Why do we need Zeno's aporia briefly? Paradoxes of Zeno of Eleatic - description, meaning and interesting facts. Achilles and the tortoise

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Zeno's aporia and their modern interpretation

Introduction. On the problem of aporia in science

Ancient philosophy has always been distinguished by a variety of ideas, as well as a large number of their adherents. All this was expressed in the diversity of teachings and, consequently, schools of ancient philosophers. One of them was the Eleatic school. Twenty-four centuries ago, its adherent Zeno of Elea pointed out the impossibility of a logically consistent understanding of the movement of bodies, although he did not doubt the sensually verifiable reality of the latter.

Zeno formulated a number of aporia related to the problem of motion. But of no less interest in epistemological (epistemology is the study of knowledge) and logical terms are the aporias that the famous Elean faced when analyzing the problem of “much in being,” the problem of obtaining an extended segment in the synthesis of so-called non-extended points (metric aporia), and other.

The difficulties reflected in Zeno’s aporias cannot be considered overcome even today. Therefore, Zeno’s aporia never ceases to interest mathematicians, physicists, philosophers, and scientists of some other directions. Interest in aporia is currently associated with the problems of scientific knowledge of space, time, movement and the structure of systems in the broadest sense, as well as with the problems of the “beginnings” of science in the sense of the history of the emergence of initial concepts about nature (“body”, “point”, “ place”, “measure”, “number”, “set”, “finite”, “infinite”, etc.) and in terms of discussions during which the meaning of these concepts was clarified and which ultimately developed into the problem of the foundation of mathematics, and also the beginnings of exact natural science.

1. Zeno of Elea

Zeno, one of the most famous ancient Greek philosophers, was born around 490 BC. in Elea, which is why it later became known as Elea. Historians note his adherence to the teachings of Parmenides and the Pythagoreans. These are the majority of reliable facts from his biography, which are confirmed not only by scientists of our time, but also by such respected people as Plato in his Parmenides.

As a representative of the Eleatic school of philosophy, Zeno discussed the problems of true and imaginary existence, by which he meant thinking and feeling, respectively. The term “being” itself was introduced into use by Zeno’s teacher Parmenides, and had such properties as immobility, homogeneity and unity. Parmenides compared being with a ball filled with some substance - complete and whole. And having rethought the ideas of his teacher, Zeno began to put forward his reasoning about the structure of being and argue that there is no movement and multiplicity in nature, which he proved with his famous aporias.

“Of the 45 aporia put forward by Zeno, 9 have reached us. Five aporia are classic, in which Zeno analyzes the concepts of set and motion” Gaidenko P.P., Evolution of the concept of science: the formation and development of the first scientific programs, M., “Urss” , 2010, p. 65-67. . Zeno's aporia has not found a satisfactory solution to this day. Moreover, modern publications, unlike Soviet ones, agree with this: “Aporias are now recognized as genuine paradoxes, associated, in particular, with the description of movement.” A.A. Ivin, A.L. Nikiforov “Dictionary of Logic”, M.: “Vlados”, 1997, p. 22. All so-called “resolutions” of aporias represent a logical error of ignorantia elenchi, consisting in the fact that the thesis that is being proven is not the one that needs to be proven. The study of Zeno's paradoxes begins with an acquaintance with the history of the Eleatic school and the interpretation of Zeno's arguments, which immediately leads us into the variety of problems associated with them and will allow us to find our own path to solving his mysteries. This requires identifying guiding viewpoints that are based on facts or stronger assumptions.

The modern interpretation of Zeno's aporias comes down to two diametrically opposed things: either to the proof of the inconsistency of Zeno himself as a formal logician, or to the establishment of the unsuitability of modern mathematical analysis, and in particular the integral system of calculus, for real life, for human existence.

The purpose of this work is to examine these very aporias, identify or not identify in them violations of the rules of formal logic and compare them with the modern life views of scientists.

2. Aporia of Zeno of Elea

2.1 "Dichotomy"

One of Zeno’s most famous arguments about the absence of motion says: “In a finite period of time it is impossible to go through an infinite number of segments, which means that the movement itself cannot begin.”

The essence of this statement is as follows. If there is a certain segment A-B, then in order to get from point A to point B, you must first get to point C, which will be the middle of the segment A-B. And in order to reach point C, you must first get to point D, which is the middle of A-C, and this will continue indefinitely. It follows from this that the movement will never begin, since the end point of even an infinitesimal displacement will never be found.

