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乌龟对阿基里斯说 - 刘易斯-卡罗尔

已有 767 次阅读 2021-10-7 15:03 |个人分类:解读哥德尔不完全性定理|系统分类:科研笔记

乌龟对阿基里斯说是刘易斯-卡罗尔(Lewis Carroll1823-1898)于1895年为哲学杂志《心智》撰写的一篇关于逻辑基础的简短寓言式对话,标题暗指芝诺的关于运动的悖论之一,根据这个悖论,阿基里斯永远不可能在与乌龟的比赛中获胜。


在此对话中,乌龟挑战阿基里斯,让阿基里斯用逻辑的力量使乌龟接受一个简单演绎论证的结论。最终,阿基里斯失败了,因为聪明的乌龟把他带入了一个无限倒退(infinite regression)。


一,译文


阿基里斯已经超过了乌龟,并舒服地坐在它的背上。


你已经到达了我们比赛的尽头了,乌龟说,尽管它是由一个系列无限的距离组成的。我以为某些聪明的人已经证明了这件事是做不到的呢? ”


可以做到,阿基里斯说,已经做到了!(Solvitur ambulando)你看,距离在不断缩短;所以,。。。


但如果一直在增加呢? ”乌龟打断了他的话,那怎么办? ”


那我就不应该在这里了,阿基里斯谦虚地回答,而你这时也已经绕地球几圈了! ”


你抬举我,或者压扁我,乌龟说,因为你是个重量级人物,没错! 好了,现在你想不想听一个比赛,大多数人都认为他们两三步就到达终点,而实际上是由无数个距离组成的,每一个距离都比前一个长?


我很乐意! ”希腊勇士说,他从他的头盔(当时很少有希腊战士拥有口袋)里拿出一个巨大的记事本和一支笔。 “继续!请慢慢说! 速记法还没有发明出来呢!


欧几里得的那个漂亮的第一命题!乌龟梦呓般地喃喃自语,你崇拜欧几里得吗? ”


当然! 至少如一个人可以欣赏在未来几个世纪内都不会出现的论著!


好了,现在,让我们来看看第一命题中的论证只是两个步骤,以及从它们得出的结论。请把它们记在你的记事本上。为了方便引述,让我们把它们称为ABZ-

(A) 等于同一事物的事物彼此相等。

(B) 这个三角形的两边等于第三边。

(Z) 这个三角形的两边彼此相等。


我想,欧几里德的读者会同意,Z是由AB的逻辑推导出来的,所以任何接受AB为真的人,必须接受Z为真的?


当然! 连高中里的最小的孩子(高中被发明出来,这要等到大约两千年后)就会同意。


如果有些读者还不接受AB是真的,仍然可能接受这个序列是有效的,我想? ”


无疑这样的读者可能存在。他可能会说我接受假设命题如果AB是真的,Z一定是真的为真;但是,我不接受AB是真的,这样放弃欧几里德的读者,最后改踢足球,才是明智之举。 »


那是不是也有一些读者会说我接受AB是真的,但我不接受假设” 


当然有可能。他也最好去踢足球。


而这二种读者在逻辑上都没有接受Z为真。乌龟继续说。


是这样的,阿基里斯同意了。


好吧,现在,我想让你把我看作是第二种读者,并从逻辑上迫使我接受Z为真。


乌龟踢足球将是, 。。。” 阿基里斯开始说道。


“—当然这不正常的,乌龟急忙打断道,不要跑题,让我们先说Z,然后再说足球!


我要强迫你接受Z,是吗?阿基里斯思索着说,而你现在的立场是,你接受AB,但你不接受假设—”


就叫它C,乌龟说。

“-但你不接受。


(C)如果AB为真,Z一定为真。 


这就是我现在的立场,乌龟说。


那么我必须让你接受C


我会这样做的,乌龟说,只要你把它记在你的笔记本上,笔记本里还有什么?


只有一些备忘录,阿基里斯说,紧张地翻动着书页,几份关于我在战斗中表现出色的备忘录!” 


