Everything Everywhere Daily: History, Science, Geography & More - Zeno's Paradoxes

Episode Date: January 14, 2023

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 Zeon and how they can possibly be resolved on this episode of Everything Everywhere Daily. Subscribe to the podcast!  https://link.chtbl.com/EverythingEverywhere?sid=ShowNotes -------------------------------- Executive Producer: Charles Daniel Associate Producers: Peter Bennett & Thor Thomsen   Become a supporter on Patreon: https://www.patreon.com/everythingeverywhere Update your podcast app at newpodcastapps.com Discord Server: https://discord.gg/UkRUJFh Instagram: https://www.instagram.com/everythingeverywhere/ Facebook Page: https://www.facebook.com/EverythingEverywhere Facebook Group: https://www.facebook.com/groups/everythingeverywheredaily Twitter: https://twitter.com/everywheretrip Website: https://everything-everywhere.com/everything-everywhere-daily-podcast/ Learn more about your ad choices. Visit megaphone.fm/adchoices

About 2,500 years ago, a Greek philosopher by the name of Zeno of Elia proposed several paradoxes about the natural world. His ideas were actually pretty simple, but they were incredibly difficult to explain away. For the last two millennia, philosophers have been trying to resolve his paradoxes, and they're still trying to explain them today. Learn more about the paradoxes of Zeno and how they could possibly be resolved on this episode of Everything Everywhere Daily. What if your perceptions about the past were wrong? ThruLine is a podcast that takes you back in time to uncover the parts of the story that may have gone unnoticed. It effectively turned day into night. And how it shaped the world now.

Time travel with us every week on the ThruLine podcast from NPR. Zeno of Elia was one of the earliest Greek philosophers. He was born before Socrates, which places him the category of pre-Socratic philosophers. This group tended to think more about the natural world. and tried to explain how the world work without resorting to mysticism. While he was born before Socrates, he was a contemporary of Socrates and probably did 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 kind of similar to each other. They all involve some sort of reductio ad absurdum argument, where you take something to an absurd conclusion.

And they also all deal with the concept of infinity at some level. Zeno was one of the first philosophers ever to 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 the ancient Greek gods. The first paradox is often called the dichotomy paradox. Let's say that the ancient Greek heroine, Adelanta, who was noted for her skill in foot races, wanted to run from one place to another.

before she can run the full distance, she first must go half the distance. But before she can go half the distance, she first must go half that distance, or one quarter the full distance. And before she can go one quarter of the full distance, she must go one-eighth the distance. And before she can go one-eighth the distance, she must go one-sixteenth the distance, et cetera, et cetera, et cetera. Because there's no end to this, she must travel in an infinite number of lengths. And this raises several problems. The first of which is, is that there's no first distance that she can travel, as any distance can be cut in half. If there's no first distance that she can travel, then it would be impossible for her to move. It's a paradox because Zeno is right. To move from A to B, you do have to go half the distance

and then 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 slow tortoise, where the tortoise has a head start. Let's say it's a hundred meter race and the tortoise gets to start at the 50 meter mark. 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. Achilles will then race to the new position of the tortoise, but the tortoise will have moved ahead again by a little bit.

Every time Achilles gets to the location of where the tortoise was, the tortoise will have moved ahead a tiny 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. However, if at every instant the arrow isn't moving,

and time is made up entirely of instance, 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're not as interesting for the purpose of this episode. So how do you resolve these paradoxes? There's clearly there has to be something wrong because obviously things move.

But it's not 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 distance has decreased in the first two paradoxes, time would decrease as well. As you go toward zero distance, you also approach zero time. In response to the Aero Paradox, Aristotle claimed that there simply 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. And 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 Adalanta is running her race, she first goes half the distance, and then a quarter of the distance, and then an eighth the distance, etc. Well, what do you get when you add up a half plus a fourth plus an eighth plus a sixteenth 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 1. If you add them all up, they equal exactly 1. The reason why this infinite number of fractions equals 1 is that there is no number whatsoever that you can possibly put between this sum and the number 1. Let's say that you picked an arbitrarily small number, less than 1, but not quite 1.

something like 0.9999,999,999, etc, etc, etc, etc. As many nines as 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 arbitrary number you picked, which was close to one. If you pick two numbers and it is impossible to find a number between them, then the two numbers have to be equal. You might have heard at some point that .999-999 repeating is equal to 1.

And this is the same reason why, because there's no number that could possibly come between it and 1. And just as you can add up distances to get a finite amount, the same would also work with time. Infinite sums are how you can mathematically resolve most of Zeno's paradoxes. But mathematics aside, there are also physical explanations. advances in physics, particularly 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 plonk units. The plonk 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 times 10 to the negative 35 meters. This length is so ridiculously small that it's hard to even conceive of. For example, to put it into perspective, if you expanded a football field to the size of the observable universe, the plonth length would still 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 Planck 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 Plank Length and Plank Time would be the equivalence of a pixel. If indeed there is a limit to space and time where neither can be made any smaller, then Zeno's paradox is resolved by physics. And if there isn't a lower limit, then it can be resolved with mathematics and infinite sums. The amazing thing is that why Zeno's arguments about motion clearly must be wrong, because motion does exist, it took 2,000 years to come up with definitive 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 you have to admit he did have some pretty good questions. The executive producer of Everything Everywhere Daily is Charles Daniel. The associate producers are Thor Thompson and Peter Bennett. I just want to thank everyone, including the show's producers, who support the show over on Patreon. If you'd like to support the show, just head over to patreon.com, which is currently the only place where you can get show merchandise. Also, if you want to talk to other listeners about the show, head over to our Facebook group or Discord server. both of which have links in the show notes.