Life

Zeno was the founder of the school of Stoicism. He was born in Cyprus around 336 BC. e. It is believed that Zeno was Phoenician by origin. He died in Athens in 264 BC. e. His father was engaged in trade, and Zeno himself apparently worked with him for some time. He came to Athens at the age of 20. Zeno read the works of Plato and Xenophon, which contain information about Socrates, and was very impressed by their memories of this great philosopher. In particular, he admired the fortitude shown by Socrates at trial, his immense calm during the death sentence, his disdain for luxury and indifference to earthly goods.

Zeno was also attracted to the philosophy of the Cynics. Nevertheless, he studied with many wise Athenian philosophy teachers and eventually founded his own school of philosophy at the age of 35. Only minor fragments of his works have survived and reached us. The word Stoicism comes from the Greek “stoa,” meaning “portico.” Zeno taught his followers in Athens under the shadow of a gallery called the stoa poilcile, or painted portico. Stoicism has had a long and varied history. Later it was easily adopted by Roman philosophers.

We know very little about Zeno's personal life. According to the information we have, he committed suicide.

Thoughts Unlike Epicureanism, Stoicism has had a large number of different variants throughout its history; within the framework of Stoicism, various philosophical directions and currents. The original form of the teachings of the Stoics changed quite a lot; there was practically no solid framework limiting these changes. Discussions of the development processes of various directions of Stoicism can actually be separated into a separate topic. For us, it will be quite sufficient to consider the foundations of Zeno’s teaching.

Materialism

In his reasoning, Zeno did not rush into metaphysical abstractions. He was a materialist and never doubted what his senses told him. The real world was tangible and material. According to Zeno's ideas, God, virtue and justice also existed in the world. Everything listed above was tangible, obvious and material. This seems a little strange. But despite this, we simply have to conclude that Zeno was a materialist, and any attempts from the outside aimed at convincing him of his own wrongness found worthy answers. However, for a philosopher such things were completely unimportant.

Stoicism's teaching on physics currently has no of great importance. However, studying the Stoics' ideas about physics can allow us to understand how the people of antiquity thought. According to the Stoics, initially there was only one element in the world, fire, all the others (air, water, earth) arose later. One of the important concepts in Zeno’s teaching is cosmic determinism. Cosmic determinism assumes that everything that happens in the world is determined by the strict laws of existence; everything that has already happened will happen again - all events, phenomena and processes move cyclically, this cycle is endless.

The philosophy of Stoicism does not go into metaphysical reflections; its essence is in no way connected with the theory of knowledge; this philosophy simply advises people on how they should live safely and with dignity. The metaphysics and logic of Stoicism in their original versions, of course, were not preserved; they underwent very significant changes. However, the ethics of Stoicism remained virtually unchanged throughout its history.

Virtue in Stoicism

Stoicism, like the philosophy of Epicurus, assumes that human life takes place in a changing and decaying world. The old world familiar to the ancient Greeks was becoming a thing of the past; power passed into other hands. The time of existence of the Greek city-state, within which all people were included in a small community with its own unique identity, has expired. City-states lost their independence; at different periods of time they were part of huge empires. Alexander the Great was the first person in the West (of course, huge empires arose in the East much earlier) to create a magnificent empire. The teaching of Stoicism in these circumstances was aimed at showing a person the need to remain indifferent to any external influence.

The Stoics said that everything in the world is controlled by something all-seeing and all-knowing. Everything that happens has a specific purpose, which is somehow connected with humanity. The omniscient principle is God; he is also the soul of the world. Each of us is gifted with sparks of Divine fire. Any human life good and prosperous when it does not contradict its essence, the nature that determined its appearance. But, on the other hand, every person must obey this nature. Virtue in this case consists in the subordination of the human will to the framework of existence established by nature. "Virtue", however, is a word whose meaning has undergone significant changes over time. When using this word, the ancient Greeks meant the real embodiment of a person’s positive qualities.

Determinism and freedom

Now we see that the main idea of ​​the philosophy of Stoicism is associated with both determinism and human freedom. Only virtue is the only element that is present in the life of every person. Health, wealth, the desire for pleasure - all this is secondary and is not necessarily included in the life priorities of absolutely every person. Virtue lies in the human will. A person may be poor, sick, persecuted by society, but all these external circumstances can only have an external impact on him. Consequently, every person has absolutely complete freedom, but he has it as long as he protects himself from false desires and shows indifference to them. None external force cannot take away a person’s morality, that is, virtue.

Thus, Stoicism teaches us to be indifferent to all external factors that influence us: good and evil depend on the person himself, on his will. If someone can understand how to be indifferent to the events happening around them, these events will lose their power over him and will not be able to have any influence on him. Only human will can be good or bad. Stoicism asserts that responsibility for good or evil lies entirely with the individual. Society cannot be blamed for the fact that a person existing in it becomes good or bad.

Indifference stoicism

Stoicism was a cold philosophy. His ethics are the ethics of indifference. Stoicism not only had a negative attitude towards all human passions, but even condemned them. The duty of a person, approved by Stoicism, is to participate in public life with the aim of spreading goodness, courage, determination, and maintaining justice in society. A person must do this properly to be virtuous. However, the ideas of helping people in need, achieving general happiness, or creating a creative, active and strong society do not fit into this picture.

However, many people who adhered to the philosophy of Stoicism were not only kind, generous and humane, but they also devoted their entire lives to serving their society. Among such personalities, the famous Roman writer Seneca (3 BC - 65 AD) and the Roman emperor Marcus Aurelius (121-180) should be especially highlighted.

conclusions

Is it possible to criticize Stoicism because of its adherence to opposing concepts - free will and determinism? On the one hand, Zeno’s teaching is based on world, or more precisely, cosmic determinism. On the other hand, the philosopher argued that virtue is the result of the embodiment of human will. This problem - the problem of the mutual exclusivity of free will and determinism - manifests itself not only in Stoicism. This is one of the most difficult problems of philosophy in general, ethics and Christian theology. For a long time it remains virtually unresolved.

Stoicism tells us that everything that happens in the world - a falling leaf, a collision between two trains, the decision to go to war in Iraq - is inevitable and predetermined. We cannot change certain circumstances. Consequently, it is not possible for us to change our own character, to restrain ourselves from committing certain actions, to change in better side. Does this match your own ideas? If not, what reasonable objection do you have to this?

It should also be noted that modern psychology concerns a similar problem. Moreover, quite often this problem arises in various cases related to the application of criminal law. Let's imagine that all our actions are really predetermined. In this case, in response to Joe Bloggs killing his neighbor, a true Stoicist must say that this event was inevitable and determined by Joe's past, the environment in which he grew up, his heredity, the situation in which he found himself at that fateful moment. A Stoicist will, of course, also say that the killer was not responsible for his actions. In other words, despite the outward appearance of freedom, Joe Bloggs was still not free. And none of us are free. However, if we touch on the concept of free will, we come to slightly different conclusions. In addition, the word “freedom” is used in the previous paragraph in different meanings. If the words “freedom” and “free” mean the ability to think and act differently in individual situations, the arguments given above are somewhat flawed.

