By Jonathan Farrow from the Thoughtful Pharaoh
It’s early in the morning. The caffeine from your morning cup of coffee has yet to fully kick in, but as you turn the corner, you see your bus. It’s just pulling in to the stop and is only 50m away. You know you can make it, so you break into a sprint.
It takes you 3.5s to travel 25m and get halfway to the bus. In that time, an old lady has gotten off. You’re halfway there and there’s still a few people who need to get off. You’ll definitely make it.
In only 1.75s you’re already halfway to the bus again (12.5m). There’s only one person left to get off.
Another 0.875s and you’ve travelled the 6.25m that gets you halfway to the bus again. There is nobody left to disembark.
In less than half a second, you’re halfway again, just over 3m from the bus. The driver must see you. He’ll wait, right?
In less time than it takes you to blink (0.22s), you’re 1.5m away, almost close enough to touch the bus. So close, and yet, somehow, you’re not quite there yet.
In order to catch the bus, you need to get halfway to the bus first. Getting to the halfway point, no matter how short a journey, will take you some finite amount of time. Unfortunately, there are an infinite number of halfway points between you and the bus. According to a grumpy Greek philosopher from the 5th century BCE named Zeno of Elea, you will never get to the bus. In fact, he argued that all motion is impossible. It is merely an illusion. This paradox, also called the Dichotomy, is one of four paradoxes that Zeno used to demonstrate this idea and it has been notoriously hard to refute.
One attempt at refutation was made early on by Diogenes the cynic, who was said to have silently stood up and walked across the room. [Incidentally, Diogenes was a hilariously stubborn man who was prone to philosophical stunts like intentionally distracting Plato’s students by obnoxiously eating food in lectures; walking around the market in daylight with a lamp in search of an “honest man”; and sleeping in a big ceramic jar in the market to prove that wealth was a corrupting influence.] While this does contradict Zeno’s conclusion that motion is impossible, it doesn’t address the argument itself. Zeno’s response would simply be that Diogenes crossing the room, just like you trying to catch your bus, is your senses tricking you into seeing motion where there was none.
Aristotle tried to refute the Dichotomy by distinguishing “actual” from “perceptual” infinities. The 50m line between you and the bus at the start of the scenario can be divided into an infinity of half-runs (therefore it is a perceptual infinity), but that is a geometrically different phenomenon than the single, undivided 50m line (the actual infinity). Aristotle conceded that Zeno found something that is impossible (running infinite half-runs), but maintained that this was not what actually happens when somebody moves (running a single finite line).
This doesn’t seem satisfactory to me. Aristotle’s distinction is an artificial one and misses the point that Zeno was trying to make. The world would need to wait for the 19th and 20th centuries for mathematicians to start talking about infinite series and to resolve Zeno’s Dichotomy paradox.
In mathematics, a series is what you get when you add up all of the numbers in a given sequence. Consider the sequence of numbers 1, 2, 3, 4…. The pattern here is that you add one to the previous number. The first three terms add to 6, the first four add to 10. Every number you count up adds to the total and as long as you keep going, the total sum will get higher and higher. This is an example of a divergent series because there is no number that the series settles on.
Now consider the sequence of numbers 25, 12.5, 6.125, 3.0625… The pattern here is that each number is half of the previous one. Unlike the sequence above, if you continue the sequence, the numbers get smaller and smaller. You will get closer and closer to 50 until you run out of space to put the 9s after 49.99999… For all intents and purposes, you will have reached 50. This solves the problem practically and is analogous to the way that we understand derivatives and integrals. Understanding how and when infinite numbers of parts can add up to finite (and known) quantities has been incredibly helpful for us. It’s the principle behind the dampening of oscillations in springs and sound waves, it lets engineers understand how wind will affect their bridges, and it lets Usain Bolt get to the finish line.
Somehow, though, this resolution still leaves me dissatisfied. It’s just a more useful and mathematically rigorous version of Diogenes’ walk across the room. In some ways, Zeno’s Dichotomy paradox still haunts modern mathematics. Kevin Brown (possibly a pseudonym for a mysterious math/physics writer), in his 2015 book “Reflections of Relativity” writes somewhat ironically of the paradox’s resolution, “it’s probably foolhardy to think we’ve reached the end. It may be that Zeno’s arguments on motion, because of their simplicity and universality, will always serve as a kind of “Rorschach image” onto which people can project their most fundamental phenomenological concerns…”
And with that, we’ve reached the end of the ABCs of interesting things. Thanks for joining me on this wonderful journey. That being said, it’s probably foolhardy to think we’ve reached the end of the Thoughtful Pharaoh.
[Featured image: Grandjean, Martin (2014); License: http://creativecommons.org/licenses/by-sa/4.0/]