Aristotle's wheel paradox

[Jon Letendre]

5 hours ago, anthony said:

It never was a "paradox", or else reality is really screwy. I wrote that this seeming puzzle is posed as a red herring (circumference of large wheel vs. small wheel) trapping those trying to 'solve' a non-existent problem of 'slippage' (with gears). Simple powers of observation confirm that any fixed inner circle conforms to the motion of its outer wheel/circle. The outer circle, alone, dictates distance. You roll a wine bottle on its side and its narrower neck rotates equally--and traverses the same distance as the bottle. Why and how? The neck must revolve slower. Inductive observation beats everything! 

Hi Tony. For sake of discussion, would you humor me and state the paradox in your own words?

I don’t want to lead the witness too much, but I’m looking for the classic sense of paradox - something that appears to be so and yet cannot be so.

What is that something?

How does it appear to be so, meaning why would anyone think it is so?

[Peter]

Tony hasn’t answered so I will display my awesome lack of knowledge.  Paradoxes are fun, if they don’t devolve into false alternatives as in, is the earth hollow or flat? The following mentioned paradoxes. Is there a finite number of them? Peter

From: "William Dwyer"  To: Atlantis Subject: ATL: Zeno's Paradoxes (Was "Argument for strong AI") Date: Wed, 11 Jul 2001 00:24:08 -0700. Nick Glover defends Zeno's paradoxes, which argue that in order to go from point A to point B, one must go half the distance and in order to go half the distance, one must go half of that, etc.  Since there are an infinite number of halves that must be traversed in order to go from A to B, Zeno argued that logically it cannot be done, and therefore that motion is impossible. Or as Nick puts it, Zeno held that "motion is impossible if reality is infinitely divisible because an object must arrive at the middle of the remaining distance before it arrives at the end, an infinite number of  times.  The common response to this is to note that the relevant sum, 1/2 + 1/4 + 1/8 + 1/16 + ..., is equal to 1 (which was not known in Zeno's time) and therefore the actual distance to be traveled (and also the time it will take) is finite. However, there is still the issue of how it possible for something to traverse an infinite number of distances in a finite amount of time even if the distances do have a finite sum." I think the answer to this problem is to recognize that the distances are "infinite" only in the sense that there is no point at which one must stop dividing them by half; nevertheless, however far along the chain of multiple divisions one goes, one will always be at some finite point  in the process.  Therefore, what is "infinite" (i.e., not inherently limited) is only the ~potential~ for continuing the process, not the ~actual~ divisions themselves; for at any point along the way, these will always constitute some finite number. Bill

Me again in an aside for Ba’al.  "Hava Nagila" (Hebrew, "Let us rejoice") is a Jewish folk song traditionally sung at Jewish celebrations. It is perhaps the first modern Israeli folk song in the Hebrew language that has become a staple of band performers at Jewish weddings and bar/bat mitzvah celebrations.

David Potts wrote to OWL about a different paradox: Bohmian mechanics is often said to provide a "causal interpretation" of quantum mechanics and to be "deterministic." But I fail to see how this is any more than rhetoric. Bohm provides no glimmer of what the deterministic hidden variables might be nor any better way of predicting the phenomena than we now possess. All the special theoretical work of Bohmian mechanics is done by nonlocality, not by hidden variables. Nor do we need Bohm's theory in order to have permission to look for hidden variables. Niels Bohr indeed declared that "a more detailed analysis of atomic phenomena [than quantum mechanics provides] . . . is _in principle_ excluded." But that is just his assertion, based on the quantum paradoxes. What does seem true is that any more detailed analysis, to be successful, will require that we rethink our assumptions about the fundamental nature of reality. And perhaps the locality of a particle's properties will be one of those assumptions we rethink. But to know this we will have to _have_ a more detailed analysis. Bohmian mechanics provides no such analysis. It is a mere promissory note . . . . By what criterion can we distinguish any of these three hypotheses, the quantum potential field, the fifth dimension, or God? What problems does any one have that aren't shared equally by the others? Nothing empirical or testable or even theoretical that I can see. What _evidence_ is there that could support any of these over the others? If there is none, then they all are simply extra baggage. I have said previously (in a response to Phil Coates) that there is no problem for the law of identity with the notion of action at a distance. So why not just say that once the first particle is measured in the Bell's inequality experiment, the second just knows the value? For, as far as _machinery_ goes, that is all that "nonlocality" amounts to. But of course that is what ordinary quantum mechanics _already says_. In short, Bohmian mechanics fails to advance us one jot beyond where Feynman said we are: absolutely clueless concerning what underlying machinery could be responsible for the quantum phenomena. What we need is a genuine theoretical advance. That means, as I have said, either some new mechanism with testable consequences or one that clearly simplifies the present theory. Bohmian mechanics would seem to have no testable consequences, since its predictions are identical to those of standard quantum mechanics. At the same time, it introduces "additional variables and equations beyond those of standard quantum mechanics" (Goldstein, section 5 of his paper). This is hardly surprising. Just like Abdelkader's hollow earth theory, Bohm's "alternative" only mathematically transforms the existing equations of the standard theory. Therefore it is almost inevitable that it complicates them. end quote

Much of what Compatibilists or Soft Determinists say, is not in contradiction to Objectivism. However Determinism is a bad concept. The same ideas are better expressed as Objectivism and its component parts.  I don’t know who wrote the following but it was either from Ellen Moore, Ellen Stuttle, or me. I think it was me. Peter

So, do we call it “Pseudo-Scientific Soft Determinism” or “Consciousness”? “Causal Motivition” or “Motivation? “Caused Tautological Thought” or “Logic?” “Caused Speech” or “Syntax?” “Causal Volition” or “Volition?” “Causal Volition” or “Reason?” “Necessitated Description” or “Concept?” “Determinism” or “Objectivism”

From: Ralph Hertle To: objectivism Subject: OWL: Defn. of Scientific Experiment Date: Mon, 04 Feb 2002 21:10:48 -0500 On scientific experiments: I suggest that the purpose, and not the function, of an experiment is to isolate the phenomena, causes and principles of interest, and to remove from consideration all factors that are not integral with the causes, etc., that are of interest. The principles, etc., that function may be directly observed, evaluated, identified and measured by means of the operation or functioning of the experiment. (It is interesting that the terms, operation and functioning, of existents and causes, are frequently found in Aristotle's scientific writings.)

