Zeno 's Paradoxes

- The
*Dichotomy*: Motion cannot exist because before that which is in motion can reach its destination, it must reach the midpoint of its course, but before it can reach the middle, it must reach the quarterpoint, but before it reaches the quarterpoint, it first must reach the eigthpoint, etc. Hence, motion can never start. - The
*Achilles*: The running Achilles can never catch a crawling tortoise ahead of him because he must first reach where the tortoise started. However, when he reaches there, the tortoise has moved ahead, and Achilles must now run to the new position, which by the time he reaches the tortoise has moved ahead, etc. Hence the tortoise will always be ahead. - The
*Arrow*: Time is made up of instants, which are the smallest measure and indivisible. An arrow is either in motion or at rest. An arrow cannot move, because for motion to occur, the arrow would have to be in one position at the start of an instant and at another at the end of the instant. However, this means that the instant is divisible which is impossible because by definition, instants are indivisible. Hence, the arrow is always at rest. - The
*Stadium*: Half the time is equal to twice the time. Take the three rows below. They start at the first position. Row A stays stationary while rows B & C move at equal speeds in opposite directions. When they have reached the second position, each B has passed twice as many C's as A's. Thus it takes row B twice as long to pass row A as it does to pass row C. However, the time for rows B & C to reach the position of row A is the same. So half the time is equal to twice the time*.*

( Quoted from http://mathforum.org/)

Zeno 's Paradoxes Solutions

Obviously the Zeno conclusions are against our experience. But we are curious about what is wrong with the deduction. The paradoxes address the basic three ideas of space, time and motion. In order to nock down the wrong notion, we must be right first, that is the idea of discussing the paradoxes. But I really doubt how many of us have the right ideas of the three, so the paradoxes may remain defiant.

The first two paradoxes of *Dichotomy and Achilles *are of the same
quality, from the math point of view, the two series set converge at two points,
one is at the beginning and the other at the end. And the conclusion comes from
that since the runner can not finish counting the series set of infinite
midpoints, so he can not find the first midpoint to start or the last point to
finish, then he will never be able to start or finish.

The assumption behind the deduction is that he must find out the final midpoint which we will never be able to pinpoint out. But this is not necessary because space is not made up of points. Space only contain points. Every step of man is a set of infinite points, so a infinite set can cover another infinite set, as long as his first step can cover (not pinpoint out) the infinite part or subset of midpoints, or any midpoint can be accounted by the covering, only the other finite part of set of midpoints is left out at the first step or last step, the set of midpoints can be fully covered by the steps.

The concept of infinite set is misapplied to the case because this is a integer problem, the step and distance should be treated as integers. So the question becomes how many integers unit of step to cover the distance.

Revelations:

1,We don't have to infinitely divide space to cover space.

2,We cover space, not points. We have no experience of a point.

3,The measurement of unit of space can be very small but it must contain some space, points are not space, therefore we don't have to cover each individual point. We only have to cover between points.

4,The space made up of points are not continuous because the next point is unknown.

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