How did Achilles ever catch the tortoise?

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A race with a head start 0:00

Here's a puzzle. Achilles, the fastest runner in Greek legend, races a tortoise and gives it a head start. Before he can pass it, he has to reach the spot where the tortoise started. But by then the tortoise has crawled a little further on. So he runs to that new spot, and it's moved again. Every time he arrives, there's one more catch-up to run. So how does Achilles ever catch the tortoise?

A book nobody has read 0:30

The puzzle comes from Zeno of Elea, a Greek philosopher born around 490 BC in what's now southern Italy. He wrote a book of paradoxes, by one later account forty of them. The book is lost; only a few lines survive, quoted by later writers. We know his arguments second-hand, mostly from Aristotle, who reported them in order to knock them down, and from Simplicius, writing about a thousand years later.

Why argue that nothing moves? 1:05

Why would anyone argue that nothing moves? Zeno was defending his teacher and friend, Parmenides, who held that reality is one unchanging whole, and that change is an illusion. Critics ridiculed the idea, so Zeno turned the tables: take their view, that things move, and follow it step by careful step to something absurd. If every step is sound and the ending is absurd, an assumption has to go, and the real game is finding which one.

Achilles, as Aristotle tells it 1:38

Here's the race as Aristotle reports it, in his book the Physics. In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point the pursued started from, so the slower must always hold a lead. He lists it as one of four arguments of Zeno's against motion.

Halfway, and halfway again 2:00

The first of the four is even simpler, and it's called the dichotomy, which means cutting in two. To reach a goal, you first have to get halfway. Then halfway through what's left, then halfway again, and the halves never run out. Run it backwards and there seems to be no first step at all. Aristotle saw that the Achilles is the same argument in disguise, with the race cut into different-sized pieces.

The arrow that never moves 2:31

The third argument aims at time itself. Take an arrow in flight. At any single moment, it fills a space exactly its own size, and something that just fills its own space is at rest. So if time is made of moments, and the arrow is at rest in every one of them, when does it ever move? Aristotle replied that time isn't built out of indivisible moments, any more than a length is built out of indivisible bits.

The step Zeno skipped 3:01

So where's the catch? Zeno leans on a hidden assumption: that infinitely many steps must take forever. But the steps aren't all the same size. Take a strip one metre long. Cut it in half, then cut the leftover piece in half, and keep going. The pieces never run out. But could they ever add up to more than a metre?

Adding forever, landing on one 3:26

They can't, because they were all cut from that one metre. Lay them end to end: a half, three-quarters, seven-eighths, and the gap left is always the size of the last piece. After ten cuts the gap is one part in 1,024, about a millimetre; after twenty, it's less than a millionth of a metre. The total gets closer to one metre than any gap you can name, and it never goes past. Mathematicians say this infinite sum converges, and its value is exactly one.

Time gets cut too 4:01

Aristotle had already spotted the key move. Zeno, he wrote, makes a false assumption: that you can't pass through infinitely many things in a finite time. Time, he said, can be divided just as finely as distance. At a steady pace, the first half of the run takes half the time, the next quarter takes a quarter of the time, and so on. The times shrink exactly as the distances do, so they add up to an ordinary, finite run.

Aristotle's second thoughts 4:33

But Aristotle wasn't satisfied with his own answer. Later in the same work he wrote that it was adequate as a reply to the questioner, but as an account of the truth, inadequate. His deeper fix was that the halfway points in a run are only possible divisions, not places the runner actually has to stop. That's called a potential infinity, and modern mathematics no longer leans on the distinction.

Achilles, with real numbers 5:00

Now let's run the race. Say Achilles covers ten metres a second, the tortoise a generous one, with a hundred metres' head start. Leg one: Achilles runs a hundred metres in ten seconds, while the tortoise crawls ten. Then ten metres in one second, then one metre in a tenth of a second. Each leg is a tenth of the one before. Add up the distances and you get a hundred and eleven point one one one, with ones forever; add up the times and you get eleven point one one one seconds.

The shortcut 5:36

Now skip the infinity altogether. Achilles gains nine metres a second, so a hundred-metre gap is gone in eleven and a ninth seconds, and by then he's run a hundred and eleven and a ninth metres, exactly what the endless sum gave. A sum where each piece is the same fraction of the one before is called a geometric series, and whenever that fraction is less than one, the total is finite. So infinitely many catch-ups fit inside eleven and a ninth seconds, and then he's past the tortoise.

