first blush or on an intuitive level, an "issue" or "problem" when presented with the kind of "description" or set-up presented for this rolling disc?

Why do you interpret it as if someone claimed that this problem is a problem in the sense that they are claiming that something is "wrong" with reality? The observation is real, and it's good to explain observations.

It's not like you would open to the back of a physics book and complain about how many problems there are. That's what it looks like: a problem designed for students to solve.

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Posted (edited)

SL,

Private discussion, fine, far as I can work it in. Sounds enjoyable. With all the pertinent information from experts in my personal library and online these days, I’ve no time for “Socratic discussions” with his armchair-planted ways. (Though I do read Plato and his man, as contributors to a long and now vast awakening to the world and the mind.)

I know some about some illusions from the scientifically informed philosophy of perception books and papers and from brain science of today. But below, I’ll leave, for anyone, samples of expert work on other sorts of errors and puzzles from the practical to the high-brow.

Edited by Boydstun
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On 5/27/2021 at 9:46 AM, StrictlyLogical said:

Is it possible to "dissect" the paradox to identify exactly where and how (perhaps even why) one is "potentially" led astray?  Clearly, some people were and possibly are of the kind who see this as being some problem or riddle... they perceive a paradox...  is it of any use for us to try to understand why?

Should we merely chalk it up to a fictional problem in a bad thinker's mind, and dismiss it completely from our thought and effort?

Fictional problem? Even Zeno's paradoxes were not fictional in the context of what was known at the time.

Maybe my attempts of understanding how involute curves of two meshing gears interact with one another contributed to my rather quick dismissal of the dilemma from the OP. Or the cut of a thread expressed in terms of 1000's of degrees, where 360° is one full revolution of the workpiece on a lathe as the cutter moves in a linear progression relative to the speed of the rotation for analyzing the production of a helix of a thread.

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If these scans show correctly, they show all of #24, the problem and its solution according to the author. If that author is Archytas, he would have been writing before Euclid. Like Euclid later on, he is stating problems and supplying a solution. In that respect, Eliuol, it's not like a modern textbook giving problems for the student to solve.

The solution the author gives to #24 is junk and not an example for right thinking. We're fortunate today on this problem: we have available for our easy understanding the solution given by Jetton (second solution) in Wikipedia, as I elaborated it above. No turning to dynamics (even were it correct), such as was done by the author of Mechanical Problems, is helpful. That turning is only distracting from the real arena of the solution, which is purely geometric and kinematic, however useful and ordinary the application of the geometry and kinematics in the world.

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Posted (edited)
2 hours ago, dream_weaver said:

Fictional problem? Even Zeno's paradoxes were not fictional in the context of what was known at the time.

Maybe my attempts of understanding how involute curves of two meshing gears interact with one another contributed to my rather quick dismissal of the dilemma from the OP. Or the cut of a thread expressed in terms of 1000's of degrees, where 360° is one full revolution of the workpiece on a lathe as the cutter moves in a linear progression relative to the speed of the rotation for analyzing the production of a helix of a thread.

"Fictional problem", in the sense that a "paradox" must involve some disconnect with reality.  Reality has no problems, the problems are thus fictional.

No hypothetical shape, event, situation, process, system, etc. which is obvious and behaves exactly as "expected" or "intuited" was ever called a "paradox".   Neither was anything which was judged too new or too complex to understand. Differential geometry is not a paradox to a musician, it's just something he/she does not have training in and does not understand, but he has no reason to suspect "paradox".  A paradox requires an experience that something is amiss... but there are no contradictions in reality (no matter how many opposing forces, collisions or disagreements) there is only existence and existence is identity.

So the "problem" is fictional, in the same way an illusion introduces a fiction... reality is what it is, but something about what we see, and should understand, is off kilter, and we know it. At least for those who experience the particular paradox...  the feeling of paradox requires a certain thinking process to get a person in the wrong place to sense that disconnect, and in truth, different people are often led in different directions...

I think in a sense the more something appears or seems opposite of what one assumes it obviously should appear or seem like, the more paradoxical it is.  Since reality is NOT at fault, our sense and assumptions of what things obviously should appear or seem like, IS.

Edited by StrictlyLogical
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Posted (edited)
2 hours ago, Boydstun said:

If these scans show correctly, they show all of #24, the problem and its solution according to the author. If that author is Archytas, he would have been writing before Euclid. Like Euclid later on, he is stating problems and supplying a solution. In that respect, Eliuol, it's not like a modern textbook giving problems for the student to solve.

The solution the author gives to #24 is junk and not an example for right thinking. We're fortunate today on this problem: we have available for our easy understanding the solution given by Jetton (second solution) in Wikipedia, as I elaborated it above. No turning to dynamics (even were it correct), such as was done by the author of Mechanical Problems, is helpful. That turning is only distracting from the real arena of the solution, which is purely geometric and kinematic, however useful and ordinary the application of the geometry and kinematics in the world.

