“That which is in locomotion must arrive at the half-way stage before it arrives at the goal.”
You know this one. An arrow shot from a bow at some guy’s head should never get there because first it must travel half the distance - then it must travel half the remaining distance (1/4), then half that distance (1/8) and blah blah blah. Going through an infinite number of halves and never actually getting to the guy’s head.
Anyway, thinking on the balcony idly again. Then it struck me. Ole’ Zeno was looking at the wrong trajectory when he was doing his halving.
When in reality:
So, the answer to the paradox is this: The arrow hits the target because the initial ‘halfway there’ point of the arrow’s ‘perfect’ flight (one infinite in nature - without obstacle or effectors) is always far beyond the actual target on the ground.
Half of something can always be proven to converge to 0 at the limit of infinity. but it can be proven to converge before that only if you understand the maths of time and distance not just distance. The key is in how we define an instant of time.
Like most paradoxes it is simply not one.
Try googling the bouncing ball and Xeno’s paradox, this is another more visceral way of visualising it, does the ball ever stop bouncing?
“Aristotle (384 BC−322 BC) remarked that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small.”
Are you able to comprehend what that means? If so can you explain it? Do that and the paradox is resolved. I’ll leave it up to you.
Clue, time and distance are related by halves only to a point where they converge at the limit of infinity. Instances in time and distance aren’t necessarily equal, they are merely conceptual, one need not reflect the other exactly by halves for us to resolve the paradox, only time needs to half.
I’m admittedly partial to St. Thomas Aquinas’ proposed solution to this paradox, but it is extremely dated now and easily dismiss-able on those grounds alone. I have come across another solution that I found interesting though–
Might sound trite but have you tried calculus? It explains the observable phenomena, it’s better than philosophy to that extent, although I must admit I am also partial to the simplicity of Aquinas, even if it is easy to dismiss, if he had the maths he would of been pretty much right. Being technically correct is the best kind of correct until absolute proof by maths stuff comes along.
Yep the axioms are flawed, it’s philosophy 101. Always kick the foundations out first.
Anyway, yeah - that’s the same page I looked for the dichotomy quote, and yeah, I read it. And wow, astonishingly enough, understood it too.
Go me.
I just thought there was an easier way to look at it. That first law of motion - “Every body remains in a state of rest or uniform motion unless it is acted upon by an external force.” Paradox over. Achilles passes the turtle, because he’s not aiming at the turtle, but at an infinite point beyond the turtle. It is his legs that stop him at the end of the race, not the motion that abstractly is still trying to ‘halve’ its way to infinity.
Same with the (perfectly elastic) ball, in a perfect vaccuum, it would never come to rest. That is does so in reality is not a property of the ball, nor of some magical zero-time bounce, but of outside forces. Zeno, and everyone else it seems, are mixing their frames of reference between real and abstract, trying to apply the logic of one to the other, and unsurprisingly, failing.
I sometimes think that “paradoxes” like this one revolve around the assumption that philosophers often create problems that don’t really exist. They create conceptual contraptions in their heads that have nothing to do with shooters, arrows and targets.
lol tab, the arrow knows and thinks things? your solution involves a conscious arrow? tsk tsk.
anyway, i’ll explain the math behind the solution.
let’s say something has to travel 1 unit. as zeno says, it must first travel half that distance.
1/2
then it must travel half the remaining distance
1/4
then half the remaining distance
1/8
so the distance it travelled so far is given by 1/2 + 1/4 + 1/8
if you kept on going, you’d get 1/16, 1/32 etc, each number is 1/2 of the previous number, which means the whole function can be described using powers of 1/2.
so, let’s do this infinitely many times, as zeno suggests. we will just say a = 1/2, for simplicity’s sake, and D is the total distance travelled.
D = a + a^2 + a^3 + a^4 + a^5 +…+ a^∞
if we multiply both sides of the equation by a, we get
Da = a^2 + a^3 + a^5 +…+ a^∞ (ever term multiplied by a)
so, D - Da = (a + a^2 + a^3 + a^4 + a^5 +…+ a^∞) - (a^2 + a^3 + a^5 +…+ a^∞)
you’ll notice that the underlined terms are the same.
since we’re subtracting them, we can just…delete them…right? if you subtract something from itself, mathematically, you get 0. let’s do that:
D - Da = a
D(1-a) = a
D = a/(1-a)
D = (1/2)/(1-1/2) = (1/2)/(1/2) = 1
I know, I know, everyone thinks I’m brain-damaged at the moment, but I’m not. Hopefully anyway.
The point I’m maybe failing to get accross, without resorting to maths, at least not very much maths is that the line everyone is busy trying to stuff an infinite number of points on, doesn’t really exist.
Look. A hundred meter runner. Let’s say he slams out of the blocks, runs himself up to full speed, then about 10ms before the line, goes over a cliff into a friction-free, gravity-free, utterly empty and infinite space. There is now no way he can actually stop. He’ll continue on forever, at whatever speed, and in whatever direction, that last push off the cliff-edge gave him. Of course he’ll never get anywhere, because there is nowhere to get to.
He still crosses the line though, and wins the race, then leaves that finishing line far behind him.
Each movement we make is, in that very initial moment of generation, essentially infinite, and it is only other forces, occurring after the event, that stop that movement. That distance D that you mention Humpty, is a trick. I take a step toward you.
But that distance is not the one relevant to Zeno’s paradox. Because that step I took, and the forward momentum it generated, was essentially ‘aimed’ at an infinite point behind you, and it will only be by force of will, and muscle, that I cut that journey short prematurely, at the point where you happen to be.
That first ‘halving’ of the (essentially infinite) distance is never reached, because it doesn’t have to be (our real destination occurs much sooner), and indeed cannot be reached, half of infinity still being infinity an’ all.
So, the answer to the paradox is this: The arrow hits the target because the initial ‘halfway there’ point of the arrow’s ‘perfect’ flight (one infinite in nature - without obstacle or effectors) is always far beyond the actual target on the ground.
[/quote]
You are about “half way” right in your assumption.
The problem is that the distance given, even if is not “half way there“, still divisible by halves.
I will give you my opinion.
The arrow reaches the target because…
The force necessary to move the arrow from a state of indifferent apathy to an active condition of conscience and awareness (target) is demonstrated in the actual flight.
The idea, as I see it, is that Zeno was trying to demonstrate the existence of “infinite divisions.” But the arrow doesn’t care about “infinite,” since it gets to its target.
If you believe in the word “infinite,” as well as on its meaning, the arrow will never hit the target… “it must arrive at the half-way stage before it arrives at the goal.”
Geometry is a poor model of the true nature of space. Zeno assumed a kind of geometric realism, that the principles of geometry and the natures of its objects were true of actual space and not just models we play around with in our heads. This entails that space is indeed composed of an infinit number of dimensionless, sizeless points, and that between any two of them exist an infinitude of others. It also entails that in order for an object to pass from one point to another, it must pass through each and every point in this infinitude.
But what if our geometric models of space are wrong? Einstein seems to have debunked Euclidian geometry as the most accurate depiction of space, and quantum physics seem to be doing something similar. I think it’s the latter in particular that might shed the necessary light on Zeno’s paradox and bring it to resolution.
i know the arrow will hit the target. i proved that already. you didn’t. you just spouted a whole bunch of babble about arrows that can think. you don’t know about my idea about the infinite. i didn’t express my own idea about the infinite, so there’s no way for you to know if it’s relevant or not.
