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第 40 期 · 核心追问 Ⅱ·07「语言还承载意义吗？」
MIT Godel Escher Bach Lecture 4
节目发布 2012-12-02章节 · 点击跳转视频
0:00
课程回顾与哥德尔编号补读
3:24
「雪是白的」：意义来自关系
7:37
漂流瓶：框架、外层、内层信息
10:23
伏尼契手稿与齐普夫定律
12:39
雪与苹果：脑中的联想网络
18:37
藏头诗解码：对话如何指向巴赫
24:32
点唱机理论：意义不是内在的
28:00
反向追问：科学作为逆向工程
34:51
方程是藏信息还是统一现象
43:27
L-系统：从字符串长出分形
59:19
两种描述一个对象：柏拉图之问
66:35
生命游戏与停机问题
77:36
投票模型、水滴与普适类
85:40
波、离散宇宙与芝诺悖论
94:09
信息熵 vs 算法复杂度
本期小问 · 档案清单
24:32
语言还承载意义吗？
59:19
数学是发现，还是发明？
77:36
整体能大于部分之和吗？
94:09
信息越多越接近真相吗？
本期讲者
贾斯汀·库里MIT 本科生，2007 年 IAP 期间开设「哥德尔、艾舍尔、巴赫」研讨课的主讲人之一，负责本讲前半段的意义理论与信息论部分；后成为应用拓扑学研究者。
柯伦·凯莱赫MIT 本科生，本课联合讲师，负责后半段的 L-系统、元胞自动机与波模拟的现场编程演示；曾与 Justin 一起参与新英格兰复杂系统研究所的冬季研讨班。
Latif课堂上最活跃的提问者，提出「意义来自与自身的关系」（被讲者称为 Latif 假说）以及「数学是否只是把信息藏起来」「空间是否离散」等关键质疑。
01课程回顾与哥德尔编号补读
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 welcome back everybody I'm glad to see more faces today um so the feedback looked pretty good um and once again I encourage you guys to brutally honest I believe wholeheartedly in this whole idea of a democratic classroom you know and having this feedback process allows to correct and make this an even better learning experience for everybody um so I handed out a couple things uh first of all I handed out kind of foremost chapter 10 from uh Douglas Hop's newest book I am a strange Loop um and that's called girdle's quintessential strange Loop and I think I'm okay and I'm not going to be sued because as long as it's less than 10% and it's for academic purposes it should be okay and considering this is
以下内容是根据知识共享许可协议提供的，你的支持将帮助 MIT 开放课件继续免费提供高质量的教育资源。若要捐款或浏览来自数百门 MIT 课程的更多资料，请访问 MIT 开放课件网站 ocw.mit.edu。好，欢迎大家回来，很高兴今天看到更多面孔。嗯，反馈看起来还不错，我还是要再次鼓励大家绝对坦诚。我是打心底里相信民主课堂这个理念的，有了这个反馈流程，我们才能不断纠正，把这门课变成对所有人来说更好的学习体验。我发了几样东西，首先发的是道格拉斯·霍夫施塔特最新著作《我是个怪圈》的第十章，那一章叫做《哥德尔的典型怪圈》。我觉得应该没问题，不会被起诉，因为只要不超过10%，而且是用于学术目的，应该就没事。考虑到这只是一本三百页书里的十五页左右，
1:06
like 15 pages out of a 300 page book uh should be under that um this is supplemental reading a lot of you raised kind of Notions of like girdle numbering does not make sense and well it takes a lot of time to wrap your head around it I mean there's a reason why there was one girdle and why he named it girdle numbering uh it wasn't like it was a trivial idea go ahead sort of like um so the fundamental idea behind girdle numbering is that you can take the formal patterns in any type of proving procedure whether it's the Miu system or whether it's proving from pianos axioms and number Theory and you can code those formal deductions into manipulations of numbers right and then once you create this link create this isomorphism into numbers you can then play around with these large numbers which then decoded give you back formal statements in the system you were playing with originally so so maybe like having an equation like no Solutions having like exactly so letif said it's like having an equation with no Solutions is
应该在那个比例之内。这是补充阅读材料。你们很多人提出了类似哥德尔编号讲不通这样的疑问，嗯，确实要真正理解它需要花很多时间。我是说，历史上只出了一个哥德尔，这是有原因的，这个编号被命名为哥德尔编号也是有原因的，它可不是个微不足道的想法。你说。差不多是这样，嗯，哥德尔编号背后最根本的思想是，你可以把任何一种证明程序中的形式模式——无论是 MIU 系统，还是从皮亚诺公理出发在数论里做证明——把这些形式推演编码成对数的操作，对吧？一旦你建立起这种联系，建立起通往数的这种同构，你就可以摆弄这些很大的数，而把它们解码回来，就又得到你原先所研究的那个系统里的形式陈述。所以，也许就像一个方程没有解一样？完全正确，所以莱蒂夫说的是，一个方程没有解，
2:17
like having an unprovable thing um that's actually going to be very close to the essential idea behind girdles proof um but if you read chapter 10 I believe Douglas hoffstead does um out of this new handout I gave you does a great job of explaining some of the key ideas behind girdle's incompleteness theorem and that's the main one that systems as powerful enough as number Theory are inherently incomplete um so that that's something I want to leave you at least two weeks to kind of chew on because eventually you're going to work on to chapter nine in GB where he does mumon and and girdle and uh there's going to be that's where he proves or does a version of um girdles incompleteness theorem in in GB but it's really not as clear as I think it's done here so um that's just kind of supplemental reading uh so now on to what I started last time and what should have been the reading assignment for today was uh the location of meaning in chapter six um I started out with this idea of of the rabbit and and gabag Guy um but I
就好比有一个不可证明的东西。嗯，这其实非常接近哥德尔证明背后的核心思想。不过如果你读第十章，我相信道格拉斯·霍夫施塔特在我发给你们的这份新材料里，非常好地解释了哥德尔不完备性定理背后的一些关键思想，而最主要的那条就是：像数论这样足够强大的系统本质上就是不完备的。嗯，所以这是我想至少留给你们两周时间慢慢消化的东西，因为你们最终会读到《哥德尔、艾舍尔、巴赫》的第九章，他在那里讲无门和哥德尔，而且，嗯，那里就是他证明，或者说给出某个版本的哥德尔不完备性定理的地方。但我觉得那部分讲得不如这里清楚，所以，嗯，这就当作补充阅读。好，现在回到我上次开的头，也就是本该是今天阅读作业的内容，第六章《意义的定位》。嗯，我上次讲到了那个兔子和「gavagai」的例子，嗯，但我
02「雪是白的」：意义来自关系
3:24
want to kind of scale back a little bit and and really bring the question down to to something which some of you will intuitively know and some of you won't have any idea and it's the uh the following sentence um and please some of you who know Spanish better correct my spelling um so Blanca who knows what this means okay so everybody raises their hand says snow is white okay but for those of you who didn't raise your hand what does this sentence mean it means nothing okay why because I can't relate to it you can't relate to it so so far we've kind of got the the Latif hypothesis which is that meaning comes from the relationship of of this to yourself um and this is an idea we're going to try to explore throughout all of today's lecture and we're going to co- te Kar and I are going to do some excellent things um so suppose you know you were dropped off in Mexico and just by hanging out with people you eventually learned Spanish and you created you created a dictionary with a set of recursive rules and you
想稍微退一步，把这个问题拉回到某个你们当中有些人会凭直觉就懂、而有些人会完全摸不着头脑的东西上，那就是下面这个句子。嗯，会西班牙语的同学请帮我纠正一下拼写。嗯，Blanca，谁知道这是什么意思？好，所有人都举手了，说「雪是白的」。好，那对于没举手的同学来说，这个句子是什么意思呢？什么意思都没有。好，为什么？因为我无法与它建立联系。你无法与它建立联系。那么，到目前为止我们已经……有了 Latif 的假说，也就是意义来自这个东西与你自身的关系，这个想法我们打算在今天整节课里都去探讨一下，我们要——我和 Kar 要一起做些很棒的事情。那么假设你被丢到墨西哥，就靠着跟人相处，你最后学会了西班牙语，然后你建立起了一部词典，配上一套递归规则，你学到说，我们有些东西
4:48
learn that well we have certain things like this is equivalent to is and you understand this in your natural language um and you create this method of of breaking down sent of sentences strings um into parts and then going to your dictionary and saying okay n s well I know s is is n I'm not sure what that is but um my dictionary tells me it's snow and Blanca um my dictionary tells me is is white okay so suppose you you you've dropped off in in the middle of Mexico and just by interacting with the people and you've created this kind of um set of recursive rules for first parsing sentences and I think that's the more languages you try to learn the first thing you're going to discover is The Stumbling Blan is the ability to actually parse what someone's saying um I know when I when I was living in in Germany this past summer basically when someone would talk to me it'd be like and I'm like excuse me like and it didn't even sound like there were words individually it was just a stream of sounds right but
比如这个等同于「是」，而你用自己的母语理解它，于是你就发明出一套方法，把句子、把字符串拆解成一个个部分，然后去查你的词典，说好，n s，嗯我知道 s 就是「是」，n 我不太确定是什么，但我的词典告诉我它是 snow（雪），而 Blanca，我的词典告诉我是white（白）。好，假设你被丢在墨西哥的正中间，仅仅靠着跟当地人互动，你就建立起了这么一套递归规则，先用来解析句子。我觉得你越是多学几门语言，你首先会发现的就是，最大的绊脚石其实是真正把别人说的话切分开的能力。我记得去年夏天我住在德国的时候，基本上人家跟我说话就是一串音，我就想，不好意思？那听上去甚至不像是一个个的词，就是一连串声音，对吧。但是
6:08
eventually by just being immersed in the culture my brain ran these kind of neural network algorithms that were were able to start breaking up those into um you know beer esot or something like that um and you you you could then actually start hearing individual things and then trying to deduce the meaning of those individual parts and then plugging them back together into your natural language to what appears to be snow as white but what do snow is white mean maybe the words don't anything okay okay so Latif claims that maybe the words don't themselves mean anything but what is that mean so all right so then the question is suppose I I wrote on a piece of paper um snow is white and I packaged it into a bottle so there's our scroll and I and I cork it and I throw it into the ocean right and it drifts up on shore what would then be the cue so then suppose it lands in this
最终，光是浸泡在那个文化里，我的大脑就跑起了某种神经网络算法，能够开始把那串声音切开，比如切成 beer、esot 之类的东西，然后你就真的能开始听到一个个独立的单元，接着试着推断这些部分各自的意思，再把它们拼回你的母语，得到看起来像「雪是白的」这样的句子。可是「雪是白的」到底是什么意思呢？也许词本身什么都不意味着。好，好，Latif 说也许词本身并不意味着任何东西。但那又是什么意思呢？好，那么问题就是，假设我在一张纸上写下「雪是白的」，然后把它装进一个瓶子里，那就是我们的卷轴，我塞上软木塞，把它扔进大海，对吧，然后它漂上了岸。那接下来的线索会是什么呢？那么假设它漂到了这个
03漂流瓶：框架、外层、内层信息
7:37
island and you see this bottle why would you suspect that the bottle meant anything why would you just pick up the bottle and throw it in your bag of trash as you go and trying to clean up your Beach seems very well constructed so it see Latif says it seems very well constructed um so is there what is it about this bottle and scroll and cork conf configuration that cues you into the idea that that there's information that there's meaning in here paper Okay so it seems deliberate in some senses so the idea that well it seems hard that accidentally you know some paper crept into a bottle corked itself and then sent it through the ocean um so so Hoffer kind of in in this past chapter breaks things down into into three layers of meaning uh he's got what's called a frame message we've got an outer
岛上，你看见了这个瓶子。你为什么会怀疑这个瓶子意味着什么？你为什么不是随手把瓶子捡起来扔进你的垃圾袋，一边走一边清理海滩呢？——它看起来做得很精心。好，Latif 说它看起来做得很精心。那么，这个瓶子、卷轴加软木塞的组合，究竟是什么东西提示了你，让你想到这里面有信息、有意义？——纸。好，所以它在某种意义上显得是有意为之的。也就是说，很难想象一张纸偶然爬进了一个瓶子里，自己塞上了软木塞，然后被送进了大海。所以侯世达在上一章里把事情分解成了三个层次的意义，他称之为框架信息（frame message），我们还有外层
8:47
message and then we've got an inner
信息（outer message），然后我们还有内层
8:57
message so h would argue that the frame message here is the fact that there's this strange configuration of a bottle of cork and a piece of paper that would cue a human being any kind of reasonably curious human being and he argues that fundamentally curiosity is a characteristic of human beings that that there's something to be solved here that there's there's information here that this is kind of screaming out to to our to our stick figure on on the uh island of tmia that decode me decode me figure out what I am so suppose then we we unpack this piece of paper and he saw this but he doesn't speak English how would he then proceed to try to extract meaning from this so the argues we could develop it into words so how would we first of all argue that these aren't just meaningless you know scraps of paper you know just marks maybe it's like a Jackson Pollock painting or something and it's just appears to be a bunch of gibberish Sandra or any like repeats W okay so Sandra said that there there's
信息（inner message）。所以他会说，这里的框架信息就是这样一个事实：存在着一个奇怪的组合——一个瓶子、一个软木塞和一张纸，它会给任何一个有相当好奇心的人一个提示。他还认为，好奇心从根本上就是人类的一个特征——这里有个东西要解，这里有信息，这东西简直是在朝着我们那个在 Tmia 岛上的火柴人大喊：解码我，解码我，弄清楚我是什么。那么假设我们把这张纸打开，他看到了这个，但他不懂英语，他接下来会怎么试着从中提取意义呢？他说我们可以把它拆成一个个词。那我们首先怎么论证这些不只是毫无意义的、纸上的涂鸦呢？也许它就像一幅波洛克的画，只是看起来像一堆胡言乱语。Sandra？有重复。好，Sandra 说这里有重复，这里似乎有
04伏尼契手稿与齐普夫定律
10:23
repetition there there's appears to be you know certain things that there're there's even this like kind of concept of a space that there should be something that's parsed here there's separation into Parts Latif May so Latif ruse maybe somebody's trying to fool you um it's interesting one of the questions I have actually in in regards to chapter six that I handed out to you so I asked you guys to Google the voyich manuscript and the voich manuscript and that's pronounced voich has been the subject of intense interest by cryptologists for you know a good 100 years or so all the guys who were you know working during WW2 to crack the German Enigma code when in their free time try to decrypt the voyich manuscript because it was this very strange medieval text had this really intricate system of writing and it had then pictures of all these sort of alien plants and weird things like this that were not at all found anywhere in Europe but the book was a European book so people were were wondering what the hell does this mean
某些东西，甚至还有一种「空格」的概念，说明这里有东西需要被切分开，有分隔成部分的痕迹。Latif？也许有人想耍你。嗯，很有意思。其实关于我发给你们的第六章，我有一个问题。我让你们去谷歌一下伏尼契手稿（Voynich manuscript），伏尼契手稿，这个读作 voich，一百多年来一直是密码学家们高度关注的对象。所有那些二战期间破解德国恩尼格玛密码的人，闲下来时都会试着去破译伏尼契手稿，因为它是一部非常奇怪的中世纪文本，有一套极其精细的书写系统，里面还画着各种像是外星植物之类的怪东西，那些东西在欧洲任何地方都找不到，可这本书偏偏是一本欧洲的书。所以人们一直在想，这他妈到底是什么意思。而这个问题其实一直没被解决，
11:32
um and the problem really wasn't solved until just about a year ago when somebody showed how you could produce almost an exact replicate of the voyich manuscript using a random generator and it was you would just have like a prefix for a word like um go and then you you would have then like a a midix I don't know what it's called um GLE GL and then and then like a suffix and then you would create like a matrix of these things and You' shift around and just randomly start generating words um and then they would appear to look like natural intelligible words but they actually meant nothing so the voyich manuscript actually contained no real meaning but it seemed to trick everybody because it had a lot of the same patterns that human language did there's some other interesting things that um that's connected to kind of parsing information but um I don't really have uh all the time to uh go into one of these things speaking of W's and letters there's actually something called Zips law which every kind of
直到大约一年前，有人展示了怎样用一个随机生成器几乎一模一样地复制出伏尼契手稿。做法就是，一个词有个前缀，比如 go，然后再有个……中缀？我不知道那叫什么，比如 GLE、GL，然后再来个后缀，然后你把这些东西排成一个矩阵，来回移位，随机地开始生成词。这样生成出来的词看上去就像是自然的、可读的词，但它们其实什么意思都没有。所以伏尼契手稿其实根本不含真正的意义，但它却骗过了所有人，因为它带有很多和人类语言一样的模式。还有一些别的有意思的东西也跟解析信息有关，不过我没有那么多时间一个个讲了。说到词和字母，其实有个东西叫齐普夫定律（Zipf's law），任何一种语言或者有意义的信息似乎
05雪与苹果：脑中的联想网络
12:39
language or meaningful message appears to obey and that's if if you rank the most common letters like I believe e and some of the vowels are the most common letters um they they follow a power law distribution which means that the second most common letter is exactly half as apparent as the first one and then so on and it follows this very strict mathematical relationship and they've even run this these these kind of power law detections on DNA and extracted that it appears to follow the same behavior as our natural languages do um and and if you feel like Googling that that's Zip's law Z PF um but you know there's a lot of things going on here and I want to return to the question well what does snow as white mean um and in breaking down that question question what is what does snow mean when I say snow what do you think think snow think of snow okay but what does that mean let's avoid circularity okay so Latif thinks of a picture of snow you know he what happens him to Latif is a visual
都遵守它。就是说，如果你把最常见的字母排个序，我记得是 e 和一些元音字母最常见，它们服从幂律分布，也就是说第二常见的字母出现频率恰好是第一名的一半，以此类推，遵循一个非常严格的数学关系。人们甚至把这种幂律检测跑到 DNA 上，发现它似乎表现得和我们的自然语言一样。而且如果你想谷歌一下，那就是 Zipf's law，Z-P-F。总之这里面有很多有意思的事情，而我想回到那个问题：「雪是白的」到底是什么意思？在拆解这个问题的时候，「雪」是什么？「雪」是什么意思？当我说「雪」的时候，你会想到什么？想「雪」，想想雪。好，但那又意味着什么呢？我们别陷入循环定义。好，Latif 想到的是一幅雪的画面，也就是说，他脑子里某个地方被唤起了一个视觉图像，
14:02