What I find fascinating about this wheel issue, is that a full enough description of the actual motion of the wheel and its parts relative to the ground and its frame of reference is, arguably (theoretically), all that is required to dispell the apparition of paradox from the mind of one capable and willing to understand fully, for when the confusion at issue is removed and reality laid bare... what else needs to be said?

There are different descriptions of that reality with different focii and different levels of completeness, which nonetheless will be sufficient to dispell the misgivings, depending upon the mind in which the irksome feeling of "paradox" resides, the particular form the paradox takes, and the particular sum of integrated and connected knowledges and intuitions of the person, which allows them to, by thier own routes, untie themselves from the Gordion knot.

It's fascinating to note that minds differ so much they will argue endlessly whether or not some particular truth told in a certain way about the non paradoxical thing (in reality) is in fact enough to dispell the misapprehensions.  That there is so much disagreement over which truth among many "really works" points mainly to the way paradox and misapprehension, errors of the mind, lodge themselves, they must be of widely varying natures and magnitudes.

Who am I to say your realization has not led you out of the labyrinth?  When I see that I require mine to escape.

Many are the different ways our minds are each led astray and knotted up, and so too, many are the different ways which work to lead us each aright and unknot our thinking.

Edited by StrictlyLogical
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3 hours ago, Boydstun said:

Like Euclid later on, he is stating problems and supplying a solution. In that respect, Eliuol, it's not like a modern textbook giving problems for the student to solve.

That makes sense. Maybe not designed for a student then, but more like those kind of problems where you know intuitively that there is a reasonable solution, but can't solve it yet because you need to be a little more creative.

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Posted (edited)

Yesterday, while pursuing my current series "Prime Movers, Immovable Movers, Self-Movers", I came across the following in Plato's Laws. In rotary motion of a disk there are "points near and far from the center describ[ing] circles of different radii in the same time; their motion varies according to these radii and is proportionately quick or slow. This motion gives rise to all sorts of wonderful phenomena, because these points simultaneously traverse circles of large and small circumference at proportionately high or low speeds---an effect one might have expected to be impossible."

There is something fascinating and, as SL termed it earlier, unexpected that the Greeks experienced in thinking about rotation of a disk, even at this elementary look mentioned by Plato. I notice the words "wonderful" and "expected to be impossible." Archytas was a personal friend of Plato, and Plato may well have seen the puzzle #24 and could have it in mind as among the "wonderful phenomena" stemming from rotating disks. The sort of showman talk of Plato and of the author of Mechanics reminds me a lot of Galileo's way of presenting mechanical things---here's an amazing thing, and I have the secret of how it comes about.

I suppose "paradox" covers a pretty wide variety of puzzles. I gather that the paradoxes of Zeno and pals were not put forth as problems having solutions, but as absurdities one enters when one denies the doctrines of their master Parmenides (doctrines folks outside that school find absurd). Those paradoxes---deeper perhaps than the wheel one we've look at---continue to be analyzed today. I've books with various resolutions to paradoxes of Zeno, but I've never pulled them all together and made an assessment.

The wheel case seems to have an element of deception to it, but I don't think it was put there in an effort to trick. Rather, there was something naturally tricking most any mind thinking about the setup. This problem involves perception/imagination, but also trains of thought. It is among cases that when intellectually resolved, understood, the illusion is dispelled. That's not true of our purely perceptual illusions, I've noticed. We continue to see the sun and moon near the horizon as larger than when they are high in the sky, even though we know they do not change size. The illusion, the dependable experience itself, is not altered by our knowing it to be an illusion (by reasoning and by taking a photograph). Also, I do not get robbed of the pleasure of that illusion, by knowing it to be an illusion; and I expect the illusion and the pleasure would continue all the same if I learned some definitive explanation of how the illusion comes about in the human visual system.

Edited by Boydstun
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Here is a diagram inverting the illustration to begin with the points on the "wheel" pointing downward. The "spoke" is provided at 10° intervals, which were then used to create the cyan and magenta splines emulating the path of the points as the "wheel" rolled horizontally.

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Posted (edited)
18 hours ago, Boydstun said:

Yesterday, while pursuing my current series "Prime Movers, Immovable Movers, Self-Movers", I came across the following in Plato's Laws. In rotary motion of a disk there are "points near and far from the center describ[ing] circles of different radii in the same time; their motion varies according to these radii and is proportionately quick or slow. This motion gives rise to all sorts of wonderful phenomena, because these points simultaneously traverse circles of large and small circumference at proportionately high or low speeds---an effect one might have expected to be impossible."

A counter-intuitive oddity, a brain-teaser rather than a paradox, imo. Quite something that Plato was then onto "tangential velocity" (the 'rotating' speed of various radii) clear above.

