PODCAST TRANSCRIPT – Season 2 – Episode 51 – Zeno’s Paradoxes
Have you ever been in a moment of your life where you wished something would change, but it doesn’t? Time passes by, and nothing changes. Sometimes graduate school can feel that way. Sometimes getting a job or working towards a job promotion can feel that way. Sometimes an experiment can feel that way. Like when you’re watching a pot of water and waiting for it to boil.
Yet when you look back on your life, you realize that it has changed. It just took what felt like infinitely small measurements of time to make that change occur. Through hindsight, when we remember those moments that felt like forever, we can see how much of our lives changed. In science and mathematics, this is known as the Quantum Zeno effect.
Zeno of Elea was one of four prominent philosophers who attended the Eleatic School of Philosophy. The other three prominent philosophers included Xenophanes, Parmenides, and Melissos. Zeno was born around 490 BCE and was close to his friend and possible lover, Parmenides. We know about Zeno because of the writings of Plato. In his work titled Parmenides, Plato alludes that Parmenides and Zeno were lovers. He writes, “Parmenides being at that time quite an old man with grey hair and a handsome and noble countenance, and certainly not over 65 years of age; Zeno about 40 years old, tall and elegant, said to have been the favorite of Parmenides.”[1] So, needless to say, Zeno was not just Parmenides’s arm candy. He was also his special someone.
But, as most women would agree with me, it is wrong to objectify Zeno because there is more than meets the eye. Zeno was also very brilliant and came to be known as a philosopher who mastered the skill of contradiction.[2] Aristotle wrote about Zeno’s methodologies and believed that Zeno never really searched for truth in his argumentation. Instead, he argued for argument’s sake. This form of argument makes sense because his paradoxes are like puzzles that do not so much require a dialect between two individuals founded on truth but rather a creative analysis based on possibilities.
However, this information doesn’t come directly from Zeno’s writing. It comes from the teachings of Socrates and the writings of Plato; we do not know exactly how many paradoxes Zeno had written. At this current time, we know of ten paradoxes. However, it is possible that Zeno wrote up to forty paradoxes.
One of Zeno’s most popular paradoxes is Achilles and the tortoise. In this paradox, Achilles is racing a tortoise. As the race begins, Achilles gives the tortoise a head start of 100 meters. Then Achille’s runs and meets the tortoise at the 100-meter mark. However, the tortoise is still ahead of him by a certain distance. In this case, let’s say the tortoise is ahead of him by fifty meters. That means Achilles now must run 50 meters to catch up with the tortoise. So, Achilles runs the 50 meters, but now the tortoise is leading by 25 meters. And so, Achilles runs the 25 meters, only to realize that the tortoise is 12 meters ahead of him. Thus, Achilles is always behind the tortoise, which leads to this paradox that Achilles will never be able to catch up to the tortoise. This idea is impossible because we all know that Achilles would be able to catch up to the tortoise. Regardless, this puzzle has been pondered and questioned by mathematicians, physicists, and philosophers for thousands of years!
The alternate to this paradox is called the Dichotomy Paradox. In this paradox, an object will never move a certain distance unless it meets the halfway point first. However, because there is no defined distance, there is no halfway defined distance. For the object to move to half the length, it would first have to move to the first half of the first half. In other words, it would have to move a quarter of a distance. But, to move to that first halfway mark, it would have to meet the halfway mark before that halfway mark. In other words, the distance between the beginning and the endpoint becomes so small that it is impossible for the object to move in the first place. And the reason this is called a dichotomy is because it requires the constant process of splitting the distance into two parts.[3]
The next paradox is called the Paradox of Place. This paradox centers around the idea of displacement and states that if everything in the universe has a place, then that place has a place, and that place has a place.[4]
Then there is the Paradox of the Grain of Millet, which states that if a single grain of millet doesn’t make a sound when it falls, but if 1,000 grains of millet make a sound, then 1,000 nothings become something. However, with technology today, we now know that even a single grain of millet makes a sound when it falls.
The Paradox of the Stadium proposes two rows of runners, with each row consisting of the same number of people. In this case, we will say four people in a row. Each row of four people runs on the same lane on a straight track. So, in this paradox, there are two lanes on a running track, and on each lane are four people. The runners in lane one run 400 meters from point A to point B. The runners in lane two also run 400 meters. However, they are going from point B to point A. In each row, they are moving at the same velocity. And the concept behind this paradox is that as each group of runners approach the center, the amount of time it takes them to reach the center will be doubled to reach the other end of the track. It’s a practical mathematical problem.
Zeno also has three paradoxes that look at plurality.[5] These include the Argument from Denseness, the Argument from Finite Size, and the Argument from Complete Divisibility. If you would like to read and hear more on these paradoxes, please come on over the www.Patreon.com/MathScienceHistory and sign up for a tier that will allow you to access ad-free, early release content!
