Her Stem Space

The argument goes as follows: Suppose that the Hare is to race the tortoise over 10 meters, and agrees to give the creature a head start, of say 5 meters. To win the race, the hare needs to cross the finish line. Before it does  that, it needs to reach the half-way point, which is where the tortoise begins. Here is the problem: by the time the Hare reaches the place where the tortoise started, a little time has elapsed and so the animal has moved on by a small amount. So, the Hare must now run to the tortoise’s new position. By the time he does this, though, it has moved on a tiny bit more. This process continues in this way: each time the Hare approaches the spot where the tortoise once stood, some time has elapsed, and the tortoise has advanced slightly further. The tortoise will always stay one step ahead of the Hare, no matter how quickly he sprints.

It sounds like comedy, but around 450 BC, Zeno of Elea advanced an argument similar to this that changed the perspective of mathematics. Little is known about this philosopher as none of his works survive, but come to be used indirectly via Plato and Aristotle. It seems, however, that Zeno of Elea has truly radical beliefs. He held that motion, and indeed any form of change, is an illusion. In support of this view, he compiled a list of paradoxes, of which Achilles, in the form of the Hare, is the most famous. Zeno also believed that it was impossible to subdivide the world into smaller components, such as earth and sky, or me and you; all such divisions too are illusory. Zeno holds that there is just one indivisible entity that exists.

This mystical view may still have an appeal to some people of a religious disposition. Not many; however, would attempt a mathematical defense to it as Zeno did. Still, his paradoxes have an importance that transcends his metaphysical beliefs, since they anticipated mathematical discoveries that would not be made for over a thousand years.

Unfortunately, in the modern era with much knowledge scattered around the land, it can be proven that Zeno was indeed wrong: great athletes can outrun tortoises, and a walking person is not safe from a rapid bullet. This should not come as a major surprise. At any rate, it is not encouraged to test the hypotheses. So, the key question is how to refute Zeno’s argument that the Hare needs to finish an endless number of chores in order to pass the turtle.

It would take an infinitely long list of all the tasks that the Hare needs to complete: first he must reach the tortoise’s initial position, then the second, the third, and then the fourth, and so on. If the list needs to be completed, it will be continuing at the end of the universe. That is what makes the problem so compelling. However, just because it would take infinitely long to say the task that the Hare must complete, it does not follow that it must take it infinitely long to do them. This is the hidden false step in the argument. 

Suppose that it takes the Hare 1 second to reach the tortoise’s starting position, and then it takes him ½ second to outrun from there to the tortoise’s next position, a further ¼ of a second to run it third position, and so on. The numbers here are chosen to illustrate the central point, rather than accurately represent the speed of the racers. The total time i will take him to catch the animal is therefore: 

1 + ½ + ¼ + ⅛ +…

This is what mathematicians call a series: a list of terms that are added up as it progresses. This particular series has an important property. Even though there are infinitely many terms in the series, as we move along them the total does not get bigger and bigger without limit. Adding up the first four term gives 1 and ⅞. Adding up the first ten gives 1 and 511/512. As those numbers are added up more, it does not increase into a larger number, instead it leans closer to the number 2. The mathematical way of saying this is that the series converges to the value of 2. It is written as:

1 + ½ + ¼ + ⅛ +…= 2

Back at the race, after 2 seconds, all of the Hare’s immediate steps have been completed. At that moment he will overtake the tortoise. The moral is that it is possible to carry out an infinite number of tasks within a finite amount of time, so long as the times for each task forms a convergent series. 

Meanwhile, a divergent series is the opposite of that. This term refers to the infinite series  in which the infinite sequence of the partial sums of its series does not have a finite limit or rather, reaches to infinity. These two series exist together in physics, engineering, and even in finance. Calculating signal processing through analyzing wavelengths and its fourier series, and in computer science with machine learning through creating solutions that creates the convergent series after the sequence of divergence.

The race between the Hare and the Turtle, therefore, is a butterfly effect that advances later branches of mathematics that most don’t see. It has not only taught moral and ethical lessons as children, but as well as references into popular mathematical concepts. Thus, if you come across a fable in the children’s section, ask this question: what kind of lessons are hidden in those pages that may change my understanding? Who knows, you might learn something unexpected.

Work Cited

Elwes, R. (2021). Mathematics 1001 : absolutely everything that matters in mathematics, in 1001 bite-sized explanations : Elwes, Richard, 1978- : Free Download, Borrow, and Streaming : Internet Archive. Internet Archive. https://archive.org/details/mathematics1001a0000elwe

TestBook. (2023). Divergent Sequences: Definition, Techniques and Solved Examples. Testbook. https://testbook.com/maths/divergent-sequences

Weisstein, E. W. (2024, August 22). Convergent Series. Mathworld.wolfram.com. https://mathworld.wolfram.com/ConvergentSeries.html

About the Author

My name is Xiu Ann, I’m a senior STEM student from the Philippines currently studying pathology. I have published a research about geodetic engineering science , and a few research synthesis from our school. I hope this article has spread knowledge, and helps in creating a cause of movement for the people searching for a science-related topics.