Posted: 6/22/2006 7:31:26 AM EDT
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The thread about .99999999 repeating infinitely equaling 1 got me thinking about something. Here's my thought.....isn't it true that nothing can ever be truely measured as to how long it is, and the length of all items is in fact infinite? Think about it...let's measure a stick that appears to be 12" long. However, as we zoom in closer, we see it's actually 12.001", or so we think, until we zoom in some more, and then it becomes 12.00100000003" long, so we zoom in some more, and so on. No matter how much more accurate you get in regards to length, there will always be more to measure (i.e. one extra atom longer, or a part of an atom longer), up to the point of being infinitely long, thus making any item impossible to measure. Am I way off base with this, or should I not have been dropped on my head as a baby? 
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You are learning, grasshopper....way to think critically! |
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You are correct. We can never meaure anything exactly. But we can get close enough for most purposes. ETA: you are incorrect about everything being infinite. in your example, as we zoom in, the amount that we find we are off by shrinks. So the measuring error is shrinking into infinify, not the object growing. |
The sad thing is I had this realization back in high school, but just now thought of posting it for the forum to debate. |
Zeno of Elea (circa 450 b.c.) is credited with creating several famous paradoxes, but by far the best known is the paradox of the Tortoise and Achilles. (Achilles was the great Greek hero of Homer's The Illiad.) It has inspired many writers and thinkers through the ages, notably Lewis Carroll and Douglas Hofstadter, who also wrote dialogues involving the Tortoise and Achilles. The original goes something like this: The Tortoise challenged Achilles to a race, claiming that he would win as long as Achilles gave him a small head start. Achilles laughed at this, for of course he was a mighty warrior and swift of foot, whereas the Tortoise was heavy and slow. “How big a head start do you need?” he asked the Tortoise with a smile. “Ten meters,” the latter replied. Achilles laughed louder than ever. “You will surely lose, my friend, in that case,” he told the Tortoise, “but let us race, if you wish it.” “On the contrary,” said the Tortoise, “I will win, and I can prove it to you by a simple argument.” “Go on then,” Achilles replied, with less confidence than he felt before. He knew he was the superior athlete, but he also knew the Tortoise had the sharper wits, and he had lost many a bewildering argument with him before this. “Suppose,” began the Tortoise, “that you give me a 10-meter head start. Would you say that you could cover that 10 meters between us very quickly?” “Very quickly,” Achilles affirmed. “And in that time, how far should I have gone, do you think?” “Perhaps a meter – no more,” said Achilles after a moment's thought. “Very well,” replied the Tortoise, “so now there is a meter between us. And you would catch up that distance very quickly?” “Very quickly indeed!” “And yet, in that time I shall have gone a little way farther, so that now you must catch that distance up, yes?” “Ye-es,” said Achilles slowly. “And while you are doing so, I shall have gone a little way farther, so that you must then catch up the new distance,” the Tortoise continued smoothly. Achilles said nothing. “And so you see, in each moment you must be catching up the distance between us, and yet I – at the same time – will be adding a new distance, however small, for you to catch up again.” “Indeed, it must be so,” said Achilles wearily. “And so you can never catch up,” the Tortoise concluded sympathetically. “You are right, as always,” said Achilles sadly – and conceded the race. Zeno's Paradox may be rephrased as follows. Suppose I wish to cross the room. First, of course, I must cover half the distance. Then, I must cover half the remaining distance. Then, I must cover half the remaining distance. Then I must cover half the remaining distance . . . and so on forever. The consequence is that I can never get to the other side of the room. What this actually does is to make all motion impossible, for before I can cover half the distance I must cover half of half the distance, and before I can do that I must cover half of half of half of the distance, and so on, so that in reality I can never move any distance at all, because doing so involves moving an infinite number of small intermediate distances first. Now, since motion obviously is possible, the question arises, what is wrong with Zeno? What is the "flaw in the logic?" If you are giving the matter your full attention, it should begin to make you squirm a bit, for on its face the logic of the situation seems unassailable. You shouldn't be able to cross the room, and the Tortoise should win the race! Yet we know better. Hmm. Rather than tackle Zeno head-on, let us pause to notice something remarkable. Suppose we take Zeno's Paradox at face value for the moment, and agree with him that before I can walk a mile I must first walk a half-mile. And before I can walk the remaining half-mile I must first cover half of it, that is, a quarter-mile, and then an eighth-mile, and then a sixteenth-mile, and then a thirty-secondth-mile, and so on. Well, suppose I could cover all these infinite number of small distances, how far should I have walked? One mile! In other words, 1= 1/2+1/4+1/8+1/16+1/32+..... At first this may seem impossible: adding up an infinite number of positive distances should give an infinite distance for the sum. But it doesn't – in this case it gives a finite sum; indeed, all these distances add up to 1! A little reflection will reveal that this isn't so strange after all: if I can divide up a finite distance into an infinite number of small distances, then adding all those distances together should just