Many philosophers, including Aristotle, Hegel and Lenin, addressed this aporia in their works, and they did not support Zeno’s point of view, but, on the contrary, gave arguments in favor of the incorrectness of the statement. Thus, Aristotle wrote that “...space and time are infinitely divisible in possibility, but not infinitely divided in reality.” Hegel, developing the thought of Aristotle, argued that divisibility is not a necessity, but only the possibility of division.

2.2 "Achilles and the Tortoise"

Another aporia is based on the principles set forth in “Dichotomy,” in which Zeno says that for Achilles to catch up with the tortoise, it is necessary for the distance separating them to disappear, and this is impossible.

The philosopher explains his idea by saying that if the tortoise and Achilles begin to move in the same direction at the same second, but the tortoise is some distance ahead, then while Achilles overcomes this path, the tortoise will move forward some distance, and the distance will shrink endlessly, but in the end there will always be a non-zero distance between them.

By writing down the equations of motion of Achilles and the tortoise, we can find out that at the moment of their supposed meeting they must go through an equal number of segments of the path, but they go through different paths: a person overcomes one “extra” segment. This, on the one hand, shows the formal incorrectness of Zeno’s statements, but, at the same time, gives modern mathematicians and philosophers enormous scope for action to prove or disprove that the part is equal to the whole.

D. Hilbert and P. Bernays note: “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 does not at all affect one paradox, which is 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), is in fact everything. must still be completed” D. Hilbert, P. Bernays “Foundations of Mathematics. Theory of evidence", 1982.

This aporia of Zeno never ceased to interest mathematicians and philosophers. However, up to the present day, there are a wide variety of opinions: from a completely dismissive attitude to the recognition that it belongs to the most important and difficult questions of the foundation of mathematics and physics.

Thus, the famous French mathematician Paul Levy considers the paradox of Achilles and the tortoise 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 me this paradox, or, more precisely, is the very answer that I summed up then with such a laconic formula: This Greek was an idiot” R. Levy, A propos du paradoxe et de la logique,. "Rev. Meta-phys. Morale", 1957, N 2, p. 130.

2.3 "Rista"

Aporia, also based on the properties of the indivisibility of space and time. It lies in the fact that an indivisible moment of time can become divisible relative to itself. To explain this, Zeno gave the example of three parallel sets of four points. The first (A) did not move, the second (B) moved to the right, and the third (C) to the left. It turned out that in one unit of time, set B covered a distance of only two points from set A, but, at the same time, to all four points of set C. This meant that this unit of time is not indivisible, or, if we take time to be constant, only the space in which matter moves is divisible. Also, this statement implied not only some kind of curvature of space, but also Zeno’s conclusion that half represents the whole.

The answer to this aporia can be considered Einstein’s special theory of relativity, which asserts the non-absoluteness of motion, time and space for bodies located in different reference systems.

2.4 "Arrow"

With this aporia, Zeno again denies the possibility of movement. In his opinion, an arrow flying at a certain speed, at each indivisible moment of time, occupies a space equal to its own length, which means it is at rest. And the time period is precisely the sum of these indivisible moments. This aporia is directed against the proposition that a continuous quantity is the sum of an infinite number of indivisible particles.

Here the philosopher does not share the ontological and mechanical concepts of movement, and therefore comes to this conclusion. Of course, from the point of view of the same special theory of relativity, in a frame of reference where only the arrow is located and nothing else, it is at rest - relative to itself. Therefore, it is impossible to call this statement of Zeno unfounded. But this shows that aporias are some kind of extremes in understanding the structure of matter.

2.5 Aporia refuting the existence of “many”

These aporias include such statements as “Plurality”, “Medimn grain”, “Measure”. They are all based on the fact that “a lot” is, on the one hand, an endless field to fill, and on the other, a limited space, because “more than a lot” cannot be. Based on this, Zeno denied the existence of “many” in general, moreover, dividing everything into infinitesimal segments.

This also includes Zeno’s famous aporia “On Place,” which tells us that if everything that exists is in one place, then this place must also be somewhere, and so on ad infinitum.

2.6 Aporia "Stadium" ("Stadium")

Let equal masses move across the stadium along parallel straight lines at equal speeds, but in opposite directions. Let the series A1, A2, A3, A4 mean stationary masses. The row B1, B2, B3, B4 means masses moving to the right, and the row G1, G2, G3, G4 means masses moving to the left.

