Darrell Hougen

Hi Jon, Perhaps I was too hasty since I haven't been reading all the posts on OL, but I just noticed that some people are quick to engage in name calling. I'm not an absolutist when it comes to being polite. If someone is being flagrantly rude and disparaging, I'll sometimes get down in the gutter and engage in a little tit-for-tat. However, I generally dislike being impolite just because someone else doesn't seem to understand something, frustration notwithstanding. Cheers, Darrell

Hi Tony, The "bias" is that one end is bigger than the other. That's the whole point. If two wheels that matched the sizes of the ends of the cup were rigidly connected to each other by means of an axle the same length as the height of the cup, the assembly would have the same "bias" as the cup. It wouldn't roll straight unless it was forced to roll straight by inducing slippage. But, Aristotle's paradox says that it will roll straight. That's the problem. You say that if the bottle rolls skew, the setup of the experiment is imprecise, but it is the fact that of the bottle rolling skew that proves the point. It isn't a bug. It's a feature. The fact that the bottle rolls skew proves that Aristotle's paradox is a paradox --- an impossible contradiction. Darrell

Hi Tony, No offense, but it does seem as though you're having trouble visualizing what is happening, so I am attempting to take you through a logical progression of steps in order to prove the point. Please forget about what your eyes are telling you and focus on the inescapable logic of the derivation. 1. You agreed that if a big wheel and a small wheel rolled independently N times, then the big wheel would roll farther than the small wheel. Correct? 2. Now, if the big wheel and small wheel were rigidly connected to opposite ends of an axle, then the big and small wheels would have to turn the same number of times. Correct? So, if the big wheel turned N times then the small wheel would also have to turn N times. Right? You said earlier that if a person were to roll a tapered glass or party cup on the table or floor, it would veer off to one side. Correct? The reason the cup would veer off to the side is that the large end rolls farther than the small end. 3. Similarly, if two wheels that were the same sizes as the ends of the cup were rigidly connected by an axle the same length as the height of the cup and both were in contact with a flat surface so that they rolled without slipping, they would veer off to the side. That follows logically from facts (1) and (2), namely (1) that the big wheel will travel farther than the small wheel and that (2) that the wheels are rigidly connected. Correct? 4. Now, if the small wheel is placed on a support so that the axle connecting the small wheel to the big wheel is parallel to the ground, the same thing would happen as in statement 3, namely the wheel assembly would veer off to the side. That follows from the fact that placing the small wheel on a support doesn't alter the fact that it won't travel as far as the big wheel nor does it change the fact that the wheels are rigidly connected. Therefore, since (1) and (2) are still valid, the pair of wheels must veer off to the side. Correct? Aristotle's paradox implies that a pair of wheels joined by an axle won't veer off to the side. It implies that they will both roll the same distance without slipping. See how that contradicts the logic of the proof? That's why it is a paradox. It asserts a contradiction. In reality, either the small wheel must slip or the big wheel must slip or the wheel assembly must veer off to the side. Darrell

Hi Merlin, I agree that Jonathan is sometimes an obnoxious jackass, but the fact of the matter is that his analysis of Aristotle's paradox is correct. He has also been very patient at times, going out of his way to produce illustrative videos. We all owe him a debt of gratitude for that. And, so far as I know, he hasn't taken this dispute outside of OL. You really should put the Wikipedia page back the way it was or let us do it. Since you laid down the gauntlet, I'll take a look at your solutions later, when I get the chance. Cheers, Darrell

I think Max was thinking of something more like this: But with two concentric circular gears and two linear gears or "pinions." Darrell

I guess the only way to make it work is to drag and drop from another window. If I click "Insert other media" it doesn't work. Darrell

What happened? https://en.wikipedia.org/wiki/Rack_and_pinion#/media/File:Rack_and_pinion_animation.gif

I need to test some things. media This isn't working. That worked.

