the most famous of the aporia:
1) Achilles will never catch up with the tortoise, because she will be able to take a step forward at the last moment
2) arrows do not fly, because at any given time they are in a state of rest
3) from point A we will never get to point B, because We will first go half the way, then another half way, then another, etc., but we will never reach point B, but only get as close as possible to it.

APORIA (from the Greek a - no and poros - way), hopelessness, impossibility, lack; seemingly insurmountable logical. difficulty, contradiction in solving a problem. In ancient Greek Zeno’s aporias (which revealed contradictions in the concepts of motion, time and space) were especially famous in philosophy”… (“Dictionary of Antiquity” M., SP “Vneshsigma”, 1992, p. 42).
Zeno of Elea (c. 490 - c. 430 BC) Greek. philosopher, student of Parmenides. Died during an unsuccessful uprising against a tyrant. Zeno tried to wittily prove the truth of his teacher’s sharply criticized teaching about the impossibility of movement and the multiplicity of things... His aporia “Achilles” is known: Achilles cannot catch up with the tortoise if it starts moving before him. The aporia is based on the underdevelopment of mathematics. concept of infinity...
His aporia, for example, “Strela” (“a flying arrow is at rest”), contributed to the development of logical. thinking (ibid., p. 213).
The most famous of Zeno's aporias are: "Measure", "Dichotomy", "Achilles", "Arrow", "Stadium" and "Millet Grain". So we're talking about the Arrow aporia.
Its essence, in general, sounds like this: an arrow fired from a bow will never reach its target. Why? Yes, because at any moment (in Zeno’s terms, at any “now”) of the arrow’s flight, it is at rest, and since the entire time of its flight consists of the sum of moments, it means the sum of rests. Therefore, from one big peace. And everything that is at rest does not move. Consequently, without being in motion, the arrow will never reach its target. Her movement is an ILLUSION of real movement...

Zeno was one of the representatives of the Eleatic school. He developed famous proofs supporting Parmenides' ideas that our senses deceive us, and only the mind can paint a true picture of the world. Zeno became famous for creating the so-called aporia, i.e. statements in which two contradictory propositions are simultaneously true. With the help of such aporia, Zeno tried to prove that the movement we observe does not actually exist, because when we begin to think about it, we encounter insurmountable difficulties and contradictions.

Below are Zeno's most famous aporias.

1. Achilles and the tortoise
Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise will crawl another ten steps, when he runs one step, the tortoise will crawl a tenth of a step, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.

They say that once Zeno told this aporia in a meeting of his colleagues, and one philosopher in response began to simply silently walk around the room, thereby as if saying: “Look, I’m moving, and Zeno claims that this is impossible!” However, by doing so, he does not refute Zeno’s aporia, because this aporia is built not on sensory (visual) understanding, but on rational understanding. Until now, many scientists are trying to refute this aporia, but it is very difficult to adequately refute such a logically correct aporia.

2. Dichotomy
Let's say a body must move from point A to point B. To overcome this path, first it must overcome half the path, and before that - a quarter. Before walking a quarter, the body will travel an eighth of the distance, before that - a sixteenth, and so on. It turns out that the body needs to go through an infinite number of sections on its path, but it cannot go through infinity. Therefore the body will never move.

Of course, we know that in fact bodies can move in space, because we constantly observe the movements of various bodies around us, but Zeno’s aporia makes us think about the truth of our observation.

3. Flying arrow
A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
Aristotle tried to refute this aporia. He wrote the following:

“Zeno argues incorrectly. If,” he says, “every body is always at rest when it is in a place equal to itself, and a moving body at the moment “now” is always in a place equal to itself, then a flying arrow is motionless. But this is incorrect, because time is not composed of indivisible “nows”, nor any other quantity.”

However, Aristotle’s position is clearly flawed, because he was never able to find logical errors in Zeno’s judgments. Indeed, his aporias are impeccable from the point of view of logic!

The Eleatic philosophical school (Eleatics) existed at the end - first half of the 5th century BC. e. , its founders are considered to be Xenophanes and Parmenides, the teacher of Zeno. The school developed a unique teaching about existence. The Eleatics defended the unity of being, believing that the idea of ​​​​the multiplicity of things in the Universe is an artificial division. The existence of the Eleatics is complete, real and knowable, but at the same time it is inseparable, unchangeable and eternal, it has neither past nor future, neither birth nor death. Knowledge of this integral world is possible only through rational reasoning, and the sensory picture of the world, including observable movements, is deceptive and contradictory. At the same time, the Eleatics also considered the geometric (and generally mathematical) method of cognition, characteristic of the Pythagoreans, a concession to sensory evidence, preferring a purely logical approach. From the same positions, for the first time in science, they raised the question of the admissibility of scientific concepts related to infinity.

The two aporias (Achilles and Dichotomy) assume that time and space are continuous and indefinitely divisible; Zeno shows that this assumption leads to logical difficulties. The third aporia (“Arrow”), on the contrary, considers time as discrete, composed of point-moments; in this case, as Zeno showed, other difficulties arise. Let us note that it is incorrect to assert that Zeno considered motion to be non-existent, because, according to Eleatic philosophy, it is impossible to prove the non-existence of anything: “what does not exist is unthinkable and inexpressible.” The purpose of Zeno's argument was more narrow: to identify contradictions in the opponent's position.

Often the “Stadium” is included among the aporias of the movement (see below), but in terms of theme this paradox is more likely to be related to the aporias of infinity. Next, the content of the aporia is retold using modern terminology.

Under the influence of the philosophical disputes that arose, two views on the structure of matter and space were formed: the first argued their infinite divisibility, and the second - the existence of indivisible particles, “atoms”. Each of these schools solved the problems posed by the Eleatics in its own way.

Achilles and the tortoise

• “In the rapid [flight] of an arrow there is a moment of absence of both movement and stop.”
• “If you take away half of a stick [length] of one chi every day, it will not be completed even after 10,000 generations.”

Aristotle's critique of aporias

Aristotle's position is clear, but not flawless - and primarily because he himself was unable to detect logical errors in the proofs or give a satisfactory explanation for the paradoxes... Aristotle was unable to refute the arguments for the simple reason that, in a logical sense, Zeno's proofs were impeccably compiled.

Atomistic approach

Epicurus of Samos

As a consequence, the observed movement changes from continuous to jerky. Alexander of Aphrodisias, a commentator on Aristotle, outlined the views of the supporters of Epicurus as follows: “Asserting that space, motion, and time are composed of indivisible particles, they also assert that a moving body moves throughout the entire extent of space, consisting of indivisible parts, and on each of the indivisible parts included in it there is no movement, but only the result of movement.” Such an approach immediately devalues ​​Zeno’s paradoxes, since it removes all infinities from there.

Discussion in New Times

The controversy surrounding Zeno's aporias continued in modern times. Until the 17th century, there was no interest in aporia, and their Aristotelian assessment was generally accepted. The first serious study was undertaken by the French thinker Pierre Bayle, author of the famous “Historical and Critical Dictionary” (). In an article on Zeno, Bayle criticized Aristotle's position and came to the conclusion that Zeno was right: the concepts of time, extension and motion involve difficulties that are insurmountable for the human mind.

