视频库 / NO.044
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MIT Godel Escher Bach Lecture 5
节目发布 2012-12-02 · jasonofthel33t
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17:39 信息越多越接近真相吗？88:39 整体能大于部分之和吗？36:19 数学是发现，还是发明？61:04 简单规则能生成生命的形态吗？
归入 Ⅰ·05 整体能大于部分之和吗？ →EDITED TRANSCRIPT · 依据现场录音编译整理，可划线生成便签
编者按：2007 年夏天，麻省理工学院两位本科生贾斯汀·柯里（Justin Curry）与柯伦·凯莱赫（Curran Kelleher）为参加 MIT 高中生暑期项目的学生开设了一门讨论课，围绕侯世达的《哥德尔、埃舍尔、巴赫》逐章展开。本文是第五讲的实录，柯里先就 TNT 一章作了「冷静的回顾」，从意义理论、信息熵谈到公理为何必须假设，随后凯莱赫上机演示随机文法如何「长出」树木，师生就自然选择与普适形态展开了讨论。本文依据 MIT 开放课程的现场录音编译整理，仅删去口语枝节，发言归属与论证脉络悉数保留。
开场：为什么这一章难读
柯里：大家好，欢迎回来，今天是第五讲。我想先道个歉，接下来的调子也会比较冷静。我会快速回顾上一讲的内容，顺便提出几句警告。但在那之前，我先为这一章道歉。有多少人读这一章时兴奋得从椅子上跳起来？没有，我看到的恰恰是举手的反面。看来大多数人不喜欢这一章，说实话，我也不喜欢。
我把它布置为阅读，唯一的理由是让大家掌握一个基本想法：像 MIU 系统、pq 系统那样，我们有一套完全形式化的排版系统来编码陈述，可以用纯机械的方式摆弄这些陈述。当然，和 MIU 与 pq 不同，TNT（Typographical Number Theory，排版数论）的优势在于，它编码的对象是我们自以为有所了解的东西，也就是数，自然数 0、1、2、3、4 的性质以及关于它们的真理。我主要是希望大家熟悉这套记号，并且熟悉这样一个想法：从几条公理这样的种子出发，只靠反复应用递归的推演规则，就能生出一整张新字符串的网，而这些字符串经过解释，恰好给出我们关于数的真理，或者我们自以为知道的东西。这才是这一章的要点。还有几件事我会强调，不过我们今天照例又是大杂烩，各种话题混着讲，最后还有一样激动人心的东西给大家看。
回顾意义理论：同构与个体语义网络
柯里：先回顾上一讲。我们当时关心的，根本上是一套意义理论（theory of meaning）。这也是我要道歉的地方之一。在学术界，语言学、哲学圈子对什么是意义理论根本没有共识，可我们在课上讲得好像给出了一套很有说服力的说法。
我们当时是这样说的：「雪是白的」这句话的意义，在于存在某种复杂的、用侯世达的话说是「奇异的」同构（exotic isomorphism），对应着你脑子里发生的活动。当你的视觉感知装置聚焦到黑板上，看见这些石灰碎屑抹在黑板表面，你的大脑就跑起一整套复杂的边缘检测算法，先分辨出这里暗、那里亮；等画面聚焦以后，你辨认出一些整体的对象，比如这一组、这一组、这一组字符；然后再对每个字母做边缘检测，读出 s、n、o、w，拼成「snow」。「snow」接着接入一张漫长而复杂的概念语义网络：你第一次看见下雪的记忆、寒冷的体感、去滑雪或滑雪板的经历，如此不断分岔下去，而且越分越成为一件非常个人的事。
我认为这一点很值得注意：「雪是白的」对我意味着什么，和它对你意味着什么，在某种根本意义上是不同的。这是个危险的说法。它意味着意义理论几乎不可能有一个形式化的、数学式的、唯一的解释，而那正是哲学家和语言学家一直想要的。他们希望「雪」直接指向外部世界，指向我们生活其中的这个世界里那种真实存在的水的结晶结构，那才是「雪」的正派公民式的意义，除此之外什么都没有。至于「有一回我们在堪萨斯遇上暴风雪」这类个人理解，他们巴不得统统清除掉。所以我当时宣扬的，是「雪是白的」这个感知输入与它在你脑中点亮的东西之间的同构。
太空唱片：外星人能听懂巴赫吗
柯里：接着我们引出了另一个想法，一个与此紧密相连的概念：信息。请大家把记忆倒回去。侯世达谈到一张唱片，绑在航天飞机之类的东西上射向宇宙深处。他提出的问题是：假如某个外星文明捡到了这张唱片，问「这是什么」，那么在完全脱离人类文化与社会背景的情况下，这张唱片还能有多少意义？一张唱片对半人马座阿尔法星的居民意味着什么？我不知道。
一上来就有一个大的论证：唱片的结构，那些整齐的同心圆纹路（更准确地说是螺旋纹路），假如外星人掏出显微镜，分析这件奇怪器物的一小段，就会注意到纹路里的某种规律性，然后他们会问，这很怪，不知道有没有什么含义。接着就有了这样的设想：他们能不能从唱片反推出唱片机大概是什么样，进而真的把音乐放出来。就算他们真的完成了这一连串不可思议的推理，就算他们把音乐放出来了，音乐对他们有意义吗？还是只是一堆混乱的杂音？拉蒂夫，你有话说？
拉蒂夫：噪音。
柯里：拉蒂夫认为音乐只会是噪音。多少人同意他？有没有人敢站出来反对？
马克斯：我们已经假设他们能听见了。
柯里：马克斯说，这里预设了他们能听。好，我们就把这一点让给他们。既然要做哲学对话，就假定他们能听，甚至假定他们的听觉频率范围与我们相近。假设宇宙中存在某种普遍的「人」的形式，在引力相近的行星上，加上碳恰好是很稳定的分子，演化在另一颗行星上几乎照样走了一遍。那么音乐还会有同样的意义吗？它会有意义吗？
学生：声音本身听起来大体是一样的。但如果有人在唱歌，歌词对他们大概毫无意义，就像我们听一首外语歌，听不懂那种语言。
柯里：那就假设放的是一段巴赫，没有人声，只有那种辉煌荣耀的声响。菲利克斯？
菲利克斯：他们不知道钢琴是什么。
柯里：对，他们大概不知道钢琴是什么。但你们觉得，即便不知道，仅凭声音本身，他们会不会仍然被触发某种美感，甚至敬畏，或者对这音乐的渴望？
学生：那些渴望从哪来？是什么触发它们？
柯里：好，我得谨慎一点，不说渴望，只说美感。你们认为这音乐里的美感和组织性有多根本？
学生：那得看别人怎么说。有人说，我可以把这个和那个联系起来，这听起来像那个，这让我有某种感受，而那种感受让我觉得它美。
柯里：所以即便对地球上的人，巴赫的音乐对每个人的意义也不一样。
学生：但感受还是在那里。我觉得听外国音乐最酷的一点就在这：你在很多意义上能辨认出绝望之类的各种情绪。所以也许关键在于，这些外星人从那些频率里能不能在脑中得到同样的感受。
柯里：正是。不过，就算撇开每个人听音乐时被触发的那些极其复杂的语义网络，也假定外星人没有这些网络，我认为仍然可以很有力地主张：模式（pattern）本身有某种根本的美，音乐里有这种东西，而且它在很大程度上是可以用数学描述的。这是个值得细想的说法：模式本身就是美，它是普遍的，任何人都能检测到模式，而且在某种意义上，模式是意义唯一可依托的东西。
模式即美：香农熵与齐普夫定律
柯里：所以上一讲我把模式和信息联系了起来。这里我又要先泼一盆冷水。我至少提到了信息的熵这个版本，公式是 H 等于负的各项 p(x) 乘以 log p(x) 求和，上次我漏了负号。我不打算把这个公式讲透，它说的无非是给某件事赋一个概率，一件事越不可能发生，它携带的熵就越大，或者反过来说。总之要点是：存在用熵来度量信息的数学办法。归根结底，如果你把这张唱片，或者我们之前讨论过的一幅画、一段音乐，描述成一串零和一，就可以给它赋一个熵值，把它当作信息的度量。
有意思的是，所有语言和某些符号系统都共享一些模式，其中一些与熵的概念密切相关。你甚至可以对字母表里的字母算出信息熵，依据仅仅是它们在语言里出现的频率。拉蒂夫，你说。
拉蒂夫：那只是因为人类心智只能处理这种东西，所以造出来的语言都是同一种结构。
柯里：拉蒂夫认为语言里出现这些模式，是因为人类心智只能处理这些。我不同意。同样的分析已经在 DNA 上做过，DNA 作为一种语言，呈现出同样的模式。除非你认为人类心智创造了 DNA，否则这个说法不成立。
与此相关的还有一个现象，大家可以自己去查：齐普夫定律（Zipf's law）。取任何一种语言，把符号按出现频率排序，会得到一条非常漂亮的幂律：第二常见的符号出现次数恰好是第一名的一半，第三常见的恰好是第一名的三分之一。只要按频率排序，语言里就会冒出这种幂律行为。齐普夫大概是四五十年代哈佛的一位语言学家，他注意到了这一点。这很有趣，因为你可以拿一样东西问：这是伪造的，还是真正的语言？我不知道它什么意思，但我注意到某些符号反复出现，于是把它们排个序，然后你就能纯粹根据符号出现频率判断它是不是一种可理解的语言。这些东西都有严格的根基。
算法信息：科学就是压缩规律
柯里：接下来我们讨论了一个相关概念，因为我说过，熵这幅图景并没有抓住我们所说的信息。如果你拍一张照片，上面全是一滩翻腾的狗呕吐物，把它转成零和一，做信息熵分析，结果可能和另一张照片差不了多少，而后者画的是谢尔宾斯基垫片（Sierpinski gasket），有明显的规律、明显的模式，可以说有明显的意义，而且无限延伸下去。所以香农式的信息熵，整个信息论这个领域，你们想学可以来 MIT 旁听，它对这两幅图的零一编码几乎分不出高下。
于是我们提出了另一个想法，并且真的动手玩了一把：谢尔宾斯基垫片显然是有意义的，因为我可以写一个很短的程序，几行语法，运行之后就画出整幅图。要编码谢尔宾斯基垫片的视觉图像，需要一串长达百万、十亿位的零一序列，这个描述长得离谱。这就像有人让你描述一个钟摆，你说「它摆过去，摆回来，摆过去，摆回来」，然后不停地这么说，想描述多长时间就说多长时间，把自己的声音录下来。这是编码这一现象规律性的极其低效的方式。而这个规律其实可以用符号的迭代来表达。我们用了林登迈尔系统（Lindenmayer system，我从来没赢过拼字比赛），像「F − F + + F − F」这样的规则，再用一个漂亮的小程序递归地展开，只需要几行代码。编码这几行代码所需的信息，远远少于一张实际的图片。
这引出了一个非常重要的概念：算法信息（algorithmic information）。我们由此真正开始琢磨一个想法：科学做的一切，无非是逆向工程。我们拿到一种现象，比如谢尔宾斯基垫片，或者一个钟摆相对某个角度 θ 来回摆动的规律，然后用极简的方式描述它。我们用符号编码这里动力学的规律，把钟摆的算法内容压到最低。向没学过微积分的同学道歉，θ 上面的两个点表示角度变化率的变化率。
然后我们开始玩元胞自动机，本质上就是一张格子，每个格子是黑是白，由它的邻居决定。仅仅摆弄这些规则，我们就撞见了一整片行为的宝藏：改变颜色数量和变化速度，可以模拟波动方程，用这么简单的有限确定性规则做出水洼里的水花；只要说「我的颜色取决于邻居在做什么、在想什么」，就能模拟投票模式，同时也暗示了从众思维的危险，如果总让自己被别人的想法左右，我们就会变成一种同质化的文化。柯伦用几行代码优雅地实现了这些极简规则，我们突然就在描述社会、物理、文化世界里大片的复杂行为。
这引出了一些有趣的念头。我记得我们偶然说到一句：因为看到行为的自相似性，同一条描述人类群体行为的投票规则，也描述了水汽凝结成珠。你往浴帘上喷水，水会结成珠，因为每个水分子都想降低自身的总能量，而抱团可以做到这一点，于是就出现了聚簇、结茧的性质，和我们在水珠里看到的一样。然后我们就开始说，哎，这是不是意味着宇宙是分形的，有些概念适用于所有层次？
泼冷水：宇宙不是分形，要可证伪
柯里：我感觉我们当时被那一刻的兴奋冲昏了头，也许把大家引上了一些虽然激动人心却根基不牢的路。如果你去问科学界的人「宇宙是不是一个巨大的分形」，很多人会说：显然不是。理由有好几条。如果你说的是跨尺度的自相似，那种「太阳系有太阳和绕着转的小行星，啊哈，原子也是同一个模型」的看法是错的。当今物理学的一件根本大事，正是弥合原子层面数学描述与行为的分歧，原子根本不长那样。学过化学翻过课本的人都见过那些奇怪的概率云图，它们受量子力学支配，与太阳系的经典力学描述完全对不上。所以在这个意义上，宇宙显然不是分形，因为规则并不在所有层次上通用。
然后我还做过一个我当时觉得很带劲的演示：拿一张纸揉成团，再展开，说这张纸的拓扑和形态，几乎与山脉、河谷、河流如出一辙，跨尺度的模式生成，我在很局部的层面做的事，就是我们在很大尺度上看到的东西。我由此暗示宇宙在概念上是分形的。但数学家或物理学家真正会告诉你的是：我描述这一现象的方程是无标度的（scale free）。无标度这个概念我们在流体力学里天天用：我用手指划过水面，能看见指头后面形成小漩涡；我们也有真实的照片，山峰刺破云层，云在山尖周围流动，形成卡门涡街（von Kármán vortex street），漩涡一个个脱落。流体力学里有太多美丽的图像。这里写「山」，我也可以擦掉写成「手指」。我真正应该提倡的，是数学方程这套形式体系的美与优雅，以及当我们能让它们无标度时，它们如何在各种层次上描述现象。所以我只想谨慎一些，不说日后会给自己惹麻烦的话。
除此之外，我鼓励大家尽情放飞，探索新想法。柯伦和我之所以从根本上相信计算机科学和数学是思考的框架，是因为你可以想得再高远、再哲学、再离谱，但到头来，你要么用证明严格地让它成立，要么在计算机上执行出来，后者在几乎所有实际意义上和证明一样好，因为它要么行，要么不行。你关于宇宙的思考，要么行，要么不行。根本上你需要可证伪性（falsifiability），你的想法要有可能不成立。做形而上学的思考很有趣，谈论高于宇宙、高于时空的东西，谈它们的结构，都很有趣，但到最后，我们要能用观察检验，或者用数学、用计算机证明。这就是我这盆冷水。对那些不得不听、又不想听的同学，抱歉。
TNT 与 ω-不完全：公理为何必须假设
柯里：现在说点更有意思的，先看一眼时间。关于数论，我特别想让大家注意一件事：什么东西可以取作公理，以及为什么我们无法证明它们。请翻到第 221 页，你们看到一座金字塔：0 + 0 = 0，0 + S0 = S0，0 + SS0 = SS0，如此往下。
首先我想指出其中的一些优雅之处，这正是数学家成为地球上最龟毛生物的原因。我们这里没有一堆数，只有一个数，就是零。其他每个数都是零的后继，或者零的后继的后继的后继。用我们的元语言说，1729 个 S 后面跟一个零，就代表 1729 这个数。这里的优雅在于，你只需要两个概念，零和后继，整个东西就白白到手了，很美。
但这个例子要凸显的是：如果不假设，我们就无法证明这条陈述：对所有 a，0 + a = a。多么平淡显然的一件事，对吧？可我们手里只有那些具体的陈述，事实上我们得到的是整座金字塔、整座真陈述之山，我们想一跃而至这条普遍性结论，觉得「这显然成立」。但我们想跳过去的欲望，根本上来自我们对数、尤其是对整数如何行为的理解，我们的心智模型（mental model）。这一点很重要：我们手头真正拥有的，从来只是心智模型，通过形式系统把它们严格化，我们才看得清自己的定义是否涵盖了我们想涵盖的一切。按 TNT 的原有设置，没有这条公理就永远证不出这条陈述，所以最终只能假设它。侯世达把这叫做 ω-不完全（ω-incomplete），ω 是我们用来一口气指称全体整数的记号。意思就是：即便你有这无限一摞真陈述，你也没有那条把它们全部囊括为一个真理的陈述。
0.999…=1：心智模型的失效
柯里：心智模型这个概念我觉得非常重要。我记得高中第一次看到下面这个证明时，简直震惊，甚至有点毛骨悚然。多少人相信 0.999…（9 无限循环）等于 1？德尔相信，娜丁相信。有人能上来证明吗？菲利克斯，你相信吗？能证明吗？先想一想，打心底里想一想。有没有人想炫一下，像年轻的伽罗瓦一样跳上黑板演示给我看？德尔，你有什么想法？
德尔：乘以 10？
柯里：好，我采纳你的建议，带大家走一遍证明。我们把这个东西叫 x，暂时忘掉它是 1，叫 x 等于 0.999…。德尔建议考虑 10x 这个量。
学生：是 9.999…。
学生：如果在无穷处停下，是 1，因为它在逼近那个数。
柯里：不用极限的概念，我们也可以更根本地证明。菲利克斯，你知道吗？
菲利克斯：减去 x。
柯里：菲利克斯说该减去 x。x 本身是 0.999…，做减法，10x 减 x 得 9。于是我们有了一条新真理：9x = 9。满足它的唯一的数是 x = 1。我永远忘不了预备微积分课上，老师第一次做这个证明时，坐我旁边的女生说：「不可能，它不是 1，看，如果它是 1 我就会写 1。」可我明明写下了一个完全不同的数，而它是同一个东西。这展示了心智模型要告诉我们的一件极其重要的事：我们可以对一个对象，比如实数，有看似正确的理解，却不断在最基本的层面被它这样吓一跳。
这些都是关于数学做什么、我们在数学里做什么的元陈述。具体到 TNT，我们的想法是，数论，至少 TNT，应该囊括我们关于数论的全部思考。这可以追溯到欧几里得对他的几何公理的说法：我们知道这些东西根本上为真，或者至少相信它们为真，它们囊括了我们关于几何的全部知识，由这些假设推出的一切也都应当确定为真。
非欧几何与希尔伯特：解释的危险
柯里：然后就出了萨凯里的事，还有高斯躲着不发表。当然，高斯总是比所有人早五十年发现，然后翻出笔记本说：「哦，抱歉，这个定理我在第一杯咖啡和第二杯咖啡之间就做过了，不过很高兴你也发现了。」这就是非欧几何（non-Euclidean geometry）的概念：打破第五公设，也就是给定一条直线和线外一点，存在唯一（唯一，加感叹号）一条直线与之不相交。但如果你在球面上工作，把直线定义为大圆，那么你能画的任何直线都是大圆，两条直线实际上有两个交点，大概是这样，抱歉画得很糟。
于是希尔伯特，就是那位极力推动数论这些东西的人，说：我们只有点和线这些根本的公理化概念，我们能从中得到的最多只是它们彼此间的逻辑关系；一旦我们试图去解释这些东西，就会惹上麻烦。当我们解释「线」这个陈述时，指的是这块平黑板上笔直的东西，而不是地球表面上弯曲的东西，我们就惹上了麻烦，因为我们提供了一种解释，而没有严格贴着形式体系走。这也显示了从你的形式体系里带走过多解释的危险。侯世达在这一点上告诫我们提防解释，我为他感到骄傲，因为解释会给你惹麻烦。
现在我先回答一些关于这一章的问题，然后介绍柯伦要做的事，再进入更新更有趣的内容。关于 TNT 有什么问题？我记得在本科的《哥德尔、埃舍尔、巴赫》研讨课上，有人说：「他那些超自然数到底在说什么鬼？」还有 ω-不一致。这些不是平凡的概念，整个数学分支都在研究它们。所以如果你们觉得完全没懂，不必自责。桑德拉？
桑德拉：我搞不清 TNT 到底有多少条规则。他谈到一些子类别，比如命题演算，然后又说把它并入 TNT。
柯里：对，命题演算被并进了 TNT。抱歉，我没布置更早那一章的阅读，所以这部分脱离了上下文。
六道 TNT 量词练习淘汰赛
柯里：你们有没有试过书里的小练习？有一节我想特别拿出来，答对的人都记一功，因为我有自己的答案，但不确定对不对。这也顺便检验大家对记号的掌握。第一道：~∀c:∃b:(SS0·b=c)。先不管波浪号，谁能给出解释，并判断真假？拉蒂夫。
拉蒂夫：对所有 c，存在 b，使得零的后继的后继乘以 b 等于 c。
柯里：不带波浪号，这句话是真是假？
拉蒂夫：假。因为二乘以任何数都不可能得到所有的数。
柯里：对。这条陈述根本上是在说，所有自然数都是偶数，都能被二整除，都是二的倍数，这显然不对。用特殊化规则，我们可以挑，比方说，三，抱歉，我们的系统里没有「三」，是 SSS0。通过解释可以看到，不存在一个 b 使得二乘以它等于三，因为三是奇数。而带上波浪号，也就是「非」，这条假陈述就变成真的了。
第二轮：∀c:~∃b:(SS0·b=c)。拉蒂夫以外的人回答。我要一个个把你们淘汰掉，最后站着的人拿最难的题。马克斯？
马克斯：对所有 c，不存在 b，使得二乘以 b 等于 c。
柯里：真假？你有一半的胜算。
马克斯：假。
柯里：对，是假的。取 c 为二，显然存在一个 b，就是一，使得二乘以一等于二。现在淘汰了两位。第三轮：∀c:∃b:~(SS0·b=c)。对所有 c，存在 b，使得二乘以 b 不等于 c。真还是假？
学生：假。
柯里：我认为其实是真的。这条陈述说，对每个数，都存在另一个数使得这个等式不成立。比方说四，你可以挑三，二乘以三不等于四，这正是它说的。我觉得波浪号放在前面很容易混淆，不如直接把它看成一个「≠」，容易得多。所以这条我认为是真的。
第四轮，说着说着我自己都绕晕了，这套记号确实笨重：~∃b:∀c:(SS0·b=c)。不是马克斯、不是拉蒂夫、不是德尔的人来。桑德拉？
桑德拉：不存在一个 b，使得对所有 c，二乘以 b 等于 c。
柯里：对。最容易的做法是先不管波浪号，考虑「存在一个 b，对所有 c 都成立」。就照写下来的样子，带波浪号，真还是假？
桑德拉：假？
柯里：等等，我写对了吗？先看不带波浪号的情形。我们有一个神奇的 b，不管放进什么数，那个数都是二与它的乘积。有没有这样一个数，使得任何数都等于二乘以它？举例，取 c 为三。
桑德拉：然后 b 取一。
柯里：对，我们找到了一个 c，就是三，使得二乘以一不等于它，所以不带波浪号是假的，带波浪号就是真的。很好。
学生：可是我们没有「一」。
柯里：不是没有一，在我们的记号里，一写作 S0，零的后继，就是这样写而已。很好，我就是想让大家试试这些题。
还有两道。前四位以外的人来。第五道：∃b:~∀c:(SS0·b=c)。
学生：存在一个 b，使得并非对所有 c，二乘以 b 等于 c。
柯里：给我一个真值判断。我们有这个 b，我们说对这个指定的 b，不管这里放什么都成立。或者说得更清楚一点，这涉及那一章里的一些符号搬运：存在一个 b，不管放进哪个 c，等式都不成立……你说是真还是假？
学生：真。
柯里：取三试试。我很确定这是真的，因为一旦固定 b，比如取四，显然并非对每个数，四乘以二都等于它，特别是对六就不成立。我没法放进任何数都让它对。还有问题吗？这条为真，大家都接受了？好。这些逻辑关系怪异而复杂，所以数学家大多数时候不用这些东西，因为会被符号绊倒，而他们脑子里早就知道自己做的是对的。
地方不够了，第六轮写在这上面，快点做完好交棒。∃b:∀c:~(SS0·b=c)。前五位以外的人，玛雅，给出解释和真值，如果你够勇敢的话。
玛雅：存在一个 b，使得对所有 c，二乘以 b 不等于 c。
柯里：对，等价地，可以把波浪号换成「≠」，这样更容易。存在一个 b，使得对所有 c，二乘以 b 不等于 c。真还是假？
玛雅：真。
柯里：存在一个 b，使得对所有 c，不管这里放什么……
玛雅：那就是假的。
柯里：对。在存在量词和全称量词之间来回翻转，真的会把人搞糊涂。这句话说的是，有一个 b，无论这里放什么，等式永远不成立。但这不对，因为你总能取到某个 c 让它成立。如果指定 b 为二，这个 b 是存在的，那么我们能找到一个 c，所以「对所有 c」不成立，具体说 c 取四，二乘以二等于四，尽管这条陈述声称我们挑的这个 b 不管放什么都不相等。
所以我的答案是：真、假、真、真、真、假。这也符合他的第二条提示：要么四真两假，要么四假两真，这与你怎么搬运波浪号有关。当然，除非你真打算当一辈子逻辑学家，否则不必花很多时间摆弄形式系统。但练习一下是好的，因为吸收这些新符号的困难，为它们在你的神经网络里腾出位置，我认为是重要的练习。这又回到意义理论。
两小时讲座给不了七年的理解
柯里：这是我交给柯伦之前最后一件要道歉的事。当我说递归、形式系统、同构、算法信息、香农熵、神经网络这些词时，它们对你们的意义，和对一位多年熬夜、折磨自己去解决这些问题的教授的意义，并不相同。反复思考、犯错、再修正自己对某样东西的理解，这个过程会迫使你大脑里的某些部分在这里、这里、这里相遇。而我只是对着你们讲，我真正想做的，只是激起你们对我所讲内容的兴趣。我没法把七年的本科和研究生功课压缩进两小时的讲座，直接钻进你们的大脑，把这个神经元接到这里，那个接到那里，让你们立刻拥有塞斯·劳埃德或任何一位专家对这些学科的理解深度。两小时的讲座给不了这个，除非我布置成百上千页的习题，让你们一周做一百小时，但我不会那么干，因为我不邪恶。
除此之外，要抓住的一点是，我注意到自己老说「根本上」，尽量少说。重要的想法是：你可以从一组基本陈述出发，你认为它们捕捉到了某种真理，然后应用一条递归规则、一个递归算法去生成新的字符串，产生新的东西。进入今天下半场之前，我希望你们想着这棵我们试图培育的真理之树。它全都始于皮亚诺公理、数论，我们应用这些规则，造出不同的陈述，就像前一两讲里我们做的 MIU 树一样。我注意到没人来挑战我那二十块钱。
我们从侯世达列出的五条基本公理开始，他实际上是这样陈述的。谁来给出解释？「精灵是精灵」，也就是零是一个数；「每个精灵都有一个元，它也是精灵」，也就是每个数都有后继，后继也是数；「精灵不是任何精灵的元」，也就是零不是任何数的后继（这是第 216 页）；「不同的精灵有不同的元」，也就是两个数不相等，它们的后继也不相等；最后，「如果精灵有 X，且每个精灵都把 X 传给它的元，那么所有精灵都有 X」，这是归纳原理：如果零有性质 P，且任何数都把性质 P 传给它的后继，那么所有数都有它，因为零传给一，一传给它的后继二，性质 P 一路传下去。数学归纳法就依赖于此。
从这些基本的东西出发，你实际上可以推出大部分数论。你只是应用这些推理规则、归纳规则，从树干出发，得到一条新定理：既然一和二不相等，那么二和三也不相等。你就这样按照规则往树上添东西，完全是局部的，完全取决于你当下手里有什么、愿意应用哪条规则。然后涌现出的模式惊人，我们今天恰好把它叫做数论。你们马上会看到一些更有趣的东西。我们休息两分钟，放松一下，然后交给柯伦。
Context Free：随机文法长出树
凯莱赫：他刚才讲的东西，也可以叫上下文无关文法（context free grammar）。上下文无关文法由符号和产生式规则组成。这里的符号是 S、0 以及各种数学运算符，从文法的角度看它们只是符号；产生式规则就是推理规则，你取一个字符串，对它的一部分做操作，得到新字符串。用类似的系统，可以定义如何在屏幕上移动圆圈。
我来解释一下。这个程序叫 Context Free，是开源项目，你们可以下载自己玩。代码里，「startshape」只是入口，不会变。我们定义一条叫「spiral」的规则，规则里有「CIRCLE」，在屏幕上画一个圆，然后再次调用 spiral。「y 2」表示每次把 y 坐标加两个单位，「size 0.9」表示每次往上走时把尺寸乘以 0.9。这条规则定义了这幅图。程序的做法是，当图形小到看不见，就停下来。所以这其实是无限递归，只是到某一点因为太小而停止。这是我们的框架，一点点改动代码，我们会得到一些惊人的图像。我边改边解释。
刚才间距是二，所以看得清是在画圆。我把间距减到 0.4，渲染，就成了这样。现在我再定义一条规则，也叫 spiral。这门语言的特性是，定义两条同名规则时，每次调用这个名字，就以相等的概率调用其中一条。先在这之前加一个旋转，「rotate 1」，每次旋转一度。你看它转了一点。如果每次尺寸乘以 0.99，缩小得慢一些，就看到螺旋出现了；乘以 0.999，螺旋更明显。就是这样，不知道为什么跑出屏幕了。
现在定义第二条 spiral 规则，「flip 90」，翻转九十度。渲染之后你们觉得会发生什么？也许我没说清 flip 是什么意思。「rotate 1」是往这个方向转一度，而调用 flip 之后，再「rotate 1」就往相反方向转。假设先调用第一条规则五次，画五个圆，然后调用第二条规则一次，翻转，再调用第一条五次，它就会往另一边转一点。两条规则一开始是等概率调用的。渲染出来就是这种蜿蜒的东西：一半时间调用第一条规则，把圆往上挪一点、旋转、缩小、画圆；另一半时间调用第二条规则，翻转旋转方向。
这门语言的另一个特性是可以改变调用概率。在第二条 spiral 规则，也就是翻转规则旁边写 0.1。不写数字默认是一，所以第一条规则的权重是一，第二条是 0.1。
学生：那是不是意味着第一条的概率是一减去那个数？因为掷骰子的话，概率不能超过一。
凯莱赫：说到概率，我想它实际做的是先求和，再取每条规则占总和的比例。不管怎样，第二条现在的概率低于一半。可以看到它连续迭代更长时间而不翻转。再减到 0.01，翻转得更少。0.001，翻转就少得多了。这么简单的规则、简单的上下文无关文法，已经生成了非常酷的图像。到目前为止有问题吗？
如果我在第二条规则里再加一个不带翻转的 spiral 调用，这意味着什么？
学生：先翻转，然后照原来的方向继续。
凯莱赫：差不多，她说先翻转，然后不翻转地继续下去。有了这条规则，它不只是翻转，而是让一支沿着切线方向出发，同时原来的那支也不翻转地继续走。渲染看看，正是这样：每次分岔时，它也继续沿原方向走。
调一调参数，就能得到一些看起来有机的形态。如果增加分岔概率，它就失控了，也许这不是我们想要的。放回去，把尺寸乘以 0.99，好了，现在增加分岔概率，就得到这些树。看，像树一样，太疯狂了。分岔概率再增加，树更茂密，因为分岔更多。是不是很酷？
这让人不禁想：大自然生长植物时，用的是这种上下文无关文法，还是林登迈尔系统那样的固定全局规则，在越来越小的尺度上反复应用？我想自然界看到的是两者的混合，因为有些植物特征非常规则，有些则不然。植物仍然是个谜。但这看起来相当有机，我对此惊叹不已。我可以摆弄参数，得到非常酷的生长形态。
把它当作植物发育的模型来看：我们模拟的是植物尺寸不断缩小，分岔时质量翻倍，这个模型有点糟。想想一棵树，它一直往上长，分岔时主干继续往上，而枝条往旁边伸出去。我们可以改文法来做到这一点，得到更像树的东西。上面这条规则是「一直走」时执行的，我把它改成只增加 y；另外两条里，一条直着走，另一条分岔并变小。直着走的尺寸取 0.9。至于 flip 在这里做什么，我稍后解释。分岔那条写「rotate 45」，尺寸 0.3。得到的是很稀疏的结构。把分岔概率提高到 0.2，就得到这些看起来像树的东西。
我再解释一遍规则，以防有人没跟上。这条规则里的第一个 spiral 编码的是：每次分岔时，尺寸按 0.9 缩小一点，然后形成分支；第二条规则里的「size 0.3」表示分支的尺寸是原主干的 0.3；「flip 90」表示下一次分岔往相反方向伸。可以把 flip 去掉，去掉之后所有分支都朝同一个方向。看，它们只往一个方向长。再玩玩参数，比如 0.95，看看会怎样。得到一些很有趣的东西。把分支稍微加粗，比如 0.4，看，像一棵树。有问题或评论吗？
如果在主规则里加一点旋转，「rotate 1」。拉蒂夫？
拉蒂夫：它在内部调用自己的时候，调用的是正在执行的那条，还是另一条？
凯莱赫：好问题。他问，规则内部调用 spiral 时，调用的是上面这条还是下面这条。这正是概率起作用的地方。spiral 内部有三处调用 spiral，每次都按一定概率调用其中一条。第一条的概率是 1 除以 1.2，很高；第二条是 0.2 除以 1.2。每次从任何地方调用它，都会进入程序问「我该调用哪一条」，程序按这些概率分配。
拉蒂夫：那放在哪里有区别吗？如果把带 flip 90 的那个 spiral 挪到上面那条规则里呢？
凯莱赫：你是说，把这个从这里拿出来放到上面？可以试试，我不知道会怎样。它必须放在某个函数里，才能调用自己。
柯里：我想这里的意思是你有两个选项：要么执行简单的 spiral 画圆例程，要么执行另一条 spiral 例程，旋转 45 度、尺寸变为 0.4，而不是翻转 90 度、尺寸变为 0.95。我不完全清楚算法怎么实现的，但我认为底下的 spiral 旁边写 0.2，意味着上面那条只以 0.8 的概率调用，显然比 0.2 高得多。计算机每次掷骰子，都在决定执行上面的 spiral 还是下面的，然后每条 spiral 的内容决定了你看到的行为。
凯莱赫：我们刚才把它挪上去之后，每次调用 spiral 都会分岔，因为现在有两个 spiral，而那一条的概率非常高，等于几乎每次都在分岔，于是得到一团乱麻，而且永远算不完。这个系统有各种有趣的模式。我把它放回原处，还在计算……好了，停了。
看起来很疯狂吧？去掉那个旋转，就变得非常规则，是一种规整的结构。可以改角度，60 度，或者 90 度，看起来像道路？
学生：像河边或湖边的树，你能看到像是河流。
凯莱赫：你是说像一条河，带着许多小支流从它分出去？这种结构在自然界随处可见，太惊人了。如果在往前走的主规则里加一点旋转，「rotate 1」，就得到这种脉络状的东西。
学生：像风吹着树。
凯莱赫：对，风吹着树的时候。或者很像藤蔓爬墙，或者植物的根，地下的根就长这样。
学生：能用这个做科赫雪花，或者你之前展示的那棵分岔树吗？
凯莱赫：我没试过，也许可以，但我觉得不行。这个程序的行为是完全随机的，它是随机过程（stochastic），临场决定执行哪条规则。它仍然是递归的，完全递归，但不是确定性的。确定性赋予科赫雪花或者我之前展示的那棵分岔树那种绝对刚性的规则性；而这里规则是随机执行的，所以得到的是不规则的分形。
柯里：不过谢尔宾斯基垫片也可以用掷骰子随机生成。
凯莱赫：对。
柯里：取三个点，随机扔一枚飞镖，然后……取到最近顶点的距离，把那个点填上？我记不清了。
凯莱赫：混沌游戏（chaos game）。用谢尔宾斯基三角形玩混沌游戏是这样的，我之前讲过，再快速讲一遍：有三个点，从某个位置出发，随机选三个点之一，从你所在的位置走到那个点的一半。比方说选了这个点，走一半；再选它，再走一半；然后选那个点，从这里走到那里的一半。这样做一千次之后，得到的所有点，在极限下，如果做无穷多次，就是谢尔宾斯基垫片。做一会儿之后就开始像谢尔宾斯基垫片了。所以这是随机的。也许我可以用这门语言写出来，也许可以，我没试过。
随机与确定：混沌游戏与稳定性
凯莱赫：我准备了一些现成的例子。我想谈的一件事是这个系统里的稳定区与不稳定区，它是一个非常动态的系统，会出现不同的局面。看这个……等等，那是 PNG，不是程序本身，打开真正的那个。
这里是怎么回事？我有这几条规则，前两条和刚才的类似，为了讲解先把第三条去掉。这挺酷的，是棵树。第一条规则前进一个单位，圆的大小也是一，所以能看到所有的小圆；前进一步，尺寸乘以 0.99，这是主规则。另一条规则的概率是 0.02 除以 1.02，是分岔规则，它再次调用 tree，旋转正 20 度或负 20 度。这就是我们的规则，它生成了这棵树。
现在加上第三条规则，它把尺寸乘以五，相当极端，但发生的概率很小。我把概率再降低，0.003，或者 0.01。它只发生了几次。可以看到通常不发生，所有这些分岔和迭代里都没发生，但在这一处，尺寸乘了五，出现一个更大的圆，然后继续传播下去；在这根末梢又发生了一次，然后又一次。
这个模式实际上也出现在演化里，非常迷人。演化中有各种物种，或者说不同的基因组谱系，它们分岔、扩散。在某个时间点，时间就是总迭代次数，假设地球上发生了一次巨大的灾变，只有这一种生物或一小群生物幸存下来，它们的权重骤然增加，随后繁衍扩散，我们就得到演化的一次新的分岔。然后同样的事在这里再次发生，砰，再分岔，再分岔。这是一个非常不稳定、不可预测的系统，因为这些小事件可以彻底改变系统的面貌。没有这条新规则时，它是稳定的：我们知道尺寸总会越来越小，直到消失，这在极限下是有保证的。但引入一条往回走的规则，系统就变得不稳定、不可预测得多。如果把这条规则的概率提高到 0.1，系统很快就失控；0.01 也一样，越来越大，直到出现这条报错信息：「图形太大」。这就是系统失控时的报错。
柯里：这几乎相当于陨石频繁砸向地球，每次只有极少数物种幸存。
凯莱赫：对，可以这么对应。
柯里：以更高的频率摧毁遗传多样性。
凯莱赫：如果回到演化的比喻，大概对应于巨大的灾变不断发生，而每次都奇迹般地有一个物种幸存，比喻有点极端，我不确定是否真的成立，但这个参数空间里的稳定区与不稳定区很有意思。再看几个例子我就结束。这棵粗树：在某些参数组合下，我们得到非常有机的形态；如果进一步提高分岔概率，就得到这些漂亮的、粗壮的、有机的树。
向光性、自然选择与普适分形
学生：生物真的用这种系统来生存吗？如果有的生物有这种系统、有的没有，有它的是不是在演化上更有优势？
凯莱赫：她是从一个完全不同的角度问：对一个给定的生物，发展出这种递归系统，在演化上是否更有利？我认为对植物来说，这是植物发展出来的主要东西之一，使它们在演化意义上更可行、更适应。这也许是个值得考虑的要点。
柯里：不过，究竟是什么让一棵树那样弯？假设这是个相对平静的地方，没有飓风级的大风把树吹弯，还有什么原因会让树那样长？
学生：阳光。
柯里：对，阳光，向光性（phototaxis）。植物有一套反馈机制，能感知光。就像我在这里放一盆植物，角落里那盏灯照着我们，植物实际上能动态地改变分岔的概率。当然机制有点不一样，是生长素在交换，细胞壁塌陷，树能很快朝另一个方向生长弯曲。所以这不单是一个概率性的上下文无关文法，它必须有某种反馈机制，再往上，演化还会改变规则本身。
凯莱赫：对。看看这些「叶枝」的数量，我说叶枝，指的是末端的那些，有这么多，它把照到的阳光最大化，把树的表面积最大化。我想这是它更适应的主要原因之一。你有问题？
学生：只是个评论。这也说得通，你说的表面积，在底部长分支没有任何意义，因为光已经被上面挡住了。
凯莱赫：对，如果有某种突变让树在底部长出分支，那棵树不会有任何优势。你说的是选择：如果有一棵树的规则让它在底部和顶部都分岔，它会被选择淘汰，因为光被上面的枝条挡住了，不可行。这就是演化的本性。
学生：这看起来也像大脑，上面是大脑皮层，底下是连接。
凯莱赫：他说这像大脑，这是大脑皮层，这些连接延伸到大脑表面，那里布满了脑细胞。
学生：或者血管结构。
凯莱赫：对，血管，从心脏出来的血管结构，分形般地铺满全身。这是一种到处都出现的普适形态，太惊人了。我们能做的只是惊叹：哇，它们都一样。但它从哪来？意味着什么？我不知道，这需要探索。
学生：会不会是它必须如此？
凯莱赫：什么意思？
学生：可以说它是最高效的算法，而演化偏向效率，所以到某个时候必然撞上这个算法，一旦得到好东西，就不会放手。
凯莱赫：你说的是整个生物系统、一切事物的演化。他说，也许这是唯一行得通的东西，它必然存在，因为它是发展我们的生物构造最高效的算法，一旦出现，就在这一长串演化事件中扎了根。我认为你说得对。而且它可以编码成一组非常简单的规则，看，这么少的文本就编码了这一切。我想我们的基因组也类似，如果演化范式找到了一种高效的方式，把一组规则编码起来，做一件让我们更适应的事，它就会留下来。分形和递归算法被编码进我们的基因组，我认为这是一个合理的假说。
学生：行为也是，比如蚂蚁，蚁群。
凯莱赫：对，天哪，无处不在，涌现性质。我讲完了，交回给贾斯汀。
收尾：宇宙是形式系统与课程去向
柯里：我来收个尾，给大家一点结论感和方向感。请把投影关掉。柯伦刚才暗示的想法，我们正站在它的边缘，也是我开课时就告诉大家的《哥德尔、埃舍尔、巴赫》的明确论题：宇宙在根本层面上是一个形式系统，遵循某些确定性的规则，或者也许是概率性的规则，但无论如何是一个形式系统。我们有「我」这个标签，往一样东西上一贴，实际上遮住了大量细节。以根本的方式理解这些，是这门课的既定目标。
