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第 39 期 · 核心追问 Ⅰ·04「数学是发现，还是发明？」
MIT Godel Escher Bach Lecture 3
节目发布 2012-12-02章节 · 点击跳转视频
0:00
开场与第四章三关键词回顾
1:28
一致性：矛盾可推出一切
4:23
完备性：真但不可证的真理盒子
8:15
希尔伯特纲领与集合论悖论
11:49
1+1=2 比天空是蓝的更真吗
14:17
哥德尔两条不完备性定理
17:06
说谎者悖论、自指与《数学原理》
21:42
哥德尔句：可证→真，真↛可证
24:37
哥德尔编号：让数论谈论自己
28:26
解释与几何：欧氏与非欧
37:12
若数学根基动摇，论证何以立足
39:49
《小和声迷宫》：嵌套层级与「我」
43:08
微积分、耶稣会与无穷的驯服
49:21
无穷的层级：ω、ℵ₀ 与对角线
53:18
递归与智能：深蓝对卡斯帕罗夫
56:55
预告第六章：Gavagai 与意义理论
本期小问 · 档案清单
28:26
数学是发现，还是发明？
56:55
会说话，就等于理解吗？
39:49
什么是「我」？
11:49
真理是客观，还是共识？
本期讲者
Justin CurryMIT 本科生时于 2007 年在 MIT 高中生暑期项目中主讲《哥德尔、埃舍尔、巴赫》读书课，课程由 MIT OpenCourseWare 发布；后成为纽约州立大学奥尔巴尼分校数学系教授，研究拓扑数据分析与层论。
01开场与第四章三关键词回顾
0:00
the following content is provided under a Creative Commons license your support will help MIT open courseware continue to offer high quality educational resources for free to make a donation or view additional materials from hundreds of MIT courses visit MIT opencourseware at ocw.mit.edu all right guys I'm going to go ahead and get started here um sorry about missing last time I had to go home and uh visit visit the family it's been couple months um we're going to do kind of a today's going to be kind of a review session of a bunch of things and we're also going to uh uh go through a dialogue probably my favorite dialogue maybe two um the little harmonic Labyrinth um but I want to start out with uh kind of just entertaining any questions that people might have burning to ask me right away confusion over the past two lectures or or what else what have you anything all right I'm sure questions will develop all right so chapter 4 which I ask you guys to have read for the previous lecture even though you had
以下内容依据知识共享许可协议提供，您的支持将帮助 MIT 开放课程继续免费提供高质量的教育资源。如需捐赠或查看来自数百门 MIT 课程的更多资料，请访问 MIT 开放课程网站 ocw.mit.edu。好了各位，我这就开始了，嗯，抱歉上次缺席，我得回家一趟，去看看家人，已经好几个月没回去了。今天我们要做的，算是对之前一堆内容的复习，然后我们还要过一遍一段对话，可能是我最喜欢的对话，也许会讲两段，嗯，就是《小小和声迷宫》。不过我想先花点时间，回答一下大家可能急着想问我的问题，对前两次课有什么困惑，或者别的什么，有什么想问的吗？好吧，我相信问题之后会冒出来的。好，那么第四章，我让大家在上一节课之前读完的那一章，尽管你们已经上了一整节讲递归的课，这一章讲的是三件事：
1:11
an entire lecture on recursion is about three things consistency
一致性、
02一致性：矛盾可推出一切
1:28
completeness and and geometry uh I'm just going to quickly kind of Define these three terms as much as they can be defined um and then kind of lead into what the whole point of uh chapter 4 was about and what the cont crossa punctus was trying to introduce you to and that's really kind of getting these two to go to girdles theorem so can anyone tell me what consistency means anyone sure said is like there like no DS cont each other yeah exactly so what Felix said was no theorems contradict each other so basically if if we were to put this in terms of a formal system if we were deriving things if we were playing with mu or or whatever whatever formal system we had and we happen to derive a proposition P we couldn't somehow simultaneously derive from our set of axioms p and this means and not P that being not P so basically if we had a set of operating assumptions and we were trying to somehow formally predict today's weather and some somehow the computer spit out well today it's going to rain and not
完备性，以及几何。嗯，我打算快速地把这三个术语尽可能地定义一下，然后再引出第四章到底想说什么，以及那段《螃蟹卡农》想向你们介绍的东西，这些其实都是在把这两点引向哥德尔定理。那么有谁能告诉我，一致性是什么意思？有人吗？当然，他说的是，就像没有定理互相矛盾，对，完全正确，Felix 说的是，没有定理互相矛盾。所以基本上，如果我们用形式系统的话来说，如果我们在推导东西，如果我们在玩mu 或者不管我们有的是什么形式系统，我们碰巧推导出了一个命题 P，而我们不可能同时从我们的公理集合中推导出公理 P，同时又推导出非 P，也就是 not P，所以基本上，如果我们有一组运作假设，我们试图用某种方式形式化地预测今天的天气，结果计算机吐出来的是：今天会下雨，并且不会下雨，
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rain at the same time um that would be kind of an example of an inconsistent system see things where you derived a contradiction directly um and the thing which is really interesting about this and I'm not going to go into all the details of it but if your formal system produces anywhere in it a contradiction you can derive anything and this is really kind of aad thing because uh and a lot of philosophers have grappled with this question why is it that if if we derive a statement say you know this table is red and not red how can we deduce from that that the universe is infinite right but somehow when you have a contradiction everything goes Haywire you can derive anything and problems abound um and this is going to be one of the kind of things which girdle's theorem girdles two theorems um and two incompleteness theems will tell us about but of course I use the word incompleteness what does completeness mean and this is kind of a harder concept to get across and it's really really counterintuitive at first um and it's
同时成立，嗯，那就算是一个不一致系统的例子了，就是那种你直接推导出矛盾的情形，嗯，而这里面真正有意思的一点是——我不打算把所有细节都讲一遍——但如果你的形式系统里任何一个地方产生了矛盾，你就可以推导出任何东西，这真是一件挺离谱的事情，因为呃，很多哲学家都纠结过这个问题：为什么如果我们推出一个陈述，比如说，这张桌子是红的并且不是红的，我们怎么能从中演绎出宇宙是无限的呢？对吧？但不知怎么的，一旦你有了矛盾，一切就乱套了，你可以推导出任何东西，问题层出不穷，嗯，而这将会是哥德尔定理——哥德尔的两个定理，嗯，两个不完备性定理——要告诉我们的内容之一。不过我当然用了“不完备”这个词，那“完备”是什么意思呢？这是个比较难讲清楚的概念，而且一开始真的非常反直觉，嗯，这也是
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what something I want to talk about today does anyone have a good definition
我今天想谈的东西。有没有人能给出一个好的定义，
03完备性：真但不可证的真理盒子
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completeness any takers Sandra do you have an idea okay um um that's all right so completeness is I think probably one of the hardest Concepts and it has to go back to go back to a picture which I which I drew hello go ahead and come on in I've Got Hand out for you um so completeness goes back to a picture I drew on the first day of lecture and I don't know if you guys remember it but it actually appears in a chapter which I didn't assign you all to read um inside idea if we had a truth box and this truth box was was somehow a a graphical display of all of our theorems and the things we could TR prove but things we also knew to be true um so let's take this to be the uh the true box and let's take this to be the not true box and as we talked you know in the past couple days if we had if we had some axioms you know kind of principal points to start building truths from um we can we can derive all sorts of uh all sorts of ideas from these from this these axioms um and this is just kind of a
关于完备性？有人想试试吗？Sandra，你有想法吗？好的，嗯，嗯，没关系，完备性我觉得大概是最难的概念之一，而且它得回到，回到我之前画的一张图，嗯，你好，进来吧，进来吧，我这儿有讲义给你。嗯，所以完备性要回到我第一节课上画的那张图，我不知道你们还记不记得，它其实出现在某一章里，那一章我没有布置给你们读，嗯，里面这个想法：假设我们有一个"真理盒子"，这个真理盒子以某种方式用图形展示出我们所有的定理，以及我们能够证证明的东西，但也包括我们知道为真的东西，嗯，那我们就把这里当作"真"的盒子，把这里当作"非真"的盒子。就像我们前几天讲的，如果我们有一些公理，你知道，就是一些出发点，用来开始构建真理，嗯，我们就可以从这些公理推导出各种各样的想法，从这些、从这些公理出发，嗯，这只是一棵有点奇怪的推演树，对吧，就像我们当时玩
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weird graphical tree of of deductions right just like when we were playing with the MU system um we we started with Mi and then that was our Axiom and then we applied all of our rules of inference to get all the possible theorems here and and these are things which are which are provable sorry if this is incomprehensibly small but it says provable um and but then we have all this space here which which we already said we're true things but provable things and this is really kind of a counterintuitive idea um and it really some of you might go yeah that's exactly captures one my what I feel um and that's the idea that there are truths things we know to be true which aren't provable and this is really kind of hard to wrap your head around like suddenly we've got we have things which we know are true but how do we know they are true it's not that we have a proof of them we just know that they are true and this really is going to go into girdle's theorem big time girdle's two incompleteness theorems um and so with
MU 系统的时候，嗯，我们从 MI 开始，那就是我们的公理，然后我们应用所有的推理规则，得到这里所有可能的定理，而这些都是可证明的东西。抱歉这写得小到看不清，但上面写的是"可证明"，嗯，但接下来我们还有这一整片空间，我们刚才说过这里是真的东西，但不可证明的东西。这真的是一个相当反直觉的想法，嗯，你们有些人可能会说，是啊，这正好说中了我的感受，嗯，那就是这个想法：存在一些真理，一些我们知道为真、却无法证明的东西。这真的挺让人绕不过弯来的，好像突然之间我们有了一些我们知道为真的东西，但我们怎么知道它们是真的呢？并不是因为我们有它们的证明，我们就是知道它们是真的，这一点会大大地引出哥德尔定理，哥德尔的两条不完备性定理。嗯，说到完备性，如果你想给它一个
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completeness if you want to give it a give it a short definition is that uh every true system derivable in the is derivable from the aums are from the system there there is no incompleteness here based on this graphical drawing here we've got obvious incompleteness we have all this space we have all these true statements which aren't re reachable from our axioms um Can anyone think of something which might be true but not provable or they have an idea go ahead uh there's anything outside the universe there's anything outside the universe exactly um that's that's one idea I I was actually just reading the other day in Seth Lloyd's uh programming the universe that the Universe if we kind of look at it expanded like so from a big bang but the rate at which it expands is four times the speed of light of course the only things we can perceive travel as Fe fast as the speed of light so we've got kind of this light cone so everything inside this shaded region are things which we can perceive but ultimately the universe is
简短的定义，那就是：每一个真的命题都可以从公理推导出来，从这个系统推导出来，这里就没有不完备性。而根据这张图，我们这里有明显的不完备性，我们有这一整片空间，我们有所有这些真命题，它们无法从我们的公理到达。嗯，有谁能想到什么东西可能是真的但不可证明的？或者有什么想法？说吧。呃，宇宙之外的任何东西。宇宙之外的任何东西，没错，嗯，这是一个想法。我前几天正好在读 Seth Lloyd 的《编程宇宙》，说宇宙，如果我们这样看，它从大爆炸这样膨胀开来，但它膨胀的速率是光速的四倍。当然，我们唯一能感知到的东西是以光速传播的，所以我们就有了这样一个光锥，阴影区域里的一切都是我们能感知到的东西。但归根结底宇宙膨胀得比这更快，所以关于宇宙有各种各样的
04希尔伯特纲领与集合论悖论
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expanding faster than that so there's all sorts of these things about the universe which we'll never know and that that's that's kind of a that's a nice physical example but it's not exactly what I mean in terms of formal systems and completeness and things of being true but not provable um and this is going to really kind of this is the heart of girdle theem so before we get there I I just want to talk briefly about geometry um we're going to meet it in kind of a variety of two settings in the chapter you met kind of ukian nonukan and I want to elaborate on that and just show you some of the cool things so I'm going to have kind of ukian and not ukian but we'll get back to that right now I kind of want to harp on what girdle's theorem is who this guy Kurt girdle was and why it is that we have one of our three titles names uh named after him so uh Kurt girdle was born in Via um he was a mathematician and he grew up in in a time where where mathematics was being directly influenced by really a
