- Who is Zenon?
- Book of paradoxes
- From the physics of Aristotle
- Meaning of the paradox
- The solution to the paradox of Achilles and the tortoise
- To infinity, not beyond
- Is this a sophism or a paradox?

The paradox of Achilles and the tortoise, put forward by the ancient Greek philosopher Zeno, defies common sense. It claims that the athletic guy Achilles will never catch up with the clumsy tortoise if it starts its movement before him. So what is it: sophism (a deliberate error in proof) or a paradox (a statement that has a logical explanation)? Let's try to understand this article.

## Who is Zenon?

Zeno was born around 488 BC in Elea (today's Velia), Italy. He lived for several years in Athens, where he devoted all his energy to explaining and developing the philosophical system of Parmenides. It is known from the writings of Plato that Zeno was 25 years younger than Parmenides and wrote a defense of his philosophical system at a very early age. Although little has been salvaged from his writings. Most of us know about him only from the writings of Aristotle, and also that this philosopher, Zeno of Elea, is famous for his philosophicalreasoning.

## Book of paradoxes

In the fifth century BC, the Greek philosopher Zeno de alt with the phenomena of movement, space and time. How people, animals, and objects can move is the basis of the Achilles-tortoise paradox. The mathematician and philosopher wrote four paradoxes or "paradoxes of motion" that were included in a book written by Zeno 2500 years ago. They supported Parmenides' position that movement was impossible. We will consider the most famous paradox - about Achilles and the tortoise.

The story is this: Achilles and the tortoise decided to compete in running. To make the contest more interesting, the tortoise was ahead of Achilles by some distance, since the latter is much faster than the tortoise. The paradox was that as long as theoretically the run continued, Achilles would never overtake the tortoise.

In one version of the paradox, Zeno states that there is no such thing as movement. There are many variations, Aristotle lists four of them, although one might essentially call them variations on two paradoxes of motion. One touches time and the other touches space.

## From the physics of Aristotle

From book VI.9 of Aristotle's physics you can learn that

In a race, the fastest runner can never overtake the slowest, as the pursuer must first reach the point where the pursuit began.

So after Achilles runs for an indefinite amount of time, he will reach a pointfrom which the tortoise started. But it will take exactly the same time for the tortoise to move forward, reaching the next point on its path, so Achilles still has to catch up with the tortoise. Again he moves forward, approaching quite quickly what the tortoise used to occupy, again "discovers" that the tortoise has crawled forward a little.

This process is repeated as long as you want to repeat it. Because dimensions are a human construct and therefore infinite, we will never reach the point where Achilles defeats the tortoise. This is precisely the paradox of Zeno about Achilles and the tortoise. Following logical reasoning, Achilles will never be able to catch up with the tortoise. In practice, of course, the sprinter Achilles will run past the slow turtle.

## Meaning of the paradox

The description is more complex than the actual paradox. That's why many people say: "I don't understand the paradox of Achilles and the tortoise." It is difficult to perceive with the mind what is actually not obvious, but just the opposite is obvious. Everything is contained in the explanation of the problem itself. Zeno proves that space is divisible, and since it is divisible, one cannot reach a certain point in space when another has moved further from that point.

Zeno, given these conditions, proves that Achilles cannot catch up with the tortoise, because space can be infinitely divided into smaller parts, where the tortoise will always be part of the space in front. It should also be noted that while time is a movement, asthis is what Aristotle did, the two runners will move indefinitely, thus being stationary. It turns out that Zenon is right!

## The solution to the paradox of Achilles and the tortoise

Paradox shows the discrepancy between how we think about the world and how the world actually is. Joseph Mazur, emeritus professor of mathematics and author of Enlightened Symbols, describes the paradox as a "trick" that makes you think about space, time, and motion in the wrong way.

Then comes the task of determining what exactly is wrong with our thinking. Movement is possible, of course, a fast human runner can outrun a turtle in a race.

The paradox of Achilles and the tortoise in terms of mathematics is as follows:

- Assuming the tortoise is 100 meters ahead, when Achilles has walked 100 meters, the tortoise will be 10 meters ahead of him.
- When it reaches those 10 meters, the turtle will be 1 meter ahead.
- When it reaches 1 meter, the turtle will be 0.1 meters ahead.
- When it reaches 0.1 meters, the turtle will be 0.01 meters ahead.

So in the same process, Achilles will suffer countless defeats. Of course, today we know that the sum 100 + 10 + 1 + 0, 1 + 0, 001 + …=111, 111 … is the exact number and determines when Achilles beats the tortoise.

## To infinity, not beyond

The confusion created by Zeno's example was primarily from an infinite number of dotsobservations and positions that Achilles first had to reach as the tortoise moved steadily. Thus, it would be almost impossible for Achilles to overtake the tortoise, let alone overtake it.

Firstly, the spatial distance between Achilles and the tortoise is getting smaller and smaller. But the time required to cover the distance decreases proportionally. The created problem of Zeno leads to the expansion of points of motion to infinity. But there was no mathematical concept yet.

As you know, only at the end of the 17th century, it was possible to find a mathematically justified solution to this problem in calculus. Newton and Leibniz approached the infinite with formal mathematical approaches.

The English mathematician, logician and philosopher Bertrand Russell said that "…Zeno's arguments in one form or another provided the basis for almost all theories of space and infinity proposed in our time to the present day…"

## Is this a sophism or a paradox?

From a philosophical point of view, Achilles and the tortoise are a paradox. There are no contradictions and errors in reasoning. Everything is based on goal setting. Achilles had a goal not to catch up and overtake, but to catch up. Goal setting - catch up. This will never allow the swift-footed Achilles to overtake or overtake the tortoise. In this case, neither physics with its laws nor mathematics can help Achilles overtake this slow creature.

Thanks to this medieval philosophical paradox,which Zeno created, we can conclude: you need to set the goal correctly and go towards it. In an effort to catch up with someone, you will always remain second, and even then at best. Knowing what goal a person sets, one can say with confidence whether he will achieve it or will waste his time, resources and energy.

In real life, there are many examples of incorrect goal setting. And the paradox of Achilles and the tortoise will be relevant as long as humanity exists.