Similar themes to aporia are addressed in Kant's antinomies. Hegel, in his History of Philosophy, emphasized that Zeno’s dialectic of matter “has not been refuted to this day” ( ist bis auf heutigen Tag unwiderlegt) . Hegel assessed Zeno as the “father of dialectics” not only in the ancient, but also in the Hegelian sense of the word dialectics. He noted that Zeno distinguishes between the sensory and conceivable movement. The latter, in accordance with his philosophy, Hegel described as a combination and conflict of opposites, as a dialectic of concepts. Hegel does not answer the question of how applicable this analysis is to real movement, limiting himself to the conclusion: “Zeno realized the definitions contained in our ideas about space and time, and discovered the contradictions contained in them.”

In the second half of the 19th century, many scientists, expressing a variety of points of view, analyzed Zeno's paradoxes. Among them :

and many others.

Modern interpretation

Quite often there have been (and continue to appear) attempts to mathematically refute Zeno’s reasoning and thereby “close the topic.” For example, by constructing a series of decreasing intervals for the aporia “Achilles and the tortoise,” one can easily prove that it converges, so that Achilles will overtake the tortoise. These “refutations,” however, change the essence of the dispute. In Zeno’s aporia, we are not talking about a mathematical model, but about real movement, and therefore it makes no sense to limit the analysis of the paradox to intramathematical reasoning - after all, Zeno precisely questions the applicability of idealized mathematical concepts to real movement. About the problem of adequacy of real movement and its mathematical model, see the next section of this article.

Usually they try to get around this paradox by arguing that the sum of an infinite number of these time intervals still converges and, thus, gives a finite period of time. However, this reasoning absolutely does not touch upon one essentially paradoxical point, namely the paradox that lies in the fact that a certain infinite sequence of events following each other, a sequence whose completion we cannot even imagine (not only physically, but at least in principle) , in fact, after all must end.

Serious studies of Zeno's aporia consider physical and mathematical models together. R. Courant and G. Robbins believe that to resolve the paradoxes it is necessary to significantly deepen our understanding of physical motion. Over time, a moving body sequentially passes through all points of its trajectory, however, if for any non-zero interval of space and time it is easy to indicate the interval following it, then for a point (or moment) it is impossible to indicate the point following it, and this violates the sequence. “There remains an inevitable discrepancy between the intuitive idea and the precise mathematical language designed to describe its main lines in scientific, logical terms. Zeno's paradoxes clearly reveal this discrepancy."

Hilbert and Bernays express the opinion that the essence of the paradoxes lies in the inadequacy of a continuous, infinitely divisible mathematical model, on the one hand, and physically discrete matter, on the other: “we do not necessarily have to believe that the mathematical space-time representation of motion has a physical value for arbitrarily small intervals of space and time.” In other words, paradoxes arise due to the incorrect application to reality of the idealized concepts of “point in space” and “moment of time”, which have no analogues in reality, because any physical object has non-zero dimensions, non-zero duration and cannot be divided infinitely.

A similar view can be found in Henri Bergson:

The contradictions pointed out by the Eleatic school concern not so much the movement itself as such, but rather the artificial transformation of movement that our mind makes.

The question of the infinite divisibility of space (undoubtedly posed by the early Pythagoreans) led, as is known, to significant difficulties in philosophy: from the Eleatics to Bolzano and Cantor, mathematicians and philosophers were unable to resolve the paradox - how a finite quantity can consist of an infinite number of points, having no size.

Bourbaki's remark means that it is necessary to explain how a physical process takes on infinitely many different states in a finite time. One possible explanation: space-time is actually discrete, that is, there are minimal portions (quanta) of both space and time. If this is so, then all paradoxes of infinity in aporia disappear. Discrete space-time was actively discussed by physicists back in the 1950s - in particular, in connection with projects of a unified field theory - but significant progress along this path could not be achieved.

S. A. Vekshenov believes that to solve paradoxes it is necessary to introduce a numerical structure that is more consistent with intuitive physical concepts than the Cantor point continuum. An example of a non-continuum theory of motion was proposed by Sadeo Shiraishi.

Adequacy of the analytical theory of motion

The general theory of motion with variable speed was developed in the late 17th century by Newton and Leibniz. The mathematical basis of the theory is mathematical analysis, initially based on the concept of an infinitesimal quantity. In the discussion about what constitutes an infinitesimal, two ancient approaches have been revived.

• The first approach, which Leibniz followed, dominated throughout the 18th century. Similar to ancient atomism, he views infinitesimals as a special kind of numbers (greater than zero, but less than any ordinary positive number). A rigorous justification for this approach (the so-called non-standard analysis) was developed by Abraham Robinson in the 20th century. Robinson's analysis is based on the extended numerical system ( hyperreal numbers). Of course, Robinson's infinitesimals bear little resemblance to ancient atoms, if only because they are infinitely divisible, but they allow us to correctly consider a continuous curve in time and space as consisting of an infinite number of infinitesimal sections.
• The second approach was proposed by Cauchy at the beginning of the 19th century. His analysis is based on ordinary real numbers, and the theory of limits is used to analyze continuous dependencies. Newton, d'Alembert and Lagrange held a similar opinion on the justification of the analysis, although they were not always consistent in this opinion.

Both approaches are practically equivalent, but from the point of view of a physicist, the first one is more convenient; Physics textbooks often contain phrases like “let dV- an infinitesimal volume...” On the other hand, the question of which approach is closer to physical reality has not been resolved. With the first approach, it is unclear what infinitesimal numbers correspond to in nature. In the second case, the adequacy of the physical and mathematical model is hampered by the fact that the operation of going to the limit is an instrumental research technique that has no natural analogue. In particular, it is difficult to talk about the physical adequacy of infinite series, the elements of which belong to arbitrarily small intervals of space and time (although such models are often and successfully used as an approximate model of reality). Finally, it has not been proven that time and space are structured in any way similar to the mathematical structures of real or hyperreal numbers.

Additional complexity was introduced into the question by quantum mechanics, which showed that in the microworld the role of discreteness is sharply increased. Thus, discussions about the structure of space, time and motion, begun by Zeno, are actively continuing and are far from completed.

Other aporias of Zeno

The above (most famous) aporia of Zeno concerned the application of the concept of infinity to motion, space and time. In other aporias, Zeno demonstrates other, more general aspects of infinity. However, unlike the three famous aporia about physical motion, other aporia are stated less clearly and relate mainly to purely mathematical or general philosophical aspects. With the advent of the mathematical theory of infinite sets, interest in them dropped significantly.

Stadium

The aporia “Stadium” (or “Rista”) in Aristotle (“Physics”, Z, 9) is not formulated entirely clearly:

The fourth [argument] is about equal bodies moving across the stadium in opposite directions parallel to bodies equal to them; some [move] from the end of the stage, others from the middle with equal speed, from which, as he thinks, it follows that half the time is equal to double.

Researchers have offered different interpretations of this aporia. L.V. Blinnikov formulated it as follows: .

S. A. Yanovskaya offers a different interpretation, based on atomistic premises:

Let time consist of indivisible extended atoms. Let us imagine two runners at opposite ends of the lists, so fast that each of them requires only one atom of time to run from one end of the lists to the other. And let both run out from opposite ends at the same time. When their meeting occurs, the indivisible atom of time will be divided in half, i.e., bodies cannot move into atoms of time, as was assumed in the aporia<Стрела>.

According to other interpretations, this aporia is similar to Galileo's paradox: an infinite set can be equal in power to its part.