When I say that I am "free", what do I mean by this expression? I can say that I am free now, at this moment in time, and I can, for example, go buy a package of chocolates. I am free to do this because I can afford to do it and because I have free time to do it. Thus I have freedom. But, on the other hand, I, for example, cannot fly. I don't have wings. It turns out I'm not free. I am not free to kill my neighbors: laws and moral principles warn me against committing such acts. Therefore, I am not free. So, in the end, am I free or not? Based on the general meaning of all that has been said, it should be concluded that not every person can violate the rules of law or morality. Can you think for yourself about other meanings of the word “free”? In what cases is it appropriate to use these values?

It seems that Stoicism's theory of indifference is a little lacking in common sense. Is it right to learn indifference and at the same time not strive to improve others? human feelings and advantages? Is it right to be indifferent, for example, when those we love and value are going through difficult times? Maybe in this case, even a wrong action, for example, theft, will become “moral” if it was done “indifferently”, if we did not invest any emotions and mental experiences into it, if we did not derive a certain benefit from it in our favor?

The philosophy of Stoicism appears to apply to emergency situations. Is this why philosophy may well turn into generally accepted ethics? Indifference, figuratively speaking, can protect us if we find ourselves in an unusual, extremely difficult situation, for example, being among the hostages. On the other hand, is it entirely correct in such a situation to simply be indifferent to what happens? Can it be called a moral success, an achievement, if people just sit at home and do not worry at all about their wives, husbands, children, friends and relatives? Should they be in a state of complete indifference towards people who are held hostage?

Be that as it may, there is a very interesting idea here, but not noticeable at first glance. If the people who are responsible for dealing with an emergency situation are not emotionally indifferent to what happened, they will not be able to act fully rationally. They must demonstrate incredible performance, composure, restraint and self-discipline in a difficult environment. It is worth noting that this applies to many everyday situations that police officers, firefighters, and medical workers often encounter.

On the other hand, if everything is good in life, it seems stupid to be indifferent, because such moments should be enjoyed.

Stoicism is a philosophy of consolation. Even the Apostle Paul, while in prison, exclaimed: “I have learned to be content with life, no matter what state I am in.” He said it like a truly brave stoic.

Supposedly ok. 465 BC he outlined his ideas in a book that has not reached us. According to tradition, Zeno died in the fight against a tyrant (probably the ruler of Elea, Nearchus). Information about him has to be collected bit by bit: from Plato, who was born 60 years later than Zeno, from the messages of Plato’s student Aristotle, from Diogenes Laertius, who in the 3rd century. AD compiled biographies of Greek philosophers. Zeno is also spoken of by later commentators of the Aristotelian school: Alexander of Aphrodisias (3rd century AD), Themistius (4th century), Simplicius and John Philoponus (both 6th century). In most cases, these sources agree so well with each other that Zeno's views can be reconstructed from them.

Historical surroundings. To appreciate the role of Zeno in the history of science and the development of logic, it is necessary to consider the state of Greek philosophy in the mid-5th century. BC. Ionian philosophers from Asia Minor sought the origin of all things, the basic element from which the Universe was formed. Each one settled on its own element: one assigned this role to water, another to air, the third to the qualityless “limitless” or “indefinite” (apeiron). The Ionians believed that all types of matter known to us arise as a result of continuously occurring processes of compression, rarefaction and condensation of the basic element. This constant change was emphasized by Heraclitus of Ephesus (6-5 centuries BC): the river we enter now is not the same as it was yesterday; everything changes; The harmony of the Universe is the harmony of opposites. Finally, the school founded by Pythagoras (6th century BC) put forward number as the main element, and numbers were considered as discrete units endowed with a spatial dimension.

Zeno's teacher Parmenides criticized all these theories - both the monism of the Ionians and the pluralism of the Pythagoreans. When we examine any basic element, we can make one of three statements about it: it exists; He does not exist; it both exists and does not exist. The third statement is internally contradictory, the second is also unthinkable, since it is impossible to talk about the absence of something using the same terms that were used to describe it. The existence of nothingness is impossible to even imagine. Therefore this element exists. Change is impossible, since this would mean that the primary element was not distributed with equal density everywhere, and there cannot be a void, since this would be a place in which the primary element does not exist. So, the Universe is a motionless, unchanging, dense and uniform ball. Everything is One.

Note that Parmenides comes to this conclusion solely with the help of logic, without resorting to speculation or intuition, characteristic of the systems of his predecessors. If the conclusion contradicts the feelings, so much the worse for the feelings: appearances are deceptive.

Zeno continued the work begun by Parmenides. His tactics amounted not to defending the teacher's point of view, but to demonstrating that even greater absurdities arose from the statements of his opponents. In this regard, Zeno developed a method of refuting opponents through a series of questions. In answering them, the interlocutor was forced to come to the most unusual paradoxes, which necessarily followed from his views. This method, called dialectical (Greek “dialegomai” - “to talk”), was subsequently used by Socrates. Since Zeno's main opponents were the Pythagoreans, most of his paradoxes are associated with the atomistic concept of Pythagoreanism. Therefore, they are especially significant for modern atomic theories of number, space, time and matter.

Paradoxes of the multitude. Since the time of Pythagoras, time and space have been viewed, from a mathematical point of view, as being composed of many points and moments. However, they also have a property that is easier to sense than to define, namely “continuity.” With the help of a series of paradoxes, Zeno sought to prove the impossibility of dividing continuity into points or moments. His reasoning boils down to the following: suppose that we have carried out the division to the end. Then one of two things is true: either we have in the remainder the smallest possible parts or quantities that are indivisible, but infinite in quantity, or division has led us to parts that have no quantity, i.e. turned into nothing, for continuity, being homogeneous, must be divisible everywhere, and not so that in one part it is divisible and in another not. However, both results are absurd: the first because the process of division cannot be considered complete while the remainder contains parts with magnitude, the second because in this case the original whole would be formed from nothing.

Simplicius attributes this reasoning to Parmenides, but it seems more likely that it belongs to Zeno. For example, in Aristotle’s Metaphysics it is said: “If the one is in itself indivisible, then, according to Zeno, it must be nothing, for he denies that that which does not increase with addition and does not decrease with subtraction could exist at all - of course, according to that reason that everything that exists has spatial dimensions." In a more complete form, this argument against the multiplicity of indivisible quantities is given by Philoponus: “Zeno, supporting his teacher, tried to prove that everything that exists must be one and immovable. He based his proof on the infinite divisibility of any continuity. Namely, he argued, if existence does not will be one and indivisible, but can be divided into many, essentially there will be no one at all (for if continuity can be divided, this will mean that it can be divided ad infinitum), and if nothing is essentially one, plurality is impossible, since a set is made up of many units.

So, existing cannot be divided into many, therefore, there is only one. This proof can be constructed in another way, namely: if there is no being that is indivisible and one, there will be no set, for the set consists of many units. But each unit is either one and indivisible, or itself is divided into many. But if it is one and indivisible, the Universe is composed of indivisible quantities, but if the units themselves are subject to division, we will ask the same question regarding each of the units subject to division, and so on ad infinitum. Thus, if existing things are multiple, the universe will appear to be composed of an infinite number of infinities. But since this conclusion is absurd, existence must be one, but it is impossible for it to be multiple, because then each unit would have to be divided an infinite number of times, which is absurd.”

