What IS Infinity? Reference: http://www.pbs.org/wgbh/nova/archimedes/contemplating.html http://www.pbs.org/wgbh/nova/archimedes/infinity.html
Words and concepts only have demonstrable meaning according to their function in a given context. Outside of a given context, infinity is just as meaningless as any other concept. Within any context, it is a notably vague term. When I talk about a chair, I can point to an example and ramble on for days about exactly why it is called a chair, pointing out each and every feature and their uses, which you are free to examine for yourself. Although people certainly ramble on for days about infinity, no one has ever been able to prove infinity exists as more than an idea. That is not to say it is a useless fantasy. Infinity possesses the properties of both the demonstrable and undemonstrable. It is a cross between the utterly paradoxical and rational. Even the paradoxical has its uses, and being a bit less extreme infinity has even more obvious uses. However, in and of itself infinity is demonstrably useless, it is only a useful concept within the context of the finite. Here is an ancient chinese poem which expresses such relationships. Tools Thirty spokes meet at a nave; Because of the hole we may use the wheel. Clay is moulded into a vessel; Because of the hollow we may use the cup. Walls are built around a hearth; Because of the doors we may use the house. Thus tools come from what exists, But use from what does not.
Wee, math trivia! We actually have a number called a googolplex which is a 1 followed by a googol zeroes.
Still not enough to win the 'who can name the bigger number' contest :-) http://www.cs.berkeley.edu/~aaronson/bignumbers.html
This is actually a huge misconcpetion on the part of the mathematical community. Calculus does not resolve Zeno's paradoxes. For two reasons,... First, if we exam the defintion of the limit in detail it never actually says that any of those infinite processes actually equal the value of the limit. It only says that as some variable approaches some value then a particular function or process approaches infinity. Well, gee, that's what Zeno was saying all along! So how is that a solution? In fact the definition of the limit is clearly set up as to not permit the variable from actually reaching the value in question. In other words, there exists a delta greater than zero 0 such that,... In other words, the calculus limit says absolutely nothing about what happens should we decide to consider delta=0. The definition of the limit is no longer valid in that form. This prevents us from actually claiming that the value is ever actually reached. This is why good mathematicians are sure to say things like "in the limit" a value is equal to something. Because to actually say flat out that it is equal is simply wrong. It denies the very conditions of the definition of the limit. Unfortunately far too many mathematicians seem to have dropped the "in the limit" phrase and think that something is actually equal to a calculus limit if they can prove that a limit exists. It is not. That is actually a misuse of the definition of a limit. Secondly, Zeno's real question is this,... "How can it be that an infinite number of tasks can be completed". Nowhere in calculus is it ever claimed that an infinite number of processes can be completed. In fact, to prove a limit all we need to do is prove things like boundedness and trends. If we look at all of the definitions and proofs for any limit we will clearly see that we haven't proven anywhere along the way that we have actually completed an infinite process, nor have we proven that it can be completed. All we have done is shown that no matter how long we continue the process we will continue to get closer and closer to the value that we call the limit. Well, again, Zeno would say, "So what? That's what I've been saying all along!" The mathematical community is absolutely incorrect to claim that it has solved Zeno's paradoxes with calculus yet this is a widely held misconception. The calculus limit really doesn't say any more than Zeno had alread pointed out. If he were alive today I am absolutely certain that he would simply say that these people just haven't truly understood the question that he is asking,... He wants to know how an infinite number of tasks ever be completed? Calculus does NOT answer that question, nor does it claim to. Yet it still claims to have solved Zeno's paradox. Clearly the mathematical community doesn't understand the question that Zeno is asking. Fortunately physics has answered Zeno's paradox. Time and space are not continuous, they are quantized. Zeno was probably right. If time and space were continuous we probably wouldn't be able to move. But since they are quantized only need to complete a finite number of tasks to move and finite distance because we do it in quantum jumps skiping over the supposed continuum. After all, Zeno's paradox is only a paradox in a continuous universe. Once the universe is known to be quantized it's no longer a paradox as to why we can move. Calculus has nothing at all to do with the paradox at all.
