Suppose, Zeno argues, that time really is a sequence of points constituting a time line. Consider the flight of an arrow. At any point in time, the arrow is at some fixed location. At a later point, it is at another fixed location. The flight of the arrow would be like the sequence of still frames that make up a movie. Since the arrow is located at a single fixed place at every time, where, asks Zeno, is the motion?
I never understood the paradox here. Isn’t the answer just the change from one frame to the next?
I think of Zeno’s paradoxes as trying to appeal to the essence of dissolved questions. Sort of like, having decomposed “does the tree make a sound?” into “does it produce vibrations” versus “does it cause auditory experiences”, somebody comes along and says “but does it make a SOUND???”, emphasizing the word “sound” to appeal to your intuitions and make you feel (incorrectly) that something in reality has yet to be resolved. Here, “motion” plays the part of “sound”, after a faulty reduction of “motion” into “series of still-frames”.
But if I were to be that guy who comes along and says “sound” until you feel uncomfortable again, I would say “What’s to say your arrow doesn’t just teleport from one frame to the next, rather than ‘move’? Can’t your ‘change from one frame to the next’ also be broken down into a series of frames?”
I wouldn’t lose any sleep over it, since you’re obviously right and anyone who denies motion is obviously wrong, but that’s at least where certain hapless philosophers are coming from.
I don’t remember how to do it well enough to explain it in detail, but the root of the problem was that people didn’t yet understand summing convergent series. For example, 1⁄2 + 1⁄4 + 1⁄8 + 1⁄16 + … = 1. It is discussed in some books on philosophy of math, I remember coming across it several times; unfortunately, a quick check of the books I have available right now can’t find a source.
That’s the solution to the Achilles and the Turtle Paradox (also Zeno’s), but the Arrow Paradox (in the comment you replied to) is different.
The Arrow Paradox is simply linguistic confusion, I think. Motion is a relation in space relative to different points of time, Zeno’s statement that the (moving) arrow is at rest at any given instant is simply false (considered in relation to instants epsilon before or after that instant) or nonsensical (considered in enforced isolation with no information about any other instant).
I never found the Arrow Paradox particularly compelling. For the Achilles and the Turtle Paradox I can at least see why someone might have found that confusing.
Remember that these people were writing long before we had calculus. With regard to the Arrow Paradox in particular: if you’re already comfortable with the notion of instantaneous rate of change, there is no paradox here. If you are not, and it seems sufficiently weird, then it may lead you to think there is.
The telling of this paradox I most remember says, “Between point A and point B, there are an infinite number of points through which the arrow must pass. So it must take the arrow an infinite amount of time to pass through those points. How can the arrow get from point A to point B?”
This is the problem with mapping a mathematical metaphor onto reality: it doesn’t always work. If the metaphor disagrees with the observation that the arrow does get from point A to point B, then it’s not doing useful work.
In fact, modern physics tells us there is a smallest possible length, the Planck length, which means there is not an infinite number of points through which the arrow must pass. Still, you don’t need modern physics to defeat this paradox; you only need the ability to observe that the arrow does get from point A to point B.
I thought the problem with the paradox was that the math was wrong. Even if we assume that there’s an infinite number of points between A and B, the more points we have, the less time the arrow would spend on each point, so if the number of point is infinite, the arrow would spend an infinitesimal amount of time at each point.
As it turns out, you need to know about time series and limits (and maybe l’Hopital’s rule) in order to correctly calculate the total flight time of the arrow (or, rather, to prove that it does not change even when the number of points is infinite), because infinity is not a number, and neither is 1 / infinity. Zeno did not know about these things, though.
Yes, you’re right. You can defeat the paradox on mathematical grounds, without having to appeal to physics. But Zeno could have defeated it on his own without using any math, simply by realizing that his metaphor was not paying rent.
I think ArisKatsaris (on the sibling comment) is right: Zeno’s whole goal was to prove that physics doesn’t work (ok, he didn’t call it “physics”, but still), so using physics to disprove his paradox would be nonsensical.
