Zeno years of life. Eleatic school

[Greek Ζήνων ὁ ᾿Ελεάτης] (V century BC), ancient Greek. philosopher, representative of the Eleatic school of philosophy, student of Parmenides, creator of the famous “Aporia of Zeno.”

Life and writings

The exact date of birth of Z. E. is unknown. According to the testimony of Diogenes Laertius, referring to the “Chronicle” of Apollodorus (Diog. Laert. IX 5), Z. E. was the natural son of Teleutagoras and the adopted son of Parmenides; Diogenes attributes his “heyday” (ἀκμή) to the 79th Olympiad (464- 461 BC), a close dating is given by Byzantium. lexicon “Suda” (78th Olympiad; see: DK. 29A2). Based on tradition. identifying “heyday” with the age of 40 years, the date of birth of Z.E. falls between 504 and 501. BC. A more probable dating is considered to be based on evidence from the dialogue “Parmenides”, in which Plato tells how Parmenides and Z.E. once visited Athens. At the same time, Z. E. was “about forty years old,” while Socrates, born c. 469 BC, “was very young,” and Parmenides was “very old... he was about 65 years old” (Plat. Parm. 127a-e). If we accept that Socrates was a little over 20 years old, then the date of birth of Z.E. turns out to be between 492 and 490. BC Plato also reports that Z. E. was a tall man, had a pleasant appearance and, according to rumors, was the “favorite” (παιδικά) of Parmenides.

Apart from Plato's message, the only relatively reliable source of information about Z. E. is the work of Diogenes Laertius, who repeats Plato's story and adds a number of details relating to the life and political activities of Z. E. (Diog. Laert. IX 5). Thus, Diogenes gives various versions of the story about Z.E.’s participation in a conspiracy against a certain tyrant and painful death at the hands of the latter. At the same time, neither the name of the tyrant is known exactly (in early sources the names of Nearchus, Diomedes and Demilus are found, in later sources the image of this tyrant merges with the famous tyrants of antiquity, in particular Dionysius I of Syracuse), nor the place of the conspiracy (mentioned as the hometown of the philosopher Elea, and other cities). According to one version, after Z.E. was captured and taken to the city square, he managed with his speeches and reproaches of cowardice to inspire the surrounding people, who stoned the tyrant. According to another version, during interrogation, in response to the demand to give up the names of his accomplices, Z. E. named all the tyrant’s loyal friends, as a result of which the tyrant executed them and lost his followers. According to a more detailed version, Z. E. was subjected to severe torture, so, being unable to endure it, he called the tyrant to him, allegedly intending to tell him the names of his accomplices, but when he approached, he grabbed his ear with his teeth and did not let go him until he was stabbed to death (cf.: Diodor. Sic. Bibliotheca. X 18). Finally, according to another version, fearing to spill the beans under torture, he bit off his own tongue and spat it out in the tyrant’s face (this version is given, in particular, by Clement of Alexandria - Clem. Alex. Strom. IV 8.56). Diogenes and the lexicon of the “Judgment” have evidence that Z. E. was subjected to a special painful execution: thrown into a mortar and there beaten to death, ground into powder (DK. 29A2; the same story is told about Anaxarchus - Diog. Laert IX 10). Tertullian provides interesting, although hardly reliable evidence about Z.E. in his treatise “Apologetics”: “When asked by Dionysius what philosophy gives, Zeno of Elea answered: “Contempt for death.” Subjected to scourging by the tyrant, he remained insensitive to suffering, confirming the truth of his saying until his death” (Tertull. Apol. adv. gent. 50). Tertullian also mentions the courage of Zeus in the face of torment in his treatise “On the Soul” (Idem. De anima 58.4); Nemesius, bishop, has evidence of this. Emesa (Nemes. De nat. hom. 30), Eusebius, bishop. Caesarea in Palestine (Euseb. Praep. evang. X 14. 15), etc. church writers. According to a certain Arab. sources, Z. E. died at the age of 78, but the general unreliability of the Arab. biographies of Z. E. and the absence of Greek. evidence makes this statement doubtful (see: Rozenthal. 1937).

Sources of information about philosophical activity, teachings and writings of Z. E. are ch. arr. treatises by Plato, Aristotle and Simplicius, in which he is mentioned in connection with the aporias he developed. From Plato’s story, the historical authenticity of which received various assessments from researchers (for a detailed analysis of the evidence about Z. E. in the dialogue “Parmenides”, see: Vlastos. 1975), we can conclude that Z. E. in his youth wrote the only essay dedicated to defense of the teaching of Parmenides on unity, and this work was not finalized by Z. E., but was stolen from him and put into circulation without his knowledge (Plat. Parm. 128d-e). It is this work of Z. E. that is discussed by the participants in the dialogue “Parmenides”: Parmenides and Z. E. who came to Athens, as well as Socrates and his students, and several are cited during the dialogue. important fragments from the work of Z. E and it is argued that it was written with the aim of “ridiculing” the opponents of Parmenides and showing that the assumption of plurality and movement “entails even more ridiculous consequences” than the assumption of a single being (Ibid. 128b-d) . Plato mentions Z. E. and his arguments also in the dialogues “The Sophist” (Ibid. 216a) and “Phaedrus” (Ibid. 261d; here Plato gives him the nickname “Elean Palamedes,” which later became famous, indicating Z.’s intellectual ingenuity. E.). Aristotle in “Metaphysics” and “Physics” examines some of Z.E.’s arguments, and in the surviving fragment of the dialogue, the “Sophist” calls him “the inventor of dialectics” (DK. 29A10), that is, the critical analysis of “opinions” by considering opposing possibilities and reducing the opponent’s argument to absurdity. He called Z. E. and St. the inventor of dialectics. Athanasius I the Great, bishop Alexandrian (Athanas. Alex. Or. contr. gent. 18). The name “eristic” (ἐριστικός - debater), which is given by Z. E. St., has a similar meaning. Epiphanius of Cyprus, who mentions him in the final treatise “Against Heresies” with the word “On the Faith of the Universal and Apostolic Church” (Epiph. De fide // GCS. Bd. 31. S. 505), and the nickname “bilingual” (ἀμφοτερόγλωσσος), about which says Timon (DK. 29A1) and Simplicius (Simplicius. In Aristotelis Physicorum libros octo Commentaria. Berolini, 1882. Vol. 1. P. 139).

Both in ancient times and today. At the time, the most common is the so-called view, according to which Z. E. wrote only one essay, the one mentioned in Parmenides and which, according to the lexicon of the Court, was called “῎Εριδες "("Competitions", "Disputes"). Apparently, it consisted of individual reasoning (λόγοι), or chains of arguments (ὑπόθεσις), dedicated to the disclosure of k.-l. one controversial issue. From ancient sources, only the “Suda” claims that Zeno had other works: “᾿Εξήγησις τῶν ᾿Εμπεδοκλέους” (Explanation of the works of Empedocles), “Πρὸς τοὺς φιλοσ όφους" (Against philosophers), "Περ φύσεως" (About nature). There is a hypothesis that the last 2 positions are just other names of the op. "῎Εριδες." Regarding op. “᾿Εξήγησις τῶν ᾿Εμπεδοκλέους” researchers note that the word ἐξήγησις in this case can mean not only a positive interpretation and explanation, but also a critical analysis of views with the aim of refuting them. Diogenes Laertius briefly mentions the “books” (βιβλία) of Earth, but does not give a single name. Although the possibility of the existence of some other works by Z. E. is recognized by some researchers (Fritz. 1972. Sp. 56), no documentary evidence about them or fragments from them have survived.

