Title: Microsoft Word - 3EDB29FE-66ED-180CE1.doc Author: www Created Date: 6/2/2003 10:32:23 AM Our ancestors were generally a lot smarter and know a lot more than we generally give them credit for; witness the construction of the pyramids. Exactly (given that our measurements must be rounded off because they are always imperfect to some degree). Do they have the same solution? (This may be realted to Goodstein’s theorem and the busy beaver problem.) Change ), You are commenting using your Facebook account. In both cases, the final answer can be found as n approaches infinity. I had no idea so many folks would enjoy a math discussion. Great, helpful writing and as others have mentioned you made learning a hostile subject (to me) a lot more pleasant. Notsurprisingly, this philosophy found many critics, who ridiculed thesuggestion; after all it flies in the fa… When you drop a ball it hits the ground, thank you gravity. I just do not understand what you are talking about. Your article brought back my studies from my college days. I understand Asymptotic, and the Tangential Curve. Ramanujan has a famous one which turned out be false–failed after a few million terms–so someone found a correction maybe 10-20 years ago (on arxiv).Sort of like an extra term in the ‘boltzmann distribution’. A length that is physically impossible to attain. Thank you for sharing such a fascinating and intelligent hub. Oh, more and more there are many paradoxes in the political scene. Utter boredom is the inevitable result, and not for just thousands of lifetimes. Depending on the circumstance it may or may not give you sensible answers. Looking at these reminds me of looking at the mountains in the Himalayas (i went to Ladack once–part of Kashmir, where again there is another conflict–i think muslims vs hindus) and Alaska/Canada along the Yukon river in winter. Are you measuring the halfway point of a halfway point? In illistrating your point one thought occurred to me... You should be a politician! Thanks to applecsmith and gunsock for your comments, and I'm glad you found it interesting. Thank you. There is a minimum size of anything in our universe. Math is really just an artificial game that we invented. Calculus to the rescue! This is the type of thing we see daily and agrees with that perception. It seems all too simple at first, but it’s got layers and layers of depth that can be brought up. Carrie Smith from Dallas, Texas on October 30, 2011: Congratulations on being the hub of the day! Zeno created several different paradoxes, but they all revolve around this concept; there are an infinite number of points or conditions that must be crossed or satisfied before a result may be seen and therefore the result cannot happen in less than infinite time. A circle for example still uses Pi, and Pi is not a precise number. To top it all off, even if you do try an infinite number of times (infinity isn’t a number, but for the sake of argument), you still wouldn’t be able to reach the door. Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion. It is kinda' like two balls if I understand correctly. If you understand the concept of mathematical limit, then this is not a problem at all. Congratulations on a well-deserved Hub of the Day! It is the application of math that causes problems, illustrated very well by Zeno's paradox.
Change ), You are commenting using your Twitter account. Congrats on getting Hub of the Day! Wish I had paid better attention in science and math classes. I have a better understanding of drag racing now relative to the lights/timers. This paradox has flaws. You're headed in the right direction -- an invisible NONLOCAL "meta-space" supports the physical visible space many incorrectly assume to be perfectly continuous. The idea was not well received in 400 BC, however, and Zeno of Elea was one of its detractors. Many of these paradoxes involve the infinite and utilize proof by contradiction to dispute, or contradict, these common-sense conclusions. An example of a falsidical paradox is Zeno’s dichotomy, which paradoxically concludes that the series 1/2, 1/4, 1/8, … sums to less than 1, when in fact (by appealing to the modern definition of limit) its sum is precisely 1. The ones i mostly look at have to do with decompositions of a number into primes. A paradox of mathematics when applied to the real world that has baffled many people over the years. Now if you set you’ll get a negative number. Zeno's paradox is no paradox at all, merely a statement of what happens under these very specific conditions of constantly decreasing velocity. Likewise, it would be completely arbitrary to claim that mathematics is unique, even if we live in a multiverse. The article is technically correct although a bit fussy. Nevertheless it usually doesn't take a genius to find out why. One concept postulated was that there are not only paralell universes, but perpendicular ones, as well, and it is when you've slipped into one of these is when you collide with a physical object, whether it's stubbing your toe, or being in a car accident. Using seemingly analytical arguments, Zeno's paradoxes aim to argue against common-sense conclusions such as "More than one thing exists" or "Motion is possible." I did that because the time numbers are much easier to find than a constant (between halfway points) negative acceleration would produce and because the time figures resulting made it easier to see just what was happening. The sum of a bunch of terms is called a series and the sum of an infinite number of terms is called an infinite series. As far as the infinite number of half to go, it is not necessary to round off as calculus gives us the correct answer. Expressed mathematically, the total time it takes for an object traveling at constant velocity to reach its target is. (The old native american idea that ‘its turtles all the way down’ –eg you reduce quarks and electrons into their component turtles, and decompose those into smaller turtles , ends with the smallest turltle known to Man. The beautiful thing is that we can compute this particular infinite series exactly, which is not