Plurality

Part of the aporia is devoted to discussing the issue of the unity and plurality of the world.

Similar issues are discussed in Plato's dialogue Parmenides, where Zeno and Parmenides explain their position in detail. In modern language, this reasoning of Zeno means that multiple being cannot be actual indefinitely and therefore must be finite, but new things can always be added to existing things, which contradicts finitude. Conclusion: being cannot be multiple.

Commentators point out that this aporia, in its scheme, is extremely reminiscent of the antinomies of set theory discovered at the turn of the 19th and 20th centuries, especially Cantor’s paradox: on the one hand, the power of the set of all sets is greater than the power of any other set, but on the other hand, for any set it is not difficult to indicate a set of greater cardinality (Cantor’s theorem). This contradiction, quite in the spirit of Zeno’s aporia, is resolved unambiguously: the abstraction of the set of all sets is recognized as unacceptable and non-existent as a scientific concept.

Measure

In other words, if dividing a thing in half preserves its quality, then in the limit we obtain that the thing is both infinitely large (since it is infinitely divisible) and infinitely small. Moreover, it is not clear how an existing thing can have infinitesimal dimensions.

These same arguments are present in more detail in the commentaries of Philoponus. Also similar reasoning of Zeno is quoted and criticized by Aristotle in his Metaphysics:

If the one-in-itself is indivisible, then, according to Zeno’s position, it must be nothing. In fact, if adding something to a thing does not make it more and taking it away from it does not make it less, then, Zeno argues, this something does not belong to the existing, clearly believing that the existing is magnitude, and since magnitude, then something corporeal: after all, corporeal is fully existing; however, other quantities, such as plane and line, if added, increase in one case and not in another; point and unit do not do this in any way. And since Zeno argues roughly and since something indivisible can exist, and, moreover, in such a way that it will be in some way protected from Zeno’s reasoning (for if such an indivisible is added, it will not, however, increase, but will multiply), then the question is how to Will one such unit or several be the same value? Assuming this is like saying that a line is made up of points.

About the place

In Aristotle's account, the aporia states: if everything that exists is placed in a certain space ( place, Greek topos), then it is clear that there will be a space of space, and so it goes to infinity. Aristotle notes to this that place is not a thing and does not need its own place. This aporia allows for an expanded interpretation, since the Eleatics did not recognize space separately from the bodies located in it, that is, they identified matter and the space occupied by it. Although Aristotle rejects Zeno’s reasoning, in his “Physics” he comes to essentially the same conclusion as the Eleatics: a place exists only in relation to the bodies in it. At the same time, Aristotle passes over in silence the natural question of how a change in place occurs when a body moves.

Medimn grains

Zeno's formulation has been criticized, since the paradox is easily explained by reference to the threshold of sound perception - an individual grain does not fall silently, but very quietly, so the sound of the fall is not heard. The meaning of aporia is to prove that a part is not similar to the whole (qualitatively different from it) and, therefore, infinite divisibility is impossible. Similar paradoxes were proposed in the 4th century BC. e. Eubulides - the “Bald” and “Heap” paradoxes: “one grain is not a heap, adding one grain does not change the matter, with what number of grains does a heap begin?”

Historical significance of Zeno's aporia

“Zeno revealed the contradictions into which thinking falls when trying to comprehend the infinite in concepts. His aporias are the first paradoxes that arose in connection with the concept of the infinite." Aristotle's clear distinction between potential and actual infinity is largely the result of understanding Zeno's aporias. Other historical merits of the Eleatic paradoxes:

As noted above, the formation of ancient atomism was an attempt to answer the questions posed by aporia. Subsequently, mathematical analysis, set theory, and new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem, but the very fact of continuous keen interest in the ancient problem shows its heuristic fruitfulness.

Various points of contact between Zeno's aporia and modern science are discussed in the article by Zurab Silagadze. At the conclusion of this article, the author concludes:

The problems posed two and a half millennia ago and studied many times since then have not yet been exhausted. Zeno's paradoxes affect fundamental aspects of reality - location, motion, space and time. From time to time new and unexpected facets of these concepts are discovered, and every century finds it useful to return again and again to Zeno. The process of reaching their final resolution seems endless, and our understanding of the world around us is still incomplete and fragmentary.

Aporias of Zeno in literature and art

In this historical anecdote, the “bold-haired sage” is a supporter of Zeno (the commentator Elias, as mentioned above, attributed the argumentation to Zeno himself), and his opponent in different versions of the anecdote is Diogenes or Antisthenes (both of them lived significantly later than Zeno, so with him couldn't argue). One version of the anecdote, mentioned by Hegel, reports that when the Eleatic found Diogenes' argument convincing, Diogenes beat him with a stick for placing too much faith in evidence.

The plot of F. Dick’s fantastic story “About the Tireless Frog” is based on the aporia “Dichotomy”.

see also

Notes

1. , With. 90
2. , part 14
3. , 29. ZENON OF ELEA
4. , With. 15-16
5. , With. 116-118
6. Ivin A. A. According to the laws of logic. - M.: Young Guard, 1983. - 208 p. - (“Eureka”).
7. Great Soviet Encyclopedia // Aporia.
8. , part 16
9. Losev A. F. Zeno of Elea // Philosophical Encyclopedia. - M.: Soviet Encyclopedia, 1962. - T. 2.
10. Asmus V.F. Eleatic school // Ancient philosophy. - M.: Higher School, 2005. - 408 p. - ISBN 5-06-003049-0
11. , part 15
12. , With. 50-52
13. , With. 18-20
14. , Yanovskaya S. A.
15. , With. 21
16. "Physics" by Aristotle.
17. , With. 29-30
18. , With. 38
19. Lurie S. Essays from the history of ancient science. - M.-L.: Publishing house. Academy of Sciences of the USSR, 1947. - P. 181. - 403 p.
20. , With. 31-35
21. , With. 35-41
22. Hegel G. W. F. Works in 14 volumes. - M.: Sotsekgiz, 1959. - T. IX. - P. 244.
23. Tannery P. The first steps of ancient Greek science. - St. Petersburg. , 1902.
24. Papa-Grimaldi, Alba. Why Mathematical Solutions of Zeno's Paradoxes Miss the Point: Zeno's One and Many Relation and Parmenides" Prohibition. The Review of Metaphysics(1996). Archived from the original on August 28, 2011.
25. Gilbert D., Bernays P. Foundations of mathematics. Logical calculus and formalization of arithmetic. - M., 1979. - P. 40.
26. Courant R., Robbins G. What is mathematics. - 3rd ed. - M.: MTsNMO, 2001. - P. 353. - 568 p. - ISBN 5-900916-45-6
27. , With. 93
28. Quote By: Danzig, Tobias. Numbers are the language of science. - M.: Technosphere, 2008. - P. 111. - ISBN 978-5-94836-172-7
29. Nicolas Bourbaki. Architecture of mathematics. Essays on the history of mathematics. - M.: Foreign literature, 1963. - P. 38.
30. van Bendegem, Jean Paul (1987). "Discussion:Zeno's Paradoxes and the Tile Argument." Philosophy of Science 54 : 295-302. Retrieved 2010-02-27.
31. Kuznetsov B. G. Einstein. Life. Death. Immortality. - 5th ed., revised. and additional.. - M.: Nauka, 1980. - P. 368-374.
32. Klein M. Mathematics. Loss of certainty. - M.: Mir, 1984. - P. 401-402.
33. Uspensky V. A. What is non-standard analysis. - M.: Nauka, 1987.
34. Gaidenko P. P. The concept of time and the problem of continuum. Archived from the original on August 14, 2011. Retrieved January 10, 2011.
35. Silagadze, Z. K. Zeno meets modern science (English). Archived
36. Blinnikov L.V. A brief dictionary of philosophical personalities. Archived from the original on August 14, 2011. Retrieved April 30, 2010.
37. , With. 127
38. Zeno's Paradoxes, Stanford Encyclopedia of Philosophy.
39. Zeno of Elea. - Encyclopedia Around the World. Archived from the original on August 14, 2011. Retrieved December 30, 2010.
40. Aristotle. Metaphysics, Book I, Chapter IV.
41. Aristotle. Physics, IV, 1, 209a.
42. , With. 124-129
43. Ivin A. A. Logics. Study Guide, Chapter 7.
44. , With. 122-124
45. , With. 27
46. , With. 89
47. MOVEMENT.
48. , With. 19
49. Carroll, Lewis. Two-part invention, or What the tortoise said to Achilles // Knowledge is power .- 1991. - No. 9. - P. 6-12.
50. Valerie, Paul. Cemetery by the sea.