Teaching: paradoxes and aporias

In the story of Z.E. Diogenes Laertius offers a bizarre picture of his teaching about nature (φύσις): “Worlds exist, but emptiness does not exist; the nature of all things came from warm, cold, dry and wet, turning into each other; people originated from the earth, and their souls are a mixture of the above-mentioned principles, in which none of them predominates” (Diog. Laert. IX 5). The affiliation of these views by Z. E. in the present day. time is rejected by most researchers. E. Zeller's hypothesis about the possible erroneous attribution of the teachings of Zeno of Citium to E. did not receive wide support, since the latter's true opinions do not coincide with the above quotation (Fritz. 1972. Sp. 57). More convincing are the various versions of the hypothesis, according to which this fragment sets out the teaching of Empedocles and his supporters - either erroneously attributed to Z. E., or actually contained in the unpreserved work of Z. E. (possibly in “᾿Εξήγησις τῶν ᾿Εμπεδοκλέους”) , directed against the teachings of Empedocles (Ibid. Sp. 57-58; see also: Longrigg J. Zeno's Cosmology? // The Classical Review. N. S. 1972. Vol. 22. N 2. P. 170-171). Finally , some researchers assumed that Z.E., like his teacher Parmenides, divided in his teaching the “path of truth” (the doctrine of the one) and the “path of opinion” (the doctrine of the many), therefore the above views are common views related to to the presentation of the “path of opinion" (Calogero. 1932. P. 98). Another option is also recognized as possible: Z. E. cited various cosmological theories in order to show their internal inconsistency and criticize the “path of opinion”, showing its logical impossibility ( Zenone: Testimonianze e frammenti. 1963. P. 15). The physical views of Z. E. are briefly mentioned by St. Epiphanius of Cyprus, according to whom Z. E. taught that “the earth is motionless and no place is empty” (τὴν γῆν ἀκίνητον κα μηδένα τόπον κενὸν εἶναι - Epiph. De fide // GCS. Bd. 31. S. 505) . According to the op. composed by John Stobaeus. “Opinions of the Philosophers,” Z. E. believed that God is “the all-one, the only eternal and infinite One” (DK. 29A30). Although these views are consistent with the general philosophy of Parmenides and his followers and may well have been shared by Z. E., there is no reliable evidence of their attribution to Z. E.

Thus, the only reliable element of the teachings of Z.E. that has survived to this day. time, are aporia (ἀπορία - impassability, difficulty, hopeless situation), also called by ancient authors “epicherems” (ἐπιχείρημα - compressed conclusion), “paralogisms” (παραλογισμός - false conclusion) and “arguments” (λό γοι). Modern researchers divide them into 2 main groups: arguments against plurality and arguments against movement. Of all the arguments conveyed by various authors, only 2 (DK. 29B1, 2, 3) are supported by genuine and verbatim quotations from the work of Z. E., while the rest are preserved in paraphrases and paraphrases of varying degrees of accuracy. The most important sources are Aristotle’s “Physics,” where the presentation of Z. E.’s arguments is accompanied by their critical analysis, as well as the works of Aristotle’s subsequent commentators (Simplicia, John Philoponus, Themistia). At the same time, among researchers there are different assessments of the accuracy and correctness of Aristotle’s transmission of the views of Z. E. The solutions to the aporias of Z. E. proposed by Aristotle were recognized as convincing and developed until the end. XIX - early XX century, when some Europeans. The researchers came to the conclusion that Aristotle’s argumentation of Z. E. is presented in a distorted form, so it is necessary to try to reconstruct the original content of the arguments. As a result of such a reconstruction, a number of scientists (V. Cousin, J. Groth, P. Tannery) concluded that the arguments of Z. E. are serious logical constructions, distorted by the sophists and deprived of their original meaning. This position was supported by B. Russell, who noted that Z. E. “invented four arguments, unusually subtle and deep,” but “the grossness of subsequent philosophers made him nothing more than an inventive swindler, and declared his arguments ordinary sophistry” (Russel B. Principles of Mathematics. L., 1937. P. 347). Many people opposed this position. researchers of Aristotle's philosophy (Zeller, D. Ross, N. Booth, etc.), who insisted on the authenticity of his interpretation of the aporias. Numerous scientific discussions have not resolved the question of the accuracy of Aristotle's evidence.

Arguments against the movement

According to the classification going back to Aristotle (“There are four discussions of Zeno about motion that cause great difficulties to those who try to solve them” - Arist. Phys. VI 9.239b), Z. E. put forward 4 arguments against the possibility of motion, which in later literature received stable names: “Dichotomy” (modern name; sometimes the names “Stages”, “Distance”, “Ristalis”), going back to Aristotle (Idem. Top. VIII 8. 160b), “Achilles and turtle”, “Arrow”, “Stadium” (sometimes also called “Moving Blocks”, “Stages”, “Rista”). All these paradoxes are united by the fact that they are based on the difficulties that arise when trying to rationally analyze the space and time continuum (Fritz. 1972. Sp. 58). The 4 paradoxes of Z. E. “represent a dilemma in which the possibility of movement is denied both from the point of view of accepting infinite divisibility, and from the point of view of accepting the absolute indivisibility” of space and time (Zeno of Elea. 1936. P. 103).

I. "Dichotomy". According to Aristotle, the essence of this aporia is that “a moving body must reach half before reaching the end” (Arist. Phys. VI 9.239b). According to a more complete version by Aristotle and Simplicius, Z. E.’s reasoning was structured as follows: in order to travel a certain path lying between 2 points, the body must first travel half of this path. But to go through this half, it must go through half of that half. Since halving can be performed an infinite number of times, the body will have to travel an infinite number of spatial segments in a limited time. This is impossible, which means that a moving body will never reach the end point of movement (DK. 29A25; cf.: Arist. Phys. VI 2.233a; 9.239b).

The first solution to this paradox was proposed by Aristotle. He recognized Z. E.'s premise about the infinite divisibility of space, but pointed out that it was incorrect to consider the time of movement to be finite - it is as infinite as space. However, this does not mean that every movement takes an infinite amount of time. According to Aristotle (Arist. Phys. VI 2.233a), both time and space are infinite in one aspect (he called this “infinity in division” - κατὰ διαίρεσιν ἀπείρων), but finite in another (“in quantity” or “in extent” " - κατὰ τὸ ποσόν). Therefore, if it takes a minute to intersect the entire segment between 2 points, then it takes half a minute to intersect half of it, a quarter of a minute to intersect half of half, etc. ad infinitum. The smaller the distances become, the shorter the time required to cross them becomes, so that the total time interval required to cross the entire distance can be divided into exactly the same (infinite) number of parts as it is divided into spatial interval between 2 points.

At the same time, although the given solution to the paradox “is sufficient to answer the one who posed the question this way” (i.e. Z. E.), it seemed to Aristotle not entirely satisfactory “for the essence of the matter and for the truth,” and a little lower in “ Physics” he again turns to the analysis of this aporia (Ibid. VIII 8. 263a-b). Aristotle saw the insufficiency of the first solution in the fact that it is unable to explain how a body, when traveling a distance, can touch an infinite number of points, even in an infinite time. To resolve this contradiction, Aristotle used the concepts of “potential infinity” and “actual infinity.” If “an infinite number of points” meant “an infinite number of actually existing points,” then the reasoning of the supporters of Z.E. would be correct, since a body cannot perform an infinite number of separate physical acts. However, in fact, according to Aristotle, an infinite number of points into which a finite distance is divisible exists only potentially (i.e., as a logical-mathematical construction) and for physical motion is a “collateral circumstance” (συμβεβηκός): “To the question , is it possible to traverse an infinite number of [parts] in time (ἐν χρόνῳ) or in length (ἐν μήκει), one should answer that... if they exist in reality (ἐντελεχείᾳ), - it is impossible if in possibility (δυνάμει), - it’s possible” (Ibidem; cf.: The Presocratic Philosophers. 1983. P. 270-272).

