In 400BC, Democritus a Greek mathematician devised a way to solve mathematical problems. His idea involved using infinitesimals i.e. little packets of distance or time without limits, as a solution. Unknown to him at the time, this approach would serve as the groundwork for Isaac Newton to build and develop calculus into what we know it today. However, when Democritus shared his idea with his peers and colleagues among which was Zeno of Elea, it was totally rejected. Not comfortable with the notion, Zeno came up with a plan on how to defeat the study of using infinitesimals. That idea became a series of paradox that employed the use of infinitesimals itself. Today we know them as Zeno’s Paradox.
Who was Zeno?
The background details of Zeno are sketchy and contain little information. The only source that chronicles his life on earth is in the first pages of Plato’s book, Parmenides. It is from there we learn that Zeno was born in Elea, southern Italy in 490BCE. He was a student and friend of Parmenides who was significantly older than him and also came from his town, Elea. Zeno was about 40years of age when Socrates was about 20. He was never a mathematician and died in 430BCE.
What is Zeno’s Paradox?
According to Plato, Zeno wrote a book on paradoxes as a means of supporting the philosophy of Parmenides. It is from there we attempt to define Zeno’s paradox.
In the simplest term, Zeno’s paradox states that two objects can never touch each other. For instance, if a ball is kept in a place and a second ball is put in motion to approach it, there must be a halfway point for the moving ball to cross before it gets to the stationary ball. But since there are an unlimited number of halfway points, the two balls can never meet. This scenario becomes an apparent paradox because in the real world two objects can touch, but Zeno discredited this idea using mathematics.
All the paradoxes Zeno created revolved around this concept that states that before a result can be seen, an infinite number of conditions or points must be satisfied or crossed. Also, the result that would be gotten must not happen in a time that is less than infinite.
Now that you have an idea of Zeno’s paradox, we would now look at the things you ought to know about them:
Paradoxes of Motion
- The dichotomy: This paradox is named dichotomy because it involves continuous division into two. Using the example of a runner, Zeno stated that he would never get to a goal line that is stationed in a race track. The reason is that the runner must reach half of the distance to the goal line first. But on arriving there, he must still cross half of the remaining distance to the goal line. And yet again, there is still half of the new remaining distance to pass, and so it continues. Therefore, according to Zeno, the runner will never get to the final destination. Why? Using Zeno’s philosophy, this is because the distance is too far, making the sum infinite. This paradox concludes that motion of any sort is impossible.
- Achilles and the tortoise: Imagine that Achilles, the fastest runner in ancient Greece, is chasing a tortoise. Both of them are moving at a constant speed on linear paths. To catch the tortoise, Achilles will have to get to the point the tortoise is at presently. However, before Achilles gets there, the tortoise will have gone to a new point. Achilles will then try to reach this new point. By the time he arrives, the tortoise has already moved on to another location. And so it continues till infinity. With this example, Zeno argued that anyone who thinks Achilles will eventually meet up with the tortoise and also believe that motion is possible is a victim of illusion.
- The arrow: The arrow paradox challenges our collective knowledge of motion and time. Using Zeno’s theory that time is made up of moments, Aristotle explains that an arrow on the move must occupy a space that is equal to itself at any given moment. Therefore, if the arrow is filling a space that is equal to itself in each moment, it means that the arrow is stationary and not moving in that moment. The reason for this is that the arrow has no time in which it can move, it is just at one place. It also cannot move at the moment because an even smaller unit of time will be required thereby making the moment indivisible.
- The stadium: This scenario is Zeno’s final paradox of motion. Consider two rows of bodies. Each is made up of an equal number of bodies that are also equal in size. These bodies pass each other with equal velocities in opposite directions. Therefore, according to Zeno, half of a time is equal and the same to the whole time.
The Paradox of Place
This assumption was very common in ancient Greece during Zeno’s time. When we are given an object, we naturally assume that we know its place. The reason is that everything on earth has a place. And since place itself exists as an entity, it too must have its own place. And this would, in turn, have its place, and so it continues ad infinitum. However, the many existences of many places lead to a contradiction.
The Grain of Millet Paradox
This paradox supports the Parmenidean theory that our senses are unreliable. Consider a bag of millet that is dropped on the floor and produces a loud sound. But this noise is generated by every grain of millet that is in the bag. And the result of the noise is also made by every part of each of the grains. Therefore, every part of the grains makes a sound when it falls to the ground. On the other hand, when you drop a tiny part of a grain on the floor, it produces no sound. Thus, this picture brings us to the conclusion that our sense of hearing is selective and deceptive and so we should not trust it.
We have gone through the history of Zeno’s paradox, how it came about. And we have also given you information on what it involves. Though modern science has refuted almost all of Zeno’s paradox, it has not ceased to be a source of inspiration to scholars.
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