东西是我们永远都不会知道的。这算是一个很好的物理例子，但这并不完全是我想说的意思，就形式系统、完备性以及"为真但不可证明"而言，嗯，这一点真的会——这就是哥德尔定理的核心。在讲到那儿之前，我想先简单聊聊几何，嗯，我们会在两种情境下遇到它。在那一章里你们接触到了欧氏几何和非欧几何，我想再展开讲讲，给你们看一些很酷的东西。所以我要分成欧氏的和非欧氏的，不过我们待会儿再回到这个。现在我想先反复强调一下哥德尔定理是什么，这位库尔特·哥德尔是谁，以及为什么我们书名里三个名字之一是以他命名的。呃，库尔特·哥德尔出生在维也纳，嗯，他是一位数学家，他成长的年代，数学正被各种冒出来的悖论
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variety of paradoxes which were popping up and a man named David Hilbert who really wanted to clear up all these ideas of paradoxes which arrived in mathematics he really felt that David Hilbert felt that mathematics was our most sure and certain source of knowledge that if there was any flaws in mathematics we were doomed as far as human beings in terms of knowing true things um and the paradoxes which I speak of really refer to two main things one is this issue kind of ukian versus not ukian this was a revolution which started happening in the 1800s and people slowly started dealing with over those hundred years entering 1900 um but then there was right towards the turn of the century the two set theory paradoxes and these were paradoxes which I talked about earlier um and that was the idea of the the barber Paradox and for those of you who aren't here for the first lecture the barber paradox says suppose we have a town where a barber shaves all people and only those people who don't shave themselves well then who
直接影响着。还有一个人叫大卫·希尔伯特，他非常想澄清所有这些出现在数学中的悖论问题。他真的觉得——大卫·希尔伯特觉得数学是我们最确定、最可靠的知识来源，如果数学中存在任何缺陷，那我们人类在认识真理这件事上就完蛋了。嗯，我说的那些悖论主要指两件事，一是欧氏几何与非欧几何的问题，这是一场从 1800 年代开始的革命，人们在那一百年里慢慢地应对它，一直到 1900 年，嗯，然后就在世纪之交前后，出现了两个集合论悖论，这些悖论我之前讲过，嗯，那就是理发师悖论。对那些没来上第一节课的同学，理发师悖论是说：假设有一个镇子，镇上的理发师给所有不给自己刮胡子的人刮胡子，而且只给这些人刮。那么谁给理发师刮胡子呢？理发师给自己刮胡子还是不刮？按照定义，按照我们设定这个镇子的方式，
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does does the barber shave himself or does he not and kind of by the definition by the way we set up the town it appears to be a contradiction because if the barber does shave himself then according to who the barber shaves he doesn't shave himself and if he does shave himself then if he doesn't shave himself then he should so it's contradiction um and this was really we thought that this was a paradox deriving from set theory we felt that set theory was going to be our sure and certain Foundation of knowledge we thought we could deal with these paradoxes David Hilbert was huge he said guys look there is no unknown mathematics has to have a sure and certain foundation and we should be able to establish as kind of a model system consistency and completeness of number Theory and he felt that number theory if anything which is true it's got to be number Theory and I I always like doing this poll but I want to I want to ask you guys to vote um and I want you to decide the truth of the following the sky sorry the sky is
这看起来是个矛盾，因为如果理发师给自己刮胡子，那么按照理发师给谁刮胡子的规定，他就不给自己刮胡子；而如果他不给自己刮胡子，那他就应该给自己刮，所以这是矛盾。嗯，当时我们真的认为这是一个源自集合论的悖论，我们觉得集合论本来会是我们确定可靠的知识基础，我们以为我们能处理这些悖论。大卫·希尔伯特影响力很大，他说：各位，看，不存在不可知的东西，数学必须有一个确定可靠的基础，我们应该能够确立一个模范系统——数论的一致性和完备性。他觉得数论，如果说有什么东西是真的，那非数论莫属。我一直喜欢做这个投票，我想问你们大家投个票，嗯，我想让你们判断以下命题的真：天空——抱歉，天空是蓝的，以及 1 + 1 等于
051+1=2 比天空是蓝的更真吗
11:49
blue and 1 + 1 equs 2 so imagine of all the Poss possible worlds um which do you feel like is more true the fact that the sky is blue or that 1+ 1 equal 2 the second one so so you you feel like the second one should be true do you feel like it's true in all possible worlds so do so you don't think there's any Universe out there where where 1 plus one could not equal two okay what about anybody else does anyone feel like no come on this isn't even perceptual what is this statement about it's about these abstract entities which I just kind of created and wrote down on a piece of paper it has absolutely no perceptual Foundation the sky is blue I look outside and what do I see but I see the sky is blue so what ises that so surely what I see has to be more true than this does any is anyone willing to defend sky is blue over 1 plus 1 equals 2 your senses true so Renee Dart was was huge on this he was like What if this is all a dream you know what if this is the Matrix Neo right like this this is
2。想象一下所有可能世界，嗯，你们觉得哪一个更真：天空是蓝的这个事实，还是 1 + 1 = 2？第二个。所以你觉得第二个应该是真的，你觉得它在所有可能世界里都为真吗？所以你不认为存在某个宇宙，在那里 1 加 1 可以不等于 2？好，其他人呢？有没有人觉得——不是吧，这甚至都不是感知层面的东西。这个命题说的是什么？它说的是那些抽象的对象，是我随手创造出来、写在一张纸上的。它完全没有任何感知上的基础。天空是蓝的——我往外一看，我看到了什么？我看到天空是蓝的。所以这是什么呢？那我看到的东西肯定应该比这个更真才对啊。有没有人愿意为"天空是蓝的"辩护，认为它比 1 + 1 = 2 更真？你的感官是真的。所以勒内·笛卡尔在这一点上影响很大，他说：万一这一切都是一场梦呢？你知道，万一这就是《黑客帝国》呢，尼奥？对吧，这
13:05
exactly what he said but what if we were to base our arithmetic on on something else suppose we based our arithmetic on raindrops so suppose we have one raindrop and another one and we we Define addition as when they meet so one raindrop plus another raindrop just gives me another raindrop so in this system 1+ 1 equal 1 um and um what's wrong I mean this is perceptually validated when I'm driving down my car you know 60 mes an hour and I have rain hitting my windshield and I see raindrops merging together maybe you like should like measure the size of it or something okay exactly so there's all sorts of problems with identity there's problems maybe with how the system is formed but you know you know even in mathematics we have to be very clear with what we're stating what we're what kind of uh field we're working over here um cuz space suppose I'm actually working over the integers mod 2 modular arithmetic and that just says if it's divisible by something I say it's zero so when I do modular arithmetic when I
正是他说的意思。但如果我们把算术建立在别的东西上呢？假设我们把算术建立在雨滴上，假设我们有一滴雨和另一滴雨，我们把加法定义为它们相遇，那么一滴雨加另一滴雨，得到的还是一滴雨，所以在这个系统里 1 + 1 = 1。嗯，那有什么问题呢？我是说这在感知上是得到验证的：当我开着车以 60 英里的时速行驶，雨打在挡风玻璃上，我看到雨滴融合在一起。也许你应该测量一下它的大小之类的。好，没错，所以这里有各种关于同一性的问题，还有可能是系统如何构建的问题。但你知道，即便在数学里我们也必须非常明确我们在陈述什么，我们在什么样的域上工作，嗯，因为——假设我实际上是在整数模 2 上工作，模算术，它就是说如果能被某个数整除，我就说它是零，所以当我做模算术、做模 2 算术时，2 模 2 就是 0，所以 1 + 1 在整数模 2 上不是 2，
06哥德尔两条不完备性定理
14:17
do mod 2 arithmetic two mod 2 is is zero so 1 + 1 over the integer is mod two it's not two but it's actually zero so I mean there's all sorts of things we have to deal with and it's some uncertainties but still like these are all rigorously defined I could say well I'm just working over the integers so I know this is fine 1 plus 1 is equal to two in all possible worlds so Hilbert felt like really this has to be it like mathematics has to be it number theory has to be so sure and certain that there's got to be no problem um but what what Kurt gerle did is he established two things um and these are going to be his two incompleteness theorems one one we're not going to really talk about so much um and I'm not going to work through all the proofs of these but we're going to try to get an idea of of what each of them mean um the first one is that any system as powerful as number
而其实是 0。所以我是说，我们有各种各样的东西要处理，也有一些不确定性，但即便如此，这些都是严格定义好的，我可以说：好吧，我就是在整数上工作，所以我知道这没问题，1 加 1在所有可能世界里都等于 2。所以希尔伯特觉得，真的，就得是这样，数学必须就是这样，数论必须确定可靠到根本不可能有问题。嗯，但库尔特·哥德尔做的是，他确立了两件事，嗯，这就是他的两条不完备性定理。其中一条我们不会讲太多，嗯，而且我也不打算把这些定理的证明都过一遍，但我们会试着理解它们各自的意思。嗯，第一条是：任何与数论一样强的
15:26
Theory and I'm just going to let NT be number Theory um which can
系统——我就用 NT 来表示数论——如果它能
15:42
prove its own
证明自身的
15:51
consistency that system is necessarily inconsistent number the exactly somehow the second system as powerful as number theory is able to talk about itself things go Haywire if it can actually prove that Here I Am Number Theory saying look I promise you guys you can actually prove from me that I'm internally consistent well any system which can talk about itself in that way is necessarily inconsistent and you know this is really kind of a nitty Gerty proof and I can't even give you all the details so the second one and I'm just going to go ahead and start it over here um is any system as powerful as number Theory so just going to kind of go ditto um is necessarily incomplete
一致性，那么这个系统必然是不一致的。数论——没错，不知怎么的，第二个……当一个和数论一样强的系统能够谈论它自己时，事情就乱套了。如果它真的能证明"我，数论，在此说：看，我向你们保证你们真的能从我这里证明我内部是一致的"——那么任何能以这种方式谈论自己的系统，就必然是不一致的。你知道，这真的是一个很琐碎细致的证明，我甚至没法把所有细节讲给你们。那么第二条，我就直接在这边开始写，嗯，是：任何与数论一样强的系统——就照抄一遍，同上，嗯——必然是不完备的。
07说谎者悖论、自指与《数学原理》
17:06
so this means that any system which is as powerful as number Theory automatically looks like this there are true statements which we can formulate which are not provable and I'm going to go ahead and give you guys an example an English language example of a statement which is yeah go ahead REI what does it mean to be as powerful as good question um and this is this is I'm going to explain this a little bit but the idea is that in order to prove girdle's incompleteness theorems he had to use a very interesting trick and that's called girdle numbering and but first I want to give you the sentence and then I'll tell you where that comes into play so here's a statement so put that here I'm going to put you star and if we forget get what the star means it's rishi's question so let's just try to remember that um so let's consider a question or the following statement I talked about the liar Paradox right I said what happens with this sentence so this statement is false that like uh in your language you
所以这意味着任何与数论一样强的系统自动就长这个样子：存在一些我们能表述出来、却无法证明的真命题。我这就给你们举个例子，一个用英语表述的例子。是的，说吧，Rishi。"一样强"是什么意思？好问题，嗯，这个我会稍微解释一下，但大意是：为了证明哥德尔的不完备性定理，他必须用一个非常有意思的技巧，那就叫做哥德尔编号。不过我先想把那个句子给你们，然后再告诉你们那个技巧在哪儿派上用场。那么这里是一个命题——我把它放这儿，我打个星号，如果我们忘了星号是什么意思，那是 Rishi 的问题，我们记住这一点就行。嗯，那我们来考虑一个问题，或者说下面这个命题。我讲过说谎者悖论，对吧？我说过这句话会怎么样："这个命题是假的。"就是……呃，在你的语言里你不能说这种话。好，非常好。那为什么我们
18:30
