Zeno's paradox, or Achilles and the tortoise. Zeno's paradoxes Method of sending to fat mathematical xxx

Movement is impossible. In particular, it is not possible to cross a room, since to do so one must first cross half the room, then half the rest of the way, then half of what's left, then half of the rest...

Zeno of Elea belonged to that Greek philosophical school that taught that any change in the world is illusory, and being is one and unchanging. His paradox (formulated in the form of four paradoxes (from the Greek. aporia"hopelessness"), which since then have given rise to about forty more different variants) shows that movement, the pattern of "visible" change, is logically impossible.

Most modern readers are familiar with Zeno's paradox precisely in the above formulation (it is sometimes called dichotomy- from the Greek. dichotomy"separation in two"). To cross the room, you first need to go half way. But then you have to overcome half of what is left, then half of what is left after that, and so on. This bisection will continue indefinitely, from which it is concluded that you will never be able to cross the room.

Aporia, known as Achilles, even more impressive. The ancient Greek hero Achilles is going to compete in running with a tortoise. If the tortoise starts a little earlier than Achilles, then in order to catch up with it, he first needs to run to the place of its start. But by the time he gets there, the tortoise will have crawled some distance that Achilles will have to cover before catching up with the tortoise. But during this time, the turtle will crawl forward some more distance. And since the number of such segments is infinite, swift-footed Achilles will never catch up with the tortoise.

Here is another aporia, in the words of Zeno:

If something moves, then it moves either in the place that it occupies, or in the place where it does not exist. However, it cannot move in the place it occupies (since at any given time it occupies all this space), but it also cannot move in the place where it does not exist. Therefore, movement is impossible.

This paradox is called arrow(at each moment of time, a flying arrow occupies a place equal to its length, therefore it does not move).

Finally, there is a fourth aporia, in which we are talking about two columns of people equal in length, moving in parallel with equal speed in opposite directions. Zeno states that the time it takes for the columns to pass each other is half the time it takes for one person to pass the entire column.

Of these four aporias, the first three are the best known and the most paradoxical. The fourth is simply related to a misunderstanding of the nature of relative motion.

The most crude and inelegant way to refute Zeno's paradox is to stand up and cross the room, outrun a tortoise, or shoot an arrow. But this does not affect the course of his reasoning. Until the 17th century, thinkers could not find the key to refuting his ingenious logic. The problem was solved only after Isaac Newton and Gottfried Leibniz presented the idea of ​​differential calculus, which operates with the concept limit; after the difference between the partitioning of space and the partitioning of time became clear; finally, after having learned how to deal with infinite and infinitesimal quantities.

Let's take the example of crossing a room. Indeed, at each point on the path you have to go half of the remaining path, but only it will take you half the time. The shorter the distance left to go, the less time it will take. Thus, when calculating the time it takes to cross a room, we add up an infinite number of infinitesimal intervals. However, the sum of all these intervals is not infinite (otherwise it would be impossible to cross the room), but is equal to some finite number - and therefore we Can cross the room in finite time.

Such a course of proof is similar to finding the limit in differential calculus. Let's try to explain the idea of ​​the limit in terms of Zeno's paradox. If we divide the distance we traveled across the room by the time it took us to do so, we get the average speed we traveled across that interval. But although both distance and time decrease (and eventually go to zero), their ratio can be finite - in fact, this is the speed of your movement. When both distance and time tend to zero, this ratio is called the speed limit. In his paradox, Zeno mistakenly assumes that when distance goes to zero, time remains the same.

But my favorite rebuttal to Zeno's paradox isn't with Newton's differential calculus, but with a quote from a Second City sketch, a comedy theater in my hometown of Chicago. In this sketch, the lecturer describes various philosophical problems. Having reached the paradox about Achilles and the tortoise, he says the following.

The problem - how, ultimately, geometric objects are arranged, what they "consist" of - was an important problem for Greek philosophy. This problem also attracted the attention of Zeno, a representative of the Elean philosophical school. The Eleatic school came up with the paradoxical doctrine that there is only one and immovable and unchanging being, the same everywhere: although it seems to people that being is plural and subject to change, this opinion leads to contradictions and therefore must be discarded.