An experiment is a demonstration in physical reality or in ideas pertaining to same, of actual or hypothesized principles regarding the functioning of metaphysical or epistemological existents. A scientific experiment, and I think that the qualification scientific is necessary, may be differentiated from a demonstration, which is the genus, in that the all the factors involved are placed in and function within a planned logical structure, procedure of events, and system of proof, that governs the type and quality of results, and which may prove or measure the existence of the principles or properties being observed. That sentence needs some work, however, the gist of a definition of the concept of scientific experiment is there.

Scientific experiments may have subsidiary purposes, e.g., to show the principle or cause of a process, or to evaluate, discover, identify or measure the properties of the selected existents. A scientific experiment is a demonstration, which has a controlled logical causal structure, which control provides for the isolation or selected of facts to be observed for the purpose of the discovery, identification and validation of the causes of those facts. Perhaps someone else has another way of conceptualizing a definition for scientific experiment. Ralph Hertle

From: Michael Hardy To: objectivism Re: A rounded view of Aristotle Date: Tue, 5 Feb 2002 22:12:09 -0500 (EST)    David Friedman wrote: >if Newton's success was due to his discovery of a philosophically correct approach to discovering truth, isn't it somewhat surprising that he devoted considerable efforts to mysticism, alchemy, et. al.?

 

Was something philosophically incorrect about Newton's work on alchemy?  Admittedly I am not familiar with it, but "work on alchemy" could just mean experimental work on the ways in which new substances are formed from old.  In Newton's day, wouldn't that have been called "alchemy"? Perhaps Newton's most important discovery was that the *same* physical laws can simultaneously explain the behavior of heavenly and earthly bodies.  Notice the non-experimental nature of all empirical observations of the former.  What is the philosophical significance of that? Mike Hardy

From: Shawn Klein To: objectivism Subject: Re: OWL: A rounded view of Aristotle Date: Tue, 05 Feb 2002 21:41:43 -0500 I think a major part of Aristotle's lack of experimenting is due to his cultural context, as John Enright pointed out.  I don't think, though I am by no means an expert in these aspects of Aristotle's philosophy, that this lack of experimentation was an outgrowth or corollary of his philosophy.  He did, to the best of my knowledge, explore his world; he got out of his proverbial armchair and do some get-your-hands-dirty philosophy.  No, he didn't experiment or develop a theory of experiment but I think he could have and would have in the right intellectual context. If I may be allowed a plug, an interesting discussion of Aristotle and science is at http://www.objectivistcenter.org/articles/sdwake_aristotle-scientist.asp Shawn Klein

From: "John Enright" To: <objectivism Subject: OWL: Dissection & Experimentation Date: Wed, 6 Feb 2002 00:38:23 -0600. Ross Barlow mentions Descartes' dissecting of a calf's eye, as indicating Descartes' attitude toward experimentation.  I feel compelled to mention in this context that Aristotle, who came from a medical family, was a skilled dissector. But I'm not sure whether dissection is properly called experimentation or not. The root sense of experiment is that of making a trial of some proposition.  The more advanced sense involves making this trial under "controlled conditions."   The controlled conditions are typically designed to eliminate all sorts of confounding factors and to allow for reliable measurement. John Enright

From: Ram Tobolski To: OWL Subject: OWL: Concepts of Science: Movement, and the Paradox of the Arrow Date: Wed, 06 Mar 2002 18:57:07 +0200 In a previous post (2/28) I argued that scientific concepts are not, in general, abstracted from things that we perceive. I gave the example of the concept of electron. The concepts of science are supposed to explain what we experience, and they have to be justified by what we experience. But they are not abstracted from experience. The concepts of science, or many of them, refer to things that we do not meet in experience. To understand these concepts we have to use imagination - and I mean conceptual imagination, not perceptual imagination. Not to imagine things which are perceivable, but to imagine things which are not perceivable, but which logically explain things which are perceivable. I've also mentioned the modern concept of momentary speed. I'm not sure how many of you are aware of how strange this concept is. By a moment I do not mean here a short period of time, as in ordinary speech, but a "point" of time, which has no duration at all, and is indivisible. But if a moment has no duration, how can we make sense of momentary speed? When I take a still picture, I capture a moment. And indeed the picture is still, that is unmoving. How can there be speed in a moment? The problem is not a new discovery. It was already known to Zeno of Elea (5th century BCE), who conceived of the famous paradoxes of movement. The relevant paradox to our concern here in the paradox of the arrow. Zeno argued: If we look at one moment within the flight of an arrow, the arrow is not moving in that moment. Therefore it is resting then. Therefore the arrow is both moving and resting, which is a paradox.

Most of what we know about Zeno's paradoxes comes from Aristotle, who discusses them in his Physics. What was Aristotle' response to the paradox of the arrow? Here is what he wrote (Physics book VI section 9, pages 239b5-239b8): "Zeno's reasoning is invalid. He claims that if it is always true that a thing is at rest when it is opposite to something equal to itself, and if a moving object is always in the now, then a moving arrow is motionless. But this is false, because time is not composed of indivisible nows, and neither is any other magnitude."

In other words, moments ("nows") do not exist. And furthermore, extensionless points do not exist, in any context! This saved Aristotle's system from Zeno's paradoxes. But the cost was, it seems to be, overwhelming. The significance of Aristotle's conclusion is that in any dimension a thing can be either in some state, or in change, but not both. And that you can never refer to the state of a thing during a change. For example: if you drop a stone to the floor, and the fall took two seconds, it is senseless, by Aristotle's logic, to ask where was the stone after one second! To speak like that would assume, that the stone would be at some _point_ in space after one second. But for Aristotle, as we saw points do not exist. And we can see again why: If the stone were at some point after one second, it would be both moving and resting at this point, as argued by Zeno, and the paradox would obtain.

The rejection of Aristotle's view about the non-existence of points was at the heart of the birth of modern science, as we can see in the writing of key figures like Galileo, Descartes and Newton. This did not come for free: The concepts of physics came again under the strain of Zeno's paradox of the arrow. I'm not aware whether the paradox has been satisfactory solved by now.