Literature

Ancient authors

• Ancient philosophers about Zeno. Archived from the original on August 14, 2011. Retrieved December 21, 2010.
• Aristotle. Physics. - In the collection: Philosophers of Greece. Fundamentals: logic, physics, ethics. - Kharkov: EKSMO, 1999. - 1056 p. - ISBN 5-04-003348-6
• Plato. Parmenides. - In the collection: Plato, Works in three volumes. - M.: Thought, 1968-1972. - (Philosophical heritage).
• Fragments of early Greek philosophers. Part I. From epic theocosmogonies to the emergence of atomism. - M.: Nauka, 1989. - 576 p.

Books by contemporary authors

• Asmus V.F. History of ancient philosophy. - M.: Higher School, 1965. - P. 40-45.
• Gaidenko P. P.. - M.: Science, 1980.
• History of mathematics / Edited by A. P. Yushkevich, in three volumes. - M.: Science, 1970. - T. I. - P. 88-93.
• Komarova V. Ya. The teachings of Zeno of Elea: an attempt to reconstruct the system of arguments // Vestnik LSU. - L., 1988.
• Kuznetsov B. G. Evolution of the world picture. - 1st ed. (2nd edition: URSS, 2010). - M.: Publishing House of the USSR Academy of Sciences, 1961. - 352 p. - (From the heritage of world philosophical thought: philosophy of science). - ISBN 978-5-397-01479-3.
• Makovelsky A. O. Pre-Socratics. In 3 volumes. - Minsk: Harvest, 1999. - 784 p. - (Classical philosophical thought)..
• Smorodinov R. A. The philosophy of consistent doubt. - Volgograd: Print, 2006. - P. 41-68.
• Grünbaum A. Modern science and Zeno's Paradoxes. - Allen & Unwin, 1968. - 153 p. - ISBN 978-0045130047
• Guenon R. Les Principes du Calcul infinitésimal. - Gallimard, 1946 and numerous reprints.- “Principles of computing infinitesimals”
• Salmon W. C. (editor) Zeno's paradoxes. - 2nd ed.. - Indianapolis: Hackett Publishing Co. Inc., 2001. - 320 p. - ISBN 978-0872205604

Brief bibliography of scientific articles with analysis of aporias

The literature is listed in chronological order.

• Svatkovsky V. P. Zeno's Paradox of a Flying Arrow // Journal of the Ministry of Public Education. - 1888. - No. 4 department. 5. - pp. 203-239.
• Khersonsky N. Kh. At the origins of the theory of knowledge. Regarding Zeno's arguments against motion // . - 1911. - No. XXXIV (August) department. 2. - pp. 207-221.
• Bolzano B. Paradoxes of the infinite. - Odessa, 1911.
• Bogomolov S. A. Arguments of Zeno of Elea in the light of the doctrine of actual infinity // Journal of the Ministry of Public Education. - 1915, new series. - No. LVI (April). - pp. 289-328.
• Dmitriev G. Once again about Zeno’s paradox “Achilles and the Tortoise” and V. Friedman’s confusion // Under the banner of Marxism. - 1928. - № 4.
• Bogomolov S. A. Actual infinity: Zeno of Elea, Isaac Newton and Georg Cantor. - L.-M., 1934.
• Yanovskaya S. A. Aporias of Zeno //

Encyclopedic YouTube

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Subtitles

Philosophy of the Eleatics

The Eleatic school of philosophy (Eleatics) existed from the end of the 6th century BC. e. until the first half of the 5th century BC. e., its ancestor is considered to be Parmenides, the teacher of Zeno. The school developed a unique doctrine of existence. Parmenides outlined his philosophical views in a poem, from which separate fragments have come down to us.

The Eleatics defended the unity of being, believing that the idea of ​​​​the multiplicity of things in the Universe is erroneous. The existence of the Eleatics is complete, real and knowable, but at the same time it is inseparable, unchangeable and eternal, it has neither past nor future, neither birth nor death. Thinking, said Parmenides’ poem, is identical in content to the subject of thinking (“thinking and what the thought is about are one and the same thing”). Further, Parmenides logically deduces the characteristics of a truly existing thing: it “has not arisen, is not destroyed, is integral [has no parts], unique, motionless and endless [in time].”

Knowledge of this integral world is possible only through reasonable (logical) reasoning, and the sensory picture of the world, including observable movements, is deceptive and contradictory. From the same positions, the Eleatics, for the first time in science, raised the question of the admissibility of scientific concepts related to infinity.

• “In the rapid [flight] of an arrow there is a moment of absence of both movement and stop.”
• “If you take away half of a stick [length] of one chi every day, it will not be completed even after 10,000 generations.”

Aristotle's critique of aporias

Aristotle's position is clear, but not flawless - and primarily because he himself was unable to detect logical errors in the proofs or give a satisfactory explanation for the paradoxes... Aristotle was unable to refute the arguments for the simple reason that, in a logical sense, Zeno's proofs were impeccably compiled.

Atomistic approach

As a consequence, the observed movement changes from continuous to jerky. Alexander of Aphrodisias, a commentator on Aristotle, outlined the views of the supporters of Epicurus as follows: “Asserting that space, motion, and time are composed of indivisible particles, they also assert that a moving body moves throughout the entire length of space, consisting of indivisible parts, and on each of the indivisible parts included in it there is no movement, but only the result of movement.” Such an approach immediately devalues ​​Zeno’s paradoxes, since it removes all infinities from there.

Discussion in New Times

The controversy surrounding Zeno's aporias continued in modern times. Until the 17th century, there was no interest in aporia, and their Aristotelian assessment was generally accepted. The first serious study was undertaken by the French thinker Pierre Bayle, author of the famous “Historical and Critical Dictionary” (). In an article on Zeno, Bayle criticized Aristotle's position and came to the conclusion that Zeno was right: the concepts of time, extension and motion involve difficulties that are insurmountable for the human mind.

