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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The paradox is that you have one line of reasoning that shows that Achilles will never reach the Tortoise, and another line of reasoning that shows that Achilles will eventually reach the Tortoise.
If both lines of reasoning are correct, you get a contradiction. But it’s not obvious where the mistake is, hence being called a paradox.
The resolution to this paradox is the realization that an infinite series can have a finite sum. That is, the first line of reasoning shows that it will take an infinite number of steps for Achilles to reach the Tortoise, but since each step gets shorter, it can be done in a finite amount of time.
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.
We use the word paradox in a couple different ways. One is in logical contradictions, i. e. “This sentence is false.” It is impossible to have the statement be either true or false, so it is a paradox. “A married bachelor” is another paradoxical statement, as you are definitionally bound by the word “bachelor” to be unmarried.
Another way we use paradoxes is in seemingly contradictory statement that do actually have an answer. Zeno’s paradox falls in this category. Like you say, there is the definite answer that Achilles does catch the tortoise, the seemingly contradictory part is the description that you have to cover an infinite number of ever smaller but never 0 distances to go anywhere, so how can you move?
The answer being that the time it takes to move those smaller increments also decreases to 0 at exactly the same rate.
The paradox is that you have one line of reasoning that shows that Achilles will never reach the Tortoise, and another line of reasoning that shows that Achilles will eventually reach the Tortoise.
If both lines of reasoning are correct, you get a contradiction. But it’s not obvious where the mistake is, hence being called a paradox.
The resolution to this paradox is the realization that an infinite series can have a finite sum. That is, the first line of reasoning shows that it will take an infinite number of steps for Achilles to reach the Tortoise, but since each step gets shorter, it can be done in a finite amount of time.
Imagine that I gave you a task:
“Start counting up in integers (1, 2, 3, etc.). When you finish counting all the integers then you can have a slice of cake.”
When are you going to have your slice of cake? You are never going to run out of integers since you can keep counting up for an infinite period of time, so in theory you should never reach the point where you can have the cake, right?
Now imagine that as you are counting the time between 1 and 2 takes a second, but you start speeding up so that the time from 2 to 3 is only a half second, from 3 to 4 a quarter of a second, etc. Conceptually this doesn’t matter since we didn’t really care about how quickly you were counting in the first example, as the issue of the integers being infinite was the real issue. But in this case somehow you “finish” and eat the cake?
This is the idea behind Zeno’s Paradox. In order for the hare to pass the tortoise it must first reach the tortoise, and in order to do that it must reach half the distance between it and the tortoise. If it reaches the halfway point then it must next reach the new halfway point, and when it does that reach the *new* new halfway point, etc. In concept this cycle can be continued infinitely since distance is infinitely divisible and so there are equally an infinite number of steps to this task as there are integers in the previous examples. Yet in this case we know the hare will be able to pass the tortoise. Somehow an infinite series of tasks was completed in finite time.
The reason this is a “paradox” is that the logic seems irrefutable although our common sense tells us otherwise. It isn’t a true paradox because it isn’t a logical contradiction but rather the reasoning seems to go against common sense.
To actually show why this isn’t a true paradox involves understanding infinite series. We can build an infinite series out of the sequence 1, 1/2, 1/4, 1/8… Now every term of the sequence is positive. “Logically” adding all the terms would result in “infinity” as there are an infinite number of positive numbers added together.
It actually isn’t obvious to a person not familiar with infinite series, why this “logic” isn’t true.
Why should we assume their speed is different if they are both advancing one point at a time? I feel like I’m missing something.
Why should we assume their speed is different if they are both advancing one point at a time? I feel like I’m missing something.
We use the word paradox in a couple different ways. One is in logical contradictions, i. e. “This sentence is false.” It is impossible to have the statement be either true or false, so it is a paradox. “A married bachelor” is another paradoxical statement, as you are definitionally bound by the word “bachelor” to be unmarried.
Another way we use paradoxes is in seemingly contradictory statement that do actually have an answer. Zeno’s paradox falls in this category. Like you say, there is the definite answer that Achilles does catch the tortoise, the seemingly contradictory part is the description that you have to cover an infinite number of ever smaller but never 0 distances to go anywhere, so how can you move?
The answer being that the time it takes to move those smaller increments also decreases to 0 at exactly the same rate.
Imagine that I gave you a task:
“Start counting up in integers (1, 2, 3, etc.). When you finish counting all the integers then you can have a slice of cake.”
When are you going to have your slice of cake? You are never going to run out of integers since you can keep counting up for an infinite period of time, so in theory you should never reach the point where you can have the cake, right?
Now imagine that as you are counting the time between 1 and 2 takes a second, but you start speeding up so that the time from 2 to 3 is only a half second, from 3 to 4 a quarter of a second, etc. Conceptually this doesn’t matter since we didn’t really care about how quickly you were counting in the first example, as the issue of the integers being infinite was the real issue. But in this case somehow you “finish” and eat the cake?
This is the idea behind Zeno’s Paradox. In order for the hare to pass the tortoise it must first reach the tortoise, and in order to do that it must reach half the distance between it and the tortoise. If it reaches the halfway point then it must next reach the new halfway point, and when it does that reach the *new* new halfway point, etc. In concept this cycle can be continued infinitely since distance is infinitely divisible and so there are equally an infinite number of steps to this task as there are integers in the previous examples. Yet in this case we know the hare will be able to pass the tortoise. Somehow an infinite series of tasks was completed in finite time.
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