Can You Ever Catch the Tortoise? 你追得上烏龜嗎？

🔊The oldest paradoxes we have come from Zeno of Elea, a Greek of the fifth century BCE. His teacher Parmenides had argued that reality is one and unchanging and that motion is an appearance. Zeno defended him by showing that the commonsense view, on which things move, leads to absurdities of its own. Aristotle preserves the most famous of his arguments: “in a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.”

我們手上最古老的悖論出自埃利亞的芝諾，西元前五世紀的希臘人。他的老師巴門尼德主張，實在是唯一而不變的，運動只是表象。芝諾替老師辯護的方式，是證明常識的看法——東西會動——自己會導出荒謬。亞里斯多德保存了他最有名的一個論證：「在賽跑中，跑得最快的永遠追不上最慢的，因為追的人必須先到達被追的人出發的那一點，所以慢的那個必定一直保持領先。」

🔊Give the slow runner a start of a hundred meters. By the time Achilles has covered the hundred, the other has crept forward ten. By the time Achilles has covered those ten, it is one meter further on, and so without end. At every stage a gap remains. The stages never run out. So it seems that Achilles is always on the point of drawing level and never does.

讓慢的那個先跑一百公尺。等阿基里斯跑完這一百公尺，對方已經往前爬了十公尺。等阿基里斯跑完這十公尺，牠又在前面一公尺了；如此下去，沒有盡頭。每一個階段都還剩一段差距，而階段永遠用不完。於是看起來，阿基里斯永遠就要追平，卻永遠沒有追平。

🔊Chinese thinkers found the same puzzle independently. The last chapter of the Zhuangzi records a saying of the debaters around Hui Shi, in the fourth century BCE: “A stick one foot long: take away half of it every day, and in ten thousand generations it will not be used up.” This is Zeno’s other great argument, the Dichotomy, turned from motion to matter. Before you reach the door you must reach the halfway point, and before that the halfway point of the first half. Run in one direction the argument says you never arrive, and run in the other it says you can never set out.

中國的思想家各自獨立地發現了同一個難題。《莊子》最後一篇記下了西元前四世紀惠施周圍那群辯者的一句話：「一尺之捶，日取其半，萬世不竭。」這是芝諾另一個重要論證「二分法」，只是從運動換成了物質。你走到門口之前，必須先走到一半；在那之前，又必須先走到前一半的一半。這個論證往一個方向推，說你永遠到不了；往另一個方向推，說你根本出發不了。

🔊The standard answer took two thousand years to state properly. Zeno assumes that infinitely many distances, or infinitely many intervals of time, must add up to an infinite amount. They need not. The distances in the race are 100, 10, 1, 0.1 and so on, and the more of them you add, the closer the total comes to 111.11… meters without ever exceeding it. Mathematicians of the nineteenth century, Cauchy foremost among them, made this exact with the concept of a limit. The times behave the same way. If Achilles runs ten meters a second, the stages take 10 seconds, then 1, then 0.1, and all of them together take 11.11… seconds. After that he is in front.

標準答案花了兩千年才被說清楚。芝諾假定，無窮多段距離，或無窮多段時間，加起來一定是無窮大。其實不必。賽跑裡的各段距離是 100、10、1、0.1……，你加進去的項愈多，總和就愈接近 111.11… 公尺，卻永遠不會超過它。十九世紀的數學家，以柯西為首，用「極限」這個概念把這件事說得精確。時間也是同樣的情形。如果阿基里斯每秒跑十公尺，各階段分別花 10 秒、1 秒、0.1 秒……，全部加起來是 11.11… 秒。在那之後，他就在前面了。

🔊Many philosophers think this dissolves the paradox, and many do not. The arithmetic tells us where and when Achilles draws level. It does not obviously tell us how anyone finishes a sequence of tasks that has no last member. In 1954 James Thomson sharpened the worry with a lamp. Switch it on for a minute, off for half a minute, on for a quarter, and so on; after two minutes infinitely many switchings are done. Is the lamp then on or off? It cannot be on, since every time it was turned on it was turned off again, and it cannot be off for the same reason. Thomson concluded that completing infinitely many tasks, a “supertask,” is impossible. His critics replied that the story merely fails to say what happens at two minutes. The argument continues, and what is at stake in it is whether space, time, and motion are really as the mathematics of the continuum describes them.