Orion1: It wasn't for 2,000 years that Zeno's paradox finally got "solved," for all intents and purposes, by the calculus. Its inventors, Isaac Newton and Gottfried Leibniz, showed us how an infinite sum can add up to a finite amount, that it can converge to a limit. As Neutron Star points out this is commonly supposed. It is a way for calculus buffs to indulge in self-congratulations. But, probably every generation could find some answer to Zeno. However, I do believe that Archimedes was aware of the finite limit of the infinite sum. Indeed he used "infinite triangle sums" to discover the area of parabola.
Physics has said no such thing. No empirically verified theory exhibits any sort of geometric quantization. While some of the promising research topics do exhibit geometric quantiziation, they still don't exhibit quantization of position.
I beg to differ. It is meaningless to speak in terms of absolute position. That might be an abstract mathematical concept, but it has no meaning with respect to the universe. In our universe any "real" particle or phenomenon exhibits some form of energetic disturbance. If it didn't how could we even claim that it exists? How could we even know of such a "invisible" entity. It would be completely undetectable in any possible way. Most particles in the universe exhibit as least some mass which associates them with gravitational energy. Any known massless particle (like say a photon) would is associated with some other form of energy (like maybe electromagnetism) So in any case, every particle that can possibly be associated with a position must necessarily take on a quantized position relative to other particles. In other words, it can only move in a way to take on, or give up, a quantum of energy. Therefore, any "real" concept of position in the universe is necessarily a quantized concept due to an objects potential energy relative to other objects. It's true that we can fool ourselves into believing that there are somehow "abstract" possible positions between these particles that aren't quantized. But isn't that putting something onto the universe that isn't really there? I mean, what good is a notion of a position that no particle can take on? So I do hold that physics has shown that position is necessarily quantized in the real universe, and it is totally meaningless to talk about some kind of continuum when the particles in the universe do not actually behave in that fashion. Why make things up that aren't a true reflection of the properties of our universe? It's going to take people a very long time to let go of this idea of the continuum. Our universe just isn't continuous. That's all there is to it. At the quantum level things "jump" around, They just don't move continuously. They also don't traverse the space between their jumping. There are either here or there, but never in between. That's the true nature of our quantized universe. This isn't just true of things that are bound to energy levels within atoms. Anything that changes position in the universe is changing its relative position to something else. Therefore it is changing the relative potential energy whether it be gravitational energy or some other form. Therefore position is quantized for everything in the universe. One could argue that a so-called "free electron", or whatever, is not restrained to quantized motions. But for the reasons I just gave an electron is never actually "free". Also, would it really even be meaningful to talk about the position of such a free particle? To speak of its position we can only do so in relative terms. Once we have set its position relative to something then in a very real sense it is bound to that reference point by mere convention and can only change its potential energy relative to that position in a quantized fashion. So how can the idea of position ever not be quantized in our universe?
Even so, you can still speak about position being quantized. But I should have been more precise: what I meant to say is that length is not quantized. Energy is also not quantized. It is true that bound particles exhibit quantized energy levels, but energy, in general, can come in any quantity. P.S. don't forget that there is no quantum theory of gravity. Furthermore, TMK, there has only been one experiment, ever, that has demonstrated quantum effects in a gravitationally bound system. (Of course there has only been one experiment that has tested it)
Oops! You're right. I forgot they didn't get there yet. But I think it's pretty obvious that this will have to be the case. I mean, do we think that quantum theory will end up giving up its quantized nature in order to agree with GR. Or is it more likely that GR will have to give up its smooth continuum to agree with QM?
I won't pretend to know what quantum geometry looks like. I will just comment that the amount of discretization inherent in quantum mechanics is vastly exaggerated in popular accounts.
