The phantom of Zenon. Achilles and the theory of relativity
	Hiromasa Noguti -------------- Cotosiro. This was written in Japanese to the homepage of Justnet in 1999 year, was translated to English in November 2000. And, this time, it, revised, was issued from Nifty December, 2003. This was rewritten in May 9, 2005. This was revised in February 15, 2009.
	That occurred at the midnight of the midsummer that it is hard to keep good sleep queerly. It is that one ghost appeared on my dream suddenly. When I turn my eyes to him, he stared at me with the sharp look. But, the appearance of him makes me feel so dignified that it is hard to talk with. "He must be Zenon!". I thought so at once. To tell the truth, I once had seen through the contradiction of his argument. So, he must have appeared with the intention to make revenge to me. Though it doesn't know why he appeared now, it is the unfounded hatred for me. I gave him my stare against the stare with unyielding attitude at once. Just then, Zenon holds a cloud on his face gloomily. And, he said "You know what surely, too, and you are not here" in sad tone. I peeped at him in bewilderment, not knowing the meaning of his words. His eyes, glimmering mysteriously, stare again at me. "Why, is a fight of dispute picked? But, it is silly idea.". I despised Zenon secretly, saying like that at the bottom of the heart. However, the opponent is the ghost anyway. If it tries to give me the harm of something or other with getting angry, it is very awkward for me. "Oh, Zenon!, You are great. Please tell me why my consideration doesn't reach the truth.". I asked to him like that modest, while taking care to not stimulate him.
			Zenon says. "Listen! To tell the truth, there is my real intention into the contradiction I stated before.". However, what a strange talk it is! "Why, isn't it strange?" I had uttered careless words, before noticing. It is because his unexpected utterance bewildered me probably. Of course, Zenon gives a sharp glance at me. I trembled in spite of oneself. However, I shouldn't show my weakness in this case. If not, I may be killed by him at once. I persistently have to take care to not make my weakness his advantage. I looked at Zenon with feigned calm. But, Zenon's attitude is strange. He glances at me sadly. His eyes had been filled with the sorrow and affection tinged with pity. I relooked at him in bewilderment. It can see the aura coming out of him like the blue fluorescence. And, its sight permeated into my heart quietly.
	What a noble sight it is! What a plaintive expression it is! At that time, I, holding my breath, regarded him in spite of oneself. But, when I blinks and again turns my eyes toward him, Zenon is nowhere no longer. "I shouldn't have come here.". Regretful voice only is heard, resounding with the dark slightly. Apparently, it seemed Zenon left from me. And, since then, Zenon doesn't appear at all. "I have won to Zenon at last!". Insolently, my heart was filled by such a delusion. However, what would his eyes filled with nobility and sorrow mean? It makes me feel the profound atmosphere though it doesn't know why. I couldn't but think of Zenon.

	It is said Zenon left many paradoxes for us. The race of Achilles and the tortoise is known as one of the paradox like that. However, this race is different from the common race a little. That is, since the quickness of the tortoise is slower than Achilles considerably, the starting line for Achilles is set in the considerably back position than the original starting line the tortoise waits in. Instead, the goal isn't set. Therefore, Achilles has to keep on running until he passes the tortoise. And, if he can pass the tortoise, Achilles is allowed the honor of the win. Conversely, if the tortoise always is running ahead than Achilles, the tortoise gains the title of the winner. But, it seems Achilles can pass the tortoise sometime if it follows the common sense. In other words, It seems Achilles gets the victory. However, Zenon says "Achilles can't catch up the tortoise, It, far from that, is impossible that Achilles passes the tortoise.". Why would he have said like that? The outline of its reason is stated in the following things.
	First, Achilles and the tortoise start to run at the same time by the signal. So, it is certain Achilles reaches to the starting line of the tortoise sometime. And, at that time, the tortoise would reach some place, that is, the position-Q. The position-Q, needless to say, is situated a little along than his starting line. After that, the tortoise reaches certain place, that is, the position-Y when Achilles reaches the position-Q. It is natural that the tortoise is advancing forward than the position-Q because the tortoise is running even if the tortoise is slow considerably. Therefore, it is natural also that the position-Y is situated a little along than the position-Q. If tortoise on the position-Y, with getting off guard at that time, does a nap or takes rest, Achilles may catch up with the tortoise.