The most famous and important for mathematics were formulated by Zeno four aporia(i.e. paradox) directed against the existence of the movement. Apparently, the first two aporias of Zeno implied that space and time are divisible to infinity, while the other two were based on the opposite idea that spatial extension and temporal duration consist of indivisible moments. Zeno tried to show that each of the two opposing views as a result leads to a contradiction, which means that the very idea of ​​movement, which is only an illusion, must be rejected.

Bisection

A moving body will never reach the end of the path, because it must first reach the middle of the path, then to the middle of the remaining path, then again to the middle of the remainder, etc. - thus, before reaching the end of the path, the body must go through an infinite number of middle, and this will take an infinite amount of time.

Achilles and the tortoise

The swift-footed Achilles will never be able to catch up with the slow tortoise if at the beginning of the movement it is at some distance ahead of Achilles: until Achilles reaches the line from which the tortoise started, she herself will crawl a certain distance, albeit less; while Achilles runs this distance, the tortoise will move even further, etc.

At each moment of time, a flying arrow occupies a space equal to itself. Therefore, it is at rest for some time. So it doesn't move at all.

Around the stadium, past a group of equal bodies A 1, A 2, A 3, A 4, two more of the same groups move in opposite directions at the same speed - B 1, B 2, B 3, B 4 and G 1, G 2, G 3 , Г 4 . Since they are moving at the same speed, they will cover the same distance at the same time. If in some time the first of the bodies B passes by all G, then during the same time the first of the bodies G will pass by half of the bodies A, which means that it will cover only half the distance that the body B has traveled, which means that since B and G move with equal speeds - it has also passed half the time for which body B has passed all bodies G. On the other hand, in the same time the first of the bodies G will pass by all B, and the first of B will pass only half of the bodies A , and hence, half the distance, spending half the time than the body D, which has passed all the bodies C. It turns out that the same time is twice as long and twice as short as itself.

Although most philosophers could not accept Zeno's strange conclusions about the non-existence of motion, the problems posed by Zeno forced a closer look at the concepts associated with space and time. So, Aristotle believed that space and time do not consist of a certain number of separate points and moments, but represent a special type of being - something continuous, or, as they say, continuum (lat. continuum - continuous). Spatial and temporal segments are in fact divisible to infinity, but only potentially divisible, in the sense that any segment can be divided by some point, what remains can also be divided, etc., but it is impossible to realize at some point an infinite number of divisions, just as it is always possible to extend the existing segment by a certain amount, but an infinite number of such extensions cannot be considered already realized. It is impossible to have an infinite line available, and it cannot be said that there are already an infinite number of points on a segment. Now, if in the first aporia a walking person every time, passing the middle of the next segment, would stop, marking this middle, then his movement would not be continuous and he would never be able to go through the entire segment. Aristotle's decision was accepted by many mathematicians: Euclid's distinction between discontinuous numbers, on the one hand, and continuous quantities, on the other hand, is related to similar considerations (see lesson 6). Nevertheless, the consideration of infinity in mathematics did not end there: for example, already in the 19th century. G. Kantor developed the theory of sets, which made it possible to consider a segment as an infinite set of points. Such a consideration made it possible to discover valuable new results, as well as to pose new interesting problems, connected, in particular, with some contradictions contained in the theory of infinite sets.

In addition, Zeno's aporias are also connected with a number of other issues related to mathematics (the summation of an infinite number of terms, the relativity of motion, the relationship between mathematical theory and physical reality, etc.).

Interesting, what do you think about these aporias?

If at any given time a flying arrow is at rest, when does it move?

Movement paradox

Zeno of Elea, a follower and admirer of Parmenides, had an accurate sense of logical form and had a gift for choosing a suitable (and witty) example to illustrate his reasonings; with such a combination of qualities, there is little that can be compared in philosophy. Zeno accepted both parts of Parmenides' philosophical thought: the conclusion that multiplicity and change are unrealistic, and the high appreciation of formal logic as a method of testing theories for correctness by testing them for logical coherence. Wanting to prove that Parmenides was right, Zeno demonstrated the absurdity of the opposite point of view (the opinion that plurality and change really exist in the world). His highest achievement in this field was a set of four riddles which he devised to prove the unreality of motion by examples which would show that neither common sense nor Pythagorean science could define motion without being confronted with contradiction or impossibility.

