A Set of Philosophical and Mathematical Problems: Zeno’s Paradoxes

Waldo Otis
Jan 13 · 10 min read

Zeno was an ancient Greek philosopher who lived in the 5th century BC, we know little about him today. Unfortunately, none of his works survived. We owe what we know about Zeno more to Plato (his handiwork; Parmenides) and Aristotle (Physics).

Zeno was a loyal student of Parmenides, and he was advocating an idea that the people of his time could not easily swallow. Parmenides defended this incredible idea: The truth is unique and unchanging. Multiplicity, change, and movement do not exist, and impressions of multiplicity, change, and our deceiving senses cause movement.

Zeno had developed four paradoxes, perhaps in response to those who were mocking his teacher’s philosophy. It is these four paradoxes that keep Zeno up to date, and I will try to explain (and defend, to show how serious they are) below. Even today, after 2,500 years, the debate over these four paradoxes has not ceased, and everyday philosophers produce more and more ideas on them.

Bertrand Russell, Henri Bergson, and Alfred North Whitehead are some of the modern philosophers who have discussed the paradoxes of Zeno. I think Hegel also mentioned them as well. Moreover, Tolstoy speaks of these paradoxes of Zeno in War and Peace.

Achilles and the Tortoise

In one of its paradoxes, Zeno compared the demigod Achilles with a turtle. Achilles is the protagonist of the Trojan War. According to a myth which can be traced back to the 1st century AD, Achilles’ mother Thetis immersed Achilles in Styx, the river of immortality on the border between hell and earth, right after his birth. Still, the Achilles’ ankle did not get wet as his mother held him there. So Achilles was just vulnerable from behind the ankle. Achilles was shot dead in the back of his ankle with an arrow of Paris. The term Achilles tendon comes from this.

In the paradox, since the turtle is much slower than Achilles, the turtle starts the race at a place which is ahead of Achilles. Zenon argues that Achilles will never catch the turtle like this. In order to overtake the turtle, Achilles must first reach the point where the turtle begins. When Achilles reaches this point, the turtle will be a little further ahead, because the turtle is also running. Now Achilles must reach this new spot where the turtle stands. But when Achilles reaches this new spot, the turtle will be a little further ahead again, and it process does not stop at all; the turtle runs continuously. This goes on and on, and Achilles is never able to catch up to the turtle. This turtle surely knows how to run!

Don’t say, “this cannot happen in life.” Parmenides and Zeno, like you, know that Achilles will catch the turtle. However, they argue that what we see is not real, that our senses are deceiving us.

Let’s think about this paradox. To have a strong foundation for our ideas, let us assume that Achilles started the race 100 meters behind the turtle and he runs at 100 meters per second. If the turtle doesn’t move, Achilles will catch the turtle in 1 second. But the turtle runs as well. Let’s say the turtle runs at 10 meters per second (Let us just Suppose it can). Let’s call the point where Achilles started the race as point A. Achilles will reach point A1, the starting point of the turtle, after 1 second. In this 1 second, the turtle will travel 10 meters and will reach point A2. Achilles will reach point A2 after 1/10 seconds. The turtle will have gone 1 meter in 1/10 seconds, to his new point, A3. Then Achilles will run this 1 meter in 1/100 seconds.

There is a paradox, and Mathematicians will just let it stay there and look away? Come on… They solved it like this:

So, mathematicians say, Achilles reaches the turtle in:

Simple arithmetic indicates that this infinite sum is 10/9. So Achilles catches the turtle after 10/9 seconds, less than 2 seconds, even less than 1.2 seconds.

Philosophers are not pleased with this answer. Philosophers do not care about mathematicians adding countless numbers in mathematics, but they oppose applying it to a problem taken from real life. How is it known that we can apply mathematics to real life? This is exactly what Zeno argues?

Mathematics tries to find the laws of nature, and I must say, quite well. For example, thanks to mathematics, planes, trains, buildings are built, and we can even go to the moon. There are many applications of mathematics. These applications show that mathematics is a useful tool to understand nature. But can mathematics be applied everywhere? For example, two apples plus three pears equal to five fruits because of 2 + 3 = 5. But if we use this mathematical truth to two liters of water and three liters of alcohol, we won’t get five liters of liquid. (I am not positive if this is the truth, but something is obviously is wrong: we need to ask the chemists.) So we should be careful in applying mathematics.

Nature is not a complete model of mathematics either. It can only be an approximate model of mathematics. Moreover, the above calculation does not show that Achilles will catch the turtle in 10/9 seconds. What the result above shows is that Achilles will reach the turtle in 10/9 seconds if he catches it at all. But, since we did not prove that Achilles had caught up with the turtle, we do not bother asking the question. Our question is not when Achilles will catch the turtle, but whether he can reach it!

Don’t get me wrong, most modern philosophers — but not all — believe that Achilles will catch the turtle. That’s not what the philosophers are arguing today. Where is the error in Zeno’s thinking? What’s wrong with the paradox? That’s the thing.

If we prove a ridiculous conclusion using our logic, this means there is an error in our reasoning and we have to find it.