有很多空白页!乌龟高兴看到,我们都要!” (阿基里斯打了个寒战),现在按我说的写:


(A) 与同一事物相等的事物彼此相等。

(B) 这个三角形的两边等于一边。

(C) 如果AB是真的,那么Z一定是真的。

(Z) 这个三角形的两边彼此相等。


你应该叫它D,而不是Z”,阿基里斯说,它排在其他三个的后面。如果你接受ABC,你必须接受Z


那我为什么必须接受? ”


因为从逻辑上讲,它是由它们得到的。如果ABC为真,Z就一定为真。我想,你对此没有异议吧? ”


如果ABC都是真的,那么Z一定是真的 乌龟若有所思地重复道。这是另一个假设,不是吗?而且,如果我没有看到它为真,我可能会接受ABC,但仍然不接受Z,不是吗? ”


你可能会 这位坦率的英雄承认, “尽管这样的愚昧肯定会是惊人的,不过,这种情况还是有可能发生的。所以我必须要求你再同意一个假设。


很好。只要你把它写下来,我就愿意同意。我们将它称为

  (D) 如果ABC为真,那么Z一定为真。


你记在你的本子上了吗? ”


记了!阿基里斯高兴地喊道,他把铅笔放进了笔鞘里,我们终于要结束这个理想的比赛了!现在你接受了AB,还有Z。既然你接受了ABCD,你当然就接受了Z


我要吗?乌龟无辜地说道,让我们把话说清楚。我接受ABCD,假设我仍然拒绝接受Z


那么逻辑会掐住你的喉咙,强迫你接受!阿基里斯胜利地回答,逻辑会告诉你你无法控制自己。既然你已经接受了ABCD,你就必须接受Z所以你没有选择,你瞧。 ”


无论逻辑能告诉我什么,都值得写下来,乌龟说, “所以,请把它记在你的书上,我们把它叫做:

 (E) 如果ABCD都为真,Z一定为真。


在我认可这一点之前,我当然不需要认可Z,所以这是很必要的一步,你明白吗? ”


我明白了,阿基里斯说,他的语气里有一丝悲伤。


到这里,讲故事的人由于在银行有急事,不得不离开这对快乐的人,直到几个月后才再次经过这个地方,阿基里斯仍然坐在那只耐人寻味的乌龟的背上,并在他的记事本上写着什么,那本子似乎已经快满了。乌龟说: ”你把最后一步写下来了吗?除非我忘了数数,否则那是一千零一个,还将有几百万个。你是否介意,作为个人恩惠,考虑到我们的这次谈话将为十九世纪的逻辑学家提供许多启发,你是否介意采用我的表弟 Mock-Turtle届时会提出的双关语,并允许自己被重新命名为Taught-Us” 


如果你愿意!疲惫的勇士用绝望的空洞音调回答说,他把脸埋在手里,除非你采用 Mock-Turtle从未用过的双关语,并允许自己被重新命名为A Kill-Ease"


二,原文



WHAT THE TORTOISE SAID TO ACHILLES.

By Lewis Carroll.



Achilles had overtaken the Tortoise, and had seated himself comfortably on its back.


"So you've got to the end of our race-course?" said the Tortoise. "Even though it does consist of an infinite series of distances? I thought some wiseacre or other had proved that the thing couldn't be done? »


"It can be done," said Achilles. "It has been done! Solvitur ambulando. You see the distances were constantly diminishing; and so—« 


"But if they had been constantly increasing?" the Tortoise interrupted. "How then? »


"Then I shouldn't be here," Achilles modestly replied; "and you would have got several times round the world, by this time! »


"You flatter me—flatten, I mean," said the Tortoise; "for you are a heavy weight, and no mistake! Well now, would you like to hear of a race-course, that most people fancy they can get to the end of in two or three steps, while it really consists of an infinite number of distances, each one longer than the previous one? »


"Very much indeed!" said the Grecian warrior, as he drew from his helmet (few Grecian warriors possessed pockets in those days) an enormous note-book and a pencil. "Proceed! And speak slowly, please! Short-hand isn't invented yet! »


"That beautiful First Proposition of Euclid!" the Tortoise murmured dreamily. "You admire Euclid? »


"Passionately! So far, at least, as one can admire a treatise that wo'n't be published for some centuries to come! »


"Well, now, let's take a little bit of the argument in that First Proposition—just two steps, and the conclusion drawn from them. Kindly enter them in your note-book. And in order to refer to them conveniently, let's call them A, B, and Z:—


(A) Things that are equal to the same are equal to each other.

(B) The two sides of this Triangle are things that are equal to the same.

(Z) The two sides of this Triangle are equal to each other.


Readers of Euclid will grant, I suppose, that Z follows logically from A and B, so that any one who accepts A and B as true, must accept Z as true? »


"Undoubtedly! The youngest child in a High School—as soon as High Schools are invented, which will not be till some two thousand years later—will grant that. »


"And if some reader had not yet accepted A and B as true, he might still accept the sequence as a valid one, I suppose? »


"No doubt such a reader might exist. He might say 'I accept as true the Hypothetical Proposition that, if A and B be true, Z must be true; but, I don't accept A and B as true.' Such a reader would do wisely in abandoning Euclid, and taking to football. »


"And might there not also be some reader who would say 'I accept A and B as true, but I don't accept the Hypothetical’?" 


