Welcome aboard, Rocks.
Be prepared to never ever have any real free time again. The Dope™ is highly addictive.
Now, something’s been bothering me… [reaches around to pull tighty whities from my ass crack]
There, that’s better.
As for the OP’s Q: most of us (if not all) simply can’t grasp a real image of what infinity is. Leave it to the mathematicians and just keep driving, is what I say…
Exapno Mapcase, why are you criticizing nicky when he has the answer absolutely correct. The point is not merely that the infinite series sums to 2. That may be true but it doesn’t get away from the fact that any finite number of terms is still going to give a value less than 2.
Rather, the point (as nicky correctly points out) is that you have made the sequence of times you are considering also be a converging series. So, the reason you never get to the place that you are considering (in a finite number of these steps) is that you never get to the time when you should get there (in a finite number of these steps).
So, the answer to the question is that you “never” get this with this sequence of times because by “never” you are implicitly limiting yourself to times of less than 2 seconds. If you had a sequence of times that did not converge to a value less than or equal to 2 seconds, you would see that you do get there in 2 seconds. (Or, by taking limits, you can even see that in this case.)
jshore, Zeno knew perfectly well that the arrow reached its target. What he couldn’t do was the math behind infinite summations. So the point is exactly, only and exclusively that “the infinite series sums to 2”. As soon as you put the word “finite” into your answer you change it into a totally different situation.
Well, I guess maybe we find different things paradoxical about it then. (I’m not sure what aspect Zeno was actually confused on.) What I see as the paradox is you know that you should be able to make it to 2m or beyond and yet the argument seems to suggest you just barely make it to 2m…and even then only by considering the limit of an infinite series. And, to understand that aspect, you have to realize that your time series is a converging sum too.
Another way to look at it is Zeno’s picked the wrong axis to step along. If you step along the time axis, you get there soon enough.
If you step along the distance axis, with the stupid rule “halving the distance each step”, you’re saddled yourself with a passel of trouble.
Then there’s the variation, where two engineers aer discussing this, only instead of an arrow, they’re postulating a “male member”, and the target is a cooperative female. One engineer gives up, saying “you’ll never get there”, the other one says “ah, but you get close enough…”.
Mathmatics is only a convienient man-made tool to model the physical universe and should not be considered absolute. For example you could never find exactly “2” physical objects, because they would different in some sub-atomic way and even the definition of the object is always subjective.
Infinitity doesn’t easily fit into this model. And in fact, you could never actually find the exact half-way point (in time and space) using math because there are infinite sub-divisions, even though you would obviously pass it.
Newton and Taylor covered this pretty well, back in their day(s). Newton was looking at differentials, where the top and the bottom of a fraction converged to zero at the same speed, and found that the fraction had a limit value, and went on with a bunch of math from this point. Taylor liked infinite sums. This kind of math will show you that you do in fact reach point a.
I expanded on this a bit, with my own theory. I state that the sooner you fall behind schedule, the more time you’ll have to cach up the slack. The second statement doesn’t translate as well, but says that well started is half finished. Thereby, by doing absolutely nothing, you’ll be half finished. By the same principles, the job should finish itself, if you continue to do nothing.
I think it is completely lacking in logic. Simply asking how something is possible does not show that it is impossible. I really don’t see any paradox. I just see a bunch of non sequitors. “Suppose I buy an orange. Then I buy another orange. And another. How will I ever get an apple? Clearly apples don’t exist.” Obviously, if you restrict your narrative to describing events that occur before one reaches one’s destination, one will never describe reaching one’s destination. But simply because one does not describe an event, that does not mean it does not occur.
One of the fundamental assertions of quantum mechanics (arguably the fundamental assertion) is that space & time are not infinitely divisible. According to this worldview, Zeno’s paradox doesn’t kick in because there is a minimum unit of time & a minimum unit of space. It just looks continuous to us. I don’t believe it myself, but it does answer the question.
Sigh, not this again. Some things in quantum mechanics (most notably, angular momentum) are not infinitely divisible, but time and space are. It’s sometimes hypothesized that when we finally work out a quantum theory of gravity, we will find a fundamental quantum of time and space, but we don’t yet have a quantum theory of gravity, and nobody has yet figured out how to quantize space or time in a self-consistent way. And even if space and time were quantized, that still doesn’t solve the problem, since one can cast Zeno’s Paradox in idealized terms which don’t necessarily correspond to the real world. And even if quantized space could solve Zeno’s Paradox, it still wouldn’t be necessary, since the concept of the sum of an infinite series already does a much better job of resolving the paradox.