In yesterday’s piece, I dove into what paradoxes are and why they’re useful.
Today, I want to talk about Zeno’s paradox of Achilles and the tortoise. Zeno was an ancient Greek philosopher who first came up with several of these puzzles we still think about today, or at least he has received the credit for doing so.
This paradox challenges our very notions of space, time, and motion. It’s easy to see why this 2500 year old thought experiment persists in our collective memory.
The Paradox of Usain and the Tortoise
The paradox starts with a simple footrace between Achilles, the fastest man in Greek mythology, and a tortoise. To give the tortoise a sporting chance, Achilles lets it start a few meters ahead of him.
I haven’t seen Achilles run, but I have seen the greatest sprinter who ever lived run.
Imagine peak Usain Bolt taking off as the gun goes off, starting our 100 meter race, the most important event in competitive sprinting. Racing against him is a tortoise, given a 20 meter head start. What’s about to happen?
As fast as Bolt is, one would assume that catching up and overtaking the tortoise would be a matter of a few seconds.
But Zeno offers a different perspective.
According to Zeno, every time Bolt reaches the point where the tortoise was, the tortoise has moved a little bit further ahead. For Achilles to reach the new position of the tortoise, he has to cover this new, albeit smaller, distance.
Again, by the time he does, the tortoise advances just a little more. Zeno argues that this process will continue infinitely, implying that Usain Bolt will never be able to overtake the tortoise.
At its core, the paradox is a question about the nature of infinity within the realms of space and time. It asserts that if space and time are infinitely divisible, then motion, as we understand it, becomes an impossibility. It's a puzzle that's not just a race on a track; it's a race in your mind, challenging our intuitive understanding of the world.
In this simple narrative, Zeno exposes the inadequacy of our intuitive grasp of concepts like distance, speed, and time. What may seem obvious is thrown into question, forcing us to engage in deeper thought about the mechanics of our reality.
Resolving the Paradox
The paradox seems airtight, but this one actually has a pretty straightforward scientific explanation.
In 1901, Max Planck realized that nature wasn’t smooth and continuous. Instead, discrete quanta—the smallest amount of energy or matter—make up the universe. Light hits us in packets of light called photons, and everything we see is made up of the tiniest imaginable pieces.
This concept radically changes our understanding of space and time. If the universe is made up of these indivisible quanta, then the idea of infinite divisibility breaks down.
There is a point where space and time become "grainy," unable to be divided any further.
So, back to Usain and our tortoise: In a universe made up of quanta, there’s a smallest unit of space Bolt and the tortoise can occupy, and a smallest unit of time in which they can move. This allows Bolt to eventually overtake the tortoise without getting stuck in Zeno's infinite loop.
It's worth considering how these ideas of limited divisibility and infinite loops apply to our modern challenges in technology and society.
The Paradox in Today's World
Moore’s Law has boldly stated:
The complexity for minimum component costs has increased at a rate of roughly a factor of two per year. Certainly over the short term this rate can be expected to continue, if not to increase.
While the definition is fluid and has evolved over time, Gordon Moore’s original conception has remained a “true north” for the semiconductor industry for five decades.
This means that ever-smaller transistors have fit onto chips, with hundreds of billions of these tiny logic gates crammed onto a single wafer. In this case, it can’t be turtles all the way down, though. These same physical limits that Usain and our tortoise have had to reckon with, also pervade the world of technological progress.
We’re not sure if we’re at that point today or not. Many pundits say so, and many others say “not yet.” Of course, there are other ways to speed up technology, like stacking chips in 3D designs, or quantum computing, but those paradigms are far from guaranteed.
Because we’re still able to eke out gains due to Moore’s Law (and the modifications that are likely to keep progress moving), the time it takes for technology to become obsolete seems to be shrinking, leaving us in a perpetual race to catch up. Much like Usain Bolt chasing the tortoise, no matter how fast we sprint toward the latest advancements, there's always something slightly beyond our reach.
This concept, that we’ll never catch up with our technology, encapsulates a singularity—a liftoff of sorts where humans are sort of left in the dust by our technology.
One last area that mirrors this paradox: the never-ending work on Democracy, and the endless debates on human rights and other social issues. Just when we think we have achieved some measure of equality or justice, we discover that much more work needs to be done, and the goal posts move.
In this way, the paradox mirrors the eternal chase for "progress," urging us to question whether we can ever truly achieve it, or if it's just a concept that keeps evolving.
The Eternal Chase
After two and a half millennia, Zeno's paradox of Achilles and the Tortoise continues to captivate us. Why?
This mental puzzle taps into questions that are deeply rooted in our understanding of the world and ourselves. Perceptions and the very nature of reality are at its core, and we are pushing the boundaries of what’s possible every day, trying to fit even more transistors onto chips, or trying to refine what our concept of justice means.
I hate to be one of those “enjoy the journey” people, but… well, the journey is everything. Unlike Bolt (or Achilles) in the footrace, we’ll never catch our tortoise.
lies. Like Bolt or Achilles in their respective races, we might not ever catch up to our elusive tortoise—whether that represents technological singularity or true social justice. But it's the pursuit, the eternal chase, that compels us to keep moving forward, to keep questioning, and to keep growing.
The chase IS a powerful motivator. Also, there was a time when we thought achieving a certain level of social justice meant we would not have to achieve it again, but we’re in a dangerous era when that’s no longer the case. We’re moving the goal posts, yes, but we are also having to defend the gains we made at its previous placement.