This is a very troubling point for many mathematicians, and brings into question whether maths is really a true description of our physical reality. 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? So, the hare does overtake the tortoise right on the finish line!īut hang on. 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. Image credit: Wikimedia Commons (modified)Īs you can see, when the hare gets to 50, 75 and 87.5 metres, it is getting closer and closer to tortoise. A graph of the hare’s and tortoise’s distances over time. 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. If we add up an infinite number of these slices, they’ll add up to the whole cake. 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. Cutting a cake with the size of the slice decreasing by half each time. A good example of a this is seen by cutting a cake. 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. Probably the most widely stated solution to the paradox came with the invention of calculus by Newton (or Leibnitz, depending on who you believe). Let’s explore some of these possible answers. Over the years, many solutions have been proposed to solve this puzzle. It shows that there is a disconnect between how we think the world works and how the world actually works. Of course, we know that in real life fast hares do overtake slow tortoises, and that’s exactly the point of the paradox. Image credit: Wikimedia Commons (modified). 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. No matter how many times the hare tries to catch up, the tortoise remains slightly ahead. The rabbit again hops to where the tortoise was, but in that time, the tortoise has moved a little bit further still. The hare then hops to the 75-metre mark, but the tortoise is still slightly ahead, at the 87.5-metre mark.
The tortoise, meanwhile, has travelled only 25 metres, so is at the 75-metre mark.
The hare, travelling at twice the speed of the tortoise, reaches the 50-metre mark where the tortoise started.
The starting pistol goes off, and they both start racing toward the finish. The hare, feeling confident, lets the tortoise have a 50-metre head start. Surely, it must be solved by now? Well, maybe not.įirst devised by the ancient Greek philosopher Zeno of Elea, this puzzle has attracted the attention of scientists and philosophers alike over the centuries.įor those who are unfamiliar with the paradox, the most popular variant goes something like this:Ī 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. You might’ve heard of Zeno’s paradox a thought experiment over 2000 years old. Zeno’s paradox: the puzzle that keeps on giving
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