Zeno, like your average ancient Greek philosopher, loved to come up with grand statements and theories that people second guess the world in which they live in. Little is known about Zeno’s life, and only a handful of his work survived to be handed down to us. He believed that we should not rely on our senses and experiences to judge the world, but rather pure logic and mathematics. This theory, albeit cold and sound, would lead Zeno down the wrong path. Let’s explore the most famous works of Zeno and how they have been debunked using modern(ish) science.
ACHILLES AND THE TORTOISE
Meet our two contestants in a race; Achilles- the most formidable warrior in Greek mythology; and the tortoise- one of the slowest land mammals around. This is similar to the Hare and the tortoise story; but rather it’s not the tendency to get lazy and overconfident that makes Achilles lose, but more the fact that he gave the tortoise a head start. According to Zeno, however fast Achilles runs, the tortoise will always win. This is because, if the tortoise gets a head start of, say x meters, then by the time Achilles reaches point x, the tortoise has moved on; and so by the time Achilles get to the tortoise’s new position, it again has moved on (but by a smaller amount). Achilles will never overtake the tortoise.
It is important to remember at this point in history, the Greeks had no idea of converging infinite series or the equation speed=distance/time. These two ideas are crucial if are to redeem Achilles’ honour and pride. The statement ‘will never overtake the tortoise’ is of course wrong. Both racers are travelling at constant speed, therefore the ever-decreasing distances between each stage (the point where Achilles catches up the tortoise’s old position), also involve ever-decreasing time intervals. And even so an infinite number of stages doesn’t equal to infinite time length. The stages all add up to an infinite time, the time it takes for Achilles to catch up to the tortoise. This idea that an infinite number of values adds up to a finite number can be explained by a geometric sequence. If you take the example:
1+1/2+1/4+1/8+1/16… It is clear that you can keep halving the fractions and making them smaller and smaller. What many people fail to realize that the longer you do this, the closer you converge to a total of 2. If you do infinite times then you do get a total of 2.
If the tortoise got a head start of 100m, and Achilles travels at 10 m/s, then we can say that it takes him 10 sec to catch up to the tortoise, using the formula s=d/t. Look at this from the point of view of Zeno, after 5 sec Achilles has covered 50m, and then another 25m in 2.5 sec, and another 12.5m in 1.25s. Of course without the knowledge of converging infinite series, Zeno had to say that Achilles couldn’t overtake the tortoise, but if you do add 5+2.5+1.25+…. You get 10s.
With the help of s=d/t and a geometric sequence, we have saved Achilles’ reputation.
‘In order to reach your destination you must cover ½ the distance first, but in order to cover ½ you must first cover ¼ of the distance and in order to cover ¼ of the distance you must cover 1/8 of the distance; and so on. If you keep chopping the distances in half forever, then you never reach the first distance marker, and so you never reach that first distance marker. This never-ending series of ever-shorter distances is infinite; so to complete the task you need to complete an infinite number of tasks, so you would never finish nor start; therefore motion itself is impossible’
Simple but elegant, this is the statement that Zeno proposed. He used pure logic to push and push an idea until it leads to a logically absurd conclusion- reduction ad absurdum. Logically you can immediately disprove Zeno, simply stand up and walk to other-end of the room. By sheer logic and the help of your body, you have proven Zeno wrong. But I like to solve it mathematically, with the water-tight application of physics- so there is no doubt left that Zeno must be wrong.
First of all, we need to convert this argument from one about distance to one about time. Assume as well you are travelling at constant speed at the moment in time when you reach the starting point of your journey. The idea of constant speed is important, for if you reduce the distance needed to cover, then the time taken to cover that distance also reduces. And the distance and time always divide to give you speed. We can think of time as a line that can be infinitely divided, but it’s too crucial to understand that time marches on regardless of what we do, we can’t take ourselves out of time’s stream, yet we can view distance from an outside perspective, distance doesn’t march on. Therefore, you travelling at constant speed will cover a distance, no matter how small the time interval, for the time interval can never be 0; time doesn’t stop.
Before I leave you, I should point out that Einstein did regard time in a similar way to space- indeed he called time the 4th dimension of what is called space-time. This would argue that maybe the flow of time is an illusion, therefore motion is an illusion, and our friend Zeno is right. But that’s not the case here. Of course, I’m not saying Einstein’s theory of relativity is wrong, but it only truly manifests when things move near the speed of light. At everyday speeds, we have every right to ignore relativistic effects.