can't say things like that okay very good and why wouldn't why wouldn't we be able to say things like this if you have like a logically perfect language you can say you shouldn't be able to say things like that then but then the problem be like how can you tell that is that you can say so what is the problem exactly with if you were to design your perfectly logical language what what would you rule oute what would you prevent sentences from doing that would pre prevent something like talking about them so there you go exactly self- reference self- reference is key and there's actually two guys two very very very smart guys Buren Russell and Alfred North Whitehead who derived a book they wrote a book and it's not a fun book to read I've heard it compared to it says about as interesting as reading the treads on a tire um so it's not interesting at all but that that book was called principia Mathematica and Douglas hoffstead will talk about this book a lot and in this book they develop a system
不能说这种话呢？如果你有一种逻辑上完美的语言，你可以说——你就不应该能说出这种话来。但那样的话问题就变成，你怎么判断哪些是能说的。那么问题究竟出在哪里，如果由你来设计你那种完美的逻辑语言，你会排除掉什么？你会禁止句子做什么，才能防止出现类似谈论它们自身的情况？答案就在这儿，没错，自指。自指是关键。而且实际上有两个人，两位非常非常非常聪明的人，伯特兰·罗素和阿尔弗雷德·诺斯·怀特海，他们推导出了一本书，他们写了一本书，那本书读起来一点也不有趣，我听人把它比作，说它大概跟读轮胎花纹一样有意思，嗯，所以完全不有趣。但那本书就叫《数学原理》，道格拉斯·侯世达会大量谈到这本书。在这本书里他们发展了一个系统，他们发展出的正是你说的那种完美语言。
19:33
they develop exactly that perfect language that you speak of [Music] um and the basic idea is that and what and if we want to formulate what you're thinking is that we we create a level language L1 and we only allow certain terms so we create kind of a a bag of terms like you know and certain sentences in L1 would be like the sky is blue Etc and snow is white these are perfectly like upstanding citizen sentences right they never break the law um but then what we do is we prevent certain terms we we prevent these sentences from talking about themselves but whenever we are in this class and we're talking about those sentences we're actually speaking in another language L2 and L2 contains L1 as a subset um and then we can start saying things like um the sentence snow is white is white let me do that [Applause] yeah so suly I've got the sentence snow is white which belongs in L1 and I'm talking about it but that's only something I can talk about in L2 because you have to in order to talk about
[音乐] 嗯，基本思路是——如果我们要把你想的东西表述出来，那就是我们创建一个层级语言 L1，我们只允许某些词项，所以我们创建一种词项集合，你知道的，L1 里的某些句子会是像"天空是蓝的"等等，还有"雪是白的"，这些都是完全守法的良民句子，对吧，它们从不违法。嗯，但接着我们要做的是禁止某些词项，我们禁止这些句子谈论它们自身。但只要我们在这个课堂上谈论那些句子，我们实际上就是在用另一种语言 L2 说话，而 L2把 L1 包含为一个子集。嗯，然后我们就可以开始说这样的话，嗯："'雪是白的'这个句子是白色的"——让我写一下 [掌声]。是的，所以现在我有了"雪是白的"这个句子，它属于 L1，而我在谈论它，但这只是我在 L2 里才能谈论的事情，因为为了谈论某样东西——你不能谈论你自己，
21:02
something you can't talk about yourself you have to LEAP outside of it and then in order to refer to it you have to stand from somewhere else right it's just like you can't really see yourself until you know you use something else to look at yourself um so they developed this system but even this had flaws and what girdle did was he actually used principia Mathematica um and he he took a statement similar to this but fact much more clever in order to prove his his incompleteness theorem and it's
你必须跳到它之外，然后为了指称它，你必须站在别的地方，对吧，这就好比你没法真正看见你自己，除非你借助别的东西来看自己。嗯，所以他们发展出了这个系统，但即使是这个系统也有缺陷。而哥德尔所做的是，他实际上用了《数学原理》，嗯，他取了一个和这个类似的命题，但事实上要巧妙得多，用来证明他的不完备性定理，那就是：
08哥德尔句：可证→真，真↛可证
21:42
this this statement is not provable and we can specify in what system and one of the things we'll talk about is in PM which means principial Mathematica but we could say this statement is not provable in number Theory but the bottom line is what does this statement say it's not like this statement because what happens if this statement's false well if it's false then whatever it says is about itself is not true so that means it's provable so if we say it's false then that means it is provable but if there's one thing which we are certain of yes go ahead it's like truthness in this case like when you're like proving since it's like complete um system just saying it's true the same thing I said is careful that only goes in One Direction so it is certainly true that a sentence okay sorry um what's your name again uh the how do you say it the Latif okay so one of the things La said was that in this in this case are we are we saying that when we're doing a derivation when we have something which
"这个命题是不可证明的。"我们还可以指明是在什么系统里，我们要讲的其中一点就是"在 PM 中"，也就是《数学原理》。但我们也可以说"这个命题在数论中是不可证明的"。不过关键在于，这个命题说的是什么？它不像前面那个命题。因为如果这个命题是假的会怎么样？如果它是假的，那么它关于自身所说的就不为真，那就意味着它是可证明的。所以如果我们说它是假的，那就意味着它是可证明的。但如果有一件事是我们确定的——好，你说。就是在这种情况下的"真"，就像当你在证明的时候，因为它是完备的，嗯，系统，就直接说它是真的，跟我说的是一回事。小心，那只在一个方向上成立。所以确实为真的是，一个句子——好，抱歉，嗯，你叫什么名字来着？呃，这个怎么念，Latif。好的，所以 La 刚才说的一点是：在这种情况下，我们是不是在说，当我们做推导时、当我们有了某个可证明的东西，我们就知道它是真的，而且
23:12
is provable we know it's true but and also vice versa but what I'm cautioning against is that's only true in One Direction so we certainly know that things which are provable are true if we can prove it if I can say I can prove to you that this is so then it automatically is so if there's and this is why people put so much trust in mathematics is that the second we have a proof of something we know it's true but what we just were asking about was what about the other way does true always imply provable and well let's ask let's ask this statement what if the statement is true true well if it's true what it says about itself must be true and that's that it's not provable so the only way that this statement is true is if it's not provable so suddenly we know that we can't go the other way and the trick that girdle use and this is why we say this is why we get to the star question is that this is not a a statement in mathematics but what girdle did is he essentially took a statement like this and he said well we're going
反过来也成立？但我要提醒的是，那只在一个方向上成立。所以我们当然知道，凡是可证明的东西都是真的：如果我们能证明它，如果我能说我可以向你证明事情就是这样，那它自动就是这样。而这正是人们如此信任数学的原因：我们一旦有了某件事的证明，我们就知道它是真的。但我们刚才问的是另一个方向：真是否总是蕴含可证明？那我们来问问，来问问这个命题：如果这个命题是确实如此——如果它是真的，那它关于自身所说的就必须是真的，也就是说它是不可证明的，所以唯一的可能就是：这个陈述为真，当且仅当它不可证明。于是我们突然知道，反过来是走不通的。而哥德尔用的那个技巧，这也是我们为什么要说、为什么会引出那个关键问题——这并不是一个数学内部的陈述。但哥德尔所做的是，他基本上拿了一个这样的陈述，然后说：好，我们让每一个字母和逻辑符号都代表一个数字，所以
09哥德尔编号：让数论谈论自己
24:37
to let every letter and logical symbol stand for a number so we're going to let P be you know 1 Z 1 0 0 or one zero and we're going to give a unique number to every symbol including spaces and then once we have this we have a girdle number for this statement and then what we can start doing is we can start giving certain operations remember when we were doing Miu we're saying well what we can always do is we can if we have three eyes we can cancel it and or if we have something a string of hyphens or whatever or a string of letters after M we can double it so what girdle did is he turned each of these rules of inference into rules of arithmetic so and I'm going to go Ahad and hop over
我们让 P 等于，比如说 1 Z 1 0 0 或者一零，然后我们给每一个符号都指定一个唯一的数字，包括空格。有了这个之后，我们就得到了这个陈述的哥德尔数。接着我们可以开始做的事情是，我们可以开始赋予某些操作——还记得我们做 MIU 系统的时候吗，我们说，我们总是可以做的事情是，我们如果有三个 I，就可以把它消掉；或者如果我们有一串连字符什么的，或者 M 后面有一串字母，我们可以把它翻倍。所以哥德尔做的就是，他把这些推理规则每一条都变成了算术的规则。那么，我这就走过去，跳到
25:40
here so what he did is he made rules of inference and he made them equivalent to or my special isomorphism symbol to rules of arithmetic so this is kind of counterintuitive um and I can't go into all the details um but we will meet them in chapter nine um and that's the idea that so suppose we have a logical logical thing like well we have the statement p and then we also have the statement that P implies Q so we know that the the statement if p is true so if it is cloudy then it's going to rain and then if we have well I'm looking outside and it's cloudy then we can immediately this is equal to Q right so if we know that if it's cloudy then it will rain is true and we have that it's it's cloudy then we can immediately deduce that it's going to rain and what gerdle did is he said well these statements I can actually make into numbers um and I can make the logical symbol and into an operation like addition and I can make implies well this would also be a symbol and I can have this total operation of
这边来。所以他做的事情是，他把推理规则做成了等价于——或者用我那个特殊的同构符号——算术规则。这有点违反直觉，嗯，我没法讲所有的细节，不过我们在第九章会碰到它们。这个想法就是，假设我们有一个逻辑上的东西，比如说我们有陈述 p，然后我们还有一个陈述P 蕴含 Q。所以我们知道，如果 p 为真——比如说如果天阴了，那就要下雨——然后如果我们有，嗯，我往外一看，天确实阴了，那我们立刻就可以得到 Q，对吧？所以如果我们知道“如果天阴了就会下雨”是真的，而且我们还有“天阴了”，那我们立刻就能推出要下雨了。而哥德尔做的是，他说，这些陈述我其实可以变成数字，而且我可以把逻辑符号“与”变成一个像加法那样的运算，我还可以把“蕴含”——嗯，这也会是一个符号——然后我可以把这整个“分离”出 Q 的操作变成一个陈述，几乎就像 1 + 1 =
27:10
Detachment of pulling out Q into a statement almost like 1 + 1 = 2 um except it then becomes and then what he did is he captured the idea of provability into a property of numbers like Prime something being Prime so then really what this this statement comes down to is like is something such and such number the number which codes for this statement does not have a property and that's why you need something as strong as number theory in order to do this numbering trick but that's just kind of a first glance at girdle's theorem and I don't want to go into too much detail about it however I do want to go back and talk a little bit more about the uh the things which we mentioned and we kind of glanced upon in chapter 4 and that was the kind of the ideas of geometry and this is this is really cool and it has something to do with uh with interpretation now I want you all to kind of remember what what we mean by interpretation and it's I think it's a term I kind of briefly defined um on the first day of lecture for those
2 一样，只不过它接着就变成……然后他做的是，他把可证明性这个概念捕捉成了数的某种性质，比如素性、某个数是素数。所以说到底，这个陈述归结为：某个如此这般的数——也就是给这个陈述编码的那个数——不具有某个性质。这也是为什么你需要像数论这样强的东西，才能玩这个编号的把戏。不过这只是对哥德尔定理的一个初步一瞥，我不想讲得太细。不过我确实想回过头来，多聊聊我们之前提到过、只是在第四章里一带而过的那些东西，也就是几何学的那些想法。这真的非常酷，而且它跟“解释”有关系。现在我想让大家回忆一下，我们说的“解释”是什么意思。我想这个词我在第一节课上大致定义过——在场的同学里，有谁能
10解释与几何：欧氏与非欧
28:26
