Zeno's Paradox- proposed solution

Philosophy Forums Philosophy
1

I haven’t done much auxiliary reading on Zeno, but have been aware of his arrow paradox for a couple of years now. So I’m not sure if someone’s already thought of this idea, but I wanted to mull it over with someone else’s input and see if people can come up with refutations.

So you’re probably familiar with the paradox wherein a marksman has to hit a target. In order for his arrow to hit the target it has to reach the halfway mark, and then the halfway mark between half and the start point, and then half of that and then etc unto infinity (cause space can always be divided).

The solution I’m proposing is to treat the arrow not as the tip of the point --which would suggest that it is a non dimensional object and therefore not itself infinitely divisible-- but as a spacial object like the area it has to pass to reach the target. So when we divide the space between the arrow and the target we mark off point A at the base of the arrow and point B at the target. Lo and behold the arrow now exists in the space that it must traverse. So now when the marksman fires it he finds that the physical space of the arrow (however small the arrow may be) physically eclipses the troublesome area of infinite diminishment that lies just before point A.
The arrow, comprised of infinite points can always force its way through the halfway barriers because some of its points will always be in front of some of the barriers and when they move they move other points forward.

Refutations?

2

Your solution is a little confusing. Sorry, could you please elaborate/explain a little clearer?

The only way for the arrow to “never reach the target due to dividing space by 2 every time” the arrow’s velocity would have to decrease by half at every distance interval (in this case 1/2, 3/4, 7/8, 15/16…). If it were traveling through a dampening medium (like a vat of honey) then the velocity would decrease as the arrow traveled. Mathematically/graphically the arrow would “approach zero velocity” as it traveled through the honey, but in reality it would eventually stop due to friction.

This cannot be said for an arrow fired in a low/zero friction environment. I think this Zeno Paradox is very poor because it can very easily be refuted with a simple demonstration. It doesn’t take into consideration constant velocity.

3

I think you’re misinterpreting the paradox itself. I didn’t phrase it very clearly. Obviously on a physical level, you’re right, the arrow will slow down. But that’s not what Zeno was talking about. What he meant was that for the arrow to reach the mark it would have to get half way to the mark first. Right? But before it could get half way to the mark it would have to get to the 1/4 point, and then the 1/8th point. And really, etc unto infinity. Zeno is suggesting that if space is infinitely divisible, movement is impossible, or at least something like that.

My solution suggests that he’s erred by trying to send a non-spacial point through to the mark. By giving the arrow no dimension and setting it outside of the space it has to travel he’s created a paradox that seems logical, but is really meaningless. If you try to draw a sketch of the paradox itself you’ll see what I mean. If you superimpose the image of an arrow on top of even part of the space it has to travel you’ll see that the arrow, by existing as a spacial object has already “moved” across the part of the paradox that nears it and that infinitely diminishes.

4

zeno’s paradox fails because time is as infinitely divisible as space… beside the fact that one doesn’t count in infinitisimals…

-Imp

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[ENTER Differential Calculus STAGE LEFT]

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Imp-- I think I can reasonably question what is meant by time at this juncture, but I might be wrong to. At any rate, I suppose you mean to put a stopwatch beside the path of the arrow and label each interval as taking so many seconds. But the problem with that, is that for the stopwatch to record time, it has to go through motion. Taking the example of a non-digital clock, you’ll see that the gears have to physically move to tick off the seconds.
So the question becomes, if time can only be recorded through motion (specifically a constant rate), if its only measurable as a sequence of events that occour in space, then yeah, it is infinitely divisible, but you can’t record the quantity if there is no possibility for motion. If the arrow can’t move from point A to point B, then you certainly can’t let anything else in the environment do the same. When Zeno stopped the path of the arrow, I think he also stopped the possibility of measuring time.

Airex: {EXIT me, STAGE RIGHT}

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That was tight.

8

i don’t even get what zenos paradox is about… i’ve tried to understand it like 2 times and gave up…

9

I sometimes have this problem when I’m trying to pick up a wireless signal with my lap-top. I’ll find one and it will say “very low”…and I get all nervous, you know? It’s like, at which point between here and there does it become very low rather than just low, and, aren’t there an infinite amount of intervals between each degree, such that it might say very low, very very low, and very very very fucking low.

At this point I hesitate to connect and I become engrossed in computations and transdimensional geostructural hypothetical modeling, where I cross reference the energy exchange rates ratios of electron points in a vacuum, thereby simulating a holographic boolean alternative logical gates system (rather than the typical venn system) which allows me to calculate the precise trajectoretical area of the probable, the probable (I didn’t say certain…this is critical, remember that) “wobble” of the quark during its exponential divisonary point before collapsing the wave function, in an attempt to determine just how “very” low the signal is.