We will now consider the masses Ai, Bi, Gi as indivisible.

At an indivisible moment of time, Bi and Gi pass through an indivisible part of space. Indeed, if at an indivisible moment of time a certain body passed through more than one indivisible part of space, then the indivisible moment of time would be divisible, but if less, then the indivisible part of space could be divided. Let us now consider the movement of indivisible Bi and Gi relative to each other: in two indivisible moments of time B4 will pass through two indivisible parts Ai and simultaneously count four indivisible parts Gi, that is, the indivisible moment will turn out to be divisible.

This aporia can be given a slightly different form. For the same time t, point B4 passes half the path of segment A1A4 and the whole segment G1G4. But each indivisible moment of time corresponds to an indivisible part of space traversed during this time. Then in some segment? and 2? contains the “same” number of points, “same” in the sense that a one-to-one correspondence can be established between the points of both segments. This was the first time such a correspondence was established between points of segments of different lengths. If we assume that the measure of a segment is obtained as the sum of indivisible measures, then the conclusion is paradoxical. The logical error at the heart of the aporia of “Stages” is hidden behind an implicit violation of the logical laws of constructing thoughts. This violation consists in the latent recognition of the mutual relativity of the movement of bodies A1 and A2, since in the aporia we are still talking about the movement of body A1 relative to body A2 (or vice versa), while simultaneously explicitly denying this relativity, since such a parameter of this movement as speed is ignored relational motion, equal to the sum of the velocity modules v1 and -v2 of the movements of bodies A1 and A2 in relation to body A0. In an explicit form, the logically contradictory structure of this aporia can be represented by the formula x (P(x) ((P(x)), where only mutually exclusive propositional functions mean simultaneously the recognition and denial of the predicates of relativity and the reality of the relational movement of bodies A1 and A2.

3. Modern interpretation

In order to conduct serious research into Zeno’s aporias, one should consider not only the physical and mathematical models - the philosopher assumed precisely the empirical, experimental component of the world, that is, he wanted to say that it is in real life, unlike the world of numbers, that such seemingly absurdities are possible . Over time, a moving body passes through all the points of its trajectory one by one, but for a randomly selected point it is impossible to indicate the next one after it, and this breaks the sequence.

D. Gilbert and P. Bernays note about “Achilles”: “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 does not at all affect one paradox, which is 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), is in fact everything. must end."

In Zeno's aporia it is assumed that microspace is structured in the same way as macrospace, and facts from the field of motion are transferred to all quantities. Meanwhile, according to modern physical views, quantities are not divisible indefinitely. Modern physics is discovering more and more new facts about the structure of the microworld. To resolve the paradoxes, it is necessary to point out that it is not at all necessary to believe that the spatiotemporal vision of motion has any physical significance for the smallest intervals of time or space.

With his four main paradoxes, Zeno achieves that he logically strictly shows that something is wrong in the Pythagorean views on motion, space and time. These demonstrations of Zeno did not force later thinkers to agree with the conclusions of Parmenides, but they did give them the opportunity to develop a respect for formal logic and open up new possibilities for its application to themselves and the world. They also, accordingly, forced them to try to formulate Pythagorean concepts in a new way, in order to eliminate the contradictions shown by Zeno. From the obvious non-standard way of Zeno’s thinking, it can be assumed that he would not have agreed with any of the arguments of modern sages and would have been able to prove that he was right with his new aporia.

Modern scientists have proven that aporias are incorrect from a mathematical point of view, but at the same time they have to admit that their very existence casts doubt on the correctness of the entire mathematical model of the world, once proposed by Aristotle and Pythagoras and largely developed by Newton and Descartes.

At the same time, on the basis of Zeno’s statements, some even now, using the latest achievements in mathematics, prove his theories and even develop them. P.V. Poluyan in his work “Non-standard analysis of non-classical motion” states that “... the body at every moment, at every single point has movement, but no speed” P.V. Poluyan “Non-standard analysis of non-classical movement” - U., 2002. He also came to the conclusion that “... for any two moments of time there are two locations of the point in space, which sets the value of the speed only for these two moments. But at the same time, any finding of a point corresponding to a moment in time located between the two initially selected ones makes it possible to find other, different from the original, relationships between the distance intervals and time.”

