In the Achilles Paradox, Achilles races to catch a slower runner—for example, a tortoise that is crawling in a line away from him. The tortoise has a head start, so if Achilles hopes to overtake it, he must run at least as far as the place where the tortoise presently is, but by the time he arrives there, it will have crawled to a new place, so then Achilles must run at least to this new place, but the tortoise meanwhile will have crawled on, and so forth. Achilles will never catch the tortoise, says Zeno. Therefore, good reasoning shows that fast runners never can catch slow ones. So much the worse for the claim that any kind of motion really occurs, Zeno says in defense of his mentor Parmenides who had argued that motion is an illusion.

Although practically no scholars today would agree with Zeno’s conclusion, we cannot escape the paradox by jumping up from our seat and chasing down a tortoise, nor by saying Zeno should have constructed a new argument in which Achilles takes better aim and runs to some other target place ahead of where the tortoise is. Because Zeno was correct in saying Achilles needs to run at least to all those places where the tortoise once was, what is required is an analysis of Zeno’s own argument.

In the Achilles Paradox, Zeno assumed distances and durations can be endlessly divided into (what modern mathematicians call a transfinite infinity of indivisible) parts, and he assumed there are too many of these parts for the runner to complete.

Aristotle‘s treatment said Zeno should have assumed instead that there are only potential infinities, so that at any time the hypothetical division into parts produces only a finite number of parts, and the runner has time to complete all these parts.

Aristotle’s treatment became the generally accepted solution until the late 19th century.

The current standard treatment or so-called “Standard Solution” implies Zeno was correct to conclude that a runner’s path contains an actual infinity of parts at any time during the motion, but he was mistaken to assume this is too many parts. This treatment employs the mathematical apparatus of calculus which has proved its indispensability for the development of modern science.  The article ends by exploring newer treatments of the paradoxes—and related paradoxes such as Thomson’s Lamp Paradox—that were developed since the 1950s.

a. His Life

Zeno was born in about 490 B.C.E. in Elea, now Velia, in southern Italy; and he died in about 430 B.C.E. He was a friend and student of Parmenides, who was twenty-five years older and also from Elea. He was not a mathematician.

There is little additional, reliable information about Zeno’s life. Plato remarked (in Parmenides 127b) that Parmenides took Zeno to Athens with him where he encountered Socrates, who was about twenty years younger than Zeno, but today’s scholars consider this encounter to have been invented by Plato to improve the story line.

Zeno is reported to have been arrested for taking weapons to rebels opposed to the tyrant who ruled Elea. When asked about his accomplices, Zeno said he wished to whisper something privately to the tyrant. But when the tyrant came near, Zeno bit him, and would not let go until he was stabbed.

Diogenes Laërtius reported this apocryphal story seven hundred years after Zeno’s death.

b. His Book

According to Plato’s commentary in his Parmenides (127a to 128e), Zeno brought a treatise with him when he visited Athens. It was said to be a book of paradoxes defending the philosophy of Parmenides. Plato and Aristotle may have had access to the book, but Plato did not state any of the arguments, and Aristotle’s presentations of the arguments are very compressed.

A thousand years after Zeno, the Greek philosophers Proclus and Simplicius commented on the book and its arguments. They had access to some of the book, perhaps to all of it, but it has not survived. Proclus is the first person to tell us that the book contained forty arguments.

This number is confirmed by the sixth century commentator Elias, who is regarded as an independent source because he does not mention Proclus. Unfortunately, we know of no specific dates for when Zeno composed any of his paradoxes, and we know very little of how Zeno stated his own paradoxes.

We do have a direct quotation via Simplicius of the Paradox of Denseness and a partial quotation via Simplicius of the Large and Small Paradox. In total we know of less than two hundred words that can be attributed to Zeno.

Our knowledge of these two paradoxes and the other seven comes to us indirectly through paraphrases of them, and comments on them, primarily by his opponents Aristotle (384-322 B.C.E.), Plato (427-347 B.C.E.), Proclus (410-485 C.E.), and Simplicius (490-560 C.E.). The names of the paradoxes were created by later commentators, not by Zeno.

c. His Goals

In the early fifth century B.C.E., Parmenides emphasized the distinction between appearance and reality. Reality, he said, is a seamless unity that is unchanging and can not be destroyed, so appearances of reality are deceptive. Our ordinary observation reports are false; they do not report what is real.

