Zeno’s paradox: the puzzle that keeps on giving

You might’ve heard of Zeno’s paradox; a thought experiment over 2000 years old. Surely, it must be solved by now? Well, maybe not.

First devised by the ancient Greek philosopher Zeno of Elea, this puzzle has attracted the attention of scientists and philosophers alike over the centuries.

For those who are unfamiliar with the paradox, the most popular variant goes something like this:

A tortoise and a hare have decided to settle once and for all who is the fastest, so they organise a race to be run over 100 metres. The hare, feeling confident, lets the tortoise have a 50-metre head start. The starting pistol goes off, and they both start racing toward the finish. The hare, travelling at twice the speed of the tortoise, reaches the 50-metre mark where the tortoise started. The tortoise, meanwhile, has travelled only 25 metres, so is at the 75-metre mark. The hare then hops to the 75-metre mark, but the tortoise is still slightly ahead, at the 87.5-metre mark. The rabbit again hops to where the tortoise was, but in that time, the tortoise has moved a little bit further still. And so on. No matter how many times the hare tries to catch up, the tortoise remains slightly ahead. And so, Zeno said, the hare can never overtake the tortoise!

Every time the hare reaches the place where the tortoise was, the tortoise has moved on a bit further. Image credit: Wikimedia Commons (modified).

Of course, we know that in real life fast hares do overtake slow tortoises, and that’s exactly the point of the paradox. It shows that there is a disconnect between how we think the world works and how the world actually works.

Over the years, many solutions have been proposed to solve this puzzle. Let’s explore some of these possible answers.

Adding up infinite sums

Probably the most widely stated solution to the paradox came with the invention of calculus by Newton (or Leibnitz, depending on who you believe). This new branch of maths gave us a rigorous way to show that the sum of an infinite combination of numbers (called a ‘series’) can sometimes add up to a finite number. A good example of a this is seen by cutting a cake.

Cutting a cake with the size of the slice decreasing by half each time. Image created by author.

If you cut a cake in half, and then a quarter, and then an eighth, and so on, you’ll see that you get closer and closer to having sliced the whole cake. If we add up an infinite number of these slices, they’ll add up to the whole cake.

To see how this relates to Zeno’s paradox, consider the graph below, showing the distance covered by the tortoise and the hare over time.

A graph of the hare’s and tortoise’s distances over time. Image credit: Wikimedia Commons (modified)

As you can see, when the hare gets to 50, 75 and 87.5 metres, it is getting closer and closer to tortoise. But like the cake, if you keep on repeating this process an infinite number of times, both the tortoise and the hare converge to the same finite distance – 100 metres. So, the hare does overtake the tortoise right on the finish line!

But hang on. We can’t physically cut a cake an infinite number of times, so why should we be able to divide the distances in a race an infinite number of times?

This is a very troubling point for many mathematicians, and brings into question whether maths is really a true description of our physical reality.

Points in space and instants in time

One way to address this problem is to avoid the issue of infinites all together. Philosophers like Aristotle and Thomas Aquinas have argued that instantaneous quantities don’t exist; distance cannot be made up of many points, and time cannot be made up of many instants. So, to consider the tortoise’s and hare’s instantaneous position and time does not make physical sense. In this way, they say the fundamental premise of Zeno’s thought experiment is flawed.

This sounds like a reasonable argument. However, what if we could just estimate the positions within an approximate period of time, with no assumptions of exact distances and times? Then the paradox would still hold, provided the estimation was precise enough.

Well it turns out this argument breaks down as well, but to see this, we need to look at the problem through a different lens: using the principles of quantum mechanics.

Length in the quantum realm

Over the past 100 years, scientists have realised that at the smallest of scales, the world behaves completely differently to what we normally experience. Our best attempt at describing this unfamiliar world is known as quantum mechanics. One counterintuitive quantum idea is that things do not have precise locations, but are rather ‘smeared out’ over some distance. So, for Zeno’s paradox, there is a physical limit on how precisely we can measure the tortoise’s and hare’s positions. This is known as ‘Heisenberg’s Uncertainty Principle’.

Quantum mechanics also challenges the paradox on the basis that space itself is infinitely divisible. It has been theorised that space is not continuous but actually is made of discrete units. The theory says that if you zoom in on a region of space, you will eventually get to a distance where you can’t zoom in any further. This is known as the ‘Planck Length’. And as you’d expect, the Planck length is mind bogglingly small – less than a trillionth the size of a proton!

Could space be made of discrete units? Image credit: Pixabay

Advocates for this theory envision the universe as a sort of flipbook, where we move from one still frame to the next, but when flicking through fast enough it seems like continuous motion.

Maybe we’re moving through space and time like pages in a flipbook? Image credit: Kristine Paulus via Flickr

This would mean that Zeno’s thought experiment is based on a false assumption: that we can divide length into infinitely small parts.

Either way, quantum physics seems to resolve the paradox by saying that there’s a physical limit on how precisely you can know something’s position.

So, has this puzzle been solved once and for all?

Quantum physics has probably given us our most convincing answer to the paradox so far, but it is not a certainty. The question of whether space is continuous or made up of discrete units is still debated among physicists (there are experiments currently trying to test this), and quantum mechanics itself is known not to be a complete theory of the universe. If history is any indication, this isn’t the last we’ll hear of Zeno’s paradox.

2 Responses to “Zeno’s paradox: the puzzle that keeps on giving”

  1. Andre Chambers says:

    Thanks Emma, I’m glad you enjoyed it!

  2. Emma Fazzino says:

    I have always been fascinated by this! Thanks for writing such an interesting post.