image is prompted in his head somewhere in in kind of his his his Tangled brain here um the visual section of his brain is activated and he has a visual memory of of snow or the first time that we had a large snow or a blizzard and the first time you ever saw snow um but about anybody else what what what does snow mean to you so then we've got kind of a in some ways a tactile sense of just of being of cold so then another part of the teeth's brain here lights up when you say snow and it goes to all the times you ever felt cold maybe just sitting in this room and having the air conditioning up a couple notches too high um you you you have a feeling like geez it's freezing in here right and freezing then triggers idea of Frozen water which triggers the idea of snow um but what else what does snow mean to you some of you might see it as oh well this is an opportunity to go you know snowboarding or something right sorry here's my bad Recreation of bindings um or go skiing or something or it's Recreation it's time it's winter
就在他这个纠缠的大脑里，他大脑的视觉区被激活了，于是他有了一段关于雪的视觉记忆，比如我们碰上一场大雪或暴风雪的那次，或者你第一次见到雪的时候。那其他人呢？「雪」对你们来说意味着什么？——于是我们又有了某种触觉上的感受，就是冷。于是当你说「雪」的时候，Latif 大脑的另一部分亮了起来，通向你所有曾经感到冷的时刻，也许就是坐在这间屋子里，空调开得高了两档，你就会有种感觉：天啊这儿冻死了，对吧。而「冻」又触发了冰冻的水这个概念，进而触发了雪的概念。那还有别的吗？「雪」对你意味着什么？你们中有些人可能会觉得，噢这是个机会，可以去滑单板之类的，对吧。抱歉，这是我画得很烂的固定器。或者去滑雪之类的，或者说它意味着娱乐、意味着冬天，
15:25
it's you know it's a break from school there's there's a whole really complex conceptual Network in everyone's brain with with the word snow just like if I were to pick apple and I ask people when I say apple completely out of context what is the first thing you think of pie apple pie okay anyone else Orchards anyone else [Applause] red Adam and Eve Adam and Eve I'm glad someone said it because I I thought I was going to have to so what what then do you do you associate with the apple and Adam and Eve say it okay so then we've got wow holy mackerel we've already gone from Apple to God pretty damn quickly um maybe I should draw that off of Adam and Eve um if we want to make it seem like that was the the track we fell um but then what what was what was it that Adam and Eve are being punished for for eating the Apple what was it the tree of knowledge how did Newton discover gravity
意味着放假不用上学。每个人脑子里都有一整张非常复杂的概念网络跟「雪」这个词连在一起。就像我要是挑「苹果」，我问大家，当我完全脱离语境地说「苹果」，你第一个想到的是什么？——派。苹果派，好。还有别人吗？——果园。还有吗？[掌声] 红色。亚当和夏娃。亚当和夏娃，我很高兴有人说了，因为我还以为得由我自己来说。那么你会把苹果和亚当夏娃跟什么联系起来？说吧。好，于是我们有了——哇，我的天，我们已经从苹果一路飞奔到上帝了。也许我该把它从亚当和夏娃那儿引出来，如果我们想让它看起来像是我们真是这么一路走过来的话。那么亚当和夏娃因为吃了苹果而受到惩罚，是因为什么？是什么呢？——知识树。牛顿是怎么发现引力的？
17:04
what computer am I using Mac excellent um or your iPod or or whatever um so immediately if if if if somehow I could look at your brain and and kind of take an ongoing you know pet scan of of what's being what's lighting up as I say the word Apple it would be this extremely complicated explosion of of electrical activity in your brain that lights up and tickles every part of the section of your brain that relates to the term apple right now of course then when I when I have a frame of message like uh this apple is tasty right then then you immediately narrow down well what what is he talking about is he eating about just a fresh Apple a red apple a granny smith apple um maybe he's talking about eating apple pie so you you you prune kind of your your ongoing neural net and you you stick to this side so I've noticed that nobody's come and claimed their dollar or two um and this is I think a convenient point to uh pull out a dialogue we did a long time ago uh does any remember the Contra cross dep
我用的是什么电脑？——Mac，太棒了。或者你的 iPod 之类的。所以说，如果我能看进你的大脑，给它做一个持续的 PET 扫描，看看当我说出「苹果」这个词时你脑子里哪些地方在亮，那会是一场极其复杂的电活动大爆发，点亮并撩动你大脑中每一个跟「苹果」这个词相关的区域。当然了，当我给出一个框架信息，比如「这个苹果很好吃」，你就会立刻收窄范围：他到底在说什么？他是在吃一个新鲜的苹果、一个红苹果、一个青苹果？也许他说的是吃苹果派。于是你就修剪掉了你那张正在运转的神经网络的一部分，只留下这一边。我注意到还没有人来领他们那一两块钱。我觉得这是个很方便的时机，可以把我们很久以前做过的那段对话拿出来。有人还记得《Contracrostipunctus》吗？
06藏头诗解码：对话如何指向巴赫
18:37
punctus and you know just as an Apple has multiple meetings this dialogue cont page 75 has a variety of meanings um I needed someone to uh start reading off the first letter of every sentence that Achilles tortoise Achilles taurise says H sorry sorry um so I think it's ho right so we go oh yeah so go Achilles tauris Achilles tortoise oh sorry h f s r s c o n r all right one second s c o n o r a s c o sorry SC o o r sorry o n t n t r a i think too many s is here um a c r r s t i there's o s t i p n u n
就像「苹果」有多重含义一样，这段对话——第 75 页——也有多种含义。我需要有人开始把阿基里斯和乌龟每句话的第一个字母念出来，念的是 H……抱歉抱歉，我觉得是 ho，对吧。那我们来，噢对，来吧，阿基里斯、乌龟、阿基里斯、乌龟，噢抱歉，h fs r s c o n r，好，等一下，s c o n o r a s c o，抱歉，SC o o r，抱歉，o n t n t r a，我觉得这儿的 s 太多了，a c r r s t i，这儿是 o s t i p n u n
20:16
c a c r s t i c a l l y a Y is
c a c r s t i c a l l y a y 是
20:38
r s s p l l s j period s a did anybody see this so and I should have asked it as a homework assignment but one of the things that the turtle and U tortoise and achilles says um is you know starting page of 81 he says you know there are many many clever ways of hiding things in music KY says or in poems poems used to do very similar things you know though it's rather out of style these days for instance Louis Carol often hid words and names in the first letters or characters of the successive lines and poems he wrote poems which conceal messages that way are called acrostics so we already had kind of a frame and outer message screaming out saying I'm talking about me sillies decode me um and it's already from the dialogue which had its one level of meaning it then within itself pointed upwards to a higher level of meaning which is this acrostic we just pulled out but what does the acrostic tell us to do sure and and what does that get us sorry acrostic Glee there shouldn't be a space here so join that over there
r s s p l l s j 句号s a有人看出来了吗？我本该把这个当作业布置下去的。乌龟和阿基里斯说的其中一件事，大概从第 81 页开始，他说，在音乐里藏东西有非常非常多巧妙的办法。KY 说，或者在诗里。诗以前也常干很类似的事，虽然现在已经相当过时了。比如说，刘易斯·卡罗尔就常常把词和名字藏在连续几行的首字母里。他写的这种藏有信息的诗叫做藏头诗（acrostic）。所以我们已经有了某种框架信息和外层信息在朝我们大喊：我在说的是我啊，笨蛋们，快解码我。而这已经是从那段本身有一层意义的对话里来的，它又在自身之内指向了更高一层的意义，也就是我们刚刚提取出来的这个藏头。可这个藏头又告诉我们要做什么呢？没错。那这又给了我们什么？抱歉，acrostic Glee，这儿不该有空格，把它连过去。
22:30
let's see um well if we then look at this as another cross stick and we take the first letters of each
我们来看看。如果我们把这个再当成一个藏头，取每个词的
22:52
word what do we get
首字母，我们会得到什么？
23:02
and then the third I think probably the highest level of meaning we can extract out of this it's jsbach so to what extent does Contra crop punctus mean jsbach right um and there's I think a lot of really neat neat stuff going on when he kind of asks about messages which can talk about themselves and so build themselves right and that's basically what we did here is we had a dialogue which gave us instructions on how to extract meaning from this and then it in turn gave us another message which told us how to extract meaning from it and in some ways it's a really inefficient things to thing to do because what was I don't know it was like a 10-page dialogue um that much was needed in order to extract the inner message or one of the enter messages here being jsbach so that's kind of a poor compression ratio here um you know it took 10 pages to tell us one name um but I think it's interesting nonetheless [Music] um but still not everybody picked this up so to what extent does this dialogue mean jsbach
然后是第三层，我想大概是我们能从中提取出来的最高一层意义：J.S. Bach。那么《Contra-crostipunctus》在多大程度上意味着 J.S. Bach 呢？我觉得当他谈到那些能谈论自身、从而构建自身的信息时，里面有很多非常精妙的东西。而这基本上就是我们刚才做的事：我们有一段对话，它给了我们如何从中提取意义的指令，然后它反过来又给了我们另一条信息，告诉我们如何从中提取意义。而在某种意义上，这么做效率非常低，因为那是——我不知道——大概十页长的一段对话，居然需要这么多篇幅才能提取出那条内层信息，或者说其中一条内层信息，也就是 J.S. Bach。所以这里的压缩比相当糟糕，你知道，花了十页只告诉我们一个名字。但我还是觉得挺有意思的。[音乐] 不过还是没有所有人看出来这一点。那么这段对话在多大程度上意味着 J.S. Bach 呢？
07点唱机理论：意义不是内在的
24:32
well I would argue not to a large extent right um and fundamentally what I'm going to argue here and this kind of goes with the go with the Latif hypothesis or conjecture is that um meaning is not inherent and to expand on this and this is what what hopster calls the Jukebox theory of meaning [Applause] uh meaning is the relationship of [Applause]
我会说，程度并不大，对吧。而从根本上说，我在这里要论证的、也跟Latif 的假说或者猜想相符的是：意义不是内在固有的。进一步展开的话，这就是侯世达所说的「意义的自动点唱机理论」。[掌声] 意义是[掌声]
25:22
things but what does that mean what I'm trying to argue here is that apple if if we wrote apple on a piece of paper and we shot it out into outer space and maybe some alien civilization Came Upon it um that there there's not really much inherent meaning in having a few ink blotches on a piece of paper or in here I have some you know Limestone broken onto a piece of rock right to what extent does limestone broken onto a piece of rock mean anything right and the only reason we say apple has meaning is because there's there's this complex isomorphism between the visual input of Apple when you read it to the electrical activity in your brain right just like and you know I don't actually know any Chinese here but um I can try to construct a character and to what extent does that mean something and you know I could have someone sitting in here and say well no that's actually the Chinese symbol for tranquility right but how was I ever supposed to know that how how how how was I supposed to discern this from
事物之间的关系。可这又是什么意思呢？我在这里想说的是，「苹果」——如果我们把 apple 写在一张纸上，然后把它射向外太空，也许某个外星文明捡到了它，那么，一张纸上的几团墨迹里其实并没有多少内在固有的意义。或者像这里，我有一块石灰岩，在一块石头上敲出些痕迹，对吧，那它在多大程度上……石灰石上刻下的痕迹本身并不意味着什么，对吧，我们之所以说 apple 这个词有意义，是因为在你读到 Apple 时的视觉输入，和你大脑里的电活动之间，存在着一种复杂的同构关系对吧，就像——你知道，其实我并不会中文，但我可以试着造一个字出来那这在多大程度上算是有意义呢？我完全可能让某个人坐在这儿说，不对，这其实是中文里表示“宁静”的那个字，对吧，可我怎么可能知道这一点呢？我到底、到底该怎么把它和一堆胡写乱画区分开？这其实说明了
26:46
nonsense um and really this this shows that meaning doesn't lie here or or here but it relies it lies here and the relationship of things so the extent to which this means anything is the extent to which it has a direct correlation in the minds of the people using the terms it's just like when you have these native languages and they die out they don't longer carry meaning or really they continue to exist in any way shape or form is that you don't have people actively using and kind of interplaying this complex feedback process of modifying what these things mean because also think about how the how these symbols how these marks on this Limestone have changed in meaning if we were to rewind a half 100 years ago I would Apple would definitely not mean laptop right and you know some I don't know if the the myth involving Newton and his discovery of gravity um which is a myth by the way um existed 100 years ago but let's suppose it did but this rewind 500 years ago um then this would absolutely have no connection with with
意义不在这边，也不在那边，而是在这里——在事物之间的关系里。所以这东西有多大程度上算有意义，取决于它在使用这些符号的人的头脑中，有多大程度的直接对应关系。这就像那些原住民语言，当它们消亡时，就不再承载意义了，或者说它们其实已经无法以任何形式继续存在，因为不再有人在积极使用它们，不再有那种不断修改这些东西含义的复杂反馈过程的互动另外也想想看，这些符号、这些刻在石灰石上的痕迹，它们的含义是怎么变化的如果我们把时间倒回五十年前，Apple 绝对不会是指笔记本电脑，对吧。我也不太确定，关于牛顿发现万有引力的那个传说——顺便说一句那是个传说——在一百年前是不是已经存在了，不过我们就假设它存在。但如果再倒回五百年前，那这东西跟
08反向追问：科学作为逆向工程
28:00
apple and you know we would have this this would be the supported neural network of of what this string meant so fundamentally I'm arguing here that this only means as much as the stuff it causes here but this brings kind of a reverse question I mean with an isomorphism you go two directions right so suppose instead and this is really what a lot of neuroscientists would love to do I just had a camera running of all the activity going on your brain and I wanted to reveal your thoughts to what extent could I say uh firing in sector B 861 5G means apple right to what extent can I say that right and this is actually going to bring forth um a really important question which will be the exploration of the rest of today's lecture um is to what extent does does physical activity have meaning how do we distinguish between random physical things and pattern physical things right what was it when Galileo was born in church and he looked at the pendulum rocking back and forth and he would use his heartbeats to keep
apple 之间就完全没有任何联系了。而这些，就构成了支撑这串字符含义的那个神经网络。所以我基本上想说的是，这串东西的意义，全在于它在这里（大脑中）引发了什么。但这也带出一个反过来的问题，我是说，同构关系是双向的，对吧。那假设反过来——这其实正是很多神经科学家梦寐以求的事——假设我用一台摄像机记录下你大脑里所有的活动，我想借此揭示你的想法，那我在多大程度上能说，B 861 5G 区域的放电就意味着 apple？我在多大程度上能这么说？这其实会引出一个非常重要的问题，也是今天剩下这堂课要探讨的：物理活动在多大程度上具有意义？我们怎么区分随机的物理现象和有规律的物理现象？就像伽利略当年在教堂里，看着来回摆动的吊灯，用自己的心跳来数摆动的周期，他
29:31
track of the period of the swings and he noticed that it was always the same and it didn't even have to do with how big this Mass was here but it seemed completely based on the length now in some ways what does what is the meaning of a of a swinging pendulum right and if I wanted to describe a a pendulum to you I could do a variety of things just like I could ask well what does the firing in these in your brain mean ideally in the future we'll be able to say well that firing that particular sequence of firings in your brain meant you were thinking about an apple but what happened you know fortunately a couple hundred years ago is that somebody was looking at this and they realized that they could do better instead of just filming the phenomenon instead of just saying okay well what happens when a pendulum swings like well the best level of description we can do is you know a streaming 256k video webcam of a pendulum right I mean that's a really bad compression right to encode the information ah key word information
发现周期总是一样的，而且甚至跟这个重物有多大无关，似乎完全取决于摆长。那么从某种意义上说，一个摆动的钟摆，它的意义是什么？如果我想向你描述一个钟摆，我可以有很多种做法。就像我也可以问，你大脑里的这些放电意味着什么？理想情况下，将来我们能够说，你大脑里那一串特定的放电序列，意味着你正在想一个苹果。但幸运的是，大概几百年前，有人在观察这个现象时意识到他们可以做得更好——不只是把现象拍下来，不只是说，好吧，钟摆摆动的时候会发生什么？我们能给出的最好描述就是用一个 256k 的网络摄像头去直播拍一个钟摆？我是说，那真是糟糕透顶的压缩方式，对吧，用来编码信息——啊，关键词，信息。我要论证的核心，
30:49
fundamentally what I'm going to argue we're pulling this down to is to what extent is there information in here and the study of theory of meaning an information theory is going to be our mathematical and rigorous approach to to what the theory of meaning is so how much information is in this right and what it took was that somebody eventually realized well I can I can describe this this complex motion actually rather simply well if I looked at the angle Theta that it makes I could say and I apologize to you who haven't had calculus but when I have a DOT above the symbol I mean I want to see how quickly it changes with time so if this were just a single dot I would mean what's the kind of just change in time of of the angle Theta but if I had two dots I asking what's the change in time of the change in time of the angle Theta and you could think of this as acceleration or whereas a single dot would correspond to Velocity I can actually write down the equation of motion of a pendulum as such Theta dot plus and this is just a
我们最终要归结到的，就是这里面到底有多少信息。而对意义理论的研究和信息论，将会是我们用数学的、严谨的方式去回答“意义理论是什么”。所以这里面有多少信息？关键在于，终于有人意识到，我其实可以相当简单地描述这个复杂的运动。如果我看它形成的角度 θ，我可以说——先跟还没学过微积分的同学道个歉——当我在符号上面加一个点，意思是我想看它随时间变化得有多快所以如果只有一个点，我说的就是角度 θ 随时间的变化率，但如果加了两个点，我问的就是“角度 θ 随时间的变化率”本身随时间的变化率，你可以把它理解成加速度，而一个点就对应速度。这样我其实就可以把一个钟摆的运动方程写成这样：θ 的二阶导数加上——这只是个常数——乘以 sin θ 等于 0。然后你们以后会学到，这个方程
32:19