This metaphysical theory is the opposite of Heraclitus’ theory, but evidently it was supported by Zeno.

Although we do not know from Zeno himself whether he accepted his own paradoxical arguments or exactly what point he was making with them, according to Plato the paradoxes were designed to provide detailed, supporting arguments for Parmenides by demonstrating that our common sense confidence in the reality of motion, change, and ontological plurality (that is, that there exist many things), involve absurdities. Plato’s classical interpretation of Zeno was accepted by Aristotle and by most other commentators throughout the intervening centuries. On Plato’s interpretation, it could reasonably be said that Zeno reasoned this way: His Dichotomy and Achilles paradoxes presumably demonstrate that any continuous process takes an infinite amount of time, which is paradoxical. Zeno’s Arrow and Stadium paradoxes demonstrate that the concept of discontinuous change is paradoxical. Because both continuous and discontinuous change are paradoxical, so is any change.

Eudemus, a student of Aristotle, offered another interpretation. He suggested that Zeno was challenging both pluralism and Parmenides’ idea of monism, which would imply that Zeno was a nihilist.

Paul Tannery in 1885 and Wallace Matson in 2001 offer a third interpretation of Zeno’s goals regarding the paradoxes of motion. Plato and Aristotle did not understand Zeno’s arguments nor his purpose, they say.

Zeno was actually challenging the Pythagoreans and their particular brand of pluralism, not Greek common sense. Zeno was not trying to directly support Parmenides.

Instead, he intended to show that Parmenides’ opponents are committed to denying the very motion, change, and plurality they believe in, and Zeno’s arguments were completely successful. This controversial issue about interpreting Zeno’s purposes will not be pursued further in this article, and Plato’s classical interpretation will be assumed.

Aristotle believed Zeno’s Paradoxes were trivial and easily resolved, but later philosophers have not agreed on the triviality.

d. His Method

Before Zeno, Greek thinkers favored presenting their philosophical views by writing poetry. Zeno began the grand shift away from poetry toward a prose that contained explicit premises and conclusions.

And he employed the method of indirect proof in his paradoxes by temporarily assuming some thesis that he opposed and then attempting to deduce an absurd conclusion or a contradiction, thereby undermining the temporary assumption.

This method of indirect proof or reductio ad absurdum probably originated with Greek mathematicians, but Zeno used it more systematically and self-consciously.

2. The Standard Solution to the Paradoxes

First published Tue Apr 30, 2002; substantive revision Mon Jun 11, 2018

Before we look at the paradoxes themselves it will be useful to sketch
some of their historical and logical significance. First, Zeno sought
to defend Parmenides by attacking his critics.

Parmenides rejected
pluralism and the reality of any kind of change: for him all was one
indivisible, unchanging reality, and any appearances to the contrary
were illusions, to be dispelled by reason and revelation.

Not
surprisingly, this philosophy found many critics, who ridiculed the
suggestion; after all it flies in the face of some of our most basic

(Interestingly, general
relativity—particularly quantum general
relativity—arguably provides a novel—if novelty
is possible—argument for the Parmenidean denial of
change: Belot and Earman, 2001.

) In response to this criticism Zeno
did something that may sound obvious, but which had a profound impact
on Greek philosophy that is felt to this day: he attempted to show
that equal absurdities followed logically from the denial of
Parmenides’ views. You think that there are many things? Then
you must conclude that everything is both infinitely small and
infinitely big! You think that motion is infinitely divisible? Then it
follows that nothing moves! (This is what a ‘paradox’ is:
a demonstration that a contradiction or absurd consequence follows
from apparently reasonable assumptions.)

As we read the arguments it is crucial to keep this method in mind.
They are always directed towards a more-or-less specific target: the
views of some person or school.

We must bear in mind that the
arguments are ‘ad hominem’ in the literal Latin sense of
being directed ‘at (the views of) persons’, but not
attacking the (character of the) people who put forward the views
rather than attacking the views themselves.

They work by temporarily
supposing ‘for argument’s sake’ that those
assertions are true, and then arguing that if they are then absurd
consequences follow—that nothing moves for example: they are
‘dialectic’ in the sense of the period).

Then, if the
argument is logically valid, and the conclusion genuinely
unacceptable, the assertions must be false after all.