許多哲學家認為這樣就化解了悖論，也有許多人不這麼想。算術告訴我們阿基里斯在哪裡、在什麼時候追平。可是，一個沒有「最後一項」的任務序列，到底是怎麼被做完的，算術並沒有明白地告訴我們。1954 年，湯姆森用一盞燈把這個疑慮磨得更尖。把燈打開一分鐘，關掉半分鐘，再開四分之一分鐘，如此下去；兩分鐘後，無窮多次開關都做完了。這時燈是亮的還是暗的？它不可能是亮的，因為每一次打開之後都又被關掉了；基於同樣的理由，它也不可能是暗的。湯姆森的結論是：完成無窮多個任務，也就是所謂的「超任務」，是不可能的。批評他的人回答說，這個故事只是沒有交代兩分鐘那一刻發生了什麼。這場爭論還在繼續；而它真正牽涉到的是：空間、時間與運動，是不是真的像連續統的數學所描述的那樣。

🔊A second family of paradoxes needs no racetrack, only a sentence. Consider: “This sentence is false.” If it is true, then what it says is the case, so it is false. If it is false, then what it says is not the case, so it is true. The ancient version is credited to Eubulides, a contemporary of Aristotle, and it is known as the Liar. It is tempting to dismiss the Liar as a trick, and for two thousand years most people did. Then, in the early twentieth century, the same structure of self-reference turned up in the foundations of mathematics. Bertrand Russell found it in the notion of a set, and Kurt Gödel used a relative of the Liar, a sentence that says of itself that it cannot be proved, to show that arithmetic contains truths no consistent system of rules can derive. Alfred Tarski’s response to the Liar was that no language can contain its own truth predicate: to say of a sentence that it is true, one has to step up into a richer language. Ordinary English falls short of that standard, which may mean that ordinary English is, strictly speaking, inconsistent.

第二個家族的悖論不需要跑道，只需要一個句子。請看：「這個句子是假的。」如果它是真的，那麼它說的就是事實，所以它是假的。如果它是假的，那麼它說的就不是事實，所以它是真的。它的古代版本相傳出自與亞里斯多德同時代的歐布利德斯，稱為「說謊者悖論」。把說謊者當成文字把戲打發掉，是很誘人的做法；兩千年來，多數人正是這樣做的。然後在二十世紀初，同一種自我指涉的結構，出現在數學的基礎裡。羅素在「集合」這個概念裡發現了它；哥德爾則用了說謊者的一個親戚——一個說自己「不可被證明」的句子——證明算術裡有些真理，是任何一致的規則系統都推導不出來的。塔斯基對說謊者的回應是：沒有任何語言能包含它自己的「真」這個謂詞；要說一個句子是真的，就得往上跨進一個更豐富的語言。日常的英語達不到這個標準；這也許表示，嚴格說來，日常的英語是不一致的。

🔊Eubulides is also credited with the paradox of the heap, the sorites. One grain of sand is not a heap. Adding a single grain to something that is not a heap never turns it into one. So no number of grains makes a heap. Each premise seems undeniable, the logic is modus ponens repeated, and the conclusion is false. What makes the sorites so hard is that the trouble is everywhere. “Heap” is vague, and so are “tall,” “adult,” “alive,” and “person,” and the same argument can be run on each of them. Law and ethics depend on such words, and on lines that have to be drawn somewhere although no particular place for them can be defended.

相傳「堆垛悖論」（sorites）也出自歐布利德斯。一粒沙不是一堆。在不成堆的東西上加一粒沙，絕不會讓它變成一堆。所以不管多少粒沙，都不成一堆。每一個前提看來都無法否認，邏輯只是反覆使用肯定前件，而結論是假的。堆垛悖論之所以這麼難，是因為問題無所不在。「堆」是模糊的，「高」「成年」「活著」「人」也都是；同一個論證可以套在每一個詞上。法律與倫理都仰賴這類詞語，也仰賴那些總得畫在某處的界線；儘管沒有哪一個特定的位置能被辯護。

🔊It would be a mistake to file all this under puzzles. A paradox is an argument that our own concepts make against us. It shows that what we believe about motion, or truth, or the meaning of ordinary words cannot all be right together, and it does so using nothing except those beliefs. The great ones are therefore never simply solved. They are answered by changing something: by inventing the limit, or separating a language from its metalanguage, or by rethinking what a set is. With this the first unit ends, on a caution. Logic can show us that our premises conflict. It does not tell us which of them to give up.

把這一切歸到「益智謎題」那一類，會是個錯誤。悖論是我們自己的概念對我們提出的論證。它證明我們對運動、對真、對日常詞語意義的種種信念，不可能同時全部正確；而且它除了這些信念之外，什麼都沒用到。所以偉大的悖論從來不是單純被「解開」的。回應它們的方式，是改變某樣東西：發明極限，或把一個語言跟它的後設語言分開，或重新思考集合是什麼。第一個單元就在這裡結束，結束在一句提醒上：邏輯能告訴我們，我們的前提互相衝突；它不告訴我們該放棄哪一個。