On the paradox of Zeno

Elea to defend his teacher. Parmenides about the "existence" is not moving, is a doctrine, put forward the famous motion paradox and many paradoxes, to show that the movement and more is impossible. His conclusions, of course, are absurd, but he gives a plausible argument, and it is often said that these arguments constitute a paradox or paradox. However, if careful scrutiny, its conclusions may not be absurd, and its argument may not be convincing, so the neutral claim that these arguments fortheArgumentis themost appropriate.

　

First, historical retrospective

The motion theory of Zeno is all from Aristotle's report in physics, there are four:

1 and dichotomy method. An object must arrive at half the full length before it reaches its destination, and the requirement can go on indefinitely, so if it is started, it will never reach the end, or it may not start at all.

2 , Achilles (a translator). The runner will never catch up with the jogger, because the Chaser must first run to the starting point of the pursuers, and when it reaches the starting point of the pursuers, there is a new starting point waiting for it, and there is an infinite starting point.

3 , flying vectors do not move. Anything that occupies a place equal to itself is stationary, and the flying Arrow always occupies a place equal to itself in any moment, so it is still.

4 , Sports Ground. Two columns of object B, c opposite to a row of stationary object A opposite movement, B crosses the number of a is half of C, so half time is equal to one times.

Four arguments can be divided into two groups, the first two assumed that space-time is continuous, the latter two assumptions are discrete, the first argument of each group absolute motion is impossible, the second argument relative motion is impossible.

On the theory of multi-Simprichu in the "physics" of the report, the main idea is: If the thing is more, then the Convention large to infinity, small will be small to zero, because any number can be infinitely divided, if the result of the division equals zero, then the sum is zero, if the result is not 0, then the infinite sum is infinite.

The above reporting from the perspective of philosophical history is too careless, but for the discussion of its philosophical meaning is almost enough.

The absolute idealist at the turn of the century 19 and 20 is like Bladley (bradley,f). h) To accept all the arguments and conclusions of Zeno in its entirety. He sees the motion, the time space as the illusion, the Zeno argumentation just conforms to his proposition, of course accepts completely. In "phenomenon and reality" he wrote: "Time and space, as the most obvious proof is not the reality, but a paradox of the illusion." "In addition to Bladley, most philosophers in the history of philosophy believe that the conclusion of Zeno is absurd and its argument is problematic." However, in the process of constantly examining the defects of their argumentation, people have discovered the profound point of the theory of Zeno. Often people think they have solved the paradox of Zeno, and it is not long before they find out.

The earliest known criticism came from Aristotle. With regard to dichotomy, he said, although it is not possible to cross an infinite point at a limited time, it is possible to see the structure of time in the same way as space, and to be able to divide infinitely, so that the infinite points of space can be crossed in infinite points of time; about Achilles, he said, "If the slow is always ahead of course, But if one is allowed to cross a distance, it can be caught; With regard to the flying vector, he said, the premise of this argument is the discontinuity of time, and if it does not recognize the premise, its conclusion is no longer established; With respect to the sports field, he said, the speed of the object relative to the stationary object is certainly not the same, The time spent crossing the same distance is of course not the same. The significance of the Assyrian criticism is to make the theory of Zeno more clear, and the preceding analysis of the various theories clearly refers to Aristotle's criticism.

Hegel's solution to the paradox of Zeno is: "The motion means that it is not in this place; This is the continuity of space and time--and that is the condition that makes the movement possible." "The main point of this solution is to emphasize the continuity of time space and to give new and unique interpretations of continuity." However, it does not seem to directly criticize the Zeno itself, and the unique interpretation of continuity is incompatible with the precision required by mathematics and logic. Influenced by Hegel, the philosophical circle in our country generally thinks that Zeno does not understand the dialectical relationship between continuity and discontinuity, which causes the contradiction of motion. The main idea is that Zeno's argument does not use dialectical logic and is therefore ineffective. This criticism is also subsumed, inconsequential.

　

Second, analysis and analysis of the dilemma

Since 19th century, many solutions have been proposed from the point of view of mathematics and logic, and I am collectively referred to as the analytical method.

1. Summation of infinite series

In the motion Paradox and multi-paradox of Zeno, the sum problem of infinite segmentation is involved, and the development of calculus makes it possible to quantify it. For the multi-paradox, it can be sure that the parts of the infinite division tend to zero but not equal to zero, the sum of which is not equal to zero, but it will not be an infinite amount.

