The Short History of Infinity #1 # Zeno’s Curse: Why Was Motion Considered Impossible?

 

The moment we start pondering infinity, we run into one of the oldest and most frustrating puzzles in history: Zeno's Paradoxes.

In the 5th century BCE, Zeno of Elea put forward an idea that has been haunting mathematicians and philosophers ever since: Motion does not actually exist in the universe; motion is entirely an illusion.

Sounds completely crazy, right? In a world where we constantly see people walking, running, and kicking balls, how could anyone claim such a thing? Zeno tried to prove this precisely by using the concept of infinity at a point where logic and mathematics refused to bend. Let's look at his most famous example: Achilles and the Tortoise.

Achilles (the legendary hero of the Trojan War, invulnerable except for his heel), the fastest runner of his time, gives a 100 meter head start to a very slow tortoise in a race. The race begins. Achilles runs and reaches the exact spot where the tortoise was (the 100 meter mark). But by that time, the tortoise has also moved forward a little bit (say, 1 meter ahead).

Achilles runs again to close that 1 meter. When he arrives, the tortoise has moved slightly further once more. You can continue this logic infinitely: No matter how much Achilles closes the gap, he always has to reach the point where the tortoise was just before, and this process consists of an infinite number of sub steps.

According to Zeno, because Achilles has to complete an infinite number of tiny steps one by one, he theoretically can never pass the tortoise, nor even catch up to it.

The fundamental reason Zeno built this paradox was the assumption that time and space can be divided infinitely. If you can mathematically slice a line or a time interval into infinite pieces, you must leave behind "infinite moments" to traverse that distance. The human mind simply cannot logically accept completing an infinite number of steps within a finite span of time.

This is precisely why, in the ancient world, infinity was seen as an uncontrolled, terrifying monster that locked up the human intellect.

It took thousands of years to untangle Zeno’s mind bending puzzle. No one could find a way out until the invention of limits and the sum of infinite series (calculus) one of the foundational pillars of modern mathematics. In the 17th century, mathematicians proved a vital truth: You can take an infinite number of steps, but the total time sum of those steps can still equal a finite number (for instance, 2 minutes). In other words, summing up infinite tiny pieces is mathematically possible.

Yet, long before he had these mathematical tools in his hands, Zeno had already forced the human mind to stare straight into the deep abyss of infinity.