Also, with the help of Zeno contradictions, W. Heisenberg formulated his “Law of Uncertainty”, which speaks of the impossibility of a material point having both momentum (velocity) and coordinates at the same time. Heisenberg's uncertainty law served as one of the postulates of quantum mechanics, and was used in the definition of “ideal measurements” by D. von Neumann and “non-ideal measurements” by L. Landau.

Zurab Silagadze in his article “Zeno meets modern science” states that “…Zeno’s paradoxes affect the fundamental aspects of reality - localization, movement, 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 achieving their final resolution seems endless, and our understanding of the world around us is still incomplete and fragmentary” Z. Silagadze “Zeno meets modern science”, 2005.

Zeno’s aporia, and, in particular, his “Arrow”, became the basis for A. Turing to put forward his paradox in 1954, which later, after its description by B. Misra and D. Sudarshan, was called “Zeno’s quantum paradox.” The essence of this paradox is that the period of decay of a certain closed system directly depends on the frequency of measurements of its state. And it is possible that an unstable particle may never decay - under conditions of frequent observation.

Largely with the help of Zeno's aporias, I. Kant was able to derive his theory. As is known, he distinguished three stages of knowledge: sensuality, reason and reason. Kant defined reason as reason that has gone beyond experience and forms ideas and the desire for absolute knowledge. And it is in the process of forming ideas that the mind encounters antinomies - a kind of paradoxes, the way out of which gives rise to the object of knowledge. Examples of Kantian antinomies are the contradictions between the divisibility and indivisibility of the world, the finitude and infinity of space and time, the existence of necessity and freedom, and, finally, the existence of a necessary being - God and his absence.

The objective idealism of G. Hegel, more precisely, the “law of contradictions” and the “law of the negation of negation”, embedded in the idea of the trinity of any cycle of events, also intersects with Zeno’s principles of “by contradiction”.

Conclusion

aporia zeno paradox philosophy

So, having analyzed the outwardly seemingly illogical statements of Zeno of Elea, we can conclude that his aporias were one of the greatest works in the history of mankind, because for almost twenty-five centuries they have been criticized by science and They themselves put the integrity of science at risk. Also, aporia gave impetus to the emergence of such philosophical movements as ancient atomism; served as a catalyst for the development of mathematical analysis and set theory.

Poems by A.S. were dedicated to aporias. Pushkin and Paul Valery, they were mentioned in “War and Peace” by L.N. Tolstoy, Kh.L. appeals to them. Borges and many other artists, telling of infinity, integrity and indivisibility.

One of the most famous sayings of Zeno: “Can motion be conceived if the division of space is allowed?” suggests that the philosopher not only thought about this problem, he sincerely believed in it, although it is known that he did not deny movement as such.

Of course, Zeno’s contribution to the development of human thought is great, and this can be illustrated even by the fact that statements created before our era cannot be definitively confirmed or refuted by modern mathematicians with much more developed computing technologies and horizons.

List of used literature

L.A. Halfin “Quantum Zeno effect” - Advances in Physical Sciences, 1990, volume 160, issue 10

V.Ya. Komarov “Teachings of Zeno of Elea” - L., 1988

D. Hilbert, P. Bernays “Foundations of Mathematics. Theory of evidence", 1982

G. Weil “On the philosophy of mathematics” - M.-L., 1934

Gaidenko P. P. “Evolution of the concept of science” - M.: Nauka, 1980

Maneev A.K. “Philosophical analysis of Zeno’s aporias” - Minsk, 1972

P.V. Poluyan “Non-standard analysis of non-classical movement” - U., 2002

Z. Silagadze “Zeno meets modern science”, 2005

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

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

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Zeno of Elea and his aporia

The task of defending Parmenides' views against the raised objections was taken upon himself by Parmenides' student and friend Zeno*. He was born at the beginning of the 5th century. BC e. (480) and died in 430 BC. e. Zeno enjoyed fame as a talented teacher and speaker. He spent his youth in quiet, solitary study, highly appreciating the superiority of mental pleasures - the only pleasures that never satiate. From Parmenides I learned to despise luxury. His reward was the voice of his own heart, beating evenly in the consciousness of his rightness. His whole life is a struggle for truth and justice. It ended tragically, but it was not fought in vain. Only numerous and small extracts made by later ancient writers have survived from his works. 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 introduced into 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 later gave Aristotle grounds to call Zeno the founder of “dialectics.” By “dialectics” Aristotle in this case understands 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 "proof by contradiction" in mathematics. 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.