Simplicius attributes to Zeno a slightly modified version of the same argument: "If a set exists, it must be exactly what it is, neither more nor less. However, if it is what it is, it will be finite. But if a set exists, things are infinite in number, because among them there will always be others, and between them more and more. Thus, things are infinite in number."

The argument about plurality was directed against a school rival to the Eleatics, most likely the Pythagoreans, who believed that magnitude or extension was composed of indivisible parts. Zeno believed that this school believes that continuous quantities are both infinitely divisible and finitely divided. The limiting elements, of which the set was supposed to consist, had, on the one hand, the properties of a geometric unit - a point; on the other hand, they possessed some properties of numerical unity - numbers. Just as a number series is constructed from repeated additions of one, a line was considered to be composed by repeated addition of point to point.

Aristotle gives the following Pythagorean definition of a point: “A unit having a position” or “A unit taken in space.” This means that Pythagoreanism adopted a kind of numerical atomism, from the point of view of which the geometric body does not differ from the physical body. Zeno's paradoxes and the discovery of incommensurable geometric quantities (c. 425 BC) led to the emergence of an insurmountable gap between arithmetical discreteness and geometric continuity. In physics, there were two somewhat similar camps: atomists, who denied the infinite divisibility of matter, and followers of Aristotle, who defended it. Aristotle again and again resolves Zeno's paradoxes for both geometry and physics, arguing that the infinitesimal exists only in potential, but not in reality. For modern mathematics, such an answer is unacceptable. Modern analysis of infinity, especially in the works of G. Cantor, has led to a definition of continuum that deprives Zeno's antinomy of paradox

Paradoxes of movement. A significant part of the extensive literature devoted to Zeno examines his proof of the impossibility of movement, since it is in this area that the views of the Eleatics come into conflict with the evidence of the senses. Four proofs of the impossibility of movement have reached us, called “Dichotomy”, “Achilles”, “Arrow” and “Stages”. It is not known whether there were only four of them in Zeno’s book, or whether Aristotle, to whom we owe their clear formulations, chose those that seemed to him the most difficult.

Dichotomy. The first paradox states that before a moving object can travel a certain distance, it must travel half that distance, then half the remaining distance, and so on. to infinity. Since when a given distance is repeatedly divided in half, each segment remains finite, and the number of such segments is infinite, this path cannot be covered in a finite time. Moreover, this argument is valid for any distance, no matter how small, and for any distance, no matter how small. high speed. Therefore, any movement is impossible. The runner is unable to even move. Simplicius, who comments on this paradox in detail, points out that here it is necessary to make an infinite number of touches in a finite time: “Whoever touches something seems to be counting, but an infinite number cannot be counted or enumerated.” Or, as Philoponus puts it, “the infinite is absolutely indefinable.” In order to traverse each of the divisions of extension, a limited time interval is necessarily required, but an infinite number of such intervals, no matter how small each of them, cannot together produce a finite duration.

Aristotle saw the “dichotomy” as a fallacy rather than a paradox, believing that its significance was negated by the “false premise ... that it is impossible to pass or touch an infinite number of points in a finite period of time.” Themistius also believes that “Zeno either really does not know or pretends when he believes that he managed to put an end to motion by saying that it is impossible for a moving body to pass through an infinite number of positions in a finite period of time.” Aristotle considers points to be only potential, and not actual being; the time or space continuum “in reality is not divided to infinity,” since this is not its nature.

Achilles. The second paradox of motion examines a race between Achilles and a tortoise, which is given a head start at the start. The paradox is that Achilles will never catch up with the tortoise, since first he must run to the place where the tortoise begins to move, and during this time it will reach the next point, etc., in a word, the tortoise will always be ahead. Of course, this reasoning resembles a dichotomy with the only difference that here the infinite division proceeds in accordance with progression, and not regression. In "Dichotomy" it was proven that the runner cannot set off because he cannot leave the place in which he is; in "Achilles" it is proven that even if the runner manages to set off, he will not run anywhere. Aristotle objects that running is not a continuous process, as Zeno interprets it, but a continuous one, but this answer returns us to the question, what is the relation of the discrete positions of Achilles and the tortoise to the continuous whole?

The modern approach to this problem is through calculations (either by the method of convergent infinite series or by simple algebraic equation), which establishes where and when Achilles will catch up with the tortoise. Suppose Achilles runs ten times faster than a tortoise, which moves 1 m per second and has a lead of 100 m. Let x be the distance in meters the tortoise has covered by the time Achilles catches up, and t be the time in seconds. Then t = x/1 = (100 + x)/10 = 111/9 s. Calculations show that the infinite number of movements that Achilles must make corresponds to a finite segment of space and time. However, calculations alone cannot resolve the paradox. After all, you first need to prove the statement that distance is speed multiplied by time, and this is impossible to do without analyzing what is meant by instantaneous speed - the concept underlying the third paradox of motion.

Most sources that present paradoxes say that Zeno denied the possibility of motion altogether, but sometimes it is argued that the arguments he defended were aimed only at proving the incompatibility of motion with the idea of ​​continuity as a multitude that he constantly challenged. In "Dichotomy" and "Achilles" it is argued that movement is impossible under the assumption of the infinite divisibility of space into points, and time into moments. The last two paradoxes of motion state that motion is equally impossible when the opposite assumption is made, namely, that the division of time and space ends in indivisible units, i.e. time and space have an atomic structure."

Arrow. According to Aristotle, in the third paradox - about the flying arrow - Zeno states: any thing either moves or stands still. However, nothing can be in motion, occupying a space that is equal in extent to it. At a certain moment, a moving body (in this case an arrow) is constantly in one place. Therefore, the flying arrow does not move. Simplicius formulates the paradox in a concise form: “An object in flight always occupies a space equal to itself, but something that always occupies an equal space does not move. Therefore, it is at rest.” Philoponus and Themistius give options close to this.

Aristotle quickly dismissed the “arrow” paradox, arguing that time does not consist of indivisible moments. “Zeno’s reasoning is erroneous when he asserts that if everything that occupies an equal place is at rest, and that which is in motion always occupies such a place at any moment, then a flying arrow will turn out to be motionless.” The difficulty is eliminated if, together with Zeno, we emphasize that at any given moment of time a flying arrow is where it is, just as if it were at rest. Dynamics does not need the concept of a "state of motion" in the Aristotelian sense, as the realization of potency, but this does not necessarily lead to the conclusion made by Zeno that since there is no such thing as a "state of motion" there is no such thing as motion itself, the arrow is inevitable is at rest.

Stages. The most controversial one is the last paradox, known as “stages,” and it is also the most difficult to explain. The form in which it is given by Aristotle and Simplicius is fragmentary, and the corresponding texts are considered not entirely reliable. A possible reconstruction of this reasoning has the following form. Let A1, A2, A3 and A4 be motionless bodies of equal size, and B1, B2, B3 and B4 be bodies of the same size as A, which uniformly move to the right so that each B passes each A in an instant, considering an instant to be the shortest possible period of time. Let C1, C2, C3 and C4 be bodies also of equal size to A and B, which uniformly move relative to A to the left so that each C passes by each A also in an instant. Let us assume that at a certain moment in time these bodies are in the following position relative to each other:

Then after two moments the position will become as follows:

From this it is obvious that C1 passed all four bodies B. The time it took C1 to pass one of the bodies B can be taken as a unit of time. In this case, the entire movement required four such units. However, it was assumed that the two moments that passed during this movement are minimal and therefore indivisible. From this it necessarily follows that two indivisible units are equal to four indivisible units.

