Understanding And Solving Zeno's Paradox
88History of Zeno.s Paradoxes
Zeno's Paradox. A paradox of mathematics when applied to the real world that has baffled many people over the years.
In about 400 BC a Greek mathematician named Democritus began toying with the idea of infinitesimals, or using infinitely small slices of time or distance to solve mathematical problems. The concept of infinitesimals was the very beginnings, the precursor if you will, to modern Calculus which was developed from it some 1700 years later by Isaac Newton and others. The idea was not well received in 400 BC, however, and Zeno of Elea was one of its detractors. Zeno came up with a series of paradoxes using the new concept of infinitesimals to discredit the whole field of study and it is those paradoxes that we will be looking at today.
In its simplest form, Zeno's Paradox says that two objects can never touch. The idea is that if one object (say a ball) is stationary and the other is set in motion approaching it that the moving ball must pass the half way point before reaching the stationary ball. As there are an infinite number of half way points the two balls can never touch - there will always be another halfway point to cross before reaching the stationary ball. A paradox because obviously two objects can touch while Zeno has used mathematics to prove that it cannot happen.
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. We will be looking at the specific example given here; all the paradoxes will have similar solutions.
First Case of Zenos Paradox
There are two ways to look at the paradox; an object with constant velocity and an object with changing velocity. In this section we will look at the case of an object with changing velocity.
Visualize an experiment consisting of ball A (the "control" ball) and ball Z (for Zeno), both paced 128 meters from a light beam of the type used in sporting events to determine the winner. Both balls are set into motion towards that light beam, ball A at a velocity of 20 meters per second and ball Z at 64 meters per second. Lets conduct our experiment in space, where friction and air resistance won't come into play.
The charts below show the distance to the light beam and the velocity at various times.
Ball A, Constant Velocity
Time since release, in seconds
| Distance from Light Beam
| Velocity, meters per second
|
|---|---|---|
1
| 108
| 20
|
2
| 88
| 20
|
3
| 68
| 20
|
4
| 48
| 20
|
5
| 28
| 20
|
6
| 8
| 20
|
6.4
| 0
| 20
|
This table shows the position of ball A when it is set into motion at 20 meters per second and that velocity is maintained at that rate.
Each second the ball will travel 20 meters, until the last time interval when it will contact the light beam in only .4 seconds from the last measurement.
As can be seen, the ball will contact the light beam at 6.4 seconds from the release time. This is the type of thing we see daily and agrees with that perception. It reaches the light beam with no trouble.
===============================================================
Ball Z, representing Zeno's Paradox
Time since release, seconds
| Distance from light beam
| Velocity, meters per second
|
|---|---|---|
1
| 64
| 64
|
2
| 32
| 32
|
3
| 16
| 16
|
4
| 8
| 8
|
5
| 4
| 4
|
6
| 2
| 2
|
7
| 1
| 1
|
8
| .5
| .5
|
9
| .25
| .25
|
10
| .125
| .125
|
This chart shows the example of a ball following Zeno's Paradox. The ball is released at a velocity of 64 meters per second, which allows it to pass the halfway point in one second.
During the next second the ball must travel half way to the light beam (32 meters) in the second one second time period and thus must undergo negative acceleration and travel at 32 meters per second. This process is repeated each second, with the ball continuing to slow down. At the 10 second mark the ball is only 1/8 of a meter from the light beam, but is also only traveling at 1/8 meter per second. The further the ball travels, the slower it goes; in 1 minute it will be traveling at .000000000000000055 (5.5*10^-17) meters per second; a very small number indeed. In just one more second it will be approaching 1 Planck length of distance (1.6*10^-35 meters) each second, the minimum linear distance possible in our universe.
If we ignore the problem created by a Planck distance it is apparent that indeed the ball will never reach the light beam. The reason, of course, is that it is continually slowing down. Zeno's paradox is no paradox at all, merely a statement of what happens under these very specific conditions of constantly decreasing velocity.