Let us consider Zeno's arguments on these three issues separately. Let's start with the question of the conceivability of emptiness, i.e. 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 so that; to exist, space must also be “somewhere,” that is, exist in a 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 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 must necessarily be thought of both as a finite number and as an infinite number.

Research on the third question - about conceivability of movement- also necessarily leads to contradictory statements. Zeno's arguments on this issue became particularly famous and became widely known. Zeno developed several such arguments, of which four have come down to us: “Dichotomy (division by two)”, “Achilles”, “Flying Arrow” and “Stadium”. Their general scheme is the same refutation “by contradiction”. Let us assume, together with the opponents of Parmenides, that motion 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,

* Not to be confused with other Greek philosophers who bore this name, such as the Stoic Zeno of Kition in Cyprus.

the body must first travel half of this distance; to go through half, it must first go through half of that half, and so on ad infinitum. 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 "Dichotomy".

The inconceivability of the movement of one, separately taken body is also proven through the argument “ Flying arrow". 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, that is, it resides within this part of space, “meaning” it is motionless in it. It turns out that the flying arrow both moves and does not move. In the “Arrow” aporia, Zeno proves that, when moving, an arrow at any given moment of time occupies a given place in space. Since each moment is indivisible (it is something like a point in time), then within its limits the arrow cannot change its position. And if it is motionless in each given unit of time, it is also motionless in a given period of time. A moving body does not move either in the place it occupies or in the place it does not occupy. Since time consists of individual moments, the movement of the arrow must be the sum of the states of rest. It also makes movement impossible. Since the arrow at each point of its path occupies a very definite place equal to its volume, and movement is impossible, if the body occupies a place equal to itself (for movement an object needs a space larger than itself), then in each place the body is at rest. In a word, from the consideration that the arrow is constantly in certain, but indistinguishable “here” and “now,” it follows that the positions of the arrow are also indistinguishable: it is at rest.

But movement is also unthinkable as the movement of two bodies relative to each other. It is inconceivable 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.

Argument "Stages" refutes the conceivability of movement, refuting one of the premises of movement accepted during Zeno's time - the assumption that space consists of indivisible parts (segments), and time also consists of indivisible parts (moments). Let's make this assumption. Let us also assume that bodies of equal size move from opposite sides along parallel lines. Let us finally assume that these bodies pass by a third body of the same size, but motionless (see figure).

A1 A2 A3 A4 B4 B3 B2 B1 ---><--- С1 С2 С3 С4

A1 A2 A3 A4 B4 B3 B2 B1 C1 C2 C3 C4

Then it turns out that the same point, moving at the same speed, will cover the same distance not at the same time, but will cover it in one case in half the time, and in the other in double the time. At the same time, the extreme points of each of the moving rows B4 B3 B2 B1 and C1 C2 C3 C4 will pass by all the other points of the other moving row. However, at the same time they will pass only half the points of the row, which remains motionless during their movement. Such a different result will depend on where we look at its movement. But as a result, we come to a contradiction, since half turns out to be equal to the whole. In other words, in the Stages argument the inconceivability of movement is demonstrated by considering time, which is supposed to be, like space, composed of a multitude of discrete but supposedly contiguous elements.

We have seen that in all of Zeno’s reasoning the question is not at all about whether we can perceive movement through the senses or not. Neither Parmenides nor Zeno doubt that movement is perceived by the senses. The question is whether it is possible to think about movement if, when thinking about movement, we assume that the space in which bodies move consists of many parts separated from one another, and if we assume that time in which all phenomena and movement occurs, consists of many moments separated from each other. The inevitability of contradictions to which thought comes under these premises proves, according to Zeno, that the conceivability of the set asserted by Parmenides' opponents is impossible. The refutation of the conceivability of empty space has the same meaning. The essence of Zeno's argument not at all to prove that space does not exist. Zeno proves otherwise. He proves that space cannot be thought of as empty space, as space existing in any part separately from matter.

Zeno's arguments provided a powerful impetus to the further development of ancient mathematics, ancient logic and ancient dialectics. These arguments revealed contradictions in the concepts of modern science to Parmenides and Zeno - in the concepts of space, the one and the many, the whole and the parts, the movement and rest, the continuous and the discontinuous. Zeno's aporia stimulated the idea to seek solutions to the difficulties he noticed. The threat of insoluble contradictions hanging over mathematical knowledge was subsequently eliminated by the atomistic materialism of Leucippus and Democritus.