According to some interpretations of "stage", Aristotle believed that Zeno made an elementary mistake here, suggesting that a body takes the same time to pass a moving body and a stationary body. Eudemus and Simplicius also interpret "stages" as merely a mixture of absolute and relative motion. But if this were so, the paradox would not deserve the attention that Aristotle paid to it. Modern commentators therefore recognize that Zeno saw a deeper problem here, affecting the structure of continuity.

Other paradoxes.

Predication. Among the more dubious paradoxes attributed to Zeno is the discussion of predication. In it, Zeno argues that a thing cannot at the same time be one and have many predicates; The Athenian sophists used exactly the same argument. In Plato's Parmenides, this reasoning goes like this: "If things are multiple, they must be both like and unlike [unlike, because they are not the same, and similar, because they have in common that they are not the same ] However, this is impossible, since dissimilar things cannot be similar, and similar dissimilar things cannot be multiple.”

Here again we see the criticism of plurality and such a characteristic indirect type of proof, and therefore this paradox was also attributed to Zeno.

Place. Aristotle attributes the “Place” paradox to Zeno; similar reasoning is given by Simplicius and Philoponus in the 6th century. AD In Aristotle's Physics, this problem is stated as follows: "Further, if place exists in itself, where is it? For the difficulty to which Zeno arrives requires some explanation. Since everything that exists has a place, it is obvious that place must also take place, etc. ad infinitum." It is believed that the paradox arises here because nothing can be contained in itself or be different from itself. Philoponus adds that, by demonstrating the self-contradiction of the concept of “place,” Zeno wanted to prove the inconsistency of the concept of plurality.

Zeno of Elea (Zenon) (c. 490-after 445 BC). Born in Elea (Southern Italy). Philosopher of the Eleatic School. He was a student and friend of Parmenides.

He probably wrote only one book in which he defended the views of Parmenides. Adkins L., Adkins R. Ancient Greece

. Encyclopedic reference book. M., 2008, p. 447. Zeno of Elea (Ζήνων) (c. 490-430 BC) - ancient Greek philosopher, born in Elea (Southern Italy). Student, who developed his doctrine of the unified, excluding for sensory perception any multiplicity of things and all their movement. Since the Eleans were natural philosophers, and Greek natural philosophy was based on a spontaneous-materialistic understanding of nature, the philosophy of Zeno of Elea (as well as the ancient Eleans) is materialistic.

Philosophical Dictionary / author's comp. S. Ya. Podoprigora, A. S. Podoprigora. - Ed. 2nd, erased - Rostov n/d: Phoenix, 2013, pp. 121-122.

Zeno of Elea (c. 490-430 BC) - ancient Greek philosopher, one of the representatives Eleatic school(Eleatics). Used dialogue as a form of presentation for the first time philosophical problems. For Zenon, being is consistent, therefore contradictory being is imaginary (apparent) being. Zenon is best known as the author of paradoxes that raised the question of the dialectical nature of movement in a negative form. The paradoxes of Zenon boil down to proof that 1) it is logically impossible to think about the multiplicity of things, 2) the assumption of movement leads to a contradiction. The most famous are his paradoxes against the possibility of movement: “Achilles and the Tortoise”, “Arrow”, etc. (Aporia).

Lenin, reflecting on Zenon's argument, emphasized the correctness of Hegel's objection to it: to move means to be in this place and at the same time not to be in it; it is the unity of discontinuity and continuity of space and time, which makes movement possible.

Zeno of Elea (c. 490 - c. 430 BC) - ancient Greek philosopher, representative of the Eleatic school (6-5 centuries BC, Elea, Southern Italy).

According to Diogenes, Laertius was a student and adopted son of Parmenides.

Aristotle considered Z. the creator of dialectics as the art of interpreting contradictions. He saw the main task of his philosophy in protecting and justifying the teaching of Parmenides about the unchanging essence of true being (“everything is one”) and the illusory nature of all visible changes and differences. The truth of existence, according to Z., is revealed only through thinking, while sensory experience leads to the discovery of the multiplicity of things, their diversity and variability, and, consequently, to unreliability. The fact of the contradiction between the data of experience, on the one hand, and their mental analysis, on the other, was expressed 3. in the form of aporia (Greek aporia - difficulty, bewilderment). All aporia 3. boil down to proof that: 1) it is logically impossible to think about the multiplicity of things; 2) the assumption of movement leads to contradictions. His most famous aporias are directed against the possibility of movement: “Dichotomy”, “Achilles”, “Arrow”, “Stages”. Thus, the aporia “Achilles” states that, in contradiction with sensory experience, the fleet-footed Achilles cannot catch up with the tortoise, because while he runs the distance separating them, she will still have time to crawl a certain segment, but when he runs this segment, she will crawl away a little more, etc. According to 3., when trying to conceive of movement, we inevitably encounter contradictions, from which the conclusion follows about the inconceivability, and thereby the impossibility of movement in general.

Zeno (5th century BC) - Student and adopted son of Parmenides. He did not tolerate restrictions either in mental or political life; opposed the tyrannical government and died during the uprising. His works were apparently destroyed, but the ingenious problems he invented - the aporia (problems) of Zeno - continue to interest scientists and philosophers. He was able to reveal contradictions in our ideas about space, time, and movement. Diogenes Laertius cited Zeno's reasoning: “A moving object does not move either in the place where it is, or in the place where it is not.” Swift-footed Achilles will never catch up with the tortoise. After all, when catching up with her, he will first run half the distance between them, but the turtle will have time to cover some space; then Achilles will again cover half the distance between them, and the tortoise will move even further...

The gap between them will decrease to a minimum, but will never become zero. Zeno's aporia shows that our reasoning largely depends on what rules we are guided by, what axioms - truths that we cannot or do not want to prove - we rely on. (This is especially clearly evident when using computers: they give us those solutions and answers that the programmers put into them in advance.)

Balandin R.K. One Hundred Great Geniuses / R.K. Balandin. - M.: Veche, 2012.

Zeno of Elea (Ζήνων δ Έλεάτης) (c. 490 - c. 430 BC, Elea, Southern Italy), ancient Greek philosopher. Representative of the Eleatic school, student of Parmenides. Aristotle considered Zeno of Elea the creator of dialectics (A 10 DK I) as the art of comprehending truth through dispute or interpretation of opposing opinions. Defending and substantiating the doctrine of Parmenides about the One, Zeno of Elea rejected the conceivability of sensory existence, the plurality of things and their movement. He argued that accepting the existence of emptiness and multiplicity leads to contradictions. The most famous are the aporia of Zeno of Elea, directed against the possibility of movement (“Dichotomy”, “Achilles”, “Arrow” and “Stadium”). The aporia of Zeno of Elea have not lost their significance for modern science, the development of which is associated with the resolution of contradictions that arise when displaying real processes of movement.

