My current write-up of the arrow paradox is here. In short, I start with a time interval being divisible into point-intervals, then I reverse that and say that an aggregate of point-intervals can cover a proper time-interval. I then suggest that this means that point-intervals can cover a space interval. I think I’m onto something here, but it still doesn’t feel completely persuasive to me.
You’re using (already-substituted) the equations for distance and velocity, which is essentially just asserting the validity of calculus and taking as given that motion is possible and both space and time are continuous. Which is fine, but wouldn’t have convinced Zeno, I think.
I’m confused b/c nominally this claims to be addressing the arrow paradox, but it sounds like you are discussing the equations in Archilles and the Tortoise. The equations are being used to determine where they will cross IF they cross. To be absolutely clear, I’m not assuming they cross. I just need to know where the point will be in order to transform it to the dichotomy paradox (actually it’s not quite the dichotomy paradox, but this paradox in reverse). And there’s absolutely no need for calculus for a constant velocity.
I’m confused b/c nominally this claims to be addressing the arrow paradox, but it sounds like you are discussing the equations in Archilles and the Tortoise.
Ok, that’s fair. That does apply more to the arrow paradox (and the dichotomy paradox, which is the same but with time reversed; if you’re allowing the rate equation later on you seem to accept the validity of algebra, which means I can reverse time by substituting a new variable t’=-t (I expect negative numbers would give Zeno a heart attack, of course)). In your arrow writeup you talk about asking what an instant is. But in order for points to form a continuous line or line segment, you need uncountably many of them, one for each real number. Or, you need space and time to only be finitely divisible, and movement to happen in discrete intervals. The former requires allowing the existence of an actually infinite set, which modern math is fine with but the ancient Greeks mostly were not.
You don’t need to know calculus to calculate distance for fixed speed (or fixed acceleration), but where do the rate equations come from in that case? How did you know to use that equation? I don’t know any mathematical (not simply empirical, which we can’t really on because if we do, we already empirically know things move and reach destinations, and Achilles catches the tortoise) basis for that other than integrating v=constant to get d=rt.
And the arrow and dichotomy paradoxes are prior to the Achilles paradox. If motion is impossible, Achilles can’t catch the tortoise. If it is possible and moving things reach destinations, maybe he can. But by the time you have the possibility of motion and the function d=rt, you have a constructive proof.
So yeah—I would say drawing those lines without claiming they cross reduces the Achilles/tortoise paradox to the dichotomy and arrow paradoxes, which in turn are the same under time reversal. But without knowing how to get d=rt without calculus (which would be admitting the existence of limits and allowing taking sums of infinite series) I can’t say if there is still a paradox left.
Well, if you assume that a line is made up of points, but that you can’t have an actual infinity, then there’s your contradiction right there—“The former requires allowing the existence of an actually infinite set, which modern math is fine with but the ancient Greeks mostly were not. ”
Good point. But if you instead take the line itself as primitive like Euclid did, you still need two points on it to define the line you mean, and while you have the starting points, I’m not sure where your second point is, since you’re not assuming there is a point where the two lines cross. How, exactly, do you draw the lines in that case?
Side note: identifying the algebraic equation d=rt with a line requires a coordinate system. What coordinate system are you using? What set of numbers are you drawing the allowed coordinate values from? Do those numbers form a field in which all the basic arithmetic operations are well defined?
The point I’m trying to get at is: Zeno’s paradox is only a paradox because 1) he didn’t accept the idea of a completed infinite set, and 2) many of his ideas about math and logic were ungrounded and underspecified. It’s very hard to still think of these as paradoxical once you’ve seen how to build up the foundations of math from set theory (for which the assumptions are very simple) including geometry, arithmetic, algebra, and so on.
I was just reading on another lesswrong thread about how T-reversal implies or not taking a complex conjugate. There was a metaphor about a stack of pancakes. If you reorder them have you flipped the stack or do you need to flip individual pancakes as well? It seemed that there were implicit time directions even in the static slices. That is if you take a static snapshot of particle and tell “now enter the next moment” it has inscribed in itself which way to go. This seems to be incontrast with the “if it is at rest, it can’t be moving” intuition. That is a particle moving to rigth if complex conjugated will move to the left and that bit of information exists even in a perfectly thin view of the particle. So a particle is not just a collection of it’s positions in different times.