of you who are here can anyone kind of tell me what interpretation is about go ahead so of like choosing like maybe like the real world real world like model of what you put in theem exactly exactly so and what what um I'm sorry I forgot your name is it Latif Latif what Latif said was that it's basically like giving a world an example of an interpretation to be giving a real world model for for what you're doing and we saw an example of an interpretation which was not true right when we assigned to our symbols 1 plus 1 equals two to raindrop Mel with raindrop it didn't give us two raindrops instead gave us just one um so that's an example of an interpretation which doesn't hold and doesn't work um but we gave some other interpretations um and when we were playing with the PQ system we didn't give it a real world interpretation but in said we gave it a mathematical interpretation of addition well we said that hyen P hyen Q hyen hyphen is 1 + 1 equal 2 um and so that's not so interesting but
告诉我“解释”是怎么回事？请说。就是像选一个，可能像现实世界的，现实世界的模型，来对应你放进系统里的东西。完全正确，完全正确。那么，嗯，不好意思我忘了你的名字，是 Latif 吗？Latif 说的是，“解释”基本上就是给出一个世界——一个“解释”的例子就是为你所做的事情给出一个现实世界的模型。我们也看过一个不成立的解释的例子，对吧？当时我们把符号 1 加 1等于 2 里的符号指派成雨滴，结果雨滴合并之后并没有给我们两滴雨，反而只给了一滴。嗯，所以那就是一个不成立、行不通的解释的例子。不过我们也给出过另外一些解释。嗯，当我们在玩 pq 系统的时候，我们没有给它一个现实世界的解释，而是给了它一个加法的数学解释——我们说，连字符 p 连字符 q 连字符连字符 就是 1 + 1 = 2。嗯，这本身没那么有意思，但有意思的是，你知道，
29:44
what is interesting is that you know several thousand or so years ago not several but at least two and a halfish uh a guy named uclid said okay okay you know what geometry is for us as true and certain as anything's going to get but what I want to do is go ahead and write down the rules write down everything we know so that we can proceed and deduce directly from from these statements and know that everything we say is true but in order to get his feet off the ground I mean he couldn't lift himself up by his own bootstraps he made a series of requests and these These are kind of known as the postulates of uid um so we got ID's postulates and I'm going to go through them all they're they're actually listed I believe in chapter 4 but there was one postulate which really got on uet's nerves um and that that was known as The Fifth postulate and he tried to derive it from the previous four but he couldn't um and that's the idea that if you have a line and a point not on the line I can give you a line there's a unique
好几千年前——不是好几千，至少两千五百年左右吧——有个叫欧几里得的人说，好吧，你知道，几何学对我们来说已经是再真、再确定不过的东西了，但我想做的是，把规则写下来，把我们知道的一切都写下来，这样我们就能从这些陈述直接往下推演，并且知道我们说的每一句话都是真的。但为了让他能起步——我是说，他没法揪着自己的鞋带把自己提起来——他提出了一系列的请求，这些就是所谓的欧几里得公设。所以我们有了欧几里得的公设，我本来打算把它们全过一遍，我记得第四章里其实列出来了。但有一条公设真的让欧几里得很不爽，嗯，那就是所谓的第五公设。他试图从前面四条推出它，但没能成功。嗯，这条公设的意思是：如果你有一条直线，还有一个不在这条直线上的点，我可以给你一条直线——存在唯一一条
31:04
line that goes to that point but never intersects this line and of course in Geometry we say that lines extend on forever and line segments kind of terminate but these are good old infinite lines um but he can never prove it and there was efforts for well over 1500 years to try to prove the fifth postulate but it's such an intuitive and obvious statement right I mean does anyone feel like this isn't right that there's that there's any reason why we can't assume this go ahead what if do like uh dealing with like different surfaces maybe the stuff can get exactly exactly so this gives us the idea of non- ucan geometry and just to give you kind of a quick example suppose we're on a the surface of a sphere and we Define our lines to be great circles so the way you make a great circle is you take your your sphere with your kind of Center o and you you cut through it and and you make a a plane slice so
直线经过那个点，却永远不与这条直线相交。当然在几何学里我们说直线是无限延伸的，而线段是有端点的，但这里说的是老老实实的无限长直线。嗯，可他始终证不出来。而且有一千五百多年的时间里，人们一直在努力证明第五公设，因为它是个如此直观、如此显然的陈述，对吧？我是说，有谁觉得这不对吗？觉得有什么理由不能这么假设吗？请说。要是，比如说，处理不同的曲面呢？也许情况就会变得……完全正确，完全正确。这就引出了非欧几何的想法。举个简单的例子：假设我们在一个球面上，而我们把“直线”定义为大圆。做大圆的方法是：你拿你的球，球心是 O，然后你从中间切开，切出一个平面截面，
32:33
that goes through the origin and you define a line to be this great circle which is formed so any line on here that you can form any line that you necessarily have and these are kind of like your lines for longitude or latitude except not necessarily um because we couldn't do something like this but if we had any any line that goes through that which has the same radius as our sphere they necessarily intersect and at least two spots so in spherical geometry things obviously don't behave the same way which and one of the things you could derive using ucd's kind of axioms was that you know a triangle the sum of the internal angles of a triangle is always 180° but on a sphere if we draw a triangle let's say we we go from somewhere on the equator we travel up to the the North Pole and then we we pit it 90° and head back down to the equator um so we've got a right angle here and a right angle here we also have a right angle here so for a spherical triangle we can actually get up to 90 Plus 90 Plus 90
这个截面通过球心，然后你把由此形成的这个大圆定义为一条“直线”。所以在这上面你能画出的任何一条线，任何一条你必然会有的线——这些有点像你的经线或纬线，不过也不完全是，嗯，因为我们没法做出像这样的东西——但如果我们有任意两条经过它的、半径与球半径相同的线，它们必然会相交，而且至少在两个地方相交。所以在球面几何里，事情显然不是按同样的方式运作的。而你用欧几里得那套公理能推出来的一件事是，你知道，一个三角形的内角和总是180 度。但在球面上，如果我们画一个三角形，比方说我们从赤道上某一点出发，一路走到北极，然后转90 度再走回赤道。嗯，这样我们这里有一个直角，这里也有一个直角，那里我们同样有一个直角。所以对一个球面三角形，我们实际上可以得到 90 加 90 加 90 等于 270 度。所以这在球面几何里显然
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270° so obviously this doesn't hold in spherical geometry and similar we have hyperbolic geometry and this is something which is a very beautiful subject um and you can you have several models of how hyperbolic geometry works you can think of them as projections but and once's kind of the upper half plane
是不成立的。类似地，我们还有双曲几何，这是一个非常美的主题。嗯，你可以有好几种关于双曲几何如何运作的模型，你可以把它们想成投影。其中一个是上半平面
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model and then one's just kind of your unit disc here and and what we Define our lines to be is these segments which end with right angles on the outside of your circle so what we can do then is actually construct two lines we can actually construct infinitely many lines um that don't intersect each other so in here you had two intersection points and here you had no if you had a line like that you would only have one intersection point but here you could have a whole family lines with no intersection points um but the weird thing is that we can give our same terms our same statements like point and line and we can do a lot of the same geometry which uclid did except if we give them different interpretations like well we'll Define the line to be like this or we'll Define a line to be like this then different things happen yes so the system is necessary but the way you interpret it is the so here is a fact it's true that with Four's original the four original postulates of uclid okay sorry what what
模型，另一个就是这里的单位圆盘。而我们把“直线”定义为这样一些弧段：它们在圆周外侧以直角相交。那么我们就可以构造出两条直线，实际上我们可以构造出无穷多条彼此不相交的直线。所以在这边（球面上）你有两个交点，而在这边你一个交点都没有。如果你有那样一条线，你就只有一个交点；但在这里，你可以有一整族没有交点的直线。嗯，但奇怪的地方在于，我们可以用同样的术语、同样的陈述，比如“点”和“线”，而且我们可以做很多欧几里得做过的同样的几何，只不过如果我们给它们不同的解释，比如说我们把“直线”定义成这样，或者定义成那样，那么发生的事情就不一样了。对，所以系统本身是必要的，但你怎么解释它才是……好，这里有个事实：以欧几里得原来的四条公设……好，抱歉，你刚才说什么？
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um what Latif said was that necessarily what's true in your formal system is the interpretation you give them um that was that is true in this example right here the truth of your statement directly depended on how you interpret your terms like a point and line and things like that and the problem was is that in the Assumption the Axiom which was ucl's fifth postulate was that what ucl's fifth postulate said only could be interpreted consistently with the other four postulates when he did it in just simple plain geometry like we're working on the top of this table but the second you interpreted all five of his postulates in this setting the fifth one was inconsistent with the previous four and you and what it said was inherently wrong um and similarly things with this uh it was It produced all sorts of it you had to be very specific about your interpretation and what you assumed otherwise you could get an inconsistent interpretation internally inconsistent interpretation so these this is kind of
嗯，Latif 说的是，你的形式系统里什么是真的，必然取决于你给它们的解释。嗯，这在眼前这个例子里确实成立——你的陈述的真假直接取决于你如何解释你的术语，比如“点”“线”之类的。而问题在于，在那条假设里，也就是欧几里得的第五公设：第五公设所说的内容，只有当他在简单平坦的几何里——像我们在这张桌面上操作那样——才能与另外四条公设一致地解释。但你一旦在这种（球面/双曲）设定下解释他全部五条公设，第五条就与前面四条不相容了，而且它所说的内容本身就是错的。嗯，这边的情况也类似——它会产生各种各样的问题，你必须非常明确地说清楚你的解释和你所假设的东西，否则你会得到一个不一致的解释，一个内部不一致的解释。所以这些东西都属于一个叫双曲几何的家族。[掌声] 而它内在上
11若数学根基动摇，论证何以立足
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all part of a a family of things called hyperbolic [Applause] geometry and inherently what this has to deal with is the beauty of of complex numbers and you can do things in hyperbolic geometry which just completely Boggle the mind which is that like suppose you had a circle a a line This is a line remember because it ends with perpendicular points on the real axis and you can find a mapping which takes it up to here and preserves the distance between these two and there's all sorts of different things you can do and it's just it's completely a gorgeous subject I encourage you all to learn more about it um but this was one of the examples because we thought that I mean what we thought for well over 15 thou 1500 years was that what uclid said was as sure and certain as any knowledge that we could have and people would often try to base their arguments and and try to derive them to Geometry um It's Kind of a Funny anecdote but um Carl Marx and and Frederick Les uh when they when they were writing their kind
跟复数之美有关。你在双曲几何里能做的一些事情简直让人脑洞大开，比如说，假设你有一个圆，一条线——记住，这是一条“直线”，因为它的两端与实轴垂直相交——你可以找到一个映射，把它变到这上面来，并且保持这两点之间的距离不变。你可以做的事情五花八门，这真的是一个极其漂亮的学科，我鼓励大家去多了解一下。嗯，但这就是那些例子之一，因为我们曾经以为——我是说，我们在一千五百多年里一直以为——欧几里得所说的东西是我们所能拥有的最确定无疑的知识。人们还经常试图把自己的论证建立在几何上、试图归结到几何上。嗯，说个挺好笑的轶事：卡尔·马克思和弗里德里希·恩格斯，他们在写他们那些著作的时候，真的试图把自己所说的内容归约成数学，因为他们觉得，如果
