Distance cannot be discrete. Suppose distance is discrete. Let quanta A and B be equidistant from quantum C but in perpendicular directions. (I.e. ABC is a quantized isoceles right triangle.) Let the distance from A to C be X quanta. What is the distance from A to B?
zeno's paradox does not need space (or indeed time) to be quantized to be solved, it has been shown on this thread nuff times that it is solvable using continous numbers.
Amen to that. Zeno's paradox tells us nothing about either quantization or relativity. It's just an easily solvable math teaser.
drnihili: <i>Suppose distance is discrete. Let quanta A and B be equidistant from quantum C but in perpendicular directions. (I.e. ABC is a quantized isoceles right triangle.) Let the distance from A to C be X quanta. What is the distance from A to B?</i> This is the picture: A CB The distance from A to C is X, and the distance from C to B is X, so the distance from A to B is, presumably, 2X. Now, in the ordinary scheme of things, the shortest distance between A and B would be sqrt(2)X. Are you worried that in the quantised space we couldn't travel directly from A to B? Or is it that you are worried about non-integral multiples of the distance quantum? These things, in themselves, don't rule out the possibility that space could be quantised like this. You wouldn't notice the quantisation if the quantum of distance was small enough - just as you don't notice the "jaggies" of slanted lines on a computer screen, provided the resolution is high enough.
I'm worried about nonintegral multiples of the distance quanta. Thanks for the clearer statement of the problem. The distance cannot be sqrt(2X). This isn't just a matter of jaggies. Whatever answer you give for small triangles, say where the distance from A to B is 1, it must be scaleable to large triangle, because large triangles can built of smaller ones. E.g. D AE CBF whatever answer you give for the distance from A to B, you must answer that the distance from D to F is twice the distance from A to B given that the triangles are all similar. This means, that for any isoceles right triangle, the ratio of the length of the hypoteneuse to the length of leg must be the same. Further, since both the hypoteneuse and leg are integral multiples of the quanta, this ratio must be a rational number. The problem is not that the hypoteneuse must be an integral multiple of the quanta, it's that the ratio of hypoteneuse to leg must be a rational. But we can mathematiclaly verify that the ratio of hypoteneuse to leg is not rational. Further we can empirically verify that the ratio is neither 1 nor 2, which are the only plausible integral values for a triangle with a leg of 1 quata. Hence distance is not quantized.
Let me further confuse the issues by offering the following. Please Register or Log in to view the hidden image! Allow me to play the heretic for a moment. If you’ll take the time to read, rather than dismissing what I say offhand, you’ll learn why many philosophers consider zeno’s paradox still worth investigating despite understanding calculus. Calculus is irrelevant to Zeno’s paradox. It is irrelevant because the answer it provides is provided by simpler methods, but also because the answer it provides does not even address the problem. I will address these two reasons in order. If the problem of Zeno’s paradox is to discover when Zeno will catch the Tortoise, then we don’t need calculus. We simply need their velocities and the amount of headstart given to the tortoise. We can then calculate how long it will take the distance Achilles runs to equal the distance Tortoise runs plus the headstart. In general, If we let A and T be Achilles and Tortoise’s velocities, and H be the amount of the head start, then we simply need to solve for y in the following equation. A(y) = H + T(y) A(y) – T(y) = H A – T = H/y H(A-T) = 1/y 1/(H(A-T)) = y Then A(y) gives the distance Achilles must run to catch the Tortoise. So we don’t need calculus to tell us when Achilles catches the Tortoise. We can do that with much simpler methods. In fact, given the sophistication of Greek geometry, it’s likely that Zeno could calculate the answer. Zeno’s paradox is not asking whether Achilles can catch the tortoise, everyone knows he can. Nor is it asking when Achilles catches the tortoise, that answer is easily found. Zeno instead wants to know how it is possible that Achilles catches the tortoise. But calculus does not answer this question. For the remainder, I’ll assume that H=1km, A=10km/hr, and T=1km/hr. Calculus tells us that how to find the sum of an infinite series. To catch the tortoise, Achilles must complete and infinite series of tasks. He must first run 1km, then .1km, then .01km, etc. Calculus tells us how to find the sum of that series. Bravo for calculus. But the problem isn’t to figure what the sum of the series is if Achilles completes it. The problem is to figure how Achilles can complete an infinite series in the first place. Think of it this way. I can complete a finite series of tasks because at each stage I have fewer tasks to do. Each task completed leaves fewer tasks yet to do. But this doesn’t happen with Achilles. He has an infinite number of tasks to do, and no matter how many of them he does, he always has infinitely many left. Calculus begs the question against Zeno. It assumes that the series is completable, and gives us the least upper bound of that completion. But how is it that Zeno ever manages to reach the end of a sequence that has no end? As long as Zeno is completing the tasks, he has not finished. IN fact as long as he is still completing tasks, he still has infinitely many to do. So he keeps going and going, never decreasing the number of remaining tasks, until suddenly, without ever finishing, they are all done. There is a point by which Zeno finishes the tasks, but there is no point at which he finishes. This is the theme that traces through all of Zeno’s paradoxes. Motion is illusory because there is not point in time or space at which it occurs. We can say when it hasn’t yet occurred, and when it has already occurred, but never when it actually occurs. Calculus is of absolutely no help in solving this problem.