constant sin theta equals 0 and and then if any of you as you will learn is that this is too hard to solve as an equation but instead we express it in a linear form and don't worry if you don't know what any of these terms mean or we approximate sin Theta which some of you have seen in pre-calculus um or even geometry as just Theta around around zero we can approximate sin Theta passing to the origin sorry as just the linear function of theta but the point is don't worry about the mathematics here I could be talking about any system and what we'll show you today is that what the project of physics is and what the project of all of science is is it's reverse engineering right all of what hoffstead has been talking about in chapter six has been going One Direction you start with a string and you ask what does it mean right and my answer was that what apple means is it means the activity your brain or and then we could say that well it actually refers to the physical apple somewhere out there in the universe but
作为方程来解太难了，所以我们改用线性形式来表达它。如果你不懂这些符号也别担心。我们把sin θ 做近似——有些人在预备微积分甚至几何课上见过——在零点附近就把它近似成 θ 本身，抱歉，是在原点附近把 sin θ 近似成 θ 的线性函数。但重点是，别纠结这里的数学。我完全可以换成任何一个系统来讲。今天我们要给你们展示的是，物理学的整个事业，乃至整个科学的事业，其实就是逆向工程。侯世达在第六章里讲的所有内容都是朝一个方向走的：你从一串符号出发，问它意味着什么，对吧。而我的回答是，apple 的意义就是它在你大脑里引发的活动。或者我们也可以说，它其实指向宇宙中某个地方的那个实体苹果，但
33:36
that becomes a difficult problem but then what about going the reverse way to what extent does activity in your brain mean this is the same question which Hoff cider doesn't talk about as to what extent does the motion of a pendulum you know just kind of rocking back and forth um let's see do I have any there you go you know to what extent is the meaning of this motion encoded in these marks on a piece of paper and this is actually in some ways a much harder question is how do we take the output and code it into the input and what we're going to be doing today is playing with some active computer simulations and showing to what extent does the output we see see on the screen can we figure out just from that what was the underlying mechanism and really what all of Science and physics has been about is showing that the complicated motions of the entire universe can be explained very shortly using symbols like this and this is going to get to an idea that we're going to call algorithmic complexity right what's beautiful about
那就变成一个难题了。那反过来走呢？你大脑里的活动在多大程度上意味着某个东西？这是同一个问题，而侯世达没有谈到它，就像问一个钟摆的运动，就这么来回晃着——我看看，我有没有……好，来了——这个运动的意义，在多大程度上被编码进了纸上的这些符号里？其实从某些角度看，这是个难得多的问题：我们怎么把输出反推回输入？而我们今天要做的，就是玩一些实时的计算机模拟，看看仅凭我们在屏幕上看到的输出，我们能在多大程度上推断出背后的机制。而整个科学和物理学一直在做的事，就是证明整个宇宙中那些复杂的运动，可以用这样的符号非常简短地解释清楚。这会引出一个概念，我们称之为算法复杂度。钟摆
09方程是藏信息还是统一现象
34:51
the pendulum is that it doesn't require a streaming feed all the time to describe its motion I can write it down in one line of mathematics and give you the complete time evolution of this system and even with the initial conditions of how fast was I swinging it initially and from what place I can get to this whirling phenomenon and get all sorts of incredibly complex Behavior out of just these marks on a Blackboard and then similarly when some of you were here two lectures ago and you looked at the mandal Bro set extremely complex fractal what if somebody had handed you that and said tell me what equation produces this for the most part you couldn't have done it what like equation of hiding the information and because you just look at the equation but in your head all these have you're talking about but the physical thing is actually sag is being sort of connected with the sy so you just adding stuff to the Sy okay so Latif said well isn't all mathematics just really hiding the information so
的美妙之处在于，描述它的运动并不需要一路不停地直播录像，我可以把它写成一行数学，就给出了这个系统完整的时间演化。再加上初始条件——一开始我摆得多快、从哪个位置放手——我就能得到这种旋转的现象，从黑板上这几个符号里，推出各种各样极其复杂的行为。同样地，你们中有些人两节课前在这儿看过曼德博集合，一个极其复杂的分形。如果有人把它交给你，说，告诉我什么方程能产生这个图形，绝大多数情况下你是做不到的。（学生：）这不就像是方程把信息藏起来了吗，因为你只看方程，但在你脑子里这些东西你都在说……不过那个物理的东西其实是跟符号连在一起的，所以你只是往符号上加东西而已。好，Latif 说，那所有的数学是不是其实都只是在把信息藏起来？我接下来要做的——
35:55
and and what I'm going to kind of do and correct me if my interpret ation what you said is wrong um is that really all we're doing is is we're taking our mental process our visual input and and we're abstracting away the details we're creating an abstraction barrier between that and this but I would argue to an extent no right um although fundamentally this sentence is only meaningful to mathematicians right and thus the models they have in their brain but this actually reveals a very simple relationship of so what happen happens when we when we talk s Theta How would how would you parse that well you might draw a circle and then say okay if I had a triangle here I would say that sin Theta is opposite over the hypotenuse right and the fact that this has a relationship with that involves some serious thinking but yes you are masking some of the details but if you wanted this to run on a computer it would be much more efficient to just code this up in a programming language than to take a film right you can do
如果我理解错了你的意思请纠正我——就是说，我们做的其实无非是把我们的心智过程、我们的视觉输入，把细节抽象掉，在两者之间建立一道抽象屏障，隔在那个和这个之间。但我要在一定程度上反驳这一点。虽然从根本上说，这个式子只对数学家才有意义，也就是只对他们脑中的模型有意义，但它其实揭示了一个非常简单的关系。当我们说 sin θ 的时候会发生什么？你会怎么解读它？你可能会画一个圆，然后说，好，如果这里有个三角形，我会说 sin θ 就是对边比斜边，对吧。而这个式子和那个式子之间存在联系，这需要相当认真的思考。但没错，你确实屏蔽掉了一些细节。可是如果你想让它在计算机上跑，把它写成一段程序代码，要比拍一段影片高效得多。你当然可以拍，
37:04
that but when you like to like find out what those symbols actually stand for it turns out that you're not really obstructing it too much you just making like these symbols just hand for like the thing so okay so latif's saying that's fundamentally still what we're doing is is we're we're saying that we're letting the symbol stand for the thing itself but I'm still going to argue against you right and the reason for that is that there's something unifying behind the looking at the arm swinging in a clock and what I'm doing with this cable and then as you'll start to study more and as we'll show today you know throwing a rock in a puddle and then just getting this entire wide class of oscillatory phenomenon what is it like to do this right and we're going to see it everywhere even when and and you can really observe this when you're driving and then we're going to get into we're going to get into wave equations the idea of when you're driving down a highway and you hit kind of a a backed up section of
（学生：）但当你想弄清楚那些符号究竟代表什么的时候，结果发现你其实并没有抽象掉多少东西，你只是让这些符号去代表那个东西而已。好，Latif 的意思是，从根本上说我们做的仍然只是让符号代表事物本身。但我还是要反驳你，原因在于，看时钟里摆动的指针，和我拿这根绳子做的这件事，背后有某种统一的东西。随着你们学得更多，也像我们今天要展示的，往水坑里扔一块石头，你就得到了一整个大类的振荡现象。这是怎么回事呢？我们会发现它无处不在，甚至在你开车的时候都能真切地观察到。接下来我们还会讲到波动方程：当你在高速公路上开车，遇到一段堵住的车流，发生的其实是
38:15
traffic what what's happening is that the guys in the front stopped and that forced you to stop and that forced the car behind you to stop and you propagated a wave backwards but still we can describe this ire large set of a seemingly unrelated physical phenomenon we're not talking about this single clock we're not talking about this single pendulum we're not talking about this single cable we're talking about a whole variety of phenomenon and just writing down this equation um and I think that's a really neat thing but this is a phenomenally hard thing to do it's hard to see all of these classes of phenomenon it's hard to see me throw this piece of chalk in the air and extract the equation with which describes it right so what we're going to spend the rest of today talking about is is not how do we go from Apple or how do we go from the equation to figuring out well what does it mean right and it's it's like the same question of does x² does the equation Y = X2 does that really mean the graph of x squared right and
前面的人停了，逼得你也停下，又逼得你后面的车停下，于是你把一个波向后传播了出去。但我们仍然可以用同一套东西描述这一大堆看似毫不相关的物理现象。我们说的不是这一个钟，不是这一个钟摆，也不是这一根绳子，而是一大类各式各样的现象，而只要写下这一个方程就够了。我觉得这真的很妙。但这件事做起来极其困难，很难看出这些现象属于同一类，很难看着我把这支粉笔扔到空中，就提取出描述它的方程，对吧。所以今天剩下的时间我们要谈的，不是我们怎么从 Apple 出发，或者怎么从方程出发去搞清楚它意味着什么。这就像是同一个问题：x²，方程 y = x²，它真的意味着 x 平方的那条曲线吗？你会看到，不，并不总是。它可以意味着各种各样的东西，我甚至不用
39:29
you're going to see that no it doesn't always it can it can mean a whole variety of things I don't even have to graph it I could be talking about a different domain and all sorts of different different phenomenon but it's going to be the relationship of these different ideas that we're going to be talking about and uh we've got some great things to show for you and for the most part I'm going to be handing this sorry yes Latif like isn't it like when you see all these similarities between different um structures and we like making stuff so are we making stuff up yes we are making stuff up because we're only modeling things here right like if you actually then measured you know does does the rate of change of the rate of change of the angle this cable um always equal minus s of the angle um and for this system obviously no right because I mean look it's bending right it's got all sorts of weird elastic properties it's actually the Rod's not stiff it is actually flexing about and doing all
把它画出来，我完全可以在谈论另一个领域、各种各样不同的现象。但我们要谈的，正是这些不同想法之间的关系。我们有一些很棒的东西要展示给你们，接下来大部分时间我要把讲台交给——不好意思，Latif 你说。（学生：）当你看到不同结构之间的这些相似性时，我们是不是在编东西？我们是不是在自己造出这些东西？是的，我们确实是在“编”，因为我们只是在给事物建模。比如你如果真的去测量，角度变化率的变化率，对这根绳子来说，是不是总是等于负的 sin(角度)？对这个系统显然不是，因为你看，它在弯曲，对吧，它有各种奇怪的弹性性质，这根杆其实并不刚硬，它在到处弯来弯去，做出各种
40:31
sorts of crazy things um and you know this isn't a massless rod here with just a big ball on the end it's this thing's got almost as much mass as the end um so no but the fact that we can model most of the behavior most of the Salient features of the phenomenon in a rather simple way is one of the remarkable things about science and about mathematics um so just to kind of convince you that complexity can come out of Simplicity you might say okay well doing this isn't really that complex I mean I probably could have figured out a simple way to describe it go ahead does that show something about the universe something about because you see maybe the universe is doing the same compated okay so Latif says maybe the universe is doing something different and we're just showing how it can be and simpler that's a good question but the problem is is that the most we can do the way that we fundamentally think as humans since we're such simple creatures and we're not the universe well we're part of it um is that we have to be able
乱七八糟的动作。而且这也不是一根无质量的杆末端挂个大球，这东西本身的质量几乎跟末端的一样多。所以不，并不精确。但我们能用相当简单的方式，把大部分行为、把现象中大部分显著特征都建模出来，这正是科学和数学了不起的地方之一。那么，为了让你相信复杂性可以从简单性中产生——你可能会说，好吧，做这个其实也没多复杂，我大概自己也能想出一个简单的描述办法。（学生：）你说。这是不是揭示了关于宇宙的某些东西，因为你看，也许宇宙其实是在做同样复杂的事情？好，Latif 说，也许宇宙在做的是另一回事，而我们只是在展示它可以被简化。这是个好问题。但问题在于，作为人类，我们思考方式的根本极限就在于——我们是如此简单的生物，我们不是宇宙，好吧，我们是它的一部分——我们只能
41:45
to simplify our phenomenon in ways like this and if you ever end up reading a book by Seth Lloyd called programming the universe you'll say that the universe itself is a quantum computer carrying out Quantum operations all the time and maybe somewhere in there at the you know beginning of the universe uh there there was a monkey typing away instructions into the into the quantum computer that is the universe that said okay well we're going to let Force always equal mass times acceleration for those of you who've done physics um and we're also going to let this and this and we're going to um and this was all hardcoded in there but the issue is we don't know that we don't know what happened exactly at the beginning the universe we don't know how exactly the universe operates there's still Mysteries out there and the fact that we can go from throwing this piece of chalk to understanding what's the law which governs that is really a remarkable achievement of humankind um but I want to really Hammer home this
用这样的方式去简化我们面对的现象。如果你哪天读到 Seth Lloyd 写的一本书，叫《编程宇宙》（Programming the Universe），书里会说宇宙本身就是一台量子计算机，无时无刻不在执行量子运算。也许在宇宙诞生之初的某个地方，有一只猴子在往那台作为宇宙的量子计算机里敲指令，说，好，我们规定力永远等于质量乘以加速度——学过物理的同学都懂——我们还规定这个、还有这个，还要……而这些全都是硬编码进去的。但问题是我们并不知道，我们并不知道宇宙诞生之初究竟发生了什么，我们并不知道宇宙究竟是怎么运作的，仍然有很多未解之谜。而我们能够从扔出这支粉笔，一路走到理解支配它的定律是什么，这真的是人类了不起的成就。但我特别想把这个想法敲进你们脑子里：复杂的现象往往有一个非常简单的
42:47
idea that that complex phenomenon have a um have a very simple underlying feature and this is going to relate to some different ideas of information Theory which I might periodically pop in and invade um on C Curran's time here but for the most part he's going to take over the show now um but should we take a quick break yeah let's take all right so let's come back in probably five minutes and you know refresh ourselves so and then car will take over all right
底层机制。这会关联到信息论里的一些不同概念，我可能会时不时冒出来，占用一下 Curran 的时间。不过接下来大部分时间就交给他了。要不我们先休息一下？好，我们休息吧。行，那我们大概五分钟后回来，让大家喘口气，然后 Curran 就接手。好的。
10L-系统：从字符串长出分形
43:27
okay so I'm GNA start talking uh so this is the serinsky triangle which I talked about last time and there are a lot of different ways to get to this shape so he's talking about how do you go from from seeing this shape to understanding the process of how to do it it's it's not like an easy thing um this is what's sort of unique to humans we can step back and say like okay what's going on here what kind of more generalized process might lead to this thing so I'm going to talk about a bunch of different systems uh Linden Meer systems and cellular automa and a bunch of examples of each um so first of all I'll just do an example of a Linden Meer system a Linden Meer system is when you start with a string a string is just a list of of symbols like characters and you apply these rewrite rules and you have this grammar so each each one of the symbols goes to some other set of symbols when you rewrite it so I'll just do an example um we start with f and we have these rules F goes to well this means it's a string when I put
好，那我开始讲了。这是谢尔宾斯基三角形，我上次讲过的，有很多不同的方法可以得到这个图形。他刚才讲的就是，你怎么从看到这个图形，走到理解产生它的过程，这并不是件容易的事。这大概是人类比较独特的地方：我们能退后一步说，好，这里到底发生了什么，什么样的更一般化的过程可能会导致这个东西。所以我要讲一堆不同的系统，比如林登迈耶系统（L-系统）、元胞自动机，以及每一类的一些例子。首先，我先举一个林登迈耶系统的例子，一个林登Meer 系统是这样的：你从一个字符串开始，字符串就是一串符号，比如字符，然后你应用这些重写规则，你有一套文法，所以每一个符号在你重写它的时候都会变成另外一组符号。我就直接举个例子吧，我们从 F 开始，我们有这些规则，F 变成……嗯，我加引号是表示它是个字符串，F 变成
44:53
it in quotes F goes [Applause] to
[掌声] F。所以每当我们遇到符号 F，我们就把它替换成 F+F--F+，这个对你们来说有什么意义吗？
45:13
F so when whenever we have this symbol F we replace it with f+ fusus f plus and like does this mean anything to you guys probably not so let's just go through it and so F when we apply the rule for the first time what we get is this we get f plus f - - F plus so here's when we really get to the recursive nature of Linden Meer systems we feed this string back into the rule and so for each F here we replace it with that so this F the first one the first F becomes F + f - - F plus so that's what this first F becomes then we add plus the second F and we do that whole string of stuff again minus and then another repetition of that and then plus and we just keep doing this um and now let's recall the cotch curve from last time what we do is the first thing we have this just a line and we apply this rule that says we uh make this little notch in it so we get to this and then we apply that same rule to each one of these things right
大概没有，那我们就一步步来走一遍。所以 F，我们第一次应用规则时得到的是这个，我们���到F+F--F+。所以到这里我们才真正看到林登迈尔系统的递归本质：我们把这个字符串再喂回规则里，于是这里的每一个 F，我们都用那个式子替换它。所以这个 F，第一个 F，就变成了 F+F--F+，这就是第一个 F 变成的东西，然后我们加上加号，第二个 F 又变成一整串同样的东西，然后减号，再来一遍那一串，然后加号，我们就这样一直做下去。现在让我们回忆一下上次讲的科赫曲线，我们做的第一步是，我们有这么一条线，然后我们应用这条规则，说我们在它上面做出这么一个小凸起，于是我们得到这个，然后我们对其中的每一段再应用同样的规则，对吧。那这是什么意思呢？有谁知道这意味着什么吗？这些符号——意义其实来自于
47:02
