Zeno's Paradox and the Invention of Limits

Partial sums of geometric series Σ rᵏ — converges to 1/(1−r) when |r| < 1

r = 0.500.95

n = 1230

Around 450 BCE, a Greek philosopher named Zeno of Elea made an argument that has bothered people for 2,500 years: motion is impossible.

His most famous version is the paradox of Achilles and the tortoise. Achilles — the great hero, fastest man alive — races a tortoise. Being sporting, he gives the tortoise a 100-metre head start. The race begins.

By the time Achilles reaches the tortoise's starting point, the tortoise has moved (say, 10 metres). By the time Achilles covers those 10 metres, the tortoise has moved 1 more metre. By the time Achilles covers that metre, the tortoise has moved 10 centimetres. And so on, forever.

Every time Achilles reaches where the tortoise was, the tortoise has crept a little further. There are infinitely many gaps to close. Surely it takes infinite time to close infinitely many gaps. Therefore Achilles can never overtake the tortoise.

You know this is wrong. Achilles obviously wins. The paradox is: why?

What's actually being added

The gaps Achilles must cross, in metres, form a geometric series:

$100 + 10 + 1 + 0.1 + 0.01 + \cdots$

The times to cross them (assuming Achilles runs at 10 m/s):

$10 + 1 + 0.1 + 0.01 + \cdots \text{ seconds}$

This is a geometric series with first term $10$ and ratio $r = 1/10$ .

$\sum_{n=0}^\infty 10 \cdot (0.1)^n = \frac{10}{1 - 0.1} = \frac{10}{0.9} \approx 11.11 \text{ seconds}$

Achilles closes all infinitely many gaps in about $11.11$ seconds. Then he passes the tortoise and keeps going.

The resolution Zeno was missing

Zeno's implicit assumption: infinitely many steps must take infinitely long. This is simply false.

Infinitely many positive numbers can have a finite sum — as long as they shrink fast enough. A geometric series $\sum_{n=0}^\infty r^n$ converges to $\frac{1}{1-r}$ when $|r| < 1$ . The terms go to zero fast enough that their sum doesn't blow up.

What Zeno was missing is the concept of a limit — the idea that an infinite process can converge to a finite answer. This concept wouldn't be made mathematically rigorous for more than 2,000 years.

Zeno wasn't confused — he was pointing at a real gap in Greek mathematics

It's tempting to dismiss Zeno as naive. But he wasn't. Greek mathematics had no way to handle infinite sums. The concept of a limit, as we now understand it, didn't exist. Zeno's paradox identified a genuine foundational problem: what does it mean for an infinite process to conclude?

The answer required the development of the real number system, the concept of convergence, and ultimately the epsilon-delta definition of a limit — all of which took until the 19th century to fully formalise.

A harder paradox: the dichotomy

Zeno had another version, simpler and more devastating. To walk across a room, you must first walk halfway. But to walk that halfway, you must first walk a quarter of the way. And a quarter requires an eighth first. And so on.

Before you can start moving, you have to complete an infinite number of tasks. How do you even begin?

The mathematical resolution is the same: $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = 1$ . The infinite sum is finite.

But the philosophical sting remains. The mathematician says the series converges. The philosopher asks: but you still have to complete infinitely many steps. How does that happen in finite time? The modern answer — that time itself is continuous, and that what we mean by "motion" is defined using limits — is essentially importing calculus into physics.

The legacy

Zeno's paradoxes forced ancient Greek mathematics to confront infinity seriously. Eudoxus of Cnidus (c. 360 BCE) developed the method of exhaustion — an early form of limits — to handle infinite processes in geometry. Archimedes used it to compute areas and volumes with remarkable precision.

But the method of exhaustion was laborious and had to be applied case-by-case. What it lacked was the general machinery of limits that Newton, Leibniz, and later Cauchy and Weierstrass would provide.

In a real sense, the 2,000 years between Zeno and Newton were spent slowly accumulating the conceptual tools needed to answer a question a Greek philosopher asked about a tortoise.