Zeno of Elea was a Greek philosopher who lived about four centuries before Christ. He is known for his paradoxes, some of which deny the existence of motion. Zeno attempted to prove that space is not made up of discontinuous elements and, in particular, that movement does not exist. The paradox of Achilles and the tortoise, and the arrow paradox, have survived to this day and continue to torment philosophy students.
Not much is known about the life of Zeno of Elea, since very few fragments remain and almost all of them are indirect references from other authors who mention facts or situations attributed to him. The date of his birth is also uncertain, but it is accepted that he was born between 490 and 485 BC. Zeno's reasonings, who was a disciple of Parmenides and had a Pythagorean education, constitute the first attempt at infinitesimal thought, which was only developed in depth about two thousand years later by Leibniz and Newton.
Zeno's paradoxes are a series of paradoxes designed to demonstrate that the sensations we obtain from the world are illusory and movement does not exist. Zeno repeated to anyone who would listen that a person could not cover any distance at all, even though the senses showed that such a thing was possible. His ideas belong to the category of false paradoxes, also known as sophisms. This means that they reach a result that is false, due to a fallacy in reasoning, generally the product of a lack of knowledge about the concept of infinity in the era in which they were formulated.
Achilles and the Tortoise
The Achilles and the tortoise is perhaps the most famous of Zeno's paradoxes. The philosopher argued that in a hypothetical race between Achilles (the warrior who killed Hector) and a tortoise, if the tortoise had an initial lead, the human would always lose. Zeno "demonstrated" that, although the warrior runs much faster than the tortoise, he could never catch up.
Let us imagine that the distance to cover in the race is a hundred meters, and that the tortoise has a fifty-meter head start. At the start signal, Achilles quickly covers the distance (fifty meters) that initially separated them. But upon arriving, he discovers that the tortoise is no longer there, but has advanced, much slower, ten or twenty centimeters. Far from being discouraged, the warrior keeps running. But, upon arriving again where the tortoise was, it has advanced a little more. Zeno maintains that this situation repeats indefinitely, and that Achilles will never catch the tortoise, which will finally win the race.
It is quite obvious that this is not so, and it is very easy to verify in practice that this reasoning is erroneous. However, it is not so easy to find where the flaw is, and it was not until the mid-17th century that the Scottish mathematician James Gregory mathematically demonstrated that a sum of infinite terms can have a finite result. The times in which Achilles covers the distance that separates him from the previous point where the tortoise was are infinite, but each time smaller and smaller. The sum of all these times, despite their infinite number, results in a finite span of time, which is the moment when Achilles will catch up with the tortoise.
Another way to approach the problem is to avoid infinitesimal analysis using discrete analysis. We can think that Achilles does not cover infinitesimal spaces, but discrete ones, which we can call "strides". Each stride corresponds to a specific distance of, for example, one meter. That way, the problem is reduced to calculating when the last stride of Achilles will cover a distance greater than what the tortoise could cover in the same time. In this way it can be demonstrated that, as we know today, movement exists.
The Arrow Paradox
This other paradox involves the launching of an arrow. Zeno claimed that, at each instant, the arrow is in a determined position in space. If the time period considered is small enough, the arrow will not have time to move, so it is at rest during that instant. Now, the same reasoning can be applied to the remaining infinite time periods, in which the arrow will also be at rest for the same reason. In this way Zeno demonstrated that the movement of an arrow is impossible, despite the fact that thousands of widows whose husbands had died of an arrow wound on the battlefield insisted on the contrary.
The paradox can be avoided in several ways. One of them is simply to think that each instant in which the arrow is perceived as "at rest" is something relative. One cannot judge, by looking only at a "photo" of an object, whether it is at rest or not. Instead, it is necessary to compare it with adjacent instants, previous and later. By seeing the "movie", we can determine that the arrow is in different positions at each instant, so it is indeed moving.
Another solution is to resort to the definition of speed, whose essence is change. Movement is the succession of the different spaces occupied by the body (the arrow), along the succession of the different moments that make up the total time considered. Thus, if we assume that the concept of speed, that is, movement, can be defined rationally, simultaneously we are admitting that movement, rationally, in theory, exists.
Twenty-five centuries later, what Zeno proposed seems ridiculous to us. However, if we think about the state of science four or five centuries before the Christian era, the reasonings of this man acquire real magnitude. It must have been extremely difficult for Zeno to reconcile the (apparent) certainty behind his reasonings with the evidence of the real world, where the tortoise was invariably the loser of the race, and arrows reached their target.