The Z ball with constant velocity
Time since release in seconds
| Distance to light beam
| Time since last halfway point
|
|---|---|---|
1
| 64
| 1
|
1.5
| 32
| 1/2
|
1.75
| 16
| 1/4
|
1.875
| 8
| 1/8
|
1.9375
| 4
| 1/16
|
1.9688
| 2
| 1/32
|
1.9843
| 1
| 1/64
|
Second Case of Zeno's Paradox
In the second case of the paradox we will approach the question in the more normal method of using a constant velocity. 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.
As can be seen, the time to each successive halfway point is decreasing while the distance to the light beam is also decreasing. While the numbers in the time column have been rounded off, the actual figures in the time column are found by the equation T = 1+{1-1/2^(n-1)} (n representing the number of halfway points that have been reached) or the sum (Tn-1 + 1/(2^(n-1))) where T0=0 and n ranges from 1 to ∞. In both cases, the final answer can be found as n approaches infinity.
Whether the first equation or the second is chosen the mathematical answer can only be found through the use of calculus; a tool that was not available to Zeno. In both cases, the final answer is T=2 as the number of halfway points crossed approaches ∞; the ball will touch the light beam in 2 seconds. This agrees with practical experience; for a constant velocity of 64 meters per second a ball will take exactly 2 seconds to travel 128 meters.
We see in this example that Zeno's Paradox can be applied to actual, real events we see every day, but that it takes mathematics not available to him to solve the problem. When this is done there is no paradox and Zeno has correctly predicted the time to contact of two objects approaching each other. The very field of mathematics he was attempting to discredit (infinitesimals, or it's descendent calculus) is used to understand and solve the paradox. A different, more intuitive, approach to understanding and solving the paradox is available at another hub on Paradoxal Mathematics, and if you have enjoyed this hub you might well enjoy another where a logic puzzle is presented; it is one of the best this author has seen.
vote upvote downshareprintflag
- Useful (3)
- Funny
- Awesome (3)
- Beautiful (1)
- Interesting (14)
CommentsLoading...
Fascinating - I'm not much good at math(s) but could follow this, so excellent writing
Nicely Done. Your professionalism is astounding. Thank You for the link to my hub as well.
As you already know, what I took away from my brief venture into Zeno's Paradox is very different from what you got from it. For me it simply is proof that mathematics isn't always correct, it doesn't 'always' add up.
In illistrating your point one thought occured to me... You should be a politician! (Grinning)
Have Fun my Friend! again Nice Hub.
@ Wesman: Thank you. There are lots of interesting "paradoxes" in math when being applied to the real world, but I usually find they happen because the math is being incorrectly applied to the "story problem" of real life. Just as in this case; the math actually turns out to correctly describe natural circumstances when applied correctly.
@SlyMJ: Math can be fascinating, and it sometimes goes against common sense just as Zeno's Paradox does. Nevertheless it usually doesn't take a genius to find out why. True, some math questions are still unanswered after centuries of effort, and do require genius to solve them, but most do not.
@Mikel: It was fun producing this, and I do find our different perspectives fascinating. Your take on it would never have occurred to me. You are more than welcome to the link; I think it adds to my hub to provide another viewpoint that is so different.
As far as being a (shudder) politician, you malign me, sir. That is one thing I stay far away from! lol
And you, too, enjoy your time on HP - it is always interesting.
HubPages is truly interesting with such hubs like yours and hubbers like you, I am learning. The paradoxes I am familiar with are in literature and yes, morals. Today, am adding Zeno's paradox in my mind. Oh, more and more there are many paradoxes in the political scene. Am sure you noticed it, too. Thanks so much for the lesson.
It is a fascinating place, isn't it? So many points of view, so many different fields of learning.
Yes, there are many paradoxes in our world, and they don't stop at morals and politics. In real life, the math ones are probably the easiest to resolve, as difficult as they can be.
You're more than welcome for the lesson, such as I can provide, and I appreciate the comment and sentiment.
I think I would have enjoyed my math classes much more had I been given the fascinating background behind things and such friendly explanations. Zeno's Paradox is fascinating! Thanks for lessening my aversion to math.
