Zeno’s Paradox Exploration By a Highschooler? A Unique Illusion that Questions Human Perception on Motion
Is the way we Measure and see Distance really Showing us the Reality of the Length between Two Points?
Zeno’s paradox has been a problem that has left mathematicians, philosophers, and scientists on the edge of their chairs. This problem may sound easy to comprehend, but it is relatively difficult to prove or find an actual solution. When I first heard about the problem myself when I was studying series in my math class, I was also stumped about the problem. Ever since that encounter, I kept wondering about an actual solution and if motion was actually an illusion to our human eye. However, after a few days of work, I have done a bit of my own exploration regarding the problem to see the mechanics of motion and what Zeno thought when he initially proposed the idea.
Before we get into my exploration, we first must understand what Zeno’s paradox actually is. Zeno’s initial point of view about motion was that it was an illusion, and that such a phenomenon in reality is impossible. Zeno’s paradox states that for an object to get from point A to point B, it must travel half that total distance, then half of the remaining distance, and so on until it gets to point B. However, no matter how small the distance, the object will always have to travel half of what’s left over, and therefore, never actually reaches the distance as it requires an infinite number of steps. In reality, it is not impossible to get to our destination. We clearly have taken vacations and have travelled distances less than an infinite number of steps, so what is Zeno’s paradox really talking about and is such an assumption true? From our human experiences, it is evident that it does not require an infinite amount of steps to get to our destination, but can we prove this mathematically?
If we let x be the original distance from object A to object B:
Now, for every time A approaches B, the distance gets halved as stated in the dichotomy. These distances can be expressed in terms of x in a geometric sequence as:
From Equation 1, we can simplify our sequence as
which allows us to find the total distance needed to travel after an nth step assuming the ratio of half holds throughout the whole journey. The problem here is not that the expression can be modelled, but the way we see the paradox or in general, motion. Since the ratio of Equation 1 is smaller than 1, by the nature of a geometric series, we can find the total distance by adding each distances up:
For now, we will assume x to be 1, and thus, gives us the total distance travelled as:
We can verify our the sum by plugging in x as 2 and 5 for instance:
Notice that the total distance is twice the initial distance, x, when starting to move! Thus, we can conclude that to travel across any distance x, we must travel twice the distance 2x to reach our destination point B assuming it’s stationary. However, what if point is not stationary? In this case, point B will also have traveled twice the desired distance by the time point A reaches point B:
For example, let us go back to when x = 1. Assuming that A starts where displacement is equal to 0 metres, A had travelled in total 2 metres from its initial point to B’s initial point.
Now since B’s displacement was already 2 metres to the right, when B starts to move towards any desired distance, it will only end up travelling an extra 2 metres landing a net 4 metres away from A’s initial point.
Therefore, mathematically, assuming that both A and B move at the same speed, it is impossible for A to catch up to B. This should make sense in reality as well, as the net displacement between point A and B will always remain constant, which in this context will be 2 metres. However, what if B was travelling slower than A? This brings up the Achilles and Tortoise paradox, which follows the same concept of the tortoise travelling much slower similar to object A. Although travelling slower, due to the net displacement, the achilles can never catch up to the tortoise. This is almost impossible in reality as we have seen in different applications where faster objects do end up catching slower objects, for example, in motor racing. Nevertheless, we can express this weird phenomenon with our trusty mathematics!
Let the following variables have the definitions. Let y1 be the speed of A (denoted by vA) and y2 be the speed of B (also denoted by vB):
For simplicity, y2 will be half of y1:
When B reaches its “final” point, at that time, t, the point had traveled 2 metres. However, if A travels to the same final point, it would have traveled twice the distance B travels to keep the net displacement 2 metres. Now since we want A to move faster, we can assume the speed of B to be slower by a factor of half.
For instance, by using simple 2D kinematics equations, we obtain the times of both objects when they reach their respective definitions to be:
(Point A)
(Point B)
Notice that the times of points when they both travel 4 metres displaced from the left are the same. We can further generalize this with experimentation:
Let d equal to the total distance for Point B to travel. Hence, point A has to travel 2d to reach the same point as B.
(Point A)
(Point B)
Setting both times equal to each other we obtain:
We can further explore this relation when the ratio between v1 and v2 is not half. Let us begin by assuming a 1:4 ratio between the two velocities of both points.
Setting both distances equal to each we obtain:
Let’s try with a velocity ratio of 1/3:
Notice how the ratio of half does not hold. As a result, it can be concluded that point A and point B can only meet when the constant velocities between the two have a ratio of 1:2. Moreover, in terms of motion, there isn’t an illusion that things in this world are motionless, but rather, if the ratio of distances or displacements over time is half, we must travel twice the distance than what it may appear to be. We can also find a similar relation with displacements!
Let us assume that instead of talking half a step, we take a 1/5th of a step:
Similar to the introduction, we can form a geometric sequence:
This sequence will give us the distance travelled on a step. Further since 0.2 < 1, we can find the sum of the converging series or the net displacement:
Therefore, for the initial distance, Point A must travel 5/4 metres to reach point B. Furthermore, for point B, assuming it needs to travel 1 metre, it also realistically has to travel 5/4 metres as well. Bringing back our displacement concept from before, for point A to reach the “final” destination at the same time for b, it has to travel twice the distance compared to the distance point B has to travel. But once again, what if include velocities?
(Point A)
(Point B — *assuming it is travelling 1/2 times slower than A*)
Setting both equations equal to each other:
Notice how we come back with the same 1:2 ratio of velocities when the displacement also share the same ratio. It can further be said that the net displacement holds.
Now imagine that instead of A travelling twice the distance, it travels 5/4 times more than the original distance.
Here, it does not follow the same 1:5/4 ratio of displacements, and therefore, does not indicate that the total distance needed to travel is 5/4. Though, we can now try to imitate the same 1:5/4 ratio with our velocities, assuming point B is 5/4 times slower than A.
(Point A)
(Point B)
Setting both times together we obtain:
This gives an interesting result. The velocity ratio becomes the same as the displacement ratio compared to the previous example! As a result, it is a very interesting pattern to see that the convergence of the distance at x = 1 is the same ratio for velocities for two objects with the same behaviour. If however, x is not equal to 1, the convergence or the ratios at x = 1 only have to be multiplied by the desired constant:
where n is our desired constant and c is the convergence of distances at x = 1.
These methods discussed in this paper are only relative to the original paradox. More specifically, the concepts considered are only relevant to point A and B when stationary, when one is moving, and when potentially both are moving. In general, for two objects A and B that have the same motion, in order to meet, the distance and velocity ratios have to be equal to match the net-displacement:
where d is distance and Sn is the sum of a distances. Since the distance ratio and velocity ratio are the same:
where v is our velocity, holds true for two objects to meet.
In the end, Zeno’s Paradox is indeed wrong, which has been proven mathematically and scientifically. Even if motion takes an “infinite” amount of steps, object A will travel to their desired point in some interval. However, it should be noted that Zeno’s Paradox does provide some valuable insights that should be built upon. As I stated before, Zeno believed that motion is an illusion, and he’s right, but for another reason. One of the reasons I showed to you above where the total distance needed to travel is twice the initial total distance. Another reason is how the velocity and displacement ratios have to be the same for two objects to meet under such circumstances provided by the dichotomy. We should not ignore such paradoxes that we may consider “irrelevant” today, because such thoughts can create unique lens to look at the world through. I myself was astonished when I figured this result, and I hope you are too!