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of their texts they actually try to reduce what they were saying to mathematics because they felt like if they could prove what there's what the system they were advocating in terms of mathematics then people would have to accept it um but what happens if mathematics itself is inconsistent if you get paradoxes like the set theory paradoxes or you get these possible interpretations where things are sometimes true or not true what happens then if if mathematics is not on short footing um and this is kind of a problem which I I want you guys to just think about as we as we move along through this book and what does it mean to provide an interpretation and things like that so what I want us to do is take a quick break because we're going to go into one of my favorite dialogues and you'll see the purpose of of this food later but because it's a long dialogue I want everyone to kind of take a break and get some food and drink and uh we'll we'll then read the dialogue but I'll need some volunteers for reading so
他们能用数学的方式证明他们所主张的那套体系，那人们就不得不接受它。嗯，但如果数学本身就是不一致的呢？如果你碰到集合论悖论那样的悖论，或者碰到这些可能的解释，让某些东西有时为真、有时不为真，那会怎样呢？如果数学本身就站不稳脚跟，那该怎么办？嗯，这算是一个我想让大家在我们读这本书的过程中一直思考的问题：提供一个“解释”到底意味着什么，诸如此类。所以我想让我们稍微休息一下，因为接下来我们要进入我最喜欢的对话之一。你们待会儿就会明白这些吃的有什么用了，不过因为这个对话很长，我想让大家先休息一下，拿点吃的喝的，然后我们再来读这个对话。不过我需要几位志愿者来朗读。那我们先休息五分钟左右，然后再开始读。
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let's go ahead and take like a a five minute break and before we start reading
12《小和声迷宫》：嵌套层级与「我」
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okay so a little harmonic Labyrinth what you guys think confusing confusing why you say it's confusing like it Swit rolls in between the stores so in what way do you mean rolls it's like the turtle becomes the so there's roll flipping yeah in in terms of like the way they treated each other or like the cage was ah okay yeah so and more interestingly though there was they had an opportunity to do that by kind of constantly going down to nested rolls of and they could kind of essentially be new people in some ways so that's good does anyone else have oh get Sandra talk about themselves right so we had this weird playing around with levels like uh and in some ways that was kind of hard to capture perfectly in terms of audio but I'm sure most of you saw as we were reading along that there was indentations um in the in the text and that kind of was a visual reminder as you reading what level of the story you were on at and and Douglas hofstead used all sorts of really nice tricks where you would have like a character's like
好，《小和声迷宫》，你们觉得怎么样？很绕，很绕。为什么说很绕？因为它在故事之间来回切换角色。你说的“角色”是指哪方面？就是乌龟变成了……所以有角色互换，对，就是在他们彼此对待的方式上，或者说那个笼子……啊，好，明白了。而更有意思的是，他们之所以有机会那样做，是因为不断往下进入一层层嵌套的角色，这样他们某种程度上就可以变成全新的人。这点很好。还有别人想说的吗？哦，还有 Sandra，讲讲他们自己。对，所以我们看到这种奇怪的对层次的玩法。嗯，从某种程度上说，这种效果用听的很难完美地传达出来，不过我相信你们大多数人在跟着读的时候都看到了，文本里有缩进。嗯，那种缩进算是一个视觉上的提醒，让你知道自己读到故事的哪一层了。而且侯世达用了各种各样很漂亮的花招，比如说你会看到一个角色说：哦，我想他说的是，我想他说的是补药（tonic）；而他们是在
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oh I think I think he means I think he means tonic and they would be talking up here on level level one and you know down on on level two uh they would say oh you know thank you yeah yes tonic is exactly what I needed even though um and this in this situation these guys don't really know about their higher levels of reality it's just like the same question of what happened to the weasel when here was sitting in our everyday normal life and he took some popping tonic and he popped up to a higher level of reality like and it kind of makes you wonder and I know for me I always get the visual image of of you know playing the universe kind of as as as a fractal and um you know we spend all our time living down in this corner we spend all our time living down in this corner of the serinsky casket and then one day somebody takes a popping tonic and something like holy mackerel there's there's actually all these levels um and then just the fact that you've had that one experience of of playing around with two levels of
上面这一层、第一层说这话的；然后在第二层，他们会说：哦，谢谢你，是的，补药正是我需要的东西。尽管——嗯，在这种情况下，这些家伙其实并不知道存在着更高层次的现实。这就跟那个问题一样：当我们坐在日常普通生活里的时候，那只鼬鼠出了什么事？他喝了一点弹出药水（popping tonic），然后就弹到了更高一层的现实。这会让你忍不住琢磨。我知道对我来说，我脑子里总会浮现出这样一个画面：把宇宙想象成一个分形。嗯，你知道，我们花了一辈子时间就活在这个角落里，我们花了一辈子时间就活在谢尔宾斯基三角垫的这个角落里，然后有一天，某个人喝了弹出药水，然后就发现：我的天，原来真的存在这么多层次。嗯，而且仅仅是因为你有过那么一次玩弄两层现实的经验，你就会开始猜想：嗯，
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reality it makes you speculate of um well why aren't there why can't there be more why can't there be kind of an infinite set of realities and how do I know what mine is and um you know there's something I I want us to kind of go ahead and just bring out into the open here um and it's the idea that when I first started this class on the first lecture I not all of you are here um I said the fundamental thing we want to answer at the end of this course is what is an eye what makes something conscious from unconscious things how do we get particles and atoms to start talking about themselves like the way we do um so fundamentally in this class we're going to be talking about a lot of really kind of deep and profound questions um and I want everybody to not feel afraid that that their their opinion might be kind of persecuted because even in this story we we managed to meet God during the middle of this dialogue um and if we can't talk about God like it's going to really narrow what we're allowed to talk
为什么不会有更多层呢？为什么不能有一整个无穷的现实集合呢？我又怎么知道我自己身处哪一层呢？嗯，你知道，有件事我想在这里摊开来讲一讲。嗯，就是这个想法：当我第一节课开始上这门课的时候——你们并不是所有人都在场——我说过，我们希望在这门课结束时回答的根本问题是：什么是“我”？是什么让某个东西有意识，而另一些东西没有？我们怎么能让粒子和原子像我们这样开始谈论它们自己？嗯，所以从根本上说，这门课上我们会讨论很多非常深刻、非常根本的问题。嗯，我希望每个人都不要担心自己的观点会受到什么攻击，因为就连在这个故事里，我们都在对话中途见到了上帝。嗯，如果我们连上帝都不能谈的话，那就……真正限制我们能讨论的范围，所以呃，同样地，当我说，你知道，什么是心智，我们如何
13微积分、耶稣会与无穷的驯服
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about so um and similarly when I'm saying you know what is the mind how do we get you know a physical brain to to then start operating in a uh with mental and conscious thoughts like that's very fundamentally asking questions about the soul and um what you guys' opinions are on that become important and I and I don't want to feel like anybody is you know learning in a hostile environment so I encourage you guys to speak actively um it's interesting because uh Douglas hoffstead presents a very uh unique picture of God right he picks uh this kind of recursive idea of a stack of Infinities um and uh just as kind of an an anecdote and this is actually a little historical fact for for you um the Jesuits right around the time of the development of calculus you know we had we kind of had Isaac Newton and livets and these guys doing doing their stuff in in the 1670s maybe a little later but it really took everyone else into like the 1730s um Newton livets for these guys that develop the calculus you know studying the
让一个物理的大脑开始以一种带有心理活动和有意识思维的方式运作，这本质上就是在追问关于灵魂的问题，而你们大家对此的看法就变得很重要，我不希望任何人觉得自己是在一个充满敌意的环境里学习，所以我鼓励你们积极发言呃，有意思的是，道格拉斯·侯世达提出了一个非常独特的上帝图景，对吧，他选择了这种关于无穷之栈的递归式想法，呃，就当作一个小故事吧，这其实也是一个小小的历史事实，讲给你们听，呃，耶稣会士，大约在微积分发展的那个时期，你知道，我们有我们差不多有艾萨克·牛顿和莱布尼茨，这些人在 1670 年代做他们的工作，也许再晚一点，但真正让其他所有人跟上来已经到了 1730 年代，呃，牛顿、莱布尼茨这些发展出微积分的人，你知道，研究的是
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infinitesimally small remember when we're playing around with Calculus uh we we're asking what happens when we kind of approximate a function originally in a finite way and then what happens is we take the limit to something which is infinitely small and what does that mean and when when you start playing around with Calculus uh you can find out find the things like the area under the curve and the way you approximate this is you know you know with kind of these blocks and taking infinitely small limits the Greeks they they were really close to developing the calculus Archimedes was in many ways conceptually just a few Stone throws away from it um but the Greeks were also much smarter than Newton and Li Nets because they said well Dumbo when you take a bunch of infinitely small things you can't get something finite right like how is it that I can take a bunch of two-dimensional circles which are infinitely thin and then stack them on top of each other in order to get you know a cylinder it doesn't make sense it
无穷小量，还记得我们摆弄微积分的时候吗，我们在问，当我们最初以有限的方式去逼近一个函数会发生什么，然后当我们对某个无穷小的东西取极限时又会发生什么那意味着什么，而当你开始摆弄微积分时，呃，你可以发现一些东西，比如曲线下的面积，而你逼近它的方式，你知道，就是用这些小方块，然后取无穷小的极限，希腊人，他们其实非常接近发展出微积分，阿基米德在很多方面从概念上讲只差几步之遥，呃，但希腊人其实也比牛顿和莱布尼茨更聪明，因为他们说，喂，笨蛋，当你把一堆无穷小的东西加起来，你不可能得到某个有限的东西，对吧，比如我怎么可能拿一堆二维的圆，它们是无限薄的，然后把它们一个个叠起来，就得到，你知道，一个圆柱体呢，这说不通
45:37
does not make sense um so really what happened here and then getting into Oiler and these guys is they essentially were trying to take our concepts of the INF infinite and making them rigorous and the reason why I go on this is that the Jesuits um around this time and getting later were one of the very first groups in schools to start teaching their students calculus because they felt that if they understood mathematically and rigorously and they could deal with concepts of the infinite they had a better understanding of God so the Jesuits deeply felt that you know you understanding calculus was essential to you understanding god um and I think this very much goes goes in spirit with the little harmonic Labyrinth where where we kind of you saw this this image of of of God over Jen and Jen is actually an Arabic word for for Genie um and then you know this going off to God itself and then coming back um so this really requires wrestling some of the conceptual tools behind dealing with the
这完全说不通，呃，所以这里真正发生的事情，再到欧拉那些人，他们本质上是在试图把我们关于无穷的概念严格化，而我讲这些的原因是，呃耶稣会士，在那个时期以及更晚一些，是最早在学校里开始教学生微积分的群体之一，因为他们觉得，如果学生能在数学上严格地理解，并且能处理无穷的概念他们就能更好地理解上帝，所以耶稣会士深信，你知道，理解微积分对于理解上帝是必不可少的，呃，我觉得这在精神上非常契合《小和声迷宫》那一章，在那里我们看到了那个意象，上帝在精灵之上，而 Jen（精灵）其实是一个阿拉伯语词，指的是神灵，呃然后，你知道，一路往上到上帝本身，然后再回来，呃，所以这真的需要我们去搏斗一番，掌握处理无穷背后的一些概念工具，呃，这不是我在这门课里能完整教给你们的东西，但我
