Zeno's Paradox
Did He or Didn't He?
(Catch Up With the Turtle)

Achilles and Friend

 

Paradoxes for Fun, Not Profit

Philosophers in Ancient Greece, when they were sitting around and not getting paid¹, liked to pose paradoxes. That is, they asked questions about situations that seemed to contradict themselves but still appeared to be true.

The most famous is the "Liar's Paradox" where you want to know if the sentence, "I am lying" is true or false². Then there's the "Arrow Paradox" which says an arrow can't really fly through the air³. Perhaps the paradox most relevant to today is the "Paradox of the Rock Group" (formerly called the "Paradox of the Ship"). Is a rock group still the same rock group when all the original members have been replaced?⁴

But one philosopher who really liked to pose paradoxes was Zeno of Elea. Elea was a city in Italy about 50 miles southeast along the coast from Naples. It was one of the Greek colonies of the Mediterranean that were called Magna Graecia, Μεγάλη Ἑλλά in Greek (pronounced "meg-AH-lay hel-LAH") and where Greek philosophers liked to sit around and philosophize. Zeno thought up a bunch of paradoxes which later philosophers like Aristotle spent a lot of time trying to explain. Sometimes they did, sometimes they didn't.

 

Zeno's Paradox

By far Zeno's most famous paradox is "Achilles and the Turtle" which in common parlance is just called "Zeno's Paradox". Zeno said that in any race the Swift Footed Achilles could never win against a turtle if the Turtle had a head start.

Ha? (To quote Shakespeare.) How is that possible?

Well, look at it this way.

1. At the start of the race the Turtle has a head start. Naturally Achilles will soon reach the place where the Turtle began the race.
2. But the Turtle will have moved on. So the Turtle also begins the second leg of the race with a head start.
3. Then when Achilles reaches the place where the Turtle started the second leg, the Turtle will again have moved further on. So the Turtle also begins the third leg of the race with a head start.
4. In fact, for any part of the race, the Turtle always begins with a head start and Achilles can never quite catch up.
5. So no matter how long the race lasts or how far Achilles and the Turtle run, the Turtle will always be ahead and so win the race.

Obviously the situation described is horse hockey, bullshine, and poppycock. But like most famous paradoxes, it's easier to think the paradox is horse hockey, bullshine, and poppycock than pinpoint the error.

 

The Answer

OK. We know Zeno's Paradox is incorrect. But exactly where is the error in Zeno's reasoning? We've always wondered about that.

No doubt you have as Captain Mephisto said to Sidney Brandy. Actually the answer is very simple.

In a nutshell, here is what Zeno does.

1. Zeno breaks up the race into a series of steps that must be contained within a finite interval
2. Then he says the steps are never contained within a finite interval.

That's the error in Zeno's reasoning.

 

A Slight Rephrasing

But hold on there, says the reader. Since when did Zeno break the race into a series of steps that must be contained within a finite interval and then say they are not contained within finite interval.

He never said THAT!

Ah, but he DID!

Or rather, what he said was equivalent to breaking the race into a series of steps that must be contained within a finite interval and then say they are not contained within a finite interval.

To prove that this rephrasing of Zeno's paradox is in fact correct - and that it proves Zeno went off the rails - we'll have to put Zeno's paradox on a strict quantitative basis. And rather than go through a general algebraic derivation - which we'll relegate to an Appendix - we'll work with an example using actual numbers.

 

And They're Off!

We'll let Achilles start off at 10 miles per hour. The Turtle will move at half that speed - 5 miles per hour. That's fast for a turtle, perhaps, but still slower than Achilles.

But we'll also give the Turtle a 10 mile head start.

We can now put our information into a nice, neat table where we list the distances and the times.

So the start of the race, we have:

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)	Total time (hours)
	Achilles (10 mph)	Turtle (5 mph)
0	0	10	10	0

Ergo, Achilles is at the starting line at Time = 0. And the Turtle has a 10 mile head start.

Now we'll let Achilles run to where the Turtle started off. So Achilles runs until he is 10 miles from the starting line.

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)	Total time (hours)
	Achilles (10 mph)	Turtle (5 mph)
0	0	10	10	0
1	10

But after Step 1, the Turtle has also run ahead. But since he's running only at five miles an hour, he's only moved half the distance as Archilles. So the Turtle has moved only 5 miles.