You see, I can literally manifest a real atomic point in definite space, and therefore divide the distances accurately between degrees of “very”. No, there are not infinite degrees of “very”. And the universe is not infinite. And God exists.

…

I’ve told you too much already. I must go.

10

how do transmisions work? i mean internet and all comunications…
it’s all quantum right?
i guess I can’t know how they work then.

I’ve tried…though… :astonished: …but my brain is blonde sometimes…for practical info.

11

Here’s a wikipedia snippet explaining it:

(((“You cannot even move.”
“ If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. ”

—Aristotle, Physics VI:9, 239b5

In the arrow paradox, Zeno asks us to imagine an arrow in flight. He then asks us to divide up time into a series of indivisible nows or moments. At any given moment if we look at the arrow it has an exact location so it is not moving. Yet movement has to happen in the present; it can’t be that there’s no movement in the present yet movement in the past or future. So throughout all time, the arrow is at rest. Thus motion can not happen.

This paradox is also known as the fletcher’s paradox—a fletcher being a maker of arrows.))

So actually, that phrasing addresses Imp’s point. Maybe. At any rate, I was discussing a different phrasing of the question. My response would be to say to this one what I posted to Imp, and that’s that any object, by existing dimensionally also exists temporally when the question of motion is brought up. So, for example, the arrow does not exist in one Now, but in many different nows. It can be seen as extending into the past on one end, and into the future on the other. If you say that the arrow occupies one Now, you are suggesting that the back end occupies the same point in space as the front tip. Any individual point along the arrow enters a specific space that motion differentiates from the previous spaces by the use of time… eh. i.e.: I was there, now I’m here.
So therefore, multiple Nows, one object, made possible by movement.
I’m not sure I’m being very clear though.
At any rate, the upshot of this is that since the arrow exists in the past present and future by the designation of different places, although it’s one object, you can see that it’s an entity in motion.

12

This is so funny, This morning, over a bowl of Cheerios, I was reading about infinite, Zeno, and infintisimals out of my Cambridge Dictionary of Philosophy!

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That’s the whole deal right there. :wink:
but then we are tripping up on TIME.
The problem is we are seeing them as separate.

After looking at some of the other posts,
all I have to say is SPACETIME my friends
(not SPACE and TIME as separate things)

Check out Russell’s ABCs of Relativity

14

Pardon me for being overly simplistic but if our senses report that the arrow has reached its mark isn’t that refutation enough?

15

Well said. But its worth mentioning that the concept of infinity is rather wondrous. Its a shame that there are other things do not require sensual confirmation. All you need to do is think about them, and they’re regarded as real.

16

With regard to the arrow, if time is suspended, then motion is suspended. If you begin to move one, you have to realize the other has moved. It isn’t a paradox really because it is misconceived. Old Zeno didn’t have Einstein and the theory of relativity around back then.

Motion, Time, and Space (not meaning to separate them) are fascinating!Everything is moving and at different speeds relative to other objects. I am not moving at all relative to the earth, yet relative to the sun I am moving incredibly fast! Relative to other stars even faster. In a way, everthing is moving, and nothing is moving. It all depends on what two points you are referencing. As you reach the speed of light, all happens in an instant! Just fascinating and mind bending!

17

Check this out!

There is a fallacy that logic students will recognize as the “quantifier switch” fallacy. The universal quantifier, “at every instant,” ranges over instants of time; the existential quantifier, “there is a place,” ranges over locations at which the arrow might be found. The order in which these quantifiers occur makes a difference!

18

Space cannot be infinitely divided. It is a self evident law IMO. If space could be divided forever there would be an infinite amount of distance between each object as Zeno’s paradox suggests. Then no-thing could ever touch anything else. We know this is untrue because the arrow does eventually hit the target.

19

On a quantum level the arrow doesn’t touch anything due to atomic repulsion. But yet there is still space there that can be traveled and divided.

Space can be infinitely divided. A finite length (which is defined relative to a predetermined standard) is made up of an infinite number of infinitesimal portions. If space were not infinitely divisible then differential calculus would not be possible. I don’t know about the other math nerds here, but Riemann Sums are a bitch. In a series of “divisions by 2”, at no point will you run out of stuff to divide, because all you have to do is divide one more time.

x/2 will never = 0 unless x=0. But if you start with a non-zero x, then you will always be able to divide again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, and again, …

20

Traveled and divided by things other than the arrow’s tip though (due to said atomic repulsion), right? Thus, in that sense, the arrow has reached the target.