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infinite um and it's not something I can teach you fully in this class but I encourage you all to pursue it um quick question you'll notice that the each of the Genies did it in half the amount of time that it took the previous Genie can anyone tell me why or at least why hoffstead went ahead and paid attention to that detail because you don't want to do it forever okay so you need something in Felix go ahead one great one moment right okay so the amount of time it took was 1 plus a half of one Genie's time plus half of the previous Genie's time plus yes and so on um yes sorry knew that it wasn't right 116th missed the term dot dot dot and do you know what this equals okay there you go we we got some winners so this is actually an example of geometric
鼓励你们都去钻研一下，呃，一个小问题，你们会注意到，每个精灵完成任务所花的时间都是前一个精灵的一半有谁能告诉我为什么吗，或者至少说说侯世达为什么要特意关注这个细节，因为你不想让它一直做下去，好的，所以你需要有个东西，菲利克斯，你来说，对，一个，很好，稍等一下，好，所以所花的时间是 1 加上某个精灵时间的一半加上前一个精灵时间的一半，加上，对，如此类推，呃，是的，抱歉，我知道那不对，是十六分之一，漏掉了那一项，点点点那你们知道这等于多少吗，好，答对了，我们有几位答对了，所以这其实是一个几何
48:10
progression and this gives you an idea of we actually know that this infinite process could converge in a finite amount of time really it took until this time and if we were to go way back let's go ahead and have kind of a logarithmic scale backwards to around I think it was 200 or 300 BC that Zeno of AA Used actually this same argument for saying that you know motion was inherently impossible and that and that um you know it we could never go anywhere and all that all motion was an illusion because we require doing an infinite amount of stuff and that you could never do an infinite amount of stuff because it would necessarily be infinite but it took us as you know a human Collective conscious well over you know 1,700 close to 2,000 years to understand and develop the tools necessary to deal with Infinities um so that's I think that's a really kind of important thing to deal with um and I could go on and give entire courses about Infinities and talk about a lot of the important characters um and there's
级数的例子，这让你们了解到，我们确实知道这个无穷的过程可以在有限的时间内收敛，实际上一直到这个时代才搞清楚，而如果我们往回追溯，我们用一种对数式的尺度往回走，大约到我想是公元前 200 年或 300 年，埃利亚的芝诺其实用的就是同样的论证，来说明，你知道运动本质上是不可能的，而且，呃，你知道，我们哪儿也去不了，一切运动都是幻觉，因为我们需要完成无穷多件事，而你永远不可能完成无穷多件事，因为那必然是无限的，但是我们人类作为一个集体意识，花了远超一千七百年、接近两千年的时间才理解并发展出处理无穷所必需的工具，呃，所以我觉得这是一件非常重要的事情，呃，我可以一直讲下去，开一整门关于无穷的课，谈很多重要的人物，呃，不过
14无穷的层级：ω、ℵ₀ 与对角线
49:21
just some there's just one term I want to introduce um and a couple Concepts really quickly that's the idea that there's never a top infinity and you can and you can always construct more Infinities from smaller ones um and you can actually carry out kind of a formal system for playing around with these Infinities and uh you know you can have you have your natural numbers three and then dot dot dot dot and then we can just say okay well let's take all of those guys and we'll call them Omega well then we can take omega and then we can take omega Plus one and dot dot and now we got two omegas and then we can actually start carrying out Cardinal arithmetic um you might also see and there's kind of a mix of notations and con and uh Concepts between different fields but you might also see this first level of infinity as a a not or a Subzero um and this refers to the level of infinity which you get from the natural numbers of course some of you may or may not know this but um we've got all sorts of Infinities we can
我只想介绍一个术语，呃，还有几个概念，很快讲一下，那就是这样一个想法永远不存在最高的那个无穷，你总是可以从更小的无穷构造出更多的无穷，呃，而且你其实可以建立起一套形式系统来摆弄这些无穷，呃，你知道，你可以有你有你的自然数，三，然后点点点点，然后我们可以直接说，好，我们把所有这些家伙拿过来，管它们叫 Omega，那么我们就可以取 omega，然后我们可以取 omega 加一，点点，然后现在我们有了两个 omega，然后我们其实就可以开始做基数算术了，呃，你可能也会看到，不同领域之间的记号和概念有些混用，但你可能也会看到第一层无穷被写作aleph-null 或者 aleph-零，呃，这指的是你从自然数得到的那一层无穷，当然，你们有些人可能知道也可能不知道，但呃，我们有各种各样的无穷，我们可以开始构造出无数个，但我们甚至可以
50:35
start constructing tons but we can even Define exponentiation so so one of the big things is that well what happens when you have two raised to the AL knot well the claim is that this is AA one which is roughly the size of the reals so and we're talking all your numbers on the real line and this was really this is kind of a paradoxical thing because in here we have this many right and what's very strange is yeah go ahead isn't it like a property of infin like amounts have like subance of them that I say as the whole thing yes exactly so you just actually stated a very rigorous form of defining something to be infinite is that you can put it into one toone correspondence with itself what Latif said is yes is one of the properties of infinity the fact that you can put it in correspondence with a subset of itself and the idea is so for example take um we can take the normal integers and fortunately we have an infinite amount of those guys and we can put them into one toone correspondence with the even
定义指数运算，所以一个重要的问题是，当你有 2 的 aleph-零次方时会怎样，主张是，这就是 aleph-一，大致相当于实数的大小，所以，我们说的是实数轴上所有的数，而这真的，这算是一件挺悖论的事情，因为在这里我们有这么多，对吧，而非常奇怪的是，对，你说这是不是无穷的一种性质，就是它的某个子集跟整体一样大，是的，正是如此，所以你其实刚才陈述了一个非常严格的定义无穷的方式，就是你可以把它跟自身建立一一对应，拉蒂夫说的是，对，这是无穷的性质之一，就是你可以把它跟自身的一个子集建立对应，这个想法是，比如说，呃，我们可以取普通的整数，幸运的是我们有无穷多个这样的数，我们可以把它们跟偶数建立一一对应，我们可以造出一个完全
51:55
numbers and we can create a completely bjective map just by division or multiplication by two but the weird thing is that we intuitively feel like there are all of these guys are inside of here so there should be less of them than than these but with infinity you can do all sorts of things um and that's what's magical and what's nice is you can also take the the interval 0 to one and you can create a map which sends this to the entire infinite real line so it's amazing because we can capture an infinite thing in a very finite way um and we also know because of a a good guy named Cantor uh that there that these are really different quantities that there's a level of infinity different between the natural numbers and the reals um and that deals with a diagonal argument which I can maybe show you one day um so you guys kind of read chapter five which was recursive structures and processes for today's lecture although we've been talking a lot about other things and this is really motivated because uh you got an excellent lecture
双射的映射，只需要除以二或乘以二，但奇怪的地方在于，我们直觉上会觉得所有这些家伙都在这里面，所以它们的数量应该比那些少，但有了无穷，你可以做各种各样的事，呃，这就是它神奇的地方，而且很妙的是，你还可以取 0 到 1 这个区间，然后造一个映射，把它映射到整条无限的实数轴上，所以这太惊人了，因为我们可以用一种非常有限的方式装下一个无限的东西，呃而且我们也知道，多亏了一位好人康托尔，呃，我们知道这些确实是不同的量，自然数和实数之间存在着不同层级的无穷，呃，这涉及对角线论证也许哪天我可以给你们演示一下，呃，所以你们大致读了第五章，也就是《递归的结构与过程》，作为今天这堂课的内容，虽然我们一直在聊很多别的东西，而这么安排其实是因为，呃，你们上次上了一堂非常精彩的课
15递归与智能：深蓝对卡斯帕罗夫
53:18
last time from Karen kellerer who showed you all the different varieties in which infinity and recursion kind of come together um but fundamentally we still have we still have a question which we're which we're pursuing um and this kind of deres back to I think our two most important tools for thinking which we'll meet in this in this first part of the book and that's recursion and isomorphism and remember isomorph morphisms um come about when you're trying to put equivalence relationships between one thing and the other and you can do it in a kind of well- defined way um and I want you to kind of think about what's what's the relationship between between these two and what happens when um and really what connects these Concepts and the way which we uh we deal with things but um I want to highlight just at least one section or two um from chapter 5 um which I don't know if you guys found interesting as well um is that idea of so we got we have recursion here and we can do all sorts of different things
是凯伦·凯勒勒讲的，她向你们展示了无穷和递归结合起来的各种不同形态，呃，但根本上我们仍然有一个问题在追问，呃，而这又要回溯到我认为这本书第一部分我们会遇到的两个最重要的思维工具，那就是递归和同构，记住，同构，呃，是在你试图在一个事物和另一个事物之间建立等价关系时出现的，而且你可以用一种定义良好的方式来做，呃，我希望你们思考一下这两者之间是什么关系，以及当，呃，真正把这些概念联系起来的是什么，还有我们处理事物的方式，但是，呃我想至少重点讲第五章里的一两节，呃我不知道你们是不是也觉得有意思，呃，就是那个想法，我们这里有递归，我们可以做各种各样的事情，对吧，呃，柯伦向你们展示了我们如何构造分形、树、山脉，以及各种各样美丽的
54:42
right um Curran showed you how we can construct fractals and trees and mountains and all sorts of beautiful things and and we can actually create all sorts of uh with the recursive transition networks we can get recursive programs to create sentences and and language so then it appears that well if recursion is at the heart of intelligence um why are humans so bad at it and two is it really going to be what leads us to creativity and creating a computer which we can't distinguish from a man or from a person um and it's it's funny just to show it kind of the date on this book um Douglas hopstad actually says you know people people said it would only take 10 years before we could create a program a computer program that would be the world champion in chess and of course from then they said it would take another 10 years then another 10 years so he's kind of alluding to the idea that it would never happen but sure enough in the early 90s IBM had developed deep blue um who beat Kasparov you know that then World CH champion at
东西，而且我们其实还能造出各种，呃，用递归转移网络，我们可以得到递归程序来生成句子和语言，那么看起来，如果递归是智能的核心，呃，为什么人类如此不擅长递归呢，第二个问题是，它真的会是引领我们走向创造力、造出一台我们无法与人区分开来的计算机的东西吗，呃，而且很有意思的是，从这本书的年代就能看出来，呃，道格拉斯·侯世达其实说，你知道，人们说只要十年我们就能造出一个程序，一个计算机程序能成为国际象棋的世界冠军，当然从那以后他们又说还要十年，然后又是十年，所以他多少在暗示这永远不会发生，但果不其然，九十年代初 IBM 造出了深蓝，呃，它击败了卡斯帕罗夫你知道，当时的国际象棋世界冠军，这算是一次胜利，表明人类思考的方式
55:57
chess and it was kind of a Triumph showing that the way in which a human thinks and the way that a human plays chess is very much more intuitive than kind of an analytical problem solving try as you might to you know think like well and this is he was talking about replying a recursive algorithm is you basically analyze all possible moves and then you choose whichever one would be worst for your opponent and then in deciding what move they're going to make in the next step you apply the same method except you take on your opponent's role and you say well analyze all the possible moves given that one and what would be the worst move for them and then you continue this and depending how far out you can search this tree right we have this conceptual tree of chess moves um really helps give you an advantage but Kasparov doesn't think like that the way Kasparov thinks is he intuits something and and that's really kind of a magical element of human intelligence which we haven't yet been able to capture with our computer