Philosophical encyclopedic dictionary. - M.: Soviet Encyclopedia. Ch. editor: L. F. Ilyichev, P. N. Fedoseev, S. M. Kovalev, V. G. Panov. 1983.

Fragments: DK I; Zeno of Elea. A text with transi. and notes by H. D. P. Lee, Camb., 1936; in Russian trans. - Makovelsky A. O., Pre-Socratics, part 2, Kazan, 1915, p. 73-87.

Literature: Tsekhmistro I. 3., Aporia 3. through the eyes of the 20th century, “VF”, 1966, No. 3; P a n chenko A.I., Aporia Z. and modern times. philosophy, ibid., 1971, No. 7; Yanovskaya S.A., Have they been overcome in modern times? science difficulties known as. “aporias 3.”?, in her book: Methodological. problems of science, M., 1972; Booth N. V., Zeno's paradoxes, "Journal of Hellenic studies", 1957, v. 77; Grün-baum A., Modern science and Zeno's paradoxes, Middletown, 1967.

Zeno (Ζήνων) from Elea (Southern Italy; according to Apollodorus, acme 464-461 BC; according to Plato, “Parmenides” 127e, ca. 450, which is less likely) - ancient Greek philosopher, representative of the Eleatic school, student of Parmenides . In the dialogue “The Sophist” (fr. 1 Ross), Aristotle called Zeno “the inventor of dialectics,” that is, the critical analysis of “accepted opinions” (...) or the refutation of an opponent’s thesis by reductio ad absurdum. Plato in the Phaedrus (261 d) speaks of the “Elean Palamedes” (synonymous with a clever inventor), whose “antilogical art” (argumentation of thesis and antithesis) is capable of instilling in his listeners that “the same thing is like and unlike, one and plural, rests and moves." The Byzantine Lexicon of the Suda lists the titles of four works of Zeno: “Disputes”, “Interpretation of Empedocles”, “Against the Philosophers”, “On Nature”; Plato in Parmenides mentions one essay written to “ridicule” Parmenides’ opponents and to show that the assumption of plurality and movement leads “to even more ridiculous conclusions” than the assumption of One being. In his commentary on this passage, Proclus (in Parm. P. 694, 23 Diehl) reports that Zeno's work contained 40 arguments (...) against the multitude. The most famous already in ancient times were 4 arguments (so-called aporia) against the possibility of movement, preserved in the paraphrases of Aristotle (Physics VI 9): 1) “Stages” (otherwise “Dichotomy”, 29 A 25 DK); 2) “Achilles and the Tortoise” (29 A 26 DK); 3) “Strela” (29 A 27 DK); 4) “Moving bodies” (otherwise “Stages”, do not mix with 1st, 29 A 28 DK). The antinomies of the set, given by Simplicius in verbatim quotations from Zeno (29 B 1-3 DK), and the paradox of place (29 A 24 DK) have also been preserved. The view that Zeno's arguments were directed against the supporters of Pythagorean "mathematical atomism", who constructed physical bodies from geometric points and accepted the atomic structure of time, is now abandoned by most researchers, since the existence of the early theory of "mathematical atomism" is not attested. Zeno's opponents could simply be adherents of common sense, to whom he wanted to show the absurdity and, therefore, the unreality of the phenomenal world of multitude and movement. At the same time, Zeno did not recognize any reality other than the spatially extended one. Zeno's aporia one way or another rests on the problem of continuum, which acquired particular relevance in connection with the set theory of G. Cantor and quantum mechanics of the 20th century.

Fragments and evidence: DK I, 247-258; Zeno. Testimonianze e frammenti, intr., trad, e comm. a cura di M. Untersteiner, Firenze, 1963; Lee H. D. P. Zeno of Elea. Cambr., 1936. Lit.: Yanovskaya S.A. Have they overcome the modern science difficulties known as the "Aporius of Zeno"? - In the collection: Problems of logic. M, 1963; Koire A. Essays on the history of philosophical thought, trans. from French M, 1985, p. 27-50; Komarova V. Ya. The teachings of Zeno of Eleica: an attempt to reconstruct the system of arguments. L., 1988; Salmon W. Ch. (ed.) Zeno's paradoxes. Indianopolis, 1970; Vlastos G. Zeno's Race Course - Furley D. J., Allen R. E. (ed.). Studies in presocratic philosophy, v. 2. L., 1975; Ferber R. Zenons Paradoxien der Bewegung und die Strukturvon Raum und Zeit. Miinch., 1981.

A.B. Lebedev

New philosophical encyclopedia. In four volumes. / Institute of Philosophy RAS. Scientific ed. advice: V.S. Stepin, A.A. Guseinov, G.Yu. Semigin. M., Mysl, 2010, vol. II, E – M, p. 44.

Zeno of Elea (c. 490-430 BC) - ancient Greek philosopher, representative of the Eleatic school. He was portrayed as a politician who fought against tyrants and died in this fight.

Zeno's teachings contain a number of arguments aimed at defending the philosophy of Parmenides. Aristotle called Zeno the founder of “dialectics,” meaning by it a way to find out the truth by identifying internal contradictions in the opponent’s thinking and avoiding these contradictions. The path that Zeno follows in justifying his views and defending the views of Parmenides is proof by contradiction. It is believed that Zeno presented 40 proofs “against the plurality” of beings and five proofs “against motion,” i.e. protecting his immobility. In the extant literature there is evidence against plurality (four) and evidence against movement (four). They are called Zeno's aporias.

Zeno proceeded from the fact that true being is motionless, it is unknowable by the senses. The movement and variety of things cannot be explained by reason, they are only “opinions”, the result of sensory perception. Without denying the reliability of sensory perception, Zeno at the same time believed that it is impossible to obtain true knowledge through sensory perception and if we recognize motion and multitude as existing, this leads to insoluble contradictions, which he sought to prove as follows.

Zeno's aporia "dichotomy". If an object is moving, it must travel halfway before it reaches the end. But before going through this half, the given object must go through half of this half, and so on ad infinitum. Those. movement can neither begin nor end.

Aporia "Achilles and the Tortoise". There is a turtle in front of Achilles, and they start running at the same time. Achilles will never catch up with the turtle, since by the time he reaches the place where the turtle was, it will crawl some distance, and this will be repeated ad infinitum.

Aporia "Flying Arrow". A flying arrow, according to Zeno, will always be at rest, since at every moment of movement it will occupy an equal place.

Aporia "Stadium". Two bodies move towards each other and relative to each other. In this case, one of them will spend the same amount of time passing by the other as it would spend passing by the one at rest. So half is equal to the whole, which is absurd. Thus, all the logical consequences arising from these aporias indicate that movement is an appearance, not a reality.

Of course, all of Zeno’s aporia are easily refuted if, when considering them, we take into account not only the discontinuity of movement and space, but also their continuity. However, they reflected the difficulty of forming the conceptual apparatus of science, as well as the inconsistency of such concepts as space, time, motion. And Zeno himself had no doubt at all that we perceive movement through our senses. He formulates his aporia in order to show that we cannot think of movement if we understand space as consisting of parts separated from each other, and time as consisting of moments separated from each other. Those. Zeno proves that there is no plurality, existence is one.