Right. I remember that. So, okay, that means that if you have a large wheel turning at N rotations per minute and a small wheel turning at N rotations per minute, the large wheel will travel farther than the small wheel. Right? Darrell

Hi Tony, The one with the longer circumference also has the greater tangential velocity. In fact, it has a greater tangential velocity because it has a longer circumference. The circumference is proportional to the radius. The greater the radius, the greater the circumference. The same thing is true of the tangential velocity. The tangential velocity is proportional to the radius. The greater the radius, the greater the tangential velocity. So, if R = 2r for example, then C = 2c and V = 2v where R, r = radius, C, c = circumference and V, v = tangential velocity of the big and small wheels respectively. If that doesn't make sense to you, stand up with your arms outstretched and turn around. Your hands move both farther and faster than your elbows or shoulders. See what I mean? Darrell

Hi Tony, Yes, I remember that you mentioned that. But, if V > v, that means that one wheel is rolling faster than the other. If one wheel is rolling faster than the other, then they can't get to the end point at the same time. Or something else has to give. Darrell

Hi Tony, You are correct that the glass will roll off to one side. But, there is nothing special about a level surface. If the surface were tilted, what do you think would happen? Imagine that the surface is tilted so that the ends of the cup are straight up and down. Won't the glass continue to curve in the same direction as long as it doesn't slip? You can create the same effect by just putting the small end of the cup on a book of the appropriate height. Try it. You don't need any fancy scientific equipment. Darrell

Or, in terms of the original statement of the paradox, it is impossible to have: X2 - X1 = R * (T2 - T1) x2 - x1 = r * (t1 - t1) X2 - X1 = x2 - x1 T2 - T1 = t2 - t1 and R > r where X2 - X1 and x2 - x1 are the distances traveled by the big and small wheels, respectively, T2 - T1 and t2 - t1 are the angles (theta) that both wheels rotate (e.g. 2pi radians) and R and r are the radii. That is a mathematical statement of Aristotle's paradox. Darrell

Hi Tony, Actually, I don't think you need laboratory conditions. The effect of having wheels of different sizes is very pronounced. In fact, you could perform an experiment with an ordinary drinking glass. Find a glass that is tapered so that the two ends have different diameters and roll it on the table or floor and watch what happens. A simple Dixie cup or party cup should work just fine. You don't have to roll it fast. Darrell

Hi Max, Maybe "resolve" (my word) or "solve" isn't the right term. Of course, Aristotle's paradox cannot be "resolved" if all of the conditions are enforced. It is impossible for two wheels that are rigidly attached to each other to turn without slipping on two different tracks if the radii are different. I was simply trying to point out that mathematically, there must be slippage somewhere in the system. Formally, it is impossible to have: V = RW v = rw V = v W = w and R > r where V, v = tangential velocities of the big and small wheels respectively, W, w = their respective angular velocities, and R, r = their respective radii. That is what Aristotle's paradox demands. Darrell

Hi Tony, After reading MSK's post from Nov. 22nd --- I'll catch up eventually --- I realized that there are two ways to resolve the paradox. Perhaps the second way is easier for you. Let R, W, and V be the radius, angular velocity and tangential velocity of the big wheel. Then V = RW. Define r, w, and v similarly for the small wheel so that v = rw. Then, if R > r either V > v or w > W. Either the tangential velocity of the big wheel is larger or the angular velocity of the small wheel is larger. So, another way of resolving the paradox is to say that the wheels are actually separate wheels that turn at different rates. If that is easier for you to visualize, that works too. Darrell

