Zeno's Paradoxes

Greek philosopher Zeno proved that movement was impossible with a few simple paradoxes.

by Brian Dunning

Filed under General Science, Logic & Persuasion

Skeptoid #267
July 19, 2011
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Zeno's Paradoxes
Parmenides of Elea
(Public domain photo)

Even if you think you haven't heard of them by name, you'll recognize them. The most familiar of Zeno's paradoxes states that I can't walk over to you because I first have to get halfway there, and once I do, I still have to cover half the remaining distance, and once I get there I have to cover half of that remaining distance, ad infinitum. There are an infinite number of halfway points, and so according to logic, I'll never be able to get there. But it's easy to prove this false by simply doing it, which we can all do. So we have a paradox, a contradiction, something that must be true but which, clearly, is not. Does there exist a solution which adequately addresses the contradicting phenomena? Some say there is; some say there is not.

Zeno of Elea was a Greek philosopher, born about 490 BCE, and was a devotee of Parmenides, founder of the Eleatic school of thought in what is now southern Italy. Zeno survives as a character in Plato's dialog titled Parmenides, and from this we know what the Eleatic school was about and where Zeno was coming from with his paradoxes. Parmenides taught (in part) that the physical world as we perceive it is an illusion, and that the only thing that actually exists is a perpeutal, unchanging whole that he called "One Being". What we perceive as movement is not physical movement at all, just different interpretations or appearances of the One Being. Personally, I think they smoked a lot of weed at the Eleatic school, but Zeno was into this and came up with his paradoxes in order to support Parmenides' view of the world. Zeno's paradoxes were intended to prove that movement must be impossible, therefore Parmenides must be right.

He is believed to have developed a total of about nine such paradoxes, but they were never published. The most famous and interesting are his three paradoxes of motion:

First is the paradox of Achilles and the tortoise, who contrived to have a footrace. Achilles, knowing he was the swifter, gave the tortoise a hundred-meter head start. In the time that it took Achilles to travel the hundred meters, the tortoise moved ten, so that when Achilles got there he found the tortoise still had a lead. In the time it took Achilles to run those ten, the tortoise moved another meter. No matter how many times Achilles advanced to the tortoise's last position, the tortoise had crept forward a bit more by the time he got there. Even though Achilles would seem to be the faster runner, it was impossible for him to ever catch the tortoise.

Second and most famous is the so-called dichotomy paradox, in which we repeatedly rend in twain every distance to be traveled. For Homer to walk to the bus stop, he must get halfway there. Once arrived, he must travel half of the remaining distance, and so on and so on, with 1/8 the distance remaining, then 1/16, then 1/32, then 1/64; he will have an infinite supply of remaining distances to travel, and thus can never arrive at the stop.

The third is the paradox of the fletcher who finds that all of his arrows are unable to move at all. At any given instant in time, the arrow is motionless in flight. During that frozen moment, the arrow cannot move at all, since it has no time in which to do it. Time consists of an infinite succession of moments, in each of which the arrow is unable to move. Nowhere can we find a given instant in which the arrow has time to move, and so no matter how many such instants we have, the arrow can neither fly nor fall to the ground.

Zeno's paradoxes are often touted by some people as evidence that physics or science are wrong. If an ancient Greek philosopher can describe a simple situation, which our intuition tells us is obviously correct, it's easy for us to assign it more significance than we do the confusing jumble that is modern science. Why should we listen to Einstein, who gives us a lot of unfathomable equations, when Zeno's elegant fables prove that the physical world is not as science tells us it should be? Given this line of reasoning, it's hardly surprising that Zeno has become something of a darling to some New Age supporters of a spiritual, not a physical, universe.

Famously, upon hearing the paradoxes, a fellow philosopher named Diogenes the Cynic simply stood up, walked around, and sat back down again. My kind of guy. His response may have been glib, but it elegantly refuted Zeno's claim. At least, it refuted the physical implications of the claim, it did not address the philosophical aspects; nor did it provide the mathematical solutions.

Zeno's paradoxes are an interesting intersection between mathematics and philosophy. Mathematically, it's trivial to calculate exactly when and where Achilles will overtake the tortoise, but the philosophical argument remains (apparently) intractable. Bertrand Russell described the paradoxes as "immeasurably subtle and profound". So philosophers have come up with some pretty interesting efforts to try and resolve this.

