Understanding and Solving Zeno's Paradox
History 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 halfway 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.
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 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 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 a few more seconds 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.
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

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+{11/2^(n1)} (n representing the number of halfway points that have been reached) or the sum (T_{n1} + 1/_{(2^(n1))}) where T_{0}=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.
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

Questions & Answers
© 2011 Dan Harmon
Comments
Is there a another single word which can describe the 1/2 way point of a 1/2 point progression other than a Zeno's Paradox?
I always considered Zeno's Paradox an interesting physical discrepancy of moving objects. Your article brought back my studies from my college days. I followed your explanation with the two balls moving towards the light beam as a way to show that this is not a paradox after all.
I found your first explanation interesting. Even though Zeno’s Paradox insists that the infinite number of half way points would cause the balls to avoid ever reaching the light beam, your introduction of Plank Length sure does seem to put the paradox to rest.
Even though Plank Length may very well be the shortest length possible in our universe, it really is only the smallest we can observe. It would require enormous energy to observe that last infinitesimal movement and that might not leave us with anything after the experiment.
Now, after I thought about all that, I continued reading your article and your second example–using constant velocity–satisfied me. Calculus to the rescue! Too back Zeno never had it as a mathematical tool.
I feel better now, having read your article Dan. Because at least you have shown a real example that conforms to the way we experience our physical world every day, getting from one place to another without any problems. I think I can get there now.
Could you explain the difference.
A circle for example still uses Pi, and Pi is not a precise number
how can Pi be correct except as a round down?
And Calculus is taking very small straight line segments of a curve that can''t possibly be precise.
That is at least what I remember.
I have trouble with this paradox.
When you drop a ball it hits the ground, thank you gravity.
I understand Asymptotic, and the Tangential Curve.
And while mathematically there is an infinite number of half to go, we just round off where it is practical.
For example, when a capacitor is being charged we use five time constants because that is effectively the answer for a full charge, but mathematically the full charge never occurs.
It is amazing that we can use Pi to make circles.
And Pythagoras gave us the square root of 2 which by itself is hard to reproduce, but easy to make a right triangle where each side is one, and the resulting hypotenuse would be the square root of 2.
It seems like the math doesn't follow nature.
Sounds interesting, but I am completely confused about the multiple halfway points. *Scratches head*
How many halfway points exist between two points? I think only 1. Are you measuring the halfway point of a halfway point? No need to elaborate or go into a long explanation. I just do not understand what you are talking about. LOL!!!
@wilderness
Your assumption "Zeno was interested in disproving the new mathematics, not in applying his work to reality" is unfounded.
We can only infer what he might have possibly sought to disprove (since there is no record of argument directly attributable to Zeno).
As I understand matters, the term "infinitesimals" was not used until around 20 centuries later.
In any case, you've not addressed the basic disconnect and the inherent paradox of physical movement (in light of the quantum evidence).
you have in my opinion, by avoiding the issue, demonstrated "cognitive dissonance".
@wilderness
"Zeno, however, was not concerned with physical practicality"
?
He queried how physical things, such as arrows, runners and hares moved, in physical space, IF (from what we know of him, reported by others) said physical things must move through infinite steps.
Hence the paradox of the impossibility (according to his thinking) of movement with that of everyday life.
The "cognitive dissonance" evident in the analysis of his paradoxes is revealed by assertions that he was not querying the detailed explanation of the everyday practically of physical movement.
@rasta1
You're headed in the right direction  an invisible NONLOCAL "metaspace" supports the physical visible space many incorrectly assume to be perfectly continuous. Obviously a discontinuous physical space embedded in a continuous metaspace is what fits the facts, and common sense.
What if Zeno is right if he examined the paradox from a quantum physics point of view. What if there is an invisible space between all matter.
Followed every word and yes, I almost got it  so many thanks. Great, helpful writing and as others have mentioned you made learning a hostile subject (to me) a lot more pleasant. I want to read it all again to get it more. I'm sure I will.
Sharing and voting.
This topic has been handled more congruently (with physical reality) on https://hubpages.com/education/Congruentsolutions...
@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 metaphysical spaces.
But for physical movement, only? No, it is quite simply wrong to say that calculus (infiniteseries) 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.
@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 subPlanck scaled increments (required by infiniteseries 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.
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/steaphenpirie/pro...
This is amazing. I shared with people at work and everyone was really fascinated. Thanks for the awesome hub.
My goodness! I'm no mathematicianin 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.
Congratulations on a welldeserved Hub of the Day! Interesting article, wilderness  wellwritten and beautifully explained!
It's really interesting.
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!
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 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.
By my understanding, the more the distance between the two objects approaches zero, the number of halfway 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 halfway points.
Wow this is so fascinating, I had no idea what Zeno's Paradox was. Thank you for sharing such a fascinating and intelligent hub.
This is a very interesting topic and very easy to understand it the way you broke it down.
Very good lecture and I enjoyed reading about this famous paradox. Thanks.
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!
Great, thought provoking hub. Thumbs up. Thank you!
Congratulations on being the hub of the day! This is a very intriguing topic, thanks for sharing.
Interesting topic for a hub! Thanks for the detailed information. Congrats on getting Hub of the Day!
Congrats! Wilderness!
Such a hub, mixing History and facts...the ride of Math becomes fun, little bump here and there. but:
Infinitesilly awesome!
LORD
fascinating!
Bravo
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... ?
that's very interesting!!! This would of made math more interesting to many!
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.
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.
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.
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 occurred to me... You should be a politician! (Grinning)
Have Fun my Friend! again Nice Hub.
Fascinating  I'm not much good at math(s) but could follow this, so excellent writing
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.
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