The concept of point is another example of a concept which is not abstracted from perceived objects. Furthermore, it involves us in strange paradoxes. Nevertheless, modern science is inconceivable without this concept of the point. What does all this imply about epistemology? What do you think? Ram

From: "David Potts" To: "OWL" Subject: OWL: Re: Concepts of Science: Movement, and the Paradox of the Arrow Date: Sat, 9 Mar 2002 19:39:27 -0600

Ram writes: >Here is what he [sc. Aristotle] wrote (Physics book VI section 9, pages 239b5-239b8): "Zeno's reasoning is invalid. He claims that if it is always true that a thing is at rest when it is opposite to something equal to itself, and if a moving object is always in the now, then a moving arrow is motionless. But this is false, because time is not composed of indivisible nows, and neither is any other magnitude." In other words, moments ("nows") do not exist. And furthermore, extensionless points do not exist, in any context!

I have already commented on this business of Aristotle and momentary speed, and I have nothing new to say about it. But I can make a brief remark on this inference of Ram's, which seems to be the key one in his post. I do not think that the correct way to interpret Aristotle's often repeated claims that "time is not composed of individual nows" and a magnitude is not composed of points is to say that moments and points do not exist. Aristotle certainly believed that moments and points do exist. His point was rather that spans of time and magnitudes of space are not _composed_ of indivisible nows or extensionless points. I don't know if this point seems strange to readers or not. (To me it seems perfectly natural and true.) Let's run this discussion on the case of lines and points; everything said about them applies similarly to spans of time and indivisible moments.

If you tried to build a line by placing points next to one another, how long would it take? The answer is, _forever_! In fact, you could never even get started. For points don't take up any space (even in only one dimension). That is what it is to say that points are extensionless. Again, suppose you began dividing a line repeatedly into smaller and smaller segments. How long before you reach points? Obviously, as before, you _never_ do.

Of course, if you include "enough" points, by numbering your line with the irrational as well as the rational numbers, there will be a point for every possible location on the line. Nevertheless, it is still impossible for any two of these points to be touching. But the segments of a line do touch. Therefore lines are not composed of points. (Maybe that's what our elementary school teachers told some of us, but they lied!) They are composed of smaller line segments (if anything). However, none of this means or implies in any way that points can't be on a line or that points do not exist. Obviously they can and do. The same goes for planes and solids, btw. How many planes do you have to stack up to make an inch high solid? Silly rabbit! There's no number that could do it; planes aren't "really thin," they have _no thickness_. -David Potts

From: "DaN"  To: <objectivism Subject: OWL: In reply to Concepts of Science: Movement, and the Paradox of the Arrow Date: Sun, 10 Mar 2002 17:10:13 -0500 FROM: Dan Gibson

To begin with, Ram answered his own argument, concepts are not percepts. Scientific _concepts_ aren't perceived, they are _conceived_ from perceptual data that has been compared, contrasted, and integrated.  The perceptual facilities serve the function of a data bus, transmitting data from either the senses directly, or first through a translator/apparatus (various sensors), to the conceptual faculties for integration into concepts. Next, there isn't any speed during a moment.  Speed is a relationship, it needs two points to exists.  In order to derive a "momentary speed", reference would need to be made to a previous moment. i.e. two snapshots, with location data and time data, then and only then can a speed be derived.

To clarify the arrow paradox, both concepts of motion and rest depend on the elapse of time.  Zeno removed time from the motion concept, but granted it to rest.  Motion is a change of location over a period of time, rest is no change in location over a period of time.  He just wasn't being thorough. Nows do exist, but such concepts that involve time, do not, they occur. Points exist, lines occur. This concept of existents and occurrences, I have been pondering for a little while, and I'm still working it out.  The essence is that things that involve time or require more than one object, occur, those that do not, exist. DaN Gibson

[merjet]

Paradox or not

Whether anybody does or doesn't consider Aristotle's wheel paradox to truly be a paradox doesn't matter to me.  At some time and place in history somebody called it a paradox and the label has stuck. Call it a puzzle or problem. Call it "strange" like in the quote from Mechanica on the Wikipedia page. Apart from the terminology, the task is to explain the motion of the inner circle. 

Slip/slipping/slippage/skidding

On 11/10/2018 at 2:39 PM, BaalChatzaf said:

Slipping is the ratio of the circumference traversed to the horizontal motion of the center.  It is a well defined concept.  If 5 feet of circumference move the center 3 feet the slippage is 5:3

Oh boy, yet another meaning of slip introduced for the first time in this thread. Maybe he meant longitudinal slip, but that's far from clear. Anyway, with that meaning of slip, the outer circle in Aristotle's wheel paradox slips (= 0 numerically), but contrary to the usual premise that the wheel rolls without slipping. Also, longitudinal slip doesn't strictly apply to an inner circle of a wheel. Also, how could Baal get a longitudinal slip = 0 the way he computed the ratio?

This video shows the common meaning of a wheel slipping. This video shows the common meaning of a wheel skidding. The wheel of Aristotle's wheel paradox does neither.

On 11/10/2018 at 2:54 PM, Jonathan said:

What I don't know is what Merlin thinks he means in calling slippage "metaphorical," or what he thinks that he means in saying that slippage is irrelevant to the "paradox," or what he means in saying that he doesn't deny that the inner wheel slips while also saying that the inner wheel does not slip.

Jonathan misrepresented my position (like many times in this thread) when he used slips without scare quotes. I have put scare quotes around slips and slipping many times to indicate my usage is non-literal, i.e. metaphoric.

Temporarily set aside assuming the two circles depict a wheel. Consider instead that the two circles depict a round bottle, the inner circle depicting the outside or inside circumference of the mouth of the bottle. Roll the bottle on a table with only air between the mouth and the table. Then try to convince an auto mechanic or wine bottler that the inner circle literally slips. That is, it slips like the wheel in the first linked video above does. Longitudinal slip wouldn't strictly apply to the mouth, because it wouldn't contact the table.

[Jonathan]

3 hours ago, merjet said:

Jonathan misrepresented my position (like many times in this thread) when he used slips without scare quotes. I have put scare quotes around slips and slipping many times to indicate my usage is non-literal, i.e. metaphoric.

Temporarily set aside assuming the two circles depict a wheel. Consider instead that the two circles depict a round bottle, the inner circle depicting the outside or inside circumference of the mouth of the bottle. Roll the bottle on a table with only air between the mouth and the table. Then try to convince an auto mechanic or wine bottler that the inner circle literally slips. That is, it slips like the wheel in the first linked video above does. Longitudinal slip wouldn't strictly apply to the mouth, because it wouldn't contact the table.

We've already covered this multiple times.

