Zeno — A Man of Paradoxes

Zeno shows the Doors to Truth and Falsity (Veritas et Falsitas). Fresco in the Library of El Escorial, Madrid | Image from Wikipedia

Consider a mighty ship that was only made of wood. Let’s call it… Sea Señor 😁. Let’s say Sea Señor is used by sailors for generations, passed down from one great sea captain to another. Just like any object, over time, the ship experiences wear and tear, namely rot. So, to keep it in working condition, rotting wood planks are replaced by metal parts one by one. Now let us consider the day that the very last piece of wood in the entire ship is replaced with a metal part. Right after this moment, is this ship still the same as the original Sea Señor? If so, what makes it the same, since the very wood that made up the original has all been replaced by the metal that makes up this current ship?

Know Your Meme

This is a popular logical conundrum known as the Ship of Theseus, and targets the concept of identity. And when I say popular, I mean so popular that it even appeared in the series finale of the Marvel Cinematic Universe (MCU) show WandaVision. Don’t worry… I won’t spoil the episode.

Now, obviously paradoxes like this one aren’t fun to solve. I mean, it can feel nice to ruminate over the concept and the paradox and try to come up with a solution, but after some time, it almost seems that there is no end to the madness… no answer.

So, could you imagine having to deal with paradoxes like this all your life? Well, let’s just say a man from fifth century B.C.E. would not have had to imagine at all… since, well, he devoted his life to inventing a bunch of famous ones himself. Who was this maniac?

Zeno of Elea.

Who Is Zeno?

Unfortunately, many of the sources that talk about Zeno don’t mention much about his life, and some are not even completely trustworthy. But, here is what we seem to know.

He was born around 490 B.C.E. in a Greek colony in southern Italy called Elea. At roughly the age of 40, Zeno went to Athens with his teacher, Parmenides, where he met the now-famous Socrates. Zeno spent a good chunk of time in Athens, where he taught others, did a lot of thinking (especially questioning reality and making paradoxes) and apparently argued with people a lot. What’s with ancient Greek philosophers and just starting an argument with random people to then completely destroy these poor people in the very argument the philosophers started?

Image from All Thats Interesting.

Regardless, there’s accounts that he might have… hold on, let me see if I’m reading this right. He tried to kill a tyrant in his homeland of Elea, and then when he was unsuccessful and captured, he bit off his own tongue and spat it at the tyrant?!?! Well, people still don’t know if the source behind this story is trustworthy or not but… I don’t even care. Because that story is awesome, and I choose to believe it.

Since there isn’t much else of a life story to delve into… let’s get into the mind-wrangling paradoxes he’s so well known for. Not much of Zeno’s work has survived, so most of the paradoxes attributed to him are actually put into writing by other philosophers, so all quotes and explanations here on out are not actually Zeno’s words, and may be different from what he had actually argued. But, here they are.

Screw Plurality

Zeno apparently had some problem with the concept of “many things”, since he developed multiple, very rigorous proofs to explain why there cannot be “many things”. One major one goes as follows:

Limited and Unlimited

“If there are many things, it is necessary that they be just so many as they are and neither greater than themselves nor fewer. But if they are just as many as they are, they will be limited. If there are many things, the things that are are unlimited; for there are always others between these entities, and again others between those. And thus the things that are are unlimited.”

Consider “many things”. It will help to keep things ambiguous here for the sake of argument, but I will put in random analogies here and there to explain specific details. Oh and by the way, when I say many things, it doesn’t have to be a large number. Zeno makes the argument that there cannot ever be more than one thing.

Now, if there are many things, then there cannot be more than the amount of things there are, and there cannot be less than the amount of things there are, right? If there’s 100 bananas, there are 100 bananas. Not 99, not 101, not 50000. So, therefore, these “many things” are limited in number.

Now, Zeno makes the argument that two things are distinct if and only if there are other things between them, separating them and making them distinct. Thing x and thing y will only be distinct and separate if there is another thing z between them. But, x and z are only different because there is another thing m between them. Keep applying this, and you see that this goes on forever, with constantly more things being necessary to make things distinct. So, Zeno argues, that between any two distinct things, there are limitless number of other things. So, if there are in fact many things, there must be an unlimited number of things.

So… for there to be many things, there must be a finite, limited amount. But also, for there to be many things, there must be an infinite, unlimited amount. So, many things must be both finite and infinite at once, which is clearly contradictory. So, in this way, Zeno uses a proof by contradiction (when both ‘x’ and ‘not x’ are true) to say that, since there cannot be a finite and an infinite amount of things at the same time, there cannot be many things.

Get Ahead And You’ll Stay There Forever

So Zeno wasn’t satisfied with simply destroying the concept of plurality. He wanted to make racers feel bad about ever getting behind in a race at all. See, according to his paradox, titled Achilles and the Tortoise, if the tortoise is ever able to get ahead of Achilles (supposedly faster than a tortoise) in a race, then it will be impossible for Achilles to ever overtake the tortoise.

Now, those of you that like underdog stories where the person in last makes a dramatic comeback to win the race are probably not very happy with this one. But, it’s still interesting the way he lays it out, so just bear with me.

Imagine our runner Achilles is at the starting line, while the tortoise, because he received a head start, has already made it to position X.

At this point, Achilles is allowed to begin running, and being a good runner, he‘s much faster than the tortoise. Now, consider that Achilles has now arrived at position X, where the tortoise initially was. In the time it took Achilles to get there, the tortoise has also been moving forward, though because he is slower, the distance he has moved is much less than Achilles. Regardless, let’s say Achilles is now at position X, where the tortoise previously was, while the tortoise is now at position Y.