give me back the finite distance I started with. (An infinite sum such as the one above is known in mathematics as an infinite series, and when such a sum adds up to a finite number we say that the series is summable.) Now the resolution to Zeno's Paradox is easy. Obviously, it will take me some fixed time to cross half the distance to the other side of the room, say 2 seconds. How long will it take to cross half the remaining distance? Half as long – only 1 second. Covering half of the remaining distance (an eighth of the total) will take only half a second. And so one. And once I have covered all the infinitely many sub-distances and added up all the time it took to traverse them? Only 4 seconds, and here I am, on the other side of the room after all. And poor old Achilles would have won his race. ADDENDUM So that you don't get to feeling too complacent about infinities in the small, here's a similar paradox for you to take away with you. THOMPSON'S LAMP: Consider a lamp, with a switch. Hit the switch once, it turns it on. Hit it again, it turns it off. Let us imagine there is a being with supernatural powers who likes to play with this lamp as follows. First, he turns it on. At the end of one minute, he turns it off. At the end of half a minute, he turns it on again. At the end of a quarter of a minute, he turns it off. In one eighth of a minute, he turns it on again. And so on, hitting the switch each time after waiting exactly one-half the time he waited before hitting it the last time. Applying the above discussion, it is easy to see that all these infinitely many time intervals add up to exactly two minutes. QUESTION: At the end of two minutes, is the lamp on, or off? ANOTHER QUESTION: Here the lamp started out being off. Would it have made any difference if it had started out being on? link |
I read once that objects never really do touch. |
Technically, they don't touch. If you look at it from a subatomic level. Edit: Xeno's Paradox is bullshit if you don't look at it from a theoretical logical standpoint. The way I heard it was a turtle and an arrow being shot at the turtle. Say the turtle moves at 5m/s and the arrow at 10m/s. The turtle starts 10m ahead of the arrow. After the first second, the arrow has moved 10m, and the turtle 5m. So the turtle is now 5m ahead of the arrow. After the second second, the arrow is at the 20m mark and so is the poor, impaired turtle. It remains, however, a good illustration of infinite sums with finite answers. |
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right idea wrong method... It has to do with the hiezenberg uncertainty principle... You can measure where something is or how it's moving but not both. Also things have different lengths depending on how fast they or the obverser is moving. Reletivity is a bitch. |
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I contemplated a separate thread for this one and decided against it: Does the Barber Shave Himself?? Suppose there is a town with just one male barber; and that every man in the town keeps himself clean-shaven: some by shaving themselves, some by attending the barber. It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men who do not shave themselves. Under this scenario, we can ask the following question: Does the barber shave himself? Correct answer anyone? Bueller? |
No, actually the items could touch, but we lack the precision with which to measure the total length of 2 items, and thus measure the exact length of any one of the items. However, as I learned from the .9bar=1 thread, by definition, a yardstick is actually 36 inches long. That is 36.0bar inches. Thus 2 yardsticks can touch by definition. |
Heisenberg Uncertainty principle deals with the electron cloud around the nucleus of an atom. IIRC, it is impossible to detect the position of an electron and it's direction of travel at the same time. We don't know where every atom is at the same time, but we can say where they are most likely to be located: this is demonstrated by a cloud in drawings of an atom. Adding new complexities to the debate: how are you measuring the length? With light? What wavelength? Or with an electron microscope? How do you account for the transfer of energy that occurs when the light/electron/whatever bounces off what you are measuring? |
It's a paradox. Or, more accurately, a contradiction. Anyway, the way I first heard it, it did have an answer, but it didn't contain references to the barber's gender. |
IMO the length of any item is not infinite. Our ability to measure any item precisely is finite. |
To understand this question, you must understand the acceleration and time used to arrive at 50 miles per hour. Also, the acceleration doesn't need to be constant, you could have an acceleration that is a function of time, or even space, or both (the AR15.com way). |
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No, according to that rule he would NOT shave himself. However, if he did NOT shave himself, he would by that rule be forced to shave himself. The poison is in both glasses. |
It isn't really that complicated. Distance and time are linear in most mathematical examples. Zeroing in on .9bar for example, or x/∞ for instance is a hyperbolic function. A limit is used to calculate. Any 2D plot where the velocity is different, two paths will intersect. Assuming -∞<x<∞ |
Trying to tie this back to something I understand I fire a bullet at another person's head. The head explodes into a pulpy mass. At the subatomic level, the bullet and the head TECHNICALLY don't touch. Eh...help me here. 
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The atoms don't actually touch. According to my understanding, the "surface" of an atom is actually it's electron cloud. It is supposedly impossible to demonstrate that the atoms in these clouds can actually physically touch. Taken as a whole, the atom can touch another atom, but from what I understand, the surface of an atom is really just a 'zone', not something solid. |