Yes, if you take the original entities described in the alleged "paradox," and then you arbitrarily eliminate the upper line/surface that the smaller wheel contacts, then of course there would be nothing there for it to contact. The same would also be true if your were to arbitrarily eliminate any other of the entities that the original setup includes. For example, if you were to arbitrarily choose to eliminate the smaller wheel, then it wouldn't slip/skid while rolling against the surface, because something that doesn't exist can't roll against something that does! Or, if your were to arbitrarily eliminate the larger wheel, then it wouldn't be there to roll on the bottom line/surface.

The idea of a thought problem, though, is not to arbitrarily eliminate any of the entities that have been included in the original setup, and to then declare that there is no interaction between the two entities because you have wished one of them out of existence.

Understand yet?

J

 

[anthony]

On 11/11/2018 at 12:02 AM, Jon Letendre said:

Hi Tony. For sake of discussion, would you humor me and state the paradox in your own words?

I don’t want to lead the witness too much, but I’m looking for the classic sense of paradox - something that appears to be so and yet cannot be so.

What is that something?

How does it appear to be so, meaning why would anyone think it is so?

A sort-of 'rule of circles' everyone learns by inductive experience is that whatever a circle or wheel does - in direction, rotation and velocity - determines what any inner points or circles will do. The integrity of a wheel/circle holds true in every real-life experience of them. You see it with a teacup in a rotating saucer, a spinning frisbee, and car wheels, and so on. Turn the wheel once, or 20 times, everything within the wheel circumference rotates likewise (or proportionately, as for rotational speed) - and must finish up in the same position relative to outer rim and to the surface, as it began. 

So here we're given a diagram which supposedly shows (or suggests) that an inner circle - of obviously smaller circumference - tracks/traces a path which is greater than its own circumference (and equal to that of the large wheel or circle). How can this be? If one fixates on this inner line, this looks paradoxical, forgetting the "rule" - by which the wheel's travel is totally dictated by the outer rim, and that everything within, obeys..

E.g. Take an empty (featureless) circle or wheel and mark a point at random anywhere: inside its perimeter, at the center, or on its circumference, that's immaterial, and then rotate the wheel once--we'll see the start to finish line of travel will ALWAYS be the same length. Do that dozens of times in this same circle and all the lines will remain equal, so the information suggested by the "Paradox's" inner line-length is misleading, while not false.

Precisely *because* it is constant for ALL points and circles inside a larger turning circle.

Iow, the same length of line would be equally true for an infinity of inner circles, tiny and large, and will always match the big circle's circumference**.  

Bob: Nothing changes if there is to be a 'track' the inner wheel is on, or if the line is an imaginary 'path'. You get the same result with two surfaces, or one. Set up or picture an experiment with any bottle (best, because it has a protruding "inner circle", the neck and cap seen from the side). Construct two parallel tracks for the bottle to rest on. One higher than the other to finely adjust for the differing diameters. When the bottle is evenly balanced and leveled on both tracks, roll it along them and observe that both bottle and neck will roll evenly, with no skipping, slipping or jamming. (I'm sure if it's not balanced well, it will veer off course).

**An extreme: Imagine the Paradox with a 2 cm diameter circle in the center of a 2-meter outer wheel. When the large wheel revolves once on a surface, it travels its circumference =  6.282 meters (d x Pi). The little circle has a circumference of just 6.282 centimeters. And it too moves over six meters! Therefore, glaringly showing there is no correlation between small(er) circle and the total distance covered.

[william.scherk]

I think that one way the 'paradox' is represented -- by a two-dimension line-animation -- shows a visual illusion.

4 hours ago, anthony said:

Set up or picture an experiment with any bottle (best, because it has a protruding "inner circle", the neck and cap seen from the side). Construct two parallel tracks for the bottle to rest on. One higher than the other to finely adjust for the differing diameters.

Montreal's Metro system uses rubber tires on its bogies. There is an inner and an outer wheel on each bogie, at slightly different levels.

This is from elsewhere, a bit prolix, but shows my understanding ...

Quote:

It's an attempt to fool the brain by fooling the eye. In the two dimension space of 'helpful' illustrations of the so-called (non) paradox, either animated or not, the eye/brain cannot measure both 'tapes' or 'unreeling' lines at once... for that is what the mind-machinery assumes when it sees a pair of lines 'unrolled' from both 'surfaces' of an inner and outer 'disk.' In 2-D the brain fills in the 3-D implications. 

The eye+mind "reads" the 2-D depiction wrongly, introducing the seeming paradox when its components are measured. The physics can't agree with the illustration's implications. The innermost 'tape' wouldn't roll and unroll smoothly in 3-D reality.  The inner disc 'edge' point would describe a coiled path in 2-D, accounted for in the math.  The 'illusion' is in the instant measurement of the two impossible 'tapes' ... during a revolution of the two illusory disks-as-cylinders.

A couple of  simplified  visuo-spatial ways to instantiate the 'reality' in a 3-D way is first to strip out the notion of 2 solid surfaces and and a thickly-flanged wheel. That would not happen in real life without metal screeching at every rotation.  It's impractical, it's stupid, it's physics.  It would be the dumbest train-like thing on earth (insert digression on the Montreal Metro double-flanged wheels later).

In your mind's eye of the "illustration," this time in reality ...

Strip out any bottom-most 'surface' and put your imagination in the bogie of a train, down there watching a big flanged wheel meeting the rail. the 'inner disk' -- for there is no way two supporting edges of a double-cylinder would not screech from the skidding because they would have two supporting edges of different speeds. 

In real life, there can be but one practical supporting surface.  There is only one wheel, and its flange is not a stupid further load-bearing attribute but a steering aid -- the  flange was invented to keep the train penned-in between the two parallel guides. 

Usually the machining of the wheels and the rails is perfected such that the guiding and supporting are accomplished without metal-on-metal screeching.  Except on most bogies on most sharp turns. An extra edge of friction is introduced by the angle of the curve.

This is where the 'slippage' or 'skid' happens in the real world. So, all the aids by Jays were okay in that they contrasted the illusion with reality.

LOBSTER, that bit was supposed to be short. Ye gawdz I am prolix. I hope that didn't undo whatever satori you had already achieved ...  the eyes LIE to the BRAIN and the BRAIN just takes it!