Hi Tony, The inner wheel is fixed relative to the outer wheel. That is true. However, either the inner wheel or outer wheel must slip relative to its track. At one point you wrote that both the inner wheel and outer wheel rotate at the same angular velocity. That is true. However, if the wheels rotate at the same angular velocities, then their tangential velocities must be different. The tangential velocity is the linear velocity of a point on the outside of the wheel. If V and v are the linear velocities of points on the big and small wheels and omega is their shared angular velocity, then V = R * omega and v = r * omega. So, if R > r then V > v. If the vehicle to which the wheel is attached is moving with velocity = V, then the outer wheel will maintain rolling contact with the road while the inner wheel will skid while rolling on its track. I like Jon's suggestion of experimenting with a bottle. However, I would like to suggest performing a slightly different experiment. After finding a book that is the right height to support the neck of the bottle, press down on both the body and neck of the bottle simultaneously while trying to roll it. In other words, apply enough force to make sure that both the body and neck of the bottle remain in rolling contact with their respective surfaces. See what happens. Darrell

Tony, Merlin edited the Wikipedia page so that it no longer contains an accurate description of Aristotle's paradox. The figure has also been edited and is no longer illustrative of the paradox. Darrell

Unbelievable! I don't know why I'm reading this thread, but I cannot believe you actually edited the Wikipedia page to support your argument. People can say whatever they want on here, but taking this fight outside of OL is way beyond the pale. No one outside of OL asked to be part of this dispute. --- Darrell

Wow! Good news anyway. I saw the banning but didn't know they had been restored. I wonder if youtube is still keeping Prager videos in the adult content section --- meaning they can't be viewed in schools, for example. Darrell

Hi Michael, I guess I was a little confused about your argument, so thank you for setting the record straight. There is certainly enough sanctimony among Objectivists, but that sort of goes with the territory. Darrell

I can't watch the video right now, but I heard the news the other day. I don't know much about the case, but my feeling is that this is another ridiculous result. Chemicals have side effects. If we ban all chemicals, we'll be overrun by weeds and insects. There are always trade-offs.

Anthony, I don't think that principles are out-of-context absolutes, even for an individual. Context always matters. For example, Ayn Rand held honesty to be a virtue. However, it is a virtue in the context of peaceful coexistence. As Tara Smith has pointed out, it is even a virtue when the person one is dealing with isn't entirely rational in his reasoning. On the other hand, it isn't hard to construct scenarios involving criminals or acts of war in which it is perfectly reasonable to lie --- where, in fact, honesty would be foolish. So honesty is a virtue within a particular context. With respect to immigration the same thing is true. While the right to liberty, or specifically, the right to freedom of movement, is a right, it is not an out-of-context absolute. One generally doesn't have the right to access another person's property, for example. There is also the right to free association --- the right to voluntarily join together with other people for moral and proper purposes. One of the proper purposes of association is for mutual self defense. So, it is right and proper that people form a country with a government and restrict the people that can enter and the purposes for which they can enter. The right to liberty can't trump the right to freedom of association. The two principles can only be understood by looking at how they interact for the purpose of protecting human life --- the act of expending one's own effort for the furtherance of one's own survival and prosperity. While it is true that people all over the world have the right to liberty, it is also true that they don't have a right to demand that other people provide the conditions necessary for the protection of that right. If mass immigration undermines the ability of a group of people to form an association for the purpose of mutual self defense, then the right to liberty cannot be interpreted as superseding the right to association. Such an interpretation would undermine the right to life. The right to liberty is not an out-of-context absolute. The only way that ARI and other objectivists can justify open borders is through massive context dropping. In the full context of human existence, it seems imminently reasonable to put limits on immigration. Darrell

Michael, I'm not saying that Cambridge Analytica had a significant effect on the election or that their use of Facebook data was even a scandal. The point is that the left considers it a scandal and it is the left that is outraged over the actions of Facebook and it is the left that is driving regulation of social media as a result of that outrage. Conservatives (and classical liberals, libertarians, objectivists, etc.) have also complained about mistreatment by platforms like Facebook, but we're used to being mistreated. Besides, the owners of Facebook don't really care about us anyway. However, they do care about how they are perceived on the left and that is why they have been cracking down on right-leaning "fake news" while ignoring the fake news pumped out daily by big left-leaning media companies. Darrell