Zeno's paradoxes
Achilles can never catch a tortoise with a head start, because each time he reaches where it was, it has moved on. Motion itself, Zeno argued, is impossible.
- Reading time
- 4 min
- Difficulty
- Intermediate
- Branch
- Logic
- First posed
- Zeno of Elea, c. 450 BCE, preserved by Aristotle, Physics, Book VI
The puzzle
Achilles, the fastest runner in Greece, races a tortoise. Being generous, he gives it a head start of 100 metres. The race begins.
By the time Achilles reaches the point where the tortoise started, the tortoise has crawled forward a little, say 10 metres. When Achilles covers those 10 metres, the tortoise has moved another metre. When he covers that metre, it has moved a tenth of a metre. And so on, forever. Every time Achilles reaches where the tortoise was, it is somewhere further ahead. There are infinitely many such catch-up stages. How can Achilles complete infinitely many tasks? It seems he can never overtake the tortoise.
Zeno of Elea, a student of Parmenides, devised this and several other paradoxes around 450 BCE, apparently to defend his teacher's view that change and plurality are illusions. His writings are lost; we know the paradoxes mainly from Aristotle, who tried to refute them.
The 'dichotomy' paradox is even simpler. To walk across a room, you must first walk half way. Before that, a quarter. Before that, an eighth. There is no first step you can take, since any distance has a half that must come before it. So you can never begin.
Why it matters
Zeno's paradoxes are among the oldest puzzles about infinity, and they shaped mathematics for more than two thousand years. They forced thinkers to ask what it means to divide space and time infinitely, and whether an infinite process can be completed.
They also raise a question that remains open: does the mathematical solution actually explain how motion is possible in the physical world, or only show that the mathematics is consistent?
Key terms
- Infinite seriesThe sum of infinitely many terms. Some such sums are finite: one half plus one quarter plus one eighth and so on adds up to exactly one.
- Limit (mathematics)The value a sequence or sum approaches ever more closely as it goes on. Modern mathematics uses limits to make sense of infinite sums and motion.
- ContinuumA space or quantity with no gaps, such as the real number line, in which between any two points there are infinitely many more.
- SupertaskAn infinite sequence of tasks completed in a finite time, for example by doing each step in half the time of the one before.
- Potential infinityAristotle's notion of a process that can always be continued, such as counting, without an infinite totality ever existing all at once. Contrasted with actual infinity.
Proposed solutions
The mathematical solution
An infinite series can have a finite sum. The infinitely many stages take a finite total time, so Achilles catches the tortoise.
Aristotle: potential, not actual, infinity
A distance is potentially infinitely divisible, but it is not actually divided into infinitely many parts. Achilles crosses a finite distance, not an infinite collection of points.
Space and time are discrete
Perhaps space and time are not infinitely divisible at all. If there is a smallest unit, Zeno's infinite regress never starts.
Each solution, tested
The mathematical solution
- How it resolves the paradox
- The distances Achilles must cover, 100 + 10 + 1 + 0.1 + ..., form a geometric series whose sum is exactly 111.11... metres. If he runs at a constant speed, the times form a series with a finite sum too. Calculus, developed in the seventeenth century and made rigorous in the nineteenth with the concept of a limit, shows precisely how infinitely many shrinking quantities can add up to something finite. At 111.11 metres, Achilles draws level, and then passes.
- Why it doesn’t fully work
- This shows when and where Achilles catches the tortoise, but some philosophers argue it does not explain how he completes infinitely many distinct actions. The paradox asks how a supertask, an infinite sequence of tasks, can be finished, and the sum of a series does not directly answer that.
- What defenders say
- Defenders reply that Achilles does not perform infinitely many separate actions; he runs one continuous stretch, which can be described as infinitely many parts. Describing something as infinitely divisible does not make it infinitely hard to do.
Aristotle: potential, not actual, infinity
- How it resolves the paradox
- Aristotle distinguished potential infinity, a process that can always be continued, from actual infinity, a completed infinite collection. A line can be divided again and again without end, but it is not made of an actual infinity of pieces. Since time is divisible in the same way as distance, the infinitely divisible distance is crossed in an infinitely divisible but finite time.
- Why it doesn’t fully work
- Modern mathematics freely uses actual infinities: the real number line contains infinitely many points, and set theory treats infinite collections as completed objects. Aristotle's distinction is hard to maintain alongside it.
Space and time are discrete
- How it resolves the paradox
- Some approaches in physics suggest a fundamental scale, around the Planck length (about 1.6 times 10 to the minus 35 metres), below which our ordinary concepts of distance may break down. If space is made of indivisible units, Achilles takes finitely many steps, and the paradox dissolves.
- Why it doesn’t fully work
- Zeno anticipated this with his paradox of the arrow: if time is made of indivisible instants, then at each instant a flying arrow is motionless, occupying a space equal to itself. If it is motionless at every instant, when does it move? Discrete theories face their own puzzles.
Who argued what
- The mathematical solution Augustin-Louis Cauchy, Karl Weierstrass
- Aristotle: potential, not actual, infinity Aristotle
- Space and time are discrete Democritus, Some approaches to quantum gravity
Variations
The arrow
At any single instant, an arrow in flight occupies exactly its own length of space and does not move during that instant, since an instant has no duration. Time is made of instants. So the arrow is never moving. Modern physics answers with the idea of instantaneous velocity, defined as a limit, but some argue this re-describes the puzzle rather than solving it.
Thomson's lamp
James Thomson imagined a lamp switched on after one minute, off after a further half minute, on after a quarter minute, and so on. After two minutes, infinitely many switches have occurred. Is the lamp on or off? The sum of the times is finite, yet there is no last switch to determine the answer. Some conclude that such supertasks are impossible.
The stadium
Zeno's fourth paradox involves rows of bodies moving past each other in opposite directions. Its interpretation is disputed, but one reading is that if space and time are made of smallest units, a body could pass another without ever being opposite it, which is absurd.
Go deeper
Why the mathematics took so long
Greek mathematicians, including Archimedes, summed some infinite series using the 'method of exhaustion', a clever way of avoiding explicit infinities. Newton and Leibniz's calculus in the late seventeenth century relied on 'infinitesimals', infinitely small quantities that George Berkeley mocked as 'ghosts of departed quantities'. Only in the nineteenth century did mathematicians like Cauchy and Weierstrass give rigorous definitions of limits that avoided infinitesimals entirely.
Related ideas
Further reading and sources
- EncyclopediaStanford Encyclopedia of Philosophy: Zeno's Paradoxes
- WikipediaZeno's paradoxes
- BookPhysics, Book VI, Aristotle
- BookParadoxes, R. M. Sainsbury
- BookThe Infinite, A. W. Moore
Open questions
- Does a mathematical proof that Achilles catches the tortoise explain how he does it?
- Can an infinite number of tasks ever really be completed?
- Is space made of points, of tiny pieces, or of something else?