视频库 / NO.094
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The Limits of Understanding | World Science Festival
节目发布 2014-12-14
本期追问 · 点击跳到视频对应位置
凡是真的，都能被证明吗？数学是发现，还是发明？意识只是物质的一种状态吗？人有自由意志吗？
章节 · 点击跳转视频
0:05
开场：理解的极限与数学的地位
2:42
短片：希尔伯特的号召与哥德尔生平
7:32
嘉宾介绍：四位不同学科的思想者
10:13
维也纳学派与沉默的哥德尔
15:45
两条不完备性定理讲的是什么
19:51
柯尼斯堡的咕哝与冯·诺依曼的领悟
22:31
Chaitin：Ω数与数学的无穷复杂性
26:30
柏拉图主义、形式主义与逻辑主义
33:48
Minsky：公众对量子与演化的误解
40:43
数学为何「不可思议地有效」
53:13
意识是一个谜还是26个问题
62:34
自由意志与哥德尔式定理的野心
65:36
哥德尔之后，纯数学到底是什么
73:39
水母思想实验：数学是发现还是发明
86:40
收尾：三种「理解的极限」
本期小问 · 档案清单
——
凡是真的，都能被证明吗？
——
数学是发现，还是发明？
——
意识只是物质的一种状态吗？
——
人有自由意志吗？
本期讲者
Gregory Chaitin数学家、计算机科学家，算法信息论创始人之一，Ω（停机概率）数的发现者。长期任职于 IBM Watson 研究中心，近年提出用「变异软件的演化」建模生命的元生物学方向。
丽贝卡·戈尔茨坦哲学家、小说家，专长科学哲学与数理逻辑。著有哥德尔传记《不完备性：库尔特·哥德尔的证明与悖论》及小说《上帝存在的36个论证》，麦克阿瑟奖得主。
Mario Livio天体物理学家，曾任太空望远镜科学研究所公共推广主任，研究超新星与暗能量。著有《黄金比例》《上帝是数学家吗？》，主张数学是「发明概念、发现关系」的交织。
01开场：理解的极限与数学的地位
0:05
[Music] well good evening ladies and gentlemen and welcome to tonight's program tonight we're going to consider a difficult subject I don't think you're surprised at that we're going to consider the limits of human understanding with a particular focus on mathematics now this is a science festival and science is the most reliable way of gaining knowledge and understanding of the world in the last four centuries we've seen great advances in our understanding of both the physical and the biological worlds but what are the limits of this understanding this is a question philosophers have pondered over the centuries traditionally worrying about areas such as the validity of induction that is inferring generalizations and general laws or the reliability or even the validity of our senses in generating information about the world now tonight we are likely to touch
[音乐] 好，女士们、先生们，晚上好，欢迎来到今晚的节目。今晚我们要探讨一个很难的主题——我想这并不会让各位感到意外——我们要探讨人类理解力的极限，尤其聚焦于数学。这是一个科学节，而科学是我们获取世界知识与理解最可靠的途径。在过去四个世纪里，我们对物理世界和生物世界的理解都取得了巨大的进展。但这种理解的极限在哪里？这是几个世纪以来哲学家们一直反复思索的问题，传统上他们担忧的是诸如归纳法的有效性——也就是推导出概括与普遍规律——或者我们的感官在生成关于世界的信息时是否可靠、甚至是否有效。今晚我们可能会触及
1:25
on such philosophical issues but our main focus will be on mathematics and on mathematical logic because mathematics and mathematical thinking underpins much that is important in science it's been the Cornerstone of physics since the 17th century particularly since Isaac Newton indeed some aspects of physics particularly 20th century physics like quantum mechanics um theories of the ultimate structure of matter only make sense in terms of mathematics they are not part of our Common Sense World Turning even to biology in the last 50 years it's gradually being realized that DNA is essentially a digital information storage device and it's become clear that the way information is managed in living systems will be key to biological understanding it's my own view view that given the complexity of biological information uh and it management I wonder if we may not end up in strange places analogous to quantum mechanics in physics but as well as underpinning science mathematics and logic appears to
这类哲学议题，但我们的主要焦点会放在数学和数理逻辑上。因为数学和数学思维支撑着科学中许多重要的东西。自17世纪以来，尤其是自艾萨克·牛顿以来，它一直是物理学的基石。事实上物理学的某些方面，特别是20世纪的物理学，比如量子力学，还有关于物质终极结构的理论，只有用数学才讲得通——它们并不属于我们的常识世界。再说到生物学，在过去50年里人们逐渐意识到，DNA本质上是一种数字化的信息存储装置，而且越来越清楚的是，生命系统中信息的管理方式将是理解生物学的关键。我个人的看法是，考虑到生物信息及其管理的复杂性，我在想我们最终会不会走到某些奇特的境地，类似于物理学中的量子力学。但除了支撑科学之外，数学和逻辑看起来还是