A modern lecturer might well begin the story of Zeno in the spirit of Zeno himself:

“If I say: today I have proof that you cannot walk from where you are sitting to the door of the auditorium, because it is impossible to reach, this may seem to you such an absurdity that all the normal people among you will immediately want to go to this door, go through it and go on! But this is exactly what I am going to prove ... I will offer you four arguments that together will show how stupid and unreasonable your common sense is and why it is absurd to try to define a movement ... "

Zeno's activities did not convince those who came after him that Parmenides was right, but made them appreciate the true value of precise formal logic. She strengthened formalism by showing that fields of knowledge as far apart as mathematics and contract law use the same logical forms. It has forced philosophers to think more carefully about the definitions of being and non-being and their relation to the definition of change. Zeno showed mathematicians once and for all that the Pythagorean construction program continuous quantities from the end rows discrete units internally contradictory and therefore impossible to implement. The Greek philosophers and scientists who lived after Zeno reacted to him in the same way as to Parmenides: they did not accept the idea that reality is an all-encompassing and immovable single absolute, but instead began to prove that formal logic can be effective and reason is reliable in the world, where multiplicity and change are possible. We can see his influence in all subsequent Greek thinkers.

Zeno took the form of puzzles in his critique of the movement because he wanted to strike common sense and the professional views of mathematicians with the same weapon. Conceived as concrete situations, his puzzles raise questions that force common sense to recognize that his own fuzzy ideas may not be reasonable at all. Conceived as illustrations for more abstract critiques, the same puzzles show that the technical assumptions about the points and moments on which their ability to puzzle the listener is based lead to a clear logical contradiction. Four cases were chosen to show the Pythagoreans, mathematicians well acquainted with the method of circumstantial proof, that their definitions of motion were unfortunate. Plato writes that Zeno "beat off the blows of those who laughed at Parmenides, and did so in an interesting way." So let's look at this counterattack.

There are four riddles about the paradoxes of movement. Zeno chose so many examples because he needed to refute four different possible definitions of motion. But first, we have four of his riddles that have continued to fascinate children, mathematicians, and the most ordinary listeners ever since Zeno first told them.

The first paradox is known as the Dichotomy, or Dividing by Two. Suppose you are standing in a stadium at some distance from the door that leads to the street. Then you will never be able to leave this stadium, because before you reach the door, you must reach the middle of the path. But before you get to the middle, you have to walk the middle of the distance to it. Since the movement from one point to another takes some finite amount of time, and there are infinitely many midpoints, it will take you an infinite amount of time to go through them all and exit the stadium. What is wrong with this argument? He looks unreasonable. But how does it happen that you get out of this door?

If you are not convinced by this division by two, Zeno has a second puzzle - the riddle of Achilles and the tortoise. In it, you have to imagine yourself in the stadium again. You are looking at a running race between Achilles and a tortoise. Since the tortoise moves much more slowly, Achilles allows it to start ahead of him. But this is a mistake: by doing this, Achilles will never catch up with the tortoise, says Zeno. By the time Achilles reaches the place where the tortoise started, it will have moved forward to some other point. By the time Achilles gets to this second point, the tortoise will have moved even further forward. So Achilles will never be able to outrun the tortoise. The Greek listeners of Zeno, undoubtedly, at the first moment reacted to this with the words: “But we know that in a real run, Achilles overtook I would have won the turtle, ”and then, after a little thought, they came to the second thought:“ Yes, of course, he would have overtaken her. But how?" Since we are not easily convinced that logical reasoning leads to a conclusion that is completely opposite to reality, Zeno's challenge prompts action - to justify why it becomes possible to outrun the tortoise. We go back to his story and even draw a diagram of the contest as Zeno described it, to see where he made some wrong assumption about distance, speed or movement. This schema looks like this:

Achilles and the Tortoise Zeno

A is the place where Achilles starts from, T is the place where the tortoise starts from. By the time Achilles runs from A to A, the tortoise has moved to T; while Achilles is running from A to A, the tortoise again moves forward from T to T; and so on ad infinitum. This scheme also seems to confirm that the turtle wins the competition.