我还在斟酌，因为我在持续地、概率性地修改这门课的走向。目前的计划是：讲完无门与哥德尔，你们会从柯伦和我这里领教一种怪异的东方风味，把禅与逻辑合在一起，最后讲哥德尔不完全性定理。然后我们跳到书里靠后的第十六章，「自指与自复制」，那一章有点过头，首先它很长，57 页，但它有一个「排版遗传学」的想法，我们会看遗传、蛋白质折叠这些造就我们的过程，如何以规定的方式对应于我们讨论过的形式系统。之后再往回跳。我意识到这门课已经变成一门专题课，接下来本质上讲大脑与思维。底线是，侯世达的思想并没有一路打通，否则问题早就解决了，我们就能说「太好了，意识解决了，去干别的吧」。它没有解决，中间有巨大的缺口：好吧，也许我接受宇宙是一个形式系统，但哥德尔不完全性定理对物理系统到底能说什么？我们先把那些放下，谈大脑、大脑中的元结构、心智、思考、人工智能，以此结束课程。
当然我也在考虑放一部电影，《半梦半醒的人生》（Waking Life），不知道你们有没有看过，算是我送给大家的礼物，感谢你们在这门课上的努力，不过可能需要家长同意书，之后再说。今天讲得节奏有些慢，抱歉，希望下一讲更精彩。请阅读无门与哥德尔那一章。我本来打算今天做那一章前面的对话，显然来不及了，下次可以做。还有《我是个怪圈》的讲义也请读一下，下一讲我有很大一部分会从那里讲起。谢谢大家来，祝好。
排版 + 横图 + 来源，粘贴即成稿
章节 · 点击跳转视频
0:00
开场道歉：TNT 一章为何难读
2:33
回顾意义理论：同构与个体语义网络
6:56
太空唱片：外星人能听懂巴赫吗
12:15
模式即美：香农熵与齐普夫定律
17:39
算法信息：科学就是压缩规律
24:36
泼冷水：宇宙不是分形，要可证伪
28:47
TNT 与 ω-不完全：公理为何必须假设
33:05
0.999…=1：心智模型的失效
36:19
非欧几何与希尔伯特：解释的危险
39:52
六道 TNT 量词练习淘汰赛
55:58
两小时讲座给不了七年的理解
61:04
Context Free：随机文法长出树
79:58
随机与确定：混沌游戏与稳定性
88:39
向光性、自然选择与普适分形
95:00
收尾：宇宙是形式系统与课程去向
本期小问 · 档案清单
17:39
信息越多越接近真相吗？
88:39
整体能大于部分之和吗？
36:19
数学是发现，还是发明？
61:04
简单规则能生成生命的形态吗？
本期讲者
贾斯汀·柯里本课主讲，时为 MIT 本科生，2007 年夏季通过 MIT 高中生项目开设「哥德尔、埃舍尔、巴赫」讨论课，后成为应用拓扑学研究者（宾夕法尼亚大学博士）。
柯伦·凯莱赫本课共同讲者，时为 MIT 本科生，负责编程演示部分（元胞自动机、Context Free 生成艺术），后从事数据可视化开发。
学生参加 MIT 高中生暑期项目的学生，在课堂上参与哲学讨论与 TNT 逻辑练习。
01开场道歉：TNT 一章为何难读
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 hello welcome back today I think the fifth lecture um so I want to start off actually with some apologies um and it's going to be a bit of a sobering note um I want to quickly recap some of the things we talked about last lecture and um kind of bring forth some words of caution um but before I do that I want to also apologize um for this chapter um how many of you were kind of out of your chair excited reading this past chapter all right No in fact I see like the opposite of shooting your hand up uh um so I take it that most people didn't enjoy this last chapter okay yeah I mean neither did I um and really the only reason I I signed it as reading is that you guys could get kind of a fundamental
以下内容依据知识共享许可协议提供，您的支持将帮助MIT开放课程继续免费提供高质量的教育资源。如需捐赠或查看来自数百门MIT课程的更多资料，请访问MIT开放课程网站 ocw.mit.edu。好，大家好，欢迎回来，今天我想应该是第五讲了。嗯，我其实想先道个歉，嗯，接下来的内容可能会让人有点清醒，嗯，我想快速地回顾一下我们上节课讲的一些内容，然后提出一些需要注意的地方。不过在那之前我还想为这一章道个歉。你们当中有多少人读上一章的时候激动得差点从椅子上跳起来？好吧，不。实际上我看到的是跟举手完全相反的反应。所以我猜大部分人都不喜欢上一章，对吧。是啊，我也不喜欢。我把它布置成阅读材料的唯一原因其实是大家可以对像 MIU 这样的系统有一个基本的概念，比如我们讲过的 PQ 系统，一个完全形式化的排版系统
1:17
idea of like Miu like the PQ system we had completely formal typographical system for encoding statements right and we could play in a very mechanical way way with these statements of course unlike PQ and Miu PQ to an extent um TNT typographical number Theory um has the advantage of coding something which we think we know things about and that being numbers and the property of natural numbers 0 1 2 3 4 and so on and truths about them um so I wanted you mainly to become familiar with the notation uh and become familiar with the idea that you can start with kind of couple seeds of of axioms and just by applying these kind of recursive rules of of development you could produce kind of a whole web of of new strings which happen to when interpreted provide truths or things which we think we know about numbers um and that was really the main point of this chapter uh they going to be a a few other things I'm going to highlight um but we're going to once again do a mish mash of uh of topics like we seem to do every lecture and um
用来编码陈述，对吧，我们可以以非常机械的方式来摆弄这些陈述，当然跟 PQ 和 MIU 不一样——PQ 在某种程度上也算——TNT，也就是排版数论，它的优势在于它编码的是我们自认为有所了解的东西，也就是数，以及自然数 0、1、2、3、4 等等的性质，还有关于它们的真理所以我主要希望你们熟悉这套记号，并且熟悉这样一个想法：你可以从几条公理的种子出发，只要不断应用这些递归式的推演规则，你就能生成出一整张由新字符串构成的网络，而这些字符串在被解释之后，恰好给出了真理，或者说我们认为我们了解了数字，这其实就是这一章的主要内容，另外还有几个别的东西我想强调一下，不过我们还是会像每节课那样，把各种话题混在一起讲，而且
02回顾意义理论：同构与个体语义网络
2:33
I think we'll have something exciting to show you so first off uh recap of yester not yesterday but yesterday in terms of learning uh last week um where we were interested in kind of fundamentally a theory of [Applause] meaning um and this is already something I want to uh you know kind of offer my apologies for is in many ways we I think we provide a pretty convincing idea of what we think a theory of meaning is even though it's at it's not at all agreed upon in you know the academic linguistic pH philosophical Community what a theory of meaning actually is um what we kind of advertised is that um the meaning of kind of snow is white
我觉得我们会有些精彩的东西给大家看。首先，回顾一下昨天——不是昨天，是我们上周学到的内容。当时我们感兴趣的，从根本上说是一种意义理论[掌声] 这一点我已经想先跟大家道个歉，因为在很多方面，我觉得我们给出了一个相当有说服力的想法，关于我们认为意义理论是什么，尽管这在学术语言学界和哲学界根本没有达成共识——意义理论到底是什么其实并没有定论。我们所宣传的是，比如“雪是白的”这句话的意义
3:36
and what we take meaning to mean is that there's there's some sort of complex and in the words of Douglas Hof hofstad um an exotic isomorphism um so I could have like you know exotic or E for exotic um for the activity which goes on in your brain when you as you're visual perceptual devices hone in on the chalkboard and we see you know these These Broken bits of lim Stone against against this shell or whatever the chalkboards made out of um and as our our brain undergos kind of a complex set of edge detection algorithms we we hint this like light and dark here and here and here and here um and then as this comes into Focus um we we then recognize we have kind of global objects I don't know if you care about the quotations but you probably notice this group here and here and here um and then once you have these these broken down you then say Okay um I recognize these you then perform Edge detection on each of the letters and you say Okay s n o w spells snow snow then feeds to this whole kind of
而我们所理解的“意义”是指，存在某种复杂的、用侯世达（Douglas Hofstadter）的话来说一种奇异的同构（exotic isomorphism）。所以我可以用 exotic 或者 E 来代表“奇异的”，用来指你大脑中发生的活动当你的视觉感知装置聚焦到黑板上时，我们看到这些碎裂的石灰石粉末落在这块板上，或者不管黑板是什么材料做的。当我们的大脑进行一套复杂的边缘检测算法时，我们会察觉到这里、这里、这里、那里的明暗变化然后当这些逐渐聚焦成形，我们就能识别出整体的对象。我不知道你们在不在意那些引号，但你们大概注意到了这里、这里、还有这里的这一组，然后一旦你把这些拆解开来于是你说，好的，嗯，我认出这些了，然后你对每个字母做边缘检测，然后你说，好的，s、n、o、w拼成 snow，snow 接着又输入到这一整套又长又复杂的概念语义网络里，就是你第一次看到的那些东西
4:50
long and complicated conceptual semantic network of you know what I see the first time the memory of when you first saw snow falling um let's see uh you might just associate it with this the sensory feeling of cold um you might also associate snow with um going skiing or snowboarding um and all sorts of things um and you know just this contining to Branch out and it becoming a very individual thing and I think that's a very important point to notice is that um really what snow is white means to me is in some ways fundamentally different than what it means for you and that's a dangerous statement that means that uh kind of a theory of meaning lacks almost a formal mathematical single interpretation which we intend it to be like what we what philosophers and linguists always wanted is we wanted we want snow to point somehow to this exterior world the world which we live in and refer to the actual crystallized structure of water we we know as snow and that was you know the you know upstanding citizen meaning of snow and
第一次看到雪落下来时的记忆，嗯，我们看看啊，你可能就把它和寒冷这种感官感受联系起来嗯，你可能还会把雪和去滑雪或者滑单板联系起来嗯，还有各种各样的东西，嗯，你知道的，就这样不断地向外分叉，最后变成一件非常个人化的事我觉得这里有个非常重要的点要注意，就是嗯，「雪是白的」对我而言的意义，在某些层面上，和它对你而言的意义有根本的不同，而这是个很危险的说法，它意味着，呃，某种意义理论几乎缺乏一个形式化的、数学式的、单一的解释，而我们本来是希望它是那样的，就是我们，就是那些哲学家和语言学家一直想要的：我们希望 snow 这个词能以某种方式指向外部世界，指向我们生活其中的这个世界，并且指称我们所知道的那种水的结晶结构，也就是雪，那才是所谓正统的、体面的 snow 的意义，而且
6:11
there was nothing else none of this you know personal understanding of like well there was that one time where we got caught in a blizzard in you know Kansas or whatever like you know we wanted to get rid of all kind of in individual interpretations um so I advertised this isomorphism between what the perceptual input of snow as white does and how and what that lights up in your brain um but then we kind of carried forth to another idea and we brought out this IDE this very very very connected notion of
除此之外别无其他，没有那些个人化的理解，比如「哎，有那么一次我们在堪萨斯还是哪儿遇上了暴风雪」之类的，你懂的，我们想要摆脱所有这些个体化的解释，嗯，所以我之前强调过这种同构关系：「雪是白的」这个感知输入所做的事，和它在你大脑里点亮的东西之间的同构，嗯，但接着我们又推进到另一个想法，我们提出了这个非常非常非常有关联的概念——
03太空唱片：外星人能听懂巴赫吗
6:56
information so if you will cat your minds backwards um hoffstead talks about this record which we strap onto a you know a space shuttle or something and send it rocketing across the universe right and he kind of poses the problem of suppose some alien civilization stumbled Upon This Record and then said okay what is this and there was this kind of argument of of to what extent would the record mean anything when it was completely out of context of this kind of human cultural sociological entity which we support um here on Earth I mean what does does a rock a record mean to someone living on Alpha C Tor I don't I don't know um and one of the the the big arguments right away was that the structure of a record with kind of these very neat concentric grooves are actually spiraling um to be a little more appropriate and if somehow you know these aliens busted out their microscopes and and started you know analyzing little sections of of this strange you know artifact then they would then notice these kind
信息。所以如果你们把思绪往回倒一倒，嗯，侯世达讲过那张唱片，我们把它绑在一艘航天飞机之类的东西上，然后让它飞向宇宙深处，对吧，然后他提出了这样一个问题：假设某个外星文明偶然发现了这张唱片，然后说，好吧，这是什么东西？于是就有了这样一种争论：当这张唱片完全脱离了我们在地球上维系的这套人类文化和社会学的语境时，它还能有多少意义？我是说，一张唱片对住在半人马座阿尔法星上的人来说意味着什么？我不知道，嗯，而马上就出现的一个重要论点是，唱片的结构，那些非常整齐的同心圆凹槽——其实是螺旋形的，嗯，这么说更准确一点——如果这些外星人不知怎么掏出了显微镜，开始分析这个奇怪的人造物的一小段一小段，那他们就会注意到这些凹槽里的某种规律性，然后他们大概会问，
8:31
of regularities in sort of [Music] grooves and then they would kind of ask well that's odd um I wonder if that means anything um and then this idea of them somehow from the record be able to reverse engineer what a record player might be or look like and then actually play the music so even assuming if they had somehow figured out this incredible like like string of ideas um even if they were to play the music would the music mean anything to them would it just sound like a bunch of garbled mish mash Latif you have a comment noise so Latif argues that music will just be noise how many of you agree with Latif you you think latif's right does anyone have the coones to stand up to Latif that they can hear at all so max is saying we're assuming that they can hear at all okay so let's go ahead and Grant them that right and if we're going to you know kind of engage in this philosophical dialogue let's grant them the ability to hear and let's even assume that they can hear in a similar frequency range that we
嗯，这就怪了，我在想这是不是有什么意义？嗯，接着就有了这个设想：他们能不能从唱片本身反向推导出唱片机是什么样子、长什么样，然后真的把音乐放出来。那么就算假设他们不知怎么把这一整串不可思议的想法都想通了，嗯，就算他们真的放出了这段音乐，这音乐对他们有意义吗？还是听起来就只是一堆乱七八糟的噪音？拉提夫，你有话要说。噪音。所以拉提夫认为，音乐就只是噪音而已。你们当中有多少人同意拉提夫？你觉得拉提夫说得对吗？有没有人有胆量反驳拉提夫？——他们究竟能不能听见。所以马克斯是说，我们是在假设他们根本能听见。好，那我们就先给他们这个前提吧。既然我们要展开这种哲学对话，那就赋予他们听觉能力，我们甚至可以假设他们能听到和我们差不多的频率范围。比方说，在很多方面，好像存在某种普适的形态，
9:49
do let's say if in many ways like somehow there's this Universal form known as a human and for planets with similar gravitational poles and things like that and just the fact that carbon happens to be a really stable molecule that Evolution carried out in almost exact Sense on this other planet would the music still mean the same thing would it have meaning yes well um The Sounds themselves would still you know pretty much sound the same but if there was somebody singing that probably wouldn't make any sense to them it would be like you know listen to any foreign music right like in a language that you don't understand okay so let's say like it was a piece of Bach right there's no vocals just kind of you know triumphant sounds of you know Glory or whatever uh Felix they wouldn't know what piano is all right so they probably wouldn't know what piano is but do you think they would still have this kind of trigger of of beauty or even kind of awe or desire over over the music even if they didn't know it just from The
就叫「人类」，在那些引力、极点之类条件相似的行星上，再加上碳恰好是一种非常稳定的分子，于是演化在那个星球上以几乎完全相同的方式进行——那这音乐还会有同样的意义吗？它会有意义吗？——会吧，嗯，声音本身听起来基本上还是一样的，但如果有人在唱歌，那对他们来说大概就毫无意义了，就像你听任何一种外语歌一样对吧，用你听不懂的语言唱的。好，那我们假设放的是一首巴赫的曲子，对吧，没有人声，就是那种，你懂的，凯旋的、荣耀的声音之类的。呃，费利克斯——他们不知道钢琴是什么。好，所以他们大概不知道钢琴是什么，但你觉得他们还会不会有那种被触发的美感，甚至是敬畏感，或者对这音乐的渴望，哪怕他们并不了解它，仅仅从声音本身？——这些渴望是从哪儿来的，是什么触发的？好，那我应该
11:04
Sound where do those desires come from what Trigg those okay so let's I should be careful not desires but um just a sense of beauty do how how fundamental do you think a sense of beauty and organization is is in this music that's that's based on your somebody else just no somebody else says okay I can associate this with that this sounds more like this all this like uh uh makes me feel a certain way and that certain way makes me feel that it is beautiful so so even for individuals on this Earth what box music means is different to for everyone okay but it's still like there's still sensation there and I think that's one of the coolest things that you can um when when you listen to foreign music mhm you can in in many ways like decipher you know Despair and all these different feelings so maybe it really depends on if these people get the same certain Sensations in their mind from those different frequencies exactly but you know even putting aside the very complex semantic networks which
谨慎一点，不说渴望，就说一种美感。那你们觉得，美感和秩序感在这音乐里有多么根本？——那是基于你的……换一个人。不，换个人说。——好吧，我可以把这个和那个联系起来，这个听起来更像那个，所有这些，呃，让我产生某种感觉，而那种感觉让我觉得它是美的。——所以说，即便是地球上的不同个体，巴赫的音乐对每个人的意义也是不同的。好，但是它仍然，那里仍然有感受在，我觉得这是最酷的事情之一，就是当你听外语音乐的时候——嗯——你在很多情况下还是能读出，你懂的，绝望之类的各种情绪——所以也许这真的取决于这些人的脑子里，会不会从那些不同的频率中产生同样的某些感受。——正是如此。不过你知道，即便先撇开每个人内部都有的那套极其复杂的语义网络，
04模式即美：香农熵与齐普夫定律
12:15
everybody has internally when they trigger with this music um and you know assuming that these aliens don't have it I think in many ways you can make a good case that there's something about pattern which is fundamentally beautiful and music and that in many ways this is mathematically describable and that's kind of a careful note to think about right just the idea that pattern is itself Beauty and it's something which is universal right that pattern is detectable by anyone and that it is in some ways the only thing which there is to meaning the idea of pattern so last time we really connected pattern where I tried to connect to information now once again before we go running off and all sorts of tangents um I wanted to provide a word of caution right um because what what this means what information and in particular I I at least mentioned the idea of this kind of this entropy version of of information um and I not really going to explain this formula that well um sorry there's the Su Place minus I forgot the minus sign last
就是被这音乐触发的那些，嗯，而且假设这些外星人没有这套东西，我觉得在很多方面你还是可以很有力地论证：模式（pattern）本身有某种东西是根本上美的，音乐也是如此，而且这在很多方面是可以用数学来描述的。这是个需要小心琢磨的点，对吧，就是「模式本身即美」这个想法，而且它是某种普适的东西，对吧，模式是任何人都能察觉到的，而且在某种意义上，它是意义所依赖的唯一东西——模式这个概念。所以上次我们真正把模式，我试图把它和信息联系起来。那么现在，在我们又开始跑题跑到各个方向之前，嗯，我想先给一句提醒，对吧，嗯，因为这意味着什么——信息意味着什么，特别是我至少提到过这种熵版本的信息，嗯，而我其实并不打算把这个公式解释得多好，嗯，抱歉，这里是求和，负号——我上次忘了写负号——log P(x)，但这不过就是
13:39
time log PX but this is just this idea of like the probability assigned to something um and in some ways the less probable something is the less the more entropy there is to it or the other way around but either way like the the main thing I want to point out is that there are these mathematical ideas of of how to measure information in an entropy form and what this fundamentally boils down to is if you were to describe either this record or as we talked about previously a picture or a piece of music and just feed it kind of as a string of zeros and ones um we could actually assign certain values of of entropy and view this as a as a measure of information um and it's it's interesting that there are certain patterns that all languages and certain symbols share um and some of these are closely related to Notions of of entropy um and you can actually produce like an information entropic measure of the letters in the alphabet and that's just just based on how commonly they occur in language but
赋予某件事的概率这个想法，嗯，在某种意义上，某件事越不可能，它的熵就越大，或者反过来说，总之，我想指出的主要一点是，确实存在这些数学化的思路，用来以熵的形式度量信息。这归根结底就是：如果你把这张唱片，或者像我们之前谈到的一幅画、一段音乐，就当作一串 0 和 1 输进去，嗯，我们其实可以给它赋予某种熵值，并把这个当作信息的度量，嗯，而且很有意思的是，所有语言和某些符号都共有某些模式，嗯，其中一些和熵的概念密切相关，嗯，你其实可以做出一种字母表中各字母的信息熵度量，这仅仅基于它们在语言中出现的频率有多高，但是——对，抱歉，你说，李，所有这些……
15:02