____________________

I was really waiting for someone who had understood the whole thing to step in an congratulate and condemn in just measure, to praise and harry, to sum up grandly and wisely, to give everyone a way to stand down from being dicks to each other, to heartily agree with a rational and helpful approach next time. A real diplomat in both senses of the word, diploma-ed up the yin and diplomatic up the yang.

It coulda been me! I coulda been a contender!

A sappy end to this tirade:  Chilliwack just added to the city budget by contracting the purchase of one of these semi-electric tricycle rickshaw wonders.  It is designed for older seniors who don't/can't ride anymore but who do want to ride. It gets them out on the trails. Duos, even!  

Ain't that sweet? Imma gonna see if can volunteer my sixty-year old legs, after the City was so spendthrift with our taxes.

_______________

Yes, digress.  The Montreal Metro has rubber-tired double-cylinder bogies that would seem to instantiate the 'paradox,' for there are actually two different load-bearing surfaces, an inner standard flanged railway wheel and an outer pneumatic tire.

-- so, how come these don't skid, drag, scrape, squeal? 

It's because there is a dynamic between the level of the concrete "runners" for the tires and the rails for the flanged wheels. The rails can always take the weight of the train, with the wheels hanging in void or flattened by blow-out, and the rails rise and fall where necessary to guide the train through a switch:

Because the pneumatic tires do not have flanges, the 'steering' and confining to the track is done by horizontal wheels engaging side-beams in full-pneumatic running, where the inner flanged wheels are just along for the ride.

End quote.   Call me Mr Prolix ...

Added:  the inner flanged wheel's flange extends down just past the level of the concrete runner. If I could find a schematic I would post it here.

[Added 12:54] Here's a bogie schematic from a civil engineering blog page on Rubber Tired Metros. 

-- the coolest thing about the rubber-tired Metro in Montreal is its speed -- an entailment of the whole system being completely underground (with its yards also under cover).  This means there is never snow or ice on the rails or concrete track.  Because of the  grip of rubber tires, the trains can ascend at more than a one-in-ten grade, and can accelerate faster than steel wheel trains. Faster braking in a shorter distance too.

The design contributes to speed: the stations are nearest the surface, whereas running tunnels are lower. So a train 'falls' out of a station, accelerating rapidly, and on approach quickly slows from top inter-station speeds of ~72kph on its 'climb' up to the platforms.  It's fun to ride.  Clean, safe, bright, faster than traffic, full of polyglot speakers from every corner of Earth. 

Next stop, Haiti and French Caribbean, Next stop, Old Money Scots & English, Next stop, Hipster Frenchies earning more than 100 grand a year, next stop, Orthodox Jewry, Next Stop Vietnam-China-Armenia ... 

 

Edited November 12, 2018 by william.scherk
Marked off quoted material; added note about flange; added notes on the Metro zoom in Montreal; added punchlines.

[Peter]

Great post. William.

Invincible superhero Stan Lee just died. So long, Stan.

From Wikipedia. The Fermi paradox, or Fermi's paradox, named after physicist Enrico Fermi, is the apparent contradiction between the lack of evidence and high probability estimates for the existence of extraterrestrial civilizations. The basic points of the argument, made by physicists Enrico Fermi (1901–1954) and Michael H. Hart (born 1932), are: There are billions of stars in the galaxy that are similar to the Sun, and many of these stars are billions of years older than the Solar system.  With high probability, some of these stars have Earth-like planets, and if the Earth is typical, some may have developed intelligent life. Some of these civilizations may have developed interstellar travel, a step the Earth is investigating now. Even at the slow pace of currently envisioned interstellar travel, the Milky Way galaxy could be completely traversed in a few million years. According to this line of reasoning, the Earth should have already been visited by extraterrestrial aliens. In an informal conversation, Fermi noted no convincing evidence of this, leading him to ask, "Where is everybody?" There have been many attempts to explain the Fermi paradox, primarily either suggesting that intelligent extraterrestrial life is extremely rare or proposing reasons that such civilizations have not contacted or visited Earth.

The Paradox of the Court, also known as the counterdilemma of Euathlus, is a very old problem in logic stemming from ancient Greece. It is said that the famous sophist Protagoras took on a pupil, Euathlus, on the understanding that the student pay Protagoras for his instruction after he wins his first court case. After instruction, Euathlus decided to not enter the profession of law, and Protagoras decided to sue Euathlus for the amount owed. Protagoras argued that if he won the case he would be paid his money. If Euathlus won the case, Protagoras would still be paid according to the original contract, because Euathlus would have won his first case. Euathlus, however, claimed that if he won, then by the court's decision he would not have to pay Protagoras. If, on the other hand, Protagoras won, then Euathlus would still not have won a case and would therefore not be obliged to pay. The question is: which of the two men is in the right?

[Jon Letendre]

2 hours ago, anthony said:

Nothing changes if there is to be a 'track' the inner wheel is on, or if the line is an imaginary 'path'. You get the same result with two surfaces, or one. Set up or picture an experiment with any bottle (best, because it has a protruding "inner circle", the neck and cap seen from the side). Construct two parallel tracks for the bottle to rest on. One higher than the other to finely adjust for the differing diameters. When the bottle is evenly balanced and leveled on both tracks, roll it along them and observe that both bottle and neck will roll evenly, with no skipping, slipping or jamming. (I'm sure if it's not balanced well, it will veer off course)

No, they will not roll evenly.

One bottle rotation will advance the base of the bottle farther than the small neck advances.

[Jonathan]

4 hours ago, william.scherk said:

I think that one way the 'paradox' is represented -- by a two-dimension line-animation -- shows a visual illusion.

 

Notice that in the animation, the red arc of the larger wheel first moves left before moving right? See how, in it's very first few moments of movement, it goes to the left of the vertical line that represents the starting point? Then, as the larger wheel's red line unrolls, it extends out beyond the vertical line on the left (the finish line), and then tucks back in at the last fraction of a second? These are the types of things that some of us notice, where others don't.

4 hours ago, william.scherk said:

Quote:

It's an attempt to fool the brain by fooling the eye. In the two dimension space of 'helpful' illustrations of the so-called (non) paradox, either animated or not, the eye/brain cannot measure both 'tapes' or 'unreeling' lines at once...

 

Um, no.

"The" eye/brain? Heh. Which brain is "the" eye/brain?

My eye/brain does just fine measuring both tapes at once. My eye/brain notices and accounts for things that "the" eye/brain doesn't.