Zeno's third paradox, the arrow paradox, is the simplest of the four, but, as history has shown, the most powerful stimulant of thought. “If a flying arrow is at rest at each moment of time and occupies a space equal to its length, then when does it move?” Indeed, when? This is a good question to ask mathematicians and physicists when they start talking to us about "states" or "moments" which are "things in an unextended span of time." How can motion be built from such static rest snapshots? This question will be interesting for them and for any other person too.

The fourth riddle of Zeno makes us return to the stadium once again. Achilles and the tortoise left - perhaps, contrary to Zeno, they nevertheless reached the door - and instead of them we have before us three moving "bodies" - wagons or chariots - lined up in a certain order. One stands, the second passes by it. How long does it take the second to travel a distance equal to the length of the chariot?

This, of course, depends on the speed of the moving chariot. But no matter what speed we imagine, we are asked to take "the time it takes to travel a distance equal to one length of a chariot" as a unit of time. (Here it should be noted that for a sensible Greek lover of chariot racing, the length of a chariot was a natural measure of the distance that one chariot overtakes another, and time where it finishes earlier.) Now imagine that the third chariot is moving at the same speed as the second, but in the opposite direction. When these two chariots pass one another, the time it takes each of them to cover a distance equal to one length of the chariot is only half the original unit. So, Zeno concludes his paradox, half a unit of time equals a whole unit of time. This argument of his, when understood, greatly perplexes any person who has always taken it for granted that movement and rest are absolute opposites. The answers that come to our own mind in this case came to our common sense from the theory of relativity. We understand that the movement, of course, always occurs relative to some coordinate system, that is, the same chariot has different speeds depending on the way the speed is measured. For listeners of Zeno, this thought was not at all familiar. If Zeno had said in his conclusion: "Therefore the same moving body has different speeds at the same time," the listeners would have considered this as absurd as what he proposed to them: that a whole segment of time is equal to half this segment.

Zeno's Stadium Paradox

AAA is at rest, BBB is moving away from the turn sign and CCC is moving towards the turn sign at the same speed. If we take "the time it takes to travel a distance equal to one chariot length" as a unit of time and measure it by the movement of B relative to A, then B will pass C in half that time. This contradicts the notion that the original unit of time chosen was indivisible. This argument can be applied to show that there cannot be a smallest indivisible length of time.

While it is clear to the modern reader that Zeno did indeed discover an important truth, our 20th-century common sense is so accustomed to the idea that speed is relative that this fourth problem is of less interest to us than the other three. However, if we look at these paradoxes as critical attacks on the "scientific" ideas about motion that the Phagoreans expounded, we find that in this last of the four paradoxes, Zeno hid another problem.

At the time when Zeno and Parmenides lived, the Pythagoreans were experts in the Western world in the natural sciences and mathematics. Do the four paradoxes of Zeno fulfill their function of criticizing the more precise definitions of space, time, and motion that were then common?

The Pythagoreans seem to have agreed that the physical world, including space and time, is made up of individual "points" and "moments." So they would define motion in much the same way that we define speed, as moving through a certain number of points in space in a certain number of moments of time. In physics and geometry, the Pythagoreans also unanimously recognized the position that any continuous object that has length - for example, a line or part of it - can be divided into two parts. But apart from this agreed general opinion, there was not a single view accepted by their entire school as to what the size of moments and points was: they might not have no size or could have a finite length and a finite duration, respectively. There was also no consensus on whether a line defined by dots should be considered as a series of dots one close to the other, or whether the points on the line mark the boundaries of the intervals, and the gaps between the points are occupied by some kind of void or space. .