But remember the Turtle had a 10 mile head start. So the distance from the starting line is:

D (Starting Line)	=	5 + 10
	=	15 miles

And so the table becomes:

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)	Total time (hours)
	Achilles (10 mph)	Turtle (5 mph)
0	0	10	10	0
1	10	15

And so the distance from Archilles and the Turtle is five miles:

D (Turtle from Achilles)	=	15 - 10
	=	5 miles

And we have:

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)	Total time (hours)
	Achilles (10 mph)	Turtle (5 mph)
0	0	10	10	0
1	10	15	5

Finally we fill out the time. That's pretty simple. It's just the distance traveled divided by the speed. Of course, we get the same time for either the Turtle or Achilles.

TACHILLES	=	10 miles/10 mph		=	1 hour
TTURTLE	=	5 miles/5 mph		=	1 hour

And at the end of Step 1 the Table is now:

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)	Total time (hours)
	Achilles (10 mph)	Turtle (5 mph)
0	0	10	10	0
1	10	15	5	1

At this point it's pretty obvious how to proceed. So we'll continue to Step 2.

For Step 2, Achilles now runs to the where the Turtle was at the end of Step 1. That's an additional 5 miles.

So Achilles is:

10 + 5 miles = 15 miles

... from the starting line.

And since Achilles has run another 5 miles, he has run for another:

^{5 miles}/_{10 mph} = ¹/₂ hours

But the Turtle has also moved ahead. Of course, he's also run for another ¹/₂ hour. So at 5 miles per hour he's traveled another:

5 mph× ¹/₂ hour = 2¹/₂ miles

... and he's:

2¹/₂ + 15 = 17¹/₂ miles

... from the starting line:

And so Achilles and the Turtle are:

17¹/₂ - 15 = 2¹/₂ miles

... apart.

So putting everything in the table we have:

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)	Total time (hours)
	Achilles (10 mph)	Turtle (5 mph)
0	0	10	10	0
1	10	15	5	1
2	15	171/2	21/2	11/2

So we keep going. And after a while we have our table fairly well filled in:

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)	Total time (hours)
	Achilles (10 mph)	Turtle (5 mph)
0	0	10	10	0
1	10	15	5	1
2	15	171/2	21/2	11/2
3	171/2	183/4	11/4	13/4
4	183/4	193/8	5/8	17/8
5	193/8	1911/16	5/16	115/16
6	1911/16	1927/32	5/32	131/32
7	1927/32	1959/64	5/64	163/64
8	1959/64	19123/128	5/128	1127/128
:	:	:	:	:
etc.	etc.	etc.	etc.	etc.

 

Simplify! Simplify
- Henry David Thoreau

At this point we must now cite some middle school math (all right, it's more than middle school math). But the pattern from the table can be simplified considerably. Although we eschew the rigorous proof, the distances and times for each step can be put into nice succinct formulas. And these are:

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)	Total time (hours)
	Achilles (10 mph)	Turtle (5 mph)
n	20 -20/2n	20 -20/2n+1	10/2n	2 -1/2n

You can test the formulas to prove it produces the correct numbers for each step. Like the commercials say, it really, really works.

So we now have the race precisely and quantifiably defined by Zeno's steps. And each step is assigned both a distance and a time.

And now, as the $200 an hour psychiatrist said, we can begin, yes?

 

Zeno the Confusticator

All right. Let's repeat what Zeno told us and check if it's correct.

Zeno has taken the race between Achilles and the Turtle where the Turtle has a head start. He breaks the race down into a series of steps. Then he claims there are an infinite number of such steps.

Zeno is correct.

There are an indeed infinite number of steps. We know this because in the table the number of any particular step - which we designated as n - has no upper limit.

Next, he says that for each of these steps, the Turtle is always ahead of Achilles by a finite amount.

Again Zeno is correct.

In fact in precise mathematical terms, for any positive integer, n, then the Turtle is ahead by ¹⁰/_2ⁿ miles.

Finally Zeno says that since there are an infinite number of steps where the Turtle is ahead, the Turtle must be ahead for the entire race of any distance or any duration.