so what is what does this mean does anybody have any idea what this means this these these symbols so the meaning comes out of the isomorphism like he said how do you interpret them so F in this case um F stands for go going forward what yeah the this and this are actually the same thing this is the power of linen Meer systems so F stands for going forward and plus stands for um changing the angle by 60° and minus stands for changing the angle by minus 60 degrees so first we do the first um I'll just use this [Applause] word the first f is just this this is f and then this is the second one so F forward plus 60° F Well f+ 60° f - 60° - 60° again f plus 60° F sorry there should be an F on the end the real wrong but uh this is what you get you can think of the way this is taught is think of a turtle and a turtle has an x y and a Thea a position and a and a rotation so each symbol is interpreted as having some sort of effect on this Turtle so I wrote a program that does exactly this so let's look at that
同构，就像他说的那样，你怎么去解释它们。所以在这个例子里，F代表……F 代表向前走。什么？对，这个和这个其实是同一个东西，这就是林登迈尔系统的威力所在。所以 F 代表向前走，加号代表把角度改变 60 度，减号代表把角度改变负 60 度。所以首先我们做第一个，我就用这个词吧 [掌声]，第一个 F 就是这个，这是 F，然后这是第二个。所以 F 向前，加 60 度，F，也就是 F、+60 度、F、-60 度、再 -60 度、F、+60 度、F，抱歉，末尾还应该有一个 F，写错了，不过大致就是你会得到这个。你可以这样理解——通常教的方式是想象一只乌龟，乌龟有 x、y 和一个theta，也就是一个位置和一个朝向。所以每个符号都被解释成对这只乌龟产生某种作用。我写了一个程序就是干这个的，我们来看看，科赫林登迈尔系统。我快速过一下代码，我们有 turtle x、turtle y、turtle theta，这是关键，这是乌龟的起点。然后我们从字符串 F 开始，所以这就是 F，第一个字符串。
49:16
uh the cotch lyen Meer system so I'll just quickly go over the code um so we have Turtle X Turtle y Turtle Theta this is the important thing this is where the turtle starts so and we start with this string F and um so this is f the first string and 4 in i i less than depth i++ this just means do this certain thing depth times and depth is a variable so when I click on the screen depth increases by one um string which is f equals string like what the what the string is now replace everything in that string by the specific thing replace F by f+ f-- f plus F and then we interpret it we give the symbol's meaning by this isomorphism how we interpret it so for each character in the string if the character is f then what we do is we store the previous values and we move forward and coine of sign is just how we interpret the angle go forward um times the increment which is the the the length that we go forward by and we add a new line with those X and Y values so this is what f means and then plus what it means is just ch turn the
然后 for i，i 小于 depth，i++，这就是把某件事做 depth 次，depth 是个变量，所以当我在屏幕上点击时，depth 就加一。然后 string，也就是 F，等于——把当前字符串里的所有东西都替换成指定的东西，把 F 替换成 F+F--F+F。然后我们解释它，我们通过这个同构给符号赋予意义，也就是我们怎么去解释它。所以对字符串里的每一个字符，如果这个字符是 F，那我们要做的就是先存下之前的值，然后向前移动，cos 和 sin 就是我们解释角度的方式：向前走，乘以增量，也就是我们每次向前走的长度，然后我们用这些 x 和 y 值加一条新的线段。这就是 F 的含义。然后加号的含义就是把乌龟稍微转一下，把乌龟的角度转 π/3；减号也一样，从乌龟的角度里减去 π/3。我们可以运行一下看看是什么样子。我还加了一些别的东西，比如鼠标监听。首先，它会
50:56
turtle a little bit turn the turtle's angle by pi over 3 and likewise with minus subtract from the turtle angle pi over 3 so we can run this and see what it looks like and I added some more things like Mouse listening so first of all and it's printing out the strings so this right here if you can see it says F so I'm going to click and it does that just the rule that we talking about and now the string says f plus f-- f plus F and when it's interpreted gives you this picture so if we click it again apply the rule feed it through we get this and it prints the string here so this is the next string and then we do it again and and the string gets longer so we can do it again and again and again and we get the cotch curve which is this strange curve that has you know infinite length which is pretty wild any any questions so far so lyen Meer system is very general yeah can you go back to the picture do the picture again sure so what if you like see backwards is it going to what what like see if you flip the picture over it's
把字符串打印出来，所以这里，如果你看得见的话，写着 F。我要点一下，它就做了刚才说的那条规则，现在字符串是 F+F--F+F，解释出来就给你这幅图。所以我们再点一次，应用规则，喂回去，我们得到这个，然后它在这里把字符串打印出来，这是下一个字符串，然后我们再来一次，字符串越来越长。我们可以一次又一次地做，最后得到科赫曲线，这条奇怪的曲线，你知道，它有无限长，相当疯狂。到目前为止有什么问题吗？所以林登迈尔系统是非常通用的。请讲。你能回到那张图吗，再放一次那张图？当然。那如果你，比如说反过来看，它会……什么？就是说，如果你把这幅图翻过来，就像你有某个形状，那如果你取另一个形状呢，你知道，边界，你想要……上来比划一下，
52:33
like you have a certain sh what if you take the other sh you have you know the boundaries take you want to come up in motion and show what you mean if you were to view this curve as starting from here and going to here be the same and it's upside down it's the exact same thing yeah that's why it's a fractal there are copies of itself inside of itself nested infinitely that's that's what it means to be fractal so for example this whole thing is this right here this thing is the same as the whole thing and likewise a smaller version of what you were saying if we flip this upside down to here it's the same thing yeah so yes Isn't that cool so here's another Linden Meer system which is a bit more confusing I think but it's the same idea it's a rewrite system Lind Meer system um so we start with the string a and then the rule the rule for grammar is a becomes B minus a plus B and B becomes a plus B minus a and the string becomes that and what these things mean is if it's a or b go forward and plus you know go turn the
说明一下你的意思。如果你把这条曲线看成是从这里开始走到这里，会不会一样？它是上下颠倒的，但完全是同一个东西。对，这就是为什么它是分形，它内部有自己的副本，无限嵌套，这就是分形的含义。比如说，这整个东西就是这里这一小块，这一块和整体是一样的。同样，按你说的，取一个更小的版本，如果我们把这个上下翻转到这里，还是同一个东西。对，所以是的。这不是很酷吗？这里还有另一个林登迈尔系统，我觉得它稍微更让人困惑一点，但思路是一样的，还是重写系统，林登迈尔系统。我们从字符串A 开始，然后规则，文法的规则是 A 变成 B-A+B，B 变成 A+B-A，字符串就变成那样。这些东西的含义是：如果是 A 或 B 就向前走，加号就是把乌龟往一边转，减号就把乌龟往另一边转。所以我们得到的是……我运行一下，看看是什么样子。首先我们从这个开始，这
54:10
turtle one way and minus turn the turtle the other way and so what we get here I'll I'll run this and we can see what it looks like so first we start with this this is just a line we apply the iteration uh this is twice actually because if you iterate it once it goes like upside down and off the screen so I just didn't do it so this is after two iterations so think carefully about this shape um yeah so I'll just it iterate it again and we'll notice that from each one of these lines this shape is going to grow again only in in alternating directions so the original line was here here here so here it grows this way here it grows that way here that way goes that way there it goes up it's like inter yeah it is like it's interlocked within itself so I'll iterate it once more and from each one of these little lines it's going to stem one of those shapes again and it's it's approaching the a Pinsky triangle so we do it again come on do it again there so it's it's just a bunch of little squiggly lines but it's it's
就是一条线，然后我们应用迭代，这个其实是两次，因为如果只迭代一次，它会上下颠倒跑到屏幕外面去，所以我就没那么做。这是两次迭代之后的样子。仔细想想这个形状。对，那我就再迭代一次，我们会注意到，从这里每一条线段上，这个形状又会长出来一次，只不过方向是交替的。原来的线段在这里、这里、这里，所以这里它往这边长，这里往那边长，这里往那边，那里往那边，那里往上长，就像是互相……对，就像是它自己嵌套交错在自己里面。我再迭代一次，从这些小线段里的每一条，又会长出那个形状，它正在逼近谢尔宾斯基三角形。我们再来一次，来吧，再来一次，好，所以它只是一堆弯弯曲曲的小线段，但它正在变成谢尔宾斯基三角形。对。那你这个用的是 F 那个字符串还是别的字符串？对，这个字符串不是
55:37
becoming the serinsky triangle so yeah did you the F string or what string did you use for this yeah so this string is not F but it it begins with a it starts with a here and it applies these rules to it so these are the this is the print out of what the string is so first it's a after two iterations becomes this and then every a in this second string every a here is going to be replaced by this thing and that's what that's what we get with this one it just goes off so but it's a similar concept so this is the string and likewise with the tree um I had a tree example last time and we can make a lenden Meer system for a tree and the string is FBR so if we look right here the string is FBR and the rule is every B becomes this so we can really think about this and give these things meaning by interpreting them so f means go forward b means there's a bud on the end of the branch and R means reverse so let's think of building a tree we're we're going forward so F and on the top there's a b bud and then R
F，它是从 A 开始的，从这里的 A 开始，然后对它应用这些规则。所以这些是……这是打印出来的字符串。一开始是 A，两次迭代之后变成这个，然后这第二个字符串里的每一个 A，这里的每一个 A，都会被这个东西替换掉，这就是我们用这个得到的结果，它就一直这么下去。差不多是类似的概念，这就是字符串。树的例子也一样。我上次讲了一个树的例子，我们可以为树做一个林登迈尔系统，字符串是 FBR。看这里，字符串是 FBR，规则是每个 B 变成这个。我们可以好好想想这个，通过解释来赋予这些东西意义。F 表示向前走，B 表示枝条末端有一个芽，R 表示往回退。那我们想象一下建一棵树：我们向前走，也就是 F，顶上有一个芽，然后 R 往回退。规则是这里每一个 B，每一个 B 都变成这个，我们看看这是什么意思：每一个 B变成 -FBR++FBR-，所以每个 B、每个芽都变成 -FBR++FBR-。嗯，那我们运行一下看看
57:22
reverse and the rule is each B right here each B becomes this and let's see what this is what this means each B becomes minus FBR plus plus FBR minus so each B each Bud becomes minus f b r+ plus f FBR R minus uh yeah so let's run this and see how it looks any anybody have questions so far yeah so why are all your strings palindromes why are all the strings paland drones um wow they're palindromes because each side of the tree is exactly the same that's why and by the way if you don't know what a palindrome is it means it's the same forward and backwards is that what it is right and it makes sense because the tree is is the same if you flip it around so I'll run it so the first thing is FBR forward Bud reverse do it again and it it's this tree there we have it so look at this picture and I'm going to show you this other program that I explained in my last lecture with this other this recursive function that says tree grow tree with with the smaller size and I'll run this and lo and behold what we get is the
效果。到现在有人有问题吗？请讲。为什么你的字符串全都是回文？为什么所有字符串都是回文，哇，它们是回文是因为树的两边完全一样，就是这个原因。顺便说一下，如果你不知道什么是回文，意思就是它正着读和反着读一样，是这个意思吧？对。这很合理，因为这棵树翻过来还是一样的。那我运行一下。第一步是 FBR，前进、芽、后退。再来一次，就成了这棵树，好了。看看这张图，我要给你们看我上一讲讲过的另一个程序，那个递归函数，说的是 tree，用更小的尺寸生长树。我运行一下，你看，我们得到的是同一个东西。这个是林登迈尔系统，那个是递归函数，它们完全一样。所以这意味着什么？就好像
11两种描述一个对象：柏拉图之问
59:19
same thing this is the Linden Meer system and this is the recursive function they're exactly the same so what what is this mean like there's something deeper sort of about fractals like you can approach them from different angles get the same thing I'm just fascinated by it so I'm going to move on to cellular cellular automa now any any any questions on Lin Meer systems so cellular aut yeah you get the same thing does l Mar they have relationship with yeah so so you're saying uh your question is like they approach different ways and they equal the same thing so those two different approaches do they equal each children in some way yes so so what she asked is so you have two different approaches and if if they approach the same thing are they in fact the same thing more or less is your question uh so is the idea that really we have two different descriptions for the same phenomenon yeah um so to what extent are those descriptions the same huh so right I mean it's it's it's sort of mindboggling we have two
分形背后有某种更深的东西，你可以从不同的角度接近它们，得到同样的结果。我就是被这个迷住了。那我要转到元胞自动机了。关于林登迈尔系统还有什么问题吗？那么元胞自动……请讲。你得到同样的东西，那 L 系统它们有关系吗？对，你是说……你的问题是，它们用不同的方式接近，最后相等，那这两种不同的方式是不是在某种意义上也相等？对。所以她问的是，你有两种不同的方法，如果它们逼近同一个东西，那它们本身是不是其实也是同一个东西？大概是这个问题吧。嗯，所以问题的意思是，我们其实是对同一个现象有两种不同的描述，对吧？那这两种描述在多大程度上是相同的？呃，对，我是说，这挺让人费解的，我们对同一个东西有两种不同的描述，那这两种描述在多大程度上是一样的？我是说，它们
60:49
different descriptions for the same thing so to what extent are the descriptions the same I mean there they're definitely different but they describe the same thing so the thing that they're describing is the same but but they themselves are different I mean it's it's hard to really say yeah is it like is it sort of like describing like say describing ah okay so he he's sort of making the analogy to describing something in in reality using words or using pictures or using sound or using text so yeah what we're what you're getting at in all these different descriptions is the same fundamental thing this is the inner message right that Justin was talking about earlier you have the frame message the the message and the inner message the inner message is the real meaning the thing itself but could also like the physical thing be like another way of describing something more Abra maybe like physical reality is like describing oh man so he he said so maybe physical reality itself is just a description of something else something
肯定是不同的，但它们描述的是同一个东西，所以它们描述的对象是同一个，但它们本身是不同的。我是说，这真的很难讲。对。这是不是有点像描述……就像……比方说描述……啊，好的，他是在打个比方，就像用文字、用图画、用声音或者用文本去描述现实中的某个东西。对，所以你说的这些不同的描述，指向的是同一个根本的东西，这就是内层信息，对吧，贾斯汀之前讲过的，你有框架信息、外层信息和内层信息，内层信息才是真正的意义，是事物本身。但会不会……那个物理的东西本身也是描述别的东西的一种方式？也许，比如物理现实就是在描述……哦天哪，他说，也许物理现实本身就是对别的东西的一种描述，而数学也是另一种描述，
62:25
else more like mathematics is also another description mathematics is also another description of maybe mathematics itself is the thing being described by the physical Universe I mean it's like we have a plist on our hand we have a plate nist on our hands yeah so it's sort of like asking the question so so all of these different descriptions of fractals all lead to fractals what does it mean is the fractal itself describing something deeper I mean I don't know we we sort of get that sense but it's like it's so elusive yeah you Mo deeper into the fractal since it's cursive itself can you go deeper into the fractal a definite it's scale free yeah it's scale free the fractal itself is containing yeah the I mean yeah you can you can go into the fractal infinitely yeah but like what is it what what are the implications of that like I don't know just yeah where do it Le to it leads to more of itself I guess it's recursive you know there's no yeah the universe almost
数学也是另一种描述，或者说也许数学本身才是被物理宇宙所描述的那个东西。我是说，这就好像我们手上有一柏拉图主义了，我们手上有个柏拉图主义。对，所以这有点像在问这个问题：所有这些对分形的不同描述最后都通向分形，那这意味着什么？分形本身是不是也在描述某种更深的东西？我不知道，我们隐约有那种感觉，但它太捉摸不定了。对。你能不能更深地进入分形，既然它本身是递归的，能不能更深地进入分形？绝对可以，它是无标度的。对，无标度，分形本身包含着……对，我是说，对，你可以无限地深入分形。对。但那又怎样呢，这意味着什么，有什么含义？我不知道，就是……对，它会通向哪里？它通向的还是它自己，我猜。它是递归的嘛，你知道，没有……对，宇宙几乎就像……
63:50
like could the universe be a fractal in a more AB abstract sense yes yeah uh I just have to put in there because before you had like kind of a quantum mechanical version of the atom what we used to say is like well here's the nucleus and here are the electrons all right and of course I could have said instead of nucleus here's the Sun and and here are planets so in this sense now once again I mean this this model isn't right but what we used to have is this kind of self-similarity across scales is that you know if we zoomed in Far Enough from our solar system and went from the solar system and just kind of kept micro just telescoping in and in and in and then visually until we got to a length scale on the level of atoms I I would I could replace this and it would look exactly the same and I would say well this is the nucleus and here are [Applause] electrons of course this isn't exactly right but what's interesting is that a lot of physical phenomenon I mean and this this is a very kind of deep
宇宙会不会在某种更抽象的意义上是一个分形？是的。对，呃，我得插一句，因为在这之前你讲过原子的某种量子力学版本，我们过去常说的是，这是原子核，这些是电子，对吧。当然我也可以不说原子核，而说这是太阳，这些是行星。所以从这个意义上说，再强调一次，这个模型并不正确，但我们过去有的是这种跨尺度的自相似性：你知道，如果我们从太阳系一直放大进去，从太阳系开始不断地往里、往里、往里推近，一直推到原子那个长度尺度，我可以把这幅图换掉，而它看起来会一模一样，我会说，这是原子核，这些是电子 [掌声]。当然这并不完全正确，但有意思的是，很多物理现象——我是说，这是个非常
65:15
