In the depressing trenches of AP studying, your eyes fixate on a half-opened bag of chips right across the living room. Naturally, you decide to move towards the chips with purpose. It seems simple—but according to Zeno of Elea, you might never actually reach it.
Of course, nothing should come between you and your chips. It turns out Zeno isn’t some random, but an incredibly famous Greek philosopher, well-known for proposing various interesting and mind-boggling paradoxes. Zeno was a student of Parmenides and because of that, many tend to believe his paradoxes were meant to defend his teacher’s idea of an unchanging reality. However, this interpretation mostly comes from later speculations, including those from Plato’s dialogues. Some of his paradoxes include The Antinomy of Large and Small, The Antinomy of Limited and Unlimited, and The Paradoxes of Motion—one of which is the Dichotomy Paradox.
The Dichotomy Paradox is explained by Zeno as follows: let’s say a runner intends to meet a goal. If the goal is one meter away, the runner must cover a distance of ½ meter, then ¼ meter, then ⅛ meter, and so on ad infinitum. Because this process continues indefinitely, Zeno argues that the runner can never reach the final goal. To expand, the regressive version of the Dichotomy Paradox states that the runner can’t even take the first step because any step may be divided conceptually into a first half and a second half. Before taking a full step, the runner must take a ½ step, but before that, he must take a ¼ step, but before that, a ⅛ step, and so forth ad infinitum. Seems convincing, right?
Well, no, not really. In fact, this paradox has been resolved in both math and physics.
To begin, let’s envision the runner with an impending goal of one meter. Zeno breaks this one meter into ½ meter, ¼ meter, ⅛ meter, and so on forever. To express the total distance traveled:
Total Distance Traveled = ½ + ¼ + ⅛ + …
By a convergent geometric series, we see that the total distance equals exactly one meter. Hint: Notice how the sum of all the individual boxes still results in the whole box. See, even though the runner is completing infinitely many subdivisions, the total distance still adds up to a finite amount.
Unfortunately for us, mathematics alone isn’t enough to provide a full solution. To fully resolve this paradox, we need to realize that this paradox isn’t simply about dividing infinite parts, but the physical concept of a rate. Zeno’s paradox feels convincing because it only takes into account distance, without factoring in time. Motion isn’t limited to how far one moves, but how far one goes in a select amount of time. “The reason objects can move from one location to another (i.e., travel a finite distance) in a finite amount of time is because their velocities are not only always finite, but because they do not change in time unless acted upon by an outside force” (Siegel 2020).
There is another detail of the Dichotomy that needs resolution. How does Zeno’s runner complete the trip if there is no final step or last member of the infinite sequence of steps (intervals and goals)? During the process of “taking a trip,” can there be an absence of the crucial “last step”? The Standard Solution answers “no,” while the intuitive answer “yes,” held by Zeno, Aristotle, and the average person today, must be rejected when embracing the Standard Solution. Even if there is no “last step,” the runner can still finish the journey because completion stems from the limit of infinitely many steps, not the final step itself.
Now, unfortunately for Zeno, you can confidently say you made it across the room and got the chips—no paradox stopping you.
Authored by Chandhana Lingam Muhilan and Katie Huang