there's yes sorry go ahead Li all these that's only sort of a thing the human mind can handle so it creates all these lunges that have the same structure okay so the chief argues that these patterns and languages emerge because that's all the human mind can handle but I would argue no because they've actually done the same analysis on DNA and DNA as far as a language presents the same patterns so unless you think the human mind created DNA um then uh I would argue no so and and once again so then I I talked about kind of also a related phenomenon and these are things for you to Google and and pursue in your own time is Zip's flaw um and it's this idea that if you were to take any language and or and you would just rank the symbols based on uh frequency of occurrence um there's actually a very nice power law behavior from the most the the first the second most common letter occurs exactly half the time as the first and the third most common letter happens exactly a third of the time in the first and there's this
——这只是人脑能处理的东西，所以人脑造出了所有这些结构相同的语言。——好，所以他认为这些语言中的模式之所以出现，是因为人脑只能处理这些，但我要说不是，因为人们其实对 DNA 做过同样的分析，而 DNA 作为一种语言呈现出相同的模式，所以除非你认为是人脑创造了 DNA，嗯，否则，呃，我会说不是。所以，再一次，接着我还谈到了一个相关的现象，这些都是你们可以自己去谷歌、自己去深入了解的东西——齐普夫定律，嗯，这个想法是说，如果你拿任何一种语言，然后仅仅按照出现频率给这些符号排序，嗯，实际上会出现一个非常漂亮的幂律行为：从最常见的开始，第二常见的字母出现的次数恰好是第一名的一半，第三常见的字母恰好是第一名的三分之一，语言里就会涌现出这样一种幂律
16:17
power law Behavior which comes out of languages if you just sort them by Common uh by frequently appearing symbols um and this is kind of an interesting argument zip was a I believe a linguist at Harvard in the' 40s um or 50s I'm not sure um and he said and he noticed this behavior and it's interesting because then you can take something and ask like is this a fake is this an actual language I have no idea what it means but I just notice these certain like reappearing symbols like that and I also notice you know that and then start ranking them and there's this idea idea that you can actually discern intelligible languages based completely on uh the the frequency of occurrence of these symbols but all this this are kind of has has a rigorous footing we talked about then a related notion um because fundamentally I said this picture doesn't capture what we mean by information um because if you take a picture where's my
行为，只要你按符号出现的频率把它们排序。嗯，这是个挺有意思的论点。齐普夫我记得是哈佛的一位语言学家，在四十年代，嗯，或者五十年代，我不太确定，嗯，他注意到了这个现象。有意思的地方在于，你可以拿某个东西来问：这是假的吗？这是一门真正的语言吗？我完全不知道它是什么意思，但我就是注意到有这些反复出现的符号，我还注意到，然后开始给它们排序，于是就有了这样一个想法：你其实完全可以根据这些符号出现的频率，来判别一门语言是否可解读。不过所有这些都算是有严格基础的。然后我们谈到了一个相关的概念，嗯，因为从根本上说，我说过这幅图并没有抓住我们所说的信息，嗯，因为如果你拿一幅画——我的板擦呢——
05算法信息：科学就是压缩规律
17:39
Eraser if you take a picture and you kind of have just a bunch of seething dog barf um and you convert it into these zeros and ones um and then then do this kind of information entropy analysis on it it might not differ all that much from you take a picture which has something like the serinsky gasket drawn on it which has obvious regularity obvious pattern and could be argued obvious meaning and so on at infinitum um so this measure this kind of Shannon in iny information these are it's all just kind of a field of information Theory which if you guys want to you can come to MIT and audit a class in if you want to um this kind of analysis on on this conversion to zeros and ones of of this versus you know the seething Dog Bar would be not that all distinguishable um so then we presented this other idea and we and we really got to play around with this is that well this obviously has meaning because I can write a short little program you know with some some syntax and Etc blah and then run it and it it's very short and
板擦，如果你拿一幅画，上面就是一堆乱七八糟的狗吐出来的东西，嗯，你把它转换成这些 0 和 1，嗯，然后再对它做这种信息熵分析，结果可能和你拿一幅画着谢尔宾斯基三角形的图差不了多少——那幅图有明显的规律性、明显的模式，甚至可以说有明显的意义，如此以至无穷。嗯，所以这个度量，这种香农信息之类的东西，这些都只是信息论这个领域的内容，如果你们想学，可以来 MIT 旁听一门课。嗯，把这个和那堆狗吐出来的东西都转换成 0 和 1 之后再做这种分析，二者其实并不怎么能区分开。嗯，所以接着我们提出了另一个想法，而且我们真的把它玩了个透：那就是，这个显然是有意义的，因为我可以写一小段短短的程序，你知道的，用点语法什么的，然后跑一下，它非常短，
19:19
it presents this entire picture like this this sequence of zeros and ones that go on for you know a million billion whatever digits um that's needed to encode the visual picture of of a serinsky gasket is actually an excessively long description it's kind of like taking a pendulum and then someone asking you to describe it and you going well it rocks there and back and there and back and there and back and you just keep saying that and then forever however long period of a time you want to describe the pendulum you just keep saying that you you record your voice saying that it's a very inefficient way to encode the regularities of the phenomenon there's the idea that this can actually be expressed mathematically using just kind of a idea of iteration of of symbols um and we had this we had this Linden Meer system I have no idea if I'm spelling this correctly l i Linden not but there has to be a Meer Linden oh so there's an NM okay NM I never won the spelling be um and was like f minus r minus or Plus+
却能呈现出整幅图。而那串一直延伸下去、有上百万上十亿位数字的 0 和 1 序列，嗯，用来编码谢尔宾斯基三角形的视觉图像，其实是一种过分冗长的描述。这有点像你拿一个单摆，然后有人让你描述它，你就说，嗯，它摆过去又摆回来，摆过去又摆回来，摆过去又摆回来，你就一直这么说下去，永远说下去，你想描述这个单摆多长时间，你就一直这么说，你把自己说这些话的声音录下来——这是一种非常低效的方式来编码这个现象的规律性。而另一个想法是，这其实可以用数学表达出来，只要用一种符号迭代的思路。嗯，我们当时有这个，我们有这个林登迈耶系统——我完全不知道我拼得对不对，L-i，Linden，不对，后面得有个 Meer，Linden，噢，中间是 nm，好吧，nm，我从来没赢过拼字比赛。嗯，就是像 F 减 R 减，或者加加 R 加 F 这样，我们可以把这些编码进去，然后
20:45
Rus F and we could encode these and then we had this very nice program for expanding this out in a recursive Manner and it just took a little few lines of code um which takes much less information to encode the lines of code um than an actual picture of what's going on so this is this leads to a very important idea of kind of algorithmic
我们有一个很不错的程序，能把它以递归的方式展开，而这只需要短短几行代码，嗯，编码这几行代码所需的信息，远远少于实际画面本身所需的信息。所以这就引出了一个非常重要的想法，也就是算法
21:23
information and we then really really started playing around with the idea idea of well all science ever does is reverse engineering we take phenomenon like the supinsky gasket or the regularities of a pendulum swinging back and forth respect to some angle Theta and we describe
信息。然后我们真的开始琢磨这样一个想法：科学所做的一切无非就是逆向工程。我们拿一个现象，比如谢尔宾斯基三角形，或者单摆绕某个角度 θ来回摆动的规律性，然后我们把它
21:56
it very simply so we make our algorithmic content for the pendulum as low as possible by using these symbols to encode the regularities of the Dynamics here and I really apologize to those of you who haven't had calculus but these double dots indicate a rate of change of the rate of change of of the angle Theta um so we then started playing around with looking at cellular automaton which is essentially this grid and we and we divided up and the value of this grid whether it's black or white is somehow governed by the neighbors and by just playing around with those rules we suddenly stumbled upon an entire you know wealth of behavior by changing the number of colors it could have and how quickly it changed we could actually emulate a wave equation we could make splashes in a puddle using this kind of very simple finite and deterministic rule we modeled voting patterns by just saying well I'm going to make my color somehow dependent on what my neighbors are doing what my neighbors are thinking
描述得非常简单。所以我们让单摆的算法内容尽可能低，办法就是用这些符号来编码这里动力学的规律性。我真的要向没学过微积分的同学们道歉，不过这两个点表示的是角度 θ 的变化率的变化率。嗯，接着我们开始玩元胞自动机，它本质上就是这样一个网格，我们把它划分开，而这个网格的取值，是黑还是白，某种程度上由它的邻居决定。而仅仅通过摆弄这些规则，我们突然就撞见了一整片丰富的行为。通过改变它可以有的颜色数量、以及它变化的快慢，我们其实可以模拟出波动方程，我们可以用这种非常简单、有限而且确定性的规则，做出水坑里的水花。我们还建模了投票模式，就是说，我要让我的颜色以某种方式取决于我的邻居在做什么、我的邻居在想什么——这某种程度上揭示了群体思维的危险：如果你总是让自己被
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kind of indicating the dangers of group think if you always let yourself be effective by what other people are thinking we just become kind of a homogeneous culture um so somehow just by these very very simple rules which which kerran gracefully coded up in just a few lines we were suddenly describing huge Realms of complex behavior in the social physical and you know cultural worlds and that kind of presented some interesting ideas we started asking like well you know what about about this and that and I remember one line we ended up stumbling upon like was well because we see this self-similarity of of of behavior our our same voting rule which describes how humans act in groups um also describes gas droplet condensation if you have a bunch of spraying water on your on your shower curtain the reason why it droplets up is because you know each molecule of water is seeking to reduce its its overall energy and it can do that by grouping up with people and suddenly you got this clustering this
别人的想法影响，我们就会变成一种同质化的文化。嗯，所以仅仅靠这些非常非常简单的规则——凯伦很漂亮地用几行代码就写出来了——我们竟然一下子就能描述社会的、物理的、还有文化世界中大片大片的复杂行为。这带出了一些有意思的想法，我们开始问，嗯，那这个呢、那个呢？我记得我们后来撞见的一条思路是：因为我们看到了这种行为上的自相似性，我们那条描述人类群体行为的投票规则，嗯，同样也描述了气体液滴的凝结。如果你往浴帘上喷一堆水，它之所以会结成水珠，是因为每个水分子都在设法降低它的总体能量，而它做到这一点的方式就是和别人抱团。于是突然之间你就得到了这种聚集、这种抱团的性质，和我们在水珠里看到的
06泼冷水：宇宙不是分形，要可证伪
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cooning property which we saw in water droplets and then we start going oh like does that mean that you know the universe is fractal and that there's this Con Concepts which apply on all levels and you know we I I've kind of felt ourselves getting caught up in the moment and maybe leading you guys down paths which um although exciting aren't exactly well founded um it's not like everyone in the scientific Community if you went to them and said you know is the universe a giant fractal uh a lot of people would probably say no obviously not um and there would be some various reasons for this um because if you're talking about self-similarity along scales uh this view of well we you know we have the solar system the Sun and you know these tiny planets going around Etc and like aha but it's the same model that the atom is wrong WR one of the fundamental things in physics nowadays is patching up the disagreement in mathematical description and behavior of things at the atomic realm which is not
一样。然后我们就开始想，噢，那这是不是意味着，宇宙是分形的，存在着一些在各个尺度上都适用的概念层面，你知道，我们——我觉得我们有点被当下的气氛带跑了，可能把你们引向了一些路径，这些路径虽然让人兴奋，但其实并没有扎实的根据。并不是说科学界的每个人，如果你去问他们说，你觉得宇宙是一个巨大的分形吗？很多人可能会说，不，显然不是。而且这背后有各种各样的原因。因为如果你说的是跨尺度的自相似性，那种观点，就是说我们有太阳系、有太阳，还有这些绕着转的小行星等等，然后说，啊哈，这跟原子是同一个模型——这个说法是错的。如今物理学最根本的课题之一，就是弥合数学描述与实际行为之间的分歧，尤其是在原子层面上的东西，那跟这个完全不一样。事实上，如果你们学过化学，翻过
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at all like this it's in fact like if you guys have done chemistry and looked through the books you see these weird drawings of probability clouds which are governed by fundamentally quantum mechanics um and it doesn't at all match the classical mechanical description of the solar system um so in that sense no at all the universe is obviously not a fractal because the rules don't apply on all the same levels um and then I did this very exciting you know you know demonstration at least I kind of thought it was exciting at the time where I took a piece of paper and I and I crumpled it up and then I unfolded it and I said well the kind of topology and and morphology of the of the piece of paper is almost identical to the kind of thing we see in mountains and valleys and rivers um and this kind of pattern formation along scale the fact that what I do here on a very local level um is what we see on very large level I suggested this idea of of the universe conceptually being a fractal but what
那些书，你会看到那些奇怪的概率云图，而它们从根本上是由量子力学支配的。这跟太阳系的经典力学描述完全对不上。所以从这个意义上说，不，完全不是——宇宙显然不是一个分形，因为规则并不在所有层面上都成立。然后我又做了那个非常令人兴奋的演示，至少我当时觉得挺兴奋的，我拿了一张纸，把它揉皱，然后再展开，我说，你看这张纸的拓扑结构和形态，几乎跟我们在山脉、山谷和河流中看到的东西一模一样。这种跨尺度的图案形成，也就是说我在非常局部的层面上做的事，跟我们在非常大的尺度上看到的是一样的。我据此提出了宇宙在概念上是分形的这个想法。但
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you know a mathematician or physicist might actually tell you is that uh my equations for describing the phenomenon are scale free and there there's this idea of being scale free and we do this all the time in fluid dynamics you know I can drag my finger through the water and see little vort vortices form behind my fingers and then we've actually got exact pictures of of mountains which pierce the clouds and and the clouds are flowing around this mountain tip and you get these Von Caren sheets where these vortices split
数学家或物理学家真正会告诉你的是，我用来描述这个现象的方程是无标度的（scale free）。有这么一个“无标度”的概念，我们在流体力学里一直这么干。你知道，我可以把手指划过水面，看到手指后面形成一个个小涡旋，然后我们其实还有一些精确的照片，是山峰刺穿云层，云流过这个山尖，你就会得到这些冯·卡门涡街，涡旋在那里脱落、
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off I think it's Von Carmen but there's so many beautiful pictures you can see in fluid dynamics and things like that so this being Mountain um or I could equally erase this and say finger right and I and really what I should be trying to do here is kind of is advocate the beauty and our and elegance and our formalism with mathematical equations and how when we can make these scale free and describe phenomenon all at all sorts of levels so I just wanted to be cautious um and not say things which will later get me in trouble um but aside from that I encourage you all to get carried away and explore new ideas and then the reason why karna and I fundamentally believe in Computer Science and Mathematics as a framework for thinking is that you can be as lofty and philosophical and out there as possible but at the end of the day you have to either be able to make it work rigorously with proof or by execution on a computer which for most intensive purposes is just as good as proof um because it either works or it doesn't
分离。我想是冯·卡门吧。流体力学之类的领域里有太多漂亮的图片。所以这里画的是山，或者我完全可以把它擦掉，写上“手指”，对吧。而我在这里真正该做的，其实是去倡导数学方程这种形式体系的美与优雅，以及当我们能让它们变成无标度的、能描述各种层面上的现象时有多了不起。所以我只是想谨慎一点，不说那些以后会让我惹上麻烦的话。但除此之外，我还是鼓励你们尽情投入、探索新想法。而卡尔纳和我之所以从根本上把计算机科学和数学当作思考的框架，原因就在于：你可以尽情地高谈阔论、充满哲学意味、天马行空，但归根到底，你要么得能用严格的证明让它成立，要么得能在计算机上运行出来。对绝大多数实用目的而言，后者跟证明一样管用，因为它要么行得通，要么行不通，对吧。你思考宇宙也一样，要么成立要么不成立。从根本上说，你需要
07TNT 与 ω-不完全：公理为何必须假设
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right and you're thinking about the universe either works or it doesn't right and fundamentally you need falsifiability so there's a chance that you're thinking might not work right um and it's fun to be metaphysical and thought and talk about you know things higher than the universe in space and time and what you know the structure of these things are but at the end of the day we want to be able to test our our observations or prove them mathematically or using computers so that's kind of my my sobering note of caution um and I'm sorry for those of you had to go through it and didn't want to hear it um now on to slightly more fun things first let me check the time so number Theory um I really want to just draw this draw your attention to the to the idea of of what we can take as to be an axiom and and why we just don't like we can't prove these things and if you you'll cast your eyes onto uh page 22 one uh you saw kind of this pyramid
可证伪性——也就是说，你的想法有可能是错的。玩形而上学是很好玩，去思考和谈论那些超越时空中宇宙的东西，谈论这些东西的结构是什么样的。但归根到底，我们希望能够检验我们的观察，或者用数学、用计算机去证明它们。所以这就是我要泼的一点冷水。对于那些不得不听这段、又不想听的人，我说声抱歉。现在说点稍微好玩一些的。先让我看一下时间。那么数论——我特别想把你们的注意力引到这个问题上：什么东西我们可以当作公理，以及为什么我们没法证明这些东西。请把目光投向第22页左右，你们看到过那样一个金字塔，
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of right and you know we had 0 plus SS s0 equals SS s0 and so on um and well first of all I want to highlight some things some elegance and this is really what makes magicians the most anal creatures on the planet um is that you know we don't have numbers here we just have one number and that number is zero um and then every other number is just the successor of zero or the successor of the successor of the successor of zero or the successor times what we like to say in our meta language you know 1729 Z S's and then a zero um and that representing the number 1,729 um but there's some also Elegance in this the fact that you only need one concept A Z well two concepts you need a zero and you need the concept of a successor and suddenly you get the whole thing for free right which is which is pretty but what was highlighted by this example was that we don't have in our if we didn't assume it we wouldn't have we couldn't prove this this statement which says that for every a where a is some variable 0 plus
对吧，我们有 0 加 SS0 等于 SS0 等等。首先我想强调几点优雅之处——这也正是让数学家成为地球上最龟毛的生物的原因：我们这里其实没有“数”，我们只有一个数，那就是零。然后其他每一个数都只是零的后继，或者是零的后继的后继的后继，或者用我们的元语言来说，就是重复若干次的后继，比如 1729 个 S 然后跟一个 0，用它来表示数字 1729。但这里面还有一种优雅之处，就是你只需要一个概念——零——好吧，是两个概念：你需要一个零，还需要一个后继的概念，然后突然之间整个体系就白送给你了，对吧，这挺漂亮的。但这个例子凸显出来的问题是，我们没有——如果我们不把它当作假设，我们就没有、就无法证明这条命题，它说的是：对于每一个 a（a 是某个变量），0 加 a 等于 a。多么迟钝而显然的一件事，对吧。但我们手头能用的只有
31:39
a is a what a dll and obvious thing right but all we have at our disposal are are kind of these things and in fact we get this whole Mountain this pyramid of of of true statements um and we want to LEAP to this generality that well this is obviously the case but fundamentally our our our desire to LEAP to this conclusion comes from our our understanding our mental models of how numbers and how specifically integers behave um and that's an important point this this idea of really all we ever have at our disposal are are Al models and by making them rigorous and trying to make them rigorous through formal systems we really kind of see whether or not our definitions Encompass as much as we want so we could never actually prove this statement with the way that TNT was set up without this axium so we eventually had to assume it um and and hoffstead calls this uh just to write it here Omega inmp complete where Omega is kind of the thing we use to refer to all the integers at once um and and it's just