experiment is that and I know curan talked about this a little bit but if you take a mountain and you look at the shape of a mountain that's kind of this crinkled prac thing and then you take a piece of paper and you crumble it up and then unfold it the geomorphology of a paper is almost exactly the same as a mountain and the landscape around it so in some senses there's a self-similarity in laws in conceptual laws between the forces that govern the shaping of mountains and valleys and rivers and the forces behind me crumbling up this piece of paper and also in biology it's very it's so inspiring in biology you see these patterns at different levels on different scales and they're all the same they all share something so I mean biology like yourself like your own body is is definitely a fractal in some sense not infinitely but but there's all these it's yeah could it be like the think is some the way to describe the O so the chief said could it be the thing itself or the way that we describe
深刻的实验——我知道柯兰之前稍微讲过一点，就是如果你拿一座山，看山的形状，那种皱皱巴巴的分形般的东西，然后你拿一张纸，把它揉成一团，再展开，这张纸的地貌形态几乎和一座山以及周围的地景一模一样。所以在某种意义上，在概念性的规律层面上存在着自相似性：塑造山川、河谷、河流的那些力，和我把这张纸揉皱背后的那些力之间。生物学里也是这样，非常……在生物学里这特别让人振奋，你会看到这些模式出现在不同层次、不同尺度上，而它们都是一样的，都有某种共同的东西。所以我是说，生物，比如你自己，你自己的身体，在某种意义上绝对是分形，不是无限的，但确实有这些……对，会不会像是……问题是，有某种……描述那个东西的方式……所以这位说，会不会是事物本身，还是我们描述那个事物的方式？
12生命游戏与停机问题
66:35
the thing yeah so then that fundamentally boils down into which do you think's realer the pendulum or the equation describing the pendulum chick so Sandra says it's the Chicken and the Egg example yeah So speaking of biology and life um life is life is complicated right life is complex societies just evolve and people do things and like you you can't really predict what's going to happen and you have all this complex Behavior but um and in society it sort of arises from complex rules of interaction between people but with computers and cellular automata particularly what we can do is very very Loosely model life and we get this um and it's it's similar to actual society and that like it's very complex behavior and we can't tell by looking at it how it's going to end what's going to end up maybe small things could could balloon into you know huge influential events like people like one person you know influences the whole future like um it's fascinating so what I'm going to do is talk about these rules and what this
对，那这归根结底就变成了：你觉得哪个更真实，是钟摆，还是描述钟摆的方程？桑德拉说这是先有鸡还是先有蛋的例子。对。说到生物和生命，嗯，生命是很复杂的，对吧，生命是复杂的，社会就这么演化，人们做各种事情，你没法真的预测会发生什么，你有这一大堆复杂的行为。但是，在社会里这种复杂行为来自于人与人之间复杂的交互规则；而有了计算机，特别是元胞自动机，我们能做的是非常非常粗略地去建模生命，然后我们得到这个，它和真实社会很像，就在于它是非常复杂的行为，我们光看它没法判断它会怎么收场、最后会变成什么样。也许一些小事情可能滚雪球般变成，你知道，巨大的、有影响力的事件，就像有人，一个人，你知道，影响了整个未来，比如……很有意思，所以我接下来要讲的就是这些规则、这个程序是什么、它是怎么运作的，嗯，这就是康威的生命游戏（Conway's Game of
67:57
program is how it works um so this is Conway's Game of Life it's really cool so we have this grid of of boxes um and each box is termed a cell cellular automa and it's automa because it evolves it keeps going based on Simple Rules autom ly like it does it by itself so we have this huge grid and you can think of each cell here as though it were its own little organism uh just sort of going through life um so let's consider this one uh if we are this cell like a cell is a place for an org organism to exist or not exist so there there are some very simple rules um I WR down you haven't written down be good to just have a piece of paper set the
Life），非常酷。我们有这样一个格子网格，嗯，每个小格子叫做一个细胞，元胞自动机，之所以叫自动机是因为它会演化，它会一直按照简单的规则自动进行下去，就是说它自己就能跑起来。所以我们有这么一大片网格，你可以把每个格子想象成一个小生命体，嗯，就是在过它自己的一生。那我们看看这一个，如果我们就是这个细胞——一个细胞就是一个位置，一个生命体可以存在或不存在的位置。所以这里有一些非常简单的规则，嗯，我写下来了，你们没写下来的话，最好拿张纸记一下
69:10
sure right so a cell is either filled in or it's not filled in so if we are alive say we are this cell right here if we're alive and there's only one cell around us we die of loneliness and so this cell would die if they're two if there are two cells it's okay it's a healthy sort of environment like those are our parents maybe and we live where this cell if there are three maybe it's our parents and a brother or something we we live uh but if there are four and these numbers I'm talking about is the total number of living cells in our neighborhood so these surrounding eight cells is the neighborhood of this particular cell so if there are four living cells in the neighborhood of this one then it dies of overpopulation of Suffocation four or more if there are more than four then the cell dies and if the cell if the cell what yeah if the cell is not alive um then it then if there are three in its neighborhood then it comes alive so if there are three cells around it that are alive then it they birth a child or
好的。所以一个细胞要么是被填充的，要么是没被填充的。如果我们是活的，比如说我们就是这里这个细胞，如果我们是活的，而周围只有一个细胞，我们就会孤独而死，所以这个细胞就会死掉。如果是两个，如果周围有两个细胞，那就没问题，这是一个比较健康的环境，比如说那是我们的父母，我们就活下来。或者这个细胞，如果周围有三个，那可能是我们的父母加一个兄弟之类的，我们也活下来。但如果有四个——我说的这些数字指的是我们邻域里活细胞的总数，所以周围这八个格子就是这个特定细胞的邻域。所以如果这个细胞的邻域里有四个活细胞，那它就会因为过度拥挤、窒息而死。四个或更多，如果超过四个，这个细胞就死了。那如果这个细胞……如果这个细胞……嗯对，如果这个细胞不是活的，嗯，那么如果它的邻域里有三个活细胞，它就会活过来。所以如果它周围有三个细胞是活的，那它们就生了个孩子之类的。这就是全部的规则集，就这么寥寥几条规则
70:42
something so this is the entire rule set these these very few number of rules that makes this thing happen so it it starts off with just r randomly filled in cells oh no what have I done uh it starts off with just randomly filled in cells and let's just look at the code quickly um cellular autom so neighboring cells is get cell X Y +1 and we go through this list of things we basically get all the cells in our surrounding neighborhood and put them into a list um this notation means array this is like a list a list a list of cells a cell is an object which I made which has a property of either being alive or dead basically it's we drawn as a square and um so if my value is one my being the cell that we're considering at the moment and mind you this is inside of a loop um note this d double for Loop for each X and for each y so the stuff inside of these double for Loops is going to be executed once for every cell so let's look on the inside so my is the Cell at XY if my value is one meaning that I'm
让这一切发生。所以它一开始就是随机填充的一些细胞……哦不，我干了什么。呃，它一开始就是随机填充的细胞。我们快速看一下代码，嗯，元胞自动机。neighboring cells 就是 get cell X Y +1，我们就这样把这一串都过一遍，基本上就是把我们周围邻域里的所有细胞都取出来放进一个列表里。嗯，这个记号表示数组，就像一个列表，一个细胞列表。细胞是我做的一个对象，它有一个属性，表示它是活着还是死了，基本上就是画成一个方块。嗯，所以如果 my value 等于 1——my 就是我们此刻正在考虑的那个细胞——而且请注意这是在一个循环里面。嗯，注意这个双重 for 循环，对每个 X 和每个 y。所以这两层 for 循环里面的东西会对每一个细胞执行一次。那我们看看里面：my 就是位于 XY 的那个细胞，如果 my value 等于 1，意思就是我是
72:20
white I'm alive there's something living in this cell if there are less than two then die of loneliness if there are more than three than die of overpopulation Suffocation so this can be encoded as um if this neighborhood sum which we calculated a few lines ago just added up for each cell in neighboring cells neighborhood sum is you add that cell's value so for example with this cell neighborhood sum would be three right the number of living cells around it if my value is one then um if the neighborhood sum is two or three so if there are two things around me or three then my next value the value that I will be in the future is going to be one I'm going to stay alive so otherwise I'm going to die value is zero my next value is zero so else if my value is zero this means if the cell is originally dead if the cell is originally black there's nothing there in that case then if the neighborhood sum is exactly three then my next value is one otherwise I stay dead so if if there are three people around me then I'm going to
白色的，我是活的，这个格子里有东西活着。那么如果周围少于两个，就孤独而死；如果多于三个，就因过度拥挤、窒息而死。所以这可以这样编码：嗯，如果这个 neighborhood sum——我们前几行刚算出来的，就是对 neighboring cells 里的每个细胞，把那个细胞的值加起来。比如说对这个细胞，neighborhood sum 就是三，对吧，就是它周围活细胞的数量。如果my value 是 1，那么嗯，如果 neighborhood sum 是 2 或 3，也就是我周围有两个或三个东西，那我的 next value，也就是我将来会变成的那个值，就是 1，我会继续活着。否则我就会死，值为 0，我的 next value 就是 0。那么else if my value 等于 0，这意思是如果这个细胞本来就是死的，如果它本来是黑色的，那里什么都没有。这种情况下，如果 neighborhood sum 恰好等于 3，那我的 next value 就是 1，否则我就继续保持死亡。所以如果我周围有三个人，我就会在这个格子里出生。嗯，基本上就是这样。然后我们还会算颜色之类的，但
73:38
be born into this cell um that's basically it and we calculate the colors and whatnot but that's the uh essential piece of the program and when we run it this is what we get conways of Life any questions yeah does it repeat itself does it ever repeat itself well we'll see uh now it's repeating itself every two frames so it's actually at a steady state now so so you can go on for it will never change at this point at this point yeah it's in a it's in a periodic cycle so it's never going to change put in random data to make it change so let's see justes my mouse click work I don't remember no it doesn't work but I mean it depends on the initial conditions right let's do it let's run it again oops if we run it again it might never do that because the initial conditions are random think of this as a computer program this is the halting problem of computer science Alan turing's halting problem so that basically said that and this is in the handout I believe is it Justin maybe not sorry but um Alan turing's halting problem says if
那才是这个程序的核心部分。我们运行一下，得到的就是这个，康威生命游戏。有什么问题吗？对，它会不会重复自己？它到底会不会重复？嗯我们看看。呃，现在它每两帧就重复一次，所以它其实已经进入稳定状态了。所以你可以一直跑下去，到这个点它就再也不会变了，到这个点，对，它进入了一个周期性循环，所以它再也不会变了。得放进随机数据才能让它变。那我们看看，嗯，我的鼠标点击能用吗？我不记得了。不能用。不过我是说，这取决于初始条件，对吧。我们来试试，再跑一次哎呀。如果我们再跑一次，它可能永远不会变成那样，因为初始条件是随机的。把这个想成一个计算机程序，这就是计算机科学里的停机问题，阿兰·图灵的停机问题。它基本上说的是……而且这个我记得在讲义里，是吧 Justin？可能没有，抱歉。但是嗯，阿兰·图灵的停机问题说的是，如果你有一个给定的计算机程序，
75:07
you have a a given computer program there is no computer computer program that you can write that will analyze that program and tell you whether or not it will stop so stopping halting coming to an end or coming to a stable point is unpredictable it's impossible to to predict given a set of inputs whether or not the cell the the system will ever come to a a stable point so this the configuration of cells initially is actually a computer program that's going to be set in motion when we start the simulation and start applying these rules so we can get like the whole touring machine thing just they almost like doing the same theing machine and a teing machine right so yes applying these rules can be reduced to a touring machine yeah and a machine can also be that what a turing machine is like the lowest level of existence of a computer program like it's right but yeah but I think it's it's equivalent I'm not sure if game I think Game of Life Is But I know some other cellular automaton rules
不存在任何一个你能写出来的计算机程序，能够分析那个程序并告诉你它究竟会不会停下来。所以停下来、停机、走到终点或者达到某个稳定状态，是不可预测的，你不可能预测在给定一组输入的情况下，这个细胞——这个系统——究竟会不会达到一个稳定状态。所以最初的这个细胞配置其实就是一个计算机程序，当我们启动模拟、开始应用这些规则时，它就被开动起来了。所以我们可以搞出整个图灵机那一套，它们几乎就是在做同样的事，图灵机，对吧。所以是的，应用这些规则可以归约成一台图灵机。对，而且图灵机也可以说是……图灵机就像是计算机程序存在的最底层，就像是……对，但我觉得它们是等价的。我不确定生命游戏是不是，我觉得生命游戏是的，但我知道有一些其他的元胞自动机规则，呃，是图灵完备或者通用图灵
76:23
uh are Turing or Universal turning machines right so you can either think of it as a strip of paper which is which is just doing very simple computations back and forth or you could also do your computation on the on the level of of squares and cellular automaton like this and and then what that means to be a universal turning machine is that you can fundamentally reproduce the basic logical operations of and not and copy and from that you get essentially all mathematics and all of anything you'd want how about like girl and can you get all of my from just okay so that that's kind of a so the touchy issue there is can you derive all true statements recursively from a set of axioms no girdle's incompleteness theorem tells us this um and that's why mathematicians will never go out of a job right but fundamentally um there's always new truths out there that aren't reachable from your set of axioms so you kind of have to go out there and meta think and then and then discover it right from a higher level
机。对吧，所以你既可以把它想成一条纸带，在上面来回做非常简单的计算，也可以在方格这个层面上，用这样的元胞自动机来做你的计算。那么，成为一台通用图灵机意味着什么呢？意味着你从根本上可以复现出基本的逻辑运算：与、非、复制。有了这些，你基本上就能得到全部数学，以及任何你想要的东西。那哥德……你能不能从这些就得到全部的……好，这就有点……这里比较微妙的问题是，你能不能从一组公理出发递归地推导出所有为真的命题？不能，哥德尔不完备定理告诉了我们这一点，嗯，这也是为什么数学家永远不会失业，对吧。但从根本上说，嗯，总是存在新的真理是你的公理集合无法触及的。所以你得跳出去，做元层面的思考，然后从更高的层面去发现它，而不只是靠作用在公理上的递归运算。嗯，但有意思的是，
13投票模型、水滴与普适类
77:36
outside of just a recursive operations acting on axioms um but interestingly enough girdles incompleteness theorem and the halting problem are really fundamentally kind of the same thing um cuz the way you prove the halting problem is is you feed the program which is supposed to decide whether not it's going to stop to itself right and it's that very recursive nature and it's fascinating we'll talk about it more so keep in mind that in uh two lectures from now I think we're going to really teach you what grd incompleteness is so just just just wait for it like it's coming so I'm going to show you another two more examples of cellular automa and then I'll be done one of them is um voting patterns voting rules that's what it's called um and what it is basically think of people uh who talk to their neighbors and their opinion about you know which political party is the best is slowly influenced over time by those people around him so think of um these are like Shades of Gray right like this person is sort
哥德尔不完备定理和停机问题在根本上其实是同一回事，嗯，因为你证明停机问题的方式，就是把那个本该判定程序会不会停下来的程序喂给它自己，对吧。就是这种递归的本质，非常迷人。我们之后会更多地讲到，所以记住大概再过两讲，我觉得我们会真正教你们哥德尔不完备到底是什么，所以就等着吧，它快来了。那我再给你们看另外两个元胞自动机的例子，然后我就讲完了。其中一个是嗯，投票模式，投票规则，就是这么叫的。嗯，它基本上就是，想象一群人，呃，他们跟邻居交谈，他们关于哪个政党最好的看法，会随着时间慢慢被周围的人影响。所以想象一下，嗯，这些就像是不同深浅的灰度，对吧，比如这个人处于中间地带，这个人
78:51
of in the middle this person's like really uh Republican or something uh and and this person is really Democratic let's just consider this so um what the rule is for this in in the in the game of life it was binary that means there were only two states on or off alive or dead but in this in this situation there are an infinite number of states there it's it's it's a real number um between negative 1 and one so what what happens is this person has a number say it's 0.5 and he every every iteration he takes stock of the people around him and adds to his number the sum of his surroundings multiplied by some small number so he's he's only influenced a little bit but he's influenced by surroundings and so what we get is is called coarsening it's a cooning effect so the simulation that I'm about to show you starts off with complete noise it's it's completely random each cell is assigned a value between negative 1 and one uh randomly and then we're going to apply this Rule and and just watch it it's going to be
非常，呃，共和党倾向之类的，呃，而这个人非常民主党倾向。我们就看这个。嗯，这里的规则是这样的：在……在生命游戏里它是二值的，意思是只有两种状态，开或关，活或死。但在这个……在这个情形里，状态有无穷多种，它是一个实数，嗯，在负一和一之间。那么会发生什么呢？这个人有一个数值，比如说 0.5，然后每一次迭代，他都会看看周围人的情况，把周围人的总和乘以某个很小的数，加到自己的数值上。所以他只被影响一点点，但确实被周围影响了。于是我们得到的就是所谓的粗化（coarsening），一种粗化效应。我马上要给你们看的这个模拟，一开始完全是噪声，完全是随机的，每个细胞被随机赋予一个负一到一之间的值，嗯，然后我们就应用这条规则，然后就看着它，会很漂亮的
80:09
beautiful
80:28
is that awesome so it's going to keep evolving and the end state will either be all black all white or half black and half white divided somewhere so and it's just this very simple rule of summing the cells around you and adding to your state a a very low weighted you know portion of their opinion so we get this complex Behavior like who who would have thought right it's really cool so I'll just go through the code quickly about this one uh the structure is pretty much the same as the other ones for these doubly nested for Loops going through each individual cell my is the cell we're at neighboring cells blah blah blah um so this time the cell's value is not 01 but it's a it's a real number um so we sum the neighborhood so and this line sums up the rule right here all of it says my next value is my value my current value plus the neighborhood sum the opinion of the neighborhood times 005 yeah if you multiply by lower numberb does that mean it's just took like first time to same yes so we can
