This is Zeno of Elea, an ancient Greek philosopher famous for inventing a number of paradoxes, arguments that seem logical, but whose conclusion is absurd or contradictory. For more than 2,000 years, Zeno's mind-bending riddles have inspired mathematicians and philosophers to better understand the nature of infinity. One of the best known of Zeno's problems is called the dichotomy paradox, which means, "the paradox of cutting in two" in ancient Greek. It goes something like this: After a long day of sitting around, thinking, Zeno decides to walk from his house to the park. The fresh air clears his mind and help him think better. In order to get to the park, he first has to get half way to the park. This portion of his journey takes some finite amount of time. Once he gets to the halfway point, he needs to walk half the remaining distance. Again, this takes a finite amount of time. Once he gets there, he still needs to walk half the distance that's left, which takes another finite amount of time. This happens again and again and again. You can see that we can keep going like this forever, dividing whatever distance is left into smaller and smaller pieces, each of which takes some finite time to traverse. So, how long does it take Zeno to get to the park? Well, to find out, you need to add the times of each of the pieces of the journey. The problem is, there are infinitely many of these finite-sized pieces. So, shouldn't the total time be infinity? This argument, by the way, is completely general. It says that traveling from any location to any other location should take an infinite amount of time. In other words, it says that all motion is impossible. This conclusion is clearly absurd, but where is the flaw in the logic? To resolve the paradox, it helps to turn the story into a math problem. Let's supposed that Zeno's house is one mile from the park and that Zeno walks at one mile per hour. Common sense tells us that the time for the journey should be one hour. But, let's look at things from Zeno's point of view and divide up the journey into pieces. The first half of the journey takes half an hour, the next part takes quarter of an hour, the third part takes an eighth of an hour, and so on. Summing up all these times, we get a series that looks like this. "Now", Zeno might say, "since there are infinitely many of terms on the right side of the equation, and each individual term is finite, the sum should equal infinity, right?" This is the problem with Zeno's argument. As mathematicians have since realized, it is possible to add up infinitely many finite-sized terms and still get a finite answer. "How?" you ask. Well, let's think of it this way. Let's start with a square that has area of one meter. Now let's chop the square in half, and then chop the remaining half in half, and so on. While we're doing this, let's keep track of the areas of the pieces. The first slice makes two parts, each with an area of one-half The next slice divides one of those halves in half, and so on. But, no matter how many times we slice up the boxes, the total area is still the sum of the areas of all the pieces. Now you can see why we choose this particular way of cutting up the square. We've obtained the same infinite series as we had for the time of Zeno's journey. As we construct more and more blue pieces, to use the math jargon, as we take the limit as n tends to infinity, the entire square becomes covered with blue. But the area of the square is just one unit, and so the infinite sum must equal one. Going back to Zeno's journey, we can now see how how the paradox is resolved. Not only does the infinite series sum to a finite answer, but that finite answer is the same one that common sense tells us is true. Zeno's journey takes one hour.
这是埃利亚的芝诺 一个古希腊哲学家 以发现了许多悖论而著名 这是指,一些看上去逻辑合理 但是结论却很荒谬或者自相矛盾的论证 两千多年来 芝诺具有欺骗性的谜题们 启发了数学家和哲学家们 更好地理解了“无穷”的本质 芝诺最著名的悖论之一 叫做两分法悖论 它在古希腊语中的意思是“分成两份的悖论” 它是这么说的 闲坐着思考了一天之后 芝诺决定从他的家走去公园 清新的空气能够使他的大脑更清醒 帮助他更好地思考 为了到达公园 他首先需要走完整段路程的前半段 这一段路程 将花费他一段有限的时间 当他到达整段路程的中点时 他又需要走完剩下路程的一半 同样的,这将花费他有限的一段时间 当他到达剩下路程的中点时,他还需要走 剩下路程的前半段 这又将花费他一段有限的时间 这个过程将会一次一次又一次地发生 你可以发现,我们可以无限地这样推导下去 将剩下的不论多少路程 分割成越来越短的路程 每一段都将花费他一段有限的时间 那么,芝诺到达公园要花多长时间? 要知道这个答案 你得将每一小段所花的时间加起来 问题是,有无限多个像这样有限长度的小段 那么,总时间不应该是无穷大吗? 顺便说一下,这个论题非常常见 它说的是从任何一个地点移动到任何另一个地点 需要花费无穷长的时间 换句话说,它的意思是,任何移动都是不可能实现的 这个结论显然很荒谬 但是,逻辑的瑕疵在哪呢? 为了解决这个悖论 把这个故事还原成一个数学问题会有所帮助 我们假设芝诺的家离公园有一英里 芝诺走路的速度是一英里每小时 常识告诉我们,整段路程的时间 应该是一小时 但是,让我们从芝诺的角度来看这个问题 把这整段路程分成许多小段 最先一半路程花费1/2小时 之后的一段花费1/4小时 第三段花费1/8小时 以此类推 把这些时间加起来 我们得到一个像这样的数列 “现在”,芝诺也许会说 “因为等式的右边 有无限项 而且每一项都是有限的 那么它们之和应该是无穷大,对吗?” 这就是芝诺论证的问题所在 数学家们后来发现 将无限个有限项加总 是有可能依然得到一个有限的数字的 “为什么?”你可能会问 让我们这样想一想 让我们从这个正方形开始,它的面积是1个单位 现在把这个正方形切成两半 然后再把剩下的一半切成两半 以此类推 当我们这么做的时候 让我们算一下每一部分的面积 第一刀分成了两份 每一份的面积是1/2 第二刀将其中的一份切成了两半 以此类推 但是,不论我们切多少次 总面积都是所有小份的面积之和 现在你可以看出我们为什么要用这样一种 切割正方形的方法 我们得到了和芝诺的路程 一样的无穷项的数列 当我们切割出一个又一个蓝色矩形的时候 用数学的行话来说 当n趋近于无限大时 整个正方形将被蓝色覆盖 但是正方形的面积就是一个单位 所以这无限项之和一定等于1 再回到芝诺的路程 我们现在就知道悖论怎么解开了 不仅仅是,无限项之和可以是有限的 这个有限的结果 还跟常识告诉我们的是相等的 芝诺的路程将花费一个小时