true of all series. When we introduce scale transformation, we’ll see the true mystery of Zeno’s paradox. Covering half of the remaining distance (an eighth of the total) will take only half a second. Mathematics, however, is not “unique”. When you walk towards a wall, you must first cross half the distance, then half the remaining distance, and so on forever. Any paradox can be treated by abandoning enough of its crucial assumptions. The more compact way to express this is. I sometimes have decided maybe i should have carried my stuff–misery now or later. I hope others find it interesting - I certainly did as I researched the history of Zeno and worked my way through finding a solution to his paradox. @ infinite; Not so. Aristotle’s answer to Zeno’s basic paradox of progress, the runner, is known to invoke a distinction between ‘actual’ and ‘potential’ infinities. (Besides, in Alaska it was –45F). where is the distance and is the velocity. This is why there are so many absurdities in the fields of mathematical physics and particularly Quantum physics. See that this new situation is exactly the previous one : I have to cross a distance at a speed of half that distance per second. In this regard, Zeno’s paradox is a sort of calibration test between two àpriori non-unique worlds, the physical and the abstract. So it makes sense that in real life the balls will collide assuming that there is any velocity to reach ball Z at all. As to Zeno - I doubt that he would have had the same opinion if he had had access to calculus. Feynman was correct -- (the rest of the populace will catch up eventually): "...(the idea) that space is continuous is, I believe, wrong.". Your "infinitesilly awesome" makes me lol. and thus we can substitute this back into the original expression for and obtain, Now, we simply solve for and I’ll actually go over all the algebraic steps. Calculus (infinite series) cannot fully account for either of said movements, be it inanimate or living organisms -- if calculus could, we could entirely dispense with quantum theory. It seems like the math doesn't follow nature. Others consider Zeno's paradoxes to be logic in nature; therefore they require logical arguing. Zeno’s Paradox of the Race Course. Zeno assumed that objects could, indeed, occupy any space between beginning and end of travel and insisted that in each and every location the object could halve the distance to the target. If we ignore the problem created by a Planck distance it is apparent that indeed the ball will never reach the light beam. So there is not just one “Zeno Paradox”, but “Zeno Paradoxes”. @ tlmntim: Glad you enjoyed it. They can be thought of as breaking down into two sub-arguments: one assumes that space and time are continuous | in the sense that between any two moments of time, or locations in space, there is another If you check the comments, you will find some about Planck's limits. (Note : you can think of it as my world getting bigger at the same rate of my walk). What if there is an invisible space between all matter. Thanks for the awesome hub. Another popularly proposed solution, particularly for the fletcher's paradox, involves time and speed. Thanks for the detailed information. Again, in such a world, the convergence is just an observed phenomenon. And now quantum mechanics and string theories. It is amazing that we can use Pi to make circles. Zeno’s paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides’ doctrine that contrary to the evidence of one’s senses, the belief in plurality and change is mistaken, and in particular that motion is … Dan Harmon (author) from Boise, Idaho on April 05, 2012: Read your hub and commented on your solution. This paradox is kind of like that - "what if" you view something like travel of a ball in a certain way? I was always somewhat interested in the biggest snakes–maybe up to 7 feet was biggest i caught–and smallest–like a 4 inch snake .I’ve also seen snapping turtels as big as maybe 3 feet including the tail and as small as maybe 2 inches. ( Log Out / @Thy Tran: I actually used a varying, instantaneous negative acceleration for the second ball. The terms are related geometrically like the volumes of n-dimensional cubes when you have halve the length of the sides (e.g. Repeat, ad infinitum; it can't move at all. Let me reformulate Zeno’s paradox : I’m at A. I start walking towards B. I first cross half of the distance. I've considered this for a long time and just now it occured to me that although it seems like a paradox, no math is required to see how in real life this is no paradox at all. The Paradox. You sort of have the ‘temperature’ of the number sequence. Simone Haruko Smith from San Francisco on October 27, 2011: I think I would have enjoyed my math classes much more had I been given the fascinating background behind things and such friendly explanations. This would require completing an infinite number of tasks, which Zeno claimed to be impossible. (I can’t even remember all the terms like arithmetic, geometric, harmonic…mean. Why consciousness? . Thanks. And you, too, enjoy your time on HP - it is always interesting. Dan Harmon (author) from Boise, Idaho on November 01, 2011: Thanks, Dzy. This distance that the second ball will have traveled my never reach the 64 meter mark because at some point, its acceleration and velocity will have reach or in this case approach zero before the 64 meter is reached. Rose Clearfield from Milwaukee, Wisconsin on October 30, 2011: Interesting topic for a hub! There is one other layer still, which I believe Zeno were perhaps thinking about it but couldn’t quite formulate it (?). gunsock from South Coast of England on October 30, 2011: Great, thought provoking hub. Joseph De Cross from New York on October 30, 2011: Such a hub, mixing History and facts...the ride of Math becomes fun, little bump here and there. In the second case of the paradox we will approach the question in the more normal method of using a constant velocity. Still useful because we can only approach Planck space via math, but that will likely change with better technology when we learn to actually make