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Zeno's Paradoxes

They baffled the ancient Greeks, and they still baffle people today


Zeno of Elea (c. 490 – 430 BCE) is not to be confused with Zeno of Citium (c. 334 – 262 BCE).  The latter is the founder of Stoicism; the former—who are discussing here–is famous for his paradoxes.

Zeno was a student of Parmenides, and the purpose of his paradoxes is to defend Parmenides’ conception of ultimate reality as an unchanging, indivisible, imperishable unity.  He aims to do this by showing how if you assume that things really do move, or that they really are divisible, or that there really is a plurality of diverse things in the world, your assumption will lead to a contradiction.

Zeno wrote a book that according to later reports laid out forty paradoxes.  But little survives of the book, and from discussions by later thinkers scholars can only piece together nine of Zeno's paradoxes.  Of these, the best known are the ones seeking to prove that motion (and hence any kind of change) is impossible.  They have been labeled the Dichotomy, Achilles and the Tortoise, the Arrow, and the Stadium.

The Racetrack  (also known as the Dichotomy)

Imagine a runner about to run along a racetrack from the start (point A) to the finish (Point B).  Before he can reach B, he must travel half the distance between A and B.  Before he does that, he must travel half of that distance (i.e. a quarter of the track); before that, an eighth, before that sixteenth, and so on.  But since there is no distance, no matter how small, that can’t again be divided in half, the runner will never actually get started.

Achilles and the Tortoise

Achilles, hero of the Trojan war, was famously fleet-footed; tortoises are notoriously slow.  But if Achilles gives the tortoise a head start, no matter how fast he runs, he’ll never catch up with the beast.  Here’s why.  In the time it take Achilles to get to where the tortoise is when the race starts, (Point A), the tortoise will have moved forward a little way  top Point B.  And in the time it takes Achilles to go from A to B, the tortoise will have reached Point C.  The gap between the two will keep closing; but every time Achilles moves to where the tortoise was, the latter will always have moved a bit further forward.  Consequently, the tortoise can never be caught, just so long as he “keeps on keeping on.”

The Arrow

Imagine an arrow shot from a bow.  During a certain span of time it moves from A to B.  But at any instant, where is the arrow?  Answer: in any instant, the arrow occupies a space exactly equal to its length.  Now an instant of time is a bit like a geometer’s point.  A point, in geometry, has no spatial magnitude: it has no length or breadth.  Similarly, an instant has no temporal length..  It isn’t merely a very short period, a fraction of a nanosecond; it takes up no time whatsoever.  But motion takes time, necessarily.  So how does the arrow ever move?  At every instant it is at rest; so it is always at rest.

The Stadium

Imagine 3 rectangles, each with sides 2 feet in length, arranged like this:

            AAAA AAAA

            BBBB BBBB

            CCCC CCCC

Now suppose Rectangle A moves one foot to the left, and Rectangle C moves one foot to the right, while Rectangle B remains stationary .  If that happens, the rightmost tip of Rectangle A will have simultaneously moved past half of Rectangle B while only moving past the whole of Rectangle C.  This implies that one half is equal to two halves, which is absurd.

Apart from the above four, one other paradox of Zeno’s is fairly well-known:

The millet seed

If a bushel of millet seed makes a noise when it falls.  The bushel is made up of thousands of individual millet seeds.  But if you just drop one of these it doesn’t makes a noise. So if each seed makes no noise when it falls, how can a lot of them make a noise?  Add together a thousand silences, and you don’t get a noise.


Zeno’s paradoxes have baffled many fine minds from ancient times to the present, particularly the Racetrack, Achilles and the Tortoise, and the Arrow.  Aristotle thought he could satisfactorily answer Zeno by rejecting the assumption that actual space and actual time are infinitely divisible.  A more modern response is to invoke the distinction between a convergent infinite series and a divergent infinite series.  Zeno assumes that if a given distance is infinitely divisible, it contains an infinite number of parts, so it can’t be crossed in a finite time.  But in a convergent series (e.g. 1/2, 1/4, 1/8…etc) the infinite number of parts adds up to 1, which is a finite number. 

Zeno may not have convinced many people that motion is impossible.  But he provokes much serious reflection on the nature of time, space, and infinity.  He also pioneered a particular  form of argument–reductio ad absurdum, also known as  "indirect proof"–that has been used extensively by mathematicians and philosophers ever since.

Online resources

Zeno's paradoxes (Internet Encyclopedia of Philosophy) - Includes a detailed discussion of how modern mathematical analyses have superseded Aristotle's critique of Zeno

Zeno's paradoxes (Stanford Encyclopedia of Philosophy)

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