If we specified a distance, for example one mile, where x is the starting point and y is the finishing point, and we measured the point that is half the distance to y, at each turn, would we ever reach y?
From a purely mathematical standpoint, no. Because you can continue to divide any remaining amount (distance) by 2 for as long as you want. Just as there is no limit to an infinitely large number, neither is there a limit on the infinitely smallest either.
There is a famous paradox about that if I remember well. Quite funny,too Please Register or Log in to view the hidden image! If you imagine to shoot an arrow,you can halve the distance between you and the target infinite times, so in theory the arrow will never hit the target. The problem is, a mathematical sum of infinites is not necessary an infinite, too.(at least this is what i've been told in justification, as sincerely I'm not that good at maths Please Register or Log in to view the hidden image! )
I suppose that we should just consider ourselves lucky that arrows (and everything else) don't travel based on half-distance jumps. Please Register or Log in to view the hidden image! Otherwise, you'd never be able to get that fork to your mouth. Please Register or Log in to view the hidden image!
i beg to differ if you take an infinite number of summations then the sum will equal the whole. how can it be less? it is when we take a finite sum do we come up with less i can not offer any mathematical proof because i'm a dummy edit please comment
I believe you misunderstood, Leo. You can divide any number by two an infinite amount of times - and still have a remainer left that can be divided again. You'll never reach the end of the series. Even if there are a billion zeros behind the decimal point, there will still be a numerical digit at the end. And on and on it can go.
i must have it seems to me that if you addup all the peices it can not be anything less than the whole or am i forgetting the peice we are dividing. my mind refuses to work when confronted with the infinite
Yes, you'd be correct IF you could ever get all the pieces - but you can't get them all because there's always going to be more. You cannot start adding them up until you get all of them and you'll never get to the last one. You can keep on dividing forever. We started with one mile so look at this number: 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000004 mile. Divide it by 2. Then divide that answer by 2. Then repeat... See what's happening?
Perhaps, in the grand scheme of things, everything is in constant motion, meaning that the arrows destination is not its target, but an infinite distance in the direction it is fired in. This would mean that it would keep on going forever if it werent for friction, gravity, and a whole larger mess of 'laws'. It is possible that other laws of physics make the half and half idea irrelevant.
If it's not pure math but physics then yep you sure would. Eventually you'll get to the smallest component (a point in space-time) and it doesn't get any smaller (can't halve it).
This is called Xeno’s paradox. The fallacy that trips people up – and I can’t believe it’s not old news by now - is that it’s implicit in the story that we are slowing down. Suppose you were to take the same length of time to go each distance – 1 mile, ½ mile, ¼ mile, etc. The result would be as you suggested – you would never get there. Do you see why? You are slowing down. That is exactly what it means to go smaller distances in the same length of time. The paradox does nothing but (implicitly) require that we slow down! It might as well say that if we start going 1 mile at 1 mph, slow down by ½ mph after the first half, and slow down by ¼ mph after ¾, we would “never get there.” Of course, this is never stated explicitly, but we do it automatically when we visualize the story in our minds. We don’t think, “I am slowing down.” Now, if someone was to ask you, “If you start going a certain distance, and keep slowing down the whole way, is it possible you would never get there?” you might think about it differently. If we slow down less than half each time, we would get there, only later than if we didn’t slow down. If we slow down more than half each time, we will never get there. If we slow down exactly half, it will take infinity to get there. If we go the same speed for each consecutive half length – meaning that it would take half as much time for each leg (each being half the length of the previous leg), we would get there in the same length of time as if we never slowed down at all.
The problem is what it consists of. What does a point in space-time consist of? Space-time itself? Then what does that consist of? The essence of reality? Or is it a mistake to assume that everything must consist of something? But if that is the case, then what is that something? If it doesn't consist of anything?
Actually, I think the fallacy is that most people intuitively (but wrongly) assume that it must take an infinite amount of time to pass through an infinite number of discrete distances.
Assuming it always takes you the same amount of time to make a measurment, no you would never reach y.
that does not apply here because it doesnt depend on the speed. it doesnt matter how fast you are going, you cannot cross the limit set by the half's half's half's half's... half, therefore, you will be covering halfs faster and faster every time, but the halfs you have to cross never end. i apologise for the redundance if this has already been replied to you, i have not read the entire thread.
If we are going to take the same length of time to do each measurement, which would, in theory, allow us to keep going halfs as long as we want, we would get down to the Planck length,and wouldn't be able to measure halves any more. In this sense, you are not stopped because you keep going halves, but becuase you can't measure any smaller. See : http://www.physlink.com/Education/AskExperts/ae281.cfm?CFID=20707539&CFTOKEN=15648611 Assuming quantum theory holds, you'd be pretty much stuck. From a mathematical point of view, yes, you could go on halving forever.
Perhaps in theory, but in practice wouldn't a finite number, divided an infinite number of times, eventually equal the whole?
