Achilles and the Tortoise: A 2,000 Year-Old Mystery
A paradox posed by math and philosophy
Meet Zeno of Elea
Referred to as the father of dialectic (a form of discourse resembling debate) by Aristotle, Zeno of Elea was a pre-Socratic philosopher and a member of the Eleatic School. This school was a group of pre-Socratic philosophers. He was also a disciple of Parmenides, a fellow Eleatic. Although his works are referenced by several ancient writers, none of have been found intact. However, that’s not to say his work hasn’t been discussed.
Coming to his work, Zeno of Elea is best known for his paradoxes — statements that contradict themselves or known physical descriptions of the world around us. What was their purpose? To support Parmenides’ arguments that motion and change were not real.
Credit: World History
His paradoxes are designed as a set of philosophical problems (though have also been said to be mathematical problems) and have amused and challenged philosophers, mathematicians, and physicists for over two millennia. One of such paradoxes is known as Achilles and the Tortoise.
Achilles and the Tortoise
This paradox revolves around two characters: Achilles, the fleet-footed hero and a tortoise at a race. The tortoise is granted a head start — let’s say 10 meters. Automatically, most people think Achilles will win. So what’s the paradox? The paradox lies in how the event is described and Zeno’s reasoning behind it. Read the excerpt below in which Aristotle recounts the paradox in his treatise titled ‘Physics’ and explains why Achilles can’t win.
In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.
— as recounted by Aristotle, Physics VI:9, 239b15
Let’s talk about how this paradox takes place per its description above. Imagine Achilles at the starting line and the tortoise at a starting line that is 10 meters ahead. They start moving at the same time, at which point the tortoise is still 10 meters ahead.
After some time, Achilles catches up to the 10-meter mark (i.e. the tortoise’s starting point). But, the tortoise has been moving too and is still ahead by a specific distance. Let’s call this point B. This goes on — when Achilles reaches B, the tortoise is still ahead again, even though the distance he has to cover keeps decreasing with each cycle. This is because Zeno assumes Achilles is running at a pace faster than that of the tortoise.
So why won’t he win? According to Zeno’s reasoning, every time Achilles reaches where the tortoise was, the tortoise will have advanced further and he’ll have to repeat the above scenario all over again. We have to note that throughout the paradox, Zeno’s presumption lies in the fact that Achilles must reach where the tortoise was, by which time the tortoise will have moved ahead. And so, irrespective of the time Achilles takes to cover each gap, the slow but steady tortoise would be just ahead.
Now, what is this paradox meant to do? As stated above, Zeno was a disciple of Parmenides. As such, he wanted his paradoxes to serve as arguments that supported Parmenides’ idea that change and motion weren’t real. So Achilles and the Tortoise attempted to prove that the swifter opponent can never overtake the slower one, suggesting that motion does not exist.
Zeno’s dichotomy paradox does the same. According to Oxford Languages, dichotomy is defined as “a division or contrast between two things that are or are represented as being opposed or entirely different.” The paradox states that any moving object must reach halfway on a course before it reaches the end. But because there are an infinite number of halfway points, a moving object never reaches the end in a finite time.
I know what you’re thinking. Why is this paradox so hard to solve? It has even been dismissed as logical nonsense by some. It’s not that motion does not exist — we know it does, it’s not Achilles not being fast enough — we know he is.
No, the problem lies in the fact that there appear to be infinite checkpoints or places where the tortoise was, that Achilles needs to reach and how the Achilles and the Tortoise paradox cuts to the root of the problem of the continuum. The problem lies in how we perceive infinity and the fact that Achilles seemingly has to reach infinite points to cover a finite distance in a finite time.
Joseph Mazur, a professor emeritus of mathematics at Marlboro College says Achilles’ task seems impossible because he “would have to do an infinite number of ‘things’ in a finite amount of time.” This is also a reiteration of what Aristotle argued — that infinite subdivisions of a finite distance do not nullify the possibility of traversing that distance, since the subdivisions do not have an actual existence, unless something is done to them, in this case stopping at them.
This is where the aforementioned mathematics come in — where the theory of convergent series and divergent series have been used to tackle this infamous paradox. If it were a divergent series (i.e. a series that goes on till infinity like the Fibonacci series), Achilles would never catch up with the tortoise. But if it were a convergent series, (i.e. a series that approaches a constant), what then? Theoretically, he would be able to win, but then again, if Achilles and the tortoise decrease their speeds in the same manner, Achilles would lose again.
Is there a solution to this paradox?
So what’s the prevailing solution? There isn’t one — at least not one that’s accepted by all. For one thing, the paradox appears to be simply illogical and for another, Aristotle himself called it ‘poor physics.’ So, can it have a solution if the problem is inherently wrong? There are several theories, of course, two of the most common ones we’ll talk about below.
One of the most common solutions given is related to the theory of convergent series mentioned above. The convergence of an infinite series is a theory in itself which was devised and perfected by those of the likes of Isaac Newton and Augustin-Louis Cauchy.
This is a proposed solution because any distance, time, or force can theoretically be broken into an infinite number of pieces (the distance Achilles has to cover, in this case) and these infinite pieces can be treated as finite suggesting that Achilles catches up to the tortoise at some point. Keep in mind that the distance he needs to cover progressively decreases as Zeno has assumed Achilles is moving faster than the tortoise.
The second solution is one rooted in physics. We know objects can travel a fixed distance, (i.e. a finite distance), and we know Newton’s first law states that objects in motion remain at motion unless acted upon by an external force. Now, the reason objects move from one point to another is precisely because of this — they do not change in time unless acted upon by an outside force.
If Achilles is moving at a constant speed, he’ll be able to cover the required distance according to the relation: distance = velocity x time. And this would work for any distance, no matter how arbitrarily small.
In a way, the proposed solution depends on how well the scenario has been perceived. As such, opinions are still divided and a google search on ‘solution to Achilles and the Tortoise’ has yet to provide concurrent results. So, what do you think? Are we fated to live through another millennium with this paradox hovering over our heads?