In the case of Achilles, although he had to reach a starting point countless times, the distance it took was not an infinite amount, and the space distance in the case of the turtle was:

(where D is the initial distance, which is the speed of the fast and the slow)

is a finite number, for a limited distance, of course, can pass through within a limited time and reach the end point.

2 , infinite machine problems

Many people, after calculating the sum of the infinite series as a finite number, think that the paradox of the Achilles de Zeno is solved, and they clearly believe that the paradox lies in confusing the point of experience with the infinite distance of experience, as long as it clarifies this, the paradox is naturally eliminated.

However, the calculation of the distance is limited and does not solve the problem. Let's take a look at how we figure out. What does the sum of infinite series ultimately mean by the method of finding the limit? It's not that we add all the items of an infinite series together exactly equal to this limit value, but that we can let the infinite series and fully close to this limit value, how close we want to be close. Note that it is still "close". In elementary mathematics we have a simpler way of finding the time to catch up with a turtle, which is to assume that it is T, and that you can list the equation:

The solution equation can be obtained

One of the prerequisites in this approach is to assume that Achilles eventually catches up with the tortoise. This hypothesis suggests that what mathematics tells us is that if it does, it will take much time, but mathematics does not solve the problem of "can".

Therefore, it is also necessary to go back to the question of "crossing the infinite point in a finite time", which, if you think of crossing a point as the completion of an action, becomes an infinite operation problem, which is named "Infinite Machine" ( Infinite Machine ), some call it a "Super task" ( Super Task ). Many people have proved that the super task is impossible to complete, the infinite machine does not exist.

The most famous infinite machine is the throw machine, it is designed to: a small ball from a to the B at the start of throwing, so that the ball from a to throw to B when the time spent One-second minutes, from B to throw a place to spend one-fourth minutes, and so on, and so on, the toss time is:

The total time taken to nth time is:

The machine is now required to stop when the time reaches 1 minutes. But the problem arose, and people found it impossible to determine where the ball eventually fell. From the above view, when n takes an odd number, Fall at B, take an even time to fall at a place, but the ball more and faster, only after an unlimited time to reach 1 minutes, but an infinite numbers is no odd and even points, therefore, do not clear 1 minutes when the small ball in what position, that is, the ball does not finish, super task.

Indeed, since there is no last item in an infinite sequence, it is impossible for Achilles to reach the end of the (infinite) point given by all Zeno.

Similarly, the infinite machine problem is not solvable, but also enhances the effectiveness of the two-point method. The logical analysis of the theory of Zeno strengthens the power of its argument.

3 , flying vectors and speed problems

If we assume that a moving object is in a space at every moment (apparently, but not necessarily true), then the only way to determine whether it is motion or stillness at this moment is to see if it has speed. Isolated investigation This point does not know whether it has the speed, must consider an interval, the velocity formula is:

Which is the differential of trajectory vectors.

This requires that the trajectory is at least continuous, but Zeno is clearly assuming the opposite condition, that the trajectory is discontinuous, that the discontinuous trajectory cannot be differentiated, and that its velocity cannot be determined.

However, even if the trajectory is continuous, to find the speed, it is still not solve the problem of the motion itself, because the speed of the size, only means that if there is motion mathematics can tell us how much speed, mathematics also do not tell us whether there is movement.

In general, if a moving object is in a position at every moment, then in this moment we really do not know whether it is moving, especially when time and space are discontinuous.

4 , sports ground and space-time discontinuous structure

On the debate about sports ground from the textual aspect has not yet been fully understood, Aristotle's reporting makes people feel that the theory is too without depth, is it not the same as the motion of the stationary object and the motion of the moving object is not the same? The more plausible explanation is that Zeno's desire to reveal movement through the movement of three-column objects in a discrete space-time structure is impossible, and is crucial to the discrete structure of space-time.

Assuming at the moment 1 , the three-column objects are arranged as follows:

Methyl armour

Ethyl ethyl B

C-propyl C

Each of these objects occupies a space unit. After a time unit is the moment 2, then the moment 3, and so on, the moment 2 and the moment 3 are arranged respectively:

methyl Armour methyl armour

　 ethyl ethyl ethyl ethyl

　C-C-propyl-C-propyl

Zeno means that at the moment of 3 o'clock, just over two time units, B and C between two objects there is four space unit displacement, at the moment 2 o'clock, just over a time unit, B and C have two space unit displacement, that corresponds to a space unit displacement moment is what? At what point does the following arrangement occur?