ERROR! ERROR!

THAT is where Zeno is wrong.

Zeno Off the Beam

Yes, yes. But we already knew that Zeno is wrong.

But exactly where is Zeno off the beam? Where is the error in his reasoning?

First we must point out that Zeno's assertions are just that. He says the Turtle will always stay ahead. But - and pardon us if we shout:

ZENO HAS MADE AN ASSERTION. HE HASN'T PROVEN IT!

To do so Zeno must show that the set of steps, n,extend to any distance and any length of time.

Or in precise mathematical terms, Zeno must show that any arbitrary distance or time - no matter how large - must be less than the distance or time assigned to some step, n.

If Zeno can do that, then and only then has Zeno proven that Achilles can never catch the Turtle.

 

The Quantification of Zeno

So look at the table again.

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)	Total time (hours)
	Achilles (10 mph)	Turtle (5 mph)
n	20 -20/2n	20 -20/2n+1	10/2n	2 -1/2n

Now we could go through all sorts of brouhaha about calculating mathematical limits and such stuff. But although we do that in the appendix, it isn't necessary.

Instead, the mathematically inclined will note that the distances and times for any step are bounded. That is the distances and times cannot extend forever.

As to why that is, it's quite simple. Now it is true that the distance the Turtle is ahead - which is ²⁰/_2ⁿ and ²⁰/_2ⁿ⁺¹ does get smaller as n gets bigger. And as Zeno said, this distance is greater than zero.

But this also means that the distances that Achilles and the Turtle run:

Achilles:	20 - 20/2n miles
Turtle:	20 - 20/2n+1 miles

BOTH MUST BE LESS THAN 20 MILES FOR ANY VALUE OF n

Similarly the time of the race:

Time:

2 - 1/2n hours

MUST BE LESS THAN 2 HOURS

So it seems there the distances and time assigned to any step n are all smaller than some value of distance or time.

Or in a bit plainer English, the steps where the Turtle is ahead cannot go on forever.

Hm. Isn't this the exact opposite of what Zeno had to prove?

 

Mr. Zeno, What Does It All Mean?

If things still seem a bit confused, we'll cite something Confucius didn't say that a picture is worth a thousand words. So we'll let Zeno use a simple device from grade school math. That is, plotting the race steps along on a line.

... and we see, yes, there are an infinite number of points.

But for the Turtle to always stay ahead then the point at infinity, n_∞, must correspond to a point which is an infinite distance from the starting point, n₀.

But we've shown that Zeno's claim isn't correct. Instead what you get is:

It can in fact be shown - and it is in the appendix - that regardless how big the Turtle's head start is, as long as Achilles runs faster, the infinity of steps as defined by Zeno where the Turtle is ahead must be contained within a finite distance.

So where Zeno went wrong was exactly what we said at the beginning.

1. Zeno breaks up the race into a series of steps that must be contained within a finite interval
2. Then he says the steps are never contained within a finite interval.

With our graph, there is even simpler ways to understand were Zeno went wrong. We see that Zeno's amazing paradox is nothing more than defining a subset of points along a line. It is an infinite subset of points, yes, but the set is necessarily contained within a finite interval.

And saying Achilles will never catch the Turtle is saying no one can run from one end of a line segment to the other.

And that clinches it. There is no joy in Elea. Mighty Zeno has struck out.

 

As We Were Saying ...

But why do people still argue about Zeno's paradox? There are even some college professors in philosophy who seem to talk of it as an unsolved problem.

First, Zeno's Paradox, although it's provably incorrect, is certainly clever. He confuses everyone by creating an infinite series of steps where the Turtle is in front by a finite distance. And he then relies on the intuitive belief that infinite set of finite distances should add up to infinity. So it seems that Achilles would have to run an infinite distance to cover those steps.

But how, some ask, can Achilles ever catch up? That is by what mechanism can he reach the last of the infinite points where the Turtle is ahead.

Again Zeno's method tosses us a red herring. Remember Zeno has created the steps - they are in fact an artificial mathematical construct. They are not some physical barrier that keeps Achilles behind the Turtle. They are simply points along a line which anyone can run past.