Although Aristotle's solution is generally recognized by most researchers as convincing within certain limits (for a detailed logical, mathematical and physical analysis of the aporia and a review of various positions, see Barnes. 1982. P. 261-273; Vlastos. Zeno's Race Course. 1966. P. 95-105; Idem. Zeno of Elea. 1995. P. 248-251; Grünbaum. 1967; Idem. 1969; Ferber. 1981), to this day there is no generally accepted answer to the raised Z. E. in Paradoxically, the question is: how can one carry out (complete) an infinite sequence of actions? Many modern scientists agree with Z. E. and postulate the impossibility of this (see, for example: Weyl H. Philosophy of Mathematics and Natural Science. Princeton (N. J.), 1949. Vol. 1. P. 42; Black. 1951; Idem. 1954. P. 95-126; Thomson J. Tasks and Super-tasks // Analysis. 1954. Vol. 15. N 1. P. 5-13 ), others, on the contrary, insist that this is possible and that this is the only way to overcome the aporia; it is noted that logically the proposition about the impossibility of an infinite sequence of actions is irrefutable (see: Barnes. 1982. P. 273). Thus, “philosophy has attempted to explain why, in a certain sense and in relation to certain sequences, the concept of a sequence that is at once infinite and complete does not contain a contradiction; but so far it has not been possible to build a theory that is convincing to all scientists” (Greek Philosophy. 2006. P. 55).

II. "Achilles and the Tortoise." Aporia is closely related to the previous one and is essentially its more complex version. Aristotle in “Physics” formulated the content of the argument as follows: “...The slowest [creature] can never be overtaken in a run by the fastest, for the pursuer must first come from where the fleeing one has already moved, so the slower one will always have to then [distance] is ahead of the pursuer” (Arist. Phys. VI 9.239b). In visual form, this argument by Z.E. can be presented as follows: it is assumed that Achilles runs 10 times faster than the tortoise, and at the start the difference between them is 100 meters. In order to win the race, Achilles must first overcome the initial distance of 100 meters and end up at the point where the turtle started. However, while he is doing this, the turtle has managed to move forward 10 meters. While Achilles is running these 10 meters, the tortoise has walked 1 meter; while Achilles overcomes this meter, the tortoise advances 1/10 of a meter, and so on ad infinitum. According to Z.E.'s conclusion, Achilles will never catch up with the tortoise, since it will always have an advantage, no matter how insignificant it may be (Black. 1951. P. 91).

The most common and traditional. The solution to this paradox of Z. E. is to point out that “it is based on a mathematical error” (Whitehead A. N. Process and Reality. N. Y., 1929. P. 107; cf. also: Descartes R. Oeuvres / Ed. C. Adam, P. Tannery. P., 1901. T. 4: Correspondance, juillet 1643 - avril 1647. P. 445-447; Peirce Ch. Collected Papers. Camb., 1960. Vol. 6. P. 176-177, 182) . If we consider the intervals of length that Achilles needs to go through in accordance with the above version of the paradox, then the complete series of distances will look like: 100+10+1+1/10+... This is a convergent geometric series, the sum of which can be represented in decimal notation as 111.1... but is exactly 1111/9. The same reasoning applies to the time it takes Achilles to catch up with the tortoise. If we assume that Achilles runs 100 meters in 10 seconds, then the number of seconds he will need to catch up with the turtle is 10 + 1 + 1/10 + 1/100 +... This is also a convergent geometric series, the sum of which in decimal expression is equal to 11.11... but exactly 111/9. From this it is obvious that there is an exact time and place where Achilles and the tortoise met. Thus, Z.E. was mistaken in being unable to see that the infinite sequence of steps that Achilles needs to take requires a finite time and a finite distance (Black. 1951. P. 92-93). Z. E.'s reasoning essentially testifies only to the trivial fact that until the moment Achilles meets the tortoise, Achilles will indeed always be behind the tortoise. At the same time, the essence of the paradox in Z. E. is the postulation that Achilles will always be behind the turtle, and this conclusion, based on the above argumentation, seems incorrect.

However, with all the rigor of tradition. of the decision it only says where and when Achilles and the tortoise will meet, if they meet. However, it is not able to prove that Z.E. was mistaken in believing that they could not meet at all. The paradox is that it is impossible to perform the addition of an infinite number of terms of a series in the same way as the addition of a finite number of terms of a series. If in the first case a finite number of addition acts are performed, then in the second a limit is established, i.e. it is postulated that the greater the number of terms of the numerical series taken, the smaller will be the difference between the sum of the finite number of terms taken and the limiting number 1111/9. In relation to the aporia “Achilles”, this means that although each time the distance that Achilles needs to travel to meet decreases, it will never become equal to zero; moreover, there is always an infinite number of smaller segments that need to be covered. Thus, the real difficulty of Z. E.’s aporia lies in the logical impossibility of carrying out an infinite series of actions.

On this basis, researchers (Barnes. 1982. P. 273-275; Vlastos. Zeno of Elea. 1995. P. 252-253; Black. 1951. P. 94; Fritz. 1972. Sp. 61-62) recognize correct judgment Aristotle that “Achilles” is a complicated version of “Dichotomy” and, in addition to greater clarity and catchiness, differs from it only in that “the taken value is not divided into two equal parts” (Arist. Phys. VI 9. 230b). According to the general opinion of researchers, “the logical complexity of Achilles lies not in the size of the distance that must be covered, but in the apparent impossibility of traveling any distance at all” (Black. 1951. P. 94). As M. Black rightly noted, Zeno had enough mathematical knowledge to understand that, having walked 1111/9 meters, Achilles would actually catch up with the tortoise. The difficulty is to understand how Achilles can even run anywhere without performing an infinite number of individual acts of movement (Ibidem). Thus, the entire argumentation of various scientists, which is built regarding the problems raised in “Dichotomy,” is applicable to “Achilles.”

III. "Arrow". Aporia is recognized by researchers as the most complex and important of the aporias related to movement (Fritz. 1972. Sp. 62). The argument has been preserved for several years. different formulations; the most concise and succinct is found in Aristotle’s “Physics” (Arist. Phys. VI 9.239b), where it is said that, according to Z.E., “an arrow released stands” (ἡ ὀϊστὸς φερομένη ἕστηκενή). According to Aristotle, this is justified by the following conclusion: “If every [body] is always at rest when it is in an equal place [ὅταν ᾖ κατὰ τὸ ἴσον], and a moving [body] is always at the moment “now” (ἐν τῷ νῦν) [in a place equal to itself], then the released arrow does not move (ἀκίνητον τὴν φερομένην εἶναι ὀϊστόν).” A more detailed formulation is given by Diogenes Laertius (Diog. Laert. IX 5; DK. 29B4) and St. Epiphanius of Cyprus. The latter conveys the course of Z. E.’s reasoning as follows: “What moves moves either in the place in which it is, or in the place in which it is not. But it cannot move either in the place in which it is, or in the place in which it is not. This means that it does not move at all” (Epiph. De fide // GCS. Bd. 31. S. 506). An important role in this version of the argumentation of Z. E. is played by the concept of “place” (τόπος), interpreted in accordance with the views of Aristotle as “the boundary of the encompassing body with which it comes into contact with the encompassed” (τὸ πέρας τοῦ περιέχοντος σώματος - Arist. Phys. IV 4 . 221b-212a; modern analysis of Aristotle’s teaching about place, see: Morison B. On Location: Aristotle's Concept of Place. Oxf., 2002). Among researchers, the question of how Aristotelian the concept of “moment of time” ( τὸ νῦν) and “place” correctly reflect Z.E.’s train of thought, but it is generally accepted that their use to a certain extent helps to trace the logic of Z.E.’s reasoning (Fritz. 1972. Sp. 62-63).

