Zeno’s paradoxes and the problem of motion: Achilles and the tortoise

Zeno was an Eleatic philosopher and an associate of Parmenides. Like Parmenides, Zeno affirmed that the answer to the problem of the one or the many is that there is One and the answer to being vs becoming is that there is only being. Consequently, there is no such thing as change or motion.

Yet, how can this claim be supported? Zeno devises 9 mathematical paradoxes, three of which were commonly discussed in the ancient world by the likes of Aristotle and continue to be taken seriously today. The most famous of these is called the paradox of Achilles and the tortoise.

Achilles is to enter a footrace with a tortoise. Achilles allows the tortoise a head-start of some distance x. Even though Achilles is faster than the tortoise, by the time he reaches x the tortoise will have moved again distance y (where x > y). By the time Achilles reaches y, the tortoise will have again moved ahead distance z (where y > z) and so on ad infinitim. Zeno argues that because every time Achilles reaches the point where the tortoise was he still has farther to go to catch up, the idea that Achilles could catch the tortoise (let alone pass it) is absurd.

Aristotle summarises:

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.

—Aristotle, Physics

Upon hearing the paradox, Diogenes the Cynic simply rose and walked to empirically demonstrate the falsehood of Zeno’s conclusions. Aristotle (who took Zeno and Parmenides more seriously than many of his contemporaries) says that as the distance between Achilles and the tortoise grows smaller, so does the amount of time that Achilles requires to cover those distances. Archimedes developed a method to derive a finite answer to the sum of infinitely many terms that become progressively smaller. As modern calculus achieves the same result as Archimedes, the paradox of Achilles and the tortoise is generally considered solved.

One response to this post.

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