Blinnikov L.V. Brief dictionary philosophical personalities. M., 2002.

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Zeno of Elea is an ancient Greek philosopher who was a student of Parmenides, a representative of the Eleatic school. He was born around 490 BC. e. in Southern Italy, in the city of Elea.

What was Zeno famous for?

Zeno's arguments made this philosopher famous as a skilled polemicist in the spirit of sophistry. The content of the teachings of this thinker was considered identical to the ideas of Parmenides. The Eleatic school (Xenophanes, Parmenides, Zeno) is the predecessor of sophistry. Zeno was traditionally considered the only "student" of Parmenides (although Empedocles was also called his "successor"). In an early dialogue called The Sophist, Aristotle called Zeno the “inventor of dialectics.” He used the concept of “dialectic,” most likely, in the sense of proof from certain generally accepted premises. Aristotle's own work, Topeka, is dedicated to him.

In the Phaedrus, Plato speaks of the “Elean Palamedes” (which means “clever inventor”), who is excellent at the “art of word debate.” Plutarch writes about Zeno using the terminology commonly used to describe sophistic practice. He says that this philosopher knew how to refute, leading to aporia through counterarguments. A hint that Zeno's studies were sophistic is the mention in the dialogue "Alcibiades I" that this philosopher charged high fees for his studies. Diogenes Laertius says that Zeno of Elea first began writing dialogues. This thinker was also considered the teacher of Pericles, a famous political figure in Athens.

Zeno's politics

You can find reports from doxographers that Zeno was involved in politics. For example, he took part in a conspiracy against Nearchus, a tyrant (there are other variations of his name), was arrested and tried to bite off his ear during interrogation. This story is related by Diogenes according to Heraclides Lembus, who, in turn, refers to the book of the Peripatetic Satyrus.

Many historians of antiquity conveyed reports of this philosopher's steadfastness at trial. Thus, according to Antisthenes of Rhodes, Zeno of Elea bit off his tongue. Hermippus says that the philosopher was thrown into the mortar in which he was pounded. This episode was subsequently very popular in the literature of antiquity. Plutarch of Chaeronea, Diodir Sicilian, Flavius ​​Philostratus, Clement of Alexandria, Tertullian mention him.

Works of Zeno

Zeno of Elea was the author of the works "Against the Philosophers", "Disputes", "Interpretation of Empedocles" and "On Nature". It is possible, however, that all of them, except for the Interpretation of Empedocles, were in fact variants of the title of one book. In Parmenides, Plato mentions an essay written by Zeno in order to ridicule his teacher’s opponents and to show that the assumption of motion and plurality leads to even more absurd conclusions than the recognition of a single being according to Parmenides. The reasoning for this is presented by later authors. This is Aristotle (work "Physics"), as well as his commentators (for example, Simplicius).

Zeno's arguments

Zeno's main work appears to have been composed of a series of arguments. They were reduced to proof by contradiction. This philosopher, defending the postulate of a motionless unified being, which was put forward by the Eleatic school (Zeno’s aporia, according to a number of researchers, were created in order to support the teaching of Parmenides), sought to show that the assumption of the opposite thesis (about movement and multitude) certainly leads to absurdity, therefore, it should be rejected by thinkers.

Zeno obviously followed that if one of two opposing statements is false, the other is true. Today we know about the following two groups of arguments of this philosopher (the aporia of Zeno of Elea): against movement and against multitude. There is also evidence that there are arguments against sense perception and against place.

Zeno's arguments against multitudes

Simplicius preserved these arguments. He quotes Zeno in his commentary on Aristotle's Physics. Proclus says that the work of the thinker we are interested in contained 40 similar arguments. We will list five of them.

1. Defending his teacher, who was Parmenides, Zeno of Elea says that if there is a multitude, then, therefore, things must necessarily be both great and small: so small that they have no size at all, and so great that they are infinite.

   The proof is as follows. The existing must have a certain magnitude. When added to something, it will increase it and decrease it when taken away. But in order to be different from some other person, you must stand apart from him, be at a certain distance. That is, between two beings there will always be a third given, thanks to which they are different. It must also be different from another, etc. In general, existence will be infinitely large, since it represents the sum of things, of which there is an infinite number. (Parmenides, Zeno, etc.) is based on this idea.

2. If there is a multitude, then things will be both limitless and limited.

   Proof: if there is a multitude, there are as many things as there are, no less and no more, that is, their number is limited. However, in this case, there will always be others between things, between which, in turn, there will be third ones, etc. That is, their number will be infinite. Since the opposite is simultaneously proven, the original postulate is incorrect. That is, there is no set. This is one of the main ideas developed by Parmenides (Eleatic School). Zeno supports her.

3. If there is a multitude, then things must be dissimilar and similar at the same time, which is impossible. According to Plato, the book of the philosopher we are interested in began with this argument. presupposes that the same thing is considered as similar to itself and different from others. In Plato it is understood as a paralogism, since dissimilarity and similarity are taken in different respects.
4. Let us note an interesting argument against the place. Zeno said that if there is a place, then it must be in something, since this applies to everything that exists. It follows that the place will also be in the place. And so on ad infinitum. Conclusion: there is no room. Aristotle and commentators classified this argument as a paralogism. It is not true that “to be” means “to be in a place,” since incorporeal concepts do not exist in a place.
5. The argument against sense perception is called the "Grain of Millet" argument. If one grain or a thousandth part of it does not make noise when falling, how can a medium do it when it falls? If the medimna grain produces noise, then it must also apply to one thousandth, which it does not. This argument touches on the problem of the threshold of our perception, although it is formulated in terms of the whole and the part. The paralogism in this formulation is that we're talking about about the “noise produced by a part”, which does not exist in reality (as Aristotle noted, it exists in possibility).

Arguments against the movement

The most famous are the four aporia of Zeno of Elea against time and motion, known from Aristotle's Physics, as well as the commentaries to it by John Philoponus and Simplicius. The first two of them are based on the fact that a segment of any length can be represented as an infinite number of indivisible “places” (parts). It cannot be passed in a finite time. The third and fourth aporia are based on the fact that time consists of indivisible parts.

"Dichotomy"

Consider the "Stages" argument ("Dichotomy" is another name). Before covering a certain distance, a moving body must first travel half of the segment, and before reaching half, it must travel half of the half, and so on ad infinitum, since any segment can be divided in half, no matter how small it is.

In other words, since movement is always carried out in space, and its continuum is considered as an infinite set of different segments, this is relevant, since any continuous quantity is divisible to infinity. Consequently, a moving body will have to cover an infinite number of segments in a finite time. This makes movement impossible.

"Achilles"

If there is movement, the fastest runner will never be able to catch up with the slowest, since it is necessary that the one who is catching up first reaches the place from which the runner began to move. Therefore, of necessity, the slower runner must always be slightly ahead.

Indeed, to move means to move from one point to another. From point A, fast Achilles begins to catch up with the tortoise, which is currently located at point B. First, he needs to go halfway, that is, distance AАБ. When Achilles is at point AB, during the time he was making the movement, the tortoise will move a little further to the segment BBB. Then the runner, who is in the middle of his journey, will need to reach point Bb. To do this, it is necessary, in turn, to walk half the distance A1Bb. When the athlete is halfway to this goal (A2), the turtle crawls a little further. And so on. Zeno of Elea in both aporias assumes that the continuum is divided to infinity, thinking of this infinity as actually existing.