use of the phenomenon. I'm sure you're right - continued advancement in mathematical theory and knowledge often produces answers that weren't available before those advancements. For instance, to live forever would be an absolute fright; no matter how many experiences we have, no matter how much we learn, there will come a day when we have done everything (many times) and learned everything there is to know. Zeno argues that it is impossible for a runner to traverse a race course. Objects in separate instantaneous frames would know how to move because each frame was being constructed by a higher reality. It just says you also have to assume that velocity is constant all the way to infinitesimal scales. His reason is that “there is no motion, because that which is moving must reach the midpoint before the end” (6=A25, Aristotle, Physics 239b11-13). How long will it take to cross half the remaining distance? Zeno’s paradox is indeed a little marvel in its own right. Zeno is a Greek philosopher who lived around the time of 490 to 430 BC. thanks! Now, after I thought about all that, I continued reading your article and your second example–using constant velocity–satisfied me. Too back Zeno never had it as a mathematical tool. Perhaps we need a new sub-field in Math. No need to elaborate or go into a long explanation. I feel better now, having read your article Dan. The charts below show the distance to the light beam and the velocity at various times. It adds an interesting perspective to the already fascinating paradox. Math is just a tool we use to help explain and model reality, it doesn't define it. Even ‘worse’ is when one has equations and theorems named after people.). This infinite series is technically called a geometric series because the ratio of two subsequent terms in the series is always the same. Eventually you will reach the point where there is no half the distance; you are trying to define something to small to exist. Answer: There are no such activities in the body capable of causing such movement. Maybe hare can’t beat all of them. The ball is released at a velocity of 64 meters per second, which allows it to pass the halfway point in one second. It would require enormous energy to observe that last infinitesimal movement and that might not leave us with anything after the experiment. Thanks. (another version has achilles racing the tortoise to the galapagos islands. There are an infinite number of halfway points between the start and the 64 meter mark you reference; it can't reach 64 meters by using the same paradoxical reasoning. There is mainly one reason not to be content with this solution, although it obviously is a solution of Zeno' s paradox. Zeno didn't know how to solve his paradox, didn't have the math tools to do it, and it thus did not represent the world as it was supposed to. So at the limits of the physical universe, Pi has no meaning at all; it is a mathematical construct only and has no relation to reality. Zeno is defeated at - nearly - any scale (the infinitelysmall left aside forthe moment), the larger the better.And this advantage of this solution is its problem. Fascinating. Dan Harmon (author) from Boise, Idaho on January 03, 2013: In a way he was. Now I see why those tuners get paid the big bucks. The solution to Zeno’s paradox stems from the fact that if you move at constant velocity then it takes half the time to cross half the distance and the sum of an infinite number of intervals that are half as long as the previous interval adds up to a finite number. Dan Harmon (author) from Boise, Idaho on March 20, 2012: Belief Doctor, I think we are on the same page. This will mean, of course, that the time to reach successive halfway points will change so lets look at another chart showing this, with the ball being released at 128 meters from the light beam and traveling at a velocity of 64 meters per second. Sadly (or maybe fortuitously) they all came to naught as calculus came into its own and was developed into a useful tool. We then have the amazing fact that, I never get tired of this. Bayesian model comparison
You are more than welcome, and I'm glad you found it interesting. Calculus (infinite series) does not resolve the Zeno's Paradoxes. Even though he tried to show that movement was impossible with the new math, his thrust was still simply to disprove the concept of infinitesimals, not to apply it. You are more than welcome to the link; I think it adds to my hub to provide another viewpoint that is so different. Math is probably the most perfect discipline man has created. If I might also address the "math" of astrology; to say that because calculus cannot correctly describe every aspect of physics and cosmology and therefore the "math" of astrology (used to find human characteristics based on the location of planets) might therefore be useful explain how and why things move is ludicrous as we both know. With the understanding of Planck space our universe is known to be digital and not continuous. Half as long—only 1 second. I realize this is an old topic, but it's one I'm interested in. Now if Newton hadn't found calculus you might not... Glenn Stok from Long Island, NY on April 18, 2018: I always considered Zeno's Paradox an interesting physical discrepancy of moving objects. We’ll begin with Zeno’s arguments that if space and time are continuous, then motion is impossible. Through short analysis of historical solutions of Zeno's paradoxes it is shown that Lynd's proposed solution is insufficient. (apparently used an axiom of infinity to prove infinity didn’t exist.). Enter your email address to follow this blog and receive notifications of new posts by email. the hare can race all the reproducing tortoises at the same time and beat all of them. In fact this generalizes to any geometric series, for any that is less than 1. Zeno had no knowledge, of course, of Planck space and his paradox is thus impossible in the real world, but if it were possible calculus would be the answer. he’s going to sell half the turtles to pet stores and eat the other half. 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