firstly, in hexadecimal, the digit for 15 is F. secondly, 0.9999..... in hexadecimal is not equal to 0.FFFFFF.... it is 9/15, whereas 0.FFFFFF.... is 1. you see, 0.9999.... is just a string of symbols, and the mathematical properties of that string depend on your definitions. and thirdly, yes, it is beside the point.
The problem that was posed on page one is not Zeno's paradox. It is part of it but is not itself a paradox. The Paradox is made of four arguments: *In motion a body must reach the middle before it reaches the end ad infinitum. (This is what is being discussed so far) *If Achilles runs ten times faster than a tortoise and gives it a ten metre head start he can never cath it because as he covers the distance between them the tortoise has moved on a tenth of that distance, ad infinitum. These two arguments show that distance can not be continuous and infinitely divisible. Saying that an infinite sequence can have a finite sum, although true, is not entirely relevant as it misses the point of Zeno's argument. Consider a photon bouncing back and forth between an infinite sequence of mirrors whose perpendicular separation decreases exponentially, e.g. by a factor of 2. The path length of the light is finite as has been discussed but how can the light ever come out, so we have a finite path with no end? Third/forth arguments: based on time being made up of discrete instants *an arrow is at rest whenever it is in a place the same size as its length. At every instant the arrow is in a place of size it's length, so at every instant it is at rest. If it is at rest at every instant in its flight how can it fly? how is causality transmitted from one instant to the next? *Two rows of bodies moving in opposite directions with equal speed. Half a time is equal to the whole time". This is an attack, it would appear, on an upper bound on velocity, which Zeno considered. Two bodies pass at the maximum speed, their relative velocity is twice the maximum which is a contradiction. Zeno's Paradox is therefore that based on the above arguments space and time can be neither continuous nor discrete. This has been resolved by modern physics by assuming relativity of space and time, but Zeno wasn't to know this. Had people taken Zeno seriously we might have been teaching relativity for centuries. I'm by no means an expert and this is all straight from stuff I've read but I hope it will throw the discussion open a bit more and get us away from the loop we seem to be stuck in.
That is very much like what I wrote on Zeno's "Arrow" paradox in another forum... Incidentally, it is said that Zeno's reason for forwarding Premise 1 is that, at any instant, there is no physical difference between a stationary arrow and a moving arrow. I don't see the logical connection there, but we now know that there is a physical difference: the moving arrow would be length contracted as per Special Relativity. If SR were known at the time, would Zeno have offered this argument? Did Zeno forsee SR? Hmmm.... to which another member responded... I don't see the connection either, but if there is one, we can also ask the question in a slightly different way: would motion be possible in a universe with no length contraction? This reminds me of the fact that Poincare was extremely close to the gist of SR. He even mentioned that there is no experimental backup for the idea that simultaneity is absolute (which is closely related to length contraction). Of course, Poincare had the context of Maxwell equations and the struggle to interpret the "Lorentz-Fitzgerald contraction", but anyway, it strikes me as impressive that he made the connection, and the sole idea that Zeno may have glanced at it so long ago is just mind-boggling.
This is only one of several paradoxes by Zeno. There is no such thing as "Zeno's paradox." He devised many.
Zero said: You obviously don't understand Kant very well. Explain his noumena without the use of rationalism or reason (hint: it can't be done). Pure reason is the axiomatic foundation for all science. Philosophy is the mother of science. The fact is, Kant was a huge critic of Hume, who was the God of British empiricism. The three critiques were written solely in response to Hume's "Treatise of Human Nature," which basically asserts that nothing can be known outside of experience. Hume and Kant did agree on a good deal of issues, however. Kant is to philosophy as Einstein's GR is to physics. Even many experts don't truly understand them and they are both wildly misunderstood. The only philosopher that I think is more controversial than Kant would have to be Nietzsche, but Nietzsche isn't difficult due to his ideas, but rather for his obscurant Heraclitean style of prose. Kant is difficult for his sheer range of thought and genius.
Alpha said: First of all, the Planck length is not a "particle." It is merely a mathematical construct often used in conjunction with N- dimensional Calbi-Yau spaces in supergravity and string theories. The Planck Length has never been observed. Secondly, the theroetical Planck length is believed to be composed of quantum undulations, which is sort of a paradox in itself. Thirdly, the Heisenberg uncertainty principle originally had nothing at all to do with the Planck length. The HUP was originally postulated as a logical result of the Schrodinger wave function and was applied to particles like electrons. I do think there is a "smallest length" and I do feel that space is atomic, but we are a long ways off from proving such experimentally. Loop quantum gravity discusses the atomic nature of space in detail. In fact, atomic space is one of the major postulates of the theory -- the theory that I think will surpass M-theory as the candidate for the TOE. Space being atomic would be a logical solution to this particular paradox of Zeno. I am quite surprised that no philosophy buffs here have supplied us with Aristotle's solution to this paradox.
I haven't seen this solution posted, so I will post it: Please Register or Log in to view the hidden image! Basically, the above is an infinte geometric series but one that will converge. As others here have noticed, the infinite number of half-steps are balanced by the decreasing amount of time needed to traverse each half distance.