Zeno’s argument was that movement was an illusion, that all was one—that was the point of his paradoxes. The fact that things seemed to move, in combination with his paradox, proved (to him) that reality was an illusion.
I never understood the paradox here. Isn’t the answer just the change from one frame to the next?
I think of Zeno’s paradoxes as trying to appeal to the essence of dissolved questions. Sort of like, having decomposed “does the tree make a sound?” into “does it produce vibrations” versus “does it cause auditory experiences”, somebody comes along and says “but does it make a SOUND???”, emphasizing the word “sound” to appeal to your intuitions and make you feel (incorrectly) that something in reality has yet to be resolved. Here, “motion” plays the part of “sound”, after a faulty reduction of “motion” into “series of still-frames”.
But if I were to be that guy who comes along and says “sound” until you feel uncomfortable again, I would say “What’s to say your arrow doesn’t just teleport from one frame to the next, rather than ‘move’? Can’t your ‘change from one frame to the next’ also be broken down into a series of frames?”
I wouldn’t lose any sleep over it, since you’re obviously right and anyone who denies motion is obviously wrong, but that’s at least where certain hapless philosophers are coming from.
I don’t remember how to do it well enough to explain it in detail, but the root of the problem was that people didn’t yet understand summing convergent series. For example, 1⁄2 + 1⁄4 + 1⁄8 + 1⁄16 + … = 1. It is discussed in some books on philosophy of math, I remember coming across it several times; unfortunately, a quick check of the books I have available right now can’t find a source.
That’s the solution to the Achilles and the Turtle Paradox (also Zeno’s), but the Arrow Paradox (in the comment you replied to) is different.
The Arrow Paradox is simply linguistic confusion, I think. Motion is a relation in space relative to different points of time, Zeno’s statement that the (moving) arrow is at rest at any given instant is simply false (considered in relation to instants epsilon before or after that instant) or nonsensical (considered in enforced isolation with no information about any other instant).
I never found the Arrow Paradox particularly compelling. For the Achilles and the Turtle Paradox I can at least see why someone might have found that confusing.
Remember that these people were writing long before we had calculus. With regard to the Arrow Paradox in particular: if you’re already comfortable with the notion of instantaneous rate of change, there is no paradox here. If you are not, and it seems sufficiently weird, then it may lead you to think there is.
Oops, cached thought—I saw “Zeno’s Paradox” and jumped to the most common one without reading the details.
The telling of this paradox I most remember says, “Between point A and point B, there are an infinite number of points through which the arrow must pass. So it must take the arrow an infinite amount of time to pass through those points. How can the arrow get from point A to point B?”
This is the problem with mapping a mathematical metaphor onto reality: it doesn’t always work. If the metaphor disagrees with the observation that the arrow does get from point A to point B, then it’s not doing useful work.
In fact, modern physics tells us there is a smallest possible length, the Planck length, which means there is not an infinite number of points through which the arrow must pass. Still, you don’t need modern physics to defeat this paradox; you only need the ability to observe that the arrow does get from point A to point B.
I thought the problem with the paradox was that the math was wrong. Even if we assume that there’s an infinite number of points between A and B, the more points we have, the less time the arrow would spend on each point, so if the number of point is infinite, the arrow would spend an infinitesimal amount of time at each point.
As it turns out, you need to know about time series and limits (and maybe l’Hopital’s rule) in order to correctly calculate the total flight time of the arrow (or, rather, to prove that it does not change even when the number of points is infinite), because infinity is not a number, and neither is 1 / infinity. Zeno did not know about these things, though.
Yes, you’re right. You can defeat the paradox on mathematical grounds, without having to appeal to physics. But Zeno could have defeated it on his own without using any math, simply by realizing that his metaphor was not paying rent.
I think ArisKatsaris (on the sibling comment) is right: Zeno’s whole goal was to prove that physics doesn’t work (ok, he didn’t call it “physics”, but still), so using physics to disprove his paradox would be nonsensical.
Zeno’s argument was that movement was an illusion, that all was one—that was the point of his paradoxes. The fact that things seemed to move, in combination with his paradox, proved (to him) that reality was an illusion.