Btw, the animation is clearly fucked up. The distance covered is too long for the small wheel, and too short for the large wheel.

 

4 hours ago, william.scherk said:

 

-- so, how come these don't skid, drag, scrape, squeal? 

It's because there is a dynamic between the level of the concrete "runners" for the tires and the rails for the flanged wheels. The rails can always take the weight of the train, with the wheels hanging in void or flattened by blow-out, and the rails rise and fall where necessary to guide the train through a switch:

Yeah, the safety wheel is not in contact at the same time as the main wheel. It's there for when the main wheel fails.

J

[anthony]

3 hours ago, Jon Letendre said:

No, they will not roll evenly.

One bottle rotation will advance the base of the bottle farther than the small neck advances.

Well, no. A smaller circle fixed within a bigger one - both will rotate evenly and in coordination, due to the rotational speed of the small one being lesser. Fairly commonsensical: The larger circumference has further to travel, circularly, to complete a revolution and must turn faster. That's why a wheel acts as a wheel - no internal contradictions like distance traveled, as is suggested by the Paradox, but keeping structural integrity.

Try rolling an even-sided bottle you have marked with a lengthwise line from the top of the neck, to the bottom, along a table top, and you'll see the entire line revolves in synch and the entire bottle will roll straight  (the neck is out of contact with the table, but that's irrelevant).

There used to be simple merry-go-rounds in parks, I've not noticed many, lately. A large round metal platform centered on a hub. You give a handle a shove to spin it, and jump on.  As kids we knew that standing near the edge, one goes faster, while relatively slower when one moves in towards the middle. I'm just demonstrating physically the variable circular velocity in a wheel - velocity dependent on the distance along the axis from center to circumference. The speeds can be calculated by a math formula.

 (As final authority, we have always got the reality of wheels and circles in motion to refer to and look at. Anything posited which contradicts what one has induced-integrated from what one sees and knows of them, HAS to be in error (at least, a non-paradox). As for the math principles, they have to be derived from the facts, not fit facts to theory).

 

[Brant Gaede]

7 hours ago, Peter said:

Great post. William.

Invincible superhero Stan Lee just died. So long, Stan.

 

 

From Wikipedia. The Fermi paradox, or Fermi's paradox, named after physicist Enrico Fermi, is the apparent contradiction between the lack of evidence and high probability estimates for the existence of extraterrestrial civilizations. The basic points of the argument, made by physicists Enrico Fermi (1901–1954) and Michael H. Hart (born 1932), are: There are billions of stars in the galaxy that are similar to the Sun, and many of these stars are billions of years older than the Solar system.  With high probability, some of these stars have Earth-like planets, and if the Earth is typical, some may have developed intelligent life. Some of these civilizations may have developed interstellar travel, a step the Earth is investigating now. Even at the slow pace of currently envisioned interstellar travel, the Milky Way galaxy could be completely traversed in a few million years. According to this line of reasoning, the Earth should have already been visited by extraterrestrial aliens. In an informal conversation, Fermi noted no convincing evidence of this, leading him to ask, "Where is everybody?" There have been many attempts to explain the Fermi paradox, primarily either suggesting that intelligent extraterrestrial life is extremely rare or proposing reasons that such civilizations have not contacted or visited Earth.

 

 

The Paradox of the Court, also known as the counterdilemma of Euathlus, is a very old problem in logic stemming from ancient Greece. It is said that the famous sophist Protagoras took on a pupil, Euathlus, on the understanding that the student pay Protagoras for his instruction after he wins his first court case. After instruction, Euathlus decided to not enter the profession of law, and Protagoras decided to sue Euathlus for the amount owed. Protagoras argued that if he won the case he would be paid his money. If Euathlus won the case, Protagoras would still be paid according to the original contract, because Euathlus would have won his first case. Euathlus, however, claimed that if he won, then by the court's decision he would not have to pay Protagoras. If, on the other hand, Protagoras won, then Euathlus would still not have won a case and would therefore not be obliged to pay. The question is: which of the two men is in the right?

E doesn't have to pay if he wins because he's not in the profession. And, regardless, winning his first case implies in the contract that there'll be cases to follow. So the contract is vitiated.

Here, however, E has a fool for a client. This creates the seeming paradox.

--Brant-

[merjet]

14 hours ago, Jonathan said:

Notice that in the animation, the red arc of the larger wheel first moves left before moving right? See how, in it's very first few moments of movement, it goes to the left of the vertical line that represents the starting point? Then, as the larger wheel's red line unrolls, it extends out beyond the vertical line on the left (the finish line), and then tucks back in at the last fraction of a second? These are the types of things that some of us notice, where others don't.

Heh. Jonathan is duped by somebody else's defective animation of supposedly rolling without slipping. Look at this animation, which doesn't have the same defects.  The red dot does not move left of its starting position, and it does not "tuck back in" at the end of a revolution. 

The defective animation shows the defects because it rolls with slipping. Another indication of this is that the horizontal distance covered by the circles is less than 2/3rds as long as it would be if it accurately represented a wheel or roll of tape rolling without slipping. A reader can easily verify the defects do not reflect a real world exemplar by using a roll of tape. Put a tick mark on it. Position the tick mark at "6 o'clock" with the roll of tape at rest on a table. Then roll the tape to the right without slipping. The tick mark will not initially move left. On the other hand, the tick mark will initially move left if you roll it with slipping or rotate it in place. 

The defects in the animation William posted to supposedly show rolling without slipping is something that I noticed, but Jonathan didn't.

[Jonathan]

3 hours ago, merjet said:

Heh. Jonathan is duped by somebody else's defective animation of supposedly rolling without slipping.

Huh??? In what way was I "duped" by it???!!!  I specifically identified what's happening in the animation that Billy posted.

Hahahaha!

 

3 hours ago, merjet said:

Look at this animation, which doesn't have the same defects.  The red dot does not move left of its starting position, and it does not "tuck back in" at the end of a revolution.

Yup. Correct. And?

 

3 hours ago, merjet said:

The defective animation shows the defects because it rolls with slipping.

Right, and in the animation that you linked to, we see that the path that is traced by a dot on the edge of a wheel that rolls without slipping is a proper cycloid. But, what would be the path of a wheel that slips/skids while rolling? Do you know the answer, Merlin? I don't think that you do. Try to figure it out.