Similar themes to aporia are addressed in Kant's antinomies. Hegel, in his History of Philosophy, emphasized that Zeno’s dialectic of matter “has not been refuted to this day” ( ist bis auf heutigen Tag unwiderlegt) . Hegel assessed Zeno as the “father of dialectics” not only in the ancient, but also in the Hegelian sense of the word dialectics. He noted that Zeno distinguishes between the sensory and conceivable movement. The latter, in accordance with his philosophy, Hegel described as a combination and conflict of opposites, as a dialectic of concepts. Hegel does not answer the question of how applicable this analysis is to real movement, limiting himself to the conclusion: “Zeno realized the definitions contained in our ideas about space and time, and discovered the contradictions contained in them.”

In the second half of the 19th century, many scientists, expressing a variety of points of view, analyzed Zeno's paradoxes. Among them :

and many others.

Modern interpretation

Quite often there have been (and continue to appear) attempts to mathematically refute Zeno’s reasoning and thereby “close the topic.” For example, by constructing a series of decreasing intervals for the aporia “Achilles and the tortoise,” one can easily prove that it converges, so that Achilles will overtake the tortoise. These “refutations,” however, change the essence of the dispute. In Zeno’s aporia, we are not talking about a mathematical model, but about real movement, and therefore it makes no sense to limit the analysis of the paradox to intramathematical reasoning - after all, Zeno precisely questions the applicability of idealized mathematical concepts to real movement. About the problem of adequacy of real movement and its mathematical model, see the next section of this article.

Usually they try to get around this paradox by arguing that the sum of an infinite number of these time intervals still converges and, thus, gives a finite period of time. However, this reasoning absolutely does not touch upon one essentially paradoxical point, namely the paradox that lies in the fact that a certain infinite sequence of events following each other, a sequence whose completion we cannot even imagine (not only physically, but at least in principle) , in fact, after all must end.

Serious studies of Zeno's aporia consider physical and mathematical models together. R. Courant and G. Robbins believe that to resolve the paradoxes it is necessary to significantly deepen our understanding of physical motion. Over time, a moving body sequentially passes through all points of its trajectory, however, if for any non-zero interval of space and time it is easy to indicate the interval following it, then for a point (or moment) it is impossible to indicate the point following it, and this violates the sequence. “There remains an inevitable discrepancy between the intuitive idea and the precise mathematical language designed to describe its main lines in scientific, logical terms. Zeno's paradoxes clearly reveal this discrepancy."

Hilbert and Bernays express the opinion that the essence of the paradoxes lies in the inadequacy of a continuous, infinitely divisible mathematical model, on the one hand, and physically discrete matter, on the other: “we do not necessarily have to believe that the mathematical space-time representation of motion has a physical value for arbitrarily small intervals of space and time.” In other words, paradoxes arise due to the incorrect application to reality of the idealized concepts of “point in space” and “moment of time”, which have no analogues in reality, because any physical object has non-zero dimensions, non-zero duration and cannot be divided infinitely.

Similar points of view can be found in Henri Bergson and Nicolas Bourbaki. According to Henri Bergson:

The contradictions pointed out by the Eleatic school concern not so much the movement itself as such, but rather the artificial transformation of movement that our mind makes.

Bergson believed that there is a fundamental difference between movement and distance traveled. The distance traveled can be arbitrarily divided, while movement cannot be arbitrarily divided. Every step of Achilles and every step of the tortoise must be considered as indivisible. The same applies to the flight of an arrow:

The truth is that if an arrow leaves point A and hits point B, then its movement AB is as simple, as indecomposable - since it is movement - as the tension of the bow that launches it.

- Bergson A. Creative evolution. Chapter Four. Cinematic mechanism of thinking and mechanistic illusion. A look at the history of systems, real formation and false evolutionism

The question of the infinite divisibility of space (undoubtedly posed by the early Pythagoreans) led, as is known, to significant difficulties in philosophy: from the Eleatics to Bolzano and Cantor, mathematicians and philosophers were unable to resolve the paradox - how a finite quantity can consist of an infinite number of points, having no size.

Bourbaki's remark means that it is necessary to explain how a physical process takes on infinitely many different states in a finite time. One possible explanation: space-time is actually discrete, that is, there are minimal portions (quanta) of both space and time. If this is so, then all paradoxes of infinity in aporia disappear. Richard Feynman stated:

The theory that space is continuous seems wrong to me because [in quantum mechanics] it leads to infinitely large quantities and other difficulties. In addition, it does not answer the question of what determines the sizes of all particles. I strongly suspect that simple representations of geometry extended to very small areas of space are incorrect.

Discrete space-time was actively discussed by physicists back in the 1950s - in particular, in connection with projects of a unified field theory - but significant progress along this path was not achieved.

S. A. Vekshenov believes that to solve paradoxes it is necessary to introduce a numerical structure that is more consistent with intuitive physical concepts than the Cantor point continuum. An example of a non-continuous theory of motion was proposed by Sadeo Shiraishi.

Adequacy of the analytical theory of motion

The general theory of motion with variable speed was developed at the end of the 17th century by Newton and Leibniz. The mathematical basis of the theory is mathematical analysis, which was initially based on the concept of an infinitesimal quantity. In the discussion about what constitutes an infinitesimal, two ancient approaches have been revived.

• The first approach, which Leibniz followed, dominated throughout the 18th century. Similar to ancient atomism, he views infinitesimals as a special kind of numbers (greater than zero, but less than any ordinary positive number). A rigorous justification for this approach (the so-called non-standard analysis) was developed by Abraham Robinson in the 20th century. Robinson's analysis is based on the extended numerical system ( hyperreal numbers). Of course, Robinson's infinitesimals bear little resemblance to ancient atoms, if only because they are infinitely divisible, but they allow us to correctly consider a continuous curve in time and space as consisting of an infinite number of infinitesimal sections.
• The second approach was proposed by Cauchy at the beginning of the 19th century. His analysis is based on ordinary real numbers, and the theory of limits is used to analyze continuous dependencies. Newton, d'Alembert and Lagrange held a similar opinion on the justification of the analysis, although they were not always consistent in this opinion.

Both approaches are practically equivalent, but from the point of view of physics, the first one is more convenient; Physics textbooks often contain phrases like “let dV- an infinitesimal volume...” On the other hand, the question of which approach is closer to physical reality has not been resolved. With the first approach, it is unclear what infinitesimal numbers correspond to in nature. In the second case, the adequacy of the physical and mathematical model is hampered by the fact that the operation of going to the limit is an instrumental research technique that has no natural analogue. In particular, it is difficult to talk about the physical adequacy of infinite series, the elements of which belong to arbitrarily small intervals of space and time (although such models are often and successfully used as an approximate model of reality). Finally, it has not been proven that time and space are structured in any way similar to the mathematical structures of real or hyperreal numbers.

Additional complexity was introduced into the question by quantum mechanics, which showed that in the microworld the role of discreteness is sharply increased. Thus, discussions about the structure of space, time and motion, begun by Zeno, are actively continuing and are far from completed.

Other aporias of Zeno

The above (most famous) aporia of Zeno concerned the application of the concept of infinity to motion, space and time. In other aporias, Zeno demonstrates other, more general aspects of infinity. However, unlike the three famous aporia about physical motion, other aporia are stated less clearly and relate mainly to purely mathematical or general philosophical aspects. With the advent of the mathematical theory of infinite sets, interest in them dropped significantly.