Based on this, Z.E.'s argument can be reconstructed as follows: movement means a change of place. But no body can be in two places at the same time. It always exists only in the place in which it exists, and this, in accordance with Aristotle’s definition, means: it always occupies a space exactly corresponding to its size. Being at a certain moment in a certain place, it does not move. Being at another moment in another place, it also does not move. This means that it does not move at all, since at any moment it is in a certain place (Ibid. Sp. 63). It was in this form that Z. E.’s argument was criticized by Aristotle, in whose opinion Z. E.’s false conclusion was due to the fact that the latter considered time as consisting of individual “now” moments. On the contrary, according to the teachings of Aristotle, time is not composed of indivisible “nows” (ἐκ τῶν νῦν τῶν ἀδιαιρέτων - Arist. Phys. VI 9. 239b). In his reasoning, Aristotle uses the analysis of the concepts of “time” and “now”, carried out by him in the 4th book. “Physicists” (Ibid. IV 10-14; for a modern presentation of Aristotle’s views, see Conen F. Die Zeittheorie des Aristoteles. Münch., 1964). According to Aristotle, every “now” (i.e., moment of time) divides (διαιρεῖ) time, but at the same time is not an extended “particle of time” (μόριον τοῦ χρόνου), but is only a “border” (πέρας - Arist. Phys. IV 12.220a), connecting the past with the future (see: Lear. 1981. P. 91). The totality of moments “now” is a totality of instants devoid of extension, which does not form a temporal value. Since the extension of time does not consist of “now,” then even if we assume that Z. E. is right and admit that at every moment of “now” the arrow does not move, it does not follow from this that the arrow is motionless throughout the time of its flight. Thus, according to Aristotle, the error of Z. E. is rooted in an incorrect understanding of the nature of time.

The rapid development of mathematics and natural sciences in the end. XIX - early XX century forced many researchers take a fresh look at the paradox of the Earth; at the same time, such scientists as Tannery, A. Bergson, A. N. Whitehead, Russell, P. Weiss, to one degree or another, recognized the correctness of certain provisions of Z.E. in the Arrow argument and tried to avoid its paradoxical conclusion by creating original theories of time and motion. Thus, Bergson believed that the paradox of Z. E. will lose force only if time is considered as pure extension, as a whole, in which there is “a sequence without division... interpenetration, interconnection and organization of elements, each of which represents a whole and cannot be separated from it except in abstract thinking” and cannot be considered as a countable set of individual elements (cited from English translation: Bergson A. Time and Free Will / Transl. F. L. Pogson L., 1910. P. 101, 105).

Serious attempts to reformulate and solve this aporia of Z. E. with the help of modern. mathematical apparatus were undertaken by G. Vlastos (Vlastos. A Note to Zeno's Arrow. 1966) and A. Grünbaum (Grünbaum. 1967). According to Vlastos' interpretation, the thesis that the arrow does not move at the moment of time, understood as unextended and indivisible whole, is fair, but it cannot be concluded from it that the arrow is generally at rest. For an unextended moment of time, talking about “movement” and “rest” makes no sense, just as it makes no sense to call a point “ straight" or "round" on the basis that straight lines and circles consist of points. Based on the mathematical formula of speed, Vlastos argued that we can talk about zero speed only in relation to a period of time that has a positive, and not zero, extension (Vlastos. A Note to Zeno's Arrow. 1966. P. 12-14). Referring to Russell's position, Vlastos concluded that it is permissible to talk about movement only during a certain time interval, but it is necessary to realize that this interval tends to zero, never going to zero (Ibid. S. 15-16; cf.: Russell B. Recent Work on the Principles of Mathematics // The International Monthly (Burlington, 1901. Vol. 4. P. 91).

Grünbaum, in his reading of this argument by Z.E. (as well as in solving other arguments), proceeded from the distinction between two types of time: mind-independent physical time and consciousness-dependent human experience time, consisting of discrete “nows”. According to Grünbaum, “Zeno's refutation will become possible if the psychological criterion of temporal sequence is replaced by a strictly physical criterion in which the definition for the proposition “event R is later than event A” does not require a discrete temporal order, but allows instead dense (dense) order” (Grünbaum. 1955. P. 237). Grünbaum believed that such a definition could be obtained by using the second law of thermodynamics, applied to classes of closed systems, to define the meaning of the term “later” (Ibid. P. 237-238). Unlike Aristotle, who interpreted the infinite as exclusively potential, Grünbaum, based on G. Cantor’s set theory, considered an infinite number of intervals of space and time to exist actually (Idem. 1967. P. 41). In general, the solution to paradoxes by Z. E. Grünbaum is based on two fundamental principles of Cantor’s continuum theory: a set of points lying on a segment or on a plane can be considered in the set-theoretic sense as an uncountable set; an infinite uncountable set of unextended points can have extension. Since this part of Cantor’s constructions still raises a number of questions among mathematicians today, the credibility of Grünbaum’s position directly depends on the willingness or unwillingness to accept Cantor’s set theory as a whole (Fritz. 1972. Sp. 67-68).

The relativity of the solutions to the paradoxes of Z.E. proposed by Vlastos and Grünbaum indicates that the very nature of human cognition contains the possibility of different understandings of continuous quantities and processes, ranging from the experimental-physical to the intellectual-logical. With all the achievements of modern times. philosophy and science in the development of these areas separately, their combination now represents a largely insoluble task (Ibid. Sp. 68-69; cf.: Fränkel. 1942. P. 8-9; Lear. 1981. P. 101 -102).

IV. "Stadium". The paradox is somewhat different in its focus from the previous ones and is not directly related to the problems of space and time continuums (Fritz. 1972. Sp. 60). In scientific literature, 2 main lines of consideration of aporia have been formed: in accordance with the 1st interpretation, its subject is the relativity of motion, in accordance with the 2nd - the problem of indivisible quantities.

Aristotle conveys the content of the argument as follows: “The fourth [argument] is about equal bodies moving along a stage in opposite directions past equal [fixed objects], some [moving] from the end of the stage, others from the middle with equal speed” (Arist. Phys. VI 9.239b). Aristotle's further explanations are quite confusing and can be understood in different ways. According to the most common interpretation, going back to Simplicius (DK. 29A28), the essence of the argument can be captured by imagining 3 rows of bodies equal in all respects, each of which contains 4 bodies. The bodies in the first row are at rest; the bodies in the second row move relative to the bodies of the first row so that at the beginning of the movement the 2 first bodies of the second row correspond to the 2 first bodies of the first row; the bodies in the third row move in the direction opposite to the direction of movement of the bodies of the second row, and at the same time, the first 2 bodies of the third row correspond to the last 2 bodies of the first row:

According to Z.E.’s reasoning, during the movement, the bodies of the third row will pass by 2 bodies of the first row in the same time, and during this time they will pass by 4 bodies of the second row. If we take into account the equality of the bodies, it turns out that in the same time the bodies of the third row traveled twice as far in relation to the bodies of the second row as in relation to the bodies of the first row. But to go twice the way in the same time is to go the same way in half the time. It turns out that the bodies traveled the same path both in the whole time and in half of this time, which is contradictory and therefore impossible. It was in this that Aristotle saw the paralogism of the argument of Z. E., who argued that, in accordance with the conclusion of Z. E., “half of the time is equal to [its] double [amount]” (Arist. Phys. VI 9. 239b). According to Aristotle, who proceeded from the idea of ​​an absolute state of rest, which serves as an objective measure of movement, Z. E.’s mistake here lies in the failure to distinguish between the concepts of “absolute movement” and “relative movement.” In the form proposed, Z.E.'s reasoning is clearly flawed, since it proceeds from the incorrect assumption that a body moving at a constant speed takes the same time to pass by 2 bodies of equal size, despite the fact that one of these bodies is moving in relation to the first body, and the other is at rest.

