What is the solution to Zenon's arrow paradox?

Question

Imagine an arrow being shot from a bow. In order to reach the target, the arrow must get midway though the distance it's being shot. But to get there, it has to reach half of that distance, and so on to an infinity of intermediary destinations. In conclusion, the arrow can never reach its final destination, or even leave the bow, for that matter.

This line of reasoning starts with apparently true hypotheses and reaches a false conclusion, by using a hidden "hole" in the logic.
What is that hole? What false assumption or deduction did Zenon make in order to reach the (obviously) false conclusion?

Answer 1

The conclusion that the arrow will never reach is the target is the false conclusion. What one is really calculating is the limit of all the calculations 1/2 ( the distance) + 1/4 + 1/8 .. etc.
An arrow and a moving animal confuses the issue because as the arrow moves ahead so does the rabbit or whichever animal.
Nevertheless if you do all the calculations or use calculus then you will come to a limit. if you take it down to a dot of time is a dot of distance for both the arrow and the rabbit and do the calculations what you have is how long the arrow took to hit.. not .. the arrow won't hit.

Answer 2

my form of the paradox is this - an arrow or anything moving in its flight cannot stay at one point for any length of time, or it would be still - but it cannot stay at one point for zero time either, or it wouldnt be there at all

and i dont know the answer - apparently our logical brain cannot really conceive motion - which raises questions/doubts about the applicability of logic to reality, and the logicalness of reality

and here is a paradox of time that may be original with me

past and future time do not exist and present time is just the point of zero dimension between past and future - so what is time? - i suggest we are actually experiencing eternity, timelessness - coloured or flavoured with the everchanging spatial situation [roman times, elizabethan times, present times] - in other words, time is the particularity of eternity - and the particulars are spatial - as a particular horse is the universal horse plus particulars, so time is eternity with [spatial] particulars

[this may go some way to explain why einstein saw space and time as two faces of one thing]

all our concepts may be limited concepts [faces, facets] of the one thing that is, ie life, existence - like the limited concepts of the elephant of the blind men

Answer 3

No solution is necessary. The paradox is its own solution.
We have facts and we have reality. As a fact, we see a paradox.
As a reality, we can see that the paradox represents a deeper reality that could be made factual if all laws inconsistent with its discovery are thrown away. We still lack the technology to throw away the facts and survive.

Answer 4

The logic fails because he did not take into consideration the zero.
If you begin at 1 and average the distance (X) then it is (X/2) +1 not (X/2) as he would indicate.
The distance actually begins at 0 (at the bow)

Answer 5

HE DIDN'T KNOW THE FIRST LAW OF PHYSICS: ONCE AN OBJECT (THE ARROW) IS PUT INTO MOTION, IT STAYS IN MOTION UNTIL STOPPED BY SOMETHING OF EQUAL OR GREATER FORCE (THE TARGET). HE ASSUMED THAT THE ARROW MUST GET "HALFWAY" TO THE TARGET. IN FACT, HE FORGOT (OR DIDN'T KNOW TO BEGIN WITH) GENERAL RULES PERTAINING TO MATTER AND ENERGY

Answer 6

Space is not infinitely divisible.

Answer 7

In my opinion, Zeno was attempting to demonstrate that "false assumption" that you seek. He wasn't himself making it, but rather was demonstrating what happens when you do make it. As you said, his line of reasoning starts with an apparently true hypothesis, and reaches a false conclusion. This false conclusion is meant to show that the apparently true hypothesis was indeed false. The false assumptions concern the ultimate status of the arrow, motion, and time, which I will explain shortly. He himself did not make these flawed assumptions, but was rather attempting to demonstrate that IF one assumes these axioms, paradoxes arises, which disprove the axioms.

Zeno's paradoxes obviously offered a difficulty for math and engineering, and calculus basically solved that difficulty by providing a way of dealing with infinite sequences. But Zeno wasn't trying to say that the world doesn't work the way it appears to work, as if by saying this he meant there would be no way of divising a mathmatical system that would accurately map to the way things work. He wasn't concerned with engineering, he was concerned with the ultimate status of things, with existence. Calculus has merely provided us with the mathmatics to represent and calculate things like motion, but it hasn't addressed what Zeno was fundamentally concerned with, and what he was trying to disprove through his paradoxes (i.e., which because they clearly don't work the way things cleary do work, are meant to show that their underlying assumptions are flawed).

You see, Zeno's primary task, in concert with the other philosophers from the ancient Greek Eleatic School (founded by Parmenides) was to show that only Existence is real (to use a more modern term for what they referred to as the One reality, or "all is one") and that everything else really is Existence, and only appear as seperate from Existence in an illusory sense.

