Welcome back to Numberholic! I’m your host, Sejari, and it’s always a pleasure to have you here. Today, we’re diving into the eighteenth episode of Proven Yet Hard to Believe.
When we talk about ancient paradoxes, there’s really no need to think twice — Zeno’s paradoxes are the ones that come to mind.
Zeno actually proposed several different paradoxes, but none of his original writings have survived. We know about them thanks to philosophers like Plato and Aristotle.
Today, I want to introduce you to Zeno’s paradoxes, focusing on four of them.
Let’s start with the most famous one: the race between Achilles and the tortoise.
So, here’s the setup.
Achilles, who’s super fast, races a tortoise, which, of course, is very slow. To give the tortoise a fair shot, it gets a head start.
Now, Achilles quickly reaches the spot where the tortoise started, but by that time, the tortoise has moved a little further ahead.
Achilles sprints to that new spot — but, again, the tortoise has moved a bit more.
No matter how fast Achilles runs, the tortoise is always just a little ahead, because it’s always moving while Achilles is trying to catch up.
And so, this process goes on forever.
According to Zeno, Achilles can never actually catch the tortoise.
When I first heard this paradox, I remember thinking, “Wow, that’s so weird!”
But since this story is pretty famous, let’s move on quickly to the conclusion.
Now, my take on this is a bit different:
I think Achilles both can and cannot catch the tortoise.
In other words, the paradox is actually valid.
Normally, the way people say this paradox got “solved” is by talking about how ideas around infinite series got properly developed in the early 1800s.
Let’s think it through:
Both Achilles and the tortoise are running at constant speeds.
If we graph their positions against time, it looks like this —
At first, Achilles reaches the tortoise’s starting point.
Then he catches up to the tortoise’s next position, and so on.
This process keeps happening.
But if you look carefully at the graph, there’s a really important thing to notice:
All of this happens within a finite amount of time.
Even though the process can be broken down into infinitely many steps, it doesn’t take forever — it just looks that way.
In the 1800s, mathematicians finally figured out how to calculate infinite sums properly — exactly what’s happening here.
One of the critiques of Zeno’s paradox that really stuck with me came from Professor Zeller, who republished a history of Greek philosophy around 1883.
Here’s his point:
He said that people were confusing the infinite divisibility of time and space with actually being infinitely divided.
In other words:
- Infinite divisibility = the ability to divide something endlessly, if you want to.
- Infinitely divided = something that’s already been split into an infinite number of parts.
So just because something can be divided infinitely, does it mean it is infinite?
Can’t something be finite, even though you could divide it endlessly?
That’s a question Zeno should be asked — and honestly, I think Zeller nailed it.
But at the same time, I also believe that the usual “solutions” to Zeno’s paradox aren’t entirely accurate either.
Let’s talk about why.
There’s actually a hidden hero in the typical “solution” to the paradox — and that’s the assumption that Achilles and the tortoise are running at constant speeds.
But what if we change that?
Imagine the tortoise is still moving at a constant speed — but Achilles starts slowing down.
Like this:
At first, Achilles runs and reaches the tortoise’s position in 1 second.
During that second, the tortoise moves forward a little.
Then Achilles catches up again, but this time it takes him half a second.
The tortoise moves forward again during that half-second.
Then Achilles catches up again in a third of a second.
He is definitely slowing down, but he’s still faster than the tortoise — he’s covering same distance in less time.
Next, he takes a quarter of a second, and so on.
Now think about it:
How much time does Achilles spend in total?
If you know about series from math, you’ll recognize this as the harmonic series.
And it diverges. It adds up to infinity.
In other words, Achilles would never catch the tortoise.
Even though Achilles is always faster than the tortoise at any given moment,
he still can’t catch it — just like in Zeno’s original paradox.
So, even though people say the paradox is “solved,” there’s actually a hidden assumption baked into the solution:
the idea of constant speed.
Without that assumption, the paradox stands.
In fact, I’d argue the paradox hasn’t been fully solved — and maybe it can’t be.
Let me give you one more reason why.
Think about rational numbers — the fractions we all know and love.
You can add, subtract, multiply, and divide them — and you can order them, too.
Same thing with real numbers, right?
But here’s the catch:
What really makes real numbers different from rationals is something called the Axiom of Completeness.
In simple terms, it says that whenever you have a sequence where the numbers get closer and closer together, there has to be a limit — a number they’re approaching.
But — and this is super important — that’s an axiom.
It’s a rule we assume to be true.
It’s not something that’s proved — it’s something we declare.
And Zeno’s paradox is the same kind of thing.
Just because Achilles gets infinitely close to the tortoise, that doesn’t automatically mean he catches it.
You can’t prove it without assuming certain rules first.
So in my view, Zeno’s paradox is still alive and kicking.
In the next video, I’ll dive into Zeno’s second paradox. Stay tuned!