是不是很棒。所以它会一直演化下去，最终状态要么是全黑，要么是全白，要么是黑白各一半，在某处分开。所以，仅仅是这么一条非常简单的规则——把周围细胞加起来，然后以一个很低的权重把他们的意见加到你自己的状态上——我们就得到了这么复杂的行为，谁能想到呢，对吧，真的很酷。那我快速过一下这个的代码，呃，结构跟前面那些基本一样，还是这些双层嵌套的 for 循环，遍历每一个细胞。my 就是我们当前所在的细胞，neighboring cells 等等等等。嗯，这次细胞的值不是 0 或 1，而是一个实数。嗯，所以我们对邻域求和。然后这一行就概括了全部规则，就在这儿，它说我的next value 等于我的 value，我当前的值，加上 neighborhood sum，也就是邻域的意见，乘以 0.05。对，如果你乘以一个更小的数，是不是就意味着它需要更长时间才……对，所以我们其实可以——这就是拥有一个
81:58
actually this is the beauty of having a live program so this number uh 05 determines the speed at which the system evolves so if we make it 05 and run it uh you'll notice it it'll go a lot faster goes really fast right fast it's the same thing just actually 10 times as fast because it was 05 instead of 05 so we can go 10 times slower if we make it 05 let's make it 01 and and it's going to go really slow but that's the essence of the system just be influenced a little bit by our neighbor so it's just going really slow now but it is still going and um interesting fact um cellular automa similar to this one are used a lot to do image processing and Graphics to do like like blurring for example is a cellular autometa applied to the pixels of your picture and I would like to hop in there if you don't mind yeah but this is kind of a and this relates to a project that Kar and I did with uh the New England complex systems Institute um and this is kind of an example of what we call a kind of a universality class of
实时程序的妙处——这个数字，呃 0.5，决定了系统演化的速度。所以如果我们把它改成 0.5 再运行，呃，你会注意到它会快很多。跑得真快，对吧。是同样的东西，只是快了十倍，因为原来是 0.05 而不是 0.5。所以我们也可以慢十倍，如果我们改成 0.05。我们改成 0.01 试试，它会跑得非常慢，但这就是这个系统的本质：只被我们的邻居影响一点点。所以现在它跑得非常慢，但它确实还在跑。嗯，还有个有意思的事实，嗯，跟这个类似的元胞自动机被大量用于图像处理和图形学，比如做模糊——模糊处理就是一种元胞自动机，作用在你图片的像素上。我想插一句，如果你不介意的话。当然。这算是一种……这跟一个项目有关，就是Kar 和我跟呃新英格兰复杂系统研究所一起做的那个。嗯，这算是我们所说的一种普适类现象的例子。嗯，但有意思的是，这同一条规则，或者说这同一种行为，
83:23
phenomenon um but the interesting thing is that this same rule or this same behavior is actually what governs gas droplet condensation so if you look at like the window of your your car and initially just kind of have Mist misting down on on the surface of your window what each water particle does is because it wants to fundamentally lower its its um energy just wants to take kind of the path of least action um and based on on surface tension and the way that um essentially the the interaction between these water molecules works is that water molecules like to be next to each other right because it takes less energy to group together than it does for a bunch of water molecules to Exist by themselves and I think it's interesting that the same rules and behavior which simple things like water lets on on on a sheet of glass behave it's the same way that people behave fundamentally it takes more work if you're just a hunter gatherer by yourself than to come together in a society and grow Collective farms and
其实也支配着气体液滴的凝结。所以如果你看看你车的车窗，一开始上面只是有一层雾气，附着在窗户表面。每个水分子做的事情，是因为它从根本上想降低自己的能量，就是想走某种最小作用量的路径。嗯，基于表面张力，以及嗯，本质上这些水分子之间相互作用的方式，水分子喜欢彼此挨在一起，对吧，因为聚集在一起所需的能量，比一堆水分子各自单独存在要少。而我觉得有意思的是，同样的规则和行为——像水滴在一块玻璃上这种简单事物的行为方式——跟人的行为方式在根本上是一样的。如果你只是个独自一人的狩猎采集者，那要费的力气，比大家聚在一起形成社会、发展集体农业、有人负责这个、
84:39
have someone responsible for this and responsible for that so in this sense we really do have a universality class of phenomena the same laws which which govern and notice this is this is pure physics when it's in terms of water molecules I mean I can actually work out the energy values yet when we're talking about human system societies we don't really have equations but with Caton we do we have we have equations of how societies interact and one of the interesting things which the new New England complex systems Institute is working on is is um actually prediction of ethnic conflicts and violence and you can use very simple models like this to to predict where where you know if you're talking Gaza Strip right we've got Palestinians and Israelis how do they mix and and form together and then segregate each other and um and what happens there and this can all be described using this same kind of very simple graphical rules do we see complex what do we see because you don't understand the basic
有人负责那个，要多得多。所以从这个意义上说，我们确实有一个普适类现象，同样的规律支配着……而且注意，当它是关于水分子的时候，这是纯粹的物理，我是说我真的可以算出能量数值。然而当我们谈人类社会系统的时候，我们并没有真正的方程。但有了元胞自动机我们就有了，我们有了描述社会如何互动的方程。而新英格兰复杂系统研究所正在做的一件有意思的事情，就是嗯，实际去预测族群冲突和暴力。你可以用这样非常简单的模型去预测在哪里……比如你说加沙地带，对吧，那里有巴勒斯坦人和以色列人，他们怎么混合、怎么聚在一起，然后又怎么彼此隔离，嗯，以及那里会发生什么。这些都可以用这同一类非常简单的图形化规则来描述。我们看到的复杂……我们看到的是不是因为你不理解底层的……因为大家都看不到……好，你问的是，是不是只是因为我们
14波、离散宇宙与芝诺悖论
85:40
of because all don't see okay so you're asking like is it just that we don't see the underlying laws behind society and behind water molecules and things like that and that really is kind of an idea of which Steven wolf from in his book A New Kind of Science voices is that the Universe really is just a giant cellular automaton um and you know it's just cranking out these rules very similar to The Game of Life yeah and I I can segue very nicely from that into the next thing which is waves now cellular autometa can do waves and keep in mind that Quantum quantum mechanics is basically wave mechanics is that right Justin a lot of ways yeah more or less and the interaction of of particles can sort of be reduced to to waves standing waves in a sense so let's take a look at this program and and imagine monkeys typing on a keyboard what's the probability that a monkey will um will will type this exact configuration of of symbols not going to happen uh but it's a lot more likely that a monkey writing like sh
看不到社会背后、水分子背后这些东西的底层规律。这其实就是史蒂芬·沃尔弗拉姆在他的书《一种新科学》里提出的一个想法：宇宙其实就是一个巨大的元胞自动机。嗯，你知道，它就是在不停地跑这些规则，跟生命游戏非常像。对，而且我可以很自然地从这里过渡到下一个东西，也就是波。元胞自动机也能做出波来。而且要记住，量子……量子力学基本上就是波动力学，对吧 Justin？在很多方面是的，对，差不多。粒子之间的相互作用某种程度上可以归结为波，某种意义上是驻波。那我们来看看这个程序，然后想象一下猴子在键盘上打字，一只猴子嗯……打出这一整套符号的确切排列，概率有多大？不会发生的。呃，但一只猴子写出比如说莎士比亚之类的，可能性要大得多。我是说我们来看看，
86:54
Shakespeare or something I mean let's look at uh is this the right one let's look at this rule this is really all that we need this rule right here so let's look at the rule first and I'm going to ask you guys to try to predict what's going to happen so in this situation each cell has a height but it also has a velocity a speed so you think of it in 3D the height is the the direction this way or that way and the velocity is the speed at which it's traveling so look at this rule um and the neighborhood sum is the sum of the uh Heights uh yeah the sum of the heights minus my height so the neighborhood sum is the sum of the differences between my height and the cells around me so yeah for each cell in the neighborhood the neighborhood sum plus equals so that means add this to the neighborhood sum my height uh well the cell height for each cell around me minus my height so if we're the same height then it's going to be zero like if if all the cells are the same height then it's going to be zero
呃，是这个吗？我们来看这条规则，其实我们需要的就只有这条规则，就是这儿这条。那我们先看规则，然后我要请你们试着预测会发生什么。在这个情形里，每个细胞有一个高度，但它同时还有一个速度，所以你可以在三维里想象，高度就是这个方向或那个方向，而速度就是它移动的快慢。那看看这条规则，嗯，neighborhood sum 是那些呃高度的和，呃，对，是各个高度的和减去我的高度。所以 neighborhood sum 就是我周围的细胞跟我的高度之差的总和。对，所以对邻域里的每一个细胞，neighborhood sum 加等于——也就是把这个加到 neighborhood sum 上——我的高度，呃不，是我周围每个细胞的高度减去我的高度。所以如果我们高度相同，那就是零，比如如果所有细胞都是同一个高度，那就是零。但如果存在差异，那它就会是这些差值的总和，然后
88:19
but if there's a difference it's going to be the the sum of the differences and then my veloc velocity uh is increased by The Neighborhood sum divided by eight so the average of the neighborhood so what does this mean what what is this going to do does anybody tell me everybody around you is like very like vared you you may have like a high one but everybody the same have like very low yeah yeah so yeah if if you're if if you're very different than the the ones the people around you then your velocity is going to be higher right so let's let's run it and see what
那么我的速度就会增加，增量是邻域的总和除以八，也就是邻域的平均值。那么这意味着什么呢？这会产生什么效果？有谁能告诉我？如果你周围的人都非常不一样，你可能有一个很高的值，但如果大家都一样，就会有很低的值。对，对，所以如果你和你周围的那些人差别很大，那你的速度就会更高，对吧？那我们来运行一下，看看会发生什么。
89:10
happens so it's a wave right it's water so it's it's fundamentally a wave and since all particles are our ways it's I mean it's it's something to to consider that the universe may just be a giant computer so I had a mouse clicks to this so we can actually click on it so I I click on like the side uh hopefully it'll work all what's going on there we go look in the middle you have things going on and off on and off on and off H it's like a check yeah it's like a checkerboard sort of so we don't need all about calculus anymore well actually what calculus is doing is approximating exactly this or or the other way around this is approximating the calculus calculus that's even more detailed closer so you like more pixels in there right so he's saying if you have more pixels and and more resolution between the possible numbers that you can have then you're going to approach actual physical reality right yeah you're right but this is the fundamental limitation of computers they're discreet so I mean there's only a finite number
这是一个波，对吧？这是水，所以它本质上是一个波。既然所有粒子都是波，我是说，这值得思考一下，宇宙可能就是一台巨大的计算机。我给它加了鼠标点击功能，所以我们可以真的点击它。我点在边上，希望能起作用。好，来了，看中间，那里有东西在一开一关、一开一关、一开一关。呃，就像一个棋盘。对，有点像棋盘格。所以我们不再需要微积分了吗？其实微积分做的正是在近似这个东西，或者反过来说，这个东西是在近似微积分。微积分更精细、更接近。所以你需要更多的像素，对吧？所以他的意思是，如果你有更多的像素、在可取的数值之间有更高的分辨率，那你就会逼近真实的物理现实，对吧？对，你说得对，但这正是计算机的根本局限：它们是离散的。我是说，每个格子只有有限种可能的高度，而我们能拥有的格子数量
90:29
of possible Heights for each cell and there's only a finite number of possible cells that we can have the universe itself is discre in no way how's how's the universe itself discret I mean time is discret space is discret you have and it has to be it can be like continuous cuz that means we didn't have like as much matter as well I mean the way I understand it is the universe is continuous because take take into account I think Zeno's Paradox right no but Zeno's Paradox doesn't make sense it only make sense when you and say okay when I work from here to there I'm adding up all the small vales of these things that I I add up like in a certain amount of time so I am moving because it's not continuous it's discret well what Zeno said is to get from here to here I have to first go halfway and then halfway halfway bottom like a single Val but it does not bottom out it goes on in infinitely yeah but that's on the line we don't have lines well see I mean from here to there it's a line and it so so
也是有限的。宇宙本身可绝不是离散的。宇宙本身怎么会是离散的？我是说，时间是离散的，空间是离散的。它必须是这样，它不可能是连续的，因为那意味着我们没有那么多的物质。呃，我的理解是宇宙是连续的，因为你想想，我觉得芝诺悖论对吧？不对，芝诺悖论说不通。它只有在你说：好，我从这里走到那里时，我是在把这些小的间隔加起来、在一定时间内加完，所以我确实在移动——这才说得通，因为它不是连续的，是离散的。芝诺说的是，要从这儿到这儿，我得先走一半，然后再一半、再一半，像单个的间隔一样触底。但它并不会触底，它会无限地分下去。对，但那是在线上，而我们没有线。呃，我是说从这儿到那儿就是一条线，所以从根本上说，问题是我们能不能
91:46
fundamentally the idea is can we continue dividing matter like infinitely small right I mean that's the question right like are strings the bottom level right I'm talking in terms of string theory but what about space though I mean itself like just like locations you don't have like know imagine bigum dots you don't see that it's just like the way of like sh like locations everything you can okay to get from like this part over here that part go like this much distance but that distance in the middle is like I have like little little distances but you can't get lower than those distances those are like the fundamental distances that's why it's like huh so you're saying there's a there's a fundamental the smallest distance possible so but then here's the question in our ukian plane which is the universe and this dot which is an atom or something or a particle when it goes from here to here does it ever not go through any point in the middle there are an infinite number of points that it goes
把物质无限地一直分下去，对吧？我是说这才是问题所在，比如说弦是不是最底层？我是在用弦理论的说法。但空间呢？我是说空间本身，就像位置一样，你并没有……想象一堆点，你看不见它们，就像是那种……位置的方式，一切你能……好吧，要从这边这一块到那一块，你要走这么多距离，但中间那段距离，是由很多小距离组成的，而你不能比那些距离更小，那些就是最基本的距离，所以这就很……啊。所以你是说存在一个最小的可能距离？但接下来问题是，在我们的欧几里得平面里——也就是宇宙——还有这个点，它是一个原子或什么粒子，当它从这儿到这儿时，它有没有可能不经过中间的任何一点？它经过的点有无穷多个。那个东西本身也许
93:13
through the thing itself may may not ever be smaller than a certain distance but see what we're modeling here is space not matter here space is defined in terms of these cell just discrete quantities but in reality like there's an infinite number of little you know places at which the particle can be so space is the thing that presents us with this huge divide between simulated things and real things discreet potentially potentially yeah potentially space could also be discret and that's kind of the computational view of of the universe which in some ways you're advocate so I mean yeah if if it's if it's so small that we could never Det it then I mean it could be the universe could be discreete right could be
永远不会小于某个距离，但你看，我们在这里建模的是空间，不是物质。这里空间被定义为这些格子，只是离散的量。但在现实中，粒子可以处在无穷多个小小的位置上。所以空间才是那个把模拟之物和真实之物隔开的巨大鸿沟。离散的——有可能，有可能。对，有可能，空间也可能是离散的，这就是那种关于宇宙的计算论观点，某种程度上你也是在支持这种观点。所以我是说，对，如果它小到我们根本无法探测，那么宇宙就有可能是离散的，对吧？有可能，
15信息熵 vs 算法复杂度
94:09
because space time so maybe more fundamental than space in time yeah I mean that's for the uh rest of your life to think about so I'm finished uh we have have 10 minutes left I think I'll hand it back to Justin so he can sort of wrap up yeah great I really wanted to pull this into the discussion we were having earlier um about kind what is information what is uh what is meaning and I want to present you actually with with a problem that um Kar and I tackled uh with a group of other people during one of these New England complex systems Institute intensive uh week-long seminars that they run in January um and it was it was the the fundamental idea of like how do we measure information how do we measure um the content of say an image right the information um content of an image and why is it that we say that there there's more information and an image and let's let's just say we're take we're feeding in to a program things like this but then what's inside of it is uh might be something like
因为时空——所以也许时空比空间和时间更基本。对，我是说，这够你想一辈子了。那我就讲完了。我们还剩十分钟，我想把话筒交回给 Justin，让他来做个收尾。好的，太好了。我很想把这个带回我们之前的讨论，就是关于什么是信息、什么是呃，什么是意义。我想给大家提出一个问题，是 Kar 和我跟一群人一起在新英格兰复杂系统研究所（NECSI）一月份举办的那种密集的、为期一周的研讨班上研究过的。呃，它的核心想法就是：我们怎么度量信息？我们怎么度量，比如说，一幅图像的内容？呃，一幅图像的信息含量。为什么我们会说这幅图像里包含更多信息呢？我们就假设我们往程序里输入这样的东西，但它里面的内容可能是这样的，对吧？那为什么我们会说这个在某种意义上比一堆乱七八糟的、
95:37
this right now why do we say that this in some ways is more meaningful than just having a bunch of kind of seething dog barf like this um or you know and although I love Jackson Pollock um like I mean why do we say that there's you maybe less meaning in these lines than than when we had the supinsky triangle drawn um and it's really a fundamental problem that's still you know people are working on and doing research on um and it involves um the way that we can measure information um and and one possible measure of information is known as uh kind of information entropy like Randomness could it be something about Randomness and well that's actually fundamentally what what a lot of information theory is about is probabilities and to what extent do you expect the outcome right um and it it it's interesting um that one mathematician named Claude Shannon back in I think the 50s basically started thinking about this problem and closed a lot of the major problems out in one publication called the mathematical theory of communication
