For those of you who aren’t familiar: Achilles and a Tortoise race, however the tortoise is given a leading start. Achilles is at Point A, whereas the tortoise is ahead at point B. The race begins, and by the time Achilles makes it to point B, where the Tortoise used to be, it has reached point C. Then Achilles arrives at point C with the Tortoise at point D. So on and so forth, with Achilles never catching up to the Tortoise as per the paradox.
But he definitely catches the Tortoise eventually, right? The tortoise has a lower velocity, hence the head start, so after a certain amount of time the distance between points is smaller than Achilles and the Tortoise’s difference in speed. What, if anything, is paradoxical about the world’s most famous paradox?
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Achilles definitely catches the tortoise.
The way the story is told, they keep describing snapshots in time that are closer and closer to the time the tortoise is caught without telling you about that moment.
Like if he catches him at noon and you tell a story describing 11:59, then 11:59:30, then 11:59:45, then 11:59:52.5, etc.
You can certainly tell a story like that where you never get to noon. But we all know how time works. Eventually noon will come.
Zeno argues that motion is impossible because in order to get from one point to another, an object must first travel half the distance, then half the remaining distance, and so on, ad infinitum. Since there are an infinite number of distances to be traveled, the object can never actually reach its destination.
In reality, Achilles will eventually catch up to the tortoise, despite the paradox’s argument that he cannot. Zeno’s paradox is based on the assumption that an infinite number of smaller and smaller distances must be covered in order to reach a destination, but in reality, there is a smallest possible distance that can be traveled, such as the Planck length in physics.
Furthermore, the paradox assumes that time is infinitely divisible, which is also not true according to modern physics. When the actual physical laws are taken into account, the paradox can be resolved, and it becomes clear that Achilles can indeed catch the tortoise, given enough time.
Zeno’s paradoxes continue to be interesting philosophical puzzles that raise questions about the nature of space, time, and motion, but their solutions lie in our understanding of modern science and mathematics, which provides a more accurate and realistic description of the world around us.
Outside of some tricks with language (“this statement is a lie”), you can’t have actual paradoxes. By definition, a real paradox is impossible. However, you can get things that seem paradoxical. Normally, these situations involve reasoning that seems good, based on common assumptions, that leads to a result that contradicts what we know to be true.
So, we know that a racer can overtake another racer that has a headstart on him. Zeno’s reasoning seems solid, but indicates that overtaking should be impossible. The common assumption that turns out to be wrong is the idea that space is infinitely divisible. We’re in a simulation, and you can’t have half-a-pixel, so you have to always move across the screen in discrete units.
Achilles definitely catches the tortoise.
The way the story is told, they keep describing snapshots in time that are closer and closer to the time the tortoise is caught without telling you about that moment.
Like if he catches him at noon and you tell a story describing 11:59, then 11:59:30, then 11:59:45, then 11:59:52.5, etc.
You can certainly tell a story like that where you never get to noon. But we all know how time works. Eventually noon will come.
Zeno argues that motion is impossible because in order to get from one point to another, an object must first travel half the distance, then half the remaining distance, and so on, ad infinitum. Since there are an infinite number of distances to be traveled, the object can never actually reach its destination.
In reality, Achilles will eventually catch up to the tortoise, despite the paradox’s argument that he cannot. Zeno’s paradox is based on the assumption that an infinite number of smaller and smaller distances must be covered in order to reach a destination, but in reality, there is a smallest possible distance that can be traveled, such as the Planck length in physics.
Furthermore, the paradox assumes that time is infinitely divisible, which is also not true according to modern physics. When the actual physical laws are taken into account, the paradox can be resolved, and it becomes clear that Achilles can indeed catch the tortoise, given enough time.
Zeno’s paradoxes continue to be interesting philosophical puzzles that raise questions about the nature of space, time, and motion, but their solutions lie in our understanding of modern science and mathematics, which provides a more accurate and realistic description of the world around us.
Outside of some tricks with language (“this statement is a lie”), you can’t have actual paradoxes. By definition, a real paradox is impossible. However, you can get things that seem paradoxical. Normally, these situations involve reasoning that seems good, based on common assumptions, that leads to a result that contradicts what we know to be true.
So, we know that a racer can overtake another racer that has a headstart on him. Zeno’s reasoning seems solid, but indicates that overtaking should be impossible. The common assumption that turns out to be wrong is the idea that space is infinitely divisible. We’re in a simulation, and you can’t have half-a-pixel, so you have to always move across the screen in discrete units.
Oh so you have a set that’s open on one end. That is really weird since you finished but there is no last number. It’s like the lamp version of this.
Oh so you have a set that’s open on one end. That is really weird since you finished but there is no last number. It’s like the lamp version of this.
Oh so you have a set that’s open on one end. That is really weird since you finished but there is no last number. It’s like the lamp version of this.
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