Lack of agreement on specific details meant that Zeno had to consider four possible cases to show that none a precise description cannot be free from controversy. He seemed to feel that Parmenides had already proved the absurdity of trying to fill the gaps between dots with some sort of emptiness. Such emptiness would be a form of non-existence, and since nothing cannot do anything and cannot have any properties, it would be illogical to think that it separates or connects dots. Therefore, from the point of view of logic, only those options in which the segments of space (and time) are closely adjacent to one another do not raise objections.

The four possible Pythagorean ways of describing motion fall into two groups: either (1) the segments of space and part of time do not resemble each other, or (2) they do. If (1) they are not similar, then either (1a) each moment of time has a certain length, but the segments of space do not, or (1b) the opposite is true: points have a finite length, and moments of time have no duration. If (2) time and space are similar, then either (2a) the elements of both have no extension, or (2b) the elements of both have some minimum finite length [that is, either T = 1, S = 1, or T = 0, S = 0].

It is these four possibilities that are considered in order in the four paradoxes of motion. A table can help you visualize this in a compact form:

First, let's return to the Divide by Two problem and note that this puzzle assumes that the space between you and the door leading outward can be divided indefinitely. For both Zeno and Pythagoras, this meant that spatial points have no length. At the same time, when Zeno said: "It takes some time to pass through each point of space," he assumed that moments of time have some kind of "length" and therefore, if you add up an infinite number of moments, the sum will be endless time. This contradiction stems from the fact that space applies the Pythagorean postulate that any continuous quantity can be divided into two parts, and to time another Pythagorean theorem applies, that a continuous quantity is a sequence of an infinite number of distinct points. (From the point of view of arithmetic, since spatial points have no length and therefore their length is zero, then when they are added, the length cannot be greater than zero. But since moments of time have duration, the sum of any number of these moments will be greater than zero. If we now describe motion as the ratio of distance to time s / t, you get 0 / t, that is, immobility.)

The Achilles paradox makes the opposite assumption. When Zeno declares that Achilles never will not be able to outrun the tortoise, he clearly speaks of time, which can be divided into two parts to infinity and which, therefore, consists of moments without extension; but he assumes that at each moment of time some finite segment of the path is passed. In this case, it turns out that the speed of any movement is equal to infinity, because the ratio of distance to time (s/t) is equal to s/0. Aristotle considered the paradox about Achilles "childish" because "it is obvious that space is divided into parts in the same way as time." What Aristotle didn't realize was that Zeno was using the Achilles paradox to refute one of the logically possible Pythagorean interpretations. (In fact, these first two cases considered by Zeno were never taken seriously as scientific hypotheses until the 20th century; but the Pythagorean could consider them, and therefore Zeno included them in his attack along the entire front.)

In the arrow paradox, the assumptions are quite simple and obvious: if neither moments of time nor segments of space have absolutely no extent, the ratio of distance to time will always be 0/0, and this expression does not make sense. The reason that the problem of the arrow creates such fundamental difficulties is that we often want to cut space and time into separate fragments, as if into pieces. A whole long and interesting chapter in the history of mathematics is filled with attempts, using various strategies, to put these fragments back together into a continuous whole.

And finally, in the fourth problem with the movement of chariots relative to each other, it is assumed that points in space and points in time have a certain extent, but it is minimal, and therefore they have length but are indivisible.(If they were divisible, then by repeated division into two they could be divided into parts, each of which would be nothing and we would again be faced with the case of an arrow.) But the assumption of indivisibility immediately turns out to be wrong: we see that the fact of the relativity of motion leads to the need to divide moments or points into smaller parts, if we do not agree with the conclusion of Zeno himself: “So , two time intervals are equal to one interval. The fact that Zeno called the objects moving around the stadium the word "onkos", which meant something voluminous, is characteristic of this philosopher: ordinary listeners immediately understood that wagons or chariots were meant here, and imagined them; but the word "onkos" still meant "physical body" among the Pythagoreans, and a more learned listener could imagine a spacious stadium and on it tiny Pythagorean dots - moving onkoy-body.