Zeno's aporia is associated with the dialectic of fractional and continuous in motion (as well as space-time itself). Analyzing the hypothetical competition between Achilles and the tortoise, Zeno represents the displacement of each of them as a set of individual finite displacements: the initial segment separating the tortoise and Achilles, the segment that the tortoise crawls while Achilles overcomes the initial gap, etc. This “yet” contains the replacement of continuous movement with individual “steps” - in reality, neither Achilles nor the tortoise wait for each other and move regardless of the conditional division of their path into imaginary segments. Then the path that Achilles has to overcome is equal to the sum of an infinite number of terms, from which Zeno concludes that no (finite) time is enough for him.

If we assume that “time” is measured by the number of segments, then the conclusion is correct. It is usually pointed out, however, that Zeno simply was not familiar with the concept of the sum of an infinite series, otherwise he would have seen that an infinite number of terms still gives a finite path, which Achilles, moving at a constant speed, will undoubtedly cover in the appropriate (finite) time .

Thus, the Eleatics failed to prove that there is no movement. With their subtle reasoning, they showed what hardly any of their contemporaries comprehended - what is movement? In their reflections, they themselves rose to a high level of philosophical search for the mystery of movement. However, they were unable to break the shackles of the historical limitations of the development of philosophical views. Zeno's aporias "Achilles" and "Arrow" reveal the deep mystery of how movement is born from stillness, the apparent lack of dimensions (“the arrow is at rest at every moment”). Some special trains of thought were needed. These moves were explored by the founders of atomism.

Subsequently, Diogenes the Cynic, in order to refute Zeno’s argument against the existence of movement, stood up and began to walk. A.S. Pushkin put it this way:

No movement!

The bearded sage said,

The other one remained silent

But he began to walk in front of him.

The crisis caused by Zeno's aporias was very deep; in order to at least partially overcome it, some special, unusual ideas were required. The ancient atomists managed to do this, the most prominent among whom were Leucippus And Democritus. An analysis of Zeno’s aporias revealed their “painful” points: infinite division (of a segment of a body's path). Atomists assumed that matter, space, and time, in principle, cannot be divided indefinitely, because there are tiny, further indivisible fragments of them - atoms of matter, amers (atoms of space), chronons (atoms of time). This or that body consists of a certain number of atoms, each of which has a finite volume; accordingly, a body composed of atoms also has a finite volume. The arrow flies because, by definition, movement is the covering of a certain distance, consisting of a certain number of amers, in a certain time, which in turn consists of a certain number of its quanta (chronons). In order to get rid of the difficulties with understanding change once and for all, it was assumed that atoms are unchanging, possessing exactly the same absolute qualities as Parmenidean being, they are indivisible and homogeneous. Atomists, as it were, “reduced” change to the unchangeable, to atoms.

Zeno's aporias were refuted many times, simply pointing to the facts: A caught up with B, C flies, etc. However, such “refutations” of the argumentation of great philosophers are not worth much. The Eleatics pose the question in a truly scientific way: if there is movement, then it must be comprehended. Of course, from a misunderstanding of movement and multiplicity it does not follow that they do not exist at all. But there is nothing particularly to be proud of if you are not able to understand from a scientific point of view seemingly quite obvious things, mechanical movement, all kinds of changes. Heraclitus irritated the Eleatics because he did not give an explanation for the fact of change. Of course, the Eleatics did not prove that there was no movement; they showed their contemporaries that they hardly understood the content of the movement. The Eleatics themselves, in their understanding of movement, were at the true pinnacle of the views that existed in their era. Now let us note that modern scientists again and again turn to Zeno’s aporias, finding in them new impulses for the development of scientific thought. In doing so, of course, they modify the logic of the Eleatics’ reasoning. It has long become clear: to understand the aporia of the Eleatics, one must turn to the most developed philosophical, mathematical and physical theories. It turns out that the Eleatics continue, in some way, to be our teachers to this day. What did they teach and teach us first of all? The role and significance of logical evidence, the need to consider - along with the world of events - the intelligible level of reality, without succumbing to the “deception of the imagination”. The Eleatics posed the problem of the relationship between the unified and the plural, the continuous and the discontinuous, movement and rest.

Zeno of Elea(about 490–430 BC) - a favorite student and follower of Parmenides." He developed logic as dialectics. The most famous refutations of the possibility of movement are the famous aporia of Zeno, whom Aristotle called the inventor of dialectics. The aporia are extremely deep and arouse interest to this day He defended the immutability of being (united and motionless), non-existence cannot be thought of, this is a region of opinion. He denied the possibility of thinking about movement, analyzing, and that which cannot be thought does not exist.

