The problem arose because Zeno never heard of calculus. Intuitively, an action requiring an infinite number of steps, each with taking up a finite amount of time, will never be completed. We now know this isn’t true.
You’re missing the point. Forget the notion of a destination. No matter what distance the hammer travels before it stops, it first has to travel the first half of that distance. And it has to travel the first quarter of the distance before it can travel the first half. You’ll agree that there is an infinite regress here, yes?
By your logic, there’s no paradox if there’s a wall between Athens and Sparta that’s closer to the starting city. Don’t you think that’s just a little absurd (if you accept the paradox in the first place)?
Isn’t this paradox becomes an intro to calculus solution by taking a limit of a linear function, that as x (number of steps) approaches infinity, the distance becomes 8 feet, therefore, the rock hits the tree?
Calculus allows you to reason that the rock will reach it’s destination, but some people feel that it doesn’t accurately describe what is actually happening in the real world.
The absurdity is trying to apply this mathematical idea to a real-life situation. I drove from L.A. to Birch Bay. I did not drive 600 miles the first day, 300 miles the next, 150 miles the third, and so on. Since I was hauling a trailer, I drove 600 miles the first day and 600 miles the second. (Had I not had the trailer, I would likely have just driven it all in one go.) So either it’s absurd to assume that anyone would do the half-half-half thing, or I never made it to my house.
If someone walked between two trees, traveling half the remaining distance each time, then theoretically he would never get to the second tree. But if the paradox is phrased such that if you were to put a wall up between the trees, then the walker would never make it to the wall. But the destination is the tree; thus he hits the wall.
Goddammit, Johnny L.A definitely has it. I have noticed before how easily the paradox is solved by considering that what the arrow is actually on its way to is even an inch further on than the target it hits. The target gets in the way of the moving object.
I’d say even Johnny L.A is hampering his thoughts with ideas of gravity, the centre of the earth and other mundanity. A propelled arrow/hammer will travel infinitely (given an infinite universe…) until afflicted by other forces.
Hasn’t the paradox been superceded by Newton*?
*or whoever…
So what if the wall is closer to the destination than the starting point? Does that make a difference?
The ‘destination’ is infinitely far away from the thrown object. That’s how you accomodate the whole halving-the-distance malarky.
You can build your walls wherever you fancy.
And what of the man walking from Athens to Sparta?
The walker better have infinitely hard-wearing shoe leather and a lot of patience, as he’s going as far as the hammer/arrow, given no obstructions.
As posted by Xema and Little Nemo, the paradox is solved by realizing that you can add an infinite number of things and still get a finite result. It wasn’t proven until the advent of calculus, but there’s an astonishingly simple and intuitive way of picturing it:
Take a square. Divide it into two halves down the middle. Now divide one of the halves in half again, forming two smaller squares. Repeat the process on one of the smaller squares; continue forever. It becomes apparent that you can add the series of 1/2 + 1/4 + 1/8 + 1/16 … forever, and yet it will only add up to 1, the whole square.
Except that he intends to stop once he’s reached Sparta, and most likely will do so. What are you saying here?
Zeno: …and so before you reach Sparta, you must reach the halfway point to Sparta. And before you reach the halfway point, you must reach the quarter point. And before you reach the quarter point, you must reach the eighth point. And so on, ad infinitum. Therefore, you can never reach Sparta.
Johnny L.A.: I’ve got it! We must always intend to travel twice as far as we actually plan! The problem is solved!
(The sound you hear is that of Zeno smacking a forehead. Exactly whose forehead is being smacked, I’m not quite sure).
I better go gas up.
Lumpy I love the square example…it makes it much clearer to me.
At some point the “half distance” is going to be less than the radius of the rock.
So? An object can move by increments smaller than its own size.
BTW I’m still not getting this ‘intended destination’ thing - seems like a massive red herring - Xeno’s paradox boils down to: ‘Any distance can be infinitely divided, how can an inifinite number of steps (no matter how small) be traversed?’
Assuming reference in the paradox to the center of the rock, at some point the distance the rock must travel will be less than the radius of the rock such that the leading edge of the rock will impact the tree before the center of the rock can move the remaining distance.
As I see it, the rock travels the first half…
…and then the second half and hits the tree. This and the tree falling in the woods are the most overthought philosophical bar chats of all time.
Isn’t there a finite limit to how slightly something can travel - a quantum leap or something? In which case, as soon the tiny, endlessly divided distance was smaller than this, impact would occur…