Zhi Nuo tried to prove the theory of Elijah School, namely? how much is it? With what? Change? It's fake, inseparable? One? And then what? Static existence? Is the only real thing, exercise is just an illusion. So he designed four examples called Zeno Paradox. These paradoxes are all put forward from the philosophical point of view. The most famous one is: Achilles (a good runner in ancient Greek mythology) can't outrun the tortoise? The problem can be explained by the concept of calculus, but it cannot be solved by calculus.
Zhi Nuo: How did the tortoise defeat Achilles?
Achilles challenged the tortoise to a race. Achilles, a runner known as Scud, knows the speed disadvantage of his opponent tortoise. He let the tortoise run 65,438+000 yards first. He entered the race at 65,438+00 times the speed of the tortoise, which should be enough to ensure his victory.
At the beginning of the game, when Achilles ran to the place where the tortoise's starting point was 100 yards away from his starting point, the tortoise had climbed 10 yards. When he ran 10 yard, the tortoise climbed 1 yard again. When Achilles ran 1 yard again, the tortoise was still ahead of 0. 1 yard ... To Achilles' surprise, this situation continued, and the tortoise was always ahead. Although the distance between them is decreasing by 0. 1 code, 0.0 1 code, 0.00 1 code ... will never be 0, because any length of distance can be divided by 10 indefinitely.
So the strange conclusion is: if the slow tortoise leads by a certain distance, then the fast Achilles will never catch up with the tortoise. This paradox is also called Achilles paradox.
The crux of Zeno paradox lies in infinity and finiteness.
The problem of Achilles' pursuit seems to be seamless in philosophy. But mathematically, the time required for catching up can be calculated by summing infinite series or simply establishing equations. So what reason do we have to say that Achilles will never catch up with the tortoise?
The problem is: if Achilles finally catches up with the tortoise, time can be found. However, the essence of Zeno's paradox is to prove how to catch up, because it is impossible to complete the infinite steps of repeated cycle in a limited time.
The solution of mathematics is from the result to the process: there is nothing wrong with the logic of paradox itself, and it is far from reality because Zhi Nuo and we adopted different time systems. Everyone is used to treating motion as a continuous function of time, while Zhi Nuo adopted a discrete-time system. That is, no matter how small the time interval is, the whole time axis is still composed of infinite time. In other words, continuous time is discrete time and the time interval is regarded as the limit of infinitesimal.
So the crux of Achilles paradox lies in whether the infinite sum is finite, and whether the infinite sum is finite.
Math: How Achilles caught up with the tortoise.
Zeno's paradox holds that one of the reasons why Achilles can never catch up with the tortoise is that in order to catch up with the tortoise, he has to complete infinite steps to run 100 yards, 10 yards, 1 yards, 0. 1 yards, etc. It also believes that nothing can complete an infinite number of steps in a limited time. In other words, completing an infinite number of steps means never catching up with the tortoise.
However, it can be done mathematically. Because the sum of series without ending is constant.
100+ 10+ 1/ 10+ 1/ 100+? = 1000/9 code
The result is not infinite, but a finite value. 1000/9 is the point where Achilles catches up with the tortoise. Achilles can complete an infinite number of steps in a limited time, because the time required for each successive step is getting smaller and smaller.
In time, let's assume that Achilles' speed is 10 yards/second, and the tortoise's speed is 1 yards/second. Then 100/9 seconds, it's time for Achilles to catch up with the tortoise. 100/9 seems to be divided into infinite time intervals, and time is endless, but it is not. The motion of an object does not lie in many discrete intervals, time is smooth and continuous, and the sum of its series is constant.
10+ 1+ 1/ 10+ 1/ 100+? = 100/9 seconds
Although it is infinite time, its interval is getting shorter and shorter, and the sum of its infinite series is also a finite value.
Concluding remarks
Zhi Nuo's so-called Achilles paradox does not exist, but what about people? Sure, okay? Hallucinations. We fall into the trap because we don't have enough understanding of world artifacts.
Like Zhi Nuo's other dichotomy paradox, flying arrow paradox and racing paradox, Achilles paradox is wrong in philosophy, but Zhi Nuo pointedly put forward the question of whether space and time are continuous or discrete, which has aroused long-term discussions among philosophers and mathematicians and made great contributions to the development of mathematics and philosophy.