I like to go running in my free time. I’m not an especially fast jogger, but I think I could keep up with a tortoise. According to Zeno of Elea, an ancient Greek scholar who lived around 450 B.C.E., however, the outcome of a race between me and a tortoise is far from clear—assuming that I give the slow-moving reptile a sporting head start. Of course, Zeno was aware that humans can easily overtake tortoises. But he wanted to point out that from a mathematical perspective, this is not necessarily obvious. The thought experiment that Zeno described was as follows: Suppose I challenge a tortoise to a 100-meter run. I give the animal a one-meter head start (not too much; after all, I want to win). In this case, Zeno concluded, I will never succeed in overtaking the tortoise.
Even if this sounds totally absurd, Zeno's reasoning is sound. If I begin my chase only when the tortoise has covered a meter, by the time I have run one meter, the animal has also moved on, let’s say by 20 centimeters. By the time I reach this point, having covered 1.2 meters, the tortoise has made further progress, namely four centimeters. And so it goes, on and on: every time I reach the spot where the turtle was, it will have moved on. That’s why the animal is always ahead of me—and it’s impossible for me to win the race.
In Zeno’s original thought experiment, he made the surprise even more stark by pitting the tortoise against Achilles, the swiftest hero in Greece, so the scenario is often called the Achilles paradox. Whatever the case, there is a contradiction between what we perceive in reality—people overtaking tortoises—and what the theoretical description suggests.
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And Zeno’s considerations went further still. Because any kind of movement can be observed at an infinite number of points in time, change in general becomes a problem. Zeno’s paradoxes—which include the Achilles paradox and others, such as the arrow paradox—state that, from a mathematical point of view, the world should, to a certain extent, remain still. These arguments kept experts busy for thousands of years—and while they later reached some mathematical resolution, they’ve gained importance once again thanks to the mysterious world of quantum physics.
Different Types of Infinity
Ancient Greek philosopher Aristotle saw Zeno’s paradoxes as support for his view that space and time cannot be divided into an infinite number of small pieces. Accordingly, Achilles would eventually reach a minimum distance from the tortoise and then overtake it. But another great Greek thinker, Archimedes, argued that in this case, a distinction had to be made between different infinities: an infinitely large category on the one hand and a finite continuum on the other.
The first category includes, for example, the set of natural numbers (1, 2, 3, 4,...), which is unlimited upward: there is no largest natural number. An interval on the number line, such as a centimeter, falls into the category of the bounded continuum. Although this interval has a finite length, it consists of an infinite number of points. Archimedes observed that I would need less and less time to cover the shrinking distances that separate me from the tortoise in a race.
Archimedes was correct, though he was unable to prove this assertion. In fact, a scientific solution was not found until the emergence of the mathematical field of calculus more than 2,000 years after Zeno’s thought experiment. The methods developed at the time by Isaac Newton and Gottfried Wilhelm Leibniz made it possible to at last prove Archimedes’ idea. Although you can divide a length into an infinite number of smaller intervals, this does not mean that it takes an infinite amount of time to traverse them.
In other words, there are an infinite number of moments in which the tortoise is ahead of me, but the sum of these moments is finite—and actually quite short.
The Moment I Overtake the Tortoise
This proof can be calculated quickly using the mathematical tools available today. Let's assume that I give the turtle a one-meter head start and then begin to run at around 12 kilometers per hour. For the sake of simplicity, let the tortoise be five times slower than me—even if the animals are actually much slower. When I have covered one meter, the tortoise has moved 20 centimeters farther; when I get there, it runs an additional four centimeters, and so on. The distance S that I cover when catching up is therefore: S = 1 + 1⁄5 + 1⁄25 + 1⁄125 + .... The sum consists of an infinite number of summands that get smaller and smaller.
Without calculus, there is no way to evaluate such a sum. Newton and Leibniz created the tools to deal with ever smaller quantities, however: the so-called infinitesimals. And as it turns out, the infinite sum mentioned above produces a finite result. This can be seen by excluding the factor 1⁄5 from part of the sum: S = 1 + 1⁄5 x (1 + 1⁄5 + 1⁄25 + 1⁄125 + ...).
Because the sum is infinitely long, the expression in the brackets corresponds exactly to the value S. This gives us the following equation: S = 1 + ⅕S. Solve for S to discover that S = 5⁄4 = 1.25.
In other words, after 1.25 meters, the turtle and I are on a par—and then I overtake it. Although there are an infinite number of moments when the tortoise is ahead of me, I only need a finite amount of time to overtake it. If I actually run 12 kilometers per hour, then I have covered the 1.25 meters in just under 0.375 second.
Zeno’s Paradoxes in the Quantum World
Zeno’s paradoxes were thus solved from the 17th century onward. With calculus, mathematicians had found a way to describe variable things. From a scientific point of view, the ancient contradictions seemed to be resolved—until quantum physics emerged.
According to quantum mechanical theory, quantum objects, such as an electron or a molecule, cannot change or move while they are being observed. They behave like an actor with extreme stage fright who freezes, as it were, under the penetrating gaze of the audience.
Quantum objects normally change their state over time: they can change from one energy to another or move from one place to another. But if you constantly measure these particles, it becomes increasingly unlikely that they will change state. They remain trapped in their original state.
And physicists were actually able to observe this behavior in experiments: if they took measurements on a quantum system often enough, its temporal development could be suppressed. In other words, the system remained in its state without changing. This so-called quantum Zeno effect is now used in commercial magnetometers. This is because the quantum systems, which measure a magnetic field very precisely, retain their desired state. It is now also assumed that the quantum Zeno effect could play a role in the magnetic sense of birds.
So even 2,500 years after the Greek philosopher’s thought experiments, Zeno’s paradoxes fascinate experts. Although the problem of movement has been solved from a mathematical perspective, the paradoxes still raise questions about our reality. How can the basic building blocks of our world move and change? What exactly is a measurement? And what does an observation mean for a quantum system? Let's see if it takes a few more millennia to answer those questions.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.