Assuming that such an obvious mistake could not have been made by Z.E., pl. researchers, starting with Tannery (1885), tried to question the accuracy of Aristotle’s presentation of the argumentation of Z. E. and proposed their own readings of the aporia (see: Owen. 1957/1958. P. 208-209; Vlastos. Zeno of Elea. 1995, pp. 254-255; Barnes, 1982, pp. 285-294). According to Tannery and his followers, Z. E. is not talking about 3 rows of bodies (AAAA, BBBB, CCCC), but about 3 indivisible quantities (A, B, C), “indivisible atoms of matter” (Vlastos. Zeno of Elea, 1995, pp. 254-255; cf. Barnes, 1982, pp. 291). Further, it is assumed that the time of movement is also an “indivisible amount of time”, or “a moment in time” (Ibidem). If we reason according to the above scheme, it turns out that body B, having traveled a distance s relative to body A in time t, will travel the same distance s relative to body C in time t/2. Thus, an indivisible moment of time turns out to be divisible. Thus, provided that the premises are accepted, “Stadium” becomes an effective argument refuting atomic ideas about space and time. Although in this version the argument really stands on a par with other arguments of Z.E. against the movement, this reading is rejected by many. modern by researchers as “not having any historical support” (Vlastos. Zeno of Elea. 1995. P. 255; Barnes. 1982. P. 291; Immerwahr. 1978. P. 23).

In modern scientific literature, original attempts to read the argument of Z. E. were also proposed by D. Furley (Furley D. J. Two Studies in the Greek Atomists. Princeton, 1967. P. 72-75) and J. Immerwahr (Immerwahr. 1978), to- Some proceeded from an alternative reading of the Greek. Aristotle's text (in particular, the expressions γίγνεσθαι παρὰ ἕκαστον) and argued that Z. E. was not talking about the time required for bodies to “pass” past each other, but about the time they were strictly opposite each other. Postulating this time as divisible and measurable leads to a paradox similar to the “Arrow” paradox and in the same way resolvable only after accepting the indivisibility of the “moment of time” (Ibid. P. 24-25). When interpreting the argument, Barnes drew attention to the fact that Z. E.’s reasoning is based on provisions based on the data of everyday experience, and therefore its solution requires a rethinking of ordinary ideas about movement. According to Barnes, only when the relativity of any movement is taken into account is this paradox of the Earth truly overcome (Barnes. 1982. P. 292-294).

Arguments against plurality

According to Proclus (DK. 29A15), Z. E. put forward 40 arguments designed to refute the doctrine of the existence of plurals. things, however in the present. The time is only known for a few times. methods of argumentation of Z. E. Although, unlike the arguments against movement, a significant part of the text of the arguments against set has been preserved in the original expressions of Z. E., quoted by Simplicius, the available quotations are far from the clarity and accuracy necessary for their unambiguous interpretation, following. which their content gave rise to many discussions among researchers, both on the question of their textual relationship and the initial course of Z. E.’s reasoning, and on the question of the interpretation of its individual premises, expressions and terms (detailed philosophical and philological analysis of the text of the arguments underlying For many subsequent works, see: Fränkel. 1942; cf. also: Makin. 1982). At the same time, it is generally accepted that the thesis against which Z. E.’s reasoning was directed was the simple assumption of the existence of plurality. of things. Z. E. built his reasoning in such a way that, provided that this thesis was accepted, his opponents inevitably fell into contradiction, being forced to recognize mutually exclusive statements as true (Simplicius. In Aristotelis Physicorum libros octo Commentaria. Berolini, 1882. Vol. 1. P. 139 ). Based on the text of Simplicius, two arguments of Z. E. lend themselves to relatively successful reconstruction, the first of which contains the opposition of “big” and “small”, and the second - “finite” and “infinite”.

I. “Big” and “small”. According to this argument by Z.E., “if there are many [beings], they are both great and small: great so much that they are infinite in magnitude, and small so much so that they have no magnitude” (DK. 29B2). Initially, Z.E.'s argument apparently consisted of 2 separate parts: in accordance with the course of presentation in Simplicius, in the 1st it was proved that things are “small”, and in the 2nd - that they are “great”. The content of the 1st part of Z. E.’s argument has not been preserved, however, based on indirect data from Simplicius, researchers conclude that Z. E. argued that “nothing has size” on the basis that “each of the many existing things is identical to itself and one" (Ibidem). Thus, correlating this evidence with the fragment of Melissa (DK. 30B9), Vlastos believes that Z. E. adhered to the position common to all Eleatics that a “single” thing should not have parts, otherwise it immediately becomes “many.” In this case, the entire 1st part of the argument looks like this: “If there were many things, each of them would have to have unity and self-identity. But nothing can be one if it has size, since everything that has size is divisible into parts, and everything that has parts cannot be one. This means that if there were many things, none of them would have size” (Vlastos. Zeno of Elea. 1995. P. 242).

If Z. E.’s reasoning really was like this, then its fallacy is obvious - the concepts “one” and “many” are semantically polysemantic, therefore a thing may well be “one” in a certain sense and “many” in another sense. This also applies to John Philoponus’s detailed discussion of Z. E. about the “one” Socrates, who at the same time is not alone: ​​he is at the same time “white,” “philosopher,” “pot-bellied,” etc. According to the conclusion Z.E., “the same cannot be one and many” (DK. 29A21). Specially analyzing this paradox (Barnes. 1982. P. 253-256), Barnes notes that it is easily resolved at the semantic level by strictly defining the meanings of the terms “one” and “many” and the principles of their predicative use. However, the problem raised by Z.E. is not completely exhausted, since the ontological question remains open about how a specific thing can simultaneously be one (self-identical) and many (changeable), so that the place of paradox is taken by the “antinomy of being” ( Ibid. P. 256).

Moving on to the 2nd part of the argument, Z.E. accepts the thesis opposite to the one just discussed: “If there are many [beings], each of them must necessarily have a certain size.” According to Z.E., “what has absolutely no size (μέγεθος), no thickness (πάχος), no volume (ὄγκος) does not exist at all” (DK. 29B2). So, if a lot exists, then it is necessary that it have certain dimensions, i.e. (for 3-dimensional space) length and thickness. However, anything that has dimensions can be divided into parts. Thus, in accordance with the reconstruction of Vlastos, the main thesis of the 2nd part of Z. E.’s argument can be formulated as follows: “A certain part of each existing thing (of many - D.S.) must lie outside (ἀπέχειν) another parts of the same being” (Vlastos. Zeno of Elea. 1995. P. 243). In any existing thing that has dimensions, you can always find 2 non-overlapping parts, in these parts you can find your own parts, and so on ad infinitum. According to Z.E., the sum of this infinite number of parts will itself be infinite, which means that many things will be “infinite in magnitude.” This results in the contradiction originally stated by Z. E. (for a formalized presentation of Z. E.’s argument, see: Barnes. 1982. P. 242-244).