Stadium

The aporia “Stadium” (or “Rista”) in Aristotle (“Physics”, Z, 9) is not formulated entirely clearly:

The fourth [argument] is about equal bodies moving across the stadium in opposite directions parallel to bodies equal to them; some [move] from the end of the stage, others from the middle with equal speed, from which, as he thinks, it follows that half the time is equal to double.

Researchers have offered different interpretations of this aporia. L.V. Blinnikov formulated it as follows:

Let time consist of indivisible extended atoms. Let us imagine two runners at opposite ends of the lists, so fast that each of them requires only one atom of time to run from one end of the lists to the other. And let both run out from opposite ends at the same time. When their meeting occurs, the indivisible atom of time will be divided in half, that is, in the atoms of time, bodies cannot move, as was assumed in the Arrow aporia.

According to other interpretations, the idea of ​​this aporia is similar to Galileo’s paradox or Aristotle’s “wheel”: an infinite set can be equal in power to its part.

Plurality

Part of the aporia is devoted to discussing the issue of the unity and plurality of the world.

Similar issues are discussed in Plato's dialogue Parmenides, where Zeno and Parmenides explain their position in detail. In modern language, this reasoning of Zeno means that multiple being cannot be actual infinitely and therefore must be finite, but new things can always be added to existing things, which contradicts finitude. Conclusion: being cannot be multiple.

Commentators point out that this aporia, in its scheme, is extremely reminiscent of the antinomies of set theory discovered at the turn of the 19th and 20th centuries, especially Cantor’s paradox: on the one hand, the power of the set of all sets is greater than the power of any other set, but on the other hand, for any set it is not difficult to indicate a set of greater cardinality (Cantor’s theorem). This contradiction, quite in the spirit of Zeno’s aporia, is resolved unambiguously: the abstraction of the set of all sets is recognized as unacceptable and non-existent as a scientific concept.

Measure

In other words, if dividing a thing in half preserves its quality, then in the limit we obtain that the thing is both infinitely large (since it is infinitely divisible) and infinitely small. Moreover, it is not clear how an existing thing can have infinitesimal dimensions.

These same arguments are present in more detail in the commentaries of Philoponus. Also similar reasoning of Zeno is quoted and criticized by Aristotle in his Metaphysics:

If the one-in-itself is indivisible, then, according to Zeno’s position, it must be nothing. In fact, if adding something to a thing does not make it more and taking it away from it does not make it less, then, Zeno argues, this something does not belong to the existing, clearly believing that the existing is magnitude, and since magnitude, then something corporeal: after all, corporeal is fully existing; however, other quantities, such as plane and line, if added, increase in one case and not in another; point and unit do not do this in any way. And since Zeno argues roughly and since something indivisible can exist, and, moreover, in such a way that it will be in some way protected from Zeno’s reasoning (for if such an indivisible is added, it will not, however, increase, but will multiply), then the question is how to Will one such unit or several be the same value? Assuming this is like saying that a line is made up of points.

About the place

In Aristotle's account, the aporia states: if everything that exists is placed in a certain space ( place, Greek topos), then it is clear that there will be a space of space, and so it goes to infinity. Aristotle notes to this that place is not a thing and does not need its own place. This aporia allows for an expanded interpretation, since the Eleatics did not recognize space separately from the bodies located in it, that is, they identified matter and the space occupied by it. Although Aristotle rejects Zeno’s reasoning, in his “Physics” he comes to essentially the same conclusion as the Eleatics: a place exists only in relation to the bodies in it. At the same time, Aristotle passes over in silence the natural question of how a change in place occurs when a body moves.

Medimn grains

Zeno's formulation has been criticized, since the paradox is easily explained by reference to the threshold of the perception of sound - an individual grain does not fall silently, but very quietly, so the sound of the fall is not heard. The meaning of aporia is to prove that a part is not similar to the whole (qualitatively different from it) and, therefore, infinite divisibility is impossible. Similar paradoxes were proposed in the 4th century BC. e. Eubulides - the “Bald” and “Heap” paradoxes: “one grain is not a heap, adding one grain does not change the matter, with what number of grains does a heap begin?”

Historical significance of Zeno's aporia

“Zeno revealed the contradictions into which thinking falls when trying to comprehend the infinite in concepts. His aporias are the first paradoxes that arose in connection with the concept of the infinite." Aristotle's clear distinction between potential and actual infinity is largely the result of understanding Zeno's aporias. Other historical merits of the Eleatic paradoxes:

As noted above, the formation of ancient atomism was an attempt to answer the questions posed by aporia. Subsequently, mathematical analysis, set theory, and new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem, but the very fact of continuous keen interest in the ancient problem shows its heuristic fruitfulness.

Various points of contact between Zeno's aporia and modern science are discussed in the article by Zurab Silagadze. At the conclusion of this article, the author concludes:

The problems posed two and a half millennia ago and studied many times since then have not yet been exhausted. Zeno's paradoxes affect fundamental aspects of reality - location, motion, space and time. From time to time new and unexpected facets of these concepts are discovered, and every century finds it useful to return again and again to Zeno. The process of reaching their final resolution seems endless, and our understanding of the world around us is still incomplete and fragmentary.

Aporias of Zeno in literature and art

In this historical anecdote, the “bold-haired sage” is a supporter of Zeno (the commentator Elias, as mentioned above, attributed the argumentation to Zeno himself), and his opponent in different versions of the anecdote is Diogenes or Antisthenes (both of them lived significantly later than Zeno, so with him couldn't argue). One version of the anecdote, mentioned by Hegel, reports that when the Eleatic found Diogenes' argument convincing, Diogenes beat him with a stick for placing too much faith in evidence.

The plot of F. Dick’s fantastic story “About the Tireless Frog” is based on the aporia “Dichotomy”.

The Achilles aporia is mentioned several times in the works of Borges. The paradoxical situation described in it is also reflected in various humorous works. Takeshi Kitano directed the film Achilles and the Tortoise in 2008.