There is another problem in this paradox of Zeno: Achilles must perform an infinite number of steps to catch the turtle. First, he should go to A1, then he should go to A2, then he should go to A3 … Can any of us perform an infinite number of steps? That’s the crucial question. The mathematician can add countless numbers in his intellectual world, but we cannot add countless numbers in real life. We can’t do an infinite number of jobs. Well, at least, It’s hard to imagine that we can.

Or is Achilles doing a finite number of works to get to the turtle? Before moving on to this question, let’s talk about Zeno’s second paradox.

The Dichotomy

Zenon doesn’t just say that Achilles can’t catch the turtle. He just says Achilles cannot go from one point to another. Let’s say Achilles can go to point A and point B. Achilles must go halfway to go from A to B. After he has gone half the way, he must go half of the rest of the way, then half of it again. This process lasts forever.

Let’s suppose the distance between A and B is 1 meter. Achilles should go 1/2 of a meter first. Then, there will be another 1/2 meter left. Now, Achilles should go half of the 1/2 meter, which means 1/4th of a meter. Then, Achilles should go half of the remaining 1/4 meters, resulting in 1/8 meters. Finally, Achilles must go a 1/16th of a meter more. Achilles cannot reach point B because he cannot do an infinite number of works.

Let’s observe the arrow flying through the air and assume that the arrow is undergoing an infinite amount of processes by passing through the unlimited points. Can our brain imagine each and every centimeter of the arrow’s movement? I don’t believe so, and it’s quite difficult to imagine it as well. While our brain is taking a finite number of photographs of the arrow, these photographs are passing like a film strip (I will come back to this soon). For now, let’s keep in mind that our brain perceives the outside world in a finite way.

You may think that we can do an infinite number of works: the first work, the second work, the third work… we can do endless works! We can also alter this second paradox of work, which is very similar to the first paradox, to prove that, Achilles can not even move. Indeed, Achilles must go halfway before he can go from A to B. He has to go to a quarter of the way before he can go halfway. But he has to go to one-eighth of the way before that, and he has to go to one-sixteenth even before that. So Achilles can’t even step beyond point A because there is no first point for him to go to! For each distance, he must first go half that distance; therefore, he can’t even move.

Is there another point between A and B? Let’s check it out!

The paradox certainly stems from being divided into two. We always split the physical distance Achilles must go into two. Therefore, we cannot divide the physical distance (space) into two endlessly. By dividing into two, again and again, we get such a small gap that it cannot be divided once again. In other words, physical space is not continuous. Space is composed of the smallest not-divisible space particles. Doesn’t the particle theory of the twentieth century tell us that we should think this way as well? Let’s call these space particles, space units.

We have proved that space is made up of space units! Each distance consists of a finite number of space units.

The arrow

According to Zeno’s third paradox, there is no movement, and nothing can move. Take a flying arrow as an example. We think the arrow is moving, right? Zeno says we’re wrong.

The arrow just stands there at any moment. If you don’t believe so, take a picture of an arrow in the air. You will see the arrow in a still state in the photo. So the arrow is still at any moment. If the arrow stands still at any moment, it always does, doesn’t it? The arrow must be moving for at least one moment to move. However, the arrow just stands still at any given moment. That means the arrow just stays still, always!

We saw above that space cannot be continuous. Space is composed of small, tiny, indivisible space units. Suppose the arrow’s length is one space unit. The one-space-unit-arrow cannot move within a space unit, because the arrow must be shorter than the space unit so that it can move in that space unit; we know that there can be no objects shorter than the space unit. Therefore, the arrow that stands still in each space unit is always stationary.

Isn’t cinema like that? Isn’t the movement of a person walking on the cinema screen is thousands of pictures which do not move in front of our eyes? Isn’t the movement in nature also stationary?

The flying arrow is also stationary at any moment. But it exists in the next space unit. As Bergson said, just like the example of a person walking on the screen of the cinema, the arrow seems to move. Yet, it stops at any moment.

Another source of our first paradox is the assumption that time is continuous. Can the turtle move all the time? Couldn’t it stay still, even for a very, very short (time unit) period?

Fourth paradox

Zeno’s latest paradox is not easy to understand. As I said above, we do not have a work written by Zeno. Aristotle tells us about the paradoxes of Zeno. Aristotle’s form of explanation is not too straightforward. Therefore, the fourth paradox has several interpretations. The interpretation I give is not Aristotle’s, but it is very close to it.

Above, we have proved that space is not continuous, it is made up of indivisible space units, or rather Zeno did.

Assume that we have squares, A and B. Each square represents a space unit. There is object A in the upper left corner and object B in the lower right corner. Let A and B “move” at the same time and the same speed. A to the right and B to the left. After a while, A is in the right square, and B is in the left square.

Now let’s ask the paradoxical question. Where did A and B meet? They never met! Because there’s no square (space) for them to meet.

However, Mathematics

Life is good for only two things, discovering mathematics and teaching mathematics.

Waldo Otis

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I know that you need some words to talk. Here, reading will be perfect for you.

However, Mathematics

Life is good for only two things, discovering mathematics and teaching mathematics.

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