以及人类下棋的方式，远比某种分析式的解题要来得直觉不管你多努力想，你知道，想着，好吧，他当时说的是应用一个递归算法，就是你基本上分析所有可能的走法，然后你选出对对手最不利的那一步，然后在决定他们下一步要走什么时，你用同样的方法，只不过你要代入对手的角色，你说，好，分析在那一步之下所有可能的走法，对他们来说最糟的一步是什么，然后你就这样继续下去，取决于你能把这棵树搜索多深，对吧，我们有这么一棵国际象棋走法的概念树，呃，这确实能给你带来优势，但卡斯帕罗夫不是那样思考的，卡斯帕罗夫的思考方式是他凭直觉领悟，而这其实是人类智能中相当神奇的一个元素，我们还没能用计算机程序捕捉到它，呃，然而深蓝还是设法
16预告第六章：Gavagai 与意义理论
56:55
programs um yet somehow deep blue was able to beat him and this is kind of a I think this is an interesting example of recursion what its role in intelligence is and whether it's going to be the final answer um just to give you kind of a quick show of things to things to do I'm not talking much about chapter 5 just because uh We've read that we've done that we you've had an entire lecture on on recursion and possible roles I want you guys to pay careful attention to CH 6 um which is your reading assignment for next time because there you're going to fundamentally ask the question of of how do we get meaning how do we know that our words mean anything so the idea of developing a theory of meaning of language goes back to the idea that suppose I were to plop you down on an island in the middle of nowhere and you know all these people are going and they're speaking their own kind of language um and you have no idea what it means right um yet you're stuck there you depend on them for survival and you figure it
击败了他，我觉得这算是，我觉得这是一个关于递归的有趣例子，关于它在智能中的角色是什么以及它是否会是最终答案，呃，简单说一下接下来要做的事，我不会太多讲第五章，只是因为，呃，我们已经读过了，我们已经，你们已经上了一整堂关于递归及其可能角色的课，我希望你们仔细关注第六章，呃，那是你们下次课的阅读作业，因为在那里你们会从根本上问这个问题，我们如何获得意义，我们怎么知道我们的词语有任何含义，所以，发展一套意义理论、语言理论的想法，可以追溯到这样一个设想，假设我把你丢到一个荒无人烟的孤岛上，你知道，那里的人来来往往，说着他们自己的某种语言，呃，你完全不知道那是什么意思，对吧，呃，可你被困在那儿，你得靠他们才能生存下去，你觉得
57:58
might be a good idea to go ahead and start learning their language how would you go about doing that so suppose you go out on your hunting missions with this you know this tribe and every time there's a rabbit which which you know gals through one of the guys says Gava guy Gava guy and you're like okay I I'm going to create an internal dictionary here I'm going to write a dictionary for this language and I know that I know that Gava guy uh I don't I don't know if that's how you spell it um that's equivalent to uh rabbit but now let's let's switch roles let's suppose you're one of those um one of those native people and you you kind of believe and part of your culture is that you never really view something as greater than the sum of its parts so when you say Gava guy you don't actually mean the whole rabbit but you just mean undetached Rabbit part so because the second when you start splitting up the rabbit and you're cooking it and you have its leg over the Open Fire it's no longer Gaba guy just like when we refer
去开始学他们的语言也许是个好主意，那你会怎么做呢，所以假设你跟着这个部落出去打猎，每次有一只兔子，你知道蹦过去的时候，其中一个人就说，Gavagai，Gavagai，你就想，好，我要在脑子里建一本内部词典我要为这门语言写一本词典，我知道，我知道 Gavagai，呃，我不知道是不是这么拼，呃，等同于，呃，兔子，但现在，我们换个角色，假设你是那些，呃那些当地人中的一个，而你相信，并且作为你文化的一部分，你从来不会把某个东西看作大于其各部分之和，所以当你说 Gavagai 时，你其实指的并不是整只兔子，而只是指未分离的兔子部件，因为一旦你开始把兔子切开，你在烹饪它，你把它的腿架在明火上，它就不再是 Gavagai 了，就像我们提到牛的时候，通常在吃它时说的是牛肉
59:06
to a cow we usually talk about beef and same with a pig we usually talk about pork once we start eating it because we have this idea of connecting Gaba guy to the full rabbit but they actually want that to be undetached Rabbit part which has a completely different conceptual Network for them than it does for us so this equivalence isn't true and this comes this actually be a very rigorous problem which presents itself in the theory of meaning and language and we're going to begin focusing on that in the role of isomorphisms and this in the next lecture so go ahead La you have a question so then how are you going to learn the language how you going to learn the language well we can obviously develop some sort of functional apparatus like I'm you know I can get close to a rabbit but how do we actually decipher meaning right we can obviously learn a language in some ways and in some ways it's the same how do you know what I say is exactly what you want me to say and know never does the same
猪也一样，我们通常说猪肉，因为我们有把 Gavagai 与整只兔子联系起来的想法但他们其实想让它表示未分离的兔子部件，这在他们那里所对应的概念网络跟我们的完全不同所以这个等价关系并不成立，而这其实是一个非常严格的问题，它出现在意义理论和语言理论中，我们下节课就要开始聚焦于这个问题以及同构在其中的角色，你来说，你有问题吗，那你要怎么学这门语言呢，你要怎么学这门语言，嗯，我们显然可以发展出某种功能性的手段，比如，你知道，我可以接近“兔子”这个意思，但我们究竟怎么破译意义呢，对吧，我们显然可以在某种程度上学会一门语言，而且在某种程度上这是同一回事，你怎么知道我说的话正好就是你想让我说的意思，而且从来不是完全同一回事，就是这样，完全正确，那么我们该如何发展一套意义
60:06
thing and that's an exact that's exactly it so then how do we develop a theory of meaning but plenty of good questions good questions for for next lecture and uh turn in the surveys and I look forward to seeing you guys next time class dismissed
理论呢，不过问得都很好，很好的问题，留到下节课，呃，把问卷交上来，我期待下次再见到你们下课
视频总结 · 一句话概括与核心要点
一句话概括
这节 MIT《哥德尔、埃舍尔、巴赫》第三讲以“一致性、完备性、几何”三个概念为线索，从希尔伯特纲领与集合论悖论讲到哥德尔两条不完备定理的核心思想（自指语句 + 哥德尔编码），再借非欧几何说明“解释”决定公理真假，最后通过《小和声迷宫》对话引出无穷、递归与意义理论等后续主题。
核心要点
- 一致性 = 不能同时推出 P 与非 P：讲者以“今天下雨且不下雨”为例；并强调爆炸原理——形式系统一旦出现任何矛盾，就能推出任意命题（如“桌子是红的且不是红的”可推出“宇宙无限”），所以矛盾会让整个系统崩溃。
- 完备性 = 所有真命题都可从公理推出：用“真理盒”图示说明——公理出发的推导树覆盖“可证”区域，但盒子里还有大片“真但不可证”的空间；这种“知道为真却无证明”的直觉正是哥德尔定理的核心。学生举的物理类比是“宇宙之外”：宇宙膨胀超过光速，光锥之外的事物永远不可感知。
- 希尔伯特纲领的动机是两类悖论：一是 19 世纪非欧几何革命，二是世纪之交的集合论悖论（理发师悖论：“只给不自己刮脸的人刮脸”的理发师是否给自己刮脸）。希尔伯特认为数学是最可靠的知识来源，必须证明数论的一致性和完备性。
- “1+1=2 比天空是蓝的更真”并非无条件：课堂投票中学生倾向选算术，但讲者反驳：若以雨滴合并定义加法，则 1+1=1；在模 2 整数中，1+1=0。数学的确定性依赖于对系统与解释的严格约定。
- 哥德尔第一/第二不完备定理的课堂表述：(1) 任何与数论同等强大的系统若能证明自身一致性，则必然不一致；(2) 任何与数论同等强大的系统必然不完备——存在可表述但不可证的真命题。
- 关键语句“此语句不可证明”与“说谎者悖论”的区别：若它为假，则它可证，而可证蕴含真，矛盾；因此它只能为真，且恰恰不可证。讲者特别提醒：“可证⇒真”成立，但“真⇒可证”不成立，方向只有一边。
- 罗素—怀特海的分层语言无法彻底封住自指：《数学原理》把语言分为 L1、L2（L2 谈论 L1 的句子，如“‘雪是白的’这句话……”），试图禁止自指；但哥德尔正是在 PM 系统内构造出上述语句，方法是哥德尔编码：给每个符号（含空格）分配唯一数字，把推理规则（如肯定前件：P、P→Q 推出 Q）同构为算术运算，把“可证性”变成数的性质（类似“是素数”）——这也回答了为何要求系统“至少与数论一样强”。
- 非欧几何说明公理真假取决于解释：欧几里得第五公设（过直线外一点有唯一平行线）1500 多年无法从前四条推出；在球面几何中以大圆为“直线”，任意两条直线必交于两点，三角形内角和可达 270°；在双曲几何（上半平面/单位圆盘模型）中，过一点可有无穷多条不相交直线。同一套术语“点、线”在不同解释下，第五公设可与前四条不相容。讲者顺带提到马克思、恩格斯曾试图把理论归约为数学以获得说服力——而数学本身若不稳固，这种做法便失去根基。
- 《小和声迷宫》与无穷收敛：对话中每个精灵用前一个一半的时间，总时间为 1+1/2+1/4+…=2，是几何级数收敛的例子；讲者将其与芝诺悖论（公元前 3 世纪左右）对照，指出人类花了近 2000 年才发展出处理无穷的工具（牛顿、莱布尼茨的微积分，欧拉等人的严格化；耶稣会甚至认为学微积分有助于理解上帝）。
- 无穷没有顶点，且可与自身真子集一一对应：整数与偶数可通过乘除 2 建立双射；区间 (0,1) 可映射到整条实数线；序数可不断构造 ω、ω+1、2ω…；基数上 2^ℵ₀ = ℵ₁≈实数的大小，康托对角线论证表明自然数与实数是不同层级的无穷。
- 递归与智能的张力：递归能生成分形、树、句子，但人类并不擅长递归；霍夫施塔特写书时暗示计算机下棋称霸“永远十年后”，而 1990 年代深蓝以穷举式递归搜索击败卡斯帕罗夫——后者靠直觉，前者靠树搜索，说明递归是否是智能的最终答案仍存疑。
结论与值得注意的细节
- 课程整体主线是“什么是‘我’、意识如何从物质中产生”，讲者鼓励学生开放讨论包括上帝、灵魂在内的话题，因为对话本身就出现了“无穷嵌套的精灵与上帝”的递归式神学意象。
- 两大思考工具被明确点名：递归与同构（isomorphism），后者将在第六章“意义从何而来”中成为核心。
- 预告下一讲的“Gavagai”问题（蒯因的翻译不确定性）：土著说 Gavagai 时可能指“未分离的兔子部分”而非“兔子”，如同英语中牛/牛肉、猪/猪肉的概念切分不同，说明词语与指称之间的同构无法仅凭观察确定——引出意义理论的难题。
- 讲者多次坦言未给出定理的完整证明，细节留待第九章；哥德尔编码的具体机制、康托对角线论证均为“以后可能演示”的内容。
核心句型 · 9
1. If A then B — but that only goes in one direction
“Careful that only goes in One Direction so it is certainly true that … things which are provable are true”
用于澄清蕴含关系的不对称性：A→B 成立不意味着 B→A。适合纠正听者把充分条件误当作充要条件的场合，是学术讨论中的常用提醒句式。
2. Suppose we … and we define X as … so in this system …
“Suppose we based our arithmetic on raindrops … we Define addition as when they meet so … in this system 1+ 1 equal 1”
构造思想实验的标准骨架：先设情境，再给定义，最后推出该系统内的结论。可用于展示「换一种解释，结论随之改变」的论证。
3. the only way that X is true is if Y
“So the only way that this statement is true is if it's not provable”
表达「当且仅当」式的必要条件，比 X is true only if Y 更强调唯一性，常用于归谬推理的收束句。
4. does anyone feel like … ? is anyone willing to defend X over Y?
“Is anyone willing to defend sky is blue over 1 plus 1 equals 2”
课堂或会议中征求反方意见的地道问法，defend X over Y 表示在两者之间为 X 辩护。可用于主持讨论、引出对立观点。
5. It took us well over N years to …
“It took us … a human Collective conscious well over you know 1,700 close to 2,000 years to understand and develop the tools”
强调某成果耗时之久，well over 表示「远超」，close to 表示「接近」，两者并用可精确又有力地给出时间跨度。
6. in order to talk about something you have to leap outside of it
“In order to talk about something you can't talk about yourself you have to LEAP outside of it”
用 in order to … you have to … 说明前提条件；leap outside of 是形象的动词短语，表达「跳出系统看系统」的元层次思想。
7. sure enough, …
“So he's kind of alluding to the idea that it would never happen but sure enough in the early 90s IBM had developed deep blue”
「果不其然」，用于承接前文预测或质疑后陈述实际发生的事，常带轻微的反转意味。适合叙事与举例。
8. X is a few stone throws away from Y
“Archimedes was in many ways conceptually just a few Stone throws away from it”
习语 a stone's throw away 表示「近在咫尺」；加 conceptually 可用于抽象距离。可仿写：The team was just a stone's throw away from a breakthrough.
9. try as you might to …, …
“Try as you might to you know think like well …”
让步句式，「无论你多努力想……」，倒装结构 try as you might 相当于 however hard you try，语气书面且地道。
生词精讲 · 109 · 按出现顺序
burning to ask
phr.
0:00
迫不及待想问
counterpoint
/ˈkaʊntərpɔɪnt/
n.
1:28
对位法（文中 cont crossa punctus 指 Crab Canon 对话）
derive
/dɪˈraɪv/
v.
1:28
（从公理）推导出
proposition
/ˌprɑːpəˈzɪʃn/
n.
1:28
命题
axioms
/ˈæksiəmz/
n.
1:28
公理
grappled with
phr.
3:04
与……角力；努力应对（难题）
deduce
/dɪˈduːs/
v.
3:04
演绎、推断出
goes Haywire
phr.
3:04
失控、乱套（口语）
abound
/əˈbaʊnd/
v.
3:04
大量存在、层出不穷
counterintuitive
/ˌkaʊntərɪnˈtuːɪtɪv/
adj.
3:04
反直觉的
any takers
phr.
4:23
有人愿意（回答/接手）吗
rules of inference
phr.
5:51
推理规则
incomprehensibly
/ɪnˌkɑːmprɪˈhensəbli/
adv.
5:51
无法理解地；（此处）小到看不清
wrap your head around
phr.
5:51
弄明白、想通（某难题）
big time
phr.
5:51
（口语）大大地、非常
light cone
n.
6:58
光锥（相对论中可观测事件的边界）
harp on
phr.
8:15
反复唠叨、不断强调
paradoxes
/ˈpærədɑːksɪz/
n.
9:34
悖论
doomed
/duːmd/
adj.
9:34
注定失败的、在劫难逃的
turn of the century
phr.
9:34
世纪之交
set theory
n.
9:34
集合论
sure and certain
phr.
10:39
确定无疑的（固定搭配）
number Theory
n.
10:39
数论
possible worlds
phr.
11:49
可能世界（模态逻辑/哲学术语）
perceptual
/pərˈseptʃuəl/
adj.
11:49
感知的、知觉的
abstract entities
phr.
11:49
抽象实体
windshield
/ˈwɪndʃiːld/
n.
13:05
挡风玻璃
identity
/aɪˈdentəti/
n.
13:05
同一性（哲学/逻辑）
modular arithmetic
n.
13:05
模算术
rigorously
/ˈrɪɡərəsli/
adv.
14:17
严格地、严密地
necessarily
/ˌnesəˈserəli/
adv.
15:51
必然地
nitty Gerty
phr.
15:51
（nitty-gritty）细枝末节、实质细节
ditto
/ˈdɪtoʊ/
adv.