The greatest interest in the analysis of this aporia of Z.E. among modern. Researchers are challenged by the problem of the divisibility of every thing to infinity. The simplest solution to aporia is to deny this divisibility - this is exactly what the ancient Greeks did. atomists who postulated the existence of indivisible elements, from which all things are composed (Ibid. P. 245-246). However, the physical indivisibility of matter does not exclude the possibility of carrying out logical division within the indivisible atoms themselves, and therefore Z. E.’s argument cannot be definitively refuted empirically. Researchers who disagree with the atomist approach to the problem often draw an analogy between this argument and the aporia “Dichotomy” - in both cases we are talking about infinite division and in both cases the aporia can be solved using modern tools. mathematics by referring to the fact that the sum of an infinite convergent sequence is a finite number (Vlastos. Zeno of Elea. 1995. P. 244-245). However, this decision was challenged by W. Abraham (Abraham. 1972) and Barnes, who pointed out on the basis of the evidence of Porphyry, Simplicius and other antiques. authors that the division in this case is not carried out according to the principle of dichotomy. On the contrary, all parts of division are divided into equal parts each time, so that the result is an infinite number of equal parts having a finite size. It is clear that the sum of such a set is also infinite (Barnes. 1982. P. 246-247). Barnes tried to solve the aporia of Z. E. with the help of a special interpretation of the infinity of the members of the set, in which the number of elements in the division carried out by Z. E. turns out to be finite (Ibid. P. 249-252). Serious attention was also paid to the fact that, in essence, Z. E. proves not that the “magnitude” of the totality of parts is infinite, but that the number of parts into which any physical quantity can be divided is infinite. Thus, the problem lies not in the area of ​​​​the size of the parts, but in the fundamental question of the possibility of carrying out infinite division. The paradox of Z. E. poses two intractable questions - physical (is there a limit after which further division of matter is impossible) and mathematical (what exactly does it mean to “get the sum of an infinite sequence”). Various attempts to solve them are necessarily philosophical, not scientific, in nature and are closely related to the general picture of the world accepted by one or another researcher (see: Greek Philosophy. 2006. P. 53; McKirahan. 2006. P. 873).

II. "Finite" and "infinite". This is the only argument that was entirely preserved in the expressions of Z. E. himself. According to the quote from Simplicius, in the 1st part of the argument, Z. E. argued that “if there are many [beings], there must necessarily be exactly so many of them, how many of them there are, and no more than themselves, and no less. If there are as many of them as there are, then they are finite” (DK. 29B3). According to Vlastos, this reasoning of Z.E., with all its apparent simplicity, could not be refuted by means of ancient Greek. science and lost force only after Cantor developed the doctrine of the properties of sets, in particular the existence of actually infinite sets (Vlastos. Zeno of Elea. 1995. P. 252; cf. Fritz. 1972. Sp. 73). Barnes sees in the words of Z. E. only sophistry, easily refuted with the help of modern. mathematical apparatus (Barnes. 1982. P. 252-253).

The 2nd part of the argument is constructed by analogy with the 1st: “If there are many [beings], then the existing ones are infinite [in number], since between the existing ones there are always other [beings], and between these last there are again other [beings] "(DK. 23B3). According to the simplest interpretation, Z. E.'s argument relies on the empirical fact that two things appear to be separate things only because there is something between them that separates them from each other. But this something, in turn, must be separated from the two named things by two other things, which will prevent the merging of the original things into one whole. This division can continue indefinitely. Among researchers, the question remains debatable about exactly what things Z. E. is talking about here - about objects of the physical world, geometric points or objects in consciousness (see: Fritz. 1972. Sp. 73; Vlastos. Zeno of Elea. 1995. P. 246; Barnes 1982: 253; McKirahan 2006: 874). Most researchers recognize as correct the indication of Simplicius that this argument is again a modification of the argument “Dichotomy” (DK. 29B3) and should be considered similarly to the latter. However, there are also original readings: for example, G. Frenkel believed that the essence of the argument is not the postulation of the physical separation of bodies, but the possibility of intellectual division of any object into 2 other objects, between which there will always be a distance sufficient to place 3 objects in it. th object, albeit as small as desired (Fränkel. 1942. P. 3-7).

Z.E. are also attributed to several. arguments against plurality, in particular, the argument about “like” and “unlike” conveyed by Plato in the dialogue “Parmenides” (Plat. Parm. 127d-e; see: McKirahan. 2006. P. 872), given by Aristotle in the treatise “On the Origin and destruction” argument “Exhaustive division” (Arist. De generat. et corrupt. 316a; see: Vlastos. Zeno of Elea. 1995. P. 246-248), however, the original content of these arguments by Z. E. is difficult to reconstruct and all attempts by researchers to reconstruct the course of his argumentation are exclusively hypothetical.

Other arguments

In addition to the arguments against motion and set, there are mentions and retellings (of varying degrees of accuracy) of 2 more arguments of Z. E. Thus, he is credited with the “Paradox of Place,” most accurately conveyed by John Philoponus: “If every being [exists] somewhere, and place If there is something [existing], then the place will be in the place, the second will be in the third, and so on ad infinitum” (DK. 29A24). From this Z.E. concluded that no place exists at all. Aristotle, trying to solve this paradox, noted that the expression “to be in something” does not necessarily indicate spatial abiding, but can indicate abiding in the sense of a property or state (Arist. Phys. IV 3.210b). However, this solution does not apply in the case when all the things in question are capable of occupying space in the spatial sense. Noting this, Barnes proposed his own solution to the aporia: one can accept that things exist in places and places exist in places if one realizes that places are also places for themselves (Barnes. 1982. P. 256-258) . The same idea was formulated with greater clarity by I. Newton: “Times and spaces are, as it were, places both for themselves and for other things” (Newton I. The Mathematical Principles of Natural Philosophy / Transl. A. Motte. N. Y., 1846. P. 79).

Another argument, preserved in the presentation of Simplicius and called “Millet grain” (DK. 29A29; cf.: Arist. Phys. VII 5 250a), is very characteristic of Z.E. This paradox is presented in the form of a dialogue between Z. E. and the sophist Protagoras. Its essence is as follows: Z. E. forces the interlocutor to admit that one millet grain or some part of it does not make any sound when falling to the ground. However, a large number of grains produce sound. Moreover, there is a proportion between one grain and many. grains, so there should be a proportion between the sound when one grain falls and the sound when many fall. grains This means, concludes Z.E., that even one grain, and even one ten-thousandth part of a grain, is noisy. The solution to this paradox depends on whether “noise” is understood as a physical or as a psychological concept. In the first case, Z.E.'s argument is correct - even one grain produces a corresponding vibration in the air. However, in the second case, taking into account a certain threshold of perception of human hearing, the “noise” of one grain cannot be caught by it (Fritz. 1972. Sp. 59).

Historical, philosophical and scientific significance of the aporias of Z. E.

The question of the significance of the teachings of Z. E. for ancient Greek. philosophy and science is closely connected with the question of what and whose views his aporias were directed against. Until the end XIX century The view that dates back to Plato dominated, according to which the aporia of Z. E. were called upon to indirectly defend the principles of Parmenides, showing the contradictory views of his opponents. It was believed that these opponents shared the everyday empirical concept of plurality and movement. According to the testimony of John Philoponus, “since those who admit plurality certified it on the basis of evidence,” Z. E. “wanted to sophistically refute the evidence” (DK. 29A21). However, in the end XIX century Tannery put forward a bold hypothesis that Z. E.’s real opponents were not supporters of the reliability of sensory perceptions, but certain representatives of the Pythagorean school, who defended the doctrine that all things consist of certain primary elements that combine the properties of an arithmetic unit, a geometric point and physical atom. According to Tannery, this teaching was completely destroyed by the argumentation of Z.E., which served as an incentive for the development of ancient Greek. mathematics. Initially enthusiastically received by scientists, Tannery's position later. was subjected to serious criticism (see, for example: Van der Waerden. 1940) and to this day. Few people accept time in its entirety (see: Vlastos. Zeno of Elea. 1995. P. 256-258). Moreover, there is no definitive evidence about the influence of Z. E.’s ideas on the development of mathematical knowledge in Dr. Greece is not found.