see also

Notes

1. , With. 90.
2. , part 14.
3. , 29. ZENON OF ELEA.
4. , With. 15-16.
5. , With. 116-118.
6. Ivin A. A. According to the laws of logic. - M.: Young Guard, 1983. - 208 p. - (“Eureka”).
7. Rozhansky I. D. Ancient science. - M.: Nauka, 1980. - P. 52. - 198 p. - (History of science and technology).
8. Great Soviet Encyclopedia// Aporia.
9. Parmenides / A. V. Lebedev // V. S. Stepin Mysl, 2010. - 2816 p.
10. Eleatic School / A. V. Lebedev // New Philosophical Encyclopedia: in 4 volumes / prev. scientific editor Council V. S. Stepin. - 2nd ed., rev. and additional - M.: Mysl, 2010. - 2816 p.
11. Rozhansky I. D. Early Greek philosophy // Fragments of early Greek philosophers
12. , part 16.
13. Losev A. F. Zeno of Elea // Philosophical Encyclopedia. - M.: Soviet Encyclopedia, 1962. - T. 2.
14. Asmus V.F. Eleatic school // Ancient philosophy. - M.: Higher School, 2005. - 408 p. - ISBN 5-06-003049-0.
15. , part 15.
16. Zeno of Elea // Stanford Encyclopedia of Philosophy.
17. , With. 50-52.
18. Diogenes Laertius. Life, teachings and sayings of famous philosophers, chapter “Pythagoras”.
19. , With. 18-20.
20. , With. 21.
21. "Physics" by Aristotle.
22. , With. 29-30.
23. , With. 38.
24. Lurie S. Essays from the history of ancient science. - M.-L.: Publishing house. Academy of Sciences of the USSR, 1947. - P. 181. - 403 p.
25. , With. 31-35.
26. , With. 35-41.
27. Hegel G. W. F. Works in 14 volumes. - M.: Sotsekgiz, 1959. - T. IX. - P. 244.
28. Tannery P. The first steps of ancient Greek science. - St. Petersburg. , 1902.
29. Papa-Grimaldi, Alba. Why Mathematical Solutions of Zeno"s Paradoxes Miss the Point: Zeno"s One and Many Relation and Parmenides Prohibition (undefined) . The Review of Metaphysics(1996). Archived from the original on August 28, 2011.
30. Gilbert D., Bernays P. Foundations of mathematics. Logical calculus and formalization of arithmetic. - M., 1979. - P. 40.
31. Courant R., Robbins G. What is mathematics. - 3rd ed. - M.: MTsNMO, 2001. - P. 353. - 568 p. - ISBN 5-900916-45-6.
32. , With. 93.
33. Quote By: Danzig, Tobias. Numbers are the language of science. - M.: Tekhnosphere, 2008. - P. 111. - ISBN 978-5-94836-172-7.
34. Nicolas Bourbaki. Architecture of mathematics. Essays on the history of mathematics. - M.: Foreign literature, 1963. - P. 38.
35. van Bendegem, Jean Paul (1987). “Discussion:Zeno"s Paradoxes and the Tile Argument”. Philosophy of Science. Belgium. 54 : 295–302. Access date 2010-02-27.
36. Feynman R. The nature of physical laws. - Ed. 2nd. - M.: Nauka, 1987. - P. 152-153. - 160 s. - (Bible Quantum, issue 62).
37. Kuznetsov B. G. Einstein. Life. Death. Immortality. - 5th ed., revised. and additional - M.: Nauka, 1980. - P. 368-374.
38. Klein M. Mathematics. Loss of certainty. - M.: Mir, 1984. - P. 401-402.
39. Uspensky V. A. What is non-standard analysis. - M.: Nauka, 1987.

Zeno is a student of Parmenides who defended his arguments.
Non-existence cannot exist, because... this is space separated from matter, this is emptiness. Everything that exists is in space, therefore, in order to exist, non-existence must be outside space. Or the space must be somewhere (in the second space, 2nd in the third, etc.). But this is impossible, because... space is inseparable from matter.
He believed that every movement is contradictory and, therefore, untrue. Zeno created a whole series of aporia (difficulties) aimed against recognizing the truth of movement.
-Dichotomy argument. Proves that movement is impossible by doing the opposite. It is assumed that movement is possible. To go from point A to point B you need to go half of this distance, to go this half you need to go half of the half and so on ad infinitum. As a result, a body cannot go from point A to point B, because there will always be some distance between them.
-Argument "Flying Arrow". The path of the arrow's movement consists of the sum of the rest points. At each point of movement the arrow is at rest. Movement cannot arise from a number of states of rest. The arrow flies, i.e. moves in space. But, at the moment of flight, it occupies a space equal to its length, i.e. resides within this space - that means she is motionless in it.
-Argument "Achilles and the Tortoise". Achilles, being at the beginning of the movement behind the turtle, must again and again cover the distance that the turtle had previously covered: during the time that Achilles covers the distance separating him from the turtle, the turtle will crawl half of this distance; Achilles will overcome half, and the tortoise will crawl another quarter, etc. to infinity. The distance between them should be zero, but this is impossible.
According to Zeno, space exists; it consists of many segments and cannot be empty.

4.Aporia of Zeno.

Zeno of Elea (c. 490 BC - c. 430 BC)- ancient Greek philosopher, student Parmenides. Born in Ella. Famous for its aporia, proving the impossibility of movement, space and multitude.

Mentioned about 40 aporia of Zeno, but only 9.

1. Achilles and the Tortoise.
2. Dichotomy.
3. Arrow.
4. Stadium

All of them are described in Aristotle.

Aporia ''Dichotomy'' and ''Arrow'' are reminiscent of the following paradoxical aphorisms attributed to the leading representative of the ancient Chinese ‘ 'Schools of Names' (Ming Wuzya) Gongsun Lunu: ''In the rapid flight [of an arrow] there is a moment of absence and movement and stop''. ‘’If you take away half of a [long] stick every day, it will not be accomplished even after 10000 generations''.

The internal contradictions of the concept of movement are clearly revealed in the aporia "Achilles''.

Achilles cannot catch up with the tortoise.

Every time, with all the speed of his running and with all the smallness of the space separating them, as soon as he steps on the place that the turtle occupied in front of him, she will move forward a little. No matter how much the space between them decreases, it is infinite in its divisibility into intervals, and they must be passed through all of them, and this requires infinite time.

A similar thing is said in the aporia of the arrow, where Zeno says the following about the arrow.

In every period of time and position in space, she is motionless and this makes her immovable. The arrow is constantly in certain positions and it follows that the positions of the arrow are not different - it is at rest. Zeno’s aporia “Achilles” and “Arrow” reveal the deep mystery of how motion is born from immobility, the apparent lack of dimensions (the arrow is at rest at every moment).

Parmenides' student and friend Zeno took upon himself the task of defending Parmenides' views against the objections put forward. He was born at the beginning of the 5th century. BC. (480) and died in 430. Only numerous and small-sized extracts made by later ancient writers survive from his writings. Of these, the first place should be given to the testimony of Aristotle in the Physics, as well as the testimony of Simplicius, a commentator on Aristotle’s Physics. They make it possible to characterize the new things that Zeno brought to Greek science in comparison with Parmenides, despite the naivety of his argumentation in detail.

Zeno developed a number of arguments in defense of the teachings of Parmenides. The method he used in these arguments subsequently gave Aristotle grounds to call Zeno the founder of “dialectics.” In this case, Aristotle understands by “dialectics” the art of clarifying the truth by detecting internal contradictions contained in the thoughts of the enemy, and by eliminating these contradictions.

Zeno's method is similar to what is called in mathematics<доказательством от противного>. Zeno accepts - conditionally - the theses of Parmenides' opponents. He accepts (1) that space can be conceived as emptiness, as separate from the substance that fills space; (2) that the existence of many things is conceivable; (3) that movement can be conceivable. Having tentatively accepted these three assumptions, Zeno proves that their recognition necessarily leads to contradictions. This proves that these assumptions are false. But if they are false, then the statements contradicting them must necessarily be true. And these are the statements of Parmenides. Therefore, Parmenides’ statements are true: emptiness, multitude and motion are inconceivable. Neither Parmenides nor Zeno deny the reality of emptiness, multitude, and movement for our senses. They only deny the possibility of thinking emptiness, thinking multitude and thinking movement, without falling into contradictions.

Let us consider Zeno's arguments on these three issues separately. Let's start with the question of the conceivability of emptiness, that is, space separated from matter. If we assume the existence of such a space, then the following reasoning comes into force. Everything that exists is somewhere in space. But in order to exist, space must also exist<где-нибудь>, i.e. exist in the second space. This second space in turn must exist in a third space, and so on ad infinitum. But this is absurd. Consequently, space as separate from matter is unthinkable.