CHALLENGE TO MERLIN: Start with a wheel which has a diameter of 5 inches, and move it on a flat surface so that it steadily slips/skids while it rolls one full rotation across a distance of 24 inches. Show the path that a dot on the edge of the wheel will trace. Also identify and describe the path. Is it a proper cycloid? Is it a curtate cycloid? Is it a prolate cycloid? Or is it some other shape? Show your work.

 

3 hours ago, merjet said:

A reader can easily verify the defects do not reflect a real world exemplar by using a roll of tape.

Which reader? Any reader? Any reader but you? Heh. YOU can't easily verify it. Several of the rest of us have verified what happens to freely rolling wheels versus slipping/skidding ones, but you can't do so. 

 

3 hours ago, merjet said:

Then roll the tape to the right without slipping. The tick mark will not initially move left. On the other hand, the tick mark will initially move left if you roll it with slipping or rotate it in place.

Uh huh, and what type of cycloid does a wheel make when it is losing traction and "spinning out" by rotating faster than free rolling? Do you know? And what type of cycloid does a wheel make when it is slipping/skidding while rolling by rotating slower than free rolling? You don't know, do you?

 

3 hours ago, merjet said:

The defects in the animation William posted to supposedly show rolling without slipping is something that I noticed, but Jonathan didn't.

Really? I didn't notice? So, when I posted my comment here about my having noticed it, that's proof to you that I didn't notice it?

Heh. And you wooda seen it right off even if dumb old Jonathan, who don't see nuthin, hadn't done sed it first, eh? You actually saw it first, and stupid ol' Jonathan didn't see it? Jonafin's just a varmint hoo's tellin' hargwarsh lies and tryna trick folks with scams and sich?

Heh. Doddering old hillbilly fool.

J

[Peter]

14 hours ago, Brant Gaede said:

E doesn't have to pay if he wins because he's not in the profession. And, regardless, winning his first case implies in the contract that there'll be cases to follow. So the contract is vitiated.

Here, however, E has a fool for a client. This creates the seeming paradox.

--Brant-

No one noticed that my saying "Stan Lee was invincible but died" was a paradox. I was worried someone would take offense.  

[Jonathan]

30 minutes ago, Peter said:

No one noticed that my saying "Stan Lee was invincible but died" was a paradox. I was worried someone would take offense.  

I don't care what you say about Stan Lee, but just don't be dropping any paradoxes on Steve Ditko, cuz he was all Objectivishistic and stuff.

J

[merjet]

3 hours ago, Jonathan said:

But, what would be the path of a wheel that slips/skids while rolling? Do you know the answer, Merlin? I don't think that you do. Try to figure it out.

CHALLENGE TO MERLIN: Start with a wheel which has a diameter of 5 inches, and move it on a flat surface so that it steadily slips/skids while it rolls one full rotation across a distance of 24 inches. Show the path that a dot on the edge of the wheel will trace. Also identify and describe the path. Is it a proper cycloid? Is it a curtate cycloid? Is it a prolate cycloid? Or is it some other shape? Show your work.

Heh. Doddering old hillbilly fool.

 

I know the answers. Why are you asking me? You can't figure it out for yourself? If you think you can, prove it. Show your work.

Heh. 

[Jonathan]

1 hour ago, merjet said:

I know the answers. Why are you asking me? You can't figure it out for yourself? If you think you can, prove it. Show your work.

Heh. 

I'm not asking you, I'm challenging you. And the answers have to be your answers, and your work. So don't hire the retarded chimp that you hired last time to draw some cycloids.

J

P.S. Besides, I've already given the answers on this thread, and shown my work several times, but you're not capable of grasping it.

[merjet]

42 minutes ago, Jonathan said:

I'm not asking you, I'm challenging you. And the answers have to be your answers, and your work. So don't hire the retarded chimp that you hired last time to draw some cycloids.

J

P.S. Besides, I've already given the answers on this thread, and shown my work several times, but you're not capable of grasping it.

I challenge you, obnoxious liar.

[Jon Letendre]

On 11/12/2018 at 11:13 AM, anthony said:

Bob: Nothing changes if there is to be a 'track' the inner wheel is on, or if the line is an imaginary 'path'. You get the same result with two surfaces, or one. Set up or picture an experiment with any bottle (best, because it has a protruding "inner circle", the neck and cap seen from the side). Construct two parallel tracks for the bottle to rest on. One higher than the other to finely adjust for the differing diameters. When the bottle is evenly balanced and leveled on both tracks, roll it along them and observe that both bottle and neck will roll evenly, with no skipping, slipping or jamming. (I'm sure if it's not balanced well, it will veer off course)..

No, Tony, bottle and neck will not roll evenly. In your setup, it will veer off course by necessity every time if no skipping or slipping is allowed.

If both bottle and neck stay in contact with their tracks with no slip or skid,  then with rotation the bottle advances farther on the track than the small neck. The neck’s circumference is smaller, so it must advance less than the bottle that has greater circumference.

Please stop repeating that the neck and bottle base both make equal rotation, I really do get that.

The bottle base advances farther than the neck, given any amount of rotation, and veers off course, not rolls straight or even.

Roll a funnel. Why doesn’t it go straight?

[Jonathan]

1 hour ago, merjet said:

I challenge you, obnoxious liar.

You're regressing into infantilism in your dotage, gramps.

Who's talking care of you? Your daughter-in-law? Well, tell her that you missed your meds yesterday. Maybe that your diaper needs changing too.

J

[Max]

On 11/10/2018 at 12:58 PM, merjet said:

The Wikipedia page about Aristotle's wheel paradox (link) has been vastly improved!

By me. I included an image that shows (a) the circles before and after rolling one revolution and (b) paths of motion for three points. The image thus helps to show why the smaller circle moves 2*pi*R -- the path of motion of every point on the smaller circle is shorter and more direct than the path of motion of any point on the larger circle.

Well, at least the article gives the correct solution:

Physically, if two joined concentric circles with different radii were rolled along parallel lines then at least one would slip; if a system of cogs were used to prevent slippage then the circles would jam.

The part with the cycloids doesn't explain the paradox. That there must be an error is trivial, 2*pi*r < 2*pi*R for r < R, that impossibility is what makes it a paradox (an apparent contradiction) but that still doesn't tell us what exactly the error in the presentation of the paradox is. That is namely the supposition that both wheels can roll without slipping. The fact that this is impossible is the easy and final solution to the paradox. Further it isn't necessary to assume that the larger wheel rotates without slipping, we can as well suppose that the smaller wheel rotates without slipping; in that case the large wheel must be slipping (skidding), as the wheels are then translated over the smaller distance 2*pi*r after one revolution. In that case a point on the rim of the large wheel will trace a prolate cycloid. But cycloids are in fact just an unnecessary distraction for explaining the paradox.