02短片：希尔伯特的号召与哥德尔生平
2:42
be the most secure sphere of human understanding it's based on a firm structure of axioms and of proofs it seems to be a most reliable and self-contained part of human knowledge but tonight we shall see that this is not quite as straightforward as it seems Godel in the first half of the last century showed that mathematics was not always provably consistent in fact some mathematical facts can never be proven to be true or to be false they are mathematical paradoxes analogous to literary paradoxes may be therefore the relationship between truth and Mathematics is more tenuous than we thought this ladies and gentlemen is our territory tonight so let's begin with a video about [Music] goo in 1920 amidst the wreckage of War torn Europe David Hilbert the most influential mathematician of his time issued a call to Scholars everywhere to prove the consistency of mathematics the country um is in ruins in a lot of ways and so part of what they're seeking is um emancipation from that and they're seeking a a vehicle to the Future where
人类理解中最稳固的领域，它建立在公理和证明的坚实结构之上，似乎是人类知识中最可靠、最自洽的部分。但今晚我们会看到，事情并不像看上去那么简单。上个世纪前半叶，哥德尔证明了数学并不总是可以被证明为一致的。事实上，有些数学事实永远无法被证明为真或为假，它们是数学上的悖论，也许类似于文学中的悖论。因此，真理与数学之间的关系，也许比我们原以为的要脆弱得多。女士们、先生们，这就是我们今晚要涉足的领域。那么，就让我们从一段关于哥德尔的视频开始吧 [音乐]。1920年，在战后满目疮痍的欧洲，当时最具影响力的数学家大卫·希尔伯特向各地的学者发出号召，要求证明数学的一致性。这个国家在很多方面都是一片废墟，所以他们所寻求的一部分，是从中获得解放，他们在寻找一条通往未来的道路，在那里
4:18
science and math was going to be the promise of um of of a golden era a lot of the mathematicians at the time really believed that they would be able to prove what Hilbert asked of them which is that mathematics and arithmetic was going to be provably consistent and so it was really quite a blow when a young and and softspoken gentlemanly student from uh the university comes and declares that actually it's not true it was a 24-year-old doctoral student named Kurt girle who exposed the limitation of mathematics his ideas would shake the very foundations of mathematics and philosophy the culmination of Good's uh ideas are called the incompleteness theorems there are mathematical facts that we can never prove our true or false there is no such thing as a mathematical Theory of Everything and they are one of the most significant uh results um ever to be proven in mathematics go wasn't completely well he was a severe hypochondriac bordering on paranoid schizophrenia he had abouts in mental institutions numbers mathematics he believed was real he wasn't so sure about the truth of his senses and perception of reality by 1939 with Europe descending into a second world war girdle fled Vienna for Princeton he was invited to
科学和数学将带来一个黄金时代的希望。当时很多数学家真心相信，他们能够证明希尔伯特所要求的东西，也就是数学和算术是可以被证明为一致的。所以，当一位来自那所大学的、年轻而说话轻声细语、颇有绅士风度的学生站出来宣称这其实并不成立时，这确实是相当沉重的一击。是一位24岁的博士生库尔特·哥德尔揭示了数学的局限性。他的思想将撼动数学与哲学的根基。哥德尔思想的顶点被称为不完备性定理：存在一些数学事实，我们永远无法证明它们是真还是假；根本不存在所谓的数学万有理论。它们是数学史上被证明出来的最重要的结果之一。哥德尔的状态并不太好，他是个严重的疑病症患者，接近偏执型精神分裂。他曾几度进出精神病院。他相信数、数学是真实的；但对于自己的感官以及对现实的知觉是否为真，他就没那么确信了。到1939年，随着欧洲滑向第二次世界大战，哥德尔逃离维也纳前往普林斯顿。他受邀
