The paradox of Achilles and the tortoise, which was put forward by the ancient Greek philosopher Zeno, challenges common sense. It states that the athletic guy Achilles will never catch up with a floppy turtle if she starts her movement before him. So what is this: sophism (a conscious mistake in the proof) or a paradox (a statement that has a logical explanation)? Let's try to understand this article.
Who is Zeno?
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. From the writings of Plato it is known that Zenon was 25 years younger than Parmenides, wrote a defense of his philosophical system at a very early age. Although little has been saved from his writings. Most of us know about him only from the works of Aristotle, and also the fact that this philosopher, Zeno of Elea, is famous for his philosophical considerations.
Book of paradoxes
In the fifth century BC, the Greek philosopher Zeno dealt with the phenomena of movement, space and time. How humans, animals and objects can move is the basis of the Achilles paradox and the tortoise. The mathematician and philosopher wrote four paradoxes, or “motion paradoxes,” which were included in a book written by Zeno 2500 years ago. They supported the position of Parmenides that movement is impossible. We will consider the most famous paradox - about Achilles and the tortoise.
The story is this: Achilles and the tortoise decided to compete on the run. To make the contest more interesting, the turtle was ahead of Achilles by a certain distance, since the latter is much faster than the turtle. The paradox was that as long as the theory continues to run, Achilles will never overtake the turtle.
In one version of the paradox, Zenon argues that there is no such thing as movement. There are many variations, Aristotle lists four of them, although in essence you can call them variations of the two paradoxes of movement. One relates to time, and the other to space.
From Aristotle's Physics
From book VI.9 of Aristotle’s physics, you can find that
In a race, the fastest runner can never catch the slowest, since the pursuer must first reach the point at which the pursuit began.
So, after Achilles runs for an indefinite period of time, he will reach the point from which the turtle began to move. But in exactly the same time, the turtle will move forward, reaching the next point of its path, so Achilles still has to catch up with the turtle. Again he moves forward, rather quickly approaching what the turtle occupied before, again “discovers” that the turtle crawled forward a little.
This process repeats as long as you want to repeat it. Due to the fact that the measurements are human construction, and therefore endless, we will never reach the point where Achilles defeats the tortoise. This is precisely the Zeno paradox of Achilles and the tortoise. Following logical reasoning, Achilles will never be able to catch up with the turtle. In practice, of course, the sprinter Achilles will run past the slow tortoise.
The meaning of the paradox
The description is more complicated than the paradox. Therefore, many say: "I do not understand the paradox of Achilles and the tortoise." It is difficult to perceive with the mind that which is actually not obvious, but just the opposite is obvious. It is all about explaining the problem itself. Zeno proves that space is divisible, and since it is divisible, it is impossible to reach a certain point in space when another has moved further from this point.
Zeno, given these conditions, proves that Achilles cannot catch up with the turtle, because the space can be infinitely divided into smaller parts, where the turtle will always be part of the space in front. It should also be noted that while time is movement, as Aristotle did, the two runners will move endlessly, thus being motionless. It turns out that Zenon is right!
Achilles paradox and tortoise solution
The paradox shows a mismatch between the way we think about the world and what the world really is. Joseph Mazur, Professor Emeritus of Mathematics and author of Enlightened Symbols, describes the paradox as a “trick”, making you think of space, time, and motion in the wrong way.
Then the task arises to determine what exactly is wrong with our thinking. Movement is possible, of course, a fast human runner can get ahead of the turtle in the race.
The paradox of Achilles and the tortoise from the point of view of mathematics is as follows:
- Assuming the turtle is 100 meters ahead, when Achilles walked 100 meters, the turtle will be 10 meters ahead of him.
- When he reaches these 10 meters, the turtle will be 1 meter ahead.
- When he reaches 1 meter, the turtle will be 0.1 meters in front.
- When he reaches 0.1 meters, the turtle will be 0.01 meters in front.
Therefore, in the same process, Achilles will suffer countless defeats. Of course, today we know that the sum of 100 + 10 + 1 + 0.1 + 0.001 + ... = 111.111 ... is an exact number and determines when Achilles is ahead of the turtle.
To infinity, not beyond
The confusion created by Zeno's example was primarily from the infinite number of observation points and positions that Achilles first had to reach when the turtle was moving steadily. Thus, it would be almost impossible for Achilles to catch up with the tortoise, not to mention overtaking it.
Firstly, the spatial distance between Achilles and the tortoise is becoming less and less. But the time required to cover the distance is proportionally reduced. The created Zeno problem leads to the expansion of the points of motion to infinity. But there was no mathematical concept yet.
As you know, only at the end of the XVII century in calculus could a mathematically sound solution to this problem be found. 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 have provided the basis for almost all theories of space and infinity proposed in our time to this day ..."
Is it sophism or paradox?
Seen from a philosophical perspective, 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 overtake and overtake, but to catch up. The goal setting is to catch up. This will never allow the swift Achilles to either catch up or overtake the turtle. In this case, neither physics with its laws, nor mathematics can help Achilles overtake this slow creature.
Thanks to this medieval philosophical paradox that Zeno created, we can conclude: you need to set a goal correctly, and go to it. Trying to catch up with someone, you will always remain second, and even then in the best case. Knowing what purpose a person sets, it is safe to say whether he will achieve it or will be wasting his energy, resources and time.
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.