像狗吐出来的东西那样的涂鸦更有意义呢？呃，或者说，虽然我很喜欢杰克逊·波洛克，呃，我是说，为什么我们会说这些线条里的意义比画出谢尔宾斯基三角形时更少呢？呃，这其实是一个很根本的问题，至今还有人在研究它、做相关的研究。呃，它涉及到呃，我们度量信息的方式。呃，而信息的一种可能的度量叫做，呃，信息熵之类的。会不会跟随机性有关？嗯，其实这从根本上说正是很多信息论的内容：概率，以及你在多大程度上预期某个结果，对吧。呃，有意思的是，呃，有一位数学家叫克劳德·香农，大概在五十年代，基本上开始思考这个问题，并在一篇叫《通信的数学理论》的论文里一举解决了很多主要问题。因为他当时想搞明白，我是说，当我们对着电话
96:58
cuz he was trying to understand I mean what does it mean when we're saying something over the phone and it's going down a transmission line and there's all this noise from the outside world how do we still extract meaning from those those pulses in that in that electrical cable and how do we how do we get sounds out of that and what's the information content there um and if you actually want kind of a rigorous definition of it if you're talking about kind of one State we can call H ofx um and we kind of take the probability of what we're expecting times the log of that probability I can't really derive all this but and then when you're talking about an ensemble of things uh you would just kind of sum over this and this would become big H um but this is the field of information Theory but the problem is is that fundamentally it's a question of if I were to take this image and translate it into a bunch of bits zeros and ones right how many zeros and ones would you need for you know this
说话，声音沿着传输线路传过去，外界还有各种噪音，这时候意味着什么？我们怎么还能从电缆里的那些脉冲中提取出意义？我们怎么从中把声音还原出来？其中的信息含量又是多少？呃，如果你想要一个比较严格的定义，如果说的是某一个状态，我们可以称它为 H(x)，呃，我们大致就是取我们所预期的概率，乘以那个概率的对数——我没法把这些都推导出来——然后当你讨论的是一个系综时，呃，你就把这些求和，这就变成大写的 H。呃，这就是信息论这个领域。但问题在于，从根本上说这是一个这样的问题：如果我把这幅图像转换成一串比特，也就是 0 和 1，对吧，那需要多少个 0 和哪些是你在这种情况下需要的，你知道的
98:05
versus maybe you could like in the future you may need more for that but when you abstract from it you need this exactly so the tee said so maybe in a picture like this you need more but when you abstract from it you need less and this is fundamentally the difference between information entropy and kind of algorithmic entropy and a really simple problem to think about is so what takes more if you were to write a program to to give you a number um let's just pick a number essentially at random um and say it's three billion digits long and it's uh you know 71743 so on so on dot dot dot since this would be a fundamentally random number by definition the shortest program which I could write to to produce this number would be print this number right but let's take a different number what about 3.141592 you know dot dot dot anybody here want to show off the number of digits of pi they know feel free um I had I had a friend who was actually failing a geometry class and on Pi Day they gave an extra credit point to every
相比之下，也许将来你可能需要更多，但当你从中抽象出来时，你需要的就是这些没错，所以说，也许在这样一幅图里你需要更多，但当你抽象出来时，你需要的就更少了，而这从根本上就是信息熵和某种算法熵之间的区别，一个非常简单的可以思考的问题是：如果你要写一个程序来给出一个数，哪个需要更多？我们随便挑一个数基本上是随机的，比如说它有三十亿位长，然后是 71743 等等等等，因为这从定义上说是一个完全随机的数，我能写出的产生这个数的最短程序就是「打印这个数」，对吧？但我们换一个数，3.141592 之类的，这里有人想炫耀一下自己能背多少位圆周率吗？请随意。我有个朋友当时几何课快挂了，在圆周率日那天，老师说你能背出圆周率的每一位就加一分加分
99:24
digit of pi you could recite so the hour before class he membered 150 digits of pi and immediately brought us grade up to an a um but if you wanted to have a computer spit out Pi it would be stupid to you know first calculate it out to a million digits and then write print that but instead we've got all sorts of different methods for for calculating Pi um and some of them in involve like well you could solve it you know using trigonometric formulas or you could use all these really beautiful series and and approach a digit of it when you have like the alternating sum of of of well for example right um if you have uh uh plus uh plus 1 9th plus dot dot dot so if you're just taking the sum of 1 over n SAR this is pi^ 2 over 6 right so in terms of algorithmic information it takes it's almost something is more meaningful when it takes fewer lines to describe it then and this and then that's the argument here is that if you wanted to code this image into zeros and ones it would this image would take roughly the
所以上课前一小时他背了 150 位圆周率，立刻就把成绩提上去了提到了 A。但如果你想让计算机吐出圆周率，先把它算到一百万位再写「打印那个」，那就太蠢了而实际上我们有各种各样计算圆周率的方法其中一些涉及，比如说你可以用三角函数公式来求解，或者你可以用那些非常漂亮的级数，一位一位地逼近，比如当你有那种交错求和的时候举个例子，对吧，如果你有加、加九分之一、加等等，所以如果你只是对 1 除以 n的平方求和，这就是 π² 除以 6，对吧。所以从算法信息的角度来说，当描述某样东西所需的行数越少，它几乎就越有意义那么这里的论点就是：如果你想把这张图像编码成 0 和 1，它这张图像所需的 0 和 1 的数量，大概和一堆乱码差不多，对吧？这就显示出
100:46
same number of zeros and ones as just a bunch of gibberish right and that shows kind of a failure of of information entropy but the fact that we can write a program like Curran showed you in just a short few lines that produces this you know very quickly shows that its algorithmic information if you're to write a program um is actually much more meaningful it's not algorithmic entropy sorry um complexity and these are just some rigorous mathematical tools which we're using to think about what meaning is and what information is um and you I know I really hope I've kind of convinced you that a lot of complex phenomenon can come out of very simple things yes Chief what if we like um look at numbers in general like real numbers yes because like when you like give like a name to a number all right give a name to number you have like so many like names to give like numers so sort of like mixing together like different like U symbols and everything but you even even you like an infinite amount of symbols you
信息熵的某种失效之处。但我们能像 Curran 给你们展示的那样，只用短短几行就写出一个程序很快地生成这个东西，这个事实说明它的算法信息——如果你要写一个程序的话——其实要有意义得多不是算法熵，抱歉，是算法复杂度。这些只是一些严格的数学工具，我们用它们来思考「意义」是什么、「信息」是什么。我真的希望我多少说服了你们很多复杂的现象可以从非常简单的东西中产生。是的，老兄？如果我们，比如说，从整体上看数字比如实数呢？是的，因为当你给一个数命名的时候，好吧，给一个数命名你有那么多名字可以给这些数，就像把不同的符号混在一起之类的。但即使你有无穷多个符号，你也只能，比如说，
101:55
can only like you can like maybe like give each one like function numbers like from one 2 3 4 5 what you end up with like an infinite amount amount of numbers but the map of the integers or something but you're trying to map to the real and there's more real than the integers so you have numbers that you cannot even say what they are so does that like have something to do with the algorithmic complex because some numbers you can't say what they are there they like not computer right so I mean what you said does have a relation to this but it's actually you pretty much struck on one of the most fundamental um theorems and paradoxes in the past like 30 or 40 years um and I can give you some more information on it but it's called the lenheim scolum theorem and it's the idea that and it's really it's really an idea on logic and fundamentally the Paradox which you just said shows how if we really want a theory of meaning like if we want to be able to point out every real number but we can only do so in a finite way how do
给每一个分配像 1、2、3、4、5 这样的编号，你最后得到的是无穷多个数，但那是整数的映射之类的，可你想映射到的是实数，而实数比整数更多所以就存在一些你甚至说不出它们是什么的数。那这跟算法复杂度有关系吗？因为有些数你说不出它们是什么，它们不是可计算的，对吧？我的意思是，你说的确实和这个有关，但其实你差不多正好击中了过去大概三四十年里最基础的定理和悖论之一嗯，我可以给你更多相关资料，它叫做 Löwenheim–Skolem 定理（勒文海姆–斯科伦定理），它的想法是这其实是一个关于逻辑的想法，而且从根本上说，你刚才提到的那个悖论显示出：如果我们真的想要一个关于意义的理论，比如我们想能够指出每一个实数，但我们只能以有限的方式做到，那我们怎么
103:01
we know that we actually mean every real number right even though we can do it only in a countable way right sorry so Latif was asking about like when you're specifying real numbers right so let's take the continuous line from 0 to one I know Latif doesn't think this Line's continuous but it is um we only have like a certain number of of well here I'll take it to two so I can specify here's one um so for example the square root of two 1.41 dot dot dot dot irrational um infinite digits does not repeat so there's no real short way but we only have a finite number of symbols a countable number of symbols in fact using integers to name this number so how can we ever say Square < TK of two when we only ever have accountable number of integers to start spelling it out all we can ever do is approximate the square root of two I could give you square root of two out up to a billion billion digits and I still wouldn't be giving you square root of two right um but it's kind of the I suppose and that
知道我们真的指称了每一个实数呢？即便我们只能以可数的方式做到，对吧。抱歉，所以Latif 问的是，当你要指定实数的时候，对吧，那我们取从 0 到 1 的连续直线，我知道 Latif 不认为这条线是连续的，但它确实是。我们只有，比如说，一定数量的——好，这里我把它画到 2，这样我可以指出，这里是 1，那么举个例子，根号 2，1.41 等等等等，无理数，无穷多位，不循环，所以没有真正简短的方式。但我们只有有限个符号，可数个符号，实际上是用整数来给这个数命名。那我们怎么可能说出「根号 2」呢，既然我们只有可数个整数来开始把它拼写出来？我们所能做的只是逼近根号 2。我可以给你根号 2 精确到亿亿位，我仍然没有真正把根号 2 给你，对吧。嗯，我想这有点像，但
104:10
but then that relates to really fundamental Paradox which I can't really talk about um because it you know it's just kind of way out over over the top um but suppose you had something like the way we name sorry sorry the way we name square root of two using mathematics is as whatever X satisfies this equation and this is really the best we can do right but I mean the idea that you're striking upon it really is a fundamental one and it shouldn't be trifled with but are there any more questions because I think I'm running desperately over time um and I'm encouraging you guys to think of these problems um yes um I'm sort of thinking when you have like a ukan space it's so continuous and everything right um would you like think of Consciousness in a similar way to us it looks continuous but is it really continuous so is consciousness continuous yeah right so that's a good question because to what extent do we reduce Consciousness just to discrete firings and neurons right supposed to like be subjected killing
这又关系到一个非常根本的悖论，我没法真的展开讲，因为你知道，那有点太超纲了。不过假设你有类似这样的东西，我们命名的方式——抱歉，我们用数学给根号 2 命名的方式，就是「满足这个方程的那个 X」，而这真的是我们能做到的最好的方式了，对吧。但我是说，你所触及的这个想法确实是很根本的，不该轻视不过还有别的问题吗？因为我觉得我已经严重超时了，我鼓励你们去思考这些问题。嗯，请说。我在想，当你有一个欧几里得空间时，它是连续的之类的，对吧？那你会不会用类似的方式来看待意识——对我们来说它看起来是连续的，但它真的连续吗？意识是不是连续的，对吧？这是个好问题，因为我们在多大程度上能把意识还原成神经元的离散放电呢，对吧？它感觉是连续的，但仅仅因为它感觉像某种东西，并不意味着它就是。所以这其实是一个
105:24
that it feels continuous but just because it feels like something that's not me it is that so that's actually a very good idea because um he said so just like Consciousness we feel like it's continuous but suppose it's not really continuous one thing to think about is that I think the refresh rate on our eyes is something like 200 frames per second um so and that's really why we think things are continuous but but like when we start whirling our hand really fast in front of us we only see it at certain bits right but it's kind of from the continuity of our exper the fact that it happens so fast that we actually say that we approximate saying that it took all the positions in between so it's kind of lazy yeah well it's the best we can do right it's not good enough it's not good enough all right fair enough all right um so I encourage you guys uh it might make the typographical number Theory chapter make more sense you glance over the prepositional calculus but really focus on the typographical number Theory
非常好的想法，因为他说，就像意识一样，我们感觉它是连续的，但假设它其实并不连续。有一点可以想想：我记得我们眼睛的刷新率大概是每秒 200 帧嗯，所以这真的就是我们为什么会认为事物是连续的。但比如说，当我们开始在面前飞快地挥手时，我们只能看到某些片段，对吧？但这有点像是出于我们经验的连续性，也就是它发生得太快了，以至于我们实际上会说、会近似地说它经过了中间所有的位置，所以这有点偷懒。是啊，不过这已经是我们能做到的最好了，对吧。这还不够好。这还不够好。好吧，说得也对。好那么我鼓励你们，这可能会让「印刷体数论」那一章更容易理解。命题演算那部分你可以粗略看一下，但真正要重点看的是「印刷体数论」那一章。然后开始消化
106:28
chapter um and then start processing that handout for chapter 10 that I gave you um and hopefully in about two lectures time or so we can make sense more of what girdle's achievements were and maybe try to connect a lot of these ideas but excellent class guys I I really think this was good thank you
我发给你们的第 10 章讲义。希望大约再过两节课左右，我们就能把它理清楚更多关于哥德尔的成就，也许还能把这些想法串联起来，不过这堂课很棒，各位，我真的觉得很不错，谢谢大家
视频总结 · 一句话概括与核心要点
一句话概括
这节课以"意义不在符号本身，而在符号与解释者之间的同构关系"为主线，从《GEB》第六章"意义的位置"出发，经由信息瓶中信、藏头诗、钟摆方程，走到 L-系统、元胞自动机等"简单规则生成复杂现象"的活体演示，最后落到信息熵与算法复杂度作为"意义"的数学度量。
核心要点
- 意义不是符号固有的，而是关系性的（"点唱机理论"）："La nieve es blanca"对不懂西班牙语的人毫无意义；写在纸上或凿在石灰岩上的"apple"射向外太空也不携带意义。意义存在于视觉输入与大脑电活动之间的复杂同构中——五百年前"apple"不会关联笔记本电脑，语言一旦无人使用便失去意义。
- 信息分三层：框架信息、外层信息、内层信息：漂流瓶中软木塞+纸+瓶子这一"刻意的"配置就是框架信息，它向任何有好奇心的人喊"来解码我"；纸上字符的重复、间隔等模式是外层信息（提示如何解码）；"snow is white"的内容才是内层信息。
- 有模式不等于有意义——伏尼契手稿的教训：这本困扰密码学家近百年的中世纪手稿，约一年前被证明可用"前缀+中缀+后缀"的随机矩阵生成器几乎完全复现。它具备人类语言的统计特征（甚至类似 Zipf 定律的幂律分布），却不含任何真实内容。
- 《螃蟹卡农》对话是三层嵌套的藏头诗：取阿基里斯和乌龟每句话的首字母得到一段文字，该文字本身指示"acrostically"再取首字母，最终得到"J S BACH"。这是一条自指的信息：对话教你如何从它自身提取更高层意义，但压缩率极差——十页对话只编码了一个名字。
- 科学是"逆向工程"：从现象反推简短描述：Hofstadter 第六章只讨论"符号→意义"的单向问题；讲者强调反方向更难——从钟摆的运动反推 θ̈ + k·sinθ = 0（小角近似 sinθ≈θ 后线性化）。这一行方程不仅描述这个钟摆，还统一了时钟、悬索、水波、高速公路上向后传播的堵车波等一整类振荡现象。
- 简单规则可以生成任意复杂的结构——L-系统演示：从字符串"F"出发，反复应用重写规则 F→F+F−−F+F，再把 F 解释为"海龟前进"、± 解释为"转 ±60°"，迭代后得到无限长的科赫曲线；另一套规则 A→B−A+B、B→A+B−A 逼近谢尔宾斯基三角；"FBR"（前进/芽/回退）加规则生成树。用 L-系统和用递归函数画出的树完全一致——同一现象有两种不同描述。
- 元胞自动机：生命游戏、舆论粗化、波动方程：生命游戏仅靠"少于2死、2或3活、多于3死、恰好3生"四条规则产生不可预测的演化，其初始格局本质上是一段程序，能否稳定即图灵停机问题——而停机问题与哥德尔不完备定理在证明结构上（把程序喂给自身的自指）是同一件事。"投票规则"让每格取值 [−1,1]，每步加上邻居之和×0.005，随机噪声迅速粗化成黑白区块——与车窗上水雾凝聚成水滴、乃至族群隔离/冲突预测（新英格兰复杂系统研究所的项目）属于同一"普适类"。给每格加上高度与速度、速度按邻域高度差的均值更新，即得到波动方程的离散近似。
- 宇宙是否离散引发了课堂争论：计算机模拟是离散的，学生质疑空间本身是否连续（芝诺悖论、是否存在最小距离）；讲者以 Seth Lloyd《编程宇宙》和 Wolfram《一种新科学》中"宇宙即量子计算机/巨型元胞自动机"作为可能的立场，但明确表示这是悬而未决的问题。
- 信息熵不能衡量"意义"，算法复杂度可以：Shannon 1950 年代的《通信的数学理论》定义熵 H = −Σ p·log p，但按它计算，谢尔宾斯基三角和一团随机噪点的比特量几乎相同。区别在于：随机的 30 亿位数字最短程序就是"print 该数"，而 π 可以用 Σ1/n² = π²/6 之类级数几行算出。"需要更少行数描述的东西更有意义"——这是 Kolmogorov 式算法复杂度的核心。
结论与值得注意的细节
- 讲者反复强调本课的两大方向：Hofstadter 讨论的"符号→意义"，以及科学与信息论关心的"现象→最短描述"，二者合起来才是完整的"意义理论"。
- 补充阅读为 Hofstadter 新书《我是一个怪圈》第10章（哥德尔的典型怪圈），讲者认为它对哥德尔不完备定理的解释比《GEB》第9章更清晰；一位学生"无解方程≈不可证命题"的类比被肯定为非常接近哥德尔证明的核心思想。
- 学生 Latif 的一个问题——只有可数个符号却要命名不可数多的实数（如 √2 只能被定义为 x²=2 的解）——被讲者指出触及了 Löwenheim–Skolem 定理这一逻辑学基本悖论。
- 课末讨论意识是否连续：讲者指出我们只是因为感知刷新率高而"偷懒地"把离散经验近似为连续。
- 课程预告：两讲之后正式讲解哥德尔不完备定理；建议重点研读《GEB》的"印符数论"（TNT）章节，命题演算可略读。
- 小花絮：讲者提到有朋友为挽救几何课成绩在一小时内背了150位 π；牛顿被苹果砸中的故事被明确指出是神话。
核心句型 · 9
1. to what extent does X mean Y?
“To what extent does Contra crop punctus mean jsbach”
把「是否」问题改成「程度」问题，避免非黑即白。适合讨论有争议的哲学或评价性命题；写作中可用 to what extent 引出论证段。
2. X doesn't lie here or here, but in the relationship of things
“Meaning doesn't lie here or here but it relies it lies here and the relationship of things”
用 lie in 表达「存在于/取决于」，配合 not…but… 强调重新定位。可仿写：The value doesn't lie in the tool itself but in how it is used.
3. the only reason we say X is (because) …
“The only reason we say apple has meaning is because there's this complex isomorphism”
限定唯一原因的强调句式，语气笃定。口语中 the reason … is because 常见，正式写作改为 the reason … is that。
4. what I'm going to argue here is that …
“Fundamentally what I'm going to argue here … is that meaning is not inherent”
学术演讲中预告论点的标准开场。前面加 fundamentally 表示「归根结底」，可替换为 the point I want to make is that…。
5. correct me if my interpretation of what you said is wrong
“Correct me if my interpret ation what you said is wrong”
复述对方观点前的礼貌铺垫，既确认理解又留出纠正空间。讨论课、会议中转述他人意见时非常实用。
6. lo and behold, what we get is …
“Lo and behold what we get is the same thing”
揭晓意外或戏剧性结果的固定表达，略带幽默。演示、讲故事时用于「你看，结果是……」。
7. who would have thought (that) …?