It makes no sense to talk about an infinitesimal change in distance. It looks like a big zero, an illusion. An infinitesimal change in time is likewise illusory. The key is to not look at either of them on their own, but to consider their ratio. That's how Newton rescued us from Zeno's madness.
As I recall, Newton loved infinitessimals. That was one of the points of Berkeley's critique. But what was your point exactly?
My point was that motion has two constituents: distance and time. If we consider distance on its own, we go crazy trying to see how anyone can cover any distance when there's always a smaller distance that they have to cover first. Likewise with time. But if we consider how they change relative to each other, we get a well-defined number for dx/dt, velocity. The weirdness cancels out and suddenly motion isn't illusory. Or maybe I'm just blowing smoke. Sometimes I say things that seem profound to me, but they're really just a bunch of bullcrap. I suspect this is one of those cases. Please Register or Log in to view the hidden image!
Xenu said: Now the major question is, how big are each of these line segments? They are as big as the label you apply to them. The 'line segment' only exists in your mind. Certainly there is a distance that exists to it in the physical world, but the definition thereof is arbitrary - merely agreed upon for common reference. Xenu said: I'd have to guess that each line segment would have to be infinitely small If you think it is so, it is. It has no effect on the objective reality of whatever scenario you present. You can apply a numerical measurement to it and mathematically chisel it into abstracted disintegration, but that has no effect on the location of your physical self in the world. With that, you can see that the paradox exists and can do so freely because it has no bearing on the physical world. As I consistently contend, paradox is only excluded from 'what is possible' in the objective world. In the world of the abstract, many contradicting truths can exist simultaneously. Since math is abstract, you can both travel the distance and never have gotten there depending on the perspective you view the problem from. At least that's the way I see it. Can anyone illustrate the problems with that reasoning?
Before Achilles start moving zeno has to define the boundaries. Where the space starts from the end of Achilles' body. If zeno could find out the exact starting point then he can talk about his paradoxes. The ball is on zeno's court. Would not he endup searching for the starting point of the distance to be covered by Achilles, infinitsmally..?!, by his own argument. (There is no motion and Achilles is yet to start, remember. Hence the question of whether motion is illusion or not does not arise still)
Xenu Since the distance is finite the time to cover it is also finite.However a finite distance can be conceptually divided ad infinitum.If the relation between space covered and time is D=s(t),a bijective function,to every infinite partition of distance can be defined also an infinite partition of time.
It seems to me that most paradoxes are caused by the limitations of human languages and/or thought. In any real system, an actual paradox cannot exist i.e. it would imply unreality. Thus, paradoxes cannot exist, they only appear to. My favorite example of this is the old joke about the three men who rent a hotel room for $30. They overpay by $5 and the bellboy is sent up to return their money. The bellboy keeps $2 for himself and gives each man back $1 each. Thus, each man pays $9=27 the bellboy keeps $2 =$29, so where's the extra buck? Please Register or Log in to view the hidden image! Obviously, it's pretty easy to create a math analogy that will seem accurate, but not reflect the real world - this just means we need to be careful to apply math to reality
Just out of curiosity, why would you think that paradoxes could not exist in reality? I mean seriously, what evidence could you have for such a claim? Please Register or Log in to view the hidden image!