The long road to calculus 6:13

Archimedes added up an infinite series to find the area of a slice of a parabola, the first known example. In 1666 Isaac Newton was writing about what he called fluxions, and in 1684 Gottfried Leibniz published his version of calculus, the maths of change. Both versions leaned on vanishingly small quantities, treated sometimes as zero and sometimes as not. In 1734 the philosopher George Berkeley attacked calculus for its lack of rigour, and a really solid footing had to wait until the 1800s.

When shrinking isn't enough 6:54

But careful, because shrinking steps don't guarantee a finite total. Try one, plus a half, plus a third, plus a quarter, and so on. The pieces shrink towards nothing. Now group them: a third plus a quarter is more than a half. The next four, a fifth through to an eighth, make more than a half again. Each time you double the number of pieces, you add at least another half, so the total grows past any number you choose. This is called the harmonic series, and it diverges.

Slow, but endless 7:30

It's sneaky, because it grows so slowly. The total passes two after four pieces and three after eleven, but it needs 12,367 pieces just to pass ten. Plot it beside Zeno's halves: one flattens out at one, the other keeps creeping up forever. So the steps getting smaller isn't enough; they have to get smaller fast enough. Zeno seems to have taken it for granted that an endless sum must be endless, and he never gave a reason why.

What does a sum even mean? 8:04

Gut feeling is no help here. Take one minus one plus one minus one, forever. Pair the terms one way and you get zero; pair them another way and you get one. In 1821, the French mathematician Augustin-Louis Cauchy published a course setting out the basics of calculus as rigorously as he could. He made the first systematic, rigorous study of when an infinite series really has a sum, and his account became the standard one. In his system the halves add up to one, and one minus one plus one, forever, has no sum at all.

A limit, not a last step 8:47

The idea that settles it is called a limit. You don't add infinitely many things in one go; you ask what single number the running totals settle towards, and that number is the sum, by definition. There's no last step, so Achilles never has to take one. The same idea answers the arrow. Its speed at an instant is the number the average speeds approach over shorter and shorter stretches of time. Inside a single instant, distance over time is zero over zero, which isn't a number, so the arrow isn't proven frozen.

What calculus didn't settle 9:25

So is Zeno solved? The mathematical puzzle, yes. But the Stanford Encyclopedia of Philosophy notes that even the philosophers who answered Zeno with this mathematics agreed it isn't the whole story, because the paradoxes are also about the real world. Around 1950, the philosopher Max Black dreamed up 'infinity machines', to test whether anyone could really finish an infinite list of tasks. And attempts to unite quantum theory with gravity suggest that space and time might not be smooth all the way down. If not, the maths of the continuum may not describe a real racetrack.

So how did he catch it? 10:08

So here's the answer to the puzzle. Achilles catches the tortoise because his endless catch-ups shrink fast enough that their times add up to a finite total, in our race eleven and a ninth seconds. Zeno's hidden assumption, that infinitely many steps must take forever, is false. The arrow moves because motion happens across stretches of time, not inside a single instant. And whether real space and time are as smooth as the mathematics is a question that's still open.

From points to bridges 10:44

Zeno turned a short race into a puzzle about infinitely many points. Next time, the opposite kind of puzzle: not endless points, but a city, a river and a handful of bridges. How did seven bridges start the maths behind Google Maps?

Sources

Every factual claim in the episode is tied to one of these. Spotted an error? Tell us.

1. Nick Huggett, "Zeno's Paradoxes", Stanford Encyclopedia of Philosophy (first published… — plato.stanford.edu
2. Aristotle, Physics, Book VI (chapters 2 and 9), trans. R. P. Hardie and R. K. Gaye,… — classics.mit.edu
3. J. J. O'Connor and E. F. Robertson, "Zeno of Elea", MacTutor History of Mathematics,… — mathshistory.st-andrews.ac.uk
4. Gilbert Strang and Edwin "Jed" Herman, Calculus Volume 2, section 5.2 "Infinite… — openstax.org
5. Aristotle, Physics, Book VIII (chapter 8), trans. R. P. Hardie and R. K. Gaye, The… — classics.mit.edu
6. J. J. O'Connor and E. F. Robertson, "A history of the calculus", MacTutor History of… — mathshistory.st-andrews.ac.uk
7. J. J. O'Connor and E. F. Robertson, "Augustin-Louis Cauchy", MacTutor History of… — mathshistory.st-andrews.ac.uk
8. Gilbert Strang and Edwin "Jed" Herman, Calculus Volume 1, section 2.2 "The Limit of a… — openstax.org
9. Gilbert Strang and Edwin "Jed" Herman, Calculus Volume 1, section 2.1 "A Preview of… — openstax.org
10. Max Black, "Achilles and the Tortoise", Analysis 11(5): 91–101 (Oxford University… — doi.org

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