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become a permanent member of The Institute for advanced study good in some ways was very charming and very gentlemanly but he was also are extremely reticent and uh very withdrawn at times he had this deep Devotion to truth and he was his own thinker in that sense nothing would sway him and one of his very few friends was was Einstein Einstein will being very different person also had a similar incredible independence of Mind there must have been a way in which they they respected each other on those grounds where they could see in each other you know that complete independence of mind though no one understood the nature of their friendship or the depth of their conversation toward the end of his life Einstein felt that his own work no longer meant much he came to the Institute merely to have the privilege of walking home with goodle in the end paranoia and an obsessive fear of being poisoned led goodal to starve himself he died in a Princeton hospital weighing just 65 lb at the age of 72 I think in his fear and anxiety about his physical life the physical pain he felt the paranoia about his health his dream was that it was a reward to in the next life be presented with a a pure more platonic reality he was a very tragic [Music] figure so could I ask ask the panelists to come out on stage
成为高等研究院的永久成员。哥德尔在某些方面非常有魅力、非常有绅士风度，但他同时也极其寡言，有时非常孤僻。他对真理有着深切的执着，在这个意义上他是个独立的思考者，没有什么能动摇他。他为数不多的朋友之一就是爱因斯坦。爱因斯坦虽然是个非常不同的人，却同样有着惊人的思想独立性。他们之间一定有某种彼此欣赏的地方，正是在这一点上，他们能在对方身上看到那种彻底的思想独立。尽管没有人真正理解他们友谊的性质，也不知道他们交谈的深度，但在爱因斯坦生命的晚期，他觉得自己的工作已经没有多大意义了，他到研究院来，只是为了享有和哥德尔一起走路回家的殊荣。最终，偏执以及对被下毒的强迫性恐惧，让哥德尔把自己活活饿死。他在普林斯顿的一家医院去世，体重只有65磅，享年72岁。我想，在他对自己肉身生命的恐惧和焦虑中，在他所感受到的身体痛苦中，在他对健康的偏执里，他的梦想是：作为一种奖赏，在来世能够面对一个更纯粹的、更柏拉图式的实在。他是个非常悲剧性的 [音乐] 人物。那么，能否请几位嘉宾上台
03嘉宾介绍：四位不同学科的思想者
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[Music] [Applause] please so I'm just going to say a few words about each of them um starting with the gentleman immediately to my left uh Gregory chaon he is a mathematician and computer scientist Chief Architect of algorithmic information Theory which relates to the Computing of complexity he's also the discoverer of the Omega number and we may hear more about that later he's at the Forefront of the emerging metabiology field which regards Life as a sort of biochemical software he's the author of a dozen books about mathematics and philosophy including thinking about godal and cheering he worked for many years at the IBM Watson Research Center in New York where he is currently a Meritus just to his left Rebecca Goldstein a philosopher and author as a philosopher her areas of speciality are philosophy of science mathematical logic and 17th century
[音乐] [掌声]，有请。我先简单介绍一下他们每一位，从我左手边紧挨着的这位先生开始。格雷戈里·蔡廷，他是数学家和计算机科学家，算法信息论的主要奠基人，该理论与计算的复杂性有关。他也是欧米茄数的发现者，稍后我们可能会听到更多关于这个的内容。他站在新兴的元生物学领域的前沿，该领域把生命看作某种生物化学软件。他著有十几本关于数学与哲学的书，包括《思考哥德尔与图灵》。他曾在纽约的IBM沃森研究中心工作多年，目前是那里的荣休研究员。他左边这位是丽贝卡·戈尔茨坦，哲学家兼作家。作为哲学家，她的专长领域是科学哲学、数理逻辑和17世纪
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rationalism her first novel um the uh Mind Body problem more novels followed but among her works of non-fiction is the biography of incompleteness the proof and Paradox of Kurt godal chosen by discover magazine as one of the 10 best science books of 2006 her latest novel and I've read this one is 36 Arguments for the existence of God a work of fiction she has been a professor of both philosophy and of writing and is currently in the department of psychology at Harvard to her left Mario Livio senior astrophysicist head of the uh Office of