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.
這位是埃利亞的芝諾, 一位古希臘哲學家, 因發明許多悖論而聞名。 悖論是指看似有道理, 但結論卻是荒謬 或矛盾的論證。 兩千多年以來, 芝諾那些誤導思路的難題 啟發了許多 數學家與哲學家 來了解「無窮」的本質。 最有名的一個芝諾難題 叫做二分法悖論, 在古希臘文的意思就是「切割為二詭辯」。 內容大約是這樣: 芝諾在漫長地坐著沉思一天後, 決定從家裡散步到公園。 清新的空氣啟發他的心靈 並讓他想得更清楚。 要走到公園, 他必須先走到路程的中點。 他這部份的旅程 要花一些有限的時間。 一旦他到達這中點, 他必須再走到剩下距離的中點。 這又花了一些有限的時間。 一旦他到那兒,他還是必須再走到 剩下距離的中點, 那也會花另一些有限的時間。 這會一次又一次的發生。 你可以見到我們 永遠都在這過程打轉, 就是不斷將剩的距離分成 更小更細的路段, 每一段都須要一些 有限的時間才能通過。 所以,芝諾要多久才能走到公園? 嗯,要得到答案,你必須把每段路段 所花的時間加起來。 而問題是,有無限個這種 「有限的時間」。 所以,全部的時間也應該要是無限大嗎? 順帶一提,這個論證是很通用的。 它說明從任何地點移動到 任何其它地點 應該要花無窮的時間。 換句話說,它說明所有運動都是不可能的。 這個結果顯然很荒謬, 但邏輯上的瑕疵在哪裡? 要解開這個悖論, 把故事轉換成數學問題 會有所幫助。 我們假設芝諾的家 距離公園有一英里, 而芝諾每小時走一英里。 常理告訴我們這趟旅程 應該要花一小時。 但是,讓我們從芝諾的觀點來看看 並把路程分程許多小段。 最初的一段路程要花 1/2 小時, 下一段要花 1/4 小時, 而第三段要花 1/8 小時, 以此類推。 將這些時間全部加起來, 我們得到一串 長成這樣的級數。 「現在」,芝諾可能會說, 「因為方程式右邊有無限項, 每項又都是有限的, 它們的總和 應該是無窮,對吧?」 這就是芝諾論證的問題了。 數學家從此明白, 把無限個有限的量相加 是有可能得到 一個有限的答案。 「怎麼會呢?」你可能會問。 嗯,我們可以這樣想。 我們考慮一個 一公尺見方的正方形。 現在把這個正方形 分成兩半, 再把剩的分半, 接著往下做。 當我們這麼做時, 我們依序記錄每塊的面積。 最初的切片有兩部份, 每部份的面積都是 1/2, 而下一次切片把其中一個 1/2 再分成兩半, 依此類推。 但,無論我們切割了幾次, 整塊面積還是所有小面積的總和。 現在你可以了解 為什麼要選這麼特別的方式 來切割正方形。 我們已經做出了那串 相同的無窮級數, 就是在芝諾的旅程中 算出來的那串。 當我們建構了更多的藍色小方塊, 用數學的行話來說, 就是當我們取 n 趨近到無窮時的極限, 整個正方形都被藍色蓋住了。 但正方形的面積就只有 1 平方單位而已, 所以無窮項的總合一定是 1。 我們回到芝諾的旅程, 我們可以看到這悖論 是如何被解決的。 不止是無限項加起來可能是有限, 而且這個有限的答案還是一樣的, 和常理告訴我們的一樣 ── 芝諾的旅程要花一小時。