这些东西。事实上我们能得到整座山、整个金字塔的真命题。而我们想要跃升到这个一般性结论：这显然是成立的嘛。但从根本上说，我们想要作出这个跃升的欲望，来自于我们的理解、我们关于数、尤其是整数如何行为的心智模型。这是很重要的一点：我们手头真正拥有的，永远只是心智模型。而通过让它们变得严格，通过形式系统去把它们严格化，我们才真正看清我们的定义是否涵盖了我们想要的那么多东西。所以按照 TNT 当初的设定方式，如果没有这条公理，我们根本无法证明这个命题。于是我们最终不得不把它当作假设。霍夫施塔特把这称为——我写在这儿——ω不完全（Omega incomplete），其中 ω 是我们用来一次性指代所有整数的那个东西。这个想法就是：尽管我们有这一整个无穷
080.999…=1：心智模型的失效
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that idea that even though we have this infinite stack of true things we don't have the thing which describes them all as a truth namely this um but you know this notion of mental model I think is really important um because I remember first seeing this proof in in high school and I remember just being kind of utterly shocked and in some ways horrified by it um so how many people believe the following 999 repeating is equal to one dur believes it naine believes it can anyone come up and Prove It Felix do you believe it can you prove it think about it first think about it first deep down um does anyone want to show off and leap to the chalkboard like a young GA and just go ahead and show this to me D is that something multiplying by 10 okay so I'm going to go ahead and take your suggestion and kind of lead you down the proof so we're going to call this thing X um we'll temporarily forget that we're it's one we're going to call X this 0.9 repeating um so dur re recommends we we take we consider the quantity 10x okay
的真命题堆栈，我们却没有那个把它们全部作为一条真理来描述的东西，也就是这个。不过，心智模型这个概念我觉得真的很重要，因为我记得我第一次在高中看到这个证明时，整个人有点彻底震惊，某种程度上还挺惊恐的。所以，有多少人相信下面这个：0.999无限循环等于 1？杜尔相信，娜婷相信。有谁能上来证明一下吗？菲利克斯，你相信吗？你能证明吗？先想一想，先好好想想，从心底里想。有谁想露一手，像年轻的伽罗瓦一样冲到黑板前，直接给我演示一下？D，是不是要乘以 10？好，那我就按你的建议来，带着大家走一遍这个证明。我们把这个东西叫做 X，我们暂时忘掉它等于 1 这件事，我们就把这个 0.9 无限循环叫做 X。杜尔建议我们考虑 10X 这个量，好，
34:35
which think it's 9.9 repeating okay a bunch of hands if you stop it at Infinity one because it's approaching number okay well without to the idea of a limit we can try to prove it more fundamentally uh I don't really know Felix do you know okay so Felix says we should subtract X okay so suddenly we get 9.0 oh sorry I mean X itself is 09 repeating dot dot dot so then when we perform the subtraction we get nine so we have now this new truth that 9x = 9 the only number which satisfies that is x = 1 and I actually will never forget the girl sitting next to me in pre-calculus when when my teacher first did this and she goes no it can't be but it's not one look cuz if it were one I would have written written one right but somehow I wrote down a completely different number which was the same thing and this kind of shows the the the really important thing which mental models have to tell us we can have what appears to be a correct understanding of an object namely real numbers um but then we're continually
那它是多少？是 9.9 无限循环，对。举了不少手。如果你在无穷处截断……是 1，因为它趋近于那个数。好，那我们先不用极限的概念，试着更根本地证明它。呃，我不太确定——菲利克斯，你知道吗？好，菲利克斯说我们应该减去 X。好，那突然我们得到 9.0——哦抱歉，我是说 X 本身是 0.9 无限循环，点点点。所以做完减法之后我们得到 9。这样我们就得到了一个新的真理：9X = 9，而唯一满足它的数就是 X = 1。我永远不会忘记当年我在微积分预备课上坐在我旁边的那个女生，老师第一次讲这个的时候，她说：不，这不可能，可它明明不是 1 啊，你看，如果它是 1，我就直接写 1 了，对吧。可不知怎么的，我写下的是一个完全不同的数字，而它却是同一个东西。这恰恰体现了心智模型能告诉我们的那件非常重要的事：我们可能自以为对某个对象——比如实数——有正确的理解，但当它们对我们做出这种事时，我们又会不断地被惊到，
09非欧几何与希尔伯特：解释的危险
36:19
surprised when they do things like this to us even at the most basic levels um so that's just kind these are just kind of meta statements about what mathematics does and what we do in mathematics um and in particular what we're doing in TNT is we we have this idea that number Theory should Encompass or at least TNT should Encompass all of our thoughts of number Theory and this goes back to what uid said about his axioms of geometry he said you know we we know these things should be fundamentally true or at least we believe they are to be they are true and that they Encompass all of our all of our knowledge of of geometry and that everything they produce as a result of those assumptions should be true and certain um but then we got the whole case with you know sacari and and you know gaus you know in hiding because of course gaus would discover it 50 years before everyone else and then show his notebooks goes oh sorry yeah I I did that theorem in between uh you know my first cup of coffee and my second cup of
哪怕是在最基础的层面上。所以这些只是关于数学在做什么、我们在数学中做什么的一些元层面的陈述。具体到 TNT，我们所做的是：我们抱有这样一个想法，即数论应该涵盖——或者至少 TNT 应该涵盖——我们关于数论的全部想法。这又要回到欧几里得关于他的几何公理所说的话。他说，我们知道这些东西从根本上应该是真的，或者至少我们相信它们是真的，而且它们涵盖了我们关于几何的全部知识，并且从这些假设推出的一切都应该是真实而确定的。但后来就出现了萨凯里那一整摊事，还有高斯——他藏着掖着，因为高斯当然总是比所有人早 50 年发现，然后把他的笔记本一亮，说：哦抱歉，是的，那个定理我在第一杯咖啡和第二杯咖啡
37:25
coffee um but um well good I'm glad you discovered it too um and just this concept of non- ucan geometry where where you break that fifth postulate that that if you have two par you have a a parallel line or a line and a point not on it that there exists unique line unique exclamation point um such that it does not intersect but if you're working on the surface of a sphere for example and you define your lines to be great circles and then in fact any possible line you draw or line is great circle um you actually have two intersection points something like that um although that's a terrible picture sorry um so you know Hilbert this very guy who who's trying to Advocate all this stuff involving number Theory uh said well we just have these fundamental axiomatic Notions of a point in line and the most we can get from them are their logical relationships with each other and the second we try to interpret these things we then get ourselves into trouble when we interpret the statement line and we we mean the straight thing
之间就做出来了。不过嘛，很好，我很高兴你也发现了。这就是非欧几何的概念——你打破了那条第五公设：如果你有两条平行线，或者说有一条直线和一个线外的点，那么存在唯一一条直线（唯一，感叹号），使得它与原直线不相交。但如果你是在球面上工作，并且把直线定义为大圆，那么事实上你画的任何一条直线都是大圆，你实际上会得到两个交点，大概是这样。虽然我这图画得很糟，抱歉。所以，希尔伯特，就是这位一直在倡导所有这些数论相关工作的人，他说：我们拥有的只是关于点和线的这些基本公理性概念，我们从中最多能得到的就是它们彼此之间的逻辑关系。而一旦我们试图去解释这些东西，我们就会陷入麻烦。当我们解释“线”这个陈述，并且我们心里想的是
38:39
on this flat board um and not the curvy thing on the surface of the Earth um we get ourselves into trouble because we're providing an interpretation and not sticking directly to the formalism shows you also one of the kind of dangers of getting away with your interpretations of what your formalities tell you um and I I'm kind of proud of hoffstead for for doing that and cautioning us against uh interpretation because it gets you in trouble I want to um and then you know finally first I'm going to field some questions about this chapter before uh I kind of introduce what Curran's going to do and um and then move on to newer and more exciting things TNT anything it I mean I remember sitting in an undergraduate seminar on girle erbach and someone going like what the was he talking about with these Supernatural numbers um and you know Omega inconsistency um because I mean these aren't trivial Concepts right I mean entire fields of mathematics have been devoted to them um so if you guys are feeling completely I understood this um
这块平板上那个笔直的东西，而不是地球表面上那个弯曲的东西时，我们就会给自己惹麻烦，因为我们是在提供一种解释，而没有严格地守住形式体系。这也向你展示了一种危险：随意地按照你对形式结果的解释去发挥，会让你惹上麻烦。而我还挺佩服霍夫施塔特这么做，提醒我们警惕解释，因为它会让你陷入麻烦。我想——然后，最后，我先来接几个关于这一章的问题，之后我再介绍柯伦要讲的内容，然后我们再进入更新、更有意思的东西。TNT 有什么问题吗？我是说，我记得我当年坐在一个关于《哥德尔、埃舍尔、巴赫》的本科研讨课上，有人说，他讲的那些超自然数到底是什么鬼？还有 ω 不一致性。因为我是说，这些都不是什么小儿科的概念，对吧。整个整个的数学分支都是专门研究这些东西的。所以如果你们觉得完全——“我睡着都懂了”——好，桑德拉。我有点困惑，就是 TNT 到底有多少条规则，比如那些子
10六道 TNT 量词练习淘汰赛
39:52
in my sleep yes Sandra I was confused like how many rules TNT have talk about like sub categories um so incompass TNT right uh so he also talked propositional calculus and how he assumed that into TNT oh yeah and I'm I'm sorry but I didn't assign that earliest reading so that was kind of out of context all right um but did any of you guys try like the little exercises in in the book itself um there's one section I want to highlight um and I give you know browning points to everybody who gives me a right answer um because I think I have my own view of answers but I'm not sure if they're right this will also test your guys knowledge of the notation um so we have Tilda upside down a c colon backwards e B colon parentheses SSO um dot b um parallel lines C okay um first of all can someone tell me what this means by giving it an interpretation and then tell me whether or not it's true or false Latif there is no C for all of C it does not that there's a well here before we do the Tilda let's do this all right say
类别。TNT 涵盖了什么，对吧。他还讲了命题演算，以及他是怎么把它假定进 TNT 里的。哦对，抱歉，我没把更早的那部分阅读材料布置下去，所以那段有点脱离上下文了。好的。不过你们有没有人试过书里的那些小练习？有一节我想重点讲一下，谁给出正确答案我就给谁加分，因为我觉得我自己有一套答案，但我也不确定它们对不对。这同时也能考一考你们对记号的掌握。那么我们有：波浪号、倒 A、c、冒号、反 E、b、冒号、括号、SS0、乘、b、等号、c。好。首先，谁能通过给它一个解释告诉我这是什么意思，然后再告诉我它是真还是假？拉蒂夫。不存在 c……对所有 c……它不……那是说有一个……好，在处理波浪号之前我们先看这部分。再说
41:33
again for all c for all C there exists no B there exists a or no B oh there exist a b there exists a b such that the successor of the sucess of Z * B is C okay so this without the Tilda is it true or [Music] false yes because if the TAA it's truth and why because two * anything is not going to the effect so fundamentally what this statement is saying is that for all numbers natural numbers um that number is even because it's divisible by two or it's a multiple of two um which is clearly not the case cuz um and using our rule of substitution or sorry specialization um we could have picked well let's say three or I'm sorry three doesn't exist in our system it's SSS o or zero um and through our interpretation we can see that well there is no number B there does not exist a b such that two times that number is three namly because three is on right okay so let's try a little harder one real quick so and then with the Tilda which means not our false statement is now made true all right so let's consider so
一遍：对所有 c，对所有 c，不存在 b……存在一个……或者不存在 b……哦，存在一个 b，存在一个 b，使得零的后继的后继乘以 b 等于 c。好。那么去掉波浪号，这句是真还是假？【音乐】是的，因为如果有波浪号它就是真的。为什么？因为 2 乘以任何数都达不到那个效果。所以从根本上说，这句话是在说：对所有自然数，那个数都是偶数，因为它能被 2 整除，或者说它是 2 的倍数。而这显然不成立，因为——呃，用我们的代入规则——不好意思，是特化规则——我们本可以取，比如说 3，哦不好意思，3在我们的系统里不是这么写的，是 SSS0，或者零。通过我们的解释我们可以看到：不存在这样的数 b，不存在一个 b 使得 2 乘以那个数等于 3，因为 3 是奇数，对吧。好，那我们来试一个稍微难一点的，很快。那么加上波浪号，也就是“非”，我们那个假命题现在就变成真的了。好，那我们来考虑，
43:14
this was round one round two for every C that not b col colon s s equals see all right anyone who is not Latif can answer I'm going to successfully knock you guys out so the last man standing actually gets the hardest question so first anybody willing to field an interpretation who's not Latif Max um for doesn't exist such that go and call it two * okay so for all C there does not exist a b such that 2 * B is C willing to you have a 50-50 shot on this oh that's f yeah there you go exactly it's false because uh we specify let's let C be two and clearly when there exists a b not not exists there exists to be namely one that 2 * 1 is one so we now have two people eliminated three all right so now let's try this uh for every C such that there exists a b damn it um not Tilda I mean s s o dob equals c d for every C there exists a b that is not 2 * b c okay and is that true or [Music] false so say again we have a there's not a c for every C for every C there exists a b there exist a b that is not such
这算第一轮，第二轮：对每一个 c，非 b 冒号，SS0 乘 b 等于 c。好，除了拉蒂夫之外，谁来回答？我要把你们一个个淘汰掉，所以坚持到最后的那个人会拿到最难的题。首先，有谁愿意给个解释？不是拉蒂夫的。马克斯。呃，对所有……不存在……使得……我们把它叫做 2 乘。好，那就是：对所有c，不存在 b 使得 2 乘 b 等于 c。愿意判断吗？你有一半的机会。哦，那是假的。对，就是这样，完全正确，它是假的。因为我们可以指定，比如让 c 等于 2，那显然存在一个 b——不是不存在，是存在一个 b，也就是 1，使得 2 乘 1 等于 2。所以现在我们淘汰了两个人，第三个。好，那我们来试试这个：对每一个 c，存在一个 b……糟糕，是波浪号——我是说，非（SS0 乘 b 等于 c）。对每一个 c，存在一个 b 使得 2 乘 b 不等于 c。好，这是真还是假？【音乐】再说一遍。我们有一个……不存在一个 c……对每一个 c，对每一个 c，存在一个 b，存在一个 b 使得不是
45:33
that this isn't such that 2 * Bal right so actually so specify let's go ahead and um so what did you say was this true or false false I think it's actually true um because if you have uh because this statement says that for every number um there there exists another number such that this doesn't work um but if you have say four um you could pick anything like three and two times three is not equal to four which is exactly what that is so instead of the tildas which I find really confusing because it's out front you could just consider it to be a not equals here um which I think is easier so this is true I believe um okay fourth round sorry even as I'm saying these things I'm getting tripped up on myself so I'm like notation is kind of cumbersome uh for all C such that successor successor of zero time b equals c all right so anyone who is neither Max nor Latif nor dur Sandra all right um there does not exist a be such that all C all numers such that that 2 * yeah exactly so first we you know it's easiest to
……使得这个不……使得 2 乘 b 不等于 c，对吧。那么我们来指定一下。呃，那你刚说这是真还是假？假。我觉得它其实是真的。因为如果你有——因为这句话说的是：对每一个数，都存在另一个数，使得这个等式不成立。那如果你取比如说 4，你可以随便挑一个数，比如 3，而 2 乘 3 不等于 4，这正是它所说的。所以，与其用那些我觉得非常容易搞混的波浪号因为它在最前面，你可以直接把它看成这里是一个不等号，我觉得这样更容易理解，所以这个是真的我相信……好，第四轮，抱歉，我自己在讲这些东西的时候都会绕晕，所以我就……这套记号有点笨重，呃，对所有的 C，使得零的后继的后继乘以 b 等于 c，好的，那么谁是既不是 Max 也不是 Latif 也不是 Sandra 的，好的，呃，不存在一个 b，使得对所有的 C，对所有的数，使得 2乘以……对，没错，所以首先呢，最容易的办法是先不看那个波浪号，就只看：存在一个 b，对
47:46
consider without the TAA and just consider you know there exists a B for all C such that this is the case um and is so with the Tilda just to make at just the way it's written is this true or false false so wait hold on did I write down the right thing um okay so first let's consider the case without the Tila so we have this magical B such that any number we put in um this number is the product of two and that so if we had is there any one number such that any number is just equal to 2 * that so as an example we could specify C to be four or three yeah exactly three and then B to be one right and clearly we have found a c namely three such that 2 * 1 is not that so this is false with the Tilda it's true excellent good work you know I'm I just so say again no no no exactly like you can't um well you would Express one in this notation as just the successor of zero um so it's not that you don't have access to one it's just that in our notation um this is how we would write one basically all right but
所有的 C，使得这个成立，呃，然后加上波浪号之后，就按它写出来的样子，这到底是真还是假？假。等一下，我是不是写对了？呃，好，那我们先考虑没有波浪号的情况。我们有这么一个神奇的 b，使得任何我们代进去的数，呃，这个数都是 2 和它的乘积。所以如果我们要问：是否存在某一个数，使得任意一个数都等于 2 乘以它？举个例子，我们可以让 C 取4 或者 3。对，没错，取 3，然后 b 取 1，对吧？很明显我们找到了一个 c，也就是 3，使得 2 乘以 1 不等于它，所以这个是假的。加上波浪号它就是真的。太好了，做得不错。我，我再说一遍，不不不，就是这样，你没法……呃嗯，在这套记号里，1 会被写成 0 的后继，呃，所以并不是你用不了 1，只是说在我们的记号里，呃，1 基本上就是这么写的。好，不过很好，我是说我很高兴，我就是想让大家
49:22
good I mean I'm glad I just want people to try these um so so two more to go I believe right um so anyone who is not any of the other four people have gone um riddle me this Batman be such that not for all C there exists ss0 * b equal C okay any be
来试试这些题。呃，所以还剩两题，对吧。呃，那么还没发过言、不是刚才那四个人的，来，给我解答一下，蝙蝠侠：存在一个 b，使得并非对所有的 C 都有 SS0 乘以 b 等于 C。好，任意的 b
50:06
na * okay can you give me a truth [Music]
呃乘……好，能给我一个真值[音乐]
50:24
evaluation so we have the B so let's if we and we're we're saying that for this specified value of B regardless of what we put in here this will hold right or to kind of say it a little more clearly and this involves some of the symbol shunting we you do in the chapter there exists a b where if you put in any c it's not going to equal and I think that's so say false or is it true there ex be such that not for every
判断吗？我们有这个 b，那么我们说的是：对于这个指定的 b 值，不管我们往这里代什么，这个都成立，对吧？或者说得更清楚一点，这里涉及到你们在那一章里做的一些符号搬移，就是：存在一个 b，你不管代进去哪个 c，它都不会等于……我觉得这个……所以说，是假的，还是真的？存在一个 b，使得并非对每一个
51:22
c 2 * B is whatever so let's pick three right and I mean yeah I mean this is this is actually I'm pretty sure true because the second you fix B let it be four right it's clear that not for every number 4 * 2 is that number right so in particular for you know six this is not the case right I can't put in any number and get it right are there still questions here last one do this one again do the next one all right do the next one so we're okay with this being true true okay yeah and it's really weird complicated logical relations that's why most of the time mathematicians don't use these things because they get tripped up on the symbols um whereas they already know up here what they're doing is right um so since we're running out of space I'm going to put the sixth round up here and let's try to do this quickly so we can pass off the show such that uh so there exists a b damn I'm not going to do the interpretation um backwards e b colon upside down a c colon um TAA parentheses SSO do B parallel lines
