How does the Achilles and the Tortoise paradox work? Isn’t Achilles bound to take over the tortoise after a certain point?
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That is indeed the paradox. The paradox is about the sum of an infinite number of infinatly small numbers which can still have a value.
The paradox is that it would take an infinite number of steps.
As many later philosophers and mathematicians pointed out, this infinite number if steps (catching up to each position the tortoise was at previously) takes a finite amount of time, so there is no paradox in Achilles passing the tortoise at some finite time.
However, a response has been made that the problem is not with the simple kinematics of the situation, but the fact that such an infinite subdivision is possible – for continuous motion to take place we are forced to assume that a completed infinity exists and we can traverse it. So if you add that premise explicitly (the infinity of the continuum cannot exist) you can still conclude that motion is impossible, and whatever we perceive is some form of illusion.
That’s why it’s a paradox, a counter intuitive or absurd complication to an otherwise pretty straightforward problem
The idea is that the you reach where the tortoise was before you reach the current position of the tortoise, however we are basically back to square one because now Achilles has to reach the position of the tortoise against, but it will have moved so you again don’t pass ut.
The idea was that since you can do that infinitely many times, it will take infinite time to pass the tortoise. Now we have knowledge about infinite converging series and we know that you can do infinitely many things in a finite amount of time.
Obviously that’s not what happens, achilles just passes the turtle, but it’s just interesting to think about.
The paradox appears to arise because if you split the race into an infinite number of slices of time and add them together it would appear that by adding together an infinite number of values you would get a result that is also infinite.
Which seems intuitive. Paraphrased – “keep adding on a little more each time for infinity times and the end result cannot be finite” – it makes sense. But it’s wrong. They knew it was wrong, but couldn’t explain why, so they had this paradox where Achilles will obviously catch up with the tortoise, but they also had an apparently logically way of describing the process that made it seem like catching the tortoise is impossible.
The solution is in an area of math called Limits, and they were asking all the right questions to get to figuring out Limits, but eventually it took around two thousand years to discover the math to explain what was happening.
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