 

[Jon Letendre]

2 minutes ago, Max said:

Well, at least the article gives the correct solution:

Physically, if two joined concentric circles with different radii were rolled along parallel lines then at least one would slip; if a system of cogs were used to prevent slippage then the circles would jam.

The part with the cycloids doesn't explain the paradox. That there must be an error is trivial, 2*pi*r < 2*pi*R for r < R, that impossibility is what makes it a paradox (an apparent contradiction) but that still doesn't tell us what exactly the error in the presentation of the paradox is. That is namely the supposition that both wheels can roll without slipping. The fact that this is impossible is the easy and final solution to the paradox. Further it isn't necessary to assume that the larger wheel rotates without slipping, we can as well suppose that the smaller wheel rotates without slipping; in that case the large wheel must be slipping (skidding), as the wheels are then translated over the smaller distance 2*pi*r after one revolution. In that case a point on the rim of the large wheel will trace a prolate cycloid. But cycloids are in fact just an unnecessary distraction for explaining the paradox.

 

Thank you for that coherent and correct breath of fresh air, Max. The misapprehensions of these simple motions are astounding, no?

[anthony]

19 hours ago, Jon Letendre said:

No, Tony, bottle and neck will not roll evenly. In your setup, it will veer off course by necessity every time if no skipping or slipping is allowed.

If both bottle and neck stay in contact with their tracks with no slip or skid,  then with rotation the bottle advances farther on the track than the small neck. The neck’s circumference is smaller, so it must advance less than the bottle that has greater circumference.

Please stop repeating that the neck and bottle base both make equal rotation, I really do get that.

The bottle base advances farther than the neck, given any amount of rotation, and veers off course, not rolls straight or even.

Roll a funnel. Why doesn’t it go straight?

I covered that, and you missed it. I said:  "One higher than the other to finely adjust for the differing diameters".

I did ask you to roll a bottle along a surface, say the floor, and observe. It runs straight - right? 

Meaning, logically, the smaller diameter of the neck has ~zilch~ to do with the rotation of the entire bottle.

Now, take that to the two tracks. If and when it is perfectly aligned and balanced on its levels on two tracks, the bottle must turn as it would on a flat surface. Yes? If not, why not? (And clearly, frictional resistance, air resistance, gravity does not fall within the bounds of the "Paradox") . Anyhow, it is not clear - and superfluous - that that line is a second "track", or simply the path of travel. But in controlled laboratory conditions, two tracks will work equally well. 

Really, I don't know why nobody gets this. I think context has constantly been dropped by focusing too closely on the inner (fixed, attached) wheel.

As I keep repeating, the inner wheel/circle must move (laterally) the identical distance the outer one does!! (Or else, we can never trust a wheel again - they will break apart).

20 meters or 0.5 meter; a partial revolution - or a dozen revolutions - no difference. Fact remains: Each, single inner point and circle within the main wheel body will traverse precisely 20m or 0.5m - or whatever - as well. The inner line of travel in the paradox diagram will be the same for every internal circle, starting and ending at the same place. 

The clever placement and length of the inner line wrt. to the inner circle's circumference creates the illusion of a paradox. But *all* lines connecting all inner points from their beginning of motion to their rest, must always be that same length! (For any given circle and forward motion).I.e. the same length as the big wheel's total travel. 

The inner wheel is not an independent entity!

Its circumference has no bearing on the larger context.

Its motion and travel is extraneous to the main wheel - and must be considered only within the greater context of the larger entity.

I think this confusion between logic and fact, or, putting the theoretical in conflict with the evidence of our senses, seems to point to the old (false) dichotomy, Analytic vs Synthetic. That's well overturned by objectivity and Objectivism. Smart fellow, Aristotle, wasn't he?

[Jonathan]

15 hours ago, Max said:

Well, at least the article gives the correct solution:

Physically, if two joined concentric circles with different radii were rolled along parallel lines then at least one would slip; if a system of cogs were used to prevent slippage then the circles would jam.

The part with the cycloids doesn't explain the paradox. That there must be an error is trivial, 2*pi*r < 2*pi*R for r < R, that impossibility is what makes it a paradox (an apparent contradiction) but that still doesn't tell us what exactly the error in the presentation of the paradox is. That is namely the supposition that both wheels can roll without slipping. The fact that this is impossible is the easy and final solution to the paradox. Further it isn't necessary to assume that the larger wheel rotates without slipping, we can as well suppose that the smaller wheel rotates without slipping; in that case the large wheel must be slipping (skidding), as the wheels are then translated over the smaller distance 2*pi*r after one revolution. In that case a point on the rim of the large wheel will trace a prolate cycloid. But cycloids are in fact just an unnecessary distraction for explaining the paradox.

 

But, what if we change the conditions of the "paradox" so that certain arbitrarily selected parts of it are ignored or treated as being "metaphorical"? Let's say that the upper line, the one that contacts the base of the smaller wheel, doesn't exist, and that the large wheel sort of exists, but a section of it that I don't like is only "metaphorical." Let's also stipulate that there might be an invisible wheel behind the two visible wheels which is rolling freely on a ledge which isn't there but which I want to believe exists.

Okay, so the invisible wheel rolls freely on the non-existent ledge that I want to believe exists, but this wheel does not create a proper cycloid, but a curtate one. Instead, the large wheel, which under these conditions in reality would create a prolate cycloid, in this case creates a proper cycloid, because I want it to.

Considering all of the above, what's your solution to the "paradox"?

J

[BaalChatzaf]

34 pages!!  This is what happens when philosophers discuss a straightforward  problem in mechanics!   We are sufficiently advanced scientifically and historically to safely ignore anything the Aristotle had to say on matter and motion.  He totally  failed in addressing these matters.  Listen to Aristotle on descriptive biology, politics, rhetorics and  literary style.   The only part of science that Aristotle did well on was descriptive biology based on naked eye observations (the Greeks never developed lenses).  No less a naturalist than Charles Darwin gives Aristotle high marks in this endeavor.

Live Long and Prosper  \\//