“We get this complex Behavior like who would have thought right”
反问句表达惊讶：「谁能想到呢」。用在简单规则产生意外结果、反直觉发现之后，增强感染力。
8. it's a lot more likely that … than …
“It's a lot more likely that a monkey writing like Shakespeare”
比较可能性的句式，a lot / far / much 修饰比较级。可仿写：It's far more likely that the bug is in the parser than in the database.
9. just because it feels like X doesn't mean it is X
“Just because it feels like something that's not me it is”
反驳「感觉即事实」的经典结构：just because … doesn't mean …。适合批判性讨论中区分主观体验与客观事实。
生词精讲 · 120 · 按出现顺序
wholeheartedly
/ˌhoʊlˈhɑːrtɪdli/
adv.
0:00
全心全意地，毫无保留地
quintessential
/ˌkwɪntɪˈsenʃəl/
adj.
0:00
典型的，最本质的
supplemental
/ˌsʌplɪˈmentəl/
adj.
1:06
补充的，附加的
wrap your head around
phr.
1:06
（口语）弄明白、理解某件复杂的事
trivial
/ˈtrɪviəl/
adj.
1:06
微不足道的；（数学）平凡的
axioms
/ˈæksiəmz/
n.
1:06
公理
isomorphism
/ˌaɪsəˈmɔːrfɪzəm/
n.
1:06
同构；结构上一一对应的映射
inherently
/ɪnˈhɪrəntli/
adv.
2:17
本质上，内在地
chew on
phr.
2:17
反复思考、慢慢消化（某个想法）
scale back
phr.
3:24
缩减规模；（此处）退一步，把问题缩小
conjecture
/kənˈdʒektʃər/
n.
3:24
猜想，推测
recursive
/rɪˈkɜːrsɪv/
adj.
3:24
递归的
parse
/pɑːrs/
v.
4:48
解析，切分（句子、字符串）
stumbling block
n.
4:48
绊脚石，障碍
immersed
/ɪˈmɜːrst/
adj.
6:08
沉浸于（某文化/语言环境）
deduce
/dɪˈduːs/
v.
6:08
推断，演绎
cork
/kɔːrk/
v./n.
6:08
用软木塞塞住；软木塞
configuration
/kənˌfɪɡjəˈreɪʃən/
n.
7:37
配置，组合方式
deliberate
/dɪˈlɪbərət/
adj.
7:37
有意的，故意的
unpack
/ʌnˈpæk/
v.
8:57
打开；（引申）拆解、分析
gibberish
/ˈdʒɪbərɪʃ/
n.
8:57
胡言乱语，无意义的文字
ruse
/ruːz/
n.
10:23
诡计，骗局
cryptologists
/krɪpˈtɑːlədʒɪsts/
n.
10:23
密码学家
intricate
/ˈɪntrɪkət/
adj.
10:23
错综复杂的，精细的
medieval
/ˌmediˈiːvəl/
adj.
10:23
中世纪的
replicate
/ˈreplɪkət/
n.
11:32
复制品（常用 replica）
intelligible
/ɪnˈtelɪdʒəbəl/
adj.
11:32
可理解的，清晰易懂的
power law distribution
n.
12:39
幂律分布
circularity
/ˌsɜːrkjəˈlærəti/
n.
12:39
循环论证，循环定义
tactile
/ˈtæktəl/
adj.
14:02
触觉的
a couple notches too high
phr.
14:02
高了两档（notch：刻度、档位）
holy mackerel
interj.
15:25
（口语感叹）我的天哪
prune
/pruːn/
v.
17:04
修剪；（引申）删减、精简（网络/搜索树）
acrostics
/əˈkrɔːstɪks/
n.
20:38
藏头诗（各行首字母拼成词）
conceal
/kənˈsiːl/
v.
20:38
隐藏，掩盖
out of style
phr.
20:38
过时的，不再流行
compression ratio
n.
23:02
压缩比
nonetheless
/ˌnʌnðəˈles/
adv.
23:02
尽管如此
inherent
/ɪnˈhɪrənt/
adj.
24:32
内在的，固有的
ink blotches
n.
25:22
墨迹，墨渍
tranquility
/træŋˈkwɪləti/
n.
25:22
宁静，安宁
discern
/dɪˈsɜːrn/
v.
25:22
辨别，分辨出
die out
phr.
26:46
灭绝，消亡
interplaying
/ˌɪntərˈpleɪɪŋ/
v.
26:46
相互作用，交互影响
reverse engineering
n.
28:00
逆向工程：从成品反推其构造原理
pendulum
/ˈpendʒələm/
n.
29:31
钟摆，单摆
period
/ˈpɪriəd/
n.
29:31
（物理）周期
rigorous
/ˈrɪɡərəs/
adj.
30:49
严谨的，严格的
approximate
/əˈprɑːksɪmeɪt/
v.
32:19
近似，逼近
algorithmic complexity
n.
33:36
算法复杂度（Kolmogorov 复杂度）：生成对象的最短程序长度
initial conditions
n.
34:51
初始条件
time evolution
n.
34:51
（系统的）时间演化
fractal
/ˈfræktəl/
n.
34:51
分形
abstraction barrier
n.
35:55
抽象屏障：隐藏实现细节的分层界面
hypotenuse
/haɪˈpɑːtənuːs/
n.
35:55
（直角三角形的）斜边
oscillatory
/ˈɑːsələtɔːri/
adj.
37:04
振荡的，摆动的
propagated
/ˈprɑːpəɡeɪtɪd/
v.
38:15
传播（波、信号）
elastic
/ɪˈlæstɪk/
adj.
39:29
有弹性的
salient
/ˈseɪliənt/
adj.
40:31
显著的，突出的
hardcoded
/ˌhɑːrdˈkoʊdɪd/
adj.
41:45
硬编码的，写死的
hammer home
phr.
41:45
反复强调，使深入人心
cellular automata
/ˈseljələr ɔːˈtɑːmətə/
n.
43:27
元胞自动机（automaton 的复数）
rewrite rules
n.
43:27
重写规则（形式文法）
notch
/nɑːtʃ/
n.
45:13
刻痕，凹口；（此处）小凸起
wild
/waɪld/
adj.
50:56
（口语）令人惊叹的，疯狂的
nested
/ˈnestɪd/
adj.
52:33
嵌套的
interlocked
/ˌɪntərˈlɑːkt/
adj.
54:10
交错咬合的
squiggly
/ˈskwɪɡli/
adj.
54:10
弯弯曲曲的
bud
/bʌd/
n.
55:37
芽，花蕾
palindromes
/ˈpælɪndroʊmz/
n.
57:22
回文（正读反读相同）
lo and behold
phr.
57:22
你瞧，果不其然（引出意外结果）
mindboggling
/ˈmaɪndˌbɑːɡlɪŋ/
adj.
59:19
令人难以置信的，脑子转不过来的
elusive
/ɪˈluːsɪv/
adj.
62:25
难以捉摸的
scale free
adj.
62:25
无标度的（各尺度下性质相同）
self-similarity
n.
63:50
自相似性
telescoping
/ˈtelɪskoʊpɪŋ/
v.
63:50
像望远镜一样逐级伸缩、推进
crumble it up
phr.
65:15
把（纸）揉成一团（标准拼写 crumple）
geomorphology
/ˌdʒiːoʊmɔːrˈfɑːlədʒi/
n.
65:15
地貌学；地表形态
boils down into
phr.
66:35
归结为（常用 boil down to）
balloon into
phr.
66:35
迅速膨胀成，滚雪球般变成
overpopulation
/ˌoʊvərˌpɑːpjəˈleɪʃən/
n.
69:10
过度拥挤，人口过剩
suffocation
/ˌsʌfəˈkeɪʃən/
n.
69:10
窒息
notation
/noʊˈteɪʃən/
n.
70:42
记号，符号表示法
steady state
n.
73:38
稳态
halting problem
n.
73:38
停机问题
be reduced to
phr.
75:07
（计算理论）归约为
derive
/dɪˈraɪv/
v.
76:23
推导出
voting patterns
n.
77:36
投票模式
takes stock of
phr.
78:51
审视、评估（周围情况）
coarsening
/ˈkɔːrsənɪŋ/
n.
78:51
粗化（物理：区域合并变大的过程）
doubly nested
adj.
80:28
双重嵌套的
universality class
n.
81:58
普适类（统计物理：临界行为相同的系统集合）
condensation
/ˌkɑːndenˈseɪʃən/
n.
83:23
凝结
path of least action
n.
83:23
最小作用量路径
surface tension
n.
83:23
表面张力
hunter gatherer
n.
83:23
狩猎采集者
segregate
/ˈseɡrɪɡeɪt/
v.
84:39
隔离，分隔
cranking out
phr.
85:40
机械地大量产出
segue
/ˈseɡweɪ/
v.
85:40
（话题）自然过渡
standing waves
n.
85:40
驻波
discreet
/dɪˈskriːt/
adj.
89:10
（此处指 discrete）离散的；注意 discreet 本义为「谨慎的」
bottom out
phr.
90:29
触底，到达最低点
Euclidean plane
n.
91:46
欧几里得平面（原文拼作 ukian）
intensive
/ɪnˈtensɪv/
adj.
94:09
密集的，强化的
seething
/ˈsiːðɪŋ/
adj.
95:37
翻腾的，沸腾的
information entropy
n.
95:37
信息熵
transmission line
n.
96:58
传输线路
ensemble
/ɑːnˈsɑːmbəl/
n.
96:58
系综；（统计）整体集合
abstract from
phr.
98:05
从…中抽象出来
recite
/rɪˈsaɪt/
v.
99:24
背诵
trigonometric
/ˌtrɪɡənəˈmetrɪk/
adj.
99:24
三角函数的
alternating sum
n.
99:24
交错和（正负号交替的级数）
struck on
phr.
101:55
偶然触及、碰到（重要想法）
countable
/ˈkaʊntəbəl/
adj.
103:01
（数学）可数的
irrational
/ɪˈræʃənəl/
adj.
103:01
（数学）无理的
trifled with
phr.
104:10
轻视，随便对待
over the top
phr.
104:10
过头的，超出范围的
refresh rate
n.
105:24
刷新率
fair enough
phr.
105:24
（口语）说得有理，好吧
glance over
phr.
105:24
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理解自测 · 11 题 · 是真懂了，还是以为自己懂
1. 侯世达（经讲者转述）把一条信息分成哪三层？漂流瓶例子里各对应什么？
三层是框架信息（frame message）、外层信息（outer message）和内层信息（inner message）。在漂流瓶例子里（第 6–8 段），框架信息是「瓶子+软木塞+卷轴」这个显得刻意的组合，它提示捡到者「这里有信息，请解码」；外层信息是纸上的重复、空格等结构特征，告诉你该如何切分和解读；内层信息则是解码后的实际内容「snow is white」。讲者强调，框架信息的识别依赖接收者的好奇心，意义并不完全在载体内部。
2. 伏尼契手稿的故事在课上用来说明什么？讲者给出的「破解」结论是什么？
它用来说明「看起来像语言的东西未必有意义」，也就是外层信息可以被伪造。讲者（第 9–10 段）说这部中世纪手稿百年来吸引了包括二战密码学家在内的无数人尝试破译，但大约一年前有人展示，用前缀、中缀、后缀的矩阵随机组合，就能生成统计特征几乎一样的文本，因此手稿可能根本不含意义。它骗过众人，是因为它模仿了人类语言的模式（如重复、词长分布），这呼应了 Latif「也许有人在耍你」的提醒。
3. 康威生命游戏的规则是什么？Curran 用什么比喻解释它们？
每个格子只有生/死两种状态，以周围八格为邻域（第 53–56 段）。活细胞邻居少于 2 个则「孤独而死」，2 或 3 个则存活，多于 3 个则「过密窒息而死」；死细胞邻居恰好 3 个则「出生」。Curran 用家庭比喻：两个邻居像父母，三个像父母加兄弟，太多则挤死。代码中用双重 for 循环遍历每格，算出 neighborhood sum 后按这几条判断 next value。他强调，就这几条规则，从随机初始状态出发就能产生无法预测的复杂演化。
4. π 和一个三十亿位的随机数，在「信息熵」和「算法复杂度」上各有什么区别？
两者的信息熵可能相当——按位编码都很长、位分布都近似随机。但算法复杂度截然不同（第 77–78 段）：随机数的最短生成程序只能是「打印这个数」，长度等于数本身；而 π 虽然无限不循环，却可以用几行代码（如 Σ1/n²=π²/6 这类级数）生成。讲者由此定义：描述所需行数越少的对象越「有意义」。这解释了为什么谢尔宾斯基三角形按像素编码与噪声一样长，但它「显然」比噪声更有内容——因为它有极短的生成程序。
5. 讲者为什么说「Apple 写在纸上射向太空」没有内在意义？这个论证依赖什么前提？
因为外星文明看到的只是墨迹，正如我们看不懂随手造出的「汉字」（第 22–23 段）。论证的前提是「同构」：符号有意义，仅仅是因为在使用者群体中，视觉输入与大脑活动之间存在稳定的对应关系。一旦使用者消失（语言消亡）或对应关系改变（Apple 从水果变成电脑），意义就随之消失或漂移。因此意义不在符号里、不在大脑里，而在两者的关系及其持续修正的反馈过程中——这就是侯世达的「点唱机理论」。
6. Latif 质疑「方程只是把信息藏起来」，讲者如何反驳？反驳是否完全成功？
Latif 的观点（第 29–31 段）是：方程之所以显得简短，是因为读者脑中已有大量模型，符号只是「代表」那个东西。讲者承认方程只对数学家有意义，且确实屏蔽了细节，但反驳说方程的价值不在于缩写单个钟摆，而在于它统一了一整类现象——钟摆、绳索、水波、交通波都服从同一形式，而且在计算机上跑方程比存视频高效得多。反驳并不完全封闭：讲者随后又承认「我们确实在编」，模型只是近似；「统一性」是否本身也是人脑投射，课堂并未解决，这正是 Latif 后续追问「宇宙是否真的如此简单」的空间。
7. L-系统和递归函数画出同一棵树，Curran 说这「更深地说明了什么」？这个讨论走向了哪个哲学立场？
Curran（第 46–49 段）说这表明分形背后有某种更深的东西，可以从不同角度逼近却得到同一对象。学生追问：两种不同描述指向同一对象，它们本身是否「相同」？Curran 用三层信息回答：描述各不相同，但内层信息（对象本身）相同。学生再推一步：物理现实是否也只是对某种更深东西（如数学）的描述？Curran 回应「我们手上有个柏拉图主义了」——即数学对象独立于物理和心智而存在，物理反倒是它的一种表达。Justin 后来把它压缩成「钟摆和方程哪个更真实」。
8. 讲者说哥德尔不完备定理和停机问题「根本上是同一回事」，理由是什么？
理由是两者的证明都依赖自指的对角线论证（第 59–60 段）。停机问题的证明是把「判定程序是否停机」的程序喂给它自己，构造矛盾；哥德尔的证明是用哥德尔编号让算术命题谈论自身的「可证性」，构造出「本句不可证」的句子。两者都表明：任何足够强的递归系统内部都存在它自己无法决定的问题。讲者由此推出「数学家永远不会失业」——总有公理集合够不到的真理，需要跳出系统做元层面的思考才能发现。
9. 如果有人反驳：「信息熵已经足够，谢尔宾斯基三角形比噪声更有意义只是人的审美偏好」，讲者会如何回应？
讲者会承认「意义」确实依赖接收者（这正是上半场的论点），但指出算法复杂度给出了一个不依赖审美的客观区分（第 75–79 段）：谢尔宾斯基三角形存在一个几行代码的生成程序，噪声不存在——这是数学事实，不是偏好。他还会引用 NECSI 研讨班的经验说明这是仍在研究的真问题。不过讲者可能也要让步：「最短程序」是相对于某种描述语言的，而语言的选择本身又回到了使用者共同体——所以客观性只是相对客观。
10. 课堂上关于「宇宙是否离散」的争论，双方论据各是什么？把这个问题迁移到「意识是否连续」时结构有何相似？
支持离散的一方（第 70–73 段）认为时间和空间都有最小单位，芝诺悖论正因空间可无限分割才「说不通」，且宇宙可能就是一台巨大的元胞自动机（沃尔弗拉姆、Seth Lloyd 的立场）。支持连续的一方（Curran）区分「模型中的空间」和「现实空间」：程序里格子有限，但现实中粒子可处于无穷多位置，这是模拟与现实的鸿沟。Justin 折中：若离散尺度小到无法探测，则无法排除。迁移到意识（第 82–83 段）时结构完全相同：意识感觉连续，但可能只是神经元离散放电的近似，就像眼睛因刷新率有限而把快速运动「补成」连续——「感觉连续」不能证明「本身连续」。
11. 本讲用「投票模型」把水滴凝结、舆论粗化和人类聚落归入同一「普适类」。这种跨领域类比放到族群冲突预测上是否成立？有什么风险？
讲者（第 64–66 段）的论据是：三者都服从「每个单元被邻居轻微影响、倾向于与相似者聚集」的规则，而 NECSI 确实用类似模型研究过族群分布与暴力的关系。类比在「宏观空间模式」层面有一定预测力——它能说明为什么会形成分隔的飞地以及哪种尺度的混合最易冲突。但风险在于：水分子没有历史、制度、身份认同和主动策略，而人有；把冲突归结为局部相互作用可能忽略政治、经济等非局部因素，甚至把结构性暴力自然化为「物理必然」。讲者自己也承认「对水分子我能算出能量，对社会我们并没有方程」，所以这种模型应视为启发式工具，而非因果解释。
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