"tangential velocity is directly proportional to the radius. It increases because tangential velocity is inversely proportional to the radius". Wiki

In the Paradox as presented, the suggestive, visual red herring is an *inner* wheel 'track' or line, exactly equalling the length of the outer - except - the wheels are different diameter/circumferences!

Of course, the larger one's circumference singly dictates the distance covered and all inner points of a moving object correspond.

Paradox explained, I reckon, by the inner wheel turning at a slower (vt) on its 'track' than the larger in order to also complete one identical revolution as the outer rim, and to traverse the identical track distance in the identical time.

Demonstrating the non-contradictory nature of a wheel's properties, how it's supposed to act and does act.

Edited by whYNOT
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Posted (edited)
18 hours ago, Boydstun said:

The wheel case seems to have an element of deception to it, but I don't think it was put there in an effort to trick. Rather, there was something naturally tricking most any mind thinking about the setup. This problem involves perception/imagination, but also trains of thought. It is among cases that when intellectually resolved, understood, the illusion is dispelled. That's not true of our purely perceptual illusions, I've noticed. We continue to see the sun and moon near the horizon as larger than when they are high in the sky, even though we know they do not change size. The illusion, the dependable experience itself, is not altered by our knowing it to be an illusion (by reasoning and by taking a photograph). Also, I do not get robbed of the pleasure of that illusion, by knowing it to be an illusion; and I expect the illusion and the pleasure would continue all the same if I learned some definitive explanation of how the illusion comes about in the human visual system.

Stephen, much on board with this, that one can hold both the enjoyment of illusions together with their objective explanations. (The song: "Both Sides Now"?)

Here are usual (psychological, Ponzo illusion) explanations given for the perception of a larger sun and moon at the horizon. I favor the magnifying, refractive atmosphere theory (not given).

Edited by whYNOT
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Posted (edited)

Tony, glad you mentioned tangential velocity in connection with the remark by Plato. Why does your quote from Wiki say in the first sentence that tangential speed is directly proportional to radius, but then says in the second sentence that tangential speed is inversely proportional to radius?

I tell the story of how uniform circular motion came to be seen as an acceleration and how circular speed and the radius of the circle came to be correctly set in quantifying and defining the force attending that acceleration in the following:

Huygens pp. 27–28

Newton pp. 52–56

Edited by Boydstun
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Posted (edited)
6 hours ago, whYNOT said:

A counter-intuitive oddity, a brain-teaser rather than a paradox, imo. Quite something that Plato was then onto "tangential velocity" (the 'rotating' speed of various radii) clear above.

"tangential velocity is directly proportional to the radius. It increases because tangential velocity is inversely proportional to the radius". Wiki

In the Paradox as presented, the suggestive, visual red herring is an *inner* wheel 'track' or line, exactly equalling the length of the outer - except - the wheels are different diameter/circumferences!

Of course, the larger one's circumference singly dictates the distance covered and all inner points of a moving object correspond.

Paradox explained, I reckon, by the inner wheel turning at a slower (vt) on its 'track' than the larger in order to also complete one identical revolution as the outer rim, and to traverse the identical track distance in the identical time.

Demonstrating the non-contradictory nature of a wheel's properties, how it's supposed to act and does act.

A good relaying of information in an intuitive manner.  I think something equivalent to this is the first thing that comes to mind when most dispel the ghostly paradox.

I think it very possible, in some people's minds, with mechanical experience and understanding, no feeling of paradox would even emerge in the first place.

Although no one has really touched on this here, nor in a similar thread elsewhere, imagine clock technicians, and factory workers, at the height of the industrial revolution who worked with gears, and wheels, and pulleys, and belts, and axles, gear boxes, and conveyors and all manner of rotating, intermeshing, winding and unwinding, force and distance multiplying components... imagine their working with all manner of these, for 16 hours a day, day in day out for years, maybe working at multiple facilities and seeing multitudes of "contraptions" of various sorts.

In such a man, I would bet (and none too little an amount of money either) that the intuitive knowledge of rotation, gear ratios, pulleys, multiple rigid wheels or gears connected to a single axle and the effect thereof, would cause the man to look at the "posed" conundrum, and scratch his head... not for the lack of any knowledge and understanding but for the lack... of any lack of it, and in particular the lack of any "strange feelings" of mystery/paradox/puzzle which others might be eager to have him reaffirm.. but which he simply does not have.

Paradox needs some ignorance... not too much (that's simply bafflement) and certainly, not too little.

Edited by StrictlyLogical
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Posted (edited)
4 hours ago, StrictlyLogical said:

A good relaying of information in an intuitive manner.  I think something equivalent to this is the first thing that comes to mind when most dispel the ghostly paradox.