One such tactic concerns the Planck length, which is the smallest possible unit of length within the Planck system. Planck units are all based on universal physical constants, such as the speed of light and the gravitational constant. Philosophically it's reasonably accurate to describe the Planck length as a quantum of distance, the smallest possible unit. This means that there are a finite number of Planck lengths (albeit a staggeringly large number of them) along the racetrack of Achilles and the tortoise, and between Homer and the bus stop. There cannot be an infinite number of points, and so Homer will eventually be able to arrive. However, while this sounds like it might elegantly solve the paradox, it doesn't. It's not possible to force a quantum solution onto a geometric problem. A simple illustration of why this is so is to imagine a very small right triangle with its two equal sides each of one Planck length. The hypotenuse would have to be √2 Planck lengths, which is not possible. Planck doesn't apply here. Despite efforts to conclude otherwise, we are dealing with infinities here. Or... are we?

Intuitively, we understand 0.9 (0.9999999...) to be a value that forever approaches 1 but never quite gets there. This is fine as a concept and a thought experiment, but it is mathematically wrong. 0.9 does in fact equal 1; they are simply two different ways of writing the same value. It's easy to prove this to most people's satisfaction by dividing both values by 3. Both 1 ÷ 3 and 0.9 ÷ 3 equal 0.3, therefore both are equal to each other. Another way of looking at it is to consider the fraction 1/9, which is equal to 0.1. 2/9 is equal to 0.2, and so on, all the way up to 8/9 = 0.8 and 9/9 = 0.9, and we all know that 9/9 = 1. When we divide the number 1 into 9 equal slices, that top slice goes all the way up to exactly 1, a finite and reachable number.

If this spins your brain inside your skull, realize that you already accept many other interpretations of the same idea. Consider any other number whose decimal value is an infinite repeating series, say 3/7. It equals 0.428571 and we all accept that it equals 3/7, not "a number approaching 3/7 but that never quite gets there." It's two ways of writing the same thing.

It's the same concept when Homer takes his final step and places his foot down, completing his journey to the bus stop. He did not take a journey of infinite length. We can write an equation that describes how his final step consists of an infinitely reiterating series of smaller and smaller fractions, just as Zeno said:

We in the brotherhood call this an absolutely converging series, and contrary to Zeno's understanding, it equals 1.

Another popularly proposed solution, particularly for the fletcher's paradox, involves time and speed. Zeno, charges his critics, only considered the distances and geometry involved; and since he left time out of his paradoxes completely, he also excluded speed, since speed is a function of distance and time. When a body is in motion, its position is always changing. Motion is fluid, it is not a ratcheted series of jumping from point to point. Consequently, at any given moment in time, a moving body has no single exact position. Zeno's conjecture, that the arrow is always frozen at some point, cannot be observed, reproduced, or computed, since that's not the way things move. Imagine taking a photograph of a moving object. There will always be some motion blur. No matter how fast is the shutter of your camera, even infinitesimally fast, there will always be some tiny amount of blur. There is no such thing as a moving arrow frozen in time.

Similarly, Zeno's computation that Achilles will never catch the tortoise also omits time. Zeno's premise assumes that each segment of the race, wherein Achilles advances to the tortoise's previous position, takes some amount of time; and since there is an infinite number of such segments, it will take Achilles an infinite amount of time. This is also wrong. As the physical length of each segment decreases exponentially, in a converging series, so does the time it takes Achilles to traverse it. Achilles' time to catch the tortoise is represented by a converging series that equals a finite number.

Achilles will catch the tortoise, because the very succession of segments proposed by Zeno add up to a finite distance that Achilles will cover in a finite amount of time.

Tip Skeptoid $2/mo $5/mo $10/mo One time

Homer will reach the bus stop, because all of those infinitely compounding fractional segments are an absolutely converging series equal to a finite distance.

The fletcher's arrow is always in motion once it is shot, at no instant in time is it ever frozen with a fixed position from which it has no time to move.

So to summarize Zeno's paradoxes, they're basically word games that play upon an easily misunderstood mathematical concept. There is no paradox, because Zeno's math was wrong.

Brian Dunning

© 2011 Skeptoid Media, Inc. Copyright information

References & Further Reading

Baez, J. "The Planck Length." John Baez. University of California, Riverside, 9 Feb. 2001. Web. 8 Jul. 2011. <http://math.ucr.edu/home/baez/planck/node2.html>

Gardner, M. Aha! Aha! Insight. New York: Scientific American, 1978. 143-144.