With four paradoxes, Zeno achieves very well what he wanted. He shows logically strictly that something is wrong in the Pythagorean ideas about movement, space and time. These demonstrations by Zeno did not convince later thinkers to accept the conclusions of Parmenides, but they did make these thinkers respect formal logic and see new possibilities for its application. They also naturally forced them to try to formulate the Pythagorean concepts in a new way, in such a way as to eliminate the contradictions shown by Zeno. These attempts took many forms: in Anaxagoras, the rejection of the concept of individual points and replacing them with a continuous sequence, in Aristotle, the complete separation of arithmetic from geometry, and in atomistic theory, the clear distinction between physical and mathematical “divisibility” underlying it.

Zeno (c. 490 BC - c. 430 BC) belonged to the Eleatic Greek school of philosophy, which proclaimed that any change in the world is illusory, and being is one and unchanging. Zeno's teacher Parmenides argued: "The universe is unchanging, because, recognizing change, we would recognize the non-existence of what exists, but only being exists" (). Zeno's point of view is more dialectical. He said: “Suppose the existence of your change; in it, as in change, its nothingness is contained, or, in other words, it does not exist. At the same time, it should be noted that for Parmenides, change meant a definite and complete movement, while Zeno spoke out and spoke out against movement as such, or, in other words, against pure movement. "Pure being is not movement; on the contrary, it is nothingness of movement."

For those who held the opposite point of view, Zeno offered to refute his paradox, formulated in the form of four aporias (from the Greek aporia "no way out"), showing that the movement (a model of "visible" change) is logically impossible. Most modern readers are familiar with Zeno's paradox precisely in the above formulation (it is sometimes called a dichotomy - from the Greek. dichotomia "dividing in two"). The first aporia declared that it was impossible to cross the room. After all, first you need to overcome half the path. But then you have to overcome half of what is left, then half of what is left after that, and so on. This bisection will continue indefinitely, from which it is concluded that you will never be able to cross the room.

Aporia, known as "Achilles", is even more impressive. The ancient Greek hero Achilles, invincible in running, is about to compete with the tortoise. If the tortoise starts a little earlier than Achilles, then in order to catch up with it, he first needs to run to the place of its start. But by the time he gets there, the tortoise will have crawled some distance that Achilles will have to cover before catching up with the tortoise. But during this time, the turtle will crawl forward some more distance. And since the number of such segments is infinite, swift-footed Achilles will never catch up with the tortoise.
And here is the third aporia in the words of Zeno himself: “If something moves, then it moves either in the place that it occupies, or in the place where it does not exist. However, it cannot move in the place it occupies (since at any given time it occupies all this space), but it also cannot move in the place where it does not exist. Therefore, movement is impossible. This paradox is called "The Arrow".
Finally, there is a fourth aporia, in which we are talking about two columns of people equal in length, moving in parallel with equal speed in opposite directions. Zeno states that the time it takes for the columns to pass each other is half the time it takes for one person to pass the entire column.

The aporias of Zeno excited creative thought from the very time they were formulated. It is known that the cynic Diogenes of Sinop, in response to the arguments of our philosopher, silently got up and began to walk back and forth; thus he refuted his paradox about the impossibility of moving things. But where there is a struggle by arguments, Hegel writes, only the same refutation by arguments is permissible; in such a case one cannot be satisfied with sensuous certainty, but one must understand. Moreover, the presence of visible movement was not disputed by Zeno. Movement has sensuous certainty, it exists, just as elephants exist; in this sense it never occurred to Zeno to deny movement. The question here is about its truth, but the movement is not true, because the idea of ​​it contains a contradiction; and therefore movement does not have true being. The refutation of this provision is a completely different level of controversy and it is not easy to rise to it, since in Zeno's paradox, for the first time, such fundamental concepts as "space", "time", "movement" and human consciousness were brought together. Accordingly, in order to prove the absurdity of his aporias, it is necessary first to determine the philosophical, physical nature of space, time and movement.