c 都有 2 乘以 b 等于它。那我们就取 3 好了，对吧？我是说，对，我很确定这其实是真的，因为你一旦把 b 固定下来，比如让它等于 4，对吧？那很明显并不是对每一个数都有 4 乘以 2 等于那个数，对吧？所以具体来说，比如 6，这就不成立，对吧？我没法随便代一个数进去就让它成立。这里还有问题吗？最后一题。再做一遍这题？做下一题？好，那做下一题。所以我们都同意这个是真的？真的。好，是的，这些逻辑关系真的又怪又复杂，这也是为什么大多数时候数学家都不用这些东西，因为他们会被符号绕晕，呃，而他们脑子里其实已经知道自己在干什么了。呃，既然我们地方快不够了，我就把第六轮写在上面，我们尽量快点做完，好把场子交出去。使得，呃，存在一个 b，该死，我不做解释了，呃，反过来的 E、b、冒号，倒过来的 A、c、冒号，呃，波浪号、括号、SS0、乘以 b、等号，
53:12
C okay so anyone who's none of the previous five people Maya yeah give me interpretation and Truth value if you're brave there exists for so that there is not [Music] two right so equivalently I could replace the TAA with what other symbol line here right so I think that's easier um so there exists a b said for all c 2 * B is not equal to C so is that true or false it's true okay um this so there exists a b such that for all C so regardless of what you put in here okay so then it's false right ex see this flipping back and forth between these these existential and quantifier existential quantifiers and things like them can really make this confusing because what you have here is your is that there's a b such that regardless of what you put in here this equal this equality never holds um but that's not true because you can take any c eventually it's going going to work right because if we if we specified B to be two that's the thing which exists um then we we can find a c so it's not true that for all C and
C。好，那么前面五位之外的谁来？Maya，对，给我一个解释和真值，如果你够勇敢的话。存在……使得不存在[音乐]两个……对，所以等价地，我可以把这个波浪号换成这里的哪个符号？这条斜线，对吧？所以我觉得那样更容易，呃，也就是：存在一个 b，使得对所有的 c，2 乘以 b 都不等于 c。那这是真的还是假的？是真的。好，呃，这个，存在一个 b，使得对所有的C，也就是说不管你往这里代什么。好，那这样它就是假的，对吧？看，这样在这些存在量词和全称量词之间来回切换，真的会让人很混乱，因为这里说的是：存在一个 b，使得不管你往这里代什么，这个等式都不成立。但这不对，因为你可以取任意的 c，最终总会成立的，对吧？因为如果我们把 b 指定为 2，也就是那个所谓存在的东西，呃，那我们就能找到一个 c，所以“对所有的 C”就不成立了，
54:50
particularly that c value being four um so 2 * 2 is equal to 4 even though this thing claims that this value of B which we picked um would never be equal regardless of what you put in here so I think that's my answer key um true false true true false um and it fits his second hint he says either there are four true and two false or four false and two true and that's related to actually how you shunt these uh tildas um but once again like we don't I mean unless you're really planning on a life as a logician you're not going to have to spend a lot of time manipulating formal systems um um and as such it's good to get practice with this um just because the difficulty which you have of of taking in these new symbols and and forming kind of you know creating more space in your in your neural network for for fitting these places in is I think an important exercise um and that really goes back to that idea of a theory of meaning um and it's one of the last things I want to apologize before I hand
具体来说那个 c 值就是 4，呃，2 乘以 2 等于 4，尽管这个式子声称我们挑的这个 b 值呃，不管你代什么进去都永远不会相等。所以我觉得我的答案是：呃，真、假、真、真、假。呃，这也符合他的第二个提示，他说要么是四真两假，要么是四假两真，而这跟你怎么搬移这些波浪号有关。呃，不过话说回来，我们其实不……我是说，除非你真打算这辈子当个逻辑学家，否则你不需要花很多时间去操作形式系统。呃，尽管如此，练一练还是有好处的，呃，就是因为接受这些新符号、在你的神经网络里腾出空间去安放它们，这个过程本身的难度，我觉得是一项重要的练习。呃，这其实又回到了“意义理论”那个想法上。呃，在我把讲台交给下一位之前，我最后想说声抱歉的一件事是：当我说“递归”这种词，或者
11两小时讲座给不了七年的理解
55:58
off to the lecture to current is that when I say things like recursion and I say things like you know formal systems isomorphisms algorithmic information you know Shannon entropy etc etc um neuron Nets like these terms don't really mean the same thing for you people that it does to a professor who's been spending years working long nights torturing himself over solving these problems late into the night and that process of thinking again and again and making mistakes and then refining your mental your understanding of something forces you know certain parts of your brain to meet here and here and here and just by talking to it at you guys what I'm trying to really do is just Inspire an interest in what I'm saying um and it's not like I can condense somehow you know seven years of you know of undergraduate and postgraduate work into a 2hour leure and just immediately you know go into your brain and you know I want to put this near on here and there and there and suddenly you have the same depth of
说“形式系统”“同构”“算法信息”“香农熵”等等等等，呃，还有“神经网络”，这些词对你们来说跟对一个花了好几年、无数个夜晚折磨自己去解决这些问题的教授来说，含义并不真的一样。那种一遍又一遍地思考、犯错、然后不断打磨自己对某个东西的理解的过程，会迫使你大脑里的某些部分在这里、这里、这里连接起来。而我在这儿对你们讲这些，真正想做的只是激发大家对我所讲内容的兴趣，呃，我并不能把七年的本科和研究生工作压缩进一堂两小时的课里，然后立刻塞进你的大脑，你知道的，我想把这个神经元放这儿、那儿、那儿，然后你就突然对这些学科有了跟 Seth Lloyd 或者其他
57:12
understanding about these subjects that you know Seth Lloyd or anyone else uh who's a specialist in these fields the kind of understanding they have I can't endow that in in just a simple two-hour lecture um unless I assign you know pages and pages of problems and you're working you know 100 hours a week um which I'm not going to do because I'm not evil but aside from that the fundamental thing to pull out here and I've noticed that I say fundamental on um I'll try not to um the important idea here is that you can start with essentially a basic set of statements which you think can capture something true and apply a recursive rule a recursive algorithm to producing new Str which can produce new things so as we go into the next part of today's lecture I want you to kind of think of this truth tree which we try to grow and it all starts kind of from piano axioms number Theory and we just apply these rules and create different statements it's just like the Miu tree that we made in the first lecture or two
这些领域的专家一样的理解深度——他们那种理解，我没办法在短短两小时的讲座里赋予你们。呃，除非我给你们布置一页又一页的习题，让你们每周做上一百个小时，而我不会那么干，因为我不是恶魔。除此之外，这里要抓住的根本一点——我发现我老是说“根本”，呃，我尽量少说——这里重要的想法是：你可以从一组你认为能抓住某种真理的基本陈述出发，然后应用一条递归规则、一个递归算法去生成新的字符串，从而产生新的东西。所以当我们进入今天讲座的下一部分时，我希望你们心里想着这棵我们试图长出来的“真理之树”，而这一切都是从皮亚诺公理、数论开始的，我们只是应用这些规则、造出各种不同的陈述，这就像我们头一两节课里做的那棵 MIU 树一样。我注意到还没人来挑战我那 20 块钱。呃，好，所以事情是这样的，这些
58:23
I noticed no one's challenged me on my 20 bucks um right so what happens is that these some of these things we start with those five basic axioms uh which which Hof stats outlines zero is not the successor of any number um let's see uh zero is not the successor of any number you have to assume that zero plus a number is just that number itself um and here we go yes he actually States them um in this form so you don't go ahead and give it an interpretation Genie is a gen right Genie is a zero J is a number every gen has a meta which is also a gen so every number has a successor which is also a number so Genie is not the meta of any gen so zero is not the successor of any other number different gens sorry this is Page 216 um different gens have different metas so if two numbers are not equal the next guy after them are also not going to be equal and then finally if Genie has X and each gen relays X to its meta then all gens get X and that's the principle of induction that if zero has a property
其中一些东西，我们从那五条基本公理开始，呃，就是侯世达列出的那些：零不是任何数的后继，呃，我看看，零不是任何数的后继；你得假设零加上一个数就等于那个数本身，呃，然后我们来看，是的，他其实是用这种形式陈述的，呃，这样你就不会急着去给它一个解释：Genie 是一个精灵(djinn)，对吧，Genie 就是零，精灵就是数。每个精灵都有一个 meta，它也是一个精灵，也就是说每个数都有一个后继，它也是一个数。Genie 不是任何精灵的 meta，也就是零不是任何其他数的后继。不同的精灵……抱歉，这是第 216 页，呃，不同的精灵有不同的 meta，也就是说如果两个数不相等，那它们后面的那个数也不会相等。最后，如果 Genie 有性质 X，并且每个精灵都把 X 传给它的 meta，那么所有精灵都会得到 X，这就是归纳原理：如果零具有性质 P，并且任何数都把性质 P 传给它的后继，那么所有
59:49
p and the successor and any number relays that property P to its successor then all numbers have it because you can give from zero to one and one gives it to its successor and it's two and it always has that property p and that's what in mathematical induction relies on and you know from from these C these basic things you can actually derive most of number Theory and you just apply these that rule these inference these rules of induction and you start with this trunk and you you get a new theorem well since one and two aren't equal then two and three aren't equal so you just kind of add things to your tree based on these rules and it's completely local it's completely based on what you have at your given point and what rule you're willing to apply and then it's amazing the kind of emergent patterns which you can get and we happen to call that number Theory today I you'll shortly see some things which are a little more interesting um but on that note I think we're going to take two-minute break to de-stress and then
数都具有这个性质。因为零可以传给一，一再传给它的后继，也就是二，而它总是具有性质 P，这就是数学归纳法所依赖的东西。你知道，从这些基本的东西出发，其实就能推出大部分数论。你只要应用这些规则、这些推理规则、这些归纳规则，从这根主干开始，就能得到一条新定理：既然一和二不相等，那么二和三也不相等。于是你就根据这些规则不断往你的树上添东西，而且这完全是局部的，完全取决于你在当下这一点手里有什么，以及你愿意应用哪条规则。然后神奇的是，你能得到各种涌现出来的模式，而我们今天恰好把它叫做数论。你们很快会看到一些更有意思的东西，呃，不过说到这儿，我想我们先休息两分钟放松一下，然后把设备弄好让 K 接手。谢谢。好，
12Context Free：随机文法长出树
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get things set up for K to take over thank you okay so what he was actually talking about can also be called a context free grammar a context free grammar is something that has um symbols and production rules and the symbols here in this case are s and zero and the various mathematical operators but they're just symbols in terms of the the grammar and the production rules are the uh the rules of inference where you can take one string and perform some manipulation on a part of it to get a new string it's a rule of inference so you can use a similar system to Define moving around circles on the screen um so I'm I'm going to explain what's going on here this is a a program called context free it's an open source project that you can download and play with yourself so what's going on here the code um start shape just is the entry point it's not going to change and we Define a rule called spiral and inside this rule we have Circle which plots a circle on the screen and then we call spiral again and
那么他刚才讲的东西其实也可以叫做上下文无关文法。上下文无关文法就是这样一种东西：呃，它有符号和产生式规则。这里的符号就是 S、零，还有各种数学运算符，但从文法的角度看它们就只是符号；而产生式规则就是那些，呃，推理规则，你可以拿一个字符串，对它的某一部分做某种操作，从而得到一个新字符串，这就是一条推理规则。所以你可以用一个类似的系统来定义屏幕上圆圈的移动。呃，我来讲讲这里在干什么。这是一个叫 Context Free 的程序，它是一个开源项目，你可以自己下载来玩。那么这里发生了什么呢？代码里，呃，startshape 就是入口点，它不会变。然后我们定义一条叫 spiral 的规则，在这条规则里我们有 CIRCLE，它会在屏幕上画一个圆，然后我们再次调用 spiral。y 空格 2 表示我们把 Y 坐标增加两个单位，size 空格 0.9 表示我们
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y space two means that we increment the Y position by two units and size space 0.9 means that we multiply the size of this by 0.9 every time we go up so this rule defines this image and what this program does is um when the thing gets too small to see it just stops doing it that's so this is actually an infinite recursion but it stops at some point because it gets so small so this is our framework and by changing this this code little by little we're going to get some amazing pictures so I'm going to just do these changes and explain them as I go so I just had this spacing to be two so it's clear that we're just drawing circles I'm going to decrease the spacing to be like 0.4 and uh render so it looks like that so I'm going to Define another rule um also called spiral and so as I go I'm going to explain the the more features of this language um when you define a rule that has the same name twice what happens is whenever you call this rule it um it calls one or the other with equal probability so what I'm going to do here
每往上走一次就把大小乘以 0.9。所以这条规则定义了这幅图像。而这个程序做的是，呃，当图形变得太小看不见的时候，它就不再画了。所以这其实是一个无限递归，但它在某个时刻会停下来，因为图形变得太小了。所以这就是我们的框架，然后通过一点一点地改这段代码，我们会得到一些很惊艳的图像。所以我就来做这些改动，一边做一边解释。刚才我把间距设成 2，是为了能清楚地看出我们画的就是一个个圆。现在我要把间距减小到比如 0.4，然后，呃，渲染，看起来就是这样。接下来我要再定义一条规则，呃，也叫 spiral。随着往下走，我会解释这门语言更多的特性。呃，当你定义两条同名的规则时，会发生的事情是：每当你调用这条规则，它，呃，会以相等的概率调用其中之一。所以我在这里要做的是，呃，写 flip 90，意思是把我们的参考坐标系翻转
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is say um flip 90 which means flip our sort of frame of reference by 90° um actually first of all before I do this sorry I'm going to add a rotation so rotate one rotate one degree each time so it rotates a little bit you see that one degree each time so if we decrease the size by 0. N9 every time it's going to get smaller a little bit slower so we see the spiral happen so if we do 0999 it'll be even more spiral so that's what we get uh I don't know why it's going off the screen there we go okay so now I'm going to Define another rule called spiral and flip by 90° so if i r this what do you guys think is going to happen what do you think it's going to flip so maybe I wasn't clear about what what it means to to call that flip thing so when we say rotate one we're rotating one degree this way so after after we call Flip when we say rotate one it's going to rotate the opposite direction so say we call the first rule like five times draws five of these circles then we call the second rule once it flips and then we call the first
90 度。呃，其实在这之前，抱歉，我先加一个旋转：rotate 1，每次旋转一度，所以它会稍微转一点，你看，每次一度。那么如果我们每次把大小乘以 0.99，它就会缩小得稍微慢一点，于是我们就能看到螺旋出现。如果我们用 0.999，螺旋就会更明显，这就是我们得到的结果。呃，我不知道它为什么跑出屏幕了，好了，行了。好，现在我要再定义一条叫 spiral 的规则，并翻转 90 度。那如果我运行它，你们觉得会发生什么？你们觉得它会怎么翻转？也许我没说清楚调用那个 flip 到底是什么意思。当我们说 rotate 1 的时候，我们是朝这个方向转一度。那么在我们调用 flip 之后，再说 rotate 1，它就会朝相反的方向转。所以比如说我们调用第一条规则五次，画出五个这样的圆，然后我们调用第二条规则一次，它就翻转了，接着再调用第一条规则五次，它就会朝另一个方向稍微弯过去。而一开始这两条规则会被
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rule five more times it's going to just turn the other way a little bit and so initially they're going to be called with equal probability so if I render it this is what we get this sort of Meandering thing so what's happening is half the time it's calling the first rule which moves the circle up a little bit and rotates it and decreases its size and draws the circle and half the time we're calling this other rule which flips the direction in which we're rotating so this is what we get um another feature of the language is we can change the probabilities at which these things are called so let say 0.1 I put 0.1 right next to the second spiral rule the flipping rule so that means well if I don't specify a number it gets one so that means that the first rule is going to get called with a probability one and the second rule is going to be called with a probability 0.1 and this is whenever maybe oneus one minus that first will all probability oneus because you're rolling a die you can never go beyond probability
以相等的概率调用。所以如果我渲染一下，得到的就是这个，一种蜿蜒游走的形状。所以发生的事情是：一半的时间它调用第一条规则，把圆往上挪一点、转一点、把大小缩小一点，然后画圆；另一半的时间我们调用另一条规则，它把我们旋转的方向翻转过来。所以我们得到的就是这个。呃，这门语言的另一个特性是，我们可以改变这些规则被调用的概率。比如说 0.1，我在第二条 spiral 规则、也就是那条翻转规则旁边写上 0.1，那意思就是——如果我不指定数字，它默认是 1——也就是说第一条规则会以概率 1 被调用，而第二条规则会以概率 0.1 被调用。而这个……可能是 1 减去……1 减去第一条的概率，因为你是在掷骰子，你不可能超过概率的上限，对吧。所以说到概率，呃，它实际做的事情我觉得
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point right so talking talking about probability um what what it actually does I think is is computes the sum and then takes the fraction of that sum that each of the rules have for probability either way this one happens less probable than 50% now so we can see that it goes for slightly longer periods of iteration without any flipp so if we do it again uh decrease it 01 it'll flip even less and the the the the uh yeah flips with even less frequency so if we make it 01 uh it flips a lot less so already we're seeing these really cool pictures being generated by such simple rules simple context reg grammars any questions so far so what happens if I add inside this rule another instance of spiral without the flip what does this mean
是：先算出总和，然后取每条规则在这个总和里所占的比例作为概率。不管怎么说，现在这条规则发生的概率低于 50% 了。所以我们能看到它会连续迭代更长一段时间而不发生翻转。那我们再来一次，呃，把它减到 0.01，翻转就更少了，而且，呃，对，翻转的频率更低了。所以如果我们设成 0.01，呃，它翻转得就少多了。所以我们已经能看到，这么简单的规则、简单的上下文无关文法，就生成了这些很酷的图像。到目前为止有问题吗？那么如果我在这条规则里再加一个 spiral 的调用、但不带翻转，会怎么样？这是什么意思？
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first it f and then just yeah so more or less she said first this flips and then it goes on without flipping so yeah with this rule instead of um just flipping what it's going to do is start this one off on its tangent and also just keep going its original way without flipping so we'll render this and see what it does so it it does exactly that see whenever it branches it it also keeps going its original Direction so we can um change some parameters around and we're going to get some organic looking forms so if we decrease the probability no increase probability of branching we get this just goes out of control so maybe that's not what we want to do so now I put it back um if we multiply size by 099 yeah there we go now we increase the probability of branching and we get these trees like look at it it looks like a tree um it's really wild so if we increase the probability of branching even more we get thicker trees because they Branch more uh it's pretty wild Isn't It Isn't that cool so it makes you wonder like does