15:51
同上、同前
formulate
/ˈfɔːrmjəleɪt/
v.
17:06
表述、构想（命题）
comes into play
phr.
17:06
开始起作用、派上用场
self- reference
n.
18:30
自指
rule oute
phr.
18:30
（rule out）排除、禁止
treads on a tire
phr.
18:30
轮胎花纹（喻极其乏味）
upstanding
/ʌpˈstændɪŋ/
adj.
19:33
正直的、守规矩的
subset
/ˈsʌbset/
n.
19:33
子集
LEAP outside of
phr.
21:02
跳到……之外
bottom line
n.
21:42
要点、关键所在
derivation
/ˌderɪˈveɪʃn/
n.
21:42
推导（过程）
vice versa
/ˌvaɪs ˈvɜːrsə/
adv.
23:12
反之亦然
cautioning against
phr.
23:12
告诫、提醒（不要……）
imply
/ɪmˈplaɪ/
v.
23:12
蕴含（逻辑）
isomorphism
/ˌaɪsəˈmɔːrfɪzəm/
n.
25:40
同构
Detachment
/dɪˈtætʃmənt/
n.
27:10
分离规则（逻辑中的肯定前件）
provability
/ˌpruːvəˈbɪləti/
n.
27:10
可证性
such and such
phr.
27:10
某某、如此这般
first glance
phr.
27:10
初步一瞥、粗略一看
lift himself up by his own bootstraps
phr.
29:44
揪着鞋带把自己提起来（喻无外力自我起步）
postulates
/ˈpɑːstʃələts/
n.
29:44
公设
got on uet's nerves
phr.
29:44
（get on one's nerves）让某人烦躁
intersects
/ˌɪntərˈsekts/
v.
31:04
相交
great circles
n.
31:04
大圆（球面上过球心平面的截线）
longitude
/ˈlɑːndʒətuːd/
n.
32:33
经度
spherical geometry
n.
32:33
球面几何
hyperbolic geometry
n.
33:52
双曲几何
upper half plane
n.
33:52
上半平面（双曲几何模型）
unit disc
n.
34:22
单位圆盘
inherently
/ɪnˈhɪrəntli/
adv.
35:46
内在地、本质上
Boggle the mind
phr.
37:12
令人难以置信、脑洞大开
perpendicular
/ˌpɜːrpənˈdɪkjələr/
adj.
37:12
垂直的
anecdote
/ˈænɪkdoʊt/
n.
37:12
轶事
advocating
/ˈædvəkeɪtɪŋ/
v.
38:32
倡导、主张
on short footing
phr.
38:32
（on shaky footing）根基不稳
nested
/ˈnestɪd/
adj.
39:49
嵌套的
indentations
/ˌɪndenˈteɪʃnz/
n.
39:49
缩进
fractal
/ˈfræktl/
n.
40:59
分形
holy mackerel
phr.
40:59
（惊叹语）我的天哪
speculate
/ˈspekjəleɪt/
v.
42:04
推测、猜想
bring out into the open
phr.
42:04
公开摊开来讲
persecuted
/ˈpɜːrsɪkjuːtɪd/
v.
42:04
迫害、打压
hostile
/ˈhɑːstl/
adj.
43:08
敌对的
Jesuits
/ˈdʒeʒuɪts/
n.
43:08
耶稣会士
infinitesimally
/ˌɪnfɪnɪˈtesɪməli/
adv.
44:28
无穷小地
take the limit
phr.
44:28
取极限
a few Stone throws away
phr.
44:28
（a stone's throw）近在咫尺、只差几步
rigorous
/ˈrɪɡərəs/
adj.
45:37
严格的、严密的
goes in spirit with
phr.
45:37
在精神上契合
wrestling
/ˈreslɪŋ/
v.
45:37
搏斗、艰难应对
geometric progression
n.
48:10
几何级数、等比数列
converge
/kənˈvɜːrdʒ/
v.
48:10
收敛
logarithmic scale
n.
48:10
对数尺度
Collective conscious
phr.
48:10
集体意识
Cardinal arithmetic
n.
49:21
基数算术
natural numbers
n.
49:21
自然数
exponentiation
/ˌekspoʊˌnenʃiˈeɪʃn/
n.
50:35
指数运算、幂运算
one toone correspondence
phr.
50:35
一一对应
bjective
/baɪˈdʒektɪv/
adj.
51:55
（bijective）双射的
interval
/ˈɪntərvl/
n.
51:55
区间
diagonal argument
n.
51:55
对角线论证
equivalence relationships
phr.
53:18
等价关系
well- defined
adj.
53:18
定义良好的、明确定义的
recursive transition networks
n.
54:42
递归转移网络
alluding to
phr.
54:42
暗指、影射
sure enough
phr.
54:42
果不其然
Triumph
/ˈtraɪʌmf/
n.
55:57
胜利、凯旋
try as you might
phr.
55:57
不管你怎么努力
intuits
/ɪnˈtuːɪts/
v.
55:57
凭直觉领悟
plop you down
phr.
56:55
把你随手扔到（某地）
in the middle of nowhere
phr.
56:55
荒无人烟的地方
undetached
/ˌʌndɪˈtætʃt/
adj.
57:58
未分离的
greater than the sum of its parts
phr.
57:58
大于各部分之和
conceptual Network
phr.
59:06
概念网络
decipher
/dɪˈsaɪfər/
v.
59:06
破译、解读
functional apparatus
phr.
59:06
功能性手段/装置
class dismissed
phr.
60:06
下课
理解自测 · 11 题 · 是真懂了，还是以为自己懂
1. 讲师给出的「一致性」和「完备性」定义分别是什么？
一致性指系统中没有互相矛盾的定理：不能从公理同时推出 P 和非 P（讲师用「今天下雨且不下雨」的天气预测举例）。完备性指系统内所有真命题都能从公理推导出来，即「真理盒子」图中没有真而不可达的空白区域。讲师在第 2–7 段依次给出这两个定义，并强调完备性更反直觉，因为它意味着可能存在「我们知道为真却无法证明」的东西，这正是哥德尔定理的核心。
2. 哥德尔的两条不完备性定理在讲师的表述中分别说了什么？
第一条：任何与数论一样强的系统必然不完备，即存在能表述出来但既无法证明也无法否证的真命题。第二条：任何与数论一样强的系统若能证明自身一致性，则该系统必然不一致——换言之，一致的系统不能证明自身一致。讲师在第 13–17 段板书这两条，并说明「一样强」的含义要靠哥德尔编号解释：系统必须强到能给符号编号并把可证性表示为数的性质。
3. 什么是 Gavagai 思想实验？它想说明什么问题？
讲师在第 50–52 段引入：把你丢到陌生部落，每次兔子跑过族人都喊「gavagai」，你自然把它译成「兔子」。但若该文化不把事物看作「大于部分之和」，gavagai 可能指「未分离的兔子部件」——兔子一旦被切开烹饪就不再是 gavagai，正如英语中牛变成 beef、猪变成 pork。这说明词与词的等价依赖于双方概念网络是否同构，仅凭行为证据无法唯一确定意义，这就是下一讲要处理的意义理论问题。
4. 《小和声迷宫》里每个精灵完成任务的时间减半，讲师让学生算出了什么结果？为什么这个细节重要？
总时间为 1+1/2+1/4+1/8+…，这是一个几何级数，和为 2。讲师在第 42–43 段指出，这说明一个无穷的递归过程可以在有限时间内收敛完成，侯世达特意设计此细节就是为了让无限嵌套的下降能够「返回」。讲师随后联系芝诺悖论：公元前 300 年左右芝诺用同样论证否认运动，而人类花了近两千年才发展出处理无穷的工具，这也呼应了微积分史的讨论。
5. 为什么「可证蕴含真」成立，而「真蕴含可证」不成立？请复述讲师用哥德尔句所做的论证。
可证蕴含真是系统可靠性的基本要求，也是人们信任数学的理由：有证明就知道为真。但反向不成立，讲师在第 21–22 段用句子 G=「本命题在 PM 中不可证」论证：若 G 假，则它所说的为假，即 G 可证；而可证的东西为真，与 G 假矛盾。故 G 只能为真；而 G 为真意味着它所说的成立，即 G 不可证。于是得到一个真而不可证的命题，直接填充了「真理盒子」中可证区域之外的空白，证明真↛可证。
6. 讲师为什么要在讲哥德尔定理时插入非欧几何的讨论？两者的共同主题是什么？
共同主题是「解释」（interpretation）：形式符号本身无真假，真值取决于赋予的解释。第 26–32 段中，同样的「点」「线」术语，在平面上第五公设为真，在球面（大圆为线，两线必交）上为假，在双曲圆盘（无穷多条不交线）上也为假。这说明第五公设与前四条独立，欧氏几何不再是唯一确定的真理。它与雨滴算术、模 2 算术一样，都在动摇「数学是确定无疑知识」的希尔伯特信念，为「数学自身可能不一致」的问题铺路。
7. 讲师为何说「矛盾可以推出任何东西」是一件「离谱」的事？这与希尔伯特的动机有什么关系？
讲师在第 3 段指出，若系统中任何地方出现矛盾（如「桌子是红的且不是红的」），逻辑上可以推出任意命题，包括「宇宙是无限的」，这与直觉相悖，哲学家对此长期困惑。正因如此，一个不一致的系统毫无价值。希尔伯特把数学视为最可靠的知识来源，认为若数学有缺陷人类就无法认识真理，因此他要求证明数论的一致性和完备性（第 9–10 段）。哥德尔的第二定理恰恰说明这种自证一致的愿望无法在系统内部实现。
8. 哥德尔编号如何让「数论谈论自身」？请按讲师的思路说明其步骤。
讲师在第 22–25 段分三步说明：第一，给每个符号（包括空格）指派唯一数字，使每个命题对应一个哥德尔数；第二，把推理规则（如由 P 和 P→Q 得 Q 的分离规则）转化为数之间的算术运算，形成推理规则与算术规则的同构；第三，把「可证性」表达为数的一种性质，就像「是素数」一样。于是「本命题不可证」就变成「某个数不具有某性质」这样的纯数论命题。这也回答了学生的问题：系统必须「与数论一样强」，才能完成这种编码。
9. 讲师用深蓝击败卡斯帕罗夫的例子想说明递归与智能之间什么样的张力？
第 48–50 段中，讲师先指出递归能生成分形、语言等复杂结构，似乎是智能的核心，但随即提出两个疑问：人类为何不擅长递归？递归能否带来创造力？深蓝用递归的极小化极大搜索遍历走法树，确实赢了棋；但卡斯帕罗夫靠直觉而非穷举思考，这种直觉是人类智能中「尚未被程序捕捉的神奇元素」。讲师的结论是开放的：深蓝证明递归可以在特定任务上胜过人，但它未必是智能的「最终答案」，这为全书关于心智本质的追问保留悬念。
10. 有人反驳说：「宇宙之外我们看不到的东西」就是「真而不可证」的例子，哥德尔定理不过是这种认识局限。讲师会如何回应？
讲师在第 7–8 段已预先回应：他承认光锥之外的宇宙是一个「不错的物理例子」，但明确说「这并不完全是我在形式系统意义上所指的」。差别在于：物理视界是因观测手段有限而不知道某事实，本质上是经验性的；而哥德尔式的「真而不可证」是在给定公理系统内、关于自然数的命题，我们能从外部（元层面）确知它为真，却能证明它在系统内部无法推出。前者可能随技术进步改变，后者是逻辑必然，任何足够强的一致系统都逃不掉。
11. 把「真值取决于解释」的论点迁移到语言学习：这能否解释 Gavagai 问题，又与哥德尔定理有何联系？
可以。几何中「线」的意义由解释决定，同样「gavagai」的意义由说话者的概念网络决定；学习者以为建立了词典等价，实则可能在两套不同构的概念系统间做了错误映射，正如把 1+1=2 解释为雨滴合并会失效（第 26 段）。与哥德尔的联系在于「跳出系统」：正如 L1 的句子只能在 L2 中被谈论，要判断一种翻译是否正确，也需要一个更高的元视角，而学习者身处系统之内往往无法获得。学生追问「那怎么学语言」正说明这一困境无法完全消除，只能靠功能性的近似，这也是第六章「意义的位置」要展开的内容。
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