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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Like all paradoxes it is only a problem because when proposed there was a lack of mathematics knowledge. Lets look at the paradox in a different way:
How long does it take Achilles to reach the tortoise?
One way to solve this is to just add up all the time intervals for each step. So it takes Achiiles some time, t^1 to get to point A, t^2 to get to point B and so on. Our total time, T, is then:
T= t^1 + t^2 + t^3 …. for an infinite number of intervals.
So how do we get a finite number from adding an infinite number of positive values together? Without calculus we can’t solve this and hence the paradox.
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