Thus when we
look at Zeno’s arguments we must ask two related questions: whom
or what position is Zeno attacking, and what exactly is assumed for
argument’s sake? If we find that Zeno makes hidden assumptions
beyond what the position under attack commits one to, then the absurd
conclusion can be avoided by denying one of the hidden assumptions,
while maintaining the position. Indeed commentators at least since
Aristotle have responded to Zeno in this way.

So whose views do Zeno’s arguments attack? There is a huge
literature debating Zeno’s exact historical target.

As we shall
discuss briefly below, some say that the target was a technical
doctrine of the Pythagoreans, but most today see Zeno as opposing
common-sense notions of plurality and motion.

We shall approach the
concerning the interpretive debate.

That said, it is also the majority opinion that—with certain
qualifications—Zeno’s paradoxes reveal some problems that
cannot be resolved without the full resources of mathematics as worked
out in the Nineteenth century (and perhaps beyond).

This is not
(necessarily) to say that modern mathematics is required to answer any
of the problems that Zeno explicitly wanted to raise; arguably
Aristotle and other ancients had replies that would—or
should—have satisfied Zeno. (Nor shall we make any particular
claims about Zeno’s influence on the history of mathematics.

)
However, as mathematics developed, and more thought was given to the
paradoxes, new difficulties arose from them; these difficulties
require modern mathematics for their resolution.

These new
difficulties arise partly in response to the evolution in our
understanding of what mathematical rigor demands: solutions that would
satisfy Zeno’s standards of rigor would not satisfy ours. Thus
we shall push several of the paradoxes from their common sense
formulations to their resolution in modern mathematics.

(Another
qualification: we shall offer resolutions in terms of
‘standard’ mathematics, but other modern formulations are
also capable of dealing with Zeno, and arguably in ways that better
represent his mathematical concepts.)

2.1 The Argument from Denseness

If there are many, they must be as many as they are and neither more
nor less than that. But if they are as many as they are, they would be
limited. If there are many, things that are are unlimited. For there
are always others between the things that are, and again others
between those, and so the things that are are unlimited.

(Simplicius(a) On Aristotle’s Physics, 140.29)

This first argument, given in Zeno’s words according to
Simplicius, attempts to show that there could not be more than one
thing, on pain of contradiction: if there are many things, then they
are both ‘limited’ and ‘unlimited’, a

On the one hand, he says that any collection must
contain some definite number of things, or in his words
‘neither more nor less’. But if you have a definite number
of things, he concludes, you must have a
finite—‘limited’—number of them; in drawing
this inference he assumes that to have infinitely many things is to
have an indefinite number of them.

On the other hand, imagine
any collection of ‘many’ things arranged in
space—picture them lined up in one dimension for definiteness.
Between any two of them, he claims, is a third; and in between these
three elements another two; and another four between these five; and
so on without end. Therefore the collection is also
‘unlimited’.

So our original assumption of a plurality
leads to a contradiction, and hence is false: there are not many
things after all. At least, so Zeno’s reasoning runs.

Zeno’s Paradox: Understanding Convergent & Divergent Series

In the fifth century B.C., the Greek philosopher Zeno of Elea attempted to demonstrate that motion is only an illusion by proposing the following paradox:

Achilles the warrior is in a footrace with a tortoise, but Achilles has given the tortoise a 100-meter head start.

If Achilles runs 10 times as fast as the tortoise, by the time he catches up to the tortoise’s starting point, the tortoise will have advanced another 10 meters.

It occurs to Achilles that the next time he catches up to where the tortoise is now, the tortoise will again have advanced … and this will be the case over and over to no end. By this logic, Achilles will never catch the tortoise!

Today we know that this paradox — Zeno created several that dealt with space and time — has nothing to do with motion being illusory, but we still talk about it because it introduced some interesting math that wouldn’t receive thorough treatment until the 17th century A.D., when Gottfried Leibniz invented calculus. Even though the number of points where Achilles catches up to where the tortoise was last is infinite, the sum between all those points is finite. We call this phenomenon a “convergent series.”

A simpler version of this problem is best told as a joke. An infinite number of mathematicians walk into a bar. The first orders half a beer; the second orders a quarter; the third an eighth. After looking down the line, the bartender exclaims “You're all idiots!” pours one beer for them all to share, and closes the tab.

In this case, it’s pretty easy to see that the total of this infinite number of orders will add up to one beer. The terms in the sum get small enough quickly enough to where the total converges on some quantity.