Next, consider that enough time has elapsed from the previous snapshot that Achilles has now arrived at position Y. During this time, once again, the tortoise has also moved forward a smaller distance, to point Z.

To any observer, and to both Achilles and the tortoise, it would seem that Achilles is catching up, but the way that Zeno has presented this, there is a catch. If the same thing as the previous two cycles was repeated ad infinitum, we would see a pattern emerge. Every time Achilles makes it to the position that the tortoise was in the previous snapshot, the tortoise has always moved slightly further. Again and again, Achilles catches up to the tortoise’s previous position, and finds that the tortoise has moved forward ever so slightly. And so, Zeno argues, that it is impossible for Achilles to every actually catch up to the tortoise, since the tortoise will always be slightly further ahead.

And there is no way to simply logic your way out of this paradox without considering the meaning of infinity and other such concepts. Someone who’s either annoyed by or doesn’t care about the paradox might say that Achilles will get forward eventually since he’s faster. But Zeno already takes this into account since he mentions that the distance between Achilles and the tortoise is constantly decreasing, but there is always a non-zero distance between them where the tortoise is slightly in front of Achilles.

If we now think about Zeno’s next paradox, then not only is it impossible for Achilles to ever catch up to the tortoise, it is impossible for either of them to ever actually reach the finish line AT ALL!

Are We There Yet?

Let’s go back to the diagrams from the previous section, but simplify it by removing the tortoise entirely. So, we have the following diagram, where Achilles stands at the starting point, and wants to reach the finish line.

BAM! Off he goes.

First, on the way to the finish line, Achilles must make it to the halfway point between the start and the finish line.

Once he’s made it there, he must then make it to halfway between his current position and the finish line.

Again and again, Achilles must travel half of the distance from his current location to the finish line. So, no matter how long this repeats, Achilles will always have a halfway point between his position and the finish line that he must first reach. By this logic, Zeno realizes that it is impossible for Achilles to ever reach the finish line at all, since he will always have the rest of the half way to go.

If you’re paying attention, you might notice a kind of similarity between the past three paradoxes. All seem to rely on the mystical concept of infinity.

As humans, our brains tend to have a very difficult time thinking about something like infinity. Everywhere around us, all we see are finite things, so thinking about an infinite number of things, or an infinite number of different sections of some distance, or just the concept of infinity requires quite a bit of imagination. And Zeno takes advantage of this poor understanding of infinity and how exactly it applies in the real world to develop these three paradoxes.

In this next paradox, Zeno similarly takes advantage of our poor understanding of what motion and time really are, and questions what it means to “move” at all.

Are We Moving?

Consider an arrow, travelling through the air towards a target. Now, throughout the duration of motion, anything that moves is in one instant of time after another. Consider any particular instant of time t. At this instant of time, A will occupy space exactly equivalent to its length, and is resting in place (not moving during that very specific instant in time). But, since t is no different than any other instant of time during the duration of the arrow, the arrow is resting in place across every instant of time during its motion. And therefore, the moving arrow is always motionless.

If you’ve taken a high-school level or higher physics class or calculus class, you’ve probably already had to deal with the concept this paradox mentions. In physics, there is the notion of average velocity and acceleration, which are directly related to the concept of average slope in calculus. Let’s take average velocity for example. To calculate this, you simply find how far an object has travelled and divide it by the time it took to do so, giving you the average velocity. In calculus, this is the average slope, obtained by dividing the change in the y-axis by the change in the x-axis to find the rate of change (slope) of a line that extends from your starting point to your ending point.

This shows the method of calculating the average slope of a graph. | Figure by Study.com

Where it gets confusing is in the concept of an instantaneous rate of change, or instantaneous velocity in physics. In trying to calculate this, one must find how far an object moves without any change in time at all, or in a graph, how much the variable on the y-axis changes without any change in the variable on the x-axis.

To find this, mathematicians use a tangent line, which intersects the graph at only the point of interest.

A tangent line intersecting the graph f(x) at point c | Figure from Brilliant

The slope of this line is calculated by making many other lines that intersect two points on the graph, and by slowly decreasing the x-axis difference between these, bringing us closer to the actual slope of the tangent. Mathematically, this is represented as follows:

Figure by ITW Performance Polymers

What mathematicians have shown to be true via this is that, even though it might not make sense, there is still a rate of movement for an object even in a single instant of time, making motion possible.

Why Does It Matter?

Alright, so what if this Greek man from hundreds of years ago made these amazing paradoxes? Why does that matter?

Well, these very paradoxes greatly challenged the understanding of the universe and common phenomena, and solving these paradoxes led to a deeper, stronger, and more concrete definition of concepts like infinity, or the nature of time. In fact, some of his paradoxes were only recently “solved”, after centuries of fascinating research in the fields of physics and mathematics.

So, yeah, his paradoxes might just be the kind of weird shower thoughts some of us might have from time to time, but the very concepts he attacked and broke down make the basis of our comprehension of the mysteries of this universe. And by targeting these, Zeno opened us to a world of mysteries that continue to baffle us today.

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19 y/o innovator working on reversing ageing and researching cancer vaccines. www.akshajdarbar.com

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Akshaj Darbar

Akshaj Darbar

19 y/o innovator working on reversing ageing and researching cancer vaccines. www.akshajdarbar.com

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