"Certainly there might. He, also, had better take to football. »


"And neither of these readers," the Tortoise continued, is as yet under any logical necessity to accept Z as true? »


"Quite so," Achilles assented.


"Well, now, I want you to consider me as a reader of the second kind, and to force me, logically, to accept Z as true. »


"A tortoise playing football would be—" Achilles was beginning


"—an anomaly, of course," the Tortoise hastily interrupted. "Don't wander from the point. Let's have Z first, and football afterwards! »


"I'm to force you to accept Z, am I?" Achilles said musingly. "And your present position is that you accept A and B, but you don't accept the Hypothetical—« 


"Let's call it C," said the Tortoise.

—but you don't accept


(C) If A and B are true, Z must be true. »


"That is my present position," said the Tortoise.


"Then I must ask you to accept C. »


"I'll do so," said the Tortoise, "as soon as you've entered it in that note-book of yours. What else have you got in it? »


"Only a few memoranda," said Achilles, nervously fluttering the leaves: "a few memoranda of—of the battles in which I have distinguished myself! »


"Plenty of blank leaves, I see!" the Tortoise cheerily remarked. "We shall need them all!" (Achilles shuddered.) "Now write as I dictate:—

 (A) Things that are equal to the same are equal to each other.

 (B) The two sides of this Triangle are things that are equal to the same.

 (C) If A and B are true, Z must be true.

 (Z) The two sides of this Triangle are equal to each other. »


"You should call it D, not Z," said Achilles. "It comes next to the other three. If you accept A and B and C, you must accept Z." 


"And why must I? »


"Because it follows logically from them. If A and B and C are true, Z must be true. You don't dispute that, I imagine? »


"If A and B and C are true, Z must be true," the Tortoise thoughtfully repeated. "That's another Hypothetical, isn't it? And, if I failed to see its truth, I might accept A and B and C, and still not accept Z, mightn't I? »


"You might," the candid hero admitted; "though such obtuseness would certainly be phenomenal. Still, the event is possible. So I must ask you to grant one more Hypothetical. »


"Very good. I'm quite willing to grant it, as soon as you've written it down. We will call it

  (D) If A and B and C are true, then Z must be true. 


Have you entered that in your note-book?


"I have!" Achilles joyfully exclaimed, as he ran the pencil into its sheath. "And at last we've got to the end of this ideal race-course! Now that you accept A and B and C and D, of course you accept Z. »


"Do I?" said the Tortoise innocently. "Let's make that quite clear. I accept A and B and C and D. Suppose I still refused to accept Z? »


"Then Logic would take you by the throat, and force you to do it!" Achilles triumphantly replied. "Logic would tell you 'You can't help yourself. Now that you've accepted A and B and C and D, you must accept Z!' So you've no choice, you see. »


"Whatever Logic is good enough to tell me is worth writing down," said the Tortoise. "So enter it in your book, please. We will call it »

   (E) If A and B and C and D are true, Z must be true. 


Until I've granted that, of course I needn't grant Z. So it's quite a necessary step, you see?


"I see," said Achilles; and there was a touch of sadness in his tone.


Here the narrator, having pressing business at the Bank, was obliged to leave the happy pair, and did not again pass the spot until some months afterwards. When he did so, Achilles was still seated on the back of the much-enduring Tortoise, and was writing in his note-book, which appeared to be nearly full. The Tortoise was saying "Have you got that last step written down? Unless I've lost count, that makes a thousand and one. There are several millions more to come. And would you mind, as a personal favour, considering what a lot of instruction this colloquy of ours will provide for the Logicians of the Nineteenth Century—would you mind adopting a pun that my cousin the Mock-Turtle will then make, and allowing yourself to be re-named Taught-Us? »


"As you please!" replied the weary warrior, in the hollow tones of despair, as he buried his face in his hands. "Provided that you, for your part, will adopt a pun the Mock-Turtle never made, and allow yourself to be re-named A Kill-Ease! »


参考文献:

【1】https://en.wikisource.org/wiki/What_the_Tortoise_Said_to_Achilles

【2】https://monsieurphi.com/2017/12/12/lewis-carroll-ce-que-la-tortue-dit-a-achille/






https://blog.sciencenet.cn/blog-2322490-1307071.html

上一篇:塔斯基与《形式化语言中的真理概念》- 第一章
下一篇:解读“乌龟对阿基里斯说” - “无限倒退”(infinite regression)

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