You're welcome, Simone. I enjoy these kind of things and am glad to see someone else that does too.
I think the fact that Zeno's paradox would even suggest such an absurdity shows that math cannot always be trusted. Math is really just an artificial game that we invented. It's not perfect. This is why there are so many absurdities in the fields of mathematical physics and particularly Quantum physics.
thats very interesting!!! This would of made math more interesting to many!
Wilderness, very interesting! Mathematics is always fascinating. Einstein used mathematics to discover theory of relativity. The human mind has difficulty to see or feel infinity... ?
fascinating!
Bravo
Congrats! Wilderness!
Such a hub, mixing History and facts...the ride of Math becomes fun, little bump here and there. but:
Infinitesilly awesome!
LORD
@ infinite; Not so. 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. Of course, that was Zeno's purpose; to show that new math fields were wrong.
@ optimus; glad it gave you some enjoyment and something to think about. Math can be enjoyed by many if they just get over their fears and spend a little time understanding it.
@ Knowing Truth; Indeed man can seldom wrap our feeble minds around the concept of infinity. 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. Utter boredom is the inevitable result, and not for just thousands of lifetimes. A billion lifetimes of boredom isn't even the very first baby step towards an infinite lifespan. Something to think about .
@ tlmntim: Glad you enjoyed it. It is an interesting paradox to dissect; it seems very intuitive, but actually is not.
@ lord de cross: Although I had heard of Zeno's paradox prior to writing this hub, I had no idea that he devised it to disprove the precursor of modern Calculus. It adds an interesting perspective to the already fascinating paradox.
Your "infinitesilly awesome" makes me lol. A great addition here - thanks!
Interesting topic for a hub! Thanks for the detailed information. Congrats on getting Hub of the Day!
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!
Congratulations on being the hub of the day! This is a very intriguing topic, thanks for sharing.
Great, thought provoking hub. Thumbs up. Thank you!
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.
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!
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.
Very good lecture and I enjoyed reading about this famous paradox. Thanks.
Thank you. 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.
This is a very interesting topic and very easy to understand it the way you broke it down.
Thank you. I tried to make it as intuitive and as applicable to everyday things we see and understand as possible.
Wow this is so fascinating, I had no idea what Zeno's Paradox was. Thank you for sharing such a fascinating and intelligent hub.
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.
You're more than welcome for the hub; it was fun writing, too.
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.
You can say that, in the last second, or in the entire trip, the ball actually *does* cross an infinite amount of half-way points.
It is amazing to me that in the year 400BC 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. Fascinating.
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.
@ emrldphx: Yes, the number of halfway points approaches infinity. That doesn't mean there isn't a solution to the problem, though; that is exactly what calculus is designed to handle and solve.
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? 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. Each time it passes a halfway point it is instantaneously decelerated to 1/2 the velocity it had. 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. Given an initial velocity V of 64m/s and an acceleration of A such that V=0 at 128 meters it could take forever to reach 128 meters. 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.
Awesome wilderness. I have a better understanding of drag racing now relative to the lights/timers. Since they use beams at 60ft, 330ft, 660ft, 1000ft and finally the 1320ft 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. thanks!
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!
It's really interesting.
Thanks. It was interesting to research and write, too.
Congratulations on a well-deserved Hub of the Day! Interesting article, wilderness - well-written and beautifully explained!
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.
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.
Voted up and interesting.
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?
This is amazing. I shared with people at work and everyone was really fascinated. Thanks for the awesome hub.
Glad that you enjoyed it - there is a lot hidden in mathematics that can be fascinating to try to understand.
Sorry to ruin the party, but it is easy to demonstrate that calculus DOES NOT solve the conundrum of Zeno's Paradoxes.
Basically, if physical movement involves infinite steps, then what chemical/electrical activity in the body can account for said movement?
Answer: There are no such activities in the body capable of causing such movement. None.
Zero.
Zilch. nil, nix, nada.