The second question is about the conceivability of the set. Let us assume that the set is conceivable. Then the questions arise: 1. How should each individual element of this set be thought of?

2. How should we think about the total number of elements of a set: will their sum be a finite or infinite number?

Zeno's research shows that there are conflicting answers to both of these questions. Regarding the first question - how should each individual element of the set be thought of - it turns out that for each such element it is necessary to answer that it simultaneously has no magnitude and is infinitely large in magnitude. Regarding the second question - how should the sum of the elements of a set be thought of - it turns out that it should be thought of both as a finite number and as an infinite number.

Research on the third question - the conceivability of movement - also necessarily leads to contradictory statements.

Zeno's arguments on this issue became widely known. Zeno developed several such arguments, of which four have come down to us:<Дихотомия (деление на два)>, <Ахилл>, <Летящая стрела>And<Стадий>

Their general scheme is a refutation<от противного>. Let us assume, together with Parmenides’ opponents, that movement is conceivable. Then it is necessary to make contradictory statements about a moving body or moving bodies:

1) that movement is possible and 2) that it is impossible. Using four arguments, Zeno proves that motion is impossible.

It is impossible, firstly, as the movement of a single body moving along a straight line from one point to another. To travel a certain distance separating point A from point B, the body must first travel half of this distance; to pass half, it must first pass half of that half, and so on. to infinity. As a result of this, the body not only cannot pass from point A to point B, but cannot even leave point A, i.e., movement from point A to point B cannot only be completed once it has begun, but cannot even begin. This is the point of the argument<Дихотомия>.

The inconceivability of the movement of one, separately taken body is also proven through the argument<Летящая стрела>. By assumption, the arrow flies, that is, moves in space. But at the same time it is necessary to assert about it that at every moment of its flight it occupies a space equal to its own length, i.e. it resides within this part of space,<значит>motionless in it. It turns out that the flying arrow both moves and does not move.

But movement is also unthinkable as the movement of two bodies relative to each other. It is unthinkable, firstly, as the movement in a straight line of two bodies, separated by a certain distance and simultaneously moving in the same direction, and the body moving behind moves faster than the one moving in front. Zeno proves that under these conditions a body moving at a higher speed will never catch up with one that is moving away from it at a lower speed. Achilles, famous for the speed of his running, will never catch up with the turtle running away from him. Let Achilles run faster than the tortoise, but after any period of time, no matter how small it may be, the tortoise will have time to cover a distance that, no matter how insignificant it may be, will never be equal to zero. Consequently, Zeno argues, at no point in the run will the entire distance separating Achilles from the tortoise become zero, and therefore Achilles will indeed never catch up with the tortoise. The same result is obtained if we apply the argument to the case of Achilles<Дихотомия>. At the initial moment of running, Achilles is separated from the turtle by a distance AB. Achilles will catch up with the turtle at the moment when this distance, decreasing, becomes zero. But for this to happen, the distance AB must first be reduced to half. In turn, in order to decrease to half, it must first decrease to half of this half, and so on ad infinitum. The result is the same as in<Дихотомии>: AB distance will never go to zero.

Zeno's paradoxes“...caused such excitement,
that even now some ripples can be observed"

D. Ya. Stroik, Brief outline of the history of mathematics,
M., “Science”, 1964, p. 53.

Zeno formulated a series aporia("unsolvable propositions"), showing - in modern language - that in them are consideredtwo processes coincide: the physical movement itself and the emergence in our consciousness of a sequence of its individual fragments, and this leads to logical contradictions.

"From 45 the aporia put forward by Zeno has reached us 9 . The classic ones are five aporia in which Zeno analyzes the concepts of set and motion. The first, called the “aporia of measure,” is stated by Simplicius as follows: “Having proved that “if a thing has no size, it does not exist,” Zeno adds: “If a thing exists, it is necessary that it have a certain size, a certain thickness.” and so that there is some distance between what represents mutual difference in it.” The same can be said about the previous one, about that part of this thing that precedes in smallness in the dichotomous division. So this previous one must also have some magnitude in its previous one. What is said once can always be repeated. Thus, there will never be an extreme limit where there would not be parts different from each other. So, if there is plurality, it is necessary for things to be at the same time great and small, and so small as to have no magnitude, and so great as to be infinite.”

Zeno's argument is most likely directed against Pythagorean the idea that bodies “are made of numbers.”

In fact, if we think of a number as a point that has no magnitude (“thickness,” extension), then the sum of such points (body) will also have no magnitude, but if we think of a “corporeal” number as having some finite magnitude, then, since the body contains an infinite number of such points (for the body, according to Zeno’s assumption, can be divided “without limit”), it must have an infinite size. It follows from this that it is impossible to think of a body as a sum of indivisible units, as we saw in Pythagoreans.

One can perhaps say, continuing Zeno’s thought: if the “unit” is indivisible, then it does not have a spatial magnitude (point); if it has a magnitude, however small, then it is divisible to infinity. The Eleatics first posed a question to science, which is one of the most important methodological questions to this day: How should we think about the continuum - discrete or continuous? consisting of indivisibles (units, “unities,” monads) or divisible to infinity?

Any quantity must now be understood from the point of view of whether it consists of units (like the arithmetic number of the Pythagoreans), indivisible “wholes,” or whether it itself is a whole, and its constituent elements do not have independent existence. This question is posed in relation to number, and in relation to spatial magnitude (line, plane, volume), and in relation to time. Depending on the solution to the problem of continuum, different methods of studying nature and man are formed, that is, different scientific programs.”

Gaidenko P.P., Evolution of the concept of science: the formation and development of the first scientific programs, M., “Urss”, 2010, p. 65-67.

"ABOUT Zeno of Elea and its paradoxes, such as, for example, the well-known riddle about the fleet-footed Achilles, who cannot catch up with the tortoise, it would seem that so much has already been written that there is hardly any need to return to those formulated by him back in the 5th century. BC e. “difficult questions” (aporia) related to the representation of movement in science and to the concept of “set” (to the relationship between the continuous and the discrete).

Since then, Zeno's aporias have never ceased to interest mathematicians and philosophers. However, up to the present day, there are a wide variety of opinions about them: from a completely dismissive attitude towards them to the recognition that they belong to the most important and difficult issues of substantiating mathematics and physics.

Thus, the famous French mathematician Paul Levy the paradox of Achilles and the tortoise appears to be an obvious absurdity.

“Why imagine,” he writes, “that time will stop moving due to the fact that a certain philosopher is enumerating the terms of a convergent series?” “I confess that I have never understood how people who are otherwise quite reasonable can become confused by this paradox, and the answer that I have just outlined is the same answer that I gave when I was eleven years old to the elder who told This paradox to me, or, more precisely, is the very answer that I summed up then with such a laconic formula: “This Greek was an idiot.”

I now know that I need to express my thoughts in a more polite form and that, perhaps, Zeno expounded his paradoxes only in order to test the intelligence of his students. But my surprise at the minds confused by the concept of a convergent series remained the same. (R. Levy, A propos du paradoxe et de la logique, “Rev. Meta-phys. Morale”, 1957, N 2, p. 130).”

Yanovskaya S. A., Methodological problems of science, M., “KomKniga”, 2006, p. 214.