Blargg: I'm going by Wikipedia's summary of the paradox. In short, the tortoise starts 100 meters ahead of Achilles. After they start, Achilles will reach the tortoise's starting point, at which point the tortoise will be at some new point farther along. Achilles must then reach this point, and we're back at the same essential situation. Repeat indefinitely; Achilles will always be reaching points the tortoise was previously at, and never catch up. My resolution is that we might imagine that each iteration of the above is forward in time by a roughly equal amount, but it's not; each iteration is going forward a smaller and smaller amount of time, such that we always "slow time down" enough that we never get to the moment he actually does pass the tortoise. When he's almost there, our next step forward in time is extremely small. He's now closer, but we'll then step an even smaller amount forward in time. It's like taking a gold coin and dividing it in two and discarding one of the halves, then dividing the remaining halve in two, indefinitely; we're always removing gold, yet we'll never run out, since we're discarding less and less on each iteration.
I'm going by Wikipedia's summary of the paradox. In short, the tortoise starts 100 meters ahead of Achilles. After they start, Achilles will reach the tortoise's starting point, at which point the tortoise will be at some new point farther along. Achilles must then reach this point, and we're back at the same essential situation. Repeat indefinitely; Achilles will always be reaching points the tortoise was previously at, and never catch up. My resolution is that we might imagine that each iteration of the above is forward in time by a roughly equal amount, but it's not; each iteration is going forward a smaller and smaller amount of time, such that we always "slow time down" enough that we never get to the moment he actually does pass the tortoise. When he's almost there, our next step forward in time is extremely small. He's now closer, but we'll then step an even smaller amount forward in time. It's like taking a gold coin and dividing it in two and discarding one of the halves, then dividing the remaining halve in two, indefinitely; we're always removing gold, yet we'll never run out, since we're discarding less and less on each iteration.
Yeah, that's exactly right. Now for a solution, though lol.
What do you mean? I've shown that it's not that Achilles never catches up to the tortoise, it's that the observer never reaches the time where that occurs. The observer keeps going some fraction of the whole way, which always leaves some other fraction of time remaining until they meet.
So Zeno's Paradox proves we never advance in time.
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It's easy to refute an argument if you first misrepresent it. William Keizer
Smiling Dave: What do you mean? I've shown that it's not that Achilles never catches up to the tortoise, it's that the observer never reaches the time where that occurs. The observer keeps going some fraction of the whole way, which always leaves some other fraction of time remaining until they meet. So Zeno's Paradox proves we never advance in time.
Exactly what I was thinking.
Can you elaborate on what you mean? I'm detecting some subtext but it'd help if you were explicit.
Zeno stated several versions of his paradox. One was the tortoise and Achilles. He also had an arrow paradox, that an arrow will never hit its target, because before it can get to the target , it has to get halfway there. And before it does that, it has to get a quarter of the way there, etc. There is a third that I don't remember.
What I'm saying is that by shifting the problem to the observer and saying he will never get to the time when Achilles catches up, you haven't solved the paradox, but merely shifted it into a different version. That the observer will never observe the moment of catching up, because before he sees that etc.
My point was that what makes it seem like Achilles never reaches the tortoise is that we choose to observe the race not in normal flowing time, but in a way that causes us to keep slowing down as we get close to the point where he does catch up. Mentally we associate the iterations with a clock of sorts, and see that we'll never see Achilles catch up. It's an explanation for it coming across as a paradox. Clearly, he catches up, so we're not asking whether this proves that he doesn't, just to resolve the seeming inconsistency.
...time doesnt pass in an infinite regression, but at a specific rate..
1. Zeno also assumes that time passes at a specific rate. What makes you think otherwise?
2. What does "passing in an infinite regression" even mean? It has nothing to do with what Zeno or Blagg or I were talking about.
To elaborate, you make it sound like the study of infinite series is invalid when it comes to time, because time passes at a specific rate. But that is wildly wrong, like a batter swinging for the umpire's head instead of the ball. Both math and physics, at all levels, admit that the infinite series point of view is valid with respect to time.
...from the perspective of the observer...
So for someone else other than observer the rate is different? Can you explain to whom it is different, and what the rate is for them?
...one second per second.
So Zeno thought that time does not pass at the rate of one second per second? What rate do you think he was assuming? Two seconds per second?
We both know that real observers actually do observe events.
So Zeno must have been a real dummy, or else never watched a foot race. He never saw someone winning a race after trailing initially. Same with Bertrand Russell, who wrote that it requires advanced calculus to solve Zeno's paradox. Those two fools never got out of their ivory towers to actually observe events, I guess. One big dunce cap for them, hey?
The paradox is not germane to reality.
The paradox is not germane to reality, but it is germane to the only tool we have in our possesion to make sense of reality, aka logical thinking. It is challenging the validity of that tool, showing it produces results that do not fit reality. That's what bothered Zeno and Russell and everyone else.
...the person who poses the paradox asserts that the oberserver never reaches that instant.
No. He asserts that although reality shows us Achilles catches the tortoise, and we all see it with our own eyes, logical thinking forces us to conclude it impossible. He is challenging the validity of the accepted rules of logic and math. This is a huge problem, for it says we cannot rely on the many formulas we constantly use to predict reality. And yet, they have worked so well. There must be some escape hatch, but what is it?
No. He asserts that although reality shows us Achilles catches the tortoise, and we all see it with our own eyes, logical thinking forces us to conclude it impossible.
He is challenging the validity of the accepted rules of logic and math. This is a huge problem, for it says we cannot rely on the many formulas we constantly use to predict reality. And yet, they have worked so well. There must be some escape hatch, but what is it?
I concur that the point of these paradoxes is to show that taking a simple situation and applying some logic to it, we arrive at something that seems like it can't happen. If our tools can seem to innocently get us into situations where it's obvious that it's absurd, what of all the situations where we can't see the actual situation but can only approach it with our tools, and when we run into absurdities like in this paradox? Will we recognize that it's not the situation, but our tools (or use of them, or something in our field) that are the source of an illusion?