The internal contradictions of the concept of movement are clearly revealed in the famous aporia “Achilles”: fleet-footed Achilles can never catch up with the turtle. Why? 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 had previously occupied, she will move forward a little. No matter how much the space between them decreases, it is infinite in its divisibility into intervals and it is necessary to go through all of them, and this requires infinite time. Both Zeno and we know perfectly well that not only Achilles is fleet-footed, but any lame-footed person will immediately catch up with the tortoise. But for the philosopher, the question was posed not in the plane of the empirical existence of movement, but in terms of the possibility of its inconsistency in the system of concepts, in the dialectics of its relationship with space and time.

Aporia “Dichotomy”: an object moving towards a goal must first go halfway to it, and in order to go through this half, it must go through half of it, etc., ad infinitum. Therefore, the body will not reach the goal, because his path is endless.

Aristotle points out that Zeno confuses the infinitely divisible with the infinitely greater. Zeno considers space as the sum of finite segments and contrasts it with the infinite continuity of time. In “The Turtle” the impossibility of movement stems from the fact that it is impossible to travel an infinite number of halves of a path in a finite time. Zeno simply was not familiar with the concept of the sum of an infinite series, otherwise he would have seen that an infinite number of terms still gives a finite path, which Achilles, moving at a constant speed, will undoubtedly overcome in the appropriate (finite) time.

Thus, the Eleatics failed to prove that there is no movement. With their subtle reasoning, they showed what hardly any of their contemporaries comprehended - what is movement? In their reflections, they themselves rose to a high level of philosophical search for the mystery of movement. However, they were unable to break the shackles of the historical limitations of the development of philosophical views. Some special trains of thought were needed. These moves were explored by the founders of atomism.

The main property of the surrounding world– not a substance, but a quality (unchangeable eternity, one can think) – this is the conclusion of the Eleatics.

16. Philosophy of atomism: the concept of the atom and causality.

No. 6 Teachings of Democritus. The concept of atom and emptiness.

Atomism- the movement of ancient thought towards the philosophical unification of the fundamental principles of existence. The hypothesis was developed by Leucippus and especially Democritus (460-370 BC).

At the heart of the infinite diversity of the world is a single arche, which has an infinite number of elements (atoms). Potential infinity -You can always add another grain of sand to a pile of sand. Actualinfinity – the presence of an infinite number of elements in a limited volume. It cannot be explained using ordinary thinking.

Being is something extremely simple, further indivisible, impenetrable—an atom. There are countless atoms, they are eternal, unchangeable, neither created nor destroyed. Atoms are separated from each other by emptiness; atom-existence, emptiness-non-existence. Atoms are forever rushing around in a boundless void that has no top, no bottom, no end, no edge, colliding, interlocking and separating. Compounds of atoms form the entire diversity of nature. Atoms have the power of self-propulsion: this is their eternal nature. Atoms are put together in different configurations, which we perceive as separate things, but the difference in the structures of these configurations, i.e. the qualitative diversity of the world depends on different types of interactions between atoms

Man, a collection of atoms, differs from other creatures in the presence of a soul. The soul is a substance consisting of small, most mobile, fiery atoms.

Democritus hesitated on the question of the nature of the gods, but was firm in recognizing the existence of God. According to Democritus, the gods consist of atoms, and God is the cosmic mind.

Atomism is one of the greatest doctrines. Unlike all the ideas of the first principle put forward so far, the idea of the atom contains, among other things, the principle of the limit of the divisibility of matter: the atom was thought of as the smallest particle, which acts as the initial material element of existence in creation and the last in decomposition. And this is a brilliant rise of thought to a fundamentally new level of philosophical comprehension of existence.

The basis of knowledge is sensation. “Visitors” - the material forms of things - are separated from things; they rush in all directions in empty space and penetrate the senses through the pores. If the pores correspond in size and shape to the “videos” penetrating into them, then an image of the object appears in the sensations, corresponding to the object itself. That. already in sensations we receive the correct image of the object. However, there are objects that, due to their small size, are inaccessible to the senses; such properties of things are comprehended by the mind, and this knowledge can also be reliable.