	However, in this case, this tortoise is honest really. So, the tortoise keeps running with full power. Therefore, the tortoise is supposed to advance a little along than the position-Y when Achilles reaches the position-Y. In other words, the tortoise reaches certain place, that is, the position-Z. In passing, at that time, the position-Z is situated a little along than the position-Y. Of course, the interval between them becomes short acceleratedly as they run. It is because Achilles is very quicker than the tortoise. However, the relationship of their position doesn't change, if the above logic is proper. In other words, Achilles always is situated behind the tortoise. Of course, in this case, assume the tortoise and Achilles have no size, though it may be superfluous comment. Therefore, it is impossible that Achilles strides over the tortoise in his last one step. Now, How do you think about this logic? Probably you would say "it is strange". If so, would it be possible that it finds out the contradiction of this logic with using only this logic? Thereupon, I made one snail participate in this race, though you may think it is doubtful method. (In passing, as for the proper method and proper conclusion, it is stated after this reasoning is finished). Needless to say, the quickness of the snail in this case is slower than the tortoise considerably. Therefore, the starting line for the snail is set considerably along than the tortoise. In other words, the starting line for the tortoise is situated far behind than the snail's starting line. And, there is Achilles on the most back starting line.
		Now, the race was started actually in the moving world in Zenon's logic. They start to run from each starting line all together when the starting gun resounds. In front of my eyes, the snail is running desperately. And, the tortoise is chasing the snail patiently too. Then, from the most far starting line, Achilles comes approaching to them with the terrific quickness like an arrow. However, that sight is a little strange. Please make your consciousness concentrate to Achilles and snail, not looking at the tortoise for the moment. By the way, I assumed their quickness are in the ratio simple as follows, to make the reasoning easy. * Achilles : tortoise : snail = 4 : 2 : 1.  If so, it can suppose their starting line and arrival points as follows.
* Starting line for Achilles --400m-- starting line for tortoise --300m-- starting line for snail --100m-- position-A --100m-- position-B --25m-- position-C.
	So, in this case, when Achilles reaches the starting line of the tortoise, the snail is supposed to reach the position-A. In fact, at that time, the snail was on the position-A, when Achilles had run 400 meters. Therefore, the distance from the starting line of the snail to the position-A is 100m. Needless to say, it is because 400m multiplied by 1/4 makes 100m. Then, when Achilles reaches the position-A, the distance from the tortoise's starting line to the position-A is 400m. At that time, the snail had run 100 meters, that is 1/4 of former distance. Therefore, the above position-B is the place where snail reached at that time. Moreover, when Achilles reaches the position-B, he had run 100 meters from the position-A. And, at that time, the snail had run 25 meters, that is 1/4 of former distance. That place is the position-C. By the way, at that time, where would the tortoise be? Since no one observed the tortoise, nobody can know the position of the tortoise. They are running at the world Zenon's logic rules.
	Therefore, it is that the observer has to observe either of the relation between Achilles and the snail or the relation between the Achilles and the tortoise. It is because Zenon's world collapses for a instant if the observer see all those relations at same time. Therefore, in this case, we had observed only the Achilles and snail. In that reason, if we try to know the position of the tortoise, we always, while imagining the relation of position of Achilles and the tortoise, have to calculate the position the tortoise passes through. If not, nobody can guess the position of the tortoise because there isn't the method to know the position of the tortoise besides the method to do so. Therefore, it is doubtful whether the aspect of the race in between the three runners is based on Zenon's logic. However, for a little while, it dares to think about the relationship of those three runners here. Thereupon, look at the next two charts, to think, obeying Achilles' logic.
	1st chart: The aspect supposed to be seen when Achilles chases the tortoise. Starting line for Achilles --400m--starting line for tortoise --200m-- position-A2--100m--position-B2--50m--position-C2--25m--position-D2--12,5m-- position-E2. 2nd chart: The aspect supposed to be seen when Achilles chases the snail. Starting line for Achilles -- 700m -- starting line for snail -- 175m -- position-A3 -- 43.75m -- position-B3 -- 21.875m -- position-C3.
	Look at the 1st chart first. It can know certain speck that Achilles and the tortoise can't pass through absolutely appears in front of their eyes sometime. Thereupon, the last speck Achilles would arrive at is decided by the next equation if the interval between the above-mentioned speck in question and his start line is expressed as the Z. Then, the speck the tortoise would arrive at is decided by the same method.