Ethyl ethyl B

C-propyl C

There is no answer to this question, so talking about movement in the discrete structure of time and space is bound to be a question of two units of time, which is what Zeno says "one time is equal to half the time". This is of course absurd.

So far, the analysis of the theory of Zeno from the point of view of mathematics and logic has strengthened the power of the argument, and the modern mathematical tools have only further expressed the difficulties posed by Zeno.

Let's take a look at the transcendence of the "analytic" approach.

　

Third, the movement can not be analyzed

The founder of the Cynics School of Greek times, Augenny, has an answer to the question of Zeno. It is said that when his students asked him how to refute the Zeno, he walked around the room without saying a word, the students still do not understand, he said, "Zeno said the movement does not exist, I am not proving that he is wrong?" The story was a joke for a long time, and most people believed that he didn't know what Zeno meant. Zeno does not mean that there is no movement in the phenomenon, he certainly admits, but what he wants to say is that while objects are flying everywhere, movement is unreasonable, and we can prove that motion is impossible by logic. Therefore, the movement we see is false, not real, because the real thing must be logical.

Is it really possible to prove the impossibility of motion with logic?

Bergson is right, the whole point of the theory of Zeno is to use the motion trajectory instead of the movement itself. Many modern analytic philosophers further point out that Zeno uses the mathematical trajectory of motion to replace the physical trajectory, and then imports the real physical motion into the infinite mathematical astray.

The point structure of time and space plays a great role in the analysis of Zeno. Although space-time in dichotomy and Achilles is a dense point structure, space-time is a discrete point structure in flying vectors and sports fields. Let's look at the problems caused by the dense point structure.

The problems caused by the discrete point structure of spacetime may be more profound. Since the discrete structure inevitably leads to the absurd conclusion that all velocities are the same, it is necessary to re-examine the structure of time and space. If space-time has intrinsic structure, it is logically possible to take a discrete form. The so-called intrinsic structure refers to the structure independent of the motion of the object, Newton's absolute space-time has intrinsic structure. The problem of "sports ground" is rooted in the premise that time and space have intrinsic structure, and this premise is wrong. Modern physics reinforces the notion that time and space are the relationships between events, not independent entities; The set theory shows that in the assumption that the space is continuous, any segment has exactly the same structure, so the length of the most basic space is not intrinsic, but agreed. the problems caused by the mathematical point structure of time and space may be solved, but even if these problems are solved, will the possibility of motion be solved? Or. There are two forms of dichotomy, one is that it never ends, and the other is that it cannot start at all. The first form of infinite sequence has no last item, so it cannot reach the end, and the second form of infinite sequence has no first item, so it cannot start at all. The second form is most instructive, because if the infinite problem is well solved, the first form of the difficulty can be eliminated, but the second form of difficulties is still a point: the first impetus. If it can have an initial start, it can start, and if it does not have an initial start, it cannot start. This is a repetition of the same language, suggesting that mathematical analysis does not prove that motion is possible. Recall the question raised by Bergson: Any analysis of the trajectory is not a complete analysis of the motion itself, so it is not possible to prove that motion is impossible, and it is impossible to prove that motion is possible. "Movement" is beyond "proof".

Once the motion event is regarded as the first position, and the time space is regarded as the abstraction of the motion event, the problem of flying vector is not difficult to solve. As a physical event, flying vectors should be used as the most basic element in the analysis, rather than as an exported thing, instead, the structure of spacetime should be derived from physical events such as flying vectors. The problem of flying vectors is caused by the reversal of the order of analysis.

The physical point is not a mathematical point, Achilles actually catches the tortoise with only a limited number of steps. Abandoning the point structure set by Zeno, the Achilles problem is no longer a problem.

Summing up the philosophical conclusion drawn from the paradox of Zeno: the movement itself is the first, and the trajectory is the second, the physical experience is the first, and the mathematical description is the second, the physical event is the first, and the space-time structure is the second. The analysis of the trajectory leads to a number of mathematical and logical problems, even though these problems can eventually be solved (and now of course cannot be said to have been solved), nor does it mean that the problem of motion itself is finally solved. Movement is more basic and can not be analyzed, it is beyond the theoretical rationality. Zeno was not able to prove the impossibility of exercise, because the movement was simply not provable. (Augenny's answer is not ridiculous, but rather profound) if only the evidence is true, then the movement is indeed "untrue". The ultimate philosophical significance of Zeno's paradox involves the Elea school's view of "reality".

On the paradox of Zeno