Zeno’s Paradoxes

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Podcast Transcript

About 2,500 years ago, a Greek philosopher by the name of Zeno of Elea proposed several paradoxes about the natural word.

His ideas were actually really simple, but they were incredibly difficult to explain away. 

For the last two millennia, philosophers have been trying to resolve his paradoxes, and they are still trying to explain them today.

Learn more about the paradoxes of Zeno and how they can possibly be resolved on this episode of Everything Everywhere Daily. 

Zeno of Elea was one of the earliest Greek Philosophers. He was born before Socrates, which places him in the category of pre-Socratic philosophers. This group tended to think more about the natural world and tried to explain how the world worked without resorting to mysticism. 

While he was born before Socrates, he was a contemporary of Socrates and did most probably meet him in person. 

We don’t know a lot about the life of Zeno. Most of what we know about him comes from mentions by other Greek philosophers. None of his original writings have survived to the present. 

Most of what we know of Zeno’s ideas comes from Aristotle, and most of his ideas are the paradoxes that I’ll be talking about in this episode. 

I’ve previously done a full episode on paradoxes. However, in this episode, I want to focus on the various paradoxes which Zeno proposed, and more importantly, I want to focus on how these paradoxes can be resolved. 

There are several paradoxes that Zeno proposed, and all of them are very similar to each other. They all involve some sort of reductio ad absurdum argument, where you take something to an absurd conclusion. They also all deal with the concept of infinity at some level. 

Zeno was one of the first philosophers to ever deal with the concept of infinity, and most of his paradoxes come from the problems of trying to deal with infinity in a finite world. 

I’m going to provide the paradoxes as they were presented by Zeno, complete with references to ancient Greek gods. 

The first paradox is called the dichotomy paradox.

Let’s say that the Greek heroine Atalanta, who was noted for her skill in footraces, wanted to run from one place to another. 

Before she can run the full distance, she must first go half the distance. But before she can do half the distance, she must first go half that distance or one-quarter the full distance. 

Before she can do one-quarter the distance, she must go one-eighth the distance, and before she can do one-eighth the distance, she must go one-sixteenth the distance. Etc, etc, etc. 

Because there is no end to this, she must travel an infinite number of lengths. 

This raises several problems. The first of which is that there is no first distance that she can travel as any distance can be cut in half. If there is no first distance that she can travel, then it would be impossible to move. 

It is a paradox because Zeno is right. To move from A to B, you do have to go half the distance and half the distance again, and again. However, Zeno is clearly wrong because….we can move. 

His next paradox is known as Achilles and the Tortoise. 

In this paradox, the Greek hero Achilles is in a race with a tortoise, where the tortoise has a head start. Let’s say it is a 100-meter race, and the tortoise gets to start at 50 meters. 

Zeno claims that despite being faster, Achilles will never be able to beat the tortoise. 

The reason why is that once the race starts, the tortoise will move ahead, and so will Achilles. Achilles will sprint to the point where the tortoise started. When he gets there, the tortoise will be some smaller distance ahead. 

Then Achilles will race to the new position of the tortoise, and the tortoise will have moved ahead again by a little bit. 

Every time Achilles gets to the location of the tortoise, the tortoise will have moved ahead a little bit. 

The distances get smaller and smaller, but again, there are an infinite number of them. 

The end result is that Achilles can never catch up to the tortoise. 

The final of his movement paradoxes is known as the Arrow Paradox. 

Let’s assume you shoot an arrow at a target. At any single moment in time, the arrow would be motionless. It wouldn’t be moving forward. 

If, at every instant, the arrow isn’t moving, and time is made up entirely of instants, there can never be any motion. It would be adding up an infinite amount of zeros. 

So, these three paradoxes are collectively known as the movement paradoxes. Zeno had some others, but they aren’t as interesting. 

So, how do you resolve these paradoxes? Clearly, there has to be something wrong because, obviously, things move. 

But it isn’t obvious why Zeno’s arguments are wrong. Technically, Zeno is right or at least seems to be right, in everything that he said. 

Philosophers have tried to resolve Zeno’s paradoxes for centuries, and several different approaches have been used. 

Aristotle had several answers to Zeno.

First, he said that as the distances decreased in the first two paradoxes, the time would decrease as well. As you go towards zero distance, you also approach zero time. 

In response to the arrow paradox, Aristotle claimed that there was no such thing as a moment in time. There was no unit of time where motion would be zero, so the entire paradox made no sense.

Thomas Aquinas agreed with Aristotle in saying that instant points of time were impossible. 

From a mathematical perspective, most of Zeno’s paradoxes have been explainable since the advent of calculus, which has allowed mathematicians to deal with infinity. 

The key to unlocking it can be found in what is known as an infinite sum. 

Let’s go back to the first dichotomy paradox and look at it in another way. When Atalanta is running her race, she first goes half the distance, then a quarter the distance, then an eighth the distance, etc. 

What do you get when you add up ½ plus ¼ plus ? plus 1/16 all the way to infinity?

The infinity part is the trick.

Your first instinct might be to say that you can’t add up an infinite number of numbers. 

But….you can. 

And the answer is one. If you add them all up, they equal exactly one. 

The reason why this infinite number of fractions equals one is that there is no number between whatever number you can possibly put between this sum and one.

Let’s say you picked some arbitrarily small number, less than one but not quite one.  Something like 0.99999999…..9, however many 9’s you want, but at some point it ends. 

No matter what number you pick, you can keep adding up those fractions to get closer to one than the number you picked. 

If you pick two numbers and it is impossible to find a number between them, then the two numbers must be equal. 

You might have heard at some point that 0.999999… repeating is equal to one. This is the same reason why. There is no number between it and one. 

Just as you can add up the distances to get a final amount, the same would also work with time. 

Infinite sums are how you can mathematically resolve most of Zeno’s paradoxes. 

Mathematics aside, there are also physical explanations. 

Advances in physics, especially quantum physics, have also provided answers to many of these paradoxes. 

One of the things which quantum physics discovered is that the universe and everything which makes it up isn’t continuous. It can’t be infinitely divided into further smaller units. Instead, things are made up of quanta, which is where the whole quantum in quantum physics comes from.

In quantum physics, there are units known as Planck Units. 

The Plank Units, for length and time, are the smallest units that can be validly measured given the physical constants of the universe. 

The Planck Length is equal to 1.616×10?35 meters. This length is so ridiculously small that it is hard even to conceive of. If you expanded a football field to the size of the observable universe, the Planck Lenght would be the size of an atom. 

Planck Time is the amount of time it takes a photon moving at the speed of light to travel the Plack Length. 

It is entirely possible, in theory, for time and length to be even shorter, but it would be impossible to measure. So, unlike Zeno’s assumption that you can keep dividing length and time indefinitely, you can’t really measure it. 

If the universe were a computer program, the Planck Lenght and Time would be the equivalent of a pixel. 

If there is a limit to space and time where neither can be made any smaller, then Zeno’s Paradox is resolved by physics. 

If there isn’t a lower limit, then it can be resolved with mathematics and infinite sums. 

The amazing thing is that while Zeno’s arguments about motion clearly must be wrong…because motion does exist, it took 2,000  years to come up with definite proof. In the 17th century, in the case of calculus, and in the 20th century, in the case of quantum physics.

Zeno didn’t have all the answers, but he did have some pretty good questions.