what is the flaw of Zeno's Paradox?

Question

Suppose that a man wants to cross to the far wall of a room that is 20-ft across. First, he crosses half of the distance to reach the 10-ft mark, Next he crosses halfway across the remaining 10 ft to arrive at the 5-ft mark. Dividing the distance in half again, he crosses to the 2.5-ft mark, and continues to cross the room in this way, dividing each distance in half and crossing to that point. Because each of the increasingly smaller distance can be divided in half, he must each an infinite number of "midpoints" in a finite amount of time, and wil never reach the wall.

Explain the error in Zeno's Paradox.?

Answer 1

The error is that the time required decreases exactly in proportion to the distance.

Zeno's paradox is the same thing as saying "It takes an infinite amount of time to do an infinite number of things. But all tasks can be divided up into an infinite number of steps. Therefore, nothing can happen." But if you divide the task up an infinite number of times, then each of those divisions requires an infinitely small amount of time to complete, so you can do an infinite number of things in a finite amount of time.

Take a look at the actual math behind Zeno's paradox. Say it takes you 8 seconds to cross the room. Then it would take 4 seconds to go halfway, 2 seconds to go a quarter, 1 second to go 1/8, etc. Since the time divides exactly by as much as the distance, you can clearly see that no matter how much you subdivide the distance and add them back together, the amount of time required to complete the task will never change.

Mathematically:

The time to cross the room (Time) = Speed/Distance
Time to cross half the room is Time/2 = Speed/(2 * Distance)
We'll call any arbitrarily small fraction of the distance across the room X. X can be the whole distance, one billionth of the distance, one divided by infinity of the distance, or whatever you like.

Time to cross the room is Time/X = Speed/(X * Distance). If you know algebra, you know that you can multiply both sides by X and they will cancel out, leaving you with the same result for the time to cross the room regardless of what X is. So no number of subdivisions have any effect on how long it takes to cross the room.

This is easy algebra, because it's a linear problem. But it's possible to sum an infinite number of infinitesimal things in many cases even when it's not so straightforward. This is what taking an integral in calculus is- a sum of an infinite quantity of infinitesimal things.

Answer 2

The "error" in the paradox is to consider that time and space are discrete entities. This means that Zeno thought that time and space were infinitely divisible in smaller and smaller parts and that there is always a point in space between any two different points in space (the same goes for time). So, if we consider time and space to be perfectly continuous (this may be hard to imagine the way the problem is set, since it makes use of our discrete numerical system), then the paradox disappears. However, there are many theories on the subject and it seems that in the quantum realm the paradox is unresolvable. Check the source to read more on it.

Answer 3

Zeno supplies the occasion of Achilles racing against the turtle. The turtle is a million 000 meters basically before Achilles and at each step, Achilles is overtaking one a million/2 of the area isolating him from the turtle. So on the 1st step, the turtle is 500 meters forward, 250 on the 2d, a hundred twenty five on the 0.33, etc... yet you may see Achilles will in no way capture as much as the turtle. it quite is incorrect as a results of fact relatively, Achilles' speed might could be diminishing for that to ensue. Newtonian physics will practice this does not artwork. The turtle is a million 000 meters forward and strikes at a million meter a 2d. Achilles strikes at a hundred meters a 2d. So on the 1st 2d, Achilles is at a hundred meters, on an identical time as the turtle is at a million 001. on the 2d, its 2 hundred meters, against a million 002, etc... so which you will then see the area isolating them diminishes consistently, and not by a fragment. Achilles will overtake the turtle in the eleventh 2d, whilst he's traveled a million a hundred meters and the turtle is at element a million 011. This sophism of Zeno relies on the assumption of 'dividing the area by a million/2'. in case you're making a chart of the area traveled in Newtonian words, you will locate Achilles' speed is relatively diminishing as you bypass alongside. (From 500, to 250, to a hundred twenty five, etc...)

Answer 4

I'm not a linguistics major, but I garuntee, the way you presented that question, there is no answer. Because you ask what is the "error in the paradox?" well, a paradox has no error in it. Maybe if you asked "what is the error in the logic?" it would make sence. And I don't think "Zenos Paradox" is a paradox, it's just a statment. He will never reach the wall because he is not walking towards the wall. He is simply disecting 10 feet into smaller and smaller sections. So in conclusion, the error is that there is a wall.