先翻转，然后就……对，她大致的意思是：先翻转，然后再不翻转地继续走下去。对，所以有了这条规则，它不是呃，只做翻转，而是会让这一支沿着切线方向岔出去，同时也不翻转地沿着原来的方向继续走。那我们就渲染一下看看效果。它做的正是这个：看，每当它分叉的时候，它也会继续沿着原来的方向走。所以我们可以，呃，改一改一些参数，就会得到一些看起来很有机的形态。所以如果我们把概率调低……不，把分叉的概率调高，我们就得到这个，完全失控了，所以这也许不是我们想要的。现在我把它放回去，如果我们把 size 乘以 0.99，对，就是这样，现在我们提高分叉的概率，就得到了这些像树一样的东西你看，它看起来就像一棵树，真的挺疯狂的。如果我们再进一步提高分叉概率，就会得到更粗壮的树，因为它们分叉更多。挺神奇的对吧？是不是很酷？这就让人不禁想问，大自然在让植物生长时是不是也用了这种上下文无关
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nature use these sort of context free grammars in in the way it grows plants or is it like a Linden Meer system is it fixed these like Global rules that get applied at smaller and smaller scales I think in nature what we see is sort of a mix a mixture of both cuz some plants have very regular features and some plants don't so it's sort of it's still a mystery like plants but it looks pretty organic so I'm just amazed by this uh sort of thing so I can you know play with the parameters and get really cool growths so let's think about this as a model of plant development um we're modeling that that the size of the plant is just constantly shrinking and when it branches it's it's Mass sort of doubles which is sort of a bad model think about a tree think about a tree just going up and then when it branches uh it all it still keeps going up but a branch like goes off to the side so we can change our grammar to do this and we'll get some tree like things so I'm going to do it so this is the rule the the rule on top is just the
文法？还是说更像一个 Lindenmayer 系统（L-system），是那种固定的、全局的规则，然后在越来越小的尺度上反复应用？我觉得我们在自然界看到的其实是两者的混合，因为有些植物具有非常规整的特征，有些植物则没有。所以植物这件事仍然是个谜，但它看起来相当有机所以我对这类东西非常着迷。我可以调整这些参数，得到非常酷的生长形态。那我们把它当作一个植物发育的模型来思考。我们建模的方式是：植物的尺寸在不断缩小，而当它分叉时，它的质量差不多翻倍，这其实是个不太好的模型。想想一棵树，一棵树往上长，然后当它分叉时，主干仍然继续往上长，但是有一根枝条会朝旁边岔出去。所以我们可以修改文法来实现这一点，就能得到一些像树的东西。我这就来做。上面这条规则就是它
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rule that it does when it's going just just going so I'm going to change this to just um increment Y and I'm going to change this to so one of these is going to be the it's just going to go straight and the other one is going to branch and get smaller so sign
直着往前长时用的规则。我把这个改成只让 Y 递增，然后把这个改成……所以其中一条是直着往前长，另一条是分叉出去并变小。所以是 sin
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um9 for for just going straight and what flip does in this context well first I'll explain it later we say rotate 45° and size is 0.3 so what we get is this very spark structure so we can increase the probability of branching to like5 maybe
0.9 用于直着往前长。至于 flip 在这里的作用嘛，我待会儿再解释。我们写上 rotate45 度，size 是0.3。于是我们得到了这种非常稀疏的结构。所以我们可以把分叉概率提高到 0.5，或者
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N2 so we get these things that sort of look like trees so I'll just explain the rules in case you didn't follow so the first spiral in this rule this part of the rule encodes for the fact that um whenever it branches the size gets a little smaller by a factor of 0.9 and you form this branch and this the second rule here says size 0.3 that means that the size of this Branch is3 the size of the original trunk um and the flip part flip 90 in this rule means that next time it branches it's going to branch that direction we can actually take it out and if we take out the flip um the branches will all just go in the same direction all the time so if you take it out see the they only go in One Direction so we could sort of play with um parameters here let's say .95 let's just see what happens
0.2，这样我们就得到了这些看起来有点像树的东西。我再解释一下这些规则，以防你没跟上。这条规则里的第一个 spiral，也就是规则的这一部分，编码的是：每次分叉时，尺寸都会缩小到原来的0.9 倍，然后形成这个分枝。而这里的第二条规则写着 size 0.3，意思是这根分枝的尺寸是原来主干的 0.3 倍。至于 flip 那部分，这条规则里的 flip 90 意味着下一次分叉时，它会朝那个方向分。我们其实可以把它去掉，如果去掉 flip，所有的分枝就会一直朝同一个方向长。所以如果你把它拿掉，看，它们就只朝一个方向长。我们也可以调调参数，比如说0.95，看看会发生什么
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uh yeah we get some pretty interesting things if we make the branches a little bit thicker maybe point4 see look at that it's like a tree or something any questions so far and comments so if we add a little bit of rotation to the main rule uh rotate one Dee yeah Latif does it also take a probability of that too say again when it calls itself inside it does it call itself the thing that calling or does it call the other one oh good question so he asked um when it calls itself like inside like this one or this one or this one which one does it call the up one or the right so which one does it call when you when you call spiral so this is where the probability comes in anytime you call it these there are these three instances where spiral is called inside of spiral so it calls either one with certain probabilities and the probability of the first one is one out of 1.02 and the probability of the second one is.2 out of 1.2 sorry the probability of the first one is one of 1.2 so it's a very high
嗯，对，我们得到了一些挺有意思的东西。如果我们把分枝弄粗一点，比如 0.4，你看这个，它就像一棵树之类的。到目前为止有什么问题或者想法吗？如果我们给主规则加上一点点旋转，比如rotate 1 度……对，Latif？它是不是也带一个概率？再说一遍？就是当它在内部调用自己的时候，它调用的是调用它的那一个还是另外一个？哦，好问题。他问的是，当它调用自己的时候，比如这一个、这一个，或者这一个，它到底调用哪一个，上面那个还是右边那个？也就是当你调用 spiral 时它调用的是哪个。这正是概率起作用的地方。任何时候你调用它，spiral 内部有三处会调用spiral，它会以一定的概率调用其中之一。第一个的概率是 1 除以 1.2，第二个的概率是0.2 除以 1.2。抱歉，第一个的概率是 1 除以 1.2，所以百分比非常高，而第二
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percentage and the probability of the second one being called is 0.2 out of 1.2 so yeah every time you call it from within anything it goes into this program and say you know tell me which one should I call and the program assigns which one it is based on these probabilities so doesn't matter if you put it put it have So you you're saying maybe we could put it outside put it inside the function so that the function I mean the thing itself the reason why I put it inside the function is so that it can yeah call itself so like what happens if you take a spiral flip 90 and you put it up in the top rope for spiral the signs cut that so you you're saying what what if I take this out of here and put it up here sure yeah I mean I don't know I don't know but it has to be inside a function in order like so I think the idea is that you have kind of two options here whether or not you're just doing the simple spiral circle routine or if um you're then doing this other spiral routine where they rotate 45 degrees and change the
个被调用的概率是 0.2 除以 1.2。所以，每次你在任何地方从内部调用它，程序都会进入这个流程，问一句“告诉我该调用哪一个”，然后程序根据这些概率决定调用哪一个所以放在哪儿是不是无所谓……你是说也许我们可以把它放到外面，或者放到函数里面，让这个函数本身……我把它放在函数里面的原因是为了让它能够调用自身。那如果你把 spiral flip 90 拿出来，放到 spiral 最上面那条规则里会怎么样？就是把符号剪切过去。所以你是说如果我把这个从这儿拿出来放到上面这里？行，可以试试。嗯，我也不知道，我不确定，但它必须在一个函数里面才行。所以我觉得这里的思路是你大概有两个选项要么你只是执行那个简单的 spiral 画圆的流程，要么你执行另外那个 spiral 流程，也就是旋转 45 度、把尺寸变成 0.4 倍的那个，而不是这个 flip 90、把尺寸变成
76:46
size by 04 instead of this spiral where you flip 90 and you change the size by 0.95 and notice that and see this was what I thought I'm not exactly sure how the algorithm is implemented but I think by having 0 2 next to the bottom spiral that means the top spiral is called only with probability 08 but that's obviously much higher than 0. 2 so every time the computer rolls its die it's trying to decide do I either execute the top spiral or do I execute the bottom spiral um and then the content of each of those those spirals um is then governs the behavior you see here so what what we did when we when we moved this up whenever we call spiral Spiral it branches we have two spirals now right and that's the probability of that one is really high so that means pretty much every time we're branching so we just get this mess of branches and it will'll never finish so yeah I mean there are all kinds of really interesting um modes that we can come come to with this so I'll take it out and put it back
0.95 倍的 spiral。注意到没有，这也是我之前想的。我不太确定这个算法具体是怎么实现的，但我觉得在下面那个 spiral 旁边写 0.2，意味着上面那个 spiral 被调用的概率是 0.8，而这显然比 0.2 高得多。所以每次计算机掷骰子的时候，它是在决定：我到底执行上面那个spiral，还是执行下面那个 spiral。然后每个 spiral 里面的内容就决定了你在这里看到的行为。所以我们刚才把这个挪上去之后，每次调用 spiral 时它就会分叉，因为现在有两个 spiral 了对吧而那一条的概率非常高，也就是说几乎每一次都在分叉，所以我们就得到这一团乱七八糟的分枝，而且它永远算不完。所以，用这个东西我们能玩出各种各样非常有趣的模式。我把它拿掉，放回原来的位置。它还在计算……好，行了，它
77:59
where it was um it's still Computing uh okay there we go it
它在哪儿来着，嗯……还在计算，呃，好，出来了
78:13
stopped oh yes so it looks crazy doesn't it so if I take out that rotate it's all very regular it's sort of regular structure and we could change the angle maybe uh 60° I don't know or 90 even and it looks sort of like um roads maybe I don't know we had some conf it's like a tree like near a river or lake or something you can see that it's like sort of an AR yeah it's like a you're saying maybe it's like a Riv a river with these little sub Rivers going off of it yeah I mean this kind of structure is found everywhere in nature it's really amazing so if we add a little bit of rotation to the to the main going forward rule rotate one we get this sort of [Music] veiny that's when the wind yeah when the wind is blowing on the tree right or it looks a lot like uh Vines right when a Vine is crawling up a wall and you have or a root a root of a plant H underground yeah underground the root sort of looks like this so yeah or the special tree if you tried to do this so can I m the make the cotch snowflake
停了。哦对，看起来很疯狂对吧。如果我把那个 rotate 去掉，它就全都很规整了，是一种比较规整的结构。我们还可以改变角度，比如说60 度，或者甚至 90 度。它看起来有点像……道路吧，我也说不好。有人说像河边或湖边的一棵树之类的你可以看出它有点像……对，就像你说的，也许像一条河，一条大河带着这些小支流从中岔出去。对，这种结构在自然界随处可见，真的很惊人。所以如果我们给主干往前走的那条规则加上一点点旋转，rotate 1，我们就得到这种[音乐] 像血管一样的东西。那是风吹的时候……对，就像风吹在树上的样子，或者它也很像藤蔓，对吧，当一根藤蔓顺着墙往上爬的时候。或者是根，植物埋在地下的根，对，地下的根差不多就长这个样子。对，或者那种特殊的树……如果你试着做的话，我能不能用这套东西做出科赫雪花，或者那种分叉的
13随机与确定：混沌游戏与稳定性
79:58
or the the branching tree with this sort of thing I haven't tried uh maybe I could do it somehow but I um I don't think so I don't think I can because the how this program acts is completely random it's stochastic it chooses which rule to do on the Fly H would be well it is still recursive it's totally recursive but it's it it's not deterministic so it's something being deterministic gives it like absolute rigid regularity that we find in the cot snowflake or that branching tree that I showed earlier but yeah they're executed randomly so you get these irregular fractals but the serinsky gasket you can also generate stochastically using random D roll exactly and you you pick your three points and then you throw a random Dart and then what you then take the distance to the closest Edge and fill that point in right think you do that I forget so the chaos game the chaos game with the serinsky triangle um what you do is you have these three dots I did explain this once before but I'll just do it again quickly you have
树？我没试过。也许我能想办法做出来，但我觉得不行，我觉得我做不到，因为这个程序的运行方式是完全随机的，它是随机的（stochastic），它在运行时临时选择用哪条规则。那它还是……它仍然是递归的，完全是递归的，但它不是确定性的。所以，确定性会带来那种绝对刚性的规整性，就是我们在科赫雪花或者我之前展示的那棵分叉树里看到的。但这里的规则是随机执行的，所以你得到的是这些不规整的分形。不过谢尔宾斯基三角（Sierpinski gasket）其实也可以用掷随机骰子的方式随机地生成——没错，你先选三个点，然后你随机扔一个飞镖，然后……你取到最近那条边的距离，把那个点填上，对吧？我觉得是这么做的，我记不清了。所以那个混沌游戏，用谢尔宾斯基三角玩的混沌游戏，做法是这样：你有三个点。我之前讲过一次，但我再快速讲一遍。你有这三个点，然后你从某个位置开始，比如说这里，然后你从当前所在的位置出发
81:24
these three dots and you start at a certain point like say here and you take where you are and you choose at random one of the three dots and go halfway from where you are to that dot so say we choose this dot we go halfway there say we choose it again we go halfway then say we choose this dot we go halfway from here to here so after doing this like a thousand times we get all these points that in the limit if you were to do it an infinite number of times it would be the serinsky gasket and so it starts looking like there serinsky gasket after a little while so that's stochastic right I mean I don't know maybe I could code that in this language maybe it's possible I haven't tried so there's some some prepackaged examples that I sort of um have prepared um and one of the things that I want to talk about is these regions of stability and instability in the um in the system it's it's a very dynamic system and you get these different situations yeah look at this oh wait that's a PNG that's not the
随机选三个点中的一个，从当前位置朝那个点走一半的距离。比如我们选了这个点，我们就走到一半的地方；再比如我们又选了它，我们再走一半；然后假设我们选了这个点，我们就从这儿到这儿走一半。这样重复大约一千次之后，我们就得到了所有这些点，在极限情况下，如果你能做无限多次，它就会是谢尔宾斯基三角。而且做一会儿之后它就开始看起来像谢尔宾斯基三角了，所以那也是随机的对吧。我也不知道，也许我能用这个语言把它写出来，也许可以，我还没试过。我这里准备了一些现成的例子。其中我想讲的一件事是这个系统里的稳定区域和不稳定区域。它是一个非常动态的系统，你会得到这些不同的情形。对，你看这个……哦等等，这是个 PNG，不是真正的那个。算了，就当……我把真正的那个
82:52
actual thing oh well just consider this open the actual one here it is so what's going on here all right what's going on here is I have this one rule um first of all this rule these two rules are similar to the ones that I'll just get rid of this one for sake of explanation so this is pretty cool right it's a tree so I'll explain the rules the first rule goes forward by unit of one and the circles are one so we can actually see all the little circles goes forward by one and degreases in size um the size is multiplied by 0 N9 so that that's that's the main Rule and then we have this other rule which has a probability 02 over 1.02 the sum of them and that's the branching rule so it just calls tree again with a rotation of 20° or minus 20° so it's 20 minus 20 so this is our rule um and it generates this tree and when we add this other rule so this is a rule that multiplies the size of the thing by five which is pretty um pretty extreme right but it happens with a very small probability so I I'll decrease the
打开。在这儿。那么这里发生了什么呢？好，这里的情况是这样：我有这么一条规则。首先，这条规则……这两条规则和之前那些类似。为了方便讲解，我先把这一条去掉。这个挺酷的对吧，是一棵树。我来解释一下这些规则第一条规则是往前走一个单位，而圆的大小是 1，所以我们能真真切切看到所有这些小圆圈，它往前走一格，然后尺寸减小，尺寸乘以 0.9。这就是主规则。然后我们还有另外一条规则它的概率是 0.2 除以 1.02，也就是两者之和，这是分叉规则。它只是再次调用 tree，并带上20 度或者 -20 度的旋转，所以是 20、-20。这就是我们的规则，它生成了这棵树。然后当我们加上另外这条规则……这条规则会把东西的尺寸乘以 5，这相当极端对吧，但它发生的概率非常小。我把概率再调小一点
84:36
probability even more 0.003 so maybe 01 okay it just happened a few times so we can see that usually it doesn't happen all these all these branching and all these iterations it didn't happen but on this one here the size of it multiplied by five right so we had this bigger Circle and then it propagated more and then it happened again on this tip this very end branch of it and it happened again so I mean M tree this pattern appears in evolution actually which is really fascinating so in evolution you have these these species and whatnot or different strains of genome which sort of diverge and and Branch out and say at this point in time and the point in time being the number of iterations overall at this point in time say like some huge catastrophic event happened on the earth and this one organism or small set of organisms is the only one that survived so their weight so suddenly increased and then they propagated and spread themselves and and this is what we get this new um branching of of evolution
0.003，或者 0.01 吧。好，它刚才发生了几次。我们可以看到，通常它是不发生的，所有这些分叉和所有这些迭代里它都没发生，但在这里这一处，它的尺寸乘以了 5，对吧？于是我们就有了这个更大的圆，然后它继续往下传播，然后在这个尖端、这根最末端的分枝上又发生了一次，而且它又发生了一次。这个模式其实在演化中也会出现，这一点非常有意思。在演化中你有这些物种之类的，或者说不同的基因组分支，它们会分化、分叉出去。假设在某个时间点——这里的时间点就是总的迭代次数——在这个时间点上，假设地球上发生了某个巨大的灾难性事件而这一个生物、或者一小群生物是唯一幸存下来的，于是它们的权重就突然增加了然后它们大量繁殖、扩散开来，于是我们就得到了这个新的演化分支
86:03
and then the same thing happened here bam and then it branched again and it branched again so it's a very unstable unpredictable system because you could have these little events that just completely change the face of uh the system like here it's a stable system well like without this new rule it's a stable system we know it's always going to go down the size is always going to get really smaller and smaller and smaller until it disappears that's guaranteed in the limit but with this system we introduce a rule which goes backwards so it makes the system unstable and much more unpredictable so if we um increase the probability of this rule to 0.1 the system goes out of control really quickly and we have 0.01 it just gets completely out of control and it just gets bigger and bigger and bigger until see see this this message here a shape got too big um so this is an error that we get when this when the thing just goes out of control so this would almost correspond to like meteorites being frequently
然后同样的事情又在这里发生了，砰，然后它又分叉了，又分叉了。所以这是一个非常不稳定、不可预测的系统，因为你可能会遇到这些小事件，它们能彻底改变整个系统的面貌。而在这里，它是一个稳定的系统……我是说，没有这条新规则的话，它是个稳定的系统，我们知道它总是会往下走，尺寸总是会变得越来越小、越来越小，直到消失，在极限意义上这是有保证的。但在这个系统里，我们引入了一条反方向的规则，于是系统变得不稳定，也更加不可预测。所以如果我们把这条规则的概率提高到 0.1，系统很快就会失控。我们试 0.01……它完全失控了，越变越大、越变越大，直到——你看，看这条消息，“形状太大了”。这就是我们得到的一个错误就是当这个东西彻底失控的时候。所以这差不多相当于陨石频繁地
87:18
