Paradox of the Tortoise and Achilles

What is the Paradox Of The Tortoise And Achilles?

The Paradox of the Tortoise and Achilles is a puzzle from a long time ago, created by a Greek thinker named Zeno. Imagine a race where Achilles, a fast runner, can’t catch up to a slow tortoise because of a twist in logic. Here’s how it goes in simple terms:

  • A tortoise is given a head start in a race against Achilles.
  • Achilles starts running, and even though he is faster, he never seems to pass the tortoise.
  • Every time Achilles reaches the spot where the tortoise was, the tortoise has moved a bit further away.
  • This keeps happening over and over, and it seems like Achilles will never catch up.

This idea messes with our heads because we know fast runners catch up to slow ones in real races, but Zeno’s logic challenges this common sense.


The brain-twisting story began around 450 B.C. Zeno made these puzzles to support his teacher’s ideas—that things we see as many are really just one, and that change is just an illusion. The Tortoise and Achilles shows us a confusing side of space and time, hinting that maybe moving isn’t real because to get somewhere, you have to cross endless half-distances.

In detail, the paradox goes like this: Achilles lets the tortoise start first. As Achilles runs to where the tortoise was, the tortoise shuffles a bit ahead. Every time Achilles reaches the new spot where the tortoise has been, the tortoise creeps a little more forward. Zeno suggests that Achilles can’t pass the tortoise ever because there’s always a new spot to reach first.

Key Arguments

  • Infinite Division: This is the tricky thought that you can keep cutting any distance into tinier bits forever.
  • No Actual Motion: If Achilles has to hit these endless spots, he’ll never catch the tortoise, which makes us wonder: Is he really moving?
  • Illusion of Speed: Achilles is quicker, but this mix-up suggests that no matter his speed, the tortoise is always a step ahead.

Definition and Explanation

  • The first definition of the paradox is like a brain game where Achilles can’t catch the tortoise no matter how much he speeds up. It’s like saying if you keep cutting a cake in half, then half again, and keep going, you’ll never actually finish the cake because there’s always a smaller piece to cut.
  • The second definition is about how this puzzle makes us question if movement is real or just something we think happens because of the way we see things. In other words, it’s like when you see wheels on a car spin so fast they look like they’re not moving or even spinning backward.

Examples and Explanation

  • A game of tag where the “it” person has to tag players who keep moving, but every time “it” gets close, the players take a step away. This is like the paradox because even though “it” is faster, there’s always just a little more ground to cover.
  • When you try to touch the horizon. You walk towards it, but no matter how far you go, you can’t reach it. This never-ending pursuit echoes the idea of the tortoise staying ahead of Achilles.
  • Trying to count to infinity. You can always add one more number, just like Achilles always has one more spot to reach before he can pass the tortoise.
  • Getting closer to a mirror. As you move towards your reflection, your image seems to move too, but you can’t ever catch it. This shows how getting closer to a goal doesn’t always mean you’ll reach it, similar to Achilles and the tortoise.
  • Downloading a big file on your computer. As the download progresses, you see it’s 99% complete, but it takes longer to finish that last 1%. It’s similar to how Achilles keeps reaching points the tortoise was, but the conclusion feels just out of reach.

Answer or Resolution

The puzzle stirred up a lot of confusion and debate until a new kind of math called calculus came along. Developed by Newton and Leibniz, calculus introduced something called limits. Limits help math folks deal with never-ending things by figuring out where they would actually end up.

Calculus showed that if you add up all the tiny bits of distance Achilles runs, it adds up to one distance he can cross in one amount of time. This means, despite the endless spots he has to hit, there is a finish line he can reach within the time he’s got because he runs each bit quicker than the last.

Major Criticism

Some people think Zeno’s brain teaser doesn’t stick when you apply it to the real world. They say that in life, unlike in math, you can’t keep dividing time and space forever. These naysayers believe Zeno was just pinpointing how humans sometimes get tripped up by their own thinking, not that motion doesn’t happen.

Practical Applications

Even though Zeno’s Paradox seems all theoretical, it’s had a big impact on how we think about other things:

Mathematics: It pushed people to make calculus and work out how to understand and add up things that can be cut up into endless pieces.

Physics: It’s made big waves in how scientists get how stuff moves and how time works, even down to tiny particles doing strange things when they’re not under the microscope.

Computer Science: The idea of breaking tasks into smaller bits is sort of like what computer programs do when they’ve got a huge job that’s too big to handle all at once.

Related Topics and Explanations

  • Calculus: The field of mathematics that helped solve Zeno’s Paradox. It deals with changing things and helps us understand movement and growth.
  • Physics: Physics looks at how everything in the universe moves and interacts. Zeno’s Paradox gave physicists a lot to think about, especially about how things might not always work how we expect.
  • Philosophy of Time: This area explores what time is and how it works. The paradox challenges the idea that time flows smoothly and has made philosophers wonder if our experience of time is all in our heads.


The Tortoise and Achilles Paradox stretches our minds by questioning the basics of movement, cutting things up, and what it means to be infinite. Though it first seems puzzling, the solution using calculus teaches us a big lesson: philosophy and math can work together to solve tricky problems. While we might not face Zeno’s infinite splits in everyday life, his stories still spark our curiosity and help us figure out the big questions about the world and how we understand it.