Another way to look at the paradox is that if Archilles were to catch the Turtle he would have to cross an infinite number of finite steps. And to cross an infinite number of finite steps - so says Zeno - would require Achilles to run an infinite distance.

After all, if shouldn't an infinite sum of the finite distances itself must be infinite? Otherwise, couldn't Achilles catch the Turtle?

As we have seen the answer is no. And that is an infinite series of finite numbers does NOT have to add up to infinity. Instead many infinite subsets of numbers DO ADD UP to a finite number. These are called convergent series. And by elementary middle school math it is possible to prove that the steps as defined in Zeno's paradox are necessarily a convergent series.

The convergent series defined by Zeno's paradox is simply is the list of the distances the Turtle is ahead. Remember for any value of Step n the distance the Turtle is ahead by

¹⁰/_2ⁿ

... miles.

And the sum of the convergent series can be proven to be:

In fact, because the sum can also be expressed as:

... and you only have to sum up ¹/_2ⁿ terms, the problem is the same as the joke about the infinite number of mathematicians who walk into a bar. The first asks for a glass of beer. The second asks for a half a glass of beer. The third for a quarter glass, the fourth for an eight, and so on.

"Why are you ordering beers like that?" the barman asks. They all reply.

"Two's our limit."

1 Beer + 1/2 Beer + 1/4 Beer + 1/8 Beer + ... + 1/∞ Beer

=

2 Beers

Of course, there are many series which diverge - that is, they _ quote - "add up to infinity" - unquote. Such series are commonsensically called divergent series. The simplest example is simply:

So we have a final explanation of Zeno's paradox. Zeno creates a convergent series and claims it's divergent.

So that settles that.

And as a consolation for Zeno's fans, in his time mathematicians didn't know how to calculate summations of infinite series (the first example is from Archimedes over 200 years later).

And yes, it possible for Zeno to be correct at least sometimes. That is, in some races, the Turtle stays ahead. Remember in our example if the race goes on for a picometer less than 20 miles, then the Turtle wins.

But in any race longer than 20 miles - even by that picometer - then Achilles will win!

So everyone should be happy.

Appendix

The examples given above show a specific case where Achilles runs at 10 miles and hour and the Turtle at 5 miles per hour but with a 10 mile head start. But to prove Zeno's Paradox is based on a logical error, the general case must be proven. So the analysis below is for a race where Achilles is at the starting line and runs at x miles per hour. The Turtle runs at a speed of y miles per hour but with a head start of H miles.

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)
	Achilles (x mph)	Turtle (y mph)
0	0	H	H

All right. In the first step, Achilles runs to where the Turtle started off. That's H miles.

D_ACHILLES = H

... which we put in the table:

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)
	Achilles (x mph)	Turtle (y mph)
0	0	H	H
1	H

So how far has the Turtle run? Well, the distance is the Turtle's speed, y, multiplied by the time it takes Achilles to run H miles. We we call this time t_{Step 1}.

D_TURTLE = yt_{Step 1}

And t_{Step 1} is calculated as the distance traveled by Achilles divided by his speed.

t_{Step 1} = ^H/_x

And so during this time the Turtle has run:

D_TURTLE = ^Hy/_x

... but with the head start, H, the distance from the starting line is:

D_TURTLE = ^Hy/_x + H

And our table is:

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)
	Achilles (x mph)	Turtle (y mph)
0	0	H	H
1	H	Hy/x + H

And the distance between Achilles and the Turtle is obviously:

DTURTLE - DACHILLES	=	Hy/x + H - H
	=	Hy/x

And we finish up the second line as:

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)
	Achilles (x mph)	Turtle (y mph)
0	0	H	H
1	H	Hy/x + H	Hy/x

And now?

Well, we keep going. Achilles now has to run to where the Turtle was at the end of the last step. From the table we see that's where he's Hy / _x + H miles from the starting line:

And la table becomes:

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)
	Achilles (x mph)	Turtle (y mph)
0	0	H	H
1	H	Hy/x + H	Hy/x
2	Hy/x + H

And since Achilles ran an extra ^Hy/_x miles, the time taken was:

		Hy/x
t	=
		x

... hours.