Public Outreach at the Space Telescope Science Institute responsible for the Hubble Space Telescope an author of uh award-winning books including the golden uh ratio the story of by the world's most astonishing number and is God a mathematician selected by the Washington Post as one of the best books for 2009 his research focuses on supernova explosions and the nature of dark energy and finally on his left Marvin Minsky one of the pioneers of artificial intelligence um who's made numerous contributions to the fields of AI
理性主义。她的第一部小说是《心身问题》，之后又有更多小说问世。而在她的非虚构作品中，有一本是《不完备性：库尔特·哥德尔的证明与悖论》，被《发现》杂志评为2006年十大最佳科学书籍之一。她最新的小说——这本我读过——是《上帝存在的36个论证：一部小说》。她曾担任哲学和写作两方面的教授，目前在哈佛大学心理学系任教。她左边这位是马里奥·利维奥，资深天体物理学家，太空望远镜科学研究所公共推广办公室主任，该所负责哈勃太空望远镜。他也是多本获奖著作的作者，包括《黄金比例：世界上最令人惊叹的数字的故事》，以及《上帝是数学家吗？》——后者被《华盛顿邮报》选为2009年最佳图书之一。他的研究聚焦于超新星爆发以及暗能量的本质。最后，他左边这位是马文·明斯基，人工智能的先驱之一，在人工智能领域做出了众多贡献
04维也纳学派与沉默的哥德尔
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cognitive psychology mathematics Linguistics Robotics and Optics um he has talked about the human intellectual structure and function in his books the emotion machine and the Society of Mind and also buil built the snark the first neural network simulator his other inventions include robotic devices and he is the recipient of numerous prizes and awards for his work Dr Minsky is a professor at MIT where he co-founded the artificial intelligence lab a great lineup and welcome to all four of you I want to start with Rebecca if I could I wonder if you could tell us a little bit um about godell and also the environment in which he was living the intellectual environment in Europe following the first world war well he came um he was born in uh BR a a cck town uh his family spoke German um and he came to um to Vienna uh in the 20s and Vienna um at that time was a a city of extraordinary intellectual and artistic uh fervent um tremendous amount of uh rethinking from the foundations up uh things had not worked out so very
认知心理学、数学、语言学、机器人学和光学，嗯，他在《情感机器》和《心智社会》这两本书里探讨过人类智力的结构与功能，还造出了 SNARC——第一台神经网络模拟器，他的其他发明还包括各种机器人装置，并因这些工作获得了众多奖项和荣誉。明斯基博士是麻省理工学院的教授，他在那里共同创立了人工智能实验室。阵容非常强大，欢迎你们四位。我想先从丽贝卡开始，如果可以的话，我想请你稍微讲一讲哥德尔，以及他所处的环境，也就是一战之后欧洲的思想氛围。嗯，他来自——他出生在布尔诺，一个捷克的城镇，家里讲德语，二十年代他来到了维也纳。当时的维也纳是一座知识和艺术都异常活跃的城市，有大量从根基上重新思考一切的努力。因为事情并没有发展得很
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well World War I um and there was uh a just a a a sense of having to begin a new and one of the um centers of this thinking was a a place that we know I actually kind of think of Vienna as sort of the New York of its day you know it was kind of um everything was happening there and and and very much um not in sync with the rest of the country well you you were vienes and uh like you're a New Yorker and it was something that we now call the Vienna Circle um and it was a group of we now associate with a movement in philosophy and in science called logical positivism um and this was a a movement that was trying to again think things through from the bottom up um make things very very clear so they had a Criterion for meaningfulness uh that the meaning of a proposition is given by the conditions that would verify it uh and if there are propositions that no experience would count for or against it was a meaningless proposition so meaning meaninglessness was a term of damnation that they used very very often and this