At the same time, the question of how much the argumentation of Z. E. is connected with the teaching of Parmenides continues to be debatable (see: Solmsen. 1971; Vlastos. 1975; Barnes. 1982. P. 231-236; Makin. 1982). Researchers paid special attention to the fact that some arguments of Z. E. (in particular, the 1st argument against set) can be used not only to refute set, but also to refute the doctrine of unity. At the same time, there are different positions on the question of what concept of unity Z. E. himself used - Parmenidean, Pythagorean or his own, whether he had in mind when arguing “a single being” or “one” as an element of multiplicity. All these difficulties, aggravated by the presence of only a very small number of genuine statements by Z. E., force scientists to cautiously assert that Z. E. defended a “modified version of the Eleatic theory,” perhaps in some parts significantly diverging from the teachings of Parmenides (Solmsen. 1971. P. 140). However, the doctrines of the One of Parmenides and Z.E., with a certain interpretation, may well be consistent (see detailed justification in the work: Kullmann. 1958), so that, according to the precise remark of A.F. Losev, there is enough reason to believe that Z. E. “not only crushed space and time to infinity, but also taught about that one thing that completely and continuously embraces all things and the whole world” (Losev A.F. History of ancient aesthetics: Early classics. M., 2000. P. 358; cf.: DK. 29A30). In this regard, the statement attributed to Z. E. acquires a special meaning: if they explain to him what one is, he will be able to teach about the plurality of existence (DK. 29A16, 21). Z. E.'s thought here, apparently, is that any plurality is possible only on the basis of an intuitively comprehended unity. Even more mysterious and intriguing is the fleeting testimony of Alexander of Aphrodisias, quoted by Simplicius (DK. 29A22), according to which Z. E. taught “that one is not one of the existing things” (μηδὲν τῶν ὄντων ἔστι τὸ ἕν). In these words it is quite possible to see the beginnings of ideas about the transcendence of the divine One, which is both “existent” and “superexistent”, “non-existent”. It is also quite possible that it was as a result of the analytical work of Z. E. in the Eleatic school that the opinion about the incorporeal nature of the One was clearly formulated, absent in Parmenides, but already found in Melissa: “If it exists, then there must be one, and if it one thing, it must not have a body” (DK. 30B9). Thus, the physical and logical studies of Z. E. had essentially a theological result - it turned out that the concept of unity and indivisibility is applicable only to God and turns out to be contradictory when used in relation to the things of this world.

The aporia of Z. E. undoubtedly had a serious influence on the atomism of Leucippus and Democritus (primarily in terms of their postulation of indivisible atoms as a means of avoiding the trap of “Dichotomy”); among the sophists, traces of the influence of Z. E. are present in the surviving fragments of Gorgias and the general philosophical methodology of Protagoras . The direct influence of the teachings of Z. E. on Plato’s thought can be traced only in “Parmenides” (discussions about the one and the many); the influence of the aporias on the formation of Aristotle’s physics, many others, was much more serious and widespread. concepts (time, motion and its continuity, etc.) were developed in harsh polemics with the views of Z. E. In the Hellenistic and late Byzantine eras, the interest of philosophers in the argumentation of Z. E. was generally limited to the area of ​​its consideration in Aristotle’s “Physics”; for example, precisely through the prism Aristotelian criticism Gennady II Scholarius, Patriarch of Constantinople, set out the aporia of Z. E. in treatises devoted to the interpretation of “Physics” and other works of Aristotle. Since the Middle Ages, the name Z.E. was practically consigned to oblivion.

In modern times, references to the paradoxes of the Earth are often found in the works of scientists and philosophers, but special interest in the teachings of the Earth awakens only from the 2nd half. XIX century, when a mathematical and logical apparatus began to take shape, which later made it possible to consider paradoxes at a qualitatively new level. A kind of summary of the centuries-long history of the aporias of the Earth was summed up by F. Cajori (Cajori. 1915), who systematized all references to the paradoxes of the movement of the Earth from Aristotle to the early Russell, considering them in the aspect of their relationship with the mathematical doctrine of the limit and Cantor's set theory. In the 20th century, after analysis in the fundamental works of Russell, Bergson, Whitehead, Frenkel, Grünbaum and many others. other authors, the ideas of Z. E. become a constant subject of development and scientific discussions in modern times. philosophy. Various studies of the aporia of Z.E., which continue to appear to this day. time (see, for example: McLaughlin, Miller. 1992; Alper, Bridger. 1997; Angel. 2001; Magidor. 2008), convincingly indicate that the difficulties recorded in paradoxes that arise when trying to combine empirical and intellectual-logical approaches to reality , continue to disturb human thought, forcing it again and again to turn to the search and creation of an adequate language for describing the surrounding reality.

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Another Note on Zeno's Arrow // Phronesis. 2008. Vol. 53. N 4/5. P. 359-372.

D. V. Smirnov

Zeno of Elea (ancient Greek: Ζήνων ὁ Ἐλεάτης). Born approx. 490 BC e. - died approx. 430 BC e. Ancient Greek philosopher, student of Parmenides, representative of the Eleatic school. Born in Elea, Lucania. He is famous for his aporias, with which he tried to prove the inconsistency of the concepts of movement, space and multitude.

Scientific discussions caused by these paradoxical reasonings have significantly deepened the understanding of such fundamental concepts as the role of discrete and continuous in nature, the adequacy of physical movement and its mathematical model, etc. These discussions continue to this day.

Zeno's works have come down to us in the presentation of Aristotle's commentators: Simplicius and Philoponus. Zeno also participates in Plato’s dialogue “Parmenides”, is mentioned in Diogenes Laertius, in the Court and many other sources.

Aristotle calls Zeno of Elea the first dialectician.

Son of Teleutagoras, studied with Xenophanes and Parmenides. As Diogenes Laertius reports, Zeno participated in a conspiracy against the then Elean tyrant, whose name was definitely unknown to Diogenes. Was arrested. During interrogation, when demanded to hand over his accomplices, he behaved steadfastly and even, according to Antisthenes, bit off his own tongue and spat it in the tyrant’s face. The citizens present were so shocked by what happened that they stoned the tyrant. According to Hermippus, Zeno was executed by the tyrant: he was thrown into a mortar and pounded in it.

Diogenes reports that Zeno was the lover of his teacher, but Athenaeus decisively refutes such a statement: “But what is most disgusting and most false is to say without any need that Parmenides’ fellow citizen Zeno was his lover.”

Contemporaries mentioned 40 aporia of Zeno, 9 have come down to us, discussed by Aristotle and his commentators. The most famous aporia about movement are: Achilles and the tortoise, Dichotomy, Arrow, Stadium.

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 Parmenides. 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" ('Εξήγησι ς 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 room” (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 motion, then one of two equal quantities moving with equal speed will in equal time cover 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 the Eleatic ontology and adhered to it was considered a sufficient explanation common sense, trusting feelings.

Fragments

• DK I, 247–258;
• Untersteiner M. (ed.). Zeno. Testimonianze e frammenti. Fir., 1963;
• Lee H.D.P.. Zeno of Elea. Camb., 1936;
• Kirk G.S., Raven J.E., Schofield M.(edd.). The Presocratic Philosophers. Camb., 1983 2;
• Lebedev A.V.. Fragments, 1989, p. 298–314.