And that means the Turtle has traveled another:

		y(Hy/x)
t	=
		x
	=	y/x(Hy/x)
	=	Hy2/x2

... miles.

So the Turtle's distance from the starting line is:

D_TURTLE = ^Hy2/_x² + ^Hy/_x + H

... and our table is now:

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)
	Achilles (x mph)	Turtle (y mph)
0	0	H	H
1	H	Hy/x + H	Hy/x
2	Hy/x + H	Hy2/x2 + Hy/x + H

And the distance between Achilles and the Turtle is simply:

DTURTLE - DACHILLES	=	Hy2/x2 + Hy/x + H - (Hy/x + H)
	=	Hy2/x2 + Hy/x + H - Hy/x - H
	=	Hy2/x2

... and putting everything in the table ....

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)
	Achilles (x mph)	Turtle (y mph)
0	0	H	H
1	H	Hy/x + H	Hy/x
2	Hy/x + H	Hy2/x2 + Hy/x + H	Hy2/x2

At this point, the discerning reader can see the emerging pattern. The distances are classical polynomials of the form:

y = a + ar + ar² + ... + arⁿ

... where ...

H = a

... and ...

^y/_x = r

... and n is the step number. The formula for the Turtle is almost the same as that for Achilles except that you add one more term, a^n + 1.

The net results which we now put in the final table - and using fancy summation signs - are:

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)
	Achilles (x mph)	Turtle (y mph)
n	n Σ i=0 H(y/x)i	n+1 Σ i=0 H(y/x)i	Hyn/xn

OK. At this point the table looks nice and neat but it really isn't. The summation sign means the actual formula gets longer and longer as n gets larger. Calculating the value of the distance for larger values of n quickly becomes cumbersome and a pain.

Fortunately, mathematicians working after Zeno came up with formulas for summing up series. In our case, with the polynomial of the form:

y = a + ar + ar² + ... + arⁿ

... provided that r < 1, we know that the summation is...

		a(1-rn)
a + ar + ar2 + ... + arn	=
		(1 - r)

So letting:

H = a

... and ...

^y/_x = r
(which is less than 1 as long as x > y):

We get:

		H(1-[y/x]n)
DACHILLES	=
		(1 - y/x)

... and ...

		H(1-[y/x]n+1)
DTURTLE	=
		(1 - y/x)

... and with some minor algebra, these equations become:

		Hx(1-[y/x]n)
DACHILLES	=
		(x - y)

 

		Hx(1-[y/x]n+1)
DTURTLE	=
		(x - y)

And to calculate the times it's easiest to use the formula for Achilles. You simply divide the distance by his speed (x) and you get:

		H(1-[y/x]n)
t	=
		(x - y)

... and adding the time to the column:

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)	Total time (hours)
	Achilles (x mph)	Turtle (y mph)
n	Hx(1-[y/x]n) (x - y)	Hx(1-[y/x]n+1) (x - y)	H(y/x)n	H(1-[y/x]n) (x - y)

... and by using calculus, we add the last line where we take the limit where n → ∞.

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)	Total time (hours)
	Achilles (x mph)	Turtle (y mph)
n	Hx(1-[y/x]n) (x - y)	Hx(1-[y/x]n+1) (x - y)	H(y/x)n	H(1-[y/x]n) (x - y)
∞	Hx (x - y)	Hx (x - y)	0	H (x - y)

Finally we let H = 10 miles, x = 10 mph, y = 5 mph. Then our table is:

Step #	Distance from Starting Line (miles)		Distance from Achilles and Turtle (miles)	Total time (hours)
	Achilles (10 mph)	Turtle (5 mph)
n	20 -20/2n	20 -20/2n+1	10/2n	2 -1/2n
lim n→∞	20	20	0	2

Exactly the table we deduced earlier.

[Return]

References and Further Reading

"Zeno’s Paradoxes", Nick Huggett, The Stanford Encyclopedia of Philosophy, 2002, 2018.

"Zeno of Elea", John Palmer The Stanford Encyclopedia of Philosophy, 2008, 2017.

"Is a Band Without Its Original Members Still the Same Band?", Geoffrey Himes, Smithsonian, June 29, 2015.

"10 Classic Rock Bands with No Original Members", KGON Radio (Portland), November 23, 2019.