Pardon, the only ones I can think of off-hand are black holes. Hmm.. maybe I've made an incorrect assumption. Any examples of non-abstract paradoxes? By that I mean something like 2 things in the same place at the same time or something like that. Actually that is what makes me think that paradox is generally dissallowed in reality. In typical human experience I was thinking paradox just doesn't happen. What do you think is a good example? (and I should have said, paradoxes in objective reality only happen at the very extremes of scale, which are simply excluded from human experience (even those we know of like some quantum mechanical stuff or black holes are only abstracts to us))
Actually I was mainly poking fun. Reason abhors paradox. When we find a paradox, say that photons are both waves and particles, we immediately revise our theory until the paradox vanishes. So now we have two theories, one contains a paradox, the other doesn't. What ground can we have for preferring one over the other? Well the most obvious reason is that one of them contains doesn't have a paradox. But if the question is whether paradoxes are ever true in objective reality, then you can't use our preference for non-paradoxical theories as evidence for the claim. That preference is decidedly not objective. Whether we see paradoxes depends on our theoretical framework as paradoxes are only formulable within a theory. Whether the real world contains paradoxes I have no idea, but I have a very strong faith that our theories need not contain them. So I guess I would say that it's human language and reason that gets rid of paradox, not the real world.
I think that's the key. Does it even make sense to ask the question? The word paradox implies that a rule has been broken. If an observed phenomenon breaks a rule, is it paradoxical or is the rule faulty? To have a real-world paradox, you must have a set of rules that are true regardless of what we observe, and I don't know where one would find such a set of rules.
OH you're MEAN! Please Register or Log in to view the hidden image! You're talking about things that attempt to describe the objective world. Of course they will try to steer clear of paradox. Paradox seems inconsistent, hence a problem. Depends on the context doesn't it? I can't think of a good for instance at the moment. Zeno's paradox seems good to me. IMO, it's really in how you look at it or how you phrase it. If you look at if from the constant halving perspective, you'll never get where you're going. If you look at if from the perspective of travelling a consistent distance repeatedly, you get to the other side of the room rather quickly. I guess in that case though it's really the problem isn't well defined. Regardless, I'm sure there are some good examples that I'm excluding at the moment.. I've talked about some others recently. I'll try to come up with something. Plausibility. Not necessarily, but as a general rule. What keeps you from imagining yourself in two places at once? What I'm gettin at is "in the abstract world there is nothing to prohibit duality except intent". In other words, sure.. if I'm trying to make a filing system or organize something or fix something or thing of a bird or whatever in my thoughts... I exclude paradox from possibility generally from the reasoning that paradoxes are a violation. I say "what violation are they in your mind?". Sure they violate reason and logic... but those are about intent... there are rigid rules that prohibit duality. in your mind there are really no rules, only those you adopt (and whatever limitations your mind has inherently). Eh, I'm rambling at this point. Evidence for that claim that paradoxes are sometimes true in objective reality? I didn't do that. I was asking if you thought black holes and quantum mechanics represented the extreme conditions of paradox in the objective world and saying that humans are completely uncomfortable with paradox because it simply doesn't work in their daily lives. practically, you can't be two places at once. It's lame. Please Register or Log in to view the hidden image! I don't think that's accurate based on your argument since you referenced descriptions of objective reality as the basis of reasoning. But what about our theories about paradox? What about things that rely on perspective.. things like relativity? Reality contains paradox BECAUSE of perpecitve for SURE at least - via relativity no?
Uh, you probably don't want to get me started on theories of paradox, at least not in this thread. Once we go down that road we have to start talking about non-classical logic, fixed points, revision theory, and even *gasp* paraconsistency. I think there are unresolved paradoxes. Zeno's is one of them when suitably understood and augmented. As I argued above, calculus is irrelevant. Space can be neither continuous nor discreet. Physics is a history of one paradox after another. We can push the paradox off a little bit by revising our theory, but there's always another one at the end. Who's to say whether we'll ever be able to get rid of all of them? However, I'm all for trying. If nothing else, it's fun.