"Arrow"

In fact, a flying arrow is at rest, believed Zeno of Elea. The philosophy of this scientist has always had a justification, and this aporia is no exception. Its proof is as follows: at each moment of time the arrow occupies a certain space, which is equal to its volume (since the arrow would otherwise be “nowhere”). However, to occupy a place equal to oneself means to be at peace. From this we can conclude that we can think of motion only as the sum of various states of rest. This is impossible, since nothing comes from nothing.

"Moving Bodies"

If there is movement, you may notice the following. One of two quantities that are equal and move at the same speed will travel twice as far in equal time as the other.

This aporia has traditionally been clarified with the help of a drawing. Two equal objects, which are designated by letter symbols, move towards each other. They walk along parallel paths and at the same time pass by a third object, which is equal in size to them. Moving at the same speed, once past a stationary object and the other past a moving object, the same distance will be covered simultaneously both in a period of time and in half of it. The indivisible moment will then be twice its size. This is logically incorrect. It must either be divisible, or an indivisible part of some space must be divisible. Since Zeno admits neither one nor the other, he therefore concludes that motion cannot be thought of without the emergence of a contradiction. That is, it does not exist.

Conclusion from all aporias

The conclusion that was drawn from all the aporia formulated in support of the ideas of Parmenides by Zeno is that the evidence of the senses that convinces us of the existence of motion and multitude diverges from the arguments of reason, which do not contain contradictions in themselves, and therefore are true. In this case, reasoning and feelings based on them should be considered false.

Who were the aporias directed against?

The question of whom Zeno’s aporias were directed against does not have a single answer. A point of view was expressed in the literature according to which the arguments of this philosopher were directed against the supporters of “mathematical atomism” of Pythagoras, who constructed physical bodies from geometric points and believed that time has an atomic structure. This view currently has no supporters.

In the ancient tradition, the assumption, dating back to Plato, that Zeno defended the ideas of his teacher was considered a sufficient explanation. His opponents, therefore, were everyone who did not share the teaching that was put forward by the Eleatic School (Parmenides, Zeno), and adhered to common sense based on the evidence of feelings.

So, we talked about who Zeno of Elea is. His aporias were briefly examined. And today, discussions about the structure of movement, time and space are far from complete, so these interesting questions remain open.

Bibliographic description:
Solopova M.A. ZENON OF ELEA // Ancient philosophy: encyclopedic Dictionary. M.: Progress-Tradition, 2008. pp. 386-390.

ZENON OF ELEA (Ζήνων ὁ ’Ελεάτης ) (born about 490 BC), ancient Greek. philosopher, representative Eleatic school, student Zeno of Elea (Ζήνων) (c. 490-430 BC) - ancient Greek philosopher, born in Elea (Southern Italy). Student. Born in the city of Eleya in South. Italy. According to Apollodorus, acme 464–461 BC. According to Plato's description in the Parmenides dialogue - ca. 449: (cf. Parm. 127b: “Parmenides was already very old ... he was about sixty-five. Zeno was then about forty”; a young Socrates, presumably no younger than twenty years old, participates in the conversation with them - hence the indicated dating). In Plato, Zeno is depicted as the famous author of a collection of arguments that he compiled “in his youth” (Parm. 128d6–7) to defend the teachings of Parmenides.

Zeno's arguments glorified him as a skilled polemicist in the spirit of the fashionable series for Greece. 5th century sophistry. The content of his teaching was believed to be identical to the teaching of Parmenides, whose only “student” (μαθητής) he was traditionally considered (“successor” of Parmenides was also called Empedocles). Aristotle, in his early dialogue "The Sophist", called Zeno "the inventor of dialectics" (Arist., fr. 1 Rose), using the term dialectics, probably in the meaning of the art of proof from generally accepted premises, to which his own opus is dedicated. Topeka. Plato in the Phaedrus speaks of the “Elean Palamedes” (synonym for a clever inventor), who is excellent at “the art of word debate” (ἀντιλογική) (Phaedr. 261d). Plutarch writes about Zeno, using the terminology adopted to describe the practice of the sophists (ἔλεγξις, ἀντιλογία): “he knew how to skillfully refute, leading through counterarguments to an aporia in reasoning.” A hint of the sophistic nature of Zeno’s studies is the mention in the Platonic dialogue “Alcibiades I” that he charged high tuition fees (Plat. Alc. I, 119a). Diogenes Laertius conveys the opinion that “Zeno of Elea first began to write dialogues” (D.L. III 48), probably derived from the opinion about Zeno as the inventor of dialectics (see above). Finally, Zeno was considered the teacher of the famous Athenian politician Pericles (Plut. Pericl. 4, 5).

Doxographers have reports of Zeno himself being involved in politics (D.L. IX 25 = DK29 A1): he participated in a conspiracy against the tyrant Nearchus (there are other variations of names), was arrested and during interrogation tried to bite off the tyrant’s ear (Diogenes recounts this story according to Heracleidou Lembu, and he, in turn, is based on the book of the peripatetic Satyr). Many ancient historians conveyed reports of Z.'s steadfastness at trial. Antisthenes of Rhodes reports that Z. bit off his tongue (FGrH III B, n° 508, fr. 11), Hermippus - that Zeno was thrown into a mortar and pounded in it (FHistGr, fr. 30). Subsequently, this episode was invariably popular in ancient literature(he is mentioned by Diodorus Siculus, Plutarch of Chaeronea, Clement of Alexandria, Flavius ​​Philostratus, see A6–9 DK, and even Tertullian, A19).

Essays. According to the Suda, Z. was the author of the op. "Disputes" (῎Εριδας), "Against the Philosophers" (Πρὸς τοὺς φιλοσόφους), "On Nature" (Περὶ φύσεως) and "Interpretation of Empedocles" ('Εξήγησ ις τῶν 'Εμπεδοκλέους), – it is possible that the first three actually represent are variants of the titles of one essay; the last work called Suda is not known from other sources. Plato in Parmenides mentions one work (τὸ γράμμα) by Z., written with the aim of “ridiculing” Parmenides’ opponents and showing that the assumption of plurality and movement leads “to even more ridiculous conclusions” than the assumption of a single being. Zeno's argumentation is known in the retelling of later authors: Aristotle (in " Physics") and its commentators (primarily Simplicia).

Z.'s main (or only) work apparently consisted of a set of arguments, the logical form of which was reduced to proof by contradiction. Defending the Eleatic postulate of a single motionless being, he sought to show that the acceptance of the opposite thesis (about multitude and movement) leads to absurdity (ἄτοπον) and therefore should be rejected. Obviously, Z. proceeded from the law of the “excluded middle”: if of two opposing statements one is false, therefore the other is true. There are two main groups of arguments known about Z. - against multitude and against movement. There is also evidence of the argument against place and against sense perception, which can be seen in the context of the development of the argument against set.

