Many many years ago, an island in the Aegean Sea was hosting an unprecedented race. Achilles, the famous semi-god, was challenged by a brave turtle, who claimed “If Achilles gives me 100m of head start, I can beat him in a 200m race”. Achilles, a bit shocked, accepted the challenge. They both got behind a line, and the turtle started the race. When our brave turtle reached 100 meters, Achilles started to run. When Achilles reached to the point that turtle used to be at, the 100th meter, turtle was already some more distance ahead of Achilles. Unbothered by this, Achilles kept running until he again reached the point that turtle used to be in when he was at 100 meters. But again, by the time he reached the old position of the turtle, the turtle was ahead of him by some more distance. Continuing to repeat this situation, Achilles was never able to reach the turtle, as whenever he reached a point that turtle used to be in, the turtle had already moved ahead…
You may say, “That’s ridiculous! Achilles should easily reach and pass the turtle in minutes if not in seconds…”. This illogical yet valid debate was first brought up by Zeno, a philosopher who lived around the same time period as Diogenes, whom I wrote about last month (See the Duality of Power). Zeno was an Eleatic philosopher, the name “Elea” coming from the name of the region he lived in, which is in modern day Napoli, Italy. Zeno used these paradoxes in hopes to support Parmenides’ doctrine of monism; that everything is one and there is no change. Monism argues that motion is merely an illusion, nothing else. In this short essay I’ll try to examine this from a Quantum physical perspective, hopefully not getting too technical.
I’ll leave the first paradox to you to think about, and I will try to resolve another paradox of Zeno: the Dichotomy Paradox. Let’s say that you are moving in between two trees who are 10 meters apart. To reach the other tree you need to cover half of that distance first, which will take you some time depending on your speed. When you get to the half, you again have a distance between you and tree that needs to be covered. Again, to reach the tree you must cover the half distance, spending some time doing so. This will go on and on forever, as any distance can be divided into two equal parts. Even though the amount of time spent during each distance gets smaller, the number of distances is infinite and when you multiply that small time interval with infinite amount of distance you reach a shocking result: You need infinite amount of time to reach to the other tree! This again makes us question if moving is possible at all or not… When heard about these paradoxes, Diogenes used the most fundamental technique to prove it wrong; he stood up and moved, clearly showing that movement indeed was not impossible. But why can’t we find logical and scientific solutions to these paradoxes?
The main reason why these paradoxes were valid until recently, is that they question an unnatural concept which is hard to comprehend: Infinity. We are not exposed to infinite things in our day to day lives, so we have hard time understanding the concept of infinity. Of course, mathematics introduced many rigorous ways to solve problems dealing with infinity, which are incredibly successful in giving exact answers. But they lack a sense of concreteness, as math is an abstract science in its foundation. Instead, I will approach this from a physical sense.
Physics was thought to be “completed” by the Newton’s Laws of motion and Maxwell’s equations for electricity and magnetism. They overlapped nicely and explained most of the phenomena around us perfectly. It was only for couple of phenomena that was unexplained by these, one of which was Photoelectric effect. To give a brief info about this effect, when light was incident on a metal plate, electrons gaining energy from the light could leave the plate and reach a receiver that was connected to a circuit. By connecting a lamp to the circuit, we were able to detect if there were any electrons reaching the circuit from the metal sheet or not. With this basic set up, we investigated how could we increase the amount of electron leaving the plate, how could we make the electrons reach the circuit even if we apply some oppositely directed electric field onto them, etc. Investigating these (and thermal radiation of a black body) the quantum physicist reached a remarkable conclusion, which shaped the entire Modern Physics: the Quantization of Energy. What does quantization of energy means, in basic terms, is that the energy around us is not a continuous thing which can take any crazy value like 5.76382 or 274.7264; but instead, it was in small packs called “quanta” that can only take integer values of hf (h being the Planck’s constant and f being the frequency of the de Broglie wave associated with that particle). In more clear terms E=nε where ε=hf and n=1,2,3…
Now connecting this new concept with our prior paradoxes, we deduce that the universe that we live in is not actually continuous, therefore infinitely divisible. Instead, it is made out of incredibly small building blocks, meaning if you continue dividing space into two, eventually you will reach a point where you are no more able to divide. This shows that from a physical standpoint, you will reach the second tree, eventually.
But of course, life is never simple (and neither it is fair, but of course that’s not our interest right now) and using the quantum theory helps us solve a problem by creating another problem somewhere else. This time investigating the quantum realm too deeply introduces us to a new paradox called the Quantum Zeno Effect. I will provide further reading links and sources for science geeks like me, but to describe this in basic terms; the quantum mathematical representation of a particle is “localized” in a given region when we observe it. Meaning the uncertainty in its location is gone when we observe a particle. And these mathematical representations of particles, called the Schrödinger’s equation, are wave-like structures describing the probability of finding a particle at a given region in space. When observed, the location is known, so the probability of finding the particle at the exact spot that it was observed is nearly 1 (out of 1). This probability gets distributed over a wider space, but it is proportional to the square of the time that passes in between two observations. When we observe the particle repetitively, in a small period of time, the
probability distribution is not changed. Continuing the repeated observations makes the particle unable to change location, preventing it from moving. This means the particle may move if we don’t observe it for long enough, but it won’t move if we observe it, again giving rise to the question “Is motion nothing but an illusion after all?”
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