顺利，一战嘛，当时有一种必须重新开始的感觉。而这种思考的中心之一，就是一个我们都知道的地方。其实我常把维也纳看作那个时代的纽约，你知道，什么事都在那儿发生，而且跟这个国家的其他地方非常不同步。就像你是维也纳人，正如你是纽约客一样。那个圈子就是我们现在所说的维也纳学派，它是一群人，我们现在把它和哲学与科学中的一场运动联系在一起，叫做逻辑实证主义。这场运动同样想把一切从头到尾重新想清楚，把事情弄得非常非常明确。所以他们有一个意义标准：一个命题的意义由能够验证它的条件给出；如果有些命题没有任何经验能支持或反驳，那它就是无意义的命题。所以“无意义”是他们非常非常常用的一个贬斥性词汇。这
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was a a center uh this circle of um uh there were scientists there were philosophers there were mathematicians and there were two young uh graduate students uh uh they had been invited by a mathematician who was a a a leader of the Vienna Circle um Hans Han um and he had two graduate students one of whom was named K G and uh who was a very very quiet man and uh didn't say very much this group was also reverentially in awe of lud VI Vicken Stein uh Vicken Stein was their hero uh he was their God and they had a certain view about the foundations of mathematics what made mathematics Jew they hated mystery uh and Mathematics presents a certain degree of mystery how do we know it's true uh how do we it it's it's it's a kind of certainty and encourage ability and infallibility how does it come to the likes of us um is it descriptive are we just making it up they had a certain view of mathematics uh given by David Hilbert formalism and that's why they attached the word logical to their positivism and they were embracing Hilbert's View and um this was the environment in which uh
是一个中心，这个圈子里有科学家、有哲学家、有数学家，还有两位年轻的研究生。他们是被一位数学家邀请来的，那人是维也纳学派的领袖之一，汉斯·哈恩。他有两个研究生，其中一个叫库尔特·哥德尔，是个非常非常安静的人，话不多。这群人还对路德维希·维特根斯坦怀有近乎虔诚的敬畏，维特根斯坦是他们的英雄，是他们的神。他们对数学基础有一套特定的看法，也就是数学何以为数学。他们憎恶神秘，而数学呈现出某种程度的神秘：我们怎么知道它是真的？它带有某种确定性、不可修正性和不可错性，这种东西怎么会落到我们这种人身上？它是描述性的吗？还是我们只是编出来的？他们对数学有一套特定的看法，来自大卫·希尔伯特的形式主义，这也是他们在实证主义前面加上“逻辑”一词的原因，他们接受了希尔伯特的观点。而这就是当时的环境，
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this young goodle sat there um take keeping his counsel not really speaking he disagreed with them entirely um but he didn't say it until he had a proof incompleteness and what was that proof um well so the big mystery of course in mathematics is uh uh what's it all about math is there's a certain degree of mystery here because it's not empirical it can't it's not going to to be uh revised in the face of um anything further uh that we're going to learn about the world that's why mathematicians are cheap to hire they don't require observatories they don't require Laboratories it's all in you know in the cranium and it's done a priori uh a a a proof from principles from first principles and so the question is um and the way I often think of it is you know that old uh Carl not Carl the other marks grouo marks um joke that I wouldn't have um any club that would have me I wouldn't want to belong to any club that would have me as a member I mean in mathematics is how did we get invited to join this exclusive
年轻的哥德尔就坐在那里，不动声色，几乎不发言。他完全不同意他们，但他没有说出来，直到他拿出了一个证明——不完备性。那个证明是什么？嗯，数学里最大的谜团当然是：数学到底是怎么回事？这里确实有某种程度的神秘，因为它不是经验性的，它不会因为我们将来对世界有更多了解而被修正。这就是为什么数学家雇起来便宜——他们不需要天文台，不需要实验室，一切都在脑袋里进行，是先验完成的，是从原理、从第一原理出发的证明。所以问题就是——我常常这样想——你知道那个老笑话，不是卡尔·马克思，是另一个马克思，格劳乔·马克思的笑话：我不会加入任何愿意接纳我的俱乐部，我不想属于任何肯把我当会员的俱乐部。我的意思是，在数学中，我们是怎么被邀请加入这个高端
05两条不完备性定理讲的是什么
15:45