Huggett, N. "Zeno's Paradoxes." Stanford Encyclopedia of Philosophy. Stanford University, 30 Apr. 2002. Web. 10 Jul. 2011. <http://plato.stanford.edu/entries/paradox-zeno/>

Lynds, P. "Zeno's Paradoxes: A Timely Solution." PhilSci Archive. Univerity of Pittsburgh, 15 Sep. 2003. Web. 9 Jul. 2011. <http://philsci-archive.pitt.edu/1197/>

Plato. Parmenides. Dublin: Hodges, Figgis, & Co., 1882.

Russell, B. Our Knowledge of the External World. Chicago: The Open Court Publishing Co., 1914. 165-181.

Whitehead, A., Russell, B. Principia Mathematica. Cambridge: University Press, 1910.

Reference this article:
Dunning, B. "Zeno's Paradoxes." Skeptoid Podcast. Skeptoid Media, Inc., 19 Jul 2011. Web. 21 Dec 2014. <http://skeptoid.com/episodes/4267>

Discuss!

10 most recent comments | Show all 122 comments

45 deg line?
The end points of the initial position of the line segment are (0, 0) and (1, 0) and not (0, 0) and (0, 1) as I stated before. Sorry for the confusion. The end points of the final position of the line segment shall be (-1, 0) and (0, 0). It's nothing but sliding a horizontal line segment in horizontal direction, which has to be done in infinitely many steps within one second. Does the question make sense now, how to accomplish that?

netzweltler, Germany
October 5, 2013 4:23am

Zeno's halfway point paradox isnt a paradox it is the confusion that there is any significance in a halfway point. Its just silly, let me explain:
To get to the end from the first halfway point is the exact distance you already traveled. from A to B there is only one halfway point. anyway...
Why would breaking distance up into points mean you cant travel through time and space? Measurement is for measuring. Even if it did matter, you cant split an atom with your foot can you?
ZEN: there are no halfway points if you dont stop walking

Now the turtle and the runner.
1st minutes end: tortoise 110, runner 100
2nd minutes end: tortoise 120, runner 200
winner: runner, done.
ZEN: not needed

The arrow paradox: (from Wikipedia): "If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless."
More rhetorical tail chasing.
An object is an object, motion is relative and not based on the object but the amount of distance between the object and the point of reference, usually the earth.
ZEN: two things get closer together.
but in reality, if there is any truth to any one of these three it is this one (but Zeno had no clue I would guess.) Since no point of reference is universally significant and movement is a comparison. Movement has no universal meaning

Zeno wasnt clever, and words dont mean anything.

Chris Kortjohn, Cincinnatti
November 17, 2013 2:03am

A runner begins running.

He moves off the start line, and moves forward 1/1,000,000th of the Planck length (note: infinite-series, used to mathematically "solve" Zeno's Paradoxes means movement through all increments, including infinitely shorter than the Planck Length).

What parts of the body are responsible for that movement, given that the ions and electrons in the body are quanta that do no move continuously? What is it within the body that is NOT comprised of electrons and ions, that could cause muscles (and thus the runner's body) to move such a distance?

The Belief Doctor, Sydney
March 13, 2014 4:29pm

See, I replied to this sarcastically earlier with 'impulses from the brain' but that is legitimately the answer to your question, an electrical impulse. Which is electrons moving from one atom to another. Also, electrons DO move constantly that's just... that's just fact.

Also, given it's infinitely smaller than planck, just vibrating on the spot would probably get you there (which is something molecules do in a solid).

Now, would you kindly explain how you have something infinitely smaller than Planck? My understanding of infinite would imply that's nonsensical.

Jimmy, England
April 16, 2014 3:55am

"an electrical impulse. Which is electrons moving from one atom to another. Also, electrons DO move constantly that's just... that's just fact."

What evidence do you have to contradict David Bohm's "according to the quantum theory, movement is not fundamentally continuous"?

Also, simply stating "electrical impulse" does nothing to explain correlation of movement with cause.

As for infinitely smaller than the Planck length -- that's required by infinite-series solutions, which involve infinitely shorter increments than any increment you care to consider. Infinite means for whatever increments, there's still infinitely more shorter ones within that increment. It's never-ending. That's the literal meaning of infinite. Never-ending, endless, limitless, boundless, forever more, etc.