Arguments against plurality preserved by Simplicius (see: DK29 B 1–3), who quotes Z. in his commentary on Aristotle’s Physics, and by Plato in Parmenides (B 5); Proclus reports (In Parm. 694, 23 Diehl = A 15) that Z.'s work contained only 40 similar arguments (λόγοι).

1. “If there is a multitude, then things must necessarily be both small and large: so small that they have no size at all, and so large that they are infinite” (B 1 = Simpl. In Phys. 140, 34). Proof: what exists must have a certain magnitude; being added to something, it will increase it, and being taken away from something, it will decrease it. But in order to be different from another, you need to stand away from him, be at some distance. Consequently, between two beings there will always be something third given, thanks to which they are different. This third, as a being, must also be different from the other, etc. In general, the existing will be infinitely large, representing the sum of an infinite number of things.

2. If there is a multitude, then things must be both limited and unlimited (B 3). Proof: if there is a set, there are as many things as there are, no more and no less, which means their number is limited. But if there is a multitude, there will always be others between things, others between them, etc. ad infinitum. This means their number will be infinite. Since the opposite has been proven at the same time, the original postulate is incorrect, therefore there is no set.

3. “If there is a multitude, then things must be both similar and dissimilar, and this is impossible” (B 5 = Plat. Parm. 127e1–4; according to Plato, Zeno’s book began with this argument). The argument involves considering the same thing as similar to itself and dissimilar to others (different from others). In Plato, the argument is understood as a paralogism, because similarity and dissimilarity are taken in different relations, and not in the same thing.

4. Argument against place (A 24): “If there is a place, then it will be in something, since every being is in something. But what is in something is in place. Consequently, the place will be in the place, and so on ad infinitum. Therefore, there is no place” (Simpl. In Phys. 562, 3). Aristotle and his commentators classified this argument as a paralogism: it is not true that “to be” means “to be in a place,” since incorporeal concepts do not exist in any place.

5. Argument against sense perception: “A grain of millet” (A 29). If one grain or one thousandth of a grain does not make noise when it falls, how can the fall of a medimn grain make noise? (Simpl. In Phys. 1108, 18). Since the fall of a medim grain makes noise, then the fall of one thousandth should make noise, which in fact it does not. The argument concerns the problem of the threshold of sensory perception, although it is formulated in terms of part and whole: just as the whole is related to the part, so the noise produced by the whole must be related to the noise produced by the part. In this formulation, the paralogism consists in the fact that the “noise produced by a part” is being discussed, which in reality does not exist (but is possible, as Aristotle noted).

Arguments against the movement. The most famous are 4 arguments against motion and time, known from Aristotle’s “Physics” (see: Phys. VI 9) and comments to the “Physics” of Simplicius and John Philoponus. The first two aporias are based on the fact that any segment of length can be represented as an infinite number of indivisible parts (“places”) that cannot be traversed in a finite time; the third and fourth – on the fact that time also consists of indivisible parts (“now”).

1. "Stages"(other name "Dichotomy", A25 DK). A moving body, before covering a certain distance, must first travel half of it, and before reaching half, it must travel half of a half, etc. to infinity, because any segment, no matter how small, can be divided in half.

In other words, since movement always occurs in space, and the spatial continuum (for example, line AB) is considered as an actually given infinite set of segments, since every continuous quantity is divisible to infinity, then a moving body will have to go through an infinite number of segments in a finite time, which makes movement impossible.

2. "Achilles"(A26 DK). If there is movement, “the fastest runner will never catch up with the slowest, since it is necessary that the one who is catching up first reaches the place from which the runner began to move, therefore the slower runner must of necessity always be slightly ahead” (Arist. Phys. 239b14; cf.: Simpl. In Phys. 1013, 31).

In fact, to move means to move from one place to another. Fast Achilles from point A begins to pursue the turtle located at point B. He first needs to cover half of the whole path - that is, distance AA1. When he is at point A1, the turtle will travel a little further to a certain segment BB1 during the time he was running. Then Achilles, who is in the middle of the path, will need to reach point B1, for which, in turn, it is necessary to cover half the distance A1B1. When he is halfway to this goal (A2), the turtle will crawl a little further, and so on ad infinitum. In both aporias Z. assumes a continuum to be divisible to infinity, thinking of this infinity as actually existing.

Unlike the “Dichotomy” aporia, the added value is not divided in half; otherwise, the assumptions about the divisibility of the continuum are the same.

3. "Arrow"(A27 DK). A flying arrow is actually at rest. Proof: at each moment of time the arrow occupies a certain place equal to its volume (for otherwise the arrow would be “nowhere”). But to occupy a place equal to oneself means to be at peace. It follows that movement can only be thought of as the sum of states of rest (the sum of “advancements”), and this is impossible, because nothing comes from nothing.

4. "Moving Bodies"(other name "Stages", A28 DK). “If there is movement, then one of two equal values, moving with equal speed, at equal time will pass twice the distance, not equal, than the other” (Simpl. In Phys. 1016, 9).

Traditionally, this aporia was explained with the help of a drawing. Two equal objects (denoted by letter symbols) move towards each other along parallel straight lines and pass by a third object of equal size. Moving with equal speed, once past a moving object, and another time past a stationary object, the same distance will be covered simultaneously both in a certain time interval t and in half the interval t/2.

Let the row A1 A2 A3 A4 mean a stationary object, row B1 B2 B3 B4 an object moving to the right, and C1 C2 C3 C4 an object moving to the left:

A 1 A2 A3 A4

After the same moment of time t, point B4 passes half of the segment A1–A4 (i.e., half of a stationary object) and the entire segment C1–C4 (i.e., an object moving towards). It is assumed that each indivisible moment in time corresponds to an indivisible segment of space. But it turns out that point B4 at one moment of time t passes (depending on where to count from) different parts of space: in relation to a stationary object, it travels a shorter distance (two indivisible parts), and in relation to a moving object, it travels a larger distance (four indivisible parts). Thus, an indivisible moment of time turns out to be twice its size. This means that either it must be divisible, or an indivisible part of space must be divisible. Since Z. does not allow either one or the other, he concludes that movement cannot be thought of without contradiction, therefore, movement does not exist.

The general conclusion from the aporia formulated by Zeno in support of the teachings of Parmenides was that the evidence of the senses, which convinces us of the existence of set and motion, is at odds with the arguments of reason, which do not contain a contradiction, and therefore are true. In this case, the feelings and reasoning based on them should be considered false. The question of who Zeno’s aporia was directed against does not have a single answer. A point of view has been expressed in the literature according to which Zeno's arguments were directed against the supporters of Pythagorean "mathematical atomism", who constructed physical bodies from geometric points and accepted the atomic structure of time (for the first time - Tannery 1885, one of the last influential monographs proceeding from this hypothesis - Raven 1948 ); at present this view has no supporters (see for more details: Vlastos 1967, p. 256–258).

In the ancient tradition, the assumption dating back to Plato that Zeno defended the teachings of Parmenides and his opponents were everyone who did not accept Eleatic ontology and adhered to common sense, trusting the feelings, was considered a sufficient explanation.

Fragments

• DK I, 247–258;
• Untersteiner M. (ed.). Zeno. Testimonianze e frammenti. Fir., 1963;
• Lee H.D.P.. Zeno of Elea. Camb., 1936;
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