20 API Paradoxes
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Most modern physicists seem to agree that there is a quantum "space", a minimum distance possible, in our universe. If that is true, then the distance to the target cannot be cut in half indefinitely.
Elements of the Paradox
Eventually you will reach the point where there is no half the distance; you are trying to define something to small to exist. A rather esoteric idea, but if true it negates the paradox. Answer: There are no such activities in the body capable of causing such movement. Glad that you enjoyed it - there is a lot hidden in mathematics that can be fascinating to try to understand.
This is amazing. I shared with people at work and everyone was really fascinated. Thanks for the awesome hub. Thanks, Dzy. Yes, there are lots of basically mathematical concepts that are fun to discuss. They won't always have much connection to reality, but can be fun "what if" scenarios. This paradox is kind of like that - "what if" you view something like travel of a ball in a certain way? What then - can it do what we already know it does do? If not, why not? My goodness!
I'm no mathematician--in fact my eyes glazed over when I came to the equations you provided. However, the article made me laugh, as it reminded me of some great fun had with some friends a number of years back. We were out having coffee, and talking off the tops of our heads and through our hats about just such complex concepts, including paralell universes. The waitress thought we'd flipped for sure, as we were illustrating our points with stacks of tableware. 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.
Thank you, RedElf. 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. Congratulations on a well-deserved Hub of the Day! Interesting article, wilderness - well-written and beautifully explained! Yes, acceleration will make a huge difference, and I would think that the acceleration of a dragster would diminish as speed builds. Indeed, it's why tuners are well paid! Awesome wilderness. Since they use beams at 60ft, ft, ft, ft and finally the ft I always just viewed it as from start to the consecutive beam.
Now I can see that the acceleration from one beam to the next comparing with the start to that same beam will give a whole new meaning. It is kinda' like two balls if I understand correctly. Or what happens when one catches the other. Now I see why those tuners get paid the big bucks. That doesn't mean there isn't a solution to the problem, though; that is exactly what calculus is designed to handle and solve.
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Be careful saying that an infinity of halfway points are crossed, however. Infinity is a mathematical concept that truly has no business in real life. It is useful only in solving problems and is not something that we can measure or even actually approach. Frannie: It goes back a long way, doesn't it?
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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. Thy Tran: I actually used a varying, instantaneous negative acceleration for the second ball. 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.
In actual practice that won't work, but the T figures could remain the same with a varying negative acceleration; it's just harder to calculate and the equations can get hairy. I'm not sure what would happen to the T time if a constant negative acceleration was used. Or some other time; it probably won't be 2 seconds. I would have to spend some time on the equations to find that one. An interesting corollary here is that the ball can never move at all. 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.
Repeat that reasoning for 32 meter mark; it can't reach 32 meters. Repeat, ad infinitum; it can't move at all.
Such is the nature of a paradox - it can be much worse than it states. This paradox has flaws. The second ball is assumed to have a constant deceleration or negative acceleration, but in fact it doesn't. If you understand the concept of mathematical limit, then this is not a problem at all. 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.
Basically the ball will have stopped moving, for all practical purposes. I hope this helps. It is amazing to me that in the year BC humans were thinking about these complexities to try to determine the rules of nature. And now quantum mechanics and string theories. Wish I had paid better attention in science and math classes. By my understanding, the more the distance between the two objects approaches zero, the number of half-way points crossed approaches infinity.
The problem, I think, is that the paradox makes the movement of one object dependent on the amount of space remaining between the two objects, when in reality, the movement is independent.
Zeno actually came up with at least nine paradoxes, all based on the ideas presented in infinitesimalism and all using the same concept presented here. Sadly or maybe fortuitously they all came to naught as calculus came into its own and was developed into a useful tool. Wow this is so fascinating, I had no idea what Zeno's Paradox was. Thank you for sharing such a fascinating and intelligent hub.
Thank you. I tried to make it as intuitive and as applicable to everyday things we see and understand as possible. This is a very interesting topic and very easy to understand it the way you broke it down. I had fun researching the history behind the paradox - I had known very little of Zeno himself and found it intriguing that he was trying to disprove the "father" of modern calculus.
I'm sure you're right - continued advancement in mathematical theory and knowledge often produces answers that weren't available before those advancements. As to Zeno - I doubt that he would have had the same opinion if he had had access to calculus. Zeno was apparently a good mathematician - he just didn't have the tools to find the answer to his paradox. Wow, Wilderness - what a great explanation of Zeno's Paradox in terms that anyone can understand!
Often the appearance of a paradox simply means that we haven't developed the proper mathematical understanding to "solve" the problem - perhaps Zeno wouldn't have been as averse to new mathematical theorems if he had access to the understanding we have in the modern world! Thanks to applecsmith and gunsock for your comments, and I'm glad you found it interesting.
I had no idea so many folks would enjoy a math discussion. Appreciate the congrats, too - I also had no idea this hub even could end up as the hub of the day. Congratulations on being the hub of the day! This is a very intriguing topic, thanks for sharing. You are more than welcome, and I'm glad you found it interesting.
Thanks for mentioning the "hub of the day" - I had no idea and couldn't figure out why this hub suddenly got so much traffic from HP. I guess I know now! Interesting topic for a hub! Thanks for the detailed information.
NetBeans Weekly Newsletter, Issue #
Congrats on getting Hub of the Day! Math is probably the most perfect discipline man has created. It is the application of math that causes problems, illustrated very well by 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.