I think it very possible, in some people's minds, with mechanical experience and understanding, no feeling of paradox would even emerge in the first place.

Although no one has really touched on this here, nor in a similar thread elsewhere, imagine clock technicians, and factory workers, at the height of the industrial revolution who worked with gears, and wheels, and pulleys, and belts, and axles, gear boxes, and conveyors and all manner of rotating, intermeshing, winding and unwinding, force and distance multiplying components... imagine their working with all manner of these, for 16 hours a day, day in day out for years, maybe working at multiple facilities and seeing multitudes of "contraptions" of various sorts.

In such a man, I would bet (and none too little an amount of money either) that the intuitive knowledge of rotation, gear ratios, pulleys, multiple rigid wheels or gears connected to a single axle and the effect thereof, would cause the man to look at the "posed" conundrum, and scratch his head... not for the lack of any knowledge and understanding but for the lack... of any lack of it, and in particular the lack of any "strange feelings" of mystery/paradox/puzzle which others might be eager to have him reaffirm.. but which he simply does not have.

Paradox needs some ignorance... not too much (that's simply bafflement) and certainly, not too little.

Yes, fine insights. It seemed to me revisiting an extensive debate that one or several individuals arrive at this paradox with different approaches: geometry and math, mechanical and experimental or as visual imagery - some as pure theoreticians and some as very practical. Those who only see 'circles' and others only 'wheels'. Each according to their cognitive strengths, education, imagination, observations and experience. E.g. I posed the paradox to an engineering trained friend, who straight off related the two co-axial wheels each on a track, to a fixed assembly of cog-and-gear trains, which will instantly jam due to their conflicting rotation speeds, in brief. (Refuting the 'slippage solution'). The double wheel-and-track as single entity can't rotate. Fascinating, and an insight into psychology seemed that many are prone to a challenge: 'To fix the problem'. Some like me maintained there simply isn't a problem nor solution needed. Don't fix what isn't broke. A wheel evidently works, and works that way in practice and therefore has to in corresponding theory and math equations. Adding a second wheel - and a second ¬suggested¬ 'track' - or not. At another level which makes the maddening, at times self-doubting exercise worthwhile, the philosophies arise. To attempt (with difficulty) to reason at once both experimentally and theoretically (and visually), felt to me a useful lesson in the empirical/rationalist/objective trichotomy. The reassuring base to return to, one's conceptions must conform to the way things are.

"Ghostly", a chimera, I think the word for what happens when imagination takes flight. Still, as it has for all this time, the Wheel debate will doubtlessly rage on.

Edited by whYNOT
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4 hours ago, Boydstun said:

Tony, glad you mentioned tangential velocity in connection with the remark by Plato. Why does your quote from Wiki say in the first sentence that tangential speed is directly proportional to radius, but then says in the second sentence that tangential speed is inversely proportional to radius?

In my haste I didn't pick that error up, apologies.

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19 hours ago, whYNOT said:

. I posed the paradox to an engineering trained friend, who straight off related the two co-axial wheels each on a track, to a fixed assembly of cog-and-gear trains, which will instantly jam due to their conflicting rotation speeds, in brief. (Refuting the 'slippage solution').

That's "affirming", not refuting.

Something jams when things cannot slip, and otherwise would: it's just a matter of friction at the contact points, gears/cogs are equivalent to very high friction which does not allow slippage, whereas normal rolling surfaces (wood, rubber, metal etc). when there is an inconsistency of speed between touching surfaces, would tend to skid or slip.

19 hours ago, whYNOT said:

The double wheel-and-track as single entity can't rotate.

True for a "track" which is single rigid object (with a sort of stair step) moving at a single speed.

But you could use a double wheel type thing, for example, to slide or feed work pieces, packages, or whatever, underneath it, shallower ones at one wheel and thicker ones at the other. Those different objects would be conveyed, at different speeds.

19 hours ago, whYNOT said:

Fascinating, and an insight into psychology seemed that many are prone to a challenge: 'To fix the problem'. Some like me maintained there simply isn't a problem nor solution needed. Don't fix what isn't broke. A wheel evidently works, and works that way in practice and therefore has to in corresponding theory and math equations. Adding a second wheel - and a second ¬suggested¬ 'track' - or not. At another level which makes the maddening, at times self-doubting exercise worthwhile, the philosophies arise. To attempt (with difficulty) to reason at once both experimentally and theoretically (and visually), felt to me a useful lesson in the empirical/rationalist/objective trichotomy. The reassuring base to return to, one's conceptions must conform to the way things are.

The key I think is to remember the purpose of the exercise is not to impress others or improve other people's thinking (when not asked to) but to clarify one's own.

I think in this case the problem is actually ill-posed, and uses a kind of fundamental misdirection.  I will try to explain this in a future post.  If anyone seems interested.

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