Finally, we get to one of my favorite paradoxes, the Arrow Paradox. This paradox argues that in order to have motion, an object must change its occupied position. Aristotle describes Zeno’s paradox stating, “that it is impossible for a thing to be moving during a period of time, because it is impossible for it to be moving at an indivisible instant. This assumes that a period of time is made up of indivisible instants, which cannot be granted.”[6] What Zeno is saying is that at a particular place on a linear trajectory for an arrow, at a specific moment that arrow is at rest. But only at that moment. After that moment, it’s in another location which is also at rest. In other words, it is not traveling during a moment. And the arrow at its designated size, let’s say two meters long, occupies an equal space for that instant. As a result, the entire length of its motion consists of moments of an arrow at rest. Thus, Zeno argues that the arrow cannot be moving. But because the arrow moves from point A to point B, time consists of a multitude of instants. This analogy helps us to understand how to measure instantaneous velocity.
Mathematically, this concept is used in calculus, and it is defined as the derivative of the object’s average velocity. In physics, this concept helps us to understand a continuous function of time and determine a particle’s velocity at any point while it is in motion.
This paradox is foundational to understanding the Turing Paradox, also known as the Quantum Zeno effect. It is called the Turing Paradox because it is also similar to a paradox proposed by British scientist Alan Turing in 1958. The idea behind this is that in quantum systems, a system cannot change if you are watching it.[7] [8] However, instead of watching it, physicists can rapidly measure a system in its known initial state. What is most exciting about this ancient concept is that it is used in science and has been verified through experimentation.
The idea behind this is that by rapidly measuring a quantum system, physicists can slow down its evolution and freeze it completely while they measure it. They can keep the system in its initial state for any chosen finite time on one condition: they must perform a multitude of rapid measurements. Back to Zeno’s paradox, like that arrow that does not move in that one instant, the quantum system cannot change since it is constantly being observed through rapid measurements while it is in its initial state, at time equals zero. By observing and measuring it in its initial state, they can apply the process of probability by projecting the system’s state at a later time and apply it to the state that they want the system to evolve to. The math behind this is not too complex. If you are an undergraduate student studying calculus or calculus-based physics, you could probably understand the math that goes into freezing a Quantum Zeno Effect. You can find the math here at a YouTube channel called Pretty Much Physics. The video is titled “How to Freeze a Quantum State.” The video is one of the best mathematical explanations I have seen on the Quantum Zeno Effect.
Sometimes life feels like it is standing still. And that is really not a bad place to be. It gives you a chance to observe and think about everything in that instant. By taking a moment to observe the world, listen carefully to what we are hearing, think about what we smell, and consider how we feel, we slow down. It freezes us for just an instant. And after that instant, you’re going to keep moving forward. And you won’t be in the same place you are now. You will be moving towards goals, plans, dreams, loved ones, maybe towards work where you can enjoy the company of your favorite co-workers, or maybe towards home where you can enjoy a warm dinner and a cuddle with your pet. And even if you don’t have any of these things at this moment, I hope you know that like that quantum state that encompasses a position in a three-dimensional environment, with momentum, angular momentum, spin, time, and energy, you are moving and evolving. It is exciting to think that as humans, we have something very much in common with an ancient paradox, an arrow, a philosophy, physics, and mathematics. That is the power of math, science, and history!
Until next time, carpe diem!
Gabrielle
[1] Thomas Davidson, The Fragments of Parmenides (St. Louis: E.P. Gray, 1869), 1.
[2] John Palmer, “Zeno of Elea,” Encyclopedia, Stanford Encyclopedia of Philosophy, 2021, https://plato.stanford.edu/entries/zeno-elea/.
[3] Aristotle, Aristotle: The Physics Vol. I, trans. Frances Cornford and Philip Wicksteed (Cambridge, MA: Harvard University Press, 1933), VI:IX http://archive.org/details/aristotlephysics0000aris.
[4] Aristotle, Aristotle : The Physics Vol. II, trans. Frances Cornford and Philip Wicksteed(Cambridge, Mass. : Harvard University Press, 1970), IV:I, http://archive.org/details/aristotlephysics0000aris.
[5] Nick Huggett, “Zeno’s Paradoxes,” Stanford Encyclopedia of Philosophy. Last modified, 2018, https://plato.stanford.edu/entries/paradox-zeno/.
[6] Aristotle, Aristotle The Physics Vol. II, trans. Frances Cornford and Philip Wicksteed, vol. 2, 1 vols. (London: William Heinemann, 1934), 179, http://archive.org/details/in.ernet.dli.2015.183610.
[7] Y. S. Patil, S. Chakram, and M. Vengalattore, “Measurement-Induced Localization of an Ultracold Lattice Gas,” Physical Review Letters 115, no. 14 (October 2, 2015): 140402, https://doi.org/10.1103/PhysRevLett.115.140402.
[8] Bill Steele, “‘Zeno Effect’ Verified: Atoms Won’t Move While You Watch,” Cornell Chronicle, October 22, 2015, https://news.cornell.edu/stories/2015/10/zeno-effect-verified-atoms-wont-move-while-you-watch.