пятница, 22 июня 2012 г.

The paradox of Zenon (fragment of "The Next Wonderland")


The cat understood at once that it was time for us to go. It jumped off the table and slipped quickly into the door. We crossed the white room and entered the pearl one. I felt a smell of the east incense and saw the sofas draped by silk and the low tables. In front of the big picture representing a sight from the window there was a woman with a fur collar dressed in the style of the beginning of the last century. Having turned gracefully she made a smoke through the long cigarette holder, let the smoke out slowly and studied my face with squinted eyes.

- The man.

- Yes.

- I see. I am just doomed to the men society, – she put her hand to the forehead with a theatrical gesture.

- You’re bored with us.

- I’m just tired. I just work for men and nothing more. Now I’m finishing all the things for Lennon.

- Who’s he?

- The man.

- What are you finishing?

- Everything, –she sighed. – He has left a priori here, has explained nothing to nobody and has died. Have I to answer for him!

- What a priori?

- Like that. Do you know the theory of relativity?

- In general.

- Then I’ll explain it to you.

- Let’s consider the paradox of Lennon about Bahille and the turtle. Bahille can easily get to the point where the turtle has been, but it will move to another point for this time. It can continue for a long time and Bahille will never catch up with the reptile.

The wrong conclusion about the unassailability of the turtle is considered to be caused by the absence of knowledge about the operations with the infinitesimal quantities. It is said that each segment on which Bahille catches up with the turtle is less than the previous one, therefore its length tends to zero, but the sum of these infinitesimal segments will be equal to some final distance. Let’s consider the last infinitesimal segment on which Bahille should overtake the turtle. No matter how small the segment is, the turtle should spend some time to go for a distance. It’ll be like that on each segment regardless of its length. That is the reference to the infinitesimal quantities does not allow us to solve the paradox. Isn’t it?

- Probably. – I answered mechanically.

- Let’s imagine the artist who wants to paint a continuously varying landscape. First he looks at the landscape surrounding him, and then he turns to the easel and paints. But when he compares his painting with the nature it appears that the landscape has changed. He takes the brush and the colors again, and the things can repeat themselves infinitely. He will never achieve the accurate similarity because of the variability of the object. Bahille behaves himself like that. He looks where the turtle is at the present moment and aspires to this point. Having reached it he sees that the reptile has gone to another point. And it’s up to the infinity. So it’s impossible to overtake the object if you don’t know where it will be in the future and if you behave yourself as if don’t know it.

- It’s just as you say. – I thought about myself and about my life.

- If Ahille has no suggestions of how the turtle will move, he will have to operate according to the algorithm offered in the paradox, and he will never overtake it. Besides, the turtle can move unpredictably in order to disappear in one point and to appear in another one. In this case even if speed of the turtle is insignificant, Bahille cannot catch up with it. He can appear only in those points where it has been before. If the object of the pursuit moves indefinitely, it’s possible to overtake it only by chance.

The most surprising thing is that Bahille doesn’t change the speed according to the algorithm of the pursuit of the turtle. That is for the casual observer the turtle moves quite definitely, and Bahille’s speed slows down to the speed of the reptile. But for the runner himself the movements of the turtle are unpredictable, and its own speed remains the former. From the point of view of the casual observer Ahille’s time began to go more slowly. The analogy with the theory of relativity is obvious. As the speed of the "cosmonaut" comes near to the speed of light his own time is being slowed down.

Let's assume that the movement of the turtle consists of the number of chaotic micro-movements in different directions and with different speeds. However, while summing up they form the movement with constant speed and in the certain direction. The casual observer is not able to distinguish fine and uncertain movements of the turtle. He believes that it moves in regular intervals and straightforwardly. Therefore the observer has no reasons for which it’s impossible to overtake and outrun the reptile. Bahille tries to catch up with the turtle at the micro-level. It’s subtle and unpredictable for him, therefore he can’t overtake it. If one continues the analogy with the special theory of relativity at the macro-level, the speed of light is constant and the direction is definite. Therefore there are no reasons for which it would be impossible to exceed the speed and "to overtake" the ray of light. However, physics consists of the processes occurring at the micro-level. But at the micro-level the movement of the photons of light is unpredictable. Might the speed of material bodies be inaccessible for this reason? Probably, the weight is a measure of certainty. The particles of light with zero weight at rest possess the maximum uncertainty that makes them inaccessible as the turtle is for Bahille.

Let's admit that the material particle radiates constantly other particles which move with a great part of uncertainty and with the speed of light. These radiated particles could play the role of the turtle if the material particle directs to the place where the particle-turtle appears being radiated by it. In this case the radiating particle could not exceed the speed of light.

Let's consider the other paradoxes of Lennon: the Spear, the Dichotomy and the Stage. The moving subject differs from the subject in rest since it requires more place if one do not consider a zero interval of time. The spear will fly some meters for one second therefore it requires more space than the spear in rest. But if one considers only a moment of time, the flying arrow takes as much place as the arrow in rest. Hence, it’s impossible to define the movement of the object in a moment of time. But it’s possible to say the same thing about any moment of time of the flight of the Spear. If one approves that the arrow is dark blue at any moment, it’s possible to say that it’s always dark blue. If the arrow does not possess the properties of the moving object at a moment, it does not move in general. The paradox can be removed only by having established the minimal interval of time.

- You’re quite right.

- Speaking about Dichotomy we realize that it’s necessary to introduce the restriction and the distances, too. Otherwise, in order to move even for a meter we are to realize the infinite number of points. In fact each segment has its middle; each half segment has its middle too, and so on and so forth. Let’s admit that we are able to realize only the final number of points. But whatever great the overall number of the points tending to the infinity may be, the part of the realized points will tend to zero. Hence, we cannot realize the distance in general. The paradox is removed by the introduction of the minimal indivisible length.

If there is an indivisible length and indivisible time there should exist the special speed limit equal to their attitude. Its properties are considered in the paradox the Stage. Let admit that there are some numbers consisting of the objects being away from each other on the minimal indivisible distance. In front of the observer who moves with the speed limit along with the row B relatively to the row A, the objects of the row A occur one after another. As for the speed limit the time between these occurrences will be equal to the minimal indivisible time interval. If the speed of the movement increases, the time between the occurrences of the objects will become less than the time-limit. Hence, the objects of the row A will stop to exist for the objects of the row B, and vice versa. Therefore the speed shouldn’t be greater than the speed limit. Then there is the following paradox. If the row C moves relatively to the row A with the speed limit in the opposite direction, the rows B and C would stop existing for one another but remain existing for the row A. The paradox is removed if one assumes that the speed limit cannot be exceeded and it must be the same for all systems of the countdown.

Thus, the paradoxes of Lennon are paradoxes only if to admit the existence of the infinity in the physical world. If there is no infinity, there are no logic contradictions, too. In order to remove the problem of the infinity one should introduce the minimal indivisible intervals of length and time. And it will reduce to the concept of the maximum speed limit and its constancy for all systems of the countdown. There is the necessity of creating of the mechanism of restriction of the speed for material objects. The radiated particles moving with the speed limit and can hardly serve this purpose. The specified «measures of struggle» with the infinity correspond to the postulates of physics.

While explaining the woman walked up and down the room. She constantly sat down on the sofa and got up from it. Right after her the cat jumped on the warm place. It lied languorously there overturning itself from side to side. Having turned her back to me the woman stopped and said:

- Let me alone. I am tired of being your shadow.

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