thrown at the face of the Earth and only very few strands of species surviving at a time well this would yeah I guess this would correspond to that yeah destroying genetic diversity at more frequent internets yeah if we were to if we were to think of the metaphor to Evolution again I don't know I guess it would correspond to like just huge catastrophic events happening all the time and miraculously every time one species survives sort of extreme metaphor I don't know if it would really hold but but it's interesting this um regions of stability and instability in this space of parameters to the system so yeah it's pretty fascinating so I'll just show some more examples and then I'll be done um this thick tree yeah when we have certain parameter sets we get these really organic looking forms so if if we increase the probability of branching even more get these really nice thick organic looking
砸向地球表面，每次只有极少数物种谱系幸存下来。嗯，这个……对，我想这大概就相当于那种情况。对，以更高的频率摧毁遗传多样性。对，如果我们再用演化来做类比的话……我也说不好，我猜这相当于巨大的灾难性事件一直不停地发生，而每一次都奇迹般地有一个物种活了下来。这个类比有点极端，我不确定它是不是真的站得住脚，不过这挺有意思的，就是这个系统参数空间里的稳定区域和不稳定区域。对，挺迷人的。我再展示几个例子就讲完了。这棵粗壮的树，对，当我们用某些参数组合时，就会得到这些看起来非常有机的形态。所以如果我们把分叉概率再提高一些，就会得到这些非常漂亮的、粗壮的、看起来很有机的
14向光性、自然选择与普适分形
88:39
trees so you asked that's the question so she asked um do organisms actually use this sort of system to live and and if like they had ones that didn't have a system and did have the system that evolutionary or not right so you're so you're asking this question like sort of from a different totally different angle she she's saying like um for a given organism if it develops this kind of recursive system is it evolutionarily more advantageous like trees I think I think for plants I think that's like the one of the main thing that plants sort of developed and that made them evolutionarily speaking more feasible more fit so that's why so maybe that's an important point to consider um however what would actually make a tree Bend like that okay let's say that it's a relatively calm area and we're not having hurricane force winds forcing our trees to bend that way what else would uh why why would a tree grow like that yeah re sunlight sunlight exactly photot taxes right and plants have this feedback mechanism where they're
树。你刚才问……对，这就是那个问题。她问的是，生物在生存中是不是真的用了这种系统，而且如果有些生物没有这种系统、有些有，那这在演化上有没有区别？对，所以你是从一个完全不同的角度在问这个问题。她的意思是，对于某个生物来说，如果它发展出这种递归系统，在演化上是不是更有优势？比如树。我觉得对植物来说，我认为这算是植物发展出来的主要东西之一，正是它让植物在演化意义上更可行、更有适应力，所以这也许是个值得考虑的重要点。不过话说回来，究竟是什么会让一棵树那样弯曲呢？好，假设这是一个相对平静的地方，没有飓风级的大风逼着树往那个方向弯，那还有什么会呃，为什么一棵树会那样长？对，阳光，阳光，没错，向光性（phototaxis），对吧。植物有这种反馈机制，它们真的能感知光。就好比我在这儿放一盆植物，而角落里那盏灯
89:54
actually able to sense light and it's just like if I were to have a potted plant here and that light in the corner Illuminating us it wouldn't it would actually be able to dynamically change the probability of splitting and of course the method for that is actually a little different you have these oxin chemicals being exchanged collapses cell walls and a tree is actually very quickly able to grow and bend in a different direction um but um so it's not exactly just a probabilistic w what happened there uh it's not exactly just a probabilistic context free grammar um it's it's got to have some sort of feedback mechanism and then with the level of evolution changing the rules even there so yeah I mean look at all the the number of um Leaf branches when I say Leaf I mean the one on the end like there's so many of them and and it maximizes is the um the amount of sunlight that hits like it maximizes the amount of surface area on the tree so I think that's why that's one of the big reasons why it's more
照着我们，它其实能够动态地改变分叉的概率。当然实际的机制有点不一样，是通过生长素（auxin）这类化学物质的转移，使细胞壁塌陷，于是树能很快地朝另一个方向生长和弯曲。所以它并不完全只是一个概率性的……呃刚才怎么了……它并不完全只是一个概率性的上下文无关文法，它必须具备某种反馈机制，而且在演化这个层面上，连规则本身也在改变。所以，你看这些叶枝的数量——我说'叶'的意思是最末端的那些——它们数量特别多，这样就能最大化……最大化照射到的阳光，就是最大化整棵树的表面积，所以我觉得这就是其中一个重要原因它更具适应优势。你有问题吗？没有，我只是想说一句。我觉得也说得通，就像你说的表面积，而且
91:05
fit you had a question no I just had a comment I also Mak sense like you said the surface area and it wouldn't make any sense to have a branch at the bottom because all the light was already blocked from top right it wouldn't make any sense yeah if you had like some uh mutation where Branch did come on the bottom that tree wouldn't have any advantage right so you're getting at selection so he said like if there were some tree that had these set of rules that made it start branching at the bottom and on the top it would be selected against because it's not feasible cuz the light would be blocked out by the higher branches yeah that's that's the nature of evolution yeah yeah and that also looks like a prank you got the Cal cortex on the top that's very like C and you got the correction connections and exactly so so he said it's sort of like a brain like this is a cerebral cortex and you have all these connections that go out to the surface of the brain which have like all these you know brain cells or
在底部长树枝是没有意义的，因为光线在上面就已经被挡住了，对吧？那样根本没有意义。是啊，如果你有某种突变，让树枝真的长在底部，那棵树也不会有任何优势，对吧？所以你说的其实是选择。他刚才说的是，如果有某棵树带着这样一套规则，让它从底部开始分枝，而顶部也有分枝，那它就会被自然选择淘汰，因为这行不通，光线会被更高处的树枝挡住。对，这就是演化的本质。是啊是啊，而且这看起来也很像大脑，你顶上有大脑皮层，非常像……而且你还有那些连接。没错，所以他说这有点像大脑，就像这是大脑皮层，你有所有这些延伸到大脑表面的连接，上面有各种脑细胞。或者血管结构也是，是的，血管，你身体里血管的结构
92:10
even a vein structure yeah veins the structure of the veins in your body coming out of your heart it just fractals out to all that your body so yeah it it's this sort of universal form that appears everywhere it's really amazing yeah so all we can do is sort of awe at it say wow they're like the same man but where does it come from what does it mean I don't know it's something that needs to be explored I think could it be that he had to be that way could it be that it had to be that way yeah what do you mean by that you can say that uh it's like the um most efficient algorithm and left um stress towards efficiency so at some point you had to like hit that algorithm and when it when it gets something good it doesn't want to let it go yeah so you're talking about like it being the like uh the whole system of like all biology and things evolving he said maybe it it's the only thing that works like it has to exist because it's the most efficient algorithm for for developing our biology our wors and so
从心脏出来，就这样分形般地延展到你的全身，所以是的，这是一种普适的形态，到处都能看到，真的很惊人。是啊，所以我们能做的就是对它感到惊叹，说'哇，它们简直一模一样'，但它从哪儿来？它意味着什么？我不知道，我觉得这是需要去探索的东西。有没有可能……它必须是那样？有没有可能它就必须是那样？是啊，你这话是什么意思？可以这么说，它就像是最高效的算法，而生命本身就朝着效率的方向发展，所以在某个时刻你必然会碰上那个算法，而一旦它找到了好东西，它就不肯放手了。对，所以你说的是，整个系统，就是所有生物学和万物的演化。他说也许这是唯一行得通的东西，它必须存在，因为它是构建我们的生物结构、我们的世界的最高效算法，所以一旦它出现，就在这一长串演化事件中
93:25
once it once it appeared it sort of took hold in this big evolutionary uh series of events and I think you're right and it it it also is coded in a very simple set of rules like look at this it's like it's very little text you know that encodes for all of this and I think similarly in our genomes like if if The evolutionary Paradigm comes up with an efficient way of encoding a certain set of rules to do something which is which is uh fit like which makes us more fit than than it sticks yeah I think I think this this notion of fractals and the recursive algorithm being encoded into our genomes is is a reasonable thing to hypothesize that also like show like behaviors like yeah ants an colonies oh man yeah yeah it's everywhere properties emerging properties yeah so I think I'm done I think this is my Spiel and I'll give it back to Justin I just want to kind of wrap things up kind of give a sense of conclusion and direction of where we're going um if you want to kill the projector so exactly kar's kind of you
站稳了脚跟。我觉得你说得对，而且它还是由一套非常简单的规则编码出来的，比如你看这个，只需要极少的文本，你知道，就编码出了这一切。我觉得同样地，在我们的基因组里，如果演化范式想出了一种高效的方式，来编码某一套规则去做某件事，而这件事是有适应优势的，能让我们更能适应环境，那它就会保留下来。是啊，我觉得这个关于分形和递归算法被编码进我们基因组的想法，是个很合理的假设。这也能解释一些行为，比如蚂蚁、蚁群。哦，天哪，是啊，到处都是。涌现属性，对，涌现属性。好，我觉得我讲完了，这就是我要说的，我把话筒交回给Justin。我只是想收个尾，给大家一个结论感，还有我们接下来要走的方向。呃，能把投影仪关掉吗？没错，Kar其实一直在暗示一个想法，而我们都已经站在这个
15收尾：宇宙是形式系统与课程去向
95:00
know hinting at an idea and we're all kind of at the brink of this this concept which I told you at the very beginning is the stated thesis of girdle eer boach which is that the Universe at a fundamental level is a formal system and that it obeys certain deter deterministic rules or perhaps probabilistic rules but kind of formal system nonetheless we have this kind of label which we just stick on something and that being the eye label which actually hides a lot of detail um just like in the way that um but trying to understand them in a fundamental way is the state goal of this course um and I'm still debating kind of because I'm kind of continuously and probabilistically modifying the course of this course um and I'm trying to decide and the stated plan right now is that after we do mumon and girdle and kind of get you'll get this weird Asian kick from both K and I um combining Zen and logic um and we talk about gerles in completeness theem finally um and then we're going to Leap Forward you know chapters in the book 16
概念的边缘了。我在最开始就告诉过你们，这正是《哥德尔、埃舍尔、巴赫》所陈述的主旨，也就是宇宙在根本层面上是一个形式系统，它遵循某些确定性的规则，或者也许是概率性的规则，但无论如何都是一个形式系统。我们有这么一个标签，随手贴到某个东西上，也就是这个'我'的标签，而它其实掩盖了大量的细节。呃，就像……但以一种根本性的方式去理解它们，才是这门课所陈述的目标。呃，我还在纠结，因为我一直在持续地、概率性地调整这门课的走向。我在做决定，目前定下的计划是，在我们讲完《无门关》和哥德尔之后——你们会从K和我这儿领到一记奇怪的东方风味，呃，把禅和逻辑结合起来——然后我们会讲哥德尔不完备定理，终于讲到了。呃，接着我们要往前跳，你知道，跳过书里好几章，直接到第16章，
96:10
to a self-re and self self-re and self-re chapter um which will be kind of a little over the top first of all it's a very long chapter 57 Pages um but it's going to have this idea of a kind of typographical genetics and we're going to look at how genetics and protein folding and kind of the processes which kind of make us um correspond to some of the formal systems we've been talking about in a prescribed way then we're going to LEAP backwards and since what I've realized this course has become is really a Topic's course of a bunch of things um then leap to essentially brains and thoughts because the bottom line is Hof st's thinking hasn't been patched through all the way otherwise we would have it solved we like ohuh good thing we solve Consciousness like we can go on and do other things it's not solved um and there are these huge kind of gaps missing between like okay I maybe I buy it that the universe is a formal system but what on Earth does girdles in completeness theorem have to say about physical
一个关于自指、自指再自指的章节。呃，那一章会有点过头，首先它非常长，有57页。呃，但它会讲到一种叫'印刷术遗传学'的想法，我们要看看遗传学、蛋白质折叠，以及那些造就我们的过程，是如何以一种规定的方式对应到我们一直在讨论的那些形式系统的。然后我们要往回跳，因为我意识到这门课其实已经变成了一门专题课，讲一堆东西。呃，然后跳到大脑和思维，因为归根结底，Hofstadter的思路并没有被彻底贯通，否则我们早就把问题解决了。我们就会说'啊哈，太好了，我们解决了意识问题，可以去干别的了'。它并没有被解决。呃，而且中间还有一些巨大的空白，比如'好吧，我也许可以接受宇宙是一个形式系统，但哥德尔不完备定理到底跟物理系统有什么关系？'呃，那我们先把这些都放一边，
97:15
systems um and Let's ignore all that and then start talking about the brain kind of these meta structures and the brain and the mind and thinking and artificial intelligence that kind of wrapping up the course um but then of course I'm also thinking about possibly showing a movie um and that being Waking Life I don't know if any of you have seen it um and that just being like essentially my gift to you guys for working so hard in this class um but I might need to get permission slips so that's yet to come and uh I I kind of apologize for whatever slow pace today's lecture was um hopefully next one will be a little more exciting um but other than that read mumon and girdle uh I want I meant to do the dialogue that precedes the chapter today but obviously we can't we can do it for next lecture if you want um and then um yeah that that should be fun and read that that handout from I'm a strange Loop because I think it'll exate and it's really what I'll be lecturing from for a large bit of next
然后开始聊大脑，聊这些元结构，聊大脑、心智、思维和人工智能，用这些来给这门课收尾。呃，不过我当然也在考虑，也许放一部电影，就是《半梦半醒的人生》（Waking Life），不知道你们有没有人看过。呃，就当作我送给大家的礼物，感谢你们在这门课上这么努力。呃，但我可能得先弄到家长许可条，所以这事还没定。还有，呃，今天这节课节奏比较慢，我有点抱歉。呃，希望下一节会更精彩一些。呃，除此之外，去读《无门关》和哥德尔那部分。呃，我本来想今天把那一章前面的对话讲了，但显然来不及了，我们可以下节课讲，如果你们想的话。呃，然后，嗯，那应该会挺有意思的。还有读一下《我是个怪圈》里的那份讲义，因为我觉得它会很有帮助，而且下节课有很大一部分我就是照着它讲的。谢谢大家来上课，
98:16
lecture so thank you all for showing up and uh have a good day
呃，祝你们今天愉快
视频总结 · 一句话概括与核心要点
一句话概括
这节课先对上一讲"宇宙是分形"之类的过度推演做了澄清和"降温"，然后通过 TNT（类型化数论）的公理、ω-不完全性和量词练习说明"从少量公理递归生成真理树"的思想，最后用 Context Free 软件演示随机上下文无关文法如何用几行规则生成树、根系、河流般的有机形态，并引申到进化和生物形态的类比。
核心要点
- 意义理论没有单一形式化解释：讲者重申"snow is white"的意义是感知输入与大脑激活之间的"奇异同构"——从边缘检测到字母识别，再到每个人独有的语义网络（初雪的记忆、寒冷、滑雪）。这意味着"雪是白的"对每个人都不同，与哲学家/语言学家希望词语直接指向外部世界的"正统"意义观相冲突。
- 模式本身可能是意义的唯一普遍基础：借霍夫施塔特"送入太空的唱片"思想实验讨论外星文明能否理解巴赫。学生 Latif 认为只是噪音；讲者的回应是，即使剥离所有文化语义网络，"模式即美"是可数学描述、任何智能都可检测的东西。
- 香农熵不能区分"狗吐"与谢尔宾斯基三角：信息熵公式 −Σp(x)log p(x) 只按符号概率衡量信息；把一张噪点图和一张谢尔宾斯基垫片图都转成 0/1 串，熵值差别不大。相关事实：字母频率有熵结构，DNA 也呈现同样的语言统计规律（反驳"这只是人脑的局限"），Zipf 定律指出第 2、3 常见符号出现频率约为第 1 的 1/2、1/3，可用于判别未知文本是否是真语言。
- 算法信息量才是捕捉"规律性"的正确度量：一段几行的 L-系统程序（F−F++F−F 之类）即可生成谢尔宾斯基垫片，比逐像素编码短得多；就像描述钟摆用微分方程 θ̈ 而不是无限重复"来回、来回"。科学本质上就是这种"逆向工程"，把现象的算法内容压到最低。
- "宇宙是分形"是不严谨的说法，正确表述是"方程是尺度无关的"：讲者收回上讲的夸张：原子不是微型太阳系（量子力学的概率云与经典力学根本不同），所以规律并非在所有尺度相同。合理的说法是流体力学等方程尺度无关——手指划水与山峰穿云产生的都是冯·卡门涡街。哲学上可以放飞，但最终必须能被证明或在计算机上运行、并且可证伪。
- TNT 只需两个概念（0 和后继 S）就能表示所有自然数，但仍是 ω-不完全的：1729 写作 1729 个 S 后接 0。系统能逐条推出 0+S0=S0、0+SS0=SS0……的无限金字塔，却推不出概括它们的 ∀a:(0+a)=a，只能把它作为公理加入——这就是"ω-不完全"。0.999…=1 的证明（10x−x=9x=9）被用来说明我们对实数的心智模型会在最基础处出错。
- 五条皮亚诺公理（"精灵"寓言，第 216 页）+ 归纳法是整棵"真理树"的树干：0 不是任何数的后继；每个数都有后继；不同的数后继不同；0 加任何数等于该数；若 0 有性质 P 且每个数把 P 传给后继，则所有数有 P（归纳原理）。推理完全局部化，却能"涌现"出整个数论。
- 量词练习揭示 ~、∀、∃ 移动的微妙性：课堂上对 ~∀c:∃b:(SS0·b)=c 等六个变体逐一求值，讲者给出的答案序列为真、假、真、真、假（及一条真），符合书中"四真两假或四假两真"的提示；核心技巧是把 ~(…=…) 当作 ≠ 来读，用具体值（如 c=3、b=1）做特化。
- 上下文无关文法+随机性=有机形态：Curran 用 Context Free 演示：一条 `spiral` 规则画圆、上移、缩放 0.9 得到直线圆链；加 rotate 1 得螺旋；同名规则重复定义即按权重随机选择（权重 0.1 意味着约 1/1.1 对 0.1/1.1）；加入 `flip 90` 得蜿蜒线；分支时既继续原方向又派生新枝（size 0.3、rotate 45）就得到树、藤蔓、根系、河网。与确定性 L-系统（科赫雪花）不同，这是随机递归，产生不规则分形，类似用"混沌游戏"随机生成谢尔宾斯基三角。
- 参数空间有稳定区与不稳定区，可类比进化：在树规则中加一条概率 0.001、把尺寸乘 5 的规则，偶尔触发后该分支重新繁茂——像大灭绝后少数幸存物种辐射演化；概率提到 0.01–0.1 时系统失控，报错"shape got too big"。学生指出真实树木的形态还由向光性反馈（生长素）和自然选择塑造，因此不只是纯概率文法。
结论与值得注意的细节
- 讲者（Justin）为 TNT 一章的枯燥道歉，明确其目的只是让学生熟悉记号并理解"公理+递归规则→真理网"的机制，并坦承两小时讲座无法替代多年训练建立的神经连接。
- 希尔伯特式的警告被反复强调：形式系统只给出点、线等原始概念的逻辑关系，一旦附加"直线在平板上是直的"这类解释就会像非欧几何（球面大圆相交于两点）那样翻车——霍夫施塔特对"解释"的谨慎值得赞赏。
- 课程主旨仍是 GEB 的论题"宇宙在根本层面是形式系统（确定性或概率性）"；后续计划：读《无门关与哥德尔》和《我是一个怪圈》讲义 → 哥德尔不完全性定理 → 跳到第 16 章"自指与自复制"（57 页，含"排版遗传学"、蛋白质折叠）→ 回到大脑、思维与人工智能，可能放映电影《Waking Life》（需家长同意书）。
- 讲者承认霍夫施塔特的思路"没有完全打通"：从"宇宙是形式系统"到"不完全性定理对物理系统意味着什么"再到意识，中间存在巨大缺口——意识问题并未解决。
- 有趣细节：讲者悬赏 20 美元的挑战（关于 MIU 系统）至今无人应战；学生把随机树的图案联想到大脑皮层、血管、蚁群，讲者认为"分形递归算法编码于基因组"是合理假说，但"它必然如此、因为是最高效算法"仅是猜想。
核心句型 · 10
1. I want to start off with … but before I do that, I want to also …
“I want to start off actually with some apologies … but before I do that I want to also apologize for this chapter”
开场组织结构：先预告主题，再插入一个前置事项。适合演讲开头或汇报中「先说 A，但在此之前先交代 B」。仿写时注意 before I do that 承接自然。
2. the only reason I … is that …
“Really the only reason I signed it as reading is that you guys could get kind of a fundamental idea of …”
强调唯一动机的句式，is that 后接完整从句。用于解释决策理由、回应质疑。可仿写：The only reason I brought this up is that …
3. what X means to me is in some ways fundamentally different than what it means for you
“What snow is white means to me is in some ways fundamentally different than what it means for you”
名词性从句作主语与表语的对比结构。in some ways 起缓冲作用，避免绝对化。适合表达「同一事物对不同人意义不同」的观点。
4. let's grant them X and let's even assume that …
“Let's grant them the ability to hear and let's even assume that they can hear in a similar frequency range”
思想实验中逐步「授予」假设条件的表达。grant 在此意为「姑且承认」，even assume 进一步加码。适合辩论或推理时设定前提。
5. you can make a good case that …
“I think in many ways you can make a good case that there's something about pattern which is fundamentally beautiful”
make a case that = 有力论证某观点。比 argue 更正式、更强调「论据充分」。可用于学术写作或讨论中提出可辩护的主张。
6. what this fundamentally boils down to is …
“What this fundamentally boils down to is if you were to describe either this record or … a picture”
boil down to 把复杂问题归结为要点。what … is 分裂句结构突出结论。用于总结、化繁为简时的过渡。
7. it's not like everyone … would say X; a lot of people would probably say Y
“It's not like everyone in the scientific Community if you went to them and said … a lot of people would probably say no obviously not”
it's not like … 是口语中否定错误预设的常用开头，「并不是说……」。后接对比句纠正听者可能的误解。
8. at the end of the day you have to either … or …
“At the end of the day you have to either be able to make it work rigorously with proof or by execution on a computer”
at the end of the day = 归根结底；either … or … 给出两条硬性标准。适合在允许发散讨论后收回到底线要求。
9. unless you're really planning on …, you're not going to have to …
“Unless you're really planning on a life as a logician you're not going to have to spend a lot of time manipulating formal systems”
unless 引导的条件句 + 否定将来时，表达「除非……否则无需……」。用于安抚听众、界定内容的必要程度。
10. it makes you wonder like does X … or is it …
“It makes you wonder like does nature use these sort of context free grammars … or is it like a Linden Meer system”
提出开放式疑问的口语句式。makes you wonder 引出思考，后接两个并列的选择疑问。适合演示后引导讨论。
词汇精讲 · 120 · 按出现顺序
sobering
/ˈsoʊbərɪŋ/
adj.
