Zeno’s paradoxes

Almost 2,500 years ago, Zeno of Elea set out to challenge the way we understand the physical world through a set of brain teasers that have stuck with us for millennia. The most powerful of Zeno’s paradoxes grapple with the concept of infinity while pitting observable reality against the scientific language we use to describe that reality, suggesting that elements of the everyday, like motion and speed, are actually illusory.

Over their thousands of years of existence, they’ve revealed weaknesses in the tools we use to understand the basics of space and time. More than just riddles, they have pushed mathematicians and philosophers to be more rigorous and precise in the logic we use to describe the concepts at work in the paradoxical traps Zeno so cleverly set.

490 BC: Zeno is born in Elea, in what is now southern Italy.

450 BC: Zeno and his mentor Parmenides (“the father of metaphysics”) visit Athens, where they meet Socrates.

435 BC: Zeno dies, reportedly stabbed to death while biting the ear of the Elean tyrant he tried to overthrow.

c. 375 BC: Plato writes Parmenides, “perhaps the most puzzling Platonic dialogue,” which tells us much of what we know about Zeno.

350 BC: Aristotle addresses Zeno’s arguments in Physics.

c. 215 AD: In his Lives and Opinions of Eminent Philosophers, Diogenes Laërtius gives a small biographical sketch, including stories of Zeno’s violent death.

1821: Augustin-Louis Cauchy’s ideas of the limit and of functions as a pairing of numbers in calculus allow a satisfactory mathematical solution.

1861: Karl Weierstrass formally defines limits with his epsilon-delta method.

1961: Abraham Robinson’s non-standard analysis redeems the concept of infinitesimals, offering an alternative to limits for problems like Zeno’s paradoxes.

Not all of Zeno’s extant paradoxes are profound or difficult. The millet paradox, which states that one falling grain of millet makes no sound but a ton of falling millet makes a big one, is more of a stoner observation than a profound question about the physical world. His paradoxes of motion and space, on the other hand, are legendary.

Achilles paradox: If a turtle gets a head start in a race against Achilles, Achilles has to cover half the distance between himself and the turtle in order to catch up. Then half that. And half again. And again. In an upset, the turtle wins!

Arrow paradox: At any given instant, an arrow in flight occupies a certain space, no more and no less. At the next instant, it occupies a different space. If you assume an instant is indivisible, the arrow is not in motion. So how does it move?

Stadium paradox: Imagine three sets of three bodies in stadium rows: three As, three Bs, three Cs. The As are stationary; the Bs are moving right; the Cs are moving left at the same speed. In the same timeframe, the Cs will pass just one of the As, but two of the Bs. Crazy, right? (It doesn’t seem like it, but if you think of space and time atomistically, they pass without passing.)

“Zeno’s arguments, in some form, have afforded grounds for almost all theories of space and time and infinity which have been constructed from his time to our own.”

—Bertrand Russell

“Sure it’s crazy to deny motion, but to accept it is worse.”

—Nick Huggett

It took more than 2,000 years to break the dichotomy and Achilles paradoxes, and the people to do it were the French mathematical prodigy Augustin-Louis Cauchy and the German Karl Weierstrass. The mathematical answer can be summed up by the intuitive answer: Eventually, you get there.

In mathematical terms, one way of putting it is “the limit of an infinite sequence of ever-improving approximations is the precise value” (pdf). Not all infinite geometric series converge to a limit, but some do (pdf). Not everyone was satisfied with limit solutions. Scientists are happy with this and other approaches, but some philosophers and logicians continue the debate (pdf).

In the 1994 romantic comedy IQ, Albert Einstein’s niece (played by Meg Ryan) explains Zeno’s dichotomy paradox by approaching her crush—halfway, then halfway again—for a dance. Watch the clip here.

There’s a joke version of this, too: “When do they meet at the center of the dance hall? The mathematician said they would never actually meet because the series is infinite. The physicist said they would meet when time equals infinity. The engineer said that within one minute they would be close enough for all practical purposes.”

Zeno’s paradoxes are examples of what are known as “supertasks”: “the idea of an infinite number of actions performed in a finite amount of time.” One of the most famous supertasks is the Hilbert Hotel, which mathematician David Hilbert used to illustrate Georg Cantor’s idea that there are multiple infinities.

In the 1970 live-action/animated movie The Phantom Tollbooth, directed by Chuck Jones, one character suggests another divide his troubles in two until they disappear.

Once you’re done with Zeno, here are some other legendary paradoxes to mull.

😡 The Liar paradox, or Epimenides’s paradox: The Cretan philosopher is still remembered centuries later for the statement “all Cretans are liars”… get it?

🌉 Buridan’s bridge: From the 14th-century Parisian philosopher and Aristotle expert Jean Buridan.

🐴 Buridan’s ass: Inspired (but not created) by Buridan, it’s more paradoxical for machines than people.

🎺 Torricelli’s trumpet: A mathematical creation outlining a cone that has finite volume but infinite surface area.

💡 Fitch’s paradox of knowability: “It tells us that if any truth can be known then it follows that every truth is in fact known.”