Paradoxes of Zeno of Elea

Zenon of Elea is a Greek logician and philosopher, who is mostly known for the paradoxes named in his honor. Little is known about his life. The hometown of Zeno is Elea. Also in the writings of Plato mentioned the meeting of the philosopher with Socrates.

Around 465 BC. E. Zeno wrote a book detailing all his ideas. But, unfortunately, it has not reached our days. According to legend, the philosopher died in battle with a tyrant (presumably, the head of Elea Neharhom). All the information about Eleisk was gathered bit by bit: from the works of Plato (born 60 years after Zeno), Aristotle and Diogenes Laertius, who wrote three centuries later a biography book of Greek philosophers. There are also references to Zenon in the works of the late representatives of the school of Greek philosophy: Themistia (4th century AD), Alexander Afrodijsky (3rd century AD), as well as Filopon and Simplicius (both lived in the 6th century AD) . And the data in these sources are so well aligned that they can reconstruct all the ideas of the philosopher. In this article we will tell you about the paradoxes of Zeno. So, let's get started.

Paradoxes of the set

Since the era of Pythagoras, space and time have been considered exclusively from the point of view of mathematics. That is, it was believed that they are composed of many points and points. However, they have a property that is easier to sense than to define, namely, "continuity". Some of the paradoxes of Zeno prove that it can not be divided into moments or points. The philosopher's reasoning boils down to the following: "Let's assume that we have divided to the end. Then only one variant of the two is correct: either we get in the remainder the minimally possible quantities or parts that are indivisible but infinite in their quantity, or division will lead us to parts without size, since continuity, being homogeneous, must be divisible under any circumstances . It can not be in one part of a dividend, and in another - no. Unfortunately, both results are quite ridiculous. The first is due to the fact that the process of division can not end, while in the remainder there are parts that have a value. And the second is because in such a situation the whole would have been formed from nothing. " Simplicius attributed this argument to Parmenides, but it is more likely that his author is Zeno. We go further.

Zeno's Paradox of Motion

They are dealt with in most of the books devoted to the philosopher, since they come into dissonance with the evidence of the feelings of the Eleatics. With reference to motion, the following paradoxes of Zeno stand out: "Arrow", "Dichotomy", "Achilles" and "Stages". And they reached us through Aristotle. Let's look at them in more detail.

"Arrow"

Another name is the quantum paradox of Zeno. The philosopher asserts that any thing either stands still or moves. But nothing stays in motion, if the occupied space is equal to it in length. At a certain moment, the moving boom is in one place. Therefore, it does not move. Simplicii formulated this paradox in a brief form: "A flying object occupies an equal place in space, and that which occupies an equal place in space does not move. Therefore, the arrow rests. " Femystia and Felopon formulated similar options.

"Dichotomy"

He takes the second place in the list of Zeno's Paradoxes. It reads: "Before an object that has started moving can pass a certain distance, it must cross half of this path, then half of the remaining path, and so on, to infinity. Since, with repeated division of the distance in half, the segment becomes finite all the time, and the number of given segments is infinite, then this distance can not be overcome in a finite time. Moreover, this argument is valid both for small distances and for high velocities. Therefore, any movement is impossible. That is, the runner will not even be able to start. "

This paradox is very detailed comment Simplicius, indicating that in this case for a finite time to make an infinite number of touches. "Anyone who touches anything can count, but an infinite number can not be counted or counted." Or, as Filopon put it, the infinite set is indefinable.

Achilles

It is also known as the paradox of the Zeno turtle. This is the philosopher's most popular reasoning. In this paradox of movement, Achilles competes in running with a turtle, which at the start is given a small handicap. The paradox is that the Greek warrior will not be able to catch up with the turtle, since he will first run to the point of its start, and she will be at the next point. That is, the tortoise will always be ahead of Achilles.

This paradox is very similar to a dichotomy, but here the infinite division is consistent with the progression. In the case of the dichotomy, there was regression. For example, the same runner can not start, because he can not leave his location. And in the situation with Achilles, even if the runner moves from the spot, he still does not run anywhere.

"Stages"

If we compare all the paradoxes of Zeno on the degree of complexity, then this one would be the winner. It is more difficult than the others to set forth. Simplicius and Aristotle described this argument fragmentarily, and one can not rely 100% on its reliability. The reconstruction of this paradox has the following form: let A1, A2, A3 and A4 be fixed bodies of equal size, and B1, B2, B3 and B4 are bodies of the same size as A. B bodies move to the right so that each B And for an instant, which is the smallest time interval of all possible. Let B1, B2, B3 and B4 be bodies identical to A and B, and move relative to A to the left, overcoming each of the bodies in an instant.

It is obvious that B1 has overcome all four bodies B. We take as unit the time required for one body B to pass one body B. In this case, all the movement required four units. However, it was believed that the two moments that passed for this movement are minimal and, therefore, are indivisible. It follows that four indivisible units are equal to two indivisible units.

"A place"

So, now you know the basic paradoxes of Zeno of Elea. It remains to tell about the latter, which is known as the "Place". Aristotle attributes this paradox to Zeno. Similar arguments were cited in the works of Philopon and Simplice in the 6th century AD. E. Here is how Aristotle talks about this problem in his Physics: "If there is some place, then how to determine where it is located? The difficulty to which Zeno came, requires an explanation. Since everything that exists does take place, it becomes obvious that both the place must have a place, etc., to infinity. " In the opinion of most philosophers, the paradox appears here only because nothing existing can be different from itself and contained in itself. Filopon believes that, focusing on the self-contradiction of the concept of "place", Zeno wanted to prove the inconsistency of the theory of multiplicity.