So let's look at it backwards, the reason we can't move would be, indirectly, that the universe is infinitely large. Since we can move, the universe must not be infinitely large but rather be composed of a near infinite amount of fundamental units of mass.
I'm not suggesting this is some groundbreaking idea, but I wonder if anyone has reason to dispute it? Or if some philosopher has already stated it (Zeno may have, I'm not sure).
Response to izubachi: I'm familiar with the limit, but it doesn't really disprove what I'm saying. What we're talking about here is the limit at an asymptote However, although it will tell what number y will move towards as x goes to infinity (for a horizontal asymptote) and at what value of x, y will move towards infinity. It doesn't change the fact that the numbers never actually reach infinity or whatever number they were approaching. So if you were trying to move an infinite distance you would also never actually reach your destination. But I'm not a mathematician, so I may be mistaken.
As for relativity, I'm not sure I really see how it fits in here. It would seem, relativity and movement aside, that any distance with in the universe is a fraction of the entire size of the universe, and if the universe is infinite, then any fraction of it is infinite. Perhaps you could clarify what you mean.
Response to Halcyon&on: Yeah, I'm not arguing that a finite number cannot be divided infinitely, it certainly can. I'm just saying that I don't think space mirrors this.
Revised statements: I'm gonna leave the previous argument up there as it makes more sense considering the name of the node. After discussing the issue I see that the proof is not there. I still think it's kinda interesting though. So I now say that the universe is composed of an infinite (or near infinite) number of fundamental units of mass (I still believe in those little fellas).
The alternate explanation, (suggested to me by my brother), is that the space is discrete at some level. If to move x, you have to move half x, and then half that half x ... ad infinitum, instead of dividing x to inifinity, you eventually hit a wall. This is the discrete space step which you can then traverse.
Discrete space isn't any more right than calculus, but its an interesting solution. Of course, the moment you take into account quantum mechanics, the need for a discrete universe to explain Zeno's paradox becomes meaningless.
Also ... in regards to the debate above, I think that a finite number can be divided into an infinite number of parts. That is what calculus is about.
Even if the universe is infinite, movement relative to a given point is still possible. For example, there are an infinite number of real numbers, but we still know the exact distance between any two of them.
I think the fundamental problem with this "paradox" is that it divides space infinitely, but not time. As we keep dividing the distance to travel, we must also divide the time that will be required to travel it. As the size of the distance to travel approaches 0, so to will the time required to travel it.
As others point out, Zeno's paradox shows that given certain common-sense implicit assumptions, movement seems impossible. Quite obviously too, movement is possible; therefore one or more of the assumptions must be false.
Others have pointed out that it may be possible to pass through an infinite number of in-between locations in a finite duration of time. We can calculate the maths of that with integration and limits, but even if we couldn't, that wouldn't rule it out: not being able to prove that it is possible is quite different from proving that it is impossible.
There has been discussion on the possibility that there are not an infinite number of in-between locations. But this is known to be the case. Distance is discrete at very small scales. So is time.
The minimum distance that one can move in any meaningful sense is called the Planck length. It is around 1.6 × 10−35 metres. This has been known for around a century - Max Planck was awarded the Nobel Prize in Physics in 1918 "In recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta." So this topic of Zeno's Paradox really is quite dead.
This says nothing about how big the universe is, or if it is infinite or not; just that it does not have infinitely small subdivisions. If the universe is a simulation, that's how big the pixels are.
If you're not into the metric measurements for the Planck length, then 6/10000000000000000000000000000000000 of an inch is about right; if you need scientific accuracy then use the metric measurement already; that's what it's for. Put it this way: the Planck length is about 10-20 of the diameter of a proton, which itself is about 1.65* 10−15 m. Shrink down to the size of a sub-atomic particle and you're not even halfway finished shrinking yet. The Planck length can be precisely described (as "1.6 × 10-35 metres") but it really doesn't fit into the human imagination.
printable version chaos
Everything2 Help
cooled by themusic