Causality. The development of the universe, the order of the world, everything is essentially determined (determined) by the mechanical movement of atoms. Therefore, in his system there is no place for the objective. the existence of "chance". And the “accident” itself is explained by the absence of a causal explanation, ignorance of the causes of a certain phenomenon. For Democritus, as Diogenes Laertius says, “everything arises out of necessity: the cause of every occurrence is a whirlwind, and he calls this whirlwind necessity.” This concept of necessity is a consequence of a certain metaphysical absolutization of mechanically understood causality. (It was this point that was the main subject of criticism by one of the outstanding representatives of ancient atomism, Epicurus.) The Democritus’ understanding of causality as an absolute necessity does not, however, as Aristotle emphasized, has nothing in common with teleology and is directed precisely against the teleological interpretation of reality. “Democritus moves away from talking about purpose and transfers everything that nature uses to necessity.”

Related information.

The philosophy of the Eleatic school (Xenophanes, Parmenides, Zeno, Melissus) is close to the traditions of spontaneous, elemental materialism, but it denies the “spontaneous dialectic” of previous philosophical schools. The polemics of the Eleatics with the dialectics of Heraclitus, although it seems paradoxical, leads to the comprehension of real, objectively existing contradictions. The negation of movement, the derivation of contradictions from the premises of its existence - in particular, in Zeno's presentation - becomes a stimulus for the further development of dialectical thinking. The great contribution of the Eleatics is the desire to comprehend reality using a conceptual apparatus. With the help of basic concepts, philosophers of that time sought to reflect and understand the objectively existing world. In the teachings of the Eleatics, we encounter a relatively clear doctrine of being and certain approaches to the question of the knowability of the world.

According to the Eleatics, being is something that always exists: it is as one and indivisible as the thought of it, in contrast to the multiplicity and divisibility of all things of the sensory world. Existence is something that can only be known by reason. Thinking and being are one and the same. Thinking is the ability to comprehend unity, while sensory perception reveals multiplicity and diversity in things and phenomena. Awareness of the nature of thinking had far-reaching consequences for the thoughts of ancient Greek philosophers. It is no coincidence that in Parmenides, his student Zeno, and later in Plato and his school, the concept of the one is in the center of attention, and the discussion of the relationship between the one and the many, the one and being stimulates the development of ancient dialectics. Zeno undertook to prove that there is no movement with his “aporias”: 1) Dichotomy - division in half. To walk any distance, you must first walk half of it. The remaining distance is also divided in half, etc. Any segment has an infinite number of points, which are impossible to count in a finite period of time. 2) Achilles and the tortoise. This aporia is also based on the assumption of the actual infinity of elements of continuous magnitude. Zeno proves that Achilles will never catch up with the tortoise, because... The turtle still moves forward. 3) Arrow. A flying arrow is actually at rest. He divides time into parts and at each moment of time the arrow is at rest. In general, Zeno proves that motion cannot be imagined theoretically.

Aporias are based on the fact that any segment is divided into an infinite number of points.

Zeno concludes that neither set nor movement can be conceived without contradictions. In Zeno's aporia the problems of continuity and infinity are discussed for the first time. The Eleatics understood being in the following way: 1) There is being, there is no non-being. 2) Being is one, indivisible 3) Being is knowable, non-being is unknowable. Zeno of Elea (about 490-430 BC) is a favorite student and follower of Parmenides." He developed logic as dialectics. The most famous refutations of the possibility of movement are the famous aporia of Zeno, whom Aristotle called the inventor of dialectics. The aporia are extremely deep and arouse interest in this day. He defended the immutability of being (united and motionless), non-existence cannot be thought of, this is a field of opinion. He denied the possibility of thinking about movement, analyzing, and that which cannot be thought does not exist.

Thus, the Eleatics failed to prove that there is no movement. With their subtle reasoning, they showed what hardly any of their contemporaries understood: what is movement? In their reflections, they themselves rose to a high level of philosophical search for the mystery of movement. However, they were unable to break the shackles of the historical limitations of the development of philosophical views. Some special trains of thought were needed. These moves were explored by the founders of atomism.

The main property of the surrounding world is not substance, but quality (unchanging eternity, one can think) - this is the conclusion of the Eleatics

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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 ( hypersubstantial 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

Having proved that “if a thing has no magnitude, it does not exist,” Zeno adds: “If a thing exists, it is necessary that it have some magnitude, some thickness, and that there be some distance between what constitutes 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 a certain value of 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 size, and so great as to be infinite... Which has absolutely no size, no thickness, no volume, that's not the case at all.

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.

Notes

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