The case of Achilles: The case of the tortoise:
	If it calculates this equation, the value of the Z draws near to 800 meters endlessly. However, as a matter of fact, this means "Z=800m" mathematically. Also, the value of Z2 is 400m. And, Z2 added to the distance between the starting line for Achilles and the starting line for the tortoise makes 800m. In other words, at that time, Achilles reaching that speck mathematically is parallel with the tortoise. However, it can't be said that Achilles and the tortoise is completely parallel yet if it follows Zenon's logic. Zenon says "the tortoise always is in front of Achilles even if its interval is little endlessly". Thereupon, look at the 2nd chart. Strangely, Achilles and the snail can run on more distance than 800m. In other words, since, at 1st chart, the tortoise almost is staying in the position of 800m from the Achilles's starting line, it seems to be possible that somewhere Achilles in the 2nd chart pass the running tortoise on 1st chart. But, at the actual race, the observer has to look at only Achilles and the snail as it stated before. Not until the observer can do so, Achilles can pass the tortoise. In other words, in a moment when Achilles passes the tortoise, nobody is looking at its scene. If so, did who on earth confirm Achilles passed the tortoise? As it thought before, this reasoning is strange somehow. By any chance, would it be the above conclusion is only the result of the mathematical trap or mere mathematical conjecture? To tell the truth, it is doubtful whether this conclusion is based strictly on Zenon's logic though apparently it looks to be based on that logic. If so, as Zenon says, would it be that Achilles can't pass the tortoise?
		However, even if my strange conclusion was the result produced by the mathematical trap, the conclusion itself may be right. I sank oneself into the thinking for a little while. And, I got caught to certain strange question. When one observer looks at only the Achilles and the snail, obeying Zenon's logic, if the other observer also looks at only Achilles and the tortoise as well, would what occur? Probably, it seems Achilles possesses two positions at the same time. However, it is said that the object in motion can't possess the two position at the same time, at least in this world. If you don't say the above-mentioned to be the contradiction, would you say what to be the contradiction? In other words, it would be that this race can materialize only in Zenon's world. Thereupon, at last as follows, it starts the refutation based strictly on Zenon's logic, against Zenon.
	If Achilles on the starting line tries to pass the tortoise, Achilles first has to reach the center(1st position) in between the tortoise and him. However, if he tries to do so, before doing so, he has to reach the center(2nd position) in between the 1st position and him. Furthermore, before doing so, he has to reach the center(3rd position) in between the second position and him. The following chart expresses that thing. The starting line for Achilles__the 3rd position____the 2nd position________the 1st position________________the tortoise. In other words, there are countless specks Achilles has to pass. And, needless to say, the specks of the center like that are lying even between Achilles and the third position endlessly. Wouldn't you notice what does it mean? I understood this means Achilles can't advance forward at all. It is because it is impossible everlastingly to pass all countless speck however long it spends times for that. In other words, the things able to move are nowhere if it follows Zenon's logic. (By any chance, it may be possible to move, if it doesn't decide the target to go to. But, since Zenon has decided the target like that in this race, it doesn't think this problem now.). In spite of that, Zenon tried to make Achilles and so on run in the race in question. This is the terrible contradiction. This conclusion seems to be proper because it is that his logic broke the premise of his argument. However, Zenon is the well-known philosopher even now. Would the paradox of this extent be in the level to be brought up for him? Or, there may be more profound meaning behind this paradox. When the phantom of Zenon appeared, he at first said "You know what, too, and you are not here.". By any chance, Zenon may have tried to say the phenomenon is caused corresponding to the relation between the observer and its object. It is said that the concept of the observer was brought into the objective recognition by Einstein first. At least, I had thought so until now. However, as a matter of fact, it may have been Zenon.
			
	Moreover, Zenon also told me "There is my real intention just in the contradiction.". Then, once Zenon said that the continuously flying arrow isn't moving at all. I before stated Achilles can't advance forward at all. As a matter of fact, it is that I imitated for its statement the logic of the flying arrow Zenon stated. Anyhow, the arrow flying the space is flying discontinuously. Probably he must have intended to say like that. If so, he may have had the idea like the quantum theory. And, if so, it also seems he imagined the world of possibility at that time. I am also imagining like that, since it heard the talk of the quantum theory. I am thinking the particle isn't in equal relation with undulation, for physics. In other words, when the perception to the undulation is wished by the cognition-subject, the undulation appears in front of eyes as the particle(substance).
			Therefore, the undulation is the actual existence existing as a possibility, not the reality. Probably, this idea would come to be recognized to be the truth, soon. By any chance, Zenon also may have been thinking like that as well. Anyhow, it seems his paradoxes include the deep meaning like that. However, his expression must have been restricted because of the mental level in the age called ancient times. In other words, it seems the suitable method for the expression wasn't found. His eyes giving off the dim pale light make me feel his deep disappointment. Needless to say, it would be because nobody noticed his real intention. I think Zenon's eyesight was exceeding Aristotle and Plato, and so on by far for finding out the truth. Therefore, Zonon's true intention wasn't understood. He must have become the ghost because of the harsh circumstances like that.
  
cotosiro