So my example of 1/1,000,000 (one one-millionth) the Planck length is but a drop in a bottomless bucket of ever shorter lengths which the body somehow causes to be traversed, with each being related to a distinct physical cause (assuming adherence to standard deterministic scientific theory: cause <-> effect (aka physical movement). Noting here that "momentum" or "inertia" cannot be assigned as the cause, since momentum (p) = h/λ

The Belief Doctor, Sydney
June 2, 2014 8:40pm

"What evidence do you have to contradict David Bohm's "according to the quantum theory, movement is not fundamentally continuous"?"

A very basic understanding of the atom? If you want to describe it in quantum terms, the electron isn't moving in a motion sense but it's occupying more than one point at a time. Perhaps it's easier to just say the electron's don't stop then start. For references, http://physics.stackexchange.com/questions/23453/how-can-one-describe-electron-motion-around-hydrogen-atom might be helpful

"As for infinitely smaller than the Planck length -- that's required by infinite-series solutions, which involve infinitely shorter increments than any increment you care to consider"

Right, and this is the problematic part, the Planck length is the fundamental limit on the accuracy of measurement. What you're describing is an abstract concept applied to a non-abstract frame. Like the problem with halving the distance, and the very point of bringing in Planck as Brian did,if you make up an alternate reality where things can be both infinitely small and infinitely huge and generally infinite then, yes, these are valid inquiries. But this seems to be the core of the problem, you say the body moves over an infinite distance whereas I say it travels a finite one with finite energy with understandable causes. If you take a balanced equation and make one of the numbers infinity then naturally it'll cease to make sense because you're applying it to a finite series.

Jimmy, England
June 5, 2014 4:33pm

you wrote: "Right, and this is the problematic part, the Planck length is the fundamental limit on the accuracy of measurement."

I'm unaware that anyone has experimentally verified that to be the case. It is merely conjecture, theoretical.

In any case, you've ignored the issue -- I did not write "infinite distance" I wrote infinite increments.

What bodily processes account for being the physical cause of the physical effect (aka physical movement) through each and EVERY one of those increments?

Infinite-series solutions, again, assuming standard deterministic science's absolute, unambiguous correspondence of every physical effect having a physical cause, requires each increment to have a clearly identifiable physical cause.

And that's for each and every one of those increments. Every last one of them. No exceptions.

Either that or you've dismissed standard cause-effect correspondence, and entered new territory -- as have many leading physicists who recognise the obvious.

The Belief Doctor, Sydney
June 6, 2014 4:22pm

I should add that any mention of measurement (or the limits thereof) is irrelevant to the issue of Zeno's Paradoxes.

The Uncertainty Principle is a principle. It stands irrespective of any need to measure anything.

Likewise, my posts here concern principles, ideas and concepts that remain independent of any need of anyone to measure anything.

The question "what bodily processes account for ... physical movement through an endless series of physical increments" does not need anyone to measure anything. No instruments required -- except for one: the mental instrument of logic, analysis and reason.

The Belief Doctor, Sydney
June 6, 2014 4:39pm

Zeno's paradoxes worked to protect the "ONE" against the "MANY".

Idealism: The side A of a square and its diagonal C are incommensurable, and that CC is equal to 2AA.
Materialism: The side A of a square and its diagonal C are commensurable, and that CC is not equal to 2AA.

The "ONE" in the ancient Greek philosophy: CC is equal to 2AA.
The "MANY" in the ancient Greek philosophy: CC is not equal to 2AA.

(Refer to "eurekainla 01" from Google)

Yun Zeng, Los Angeles
August 7, 2014 8:01pm

The double slit experiment has been performed many times during the last decades. In short it shows that a particle's behaviour depends on wether it is observed or not. If you don't have a scientific beliefsystem in front of your eyes it is easy to understand what is going on in this experiment. Since particles dont have eyes or brains the only way to explain this behaviour is to accept that they are "computergenerated" when we (our consciousness) observe them.

When we accept that the particle we observe is computergenerated we also have to accept that every object we observe also is computergenerated, since objects are supposed to be made of particles.

There are other indicators that tells us this is a VR too. Example: just look up what "maximum frames per second" refers to, and compare it with the fact that the universe has its speed of light, Plancklength and Plancktime. If it looks like a duck and walks and quacks like a duck, it probably is a duck.

In short what the Achilles and tortoise paradox says is that an object cannot move through an infinite number of points on its way from A to B. In other words the number of points has to be finite, which is just another way of saying that this is a virtual reality. How else could objects skip between points? Every object here is moved just like an object in a computergame, and there is no paradox.

This is my understanding of what Tom Campbell (My big Toe) says, but you should Google him and check it out yourself.

hmmm, Norway
December 11, 2014 7:37am

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