It occurs to me that the argument about choosing a point not the point we want to be at but a little further away, and the original question is about one and the same thing. It goes like this I think. If this was a motion that could only occur in one dimension, than the answer is that you are everywhere at the same 'time', in the first place. By adding another dimension, the answer could be the same, or even you were there before you started unless the second dimension was restricted to only going forward as it were (I forget the term for this). You never actually reach the point you set out to go to in the first place as when you get there, it is actually a different point. So if we take one dimension as distance, and the other as time, and time is always moving towards the future, than when you get from point A to point B, you are not really at the original point B, but at some time T0 + X, where X is the difference in time T0, the start of the journey and T1, the end of the journey. In a pure mathematical sense then, the argument of why you actually reach the point in the real world is that you are not actually there but somewhere else. It is a paradox because we assume that where we are is where we wanted to be. In reality, things are stranger than that. Might this imply that the place we ended up is the point before the point we wanted to be? Could well be if we are fooled by what we think time and distance is to my mind.
AndersHermansson: Okaaay- Since I have been under the impression, for some time, that EVERY repeating decimal is rational, could you give some support for the statement that "0.9999... is irrational"?
At the risk of beating a dead horse even more.... My favorite variant on Zeno's paradox is the following: Imagine that Achilles wants to proceed from point A to point B, but that an infinite number of Gods have vowed to prevent him. God 1 vows that if Achilles makes it 1/2 way, he will kill him with a Lightning bolt. God 2 vows that if Achilles makes it even as far as 1/4 of the way he will kill him with a lightning bolt. God 3 vows that if he makes it 1/8 of the way .... In general, God N vows that if Achilles makes it 1/2^Nth of the way, he will be killed with a lightning bolt before proceeding further. It turns out, that Achilles can't start across the road. For suppose that he travels some distance. In that case he must have already traveled 1/N^2 of the distance for some N. But that means that God N+1 would have already killed him. In turn that means that he wouldn't have made it 1/N^2 of the distance. Since the assumption that Achilles travels some distance across the road yields a contradiction, it can't be right. So Achilles doesn't travel any portion of the distance. Now here's the clincher. Since Achilles doesn't travel any portion of the distance, none of the God's have to fulfill their vow. Their vows alone make it logically impossible for him to cross the road. But it seems absurd that Achilles can be prevented from crossing the road without any of the gods having to carry out their vow. WHat, after all, is stopping him?
huh. that s a good one. do you know the resolution? i haven t figured it out yet. the first thought that popped into my head, was that it sounds like achilles just decides to stay still, in order to not ket killed. but no: it is a contradiction for him to make any motion at all. if he takes the first step, which of the infinite gods that he just pissed off gets to kill him?
There's no resolution that I know of, certainly no mathematical one. in general, zeno's paradox is harder to understand when it's done in reverse. WHen we consider the series, 0, 1/2, 3/4, 7/8, ... we're pretty comfortable thinking. As Achilles crosses the road (this time without the gods' interference) we can see that he approaches the opposite side. Even though we can't say precisely when he reaches the other side, we can specify a least upper bound to his crossing - we know he makes it. THere is no last stage that But when you consider his starting to cross, rather than his finishing, things get murkier. Now we're dealing with the series ..., 1/8, 1/4, 1/2, 1. How does Achilles start out to cross the road? Any movement he makes must be a finite movement. But that means that it cannot be the starting point since it is preceded by an infinite series of prior finite movements. The limit of the backwards series is 0. But 0 here represents no movement at all, it can't be the beginning of movement. If you stare at the backwards case long enough, you can begin to see problems that are unresolved even by the application of calculus to the forward case.
This paradox is aimed at proving zeno's point that motion is illusion. It is as simple as Achilles cannot move.
Actually Zeno had several related paradoxes. One of them was often phrased as "Achilles cannot move FROM HERE TO THERE because: He would have to first pass over the point half way between, then the point halfway between that and THERE, then halfway between THAT...". Another, more often thought of as "Achilles can't move", asserted that at any specific time, Achilles is AT a specific point. Since there is no change in position at that time, there is no velocity, hence no motion! Yet a third, pitted Achilles, rather than the hare, against the tortoise. Give the tortoise a short lead- then Achilles can never catch up to it: He would first have to reach the point the tortoise was at the start- but the tortoise has moved slightly ahead. By the time Achilles has reached THAT point, the tortoise has moved again. By the time Achilles has reached THAT point,... Of course, Zeno was not saying that motion is impossible- he was capable of moving himself! His point was that mathematics was not capable of treating problems like this- and he was right. Mathematics of his day, any mathematics before calculus, could not handle problems like this. The "paradox" in paragraph 2 is precisely the question of "instantaneous speed" that calculus was developed to answer while the other two are problems in infinite sums- another part of calculus.
I sorry that I had not catched this debate earlier. The third Zeno paradox , I think, shows that space is discreet. There would be an end to your division to 2 when you will reach distance that can not be divide any more - the Elementary Space Unit. This is the only usefull conclusion from Zeno paradoxes. I had argue in several forums about the need from such distance. There is many others way to reach to this conclusion (I mean quantization of Space and Time). I will give you example - a basic postulate in mechanicks and all the physics is that one system passes from a state A to the state B in such a way that it spends the least possible energy. In mechanicks this is called The variational principle ", the integral of variation of free energy over elapsed time should be minimal. This means that the system chooses from all possible paths this with minimal variation. However suppose what will happen if there is infinity number of paths between the two points. The system would have a infinity of choices and will never peak one. In the case of your room this means that if you had infinity number of ways to go from the one part of the room to the other you will never be able to reach it because the atoms composing your body will have infinity number of paths and it will not be possible to find which is with least variation, hence it will be not possible to be received lows of mechaniks.