Literature

• Raven J.E. Pythagoreans and Eleatics: An Account of the Interaction Between the Two Opposed Schools During the Fifth and Early Fourth Centuries B. C. Camb., 1948;
• Guthrie, HistGrPhilos II, 1965, p. 80–101;
• Vlastos G. Zeno's Race Course (= JHP 4, 1966);
• Idem. Zeno of Elea;
• Idem. A Zenonian Argument Against Plurality;
• Idem. Plato's Testimony Concerning Zeno of Elea, repr.:
• Vlastos G. Studies in Greek Philosophy. Vol. 1. The Presocratics. Princ., 1993;
• Grunbaum A. Modern Science and Zeno's Paradoxes. Middletown, 1967;
• Salmon W.Ch.(ed.). Zeno's Paradoxes. Indnp., 1970 (2001);
• Ferber R. Zenons Paradoxien der Bewegung und die Struktur von Raum und Zeit. Münch., 1981. Stuttg., 1995 2;
• Yanovskaya S.A.. Have you overcome modern science difficulties known as the "Aporius of Zeno"? – Problems of logic. M., 1963;
• Koyre A. Essays on the history of philosophical thought (translated from French). M., 1985, p. 27–50;
• Komarova V.Ya. The Teachings of Zeno of Elea: An Attempt to Reconstruct a System of Arguments. L., 1988.

Aristotle called Zeno the creator of dialectics, the art of putting forward arguments and refuting other people's opinions. To defend Parmenides' doctrine of a single, motionless being, Zeno formulated a series of aporia ("undecidable propositions"), showing that recognition of the reality of multiplicity and movement leads to logical contradictions. Of the four dozen aporias, the most famous are those about motion: Dichotomy,Achilles and the tortoise,Arrow And Stages(Moving bodies). All these aporia are proofs by contradiction. Together with a variant of their solution, they are presented in Aristotle ( Physics, VI, 9).

In the first two ( Dichotomy And Achilles and the tortoise) the infinite divisibility of space is assumed. So, no matter how fast Achilles runs, he will never catch up with the slow turtle, because during the time it takes him to run half of the intended path, the turtle, moving without stopping, will always crawl away a little more, and this process does not has completions, for space is divisible to infinity. The other two aporia consider the irreducibility of the continuity of space and time to indivisible “places” and “moments.” A flying arrow at any fixed moment of time occupies a certain place equal to its size - it turns out that within the framework of the indivisible moment itself it is “at rest”, and then it turns out that the movement of the arrow consists of the sum of states of rest, which is absurd. Therefore, the arrow is not actually moving. Throughout subsequent history, Zeno's aporias have been the subject of attention and debate among philosophers, logicians, and mathematicians (Leibniz, Kant, Cauchy, Cantor's set theory).

Maria Solopova

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 Metaphysics Aristotle says: “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, for the reason that everything what exists has spatial dimensions.” In a more complete form, this argument against the plurality of indivisible quantities is given by Philoponus: “Zeno, supporting his teacher, tried to prove that everything that exists must be one and motionless. He based his proof on the infinite divisibility of any continuity. Namely, he argued, if existence is not one and indivisible, but can be divided into many, there will essentially be no one at all (for if continuity can be divided, this will mean that it can be divided ad infinitum), and if nothing will be being essentially one, it is also impossible to have many, since many are 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 multitude exists, things are infinite in number, because between them there will always be more others, and between those 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 one. 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 speed, no matter how high. 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 spatial 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, because 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 proved that the runner cannot set off because he cannot leave the place in which he is; in "Achilles" it is proved 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 to calculate (either by the method of convergent infinite series or by a simple algebraic equation) to determine where and when Achilles will catch up with the tortoise. Suppose Achilles runs ten times faster than a tortoise, which travels 1 m per second and has a lead of 100 m. Let X– the distance in meters covered by the turtle by the time Achilles catches up with it, and t– time in seconds. Then t = x/1 = (100+x)/10 = 11 1/9 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 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 A 1, A 2, A 3 and A 4 be stationary bodies of equal size, and B 1, B 2, B 3 and B 4 be bodies of the same size as A, which uniformly move to the right so that each B passes each A in one instant, considering an instant to be the shortest possible period of time. Let C 1, C 2, C 3 and C 4 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 here it is obvious that C 1 passed all four bodies B. The time it took C 1 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 Parmenides Plato's reasoning goes like this: “If things are multiple, they must be both similar and dissimilar [dissimilar because they are not the same thing, 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 things cannot be dissimilar. Therefore 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 Physics Aristotle puts this problem as follows: “Further, if a place exists in itself, where is it located? After all, the difficulty to which Zeno arrives requires some explanation. Since everything that exists has a place, it is obvious that a place must also have a place, etc. to infinity". 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.

Nearchus (or Diomedon - the history of the Eleatic tyrants is unclear). They say that Zeno led a conspiracy against tyranny, but was captured, under torture he did not betray his friends, but slandered the tyrant’s friends. Unable to withstand further torment, he promised to tell the truth, and when the tyrant approached him, he sank his teeth into his ear, for which he was immediately killed by the servants. According to another version, Zeno bit off his own tongue, spat it in the tyrant's face, was thrown into a large mortar and crushed to death.

Philosophy of Zeno of Elea

Diogenes Laertius(IX, 29) reports that “the opinions of [Zeno of Elea] are as follows: worlds exist, but there is no void; the nature of all things came from warm, cold, dry and wet, turning into each other; people originated from the earth, and their souls are a mixture of the above-mentioned principles, in which none predominates.” If Diogenes did not confuse Zeno with someone else, then it can be assumed that the Eleanus considered it necessary to present not only the “truth”, but also an “opinion” similar to that of which Parmenides spoke. But the main thing in his teaching is that it substantiates Parmenides’ system “by contradiction.” The ancients attributed to Zeno of Elea 40 proofs “against plurality,” that is, in defense of the doctrine of the unity of existence, and 5 proofs “against motion,” in defense of its immobility. These proofs are called aporias, insoluble difficulties. Zeno's proofs against motion and four proofs against multiplicity, including both arithmetical, numerical, and spatial aspects, survive.

The meaning of the aporia of Zeno of Elea is that he explores the logical structure of the “world of opinion”, in which number and motion dominate, and draws consequences from these concepts. Since the consequences turn out to be contradictory, the concepts themselves are reduced to absurdity and discarded. In other words, the discovery of contradictions in strictly logically derived consequences of basic concepts ancient philosophy and ordinary consciousness is considered by Zeno as a sufficient basis for their elimination from the sphere of true knowledge, from the “path of truth.” The result is “negative dialectics”, based on the use of the laws of formal logic in application to existence. It is difficult to say who owns the explicit formulation of formal logical laws, but there is no doubt that Parmenides uses the laws of identity and contradiction, and Zeno also uses the law of the excluded middle. The aporia of Zeno of Elea clearly proceed from the idea that if A and not-A are given at the same time, and if not-A is contradictory, then it is false and A is true. This is the structure of all aporias. Let's look at them separately.

Aporia of Zeno of Elea against the plurality of beings

“So, if there is a multitude, then [things] must necessarily be both small and great: so small that they have no size at all, and so great that they are infinite.” Zeno's aporia relates to magnitude, and its justification if we compare it with famous teaching Pythagoreans that a thing is a sum of material points (“things”) will be as follows. If you add another thing that has magnitude to something that has magnitude, it will increase it. But in order to differ from another thing, the added thing must be distant from it, i.e. (since Zeno does not recognize emptiness!) between every two things there must lie another thing, between it and the first two - also in thing, etc. ... ad infinitum. This means that a thing composed of extended things is infinite in size. If it is composed of unextended things, then it does not exist at all. We can take this argument of Zeno from the quantitative side: if there are many things, then there are as many of them as there are, that is, a finite number. But if there are many of them, then, according to what was said above, a third is placed between each two of them, and so on ad infinitum. The source of the contradiction is the very concept of number or set: if there are many things, then the finite thing is both infinitely large and small, and the number of things in the world is both finite and infinite.