So his task was not to say that time, space, motion, and objects do not exist and that they do not do what they appear to do. Rather, his task was to show that these things cannot have absolute existence. (A poor analogy would be the pictures on a movie screen. They exist, but in an illusory manner.)

Zeno wants to say that objects, in this case the arrow, ultimately depend on Existence for their existence. More accurately, that they are Existence (or the One) because nothing can be found in them that exists independently from Existence.

To do this, he must show that objects (and space and time) cannot be found to have any ultimate existence by virtue of themselves, because only this would grant objects the ability to stand apart from Existence. Put differently, we must either say that everything exists because of Existence, such that if Existence itself disappeared (so to speak), everything else would also disappear . . . or, we must argue that there is no such thing as Existence independent of things which must then have self-existence in their own right. Both ancient views, still valid to this day, rest upon the fundamental intuition that relative existence cannot account for existence per se. There must be something fundamental in existence, and that must either be in existence itself, or in existents (objects, phenomena).

Because, if one concludes that everything ultimately depends upon existence, then there can be no other conclusion than that everything really is existence, for there would not exist anything anywhere to stand apart from existence to give things an independent status. And that would mean that ultimately all is One.

That is what Zeno and company were trying to show.

(Calculus has absolutely nothing to do with this. They already knew that objects moved. They weren't idiots! What calculus has done is provide us with the math for what we already know happens. But that sheds no light on the question of whether existents (phenomena, objects, etc) have any absolute status of existence.)

So Zeno's task was to demonstrate that it is impossible that existents, such as an arrow or motion, have absolute existence, or existence independent of existence itself. IF an arrow really had absolute existence we would expect that it would be inherently what it is. And it is precisely when we assume that, that the paradoxes present themselves.

In otherwords, the paradox is meant to demonstrate that IF the arrow existed according to a certain view of existence in which objects posses that existence inherently, then motion would not be possible. This is because if an object, such as an arrow -- and the arrow was merely meant to be an example for all objects alike, whether that be an arrow or an electron (of course, they did not know of electrons then, but the point was the example was meant to speak of the status of objects in general, not just an arrow in particular) -- possessed absolute existence, then it would by definition exist by virtue of itself, because if it didn't, then there would be another existence upon which it depended. In otherwords, it would have to be discrete, an independent unit in a universe filled with independent units (again, the argument he was trying to disprove was that Existence, per se, does not exist, but rather existents themselves self-possess existence). It would have to be independent because if it weren't so, then it would be dependent upon something else for its existence, so that its existence by definition wouldn't be absolute. The object itself accounts for existence, just because it exists, but if there is no Existence in which this occurs, then that object has nothing in which to move unless motion is itself an independent existent. That is, unless motion comes from within the object itself. But when the arrow is not moving, it has no motion, so if it exist by virtue of itself and motion is an inherent part of it, when it is not moving it exists as a non-moving thing which would contradict what it is. So motion must itself be an independent existent which acts upon the arrow. But if motion is an independent existent which acts upon the arrow, then the arrow must always be at rest with motion acting upon it. Thus arises his paradox, which is meant to show what happens when you attempt to explain how an object that is inherently at rest -- assumed by assuming that the object exist by virtue of itself -- moves. For, if it is inherently at rest, then at one moment it must be resting at one point, and the next moment it must be resting at another point, and at no point is it never not resting. But such a motion takes place over a discrete series of points, and between each point lies an infinite number of other points.

(One argument against this is that space is not infinitely divisible. But this makes no sense with regard to the concern over the absolute existence of existents, because if space exists as that which lies between existents, then if it consists of a finite series of intervals those intervals are necessarily separated. If space is itself made up of these intervals, then these intervals must be existents, for where else would these intervals -- assuming a universe of nothing but existents [i.e., without Existence] -- get their reality? But if space IS the separation between existents, and the finite parts or intervals of space are themselves existents, then that means these finite parts of space are separated, and we have already defined space as the separation between existents. If, on the otherhand, the intervals are themselves adjacent to one another, then the problem still persists, only we must replace "space" with length, for each interval would have to be a certain length in order for them to all be adjacent. And what would constitute this length? Length can be divided into shorter lengths, and so we would need some smaller unit to account for how these intervals of space aquired length, and so forth, thus we are back to Zeno's paradox.)

I'm positive I haven't remotely done him service But the point I am trying to make, again, is that the "flaw" in his reasoning was precisely what he was trying to show. He was trying to show that IF you view the universe a certain way, motion is impossible, which shows that the universe cannot really work that way.

Namely, one must assume Existence in which all phenomena appears, for assuming rather that existence is something inherent in objects/existents leads to paradox.