My current write-up of the arrow paradox is here. In short, I start with a time interval being divisible into point-intervals, then I reverse that and say that an aggregate of point-intervals can cover a proper time-interval. I then suggest that this means that point-intervals can cover a space interval. I think I’m onto something here, but it still doesn’t feel completely persuasive to me.
I’m confused b/c nominally this claims to be addressing the arrow paradox, but it sounds like you are discussing the equations in Archilles and the Tortoise. The equations are being used to determine where they will cross IF they cross. To be absolutely clear, I’m not assuming they cross. I just need to know where the point will be in order to transform it to the dichotomy paradox (actually it’s not quite the dichotomy paradox, but this paradox in reverse). And there’s absolutely no need for calculus for a constant velocity.
Ok, that’s fair. That does apply more to the arrow paradox (and the dichotomy paradox, which is the same but with time reversed; if you’re allowing the rate equation later on you seem to accept the validity of algebra, which means I can reverse time by substituting a new variable t’=-t (I expect negative numbers would give Zeno a heart attack, of course)). In your arrow writeup you talk about asking what an instant is. But in order for points to form a continuous line or line segment, you need uncountably many of them, one for each real number. Or, you need space and time to only be finitely divisible, and movement to happen in discrete intervals. The former requires allowing the existence of an actually infinite set, which modern math is fine with but the ancient Greeks mostly were not.
You don’t need to know calculus to calculate distance for fixed speed (or fixed acceleration), but where do the rate equations come from in that case? How did you know to use that equation? I don’t know any mathematical (not simply empirical, which we can’t really on because if we do, we already empirically know things move and reach destinations, and Achilles catches the tortoise) basis for that other than integrating v=constant to get d=rt.
And the arrow and dichotomy paradoxes are prior to the Achilles paradox. If motion is impossible, Achilles can’t catch the tortoise. If it is possible and moving things reach destinations, maybe he can. But by the time you have the possibility of motion and the function d=rt, you have a constructive proof.
So yeah—I would say drawing those lines without claiming they cross reduces the Achilles/tortoise paradox to the dichotomy and arrow paradoxes, which in turn are the same under time reversal. But without knowing how to get d=rt without calculus (which would be admitting the existence of limits and allowing taking sums of infinite series) I can’t say if there is still a paradox left.
Well, if you assume that a line is made up of points, but that you can’t have an actual infinity, then there’s your contradiction right there—“The former requires allowing the existence of an actually infinite set, which modern math is fine with but the ancient Greeks mostly were not. ”
Good point. But if you instead take the line itself as primitive like Euclid did, you still need two points on it to define the line you mean, and while you have the starting points, I’m not sure where your second point is, since you’re not assuming there is a point where the two lines cross. How, exactly, do you draw the lines in that case?
Side note: identifying the algebraic equation d=rt with a line requires a coordinate system. What coordinate system are you using? What set of numbers are you drawing the allowed coordinate values from? Do those numbers form a field in which all the basic arithmetic operations are well defined?
The point I’m trying to get at is: Zeno’s paradox is only a paradox because 1) he didn’t accept the idea of a completed infinite set, and 2) many of his ideas about math and logic were ungrounded and underspecified. It’s very hard to still think of these as paradoxical once you’ve seen how to build up the foundations of math from set theory (for which the assumptions are very simple) including geometry, arithmetic, algebra, and so on.
I was just reading on another lesswrong thread about how T-reversal implies or not taking a complex conjugate. There was a metaphor about a stack of pancakes. If you reorder them have you flipped the stack or do you need to flip individual pancakes as well? It seemed that there were implicit time directions even in the static slices. That is if you take a static snapshot of particle and tell “now enter the next moment” it has inscribed in itself which way to go. This seems to be incontrast with the “if it is at rest, it can’t be moving” intuition. That is a particle moving to rigth if complex conjugated will move to the left and that bit of information exists even in a perfectly thin view of the particle. So a particle is not just a collection of it’s positions in different times.