More explained at http://beliefinstitute.com/blog/steaphen-pirie/pro
You are correct, but not in the sense you have proposed. This paradox has nothing to do with the human body, but rather with an inanimate ball approaching a "target". As such, there is no chemical or electrical activity that plays any part at all.
However, the paradox is based on the premise that the distance to the target can always be cut in half, with a finite amount of time necessary to traverse the first half. 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. 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.
@wilderness
It doesn't matter whether you single our an inanimate object (such as an arrow on its way to the target), or the physical movement of the human body.
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.
Calculus (infinite series) does not resolve the Zeno's Paradoxes.
Achilles will never and can never catch the tortoise in an infinitely divisible universe.
Feynman was correct -- (the rest of the populace will catch up eventually):
"...(the idea) that space is continuous is, I believe, wrong."
— Professor Richard Feynman
The Messenger Series: Seeking New Laws
Presumably you are aware that if objects move continuously, Maxwell's theories require that the universe will quickly disappear in a flash of energy.
If however you cite quantum theory, then again, "according to the quantum theory, movement is not fundamentally continuous" (David Bohm).
As for the ball example, we can simply conduct a thought experiment in which a person holds a ball, and attempts to move it, in whatever manner -- the same conundrum will arise.
When moving the ball through sub-Planck scaled increments (required by infinite-series solutions), then your calculus fails, quite spectacularly.
If you have evidence to the contrary, I suggest you submit your work to the various physics organisations and await your Nobel Prize.
Belief Doctor, I think we are on the same page. With the understanding of Planck space our universe is known to be digital and not continuous. The web site you gave earlier did not address this issue, but rather that biological movement is discrete and not continuous because there is an upper limit to the number of possible "causes" for movement, such as the number of neurons in the brain.
Nevertheless, calculus does indeed solve Zeno's paradoxes; those studies are based on infinitesimals whether they can occur in nature or not. 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.
It should also be noted that mathematics in general is very useful and accurate in describing the world in terms of Newtonian physics, but often fails pretty badly in discussions of quantum mechanics. Perhaps we need a new sub-field in Math. The calculus of infinite series is quite accurate and correct as far as it goes, but does not fit the world as we now understand it. 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.
@wilderness
With respect if "The calculus of infinite series is quite accurate and correct as far as it goes, but does not fit the world as we now understand it." one might question if the mathematics, or calculations of Astrology might explain movement, as well.
Since both cannot account or provide 1:1 correspondences, it is a "superstition" (in quite literal terms) to assert the calculus solves Zeno's Paradoxes.
Zeno questioned how physical things moved. Unless the mathematics provides a 'blow by blow' correspondence it is not valid (or even vaguely scientific) to affirm that it does.
What you will find is that calculus "maps" the superposition states of movement. In that sense it maps 100% the movement of things through meta-physical spaces.
But for physical movement, only? No, it is quite simply wrong to say that calculus (infinite-series) solves Zeno's Paradoxes (in the context of moving through infinite PHYSICAL steps).
Again, if you have physical evidence to the contrary, the Nobel is all yours.
I perhaps see our difference. Zeno did not question how physical things moved; he questioned the "new" field of infinitesimals in mathematics. 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. His entire thrust was to "prove" that the new math field was wrong by using that assumption of movement.
Zeno was not, therefore, the first proponent of Planck space as you indicate; he was mathematician (not a physicist) trying to disprove a new field of math. The physics of space did not enter his mind.
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.
If you don't understand calculus or don't think it useful to, say, calculate the path of a lunar orbital mission from earth, say so. There is no need to bring out the bogus "math" being used by frauds in their schemes to separate the superstitious from the contents of their wallets.
This topic has been handled more congruently (with physical reality) on http://thebeliefdoctor.hubpages.com/hub/Congruent-
Read your hub and commented on your solution.
Wesman Todd Shaw 7 months ago
Very interesting!
Thanks for the hub about something that I'd never heard of - I'm pretty sure that I've got some friends that would love this, and could talk at great length about it.
Interesting that mathematics can seem to prove that something can't happen - when we can obviously see that it CAN happen.