0:00
令人清醒的、发人深省的（常搭配 a sobering note/thought）
typographical
/ˌtaɪpəˈɡræfɪkəl/
adj.
1:17
印刷的、排版的；此处指纯符号层面的形式系统
axioms
/ˈæksiəmz/
n.
1:17
公理（无需证明而被接受的基本命题）
mish mash
/ˈmɪʃ mæʃ/
n.
1:17
大杂烩、杂乱的混合（口语，常作 mishmash）
isomorphism
/ˌaɪsəˈmɔːrfɪzəm/
n.
3:36
同构（两个结构之间保持关系的一一对应）
hone in on
phr.
3:36
聚焦于、锁定（home in on 的常见变体）
edge detection
n.
3:36
边缘检测（计算机视觉/视觉神经处理的基本步骤）
crystallized
/ˈkrɪstəlaɪzd/
adj.
4:50
结晶的；使成形的
upstanding
/ʌpˈstændɪŋ/
adj.
4:50
正直的、体面的（an upstanding citizen 正派公民，此处为幽默用法）
blizzard
/ˈblɪzərd/
n.
6:11
暴风雪
stumbled Upon
phr.
6:56
偶然发现、无意中碰到
concentric
/kənˈsentrɪk/
adj.
6:56
同心的（concentric grooves 同心圆凹槽）
artifact
/ˈɑːrtɪfækt/
n.
6:56
人工制品、文物
reverse engineer
v. phr.
8:31
逆向工程、反推（从成品推出原理或设计）
garbled
/ˈɡɑːrbəld/
adj.
8:31
混乱不清的、失真的（信息、声音）
coones
/kəˈhoʊniz/
n.
8:31
（俚语，原为 cojones）胆量、勇气
triumphant
/traɪˈʌmfənt/
adj.
9:49
凯旋的、欢庆胜利的
awe
/ɔː/
n.
9:49
敬畏、惊叹
decipher
/dɪˈsaɪfər/
v.
11:04
破译、解读
Despair
/dɪˈsper/
n.
11:04
绝望
tangents
/ˈtændʒənts/
n.
12:15
离题、岔开的话题（go off on a tangent 跑题）
entropy
/ˈentrəpi/
n.
12:15
熵（信息论中度量不确定性/信息量的指标）
boils down to
phr.
13:39
归结为、本质上就是
emerge
/ɪˈmɜːrdʒ/
v.
15:02
涌现、显现（emergent 涌现的）
power law
n.
15:02
幂律（一个量与另一个量的幂成正比的分布规律）
discern
/dɪˈsɜːrn/
v.
16:17
辨别、识别出
intelligible
/ɪnˈtelɪdʒəbəl/
adj.
16:17
可理解的、可解读的
rigorous footing
phr.
16:17
严格的基础、扎实的根据
seething
/ˈsiːðɪŋ/
adj.
17:39
翻腾的、沸腾般的（此处形容杂乱噪点）
ad infinitum
/ˌæd ɪnfɪˈnaɪtəm/
adv.
17:39
无限地、永无止境地（拉丁语）
audit a class
phr.
17:39
旁听课程（不计学分）
excessively
/ɪkˈsesɪvli/
adv.
19:19
过度地、过分地
pendulum
/ˈpendʒələm/
n.
19:19
单摆、钟摆
iteration
/ˌɪtəˈreɪʃən/
n.
19:19
迭代、重复执行
recursive
/rɪˈkɜːrsɪv/
adj.
20:45
递归的（规则调用自身）
algorithmic
/ˌælɡəˈrɪðmɪk/
adj.
21:23
算法的（algorithmic information 算法信息）
cellular automaton
/ˈseljələr ɔːˈtɑːmətɑːn/
n.
21:56
元胞自动机（网格上按局部规则演化的离散模型）
emulate
/ˈemjəleɪt/
v.
21:56
模拟、仿效
deterministic
/dɪˌtɜːrmɪˈnɪstɪk/
adj.
21:56
确定性的（同样输入必得同样输出）
group think
n.
23:23
群体思维（为求一致而放弃独立判断，常作 groupthink）
homogeneous
/ˌhoʊməˈdʒiːniəs/
adj.
23:23
同质的、单一的
condensation
/ˌkɑːndenˈseɪʃən/
n.
23:23
凝结、冷凝
getting caught up in the moment
phr.
24:36
被当下气氛冲昏头脑、一时激动
well founded
adj.
24:36
有充分根据的
patching up
phr.
24:36
修补、弥合（分歧）
topology
/təˈpɑːlədʒi/
n.
25:39
拓扑结构、拓扑学
morphology
/mɔːrˈfɑːlədʒi/
n.
25:39
形态、形态学
scale free
adj.
26:48
无标度的（在不同尺度下形式不变）
vortices
/ˈvɔːrtɪsiːz/
n.
26:48
涡旋（vortex 的复数）
pierce
/pɪrs/
v.
26:48
刺穿、穿透
advocate
/ˈædvəkeɪt/
v.
27:35
倡导、主张
formalism
/ˈfɔːrməlɪzəm/
n.
27:35
形式体系、形式化方法
lofty
/ˈlɔːfti/
adj.
27:35
高远的、崇高的（有时含「不切实际」意味）
for most intensive purposes
phr.
27:35
（口误，应为 for all intents and purposes）实际上、在各种意义上
falsifiability
/ˌfɔːlsɪˌfaɪəˈbɪləti/
n.
28:47
可证伪性（波普尔提出的科学理论判据）
metaphysical
/ˌmetəˈfɪzɪkəl/
adj.
28:47
形而上学的
anal
/ˈeɪnəl/
adj.
30:18
（口语）吹毛求疵的、过分讲究细节的
successor
/səkˈsesər/
n.
30:18
后继（数论中 n 的后继为 n+1）
meta language
n.
30:18
元语言（用来谈论另一种语言的语言）
dull
/dʌl/
adj.
31:39
乏味的、显而易见到无趣的（原文拼作 dll）
at our disposal
phr.
31:39
可供我们支配、手头可用
Encompass
/ɪnˈkʌmpəs/
v.
31:39
包含、涵盖
utterly
/ˈʌtərli/
adv.
33:05
完全地、彻底地
show off
phr. v.
33:05
炫耀、露一手
postulate
/ˈpɑːstʃələt/
n.
36:19
公设、假定（几何学中与公理同义）
great circles
n.
37:25
大圆（球面上以球心为圆心的圆，球面几何中的「直线」）
stick to
phr. v.
38:39
坚守、不偏离
trivial
/ˈtrɪviəl/
adj.
38:39
琐碎的、微不足道的；数学中指显而易见的
propositional calculus
n.
39:52
命题演算（处理命题之间逻辑关系的形式系统）
browning points
n.
39:52
（口误，应为 brownie points）讨好分、印象分
specialization
/ˌspeʃələˈzeɪʃən/
n.
41:33
特化（逻辑规则：从全称命题推出具体实例）
divisible
/dɪˈvɪzəbəl/
adj.
41:33
可整除的
last man standing
phr.
43:14
最后的幸存者、坚持到最后的人
cumbersome
/ˈkʌmbərsəm/
adj.
45:33
笨重的、繁琐的
tripped up
phr. v.
45:33
绊倒；（比喻）出错、被难住
riddle me this
phr.
49:22
「猜猜这个」（源自蝙蝠侠中谜语人的口头禅，出题时的幽默用语）
symbol shunting
n.
50:24
符号搬移（侯世达用语，指机械地移动/变换逻辑符号）
pass off
phr. v.
51:22
移交、把（场子）交出去
existential quantifiers
n.
53:12
存在量词（∃，「存在」）
logician
/loʊˈdʒɪʃən/
n.
54:50
逻辑学家
torturing himself
phr.
55:58
折磨自己（此处指苦思冥想）
condense
/kənˈdens/
v.
55:58
压缩、浓缩
endow
/ɪnˈdaʊ/
v.
57:12
赋予（endow sb with sth）
relays
/ˈriːleɪz/
v.
58:23
传递、转达
inference
/ˈɪnfərəns/
n.
59:49
推理、推论（rules of inference 推理规则）
emergent
/ɪˈmɜːrdʒənt/
adj.
59:49
涌现的（整体呈现出部分所没有的性质）
context free grammar
n.
61:04
上下文无关文法（形式语言理论中的一类产生式文法）
production rules
n.
61:04
产生式规则（文法中把一个符号替换为符号串的规则）
increment
/ˈɪŋkrəmənt/
v.
62:25
递增、增加一个单位
render
/ˈrendər/
v.
62:25
渲染、生成图像
Meandering
/miˈændərɪŋ/
adj.
65:31
蜿蜒曲折的、漫无目的的
tangent
/ˈtændʒənt/
n.
67:58
切线；切线方向
out of control
phr.
67:58
失控
feasible
/ˈfiːzəbəl/
adj.
69:37
可行的
sparse
/spɑːrs/
adj.
71:31
稀疏的（原文拼作 spark）
stochastic
/stəˈkæstɪk/
adj.
79:58
随机的、概率性的
on the Fly
phr.
79:58
即时地、在运行过程中临时（决定）
rigid
/ˈrɪdʒɪd/
adj.
79:58
刚性的、僵硬的、严格的
prepackaged
/priːˈpækɪdʒd/
adj.
81:24
预先准备好的、现成的
catastrophic
/ˌkætəˈstrɑːfɪk/
adj.
84:36
灾难性的
strains
/streɪnz/
n.
84:36
（生物）品系、株系
diverge
/daɪˈvɜːrdʒ/
v.
84:36
分化、分岔
propagated
/ˈprɑːpəɡeɪtɪd/
v.
84:36
繁殖、传播、扩散
genetic diversity
n.
87:18
遗传多样性
miraculously
/mɪˈrækjələsli/
adv.
87:18
奇迹般地
hurricane force winds
n.
88:39
飓风级强风
photot taxes
/ˌfoʊtoʊˈtæksɪs/
n.
88:39
趋光性（phototaxis；讲者此处应指植物的 phototropism 向光性）
oxin
/ˈɔːksɪn/
n.
89:54
生长素（auxin，调节植物生长方向的激素）
selected against
phr.
91:05
被自然选择淘汰
cerebral cortex
/səˈriːbrəl ˈkɔːrteks/
n.
91:05
大脑皮层
took hold
phr. v.
92:10
站稳脚跟、确立下来
Paradigm
/ˈpærədaɪm/
n.
93:25
范式、模式
hypothesize
/haɪˈpɑːθəsaɪz/
v.
93:25
假设、提出假说
Spiel
/ʃpiːl/
n.
93:25
（口语）一套说辞、长篇讲话
at the brink of
phr.
95:00
处于……的边缘
thesis
/ˈθiːsɪs/
n.
95:00
论点、主旨
over the top
phr.
96:10
过头的、夸张的
protein folding
n.
96:10
蛋白质折叠
bottom line
n.
96:10
要点、底线、归根结底的结论
permission slips
n.
97:15
（家长）许可单、同意书
理解自测 · 11 题
1. 讲者为什么说 TNT（印刷数论）比 MIU 和 pq 系统「更有优势」？
因为 TNT 编码的对象是我们自认为有所了解的东西——自然数及其性质，而 MIU 和 pq 系统只是无明确外部指涉的符号游戏。讲者在开场回顾（第 1 段）指出，TNT 让人看到：从几条公理种子出发、反复应用递归规则，就能生成一整张字符串之网，这些字符串经解释后恰好是关于数的真命题。这正是布置这一章的目的——熟悉记号，并体会「公理+递归规则→真理之树」的思想，尽管大家读起来并不愉快。
2. 太空唱片思想实验中，讲者最终提出了什么关于「意义」的普适成分？
讲者提出「模式（pattern）本身即美，且是意义中唯一普适的成分」。在第 6–10 段的讨论中，学生 Latif 认为外星人只会听到噪音，Max 指出前提是外星人有听觉；讲者逐步授予假设后，承认人声歌词、钢琴等文化知识无法传递，但坚持即便剥离个体语义网络，音乐中的模式仍可被任何智能体探测，且可用数学描述。这把上一讲的「意义即同构」推进为「意义的核心是可被数学化的结构」。
3. 讲者如何反驳「语言中的统计规律只是人脑局限的产物」这一说法？
讲者用 DNA 作反例。第 12 段中学生 Li 提出，语言呈现相同结构是因为人脑只能处理这些；讲者回应说，人们对 DNA 序列做过同样的频率分析，DNA 作为一种「语言」也呈现出相同的模式（如齐普夫式的幂律分布），而 DNA 显然不是人脑创造的。因此这些统计规律不是人类认知的副产品，而是更普遍的信息组织方式，这也为后文「模式是普适的」提供了经验支持。
4. 为什么香农熵无法区分「随机噪点」和「谢尔宾斯基三角」？这个局限引出了什么新概念？
香农熵只度量比特串的统计不确定性，不度量结构。第 14 段指出，把一幅随机噪点图和一幅谢尔宾斯基三角图都转成 0/1 串后做熵分析，两者的数值可能相差无几，尽管后者有明显规律和「意义」。这一局限引出了算法信息（Kolmogorov 复杂度）：谢尔宾斯基三角可用几行 L-system 代码生成，其「最短描述」极短，而噪点无法压缩。讲者进一步把这推广为科学的本质——用最短的方程（如单摆方程 θ̈）编码现象的规律，即「科学就是逆向工程/压缩」。
5. 讲者为什么在本讲中反复「道歉」并「泼冷水」？他修正了上一讲的哪个具体说法？
讲者担心上一讲被课堂气氛带动，向学生灌输了「令人兴奋但缺乏根据」的想法，最典型的是「宇宙是分形」。第 20–22 段中他明确指出：科学界大多数人会否定这一说法，因为原子并非微缩太阳系——量子力学的概率云与经典力学的行星轨道完全不同，规则不跨尺度重复。更严谨的表述是某些方程是「无标度的」（scale-free），如流体力学中手指划水与山峰绕云都产生冯·卡门涡街。他借此确立课程方法论：可以天马行空，但最终必须落到证明或可运行程序上，且必须具备可证伪性。
6. 什么是「ω-不完全」？它与 0.999…=1 的例子在论证上有什么共同点？
ω-不完全指 TNT 能逐一证明 0+0=0、0+S0=S0、0+SS0=SS0……这无穷多个具体命题，却无法证明「对所有 a，0+a=a」这条概括命题，除非把归纳原理作为公理加入（第 25–26 段）。0.999…=1 的例子（第 27–28 段）则展示同学「这不可能」的震惊反应。两者的共同点是揭示「心智模型」与「形式系统」之间的落差：我们直觉上认为显然的事，形式系统未必能推出；我们自以为理解的对象（实数），也会在最基础处颠覆直觉。讲者由此说明形式化的价值——它检验我们的定义是否真的涵盖了我们想要的东西。
7. 从欧几里得到希尔伯特，讲者讲述的几何学历史想说明什么教训？
教训是「解释」（interpretation）是危险的，应严守形式体系。第 29–31 段回顾：欧几里得认为公理是自明真理且涵盖全部几何知识；但萨凯里、高斯等人发现打破第五公设（平行公设）后可得到自洽的非欧几何，例如球面上以大圆为「直线」时任意两条都相交。希尔伯特由此主张公理只规定点、线之间的逻辑关系，一旦把「线」解释为平板上的直线而非球面上的曲线，就会出错。讲者称赞侯世达在书中同样警告读者不要急于解释形式符号，这与 TNT 练习中「量词顺序不同意义完全不同」的困难相呼应。
8. Curran 的 Context Free 演示中，仅改动哪一行就把「蜿蜒曲线」变成了「树」？这说明了什么？
在带 flip 90 的第二条 spiral 规则中，额外加入一个不带翻转的 spiral 调用（第 52 段）。原本翻转规则只是改变旋转方向，加上这一行后，每次触发时一支沿切线岔出、另一支继续原方向，于是曲线变成了分叉的树。这说明极少的规则改动能带来形态上的质变，也直观展示了「简单递归规则→复杂有机形态」的主题。后续通过调整分叉概率（0.1→0.5）、缩放比（0.9/0.3）、角度（45°/60°/90°）与旋转，同一套文法被学生联想为树、河流、藤蔓、根系、血管与大脑皮层。
9. Curran 用「size 5」规则类比演化中的什么现象？参数从 0.003 调到 0.01 时发生了什么？
他类比的是大灭绝事件后少数幸存谱系的辐射扩散（第 64–67 段）。正常规则中 size 恒小于 1，系统必然收敛；「size 5」以极小概率把某分支放大 5 倍，相当于某个物种在灾变后突然获得巨大「权重」并大量繁殖分化。放大概率为 0.003 时系统偶有爆发但整体稳定；调到 0.01 时形状不断增大直至程序报错「shape got too big」，相当于陨石频繁撞击、遗传多样性被反复摧毁。这展示了参数空间中的稳定区与不稳定区——微小参数变化导致系统行为质变，是动力系统中分岔现象的直观例子。
10. 如果有人反驳说「真实的树不是随机文法生成的，因为树会朝光生长」，讲者会如何回应？
讲者实际上已经主动承认了这一点。第 68–69 段中 Justin 引导学生说出「阳光」，并解释植物具有向光性（他误说为 phototaxis）：通过生长素（auxin）的分布使背光侧细胞伸长，树能动态改变分叉方向和概率。他明确说「这不完全只是一个概率性的上下文无关文法，它必须有某种反馈机制」，而且演化层面还会改写规则本身。因此讲者的立场并非「树就是随机文法」，而是「随机文法抓住了分枝形态的生成逻辑，真实生物在此基础上叠加了环境反馈与自然选择」。学生随后补充：底部分枝会被遮光而被选择淘汰，这正是规则本身被演化筛选的例子。
11. 把本讲「宇宙是形式系统」的论点放到意识问题上，讲者认为这一论证链条完整吗？这对读《GEB》有什么启示？
讲者认为不完整，并在第 73–74 段坦率承认。他指出全书主旨是宇宙在根本层面是一个（确定性或概率性的）形式系统，「我」只是掩盖了大量底层细节的标签；但侯世达的思路「并未贯通」，否则意识问题早已解决。具体空白在于：即便接受宇宙是形式系统，哥德尔不完备定理对物理系统究竟意味着什么，从形式系统到大脑、心智的过渡缺乏严格论证。这与本讲反复强调的「可证伪性」「必须落到证明或程序」的标准一致——讲者把 GEB 当作激发思考的框架而非已证成的理论。启示是：读 GEB 时应区分「有严格基础的部分」（TNT、不完备定理、递归生成）与「类比性的推广」（宇宙分形、意识即怪圈），后者需要保持批判性。
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