NeutronStar I thought what you said about the notion of infinitessimals and limits not solving Zeno's paradoxes was spot on. Especially: "The mathematical community is absolutely incorrect to claim that it has solved Zeno's paradoxes with calculus yet this is a widely held misconception. The calculus limit really doesn't say any more than Zeno had alread pointed out." However you go on to say: "Fortunately physics has answered Zeno's paradox. Time and space are not continuous, they are quantized. Zeno was probably right. If time and space were continuous we probably wouldn't be able to move. But since they are quantized only need to complete a finite number of tasks to move and finite distance because we do it in quantum jumps skiping over the supposed continuum." I see this as the reverse of the truth for two reasons. Firstly, I can't agree that science has shown that time and space are quantised. How has science done this? Second, Zeno's point was that if one takes spacetime to be quantised, or, equivalently, takes the number line to be a series of points, then motion is (mathematically) paradoxical. If one takes spacetime as a continuum then the paradox disappears. This is consistent with the effectiveness of the calculus, since a 'point' in a continuum is actually an infinitely divisible range. By that I mean, the only thing that is infinitewly divisible is a point in a continuum. If spacetime (or the number line) is quantised then why do we need to use infinitessimals to calculate motion? If you bring Zeno up to date and imagine Achilles and the Tortoise to be two particles, each being one fundamental quanta in diameter, moving in a spacetime that is quantised into fundamental quanta of time and space (discrete moments and discrete positions), then relative motion between A and T makes no sense at all. That is, not unless you allow length contraction and time dilation.
Well you could define position change; you observe particle T at time 0 and again later at time t, and get different postions, and likewise for particle A.
Yes, that is true if it is also assumed that a finite line segment contains an infinite number of points. Which is what was believed in Zeno's day. And is still believed to be true today. But actually that can't possibly be the case. That situation is a logical contradiction. That is to say that such a concept is a logical inconsistency. I'm afraid I don't see why it would disappear if spacetime is a continuum. How does this change Zeno's paradox or resolve it? I'm afraid that you've lost me here. How can a 'point' be an infinitely divisible range? I can see the space between points as being an infinitely divisible range. But what sense does it make to say that any given point itself is an infinitely divisible range? And more to the point (no pun intended) where in the formalism of calculus is there any mention of a point being an infinitely divisible range, or that such a concept has anything at all to do with infinitesimals? It's not in the formal definition of the limit I can say that much with certainty.
Why can't we complete an infinite number of tasks? This is an assumption. If it is taken true, and a "task" is defined as anything that you can describe in words, such as "going half the distance left," then Zeno has a point. If not, then his argument is meaningless. What reasons do you have for believing it?
True. But try working out the relative motion one instant at a time. Particle A can go no faster than one quanta of distance per instant (otherwise it would be in more than one place at the same time) and particle B can go no slower than one quanta of distance per instant, (otherwise it would be stationary). The only solution is to allow A to be at more than one place at a time (changing its length) or for the instants of A to be longer than the instants of B (changing its clock). I can't see a way around this problem except to say that spacetime is not quantised. This doesn't seem to disagree with any evidence as far as I can tell. (We seem to having this discussion in two threads so sorry if I'm repeating stuff).
I agree. However the problem does not arise if the number line is conceived of as a continuum (as Charles Sanders Peirce argued it should be). It allows points to be treated as if they are infinitely divisible ranges, and thus they can be treated non-paradoxically by mathematics. An infinitely divisible point is by definition not a point but a range. If it was a mathematical point it would not be divisible. You're right, a point cannot be a range. However an infinitely divisible point (or rather, the concept of an infinitely divisible point) is not a point. There's no mention of it because it's taken for granted that such points have extension (conceptually at least). If they did not then how could they be divided? A limit is defined according to some specified tolerance of error. That is, at the limit .9999... equals 1. But only because we decide to stop calculating and accept some innacuracy.