Hegel himself, who gave Zeno's paradox an important place in his Lectures on the History of Philosophy, builds his arguments as follows. The first form of refutation consists in the statement: "Movement does not have truth, since the moving must reach half of the space before it reaches the goal." That is, we must recognize, as a premise, the continuity of space. The mover must reach a certain destination; this path is the whole. To pass through the whole, the mover must first pass through the half; now the end point is the end of this half, but this half of space is in turn the whole, which thus also has halves in it; what is moving, therefore, must first reach half of this half - and so on. to infinity. Zeno here points to the infinite divisibility of space: since space and time are absolutely continuous, it is impossible to stop with division anywhere. Each magnitude (and each time and space always has a magnitude) is in turn divisible into two halves, which must be traversed, and this is always the case, no matter how small a space we take. Movement turns out to be the passage of this infinite number of moments; it therefore never ends; what is in motion, therefore, cannot reach its final point.

The general resolution of this contradiction, given by Aristotle, is that space and time are not infinitely separated, but only infinitely divisible. But it may seem and indeed it seems that, being divisible, i.e. divided in possibility, they must also be divided in reality, for otherwise they could not be divided indefinitely. Proceeding from this consideration, we, without hesitation, agree, as with something innocent, with the assertion that what is moving must reach half; but thus, writes Hegel, we have already agreed with everything else, i.e. agreed that it would never reach, for to say it once is tantamount to repeating the same statement an innumerable number of times. It is objected that in a large space one can recognize the need to reach half, but at the same time they imagine that in a very small space they reach a point where dividing in half is no longer possible, i.e. they reach the indivisible, not continuous, they reach what is not space. But this is not true, for continuity is the essential definition; the assumption that half is present already contains a break in continuity. It should be said: there is no half space, for space is continuous; you can break a piece of wood into two halves, but not space, and in motion there is only space. One could say: space consists of infinitely many points. It is commonly imagined that it is possible to pass from one such indivisible point to another, but in this way one cannot go further, for there are an innumerable number of such points. With his seemingly innocent assumption, Zeno forces us to split the continuous into its opposite, into an indefinite set, as a result of which we do not accept continuity and, therefore, do not accept the presence of movement. It is erroneous to assert that it is possible if you reach one such point, which is no longer continuous; this is erroneous, because movement is connection.

The same is true of the second aporia. Faster movement, says Zeno, does not help Achilles to run the distance he is behind; the time it uses for this is always used by the slower one, so that in its continuation it again outstrips the first, albeit by a smaller and smaller distance, which, however, due to the continuous division in half, never completely disappears. Aristotle, considering this argument, says briefly about it: “This proof represents the same infinite division; it is, however, false, for the fast-moving one will still catch up with the slow one, if it is allowed to cross the border. His answer, Hegel writes, is correct and contains everything necessary: ​​in this representation, it is precisely two points of time and two spaces that are separated from each other, i.e. separated from each other; if, on the contrary, we assume that time and space are continuous, so that two points of time or space, as continuous, are related to each other, then they are two points and equally not two points, but identical. In representation, we resolve this question most easily by saying: “Since the second body is faster, it passes through more space at the same time than a slowly moving one; it can therefore reach the place where the first body begins its movement, and then go still further. Time, therefore, is that limited, beyond which, according to Aristotle, we must go, through which we must penetrate further; since it is continuous, we, in order to solve the difficulty, must say that what th; we distinguish as two parts of time, must be taken as one part of time. In motion, two points of time, as well as two points of space, are in fact one point. For when we want to understand movement in general, we say that the body is in one place and then goes to another place. During the movement, it is no longer in the first place, but at the same time it is not yet in the second place; if it were in one of these places, it would be at rest. But where is it? If we say that it is between these two places, then we will not actually say anything, for in that case it would also be in the same place, and, therefore, the same difficulty would arise before us. But to move means to be in a given place and at the same time not to be in it, and therefore to be in both places at the same time; this is the continuity of time and space, which alone makes motion possible. Zeno, in his conclusion, strictly separated these two points from each other. We also recognize the discreteness of time and space, but they should equally be allowed to transcend the boundary, i.e. to posit a boundary as something that is not a boundary, or to posit divided parts of time, which at the same time are also undivided parts.

From what has been said, it is obvious how Zeno's third aporia can be refuted when he says: “The flying arrow is at rest, and precisely because the moving one is always in the “now” equal to itself and the “here” equal to itself, in the indistinguishable”; arrow - here and here and here. We can say of an arrow that it is always the same, since it is always in the same space and in the same time; it does not go beyond its own space, does not occupy another, i.e. more or less space, but we call this not movement, but rest. In "here" and "now" becoming different is abolished; in them, it is true, limitation is posited in general, but it is posited only as a moment; for in "here" and "now" as such there is no distinction. Aristotle says of this third proof: "It arises from what Zeno assumes that time consists of now, but if we do not agree with this, there will be no conclusion."

As for Zeno's fourth objection, it is built on the contradiction that results from moving in opposite directions; the general movement receives one body entirely, while in itself it does only a part. But in fact, the distance traveled by one body is the sum of the distances traveled by both.

Philosophy in the Age of the Greco-Persian Wars

Neither knowledge nor thinking ever begins with complete truth - that is their goal; thinking would be unnecessary if truth were ready.

The paradoxes of Zeno (circa 425 BC) are one of the most interesting problems in the world of mathematics. At first glance, incredibly simple, they actually touch on issues such as infinity, the indivisibility of space and time.

First of all, I would like to say that Zeno of Elea is an ancient Greek philosopher, a representative of the Elea school, a student of Parmenides. And, like a good student, Zeno formulated a number of aporias (paradoxes, unresolvable positions) to prove the teachings of Parmenides about a single immovable being. It is also believed that these paradoxes were directed against a rival school with the Eleatics, most likely against the Pythagoreans, who believed that the size or extent is made up of indivisible parts. Aristotle gives the following Pythagorean definition of a point: "A unit having a position" or "A unit taken in space."

The most famous aporias about movement are: Achilles and the tortoise, Dichotomy, Arrow, Stages. For the first time they with solutions were mentioned in the work of Aristotle.

Achilles and the tortoise

In this paradox, Zeno claims that Achilles will never be able to catch up with the tortoise walking in front of him.

At first glance, the answer is obvious. Achilles will catch up with the tortoise. But there is a snag in the decision, over which many scientists are still arguing and cannot agree.

This problem can be solved using a geometric progression. So, in order to catch up with the tortoise, Achilles from the beginning needs to reach the place where she started her journey. To do this, he needs to go half the way, then half the half way, and so on. That is, each time the path will decrease by 0.5 times. This is our geometric progression with the denominator q=0.5. By the denominator, we see that the progression is infinitely decreasing. Therefore, Achilles' path will tend to zero, but then he will never catch up with the tortoise!

Dichotomy

This paradox states that before a moving object can travel a certain distance, it must travel half that distance, then half the rest of the distance, and so on ad infinitum. Since every segment remains finite upon repeated divisions of a given distance in half, and the number of such segments is infinite, this path cannot be traveled in a finite time. Moreover, this argument is valid for any, arbitrarily small distance, and for any, arbitrarily large speed. Therefore, no movement whatsoever is possible. The runner is not even able to start.

Arrow

In this aporia, Zeno argues that any thing is either at rest or in motion. He proved this by the example of an arrow. Zeno argued that a flying arrow does not move, since at a certain point in time it occupies a space equal to itself, and such a space can only be occupied by an immovable object.

For many years, among mathematicians and philosophers, disputes about Zeno's aporias did not subside. They are still trying to logically substantiate, prove. The essence of Zeno's paradoxes lies in the fact that neither space nor time can be considered as a set of infinite numbers not interconnected with each other. Many experts agreed with Bertrand Russell's famous analysis of Zeno's paradoxes. According to Russell, Zeno's paradoxes were not satisfactorily resolved until the advent of Georg Cantor's theory of infinite sets.

Cantor's theory allows us to consider infinite sets (whether it be sets of points on a line or instants of time) not as a set of isolated individual points and events, but as something whole. Solving Zeno's paradoxes requires a theory like Cantor's set theory, in which our intuitions about individual points and individual events are combined into a system - a consistent theory of infinite sets.