cognitive Club how do the likes of us have this kind of certainty um are we just making it up are these just stipulated rules is it just a higher form of Chess that was the formalist school or is it descript of something girle tried to put forth a theorem a mathematically proved theorem R rigorously proved that would address this meta mathematical question um here's so so that's the interest of this thing that's why it seems it seems to be mathematics that spills out Beyond mathematics that tells us something perhaps about the nature of mathematical truth the nature nature of knowledge the nature of of us as knowers perhaps right here's what it actually says I mean there are two of them and the the second one follows as a coraly from the first so here's the first um in any formal system you may might want me to go on and Define what a formal system is yeah tell us what a formal system is formal system is something that is stipulated by rules you um you stipulate the axioms you I mean you stipulate the alphabet here's the alphabet of this thing you stipulate how you can combine uh the uh alphabet into formal basball is a formal system and you don't worry about what it all means it's
认知俱乐部的？我们这样的人怎么会拥有这种确定性？我们只是编出来的吗？这些只是约定的规则吗？它只是一种更高级的国际象棋吗？这就是形式主义学派的看法。还是说它在描述某种东西？哥德尔试图提出一个定理，一个用数学严格证明的定理，来回答这个元数学的问题。所以这就是这件事的趣味所在，这就是为什么它看起来像是数学溢出到了数学之外，也许告诉了我们一些关于数学真理的本质、知识的本质、以及我们作为认知者的本质的东西。它实际说的是这样：其实有两个定理，第二个作为推论从第一个推出。第一个是：在任何形式系统中——你可能想让我接着定义什么是形式系统。对，请说说什么是形式系统。形式系统是由规则规定出来的东西，你规定公理，我是说你规定字母表，这个系统的字母表是这些，你规定怎么把字母组合起来。形式棒球就是一个形式系统，你不用管它到底意味着什么。它是
17:11
something that's purely computational a formal system is something that's mechanical its rules it could be programmed into a a computer perhaps um it's it's a purely computational mechanical way you don't have to worry about what what it all means that's a very vague informal uh description of what a formal system is in any formal system um and what formalism wanted to do was to reduce all of mathematics to formal systems uh to just show it's just a purely mechanical thing following from stipulated rules uh in that way dispel the mystery okay girdle's first and completeness Theorem in any formal system that is Rich enough to express arithmetic there will exist a proposition which is you can express which is undecidable meaning neither it nor its negation can be proved in that system um uh another way of say saying this uh a little more controversial is that in any formal system Rich enough to express arithmetic there are true propositions that can't be proved um and the second incompleteness theorem which uh was immediate which everybody saw the
纯粹计算性的东西。形式系统是机械的，是规则，也许可以编进计算机里。它是一种纯粹计算的、机械的方式，你不必操心它意味着什么。这是对形式系统一个很含糊、很不正式的描述。在任何形式系统中——形式主义想做的，就是把全部数学都归约成形式系统，表明它只是从规定的规则出发的纯机械的东西，用这种方式消除神秘感。好，哥德尔第一不完备性定理：在任何足以表达算术的形式系统中，都存在一个你能表达出来的命题，它是不可判定的，意思是它和它的否定都无法在该系统内被证明。另一种说法，稍微更有争议一些，是：在任何足以表达算术的形式系统中，都存在无法被证明的真命题。而第二不完备性定理，是紧接着的、大家都看出来的
18:31
relevance of immediately because of what Hilbert had said needed to be done proving the consistency um of arithmetic the second incompleteness theorem which is a coral area of the first says one of the things that you can't prove in a formal system is the consistency of that system um and consistency again inally what this means a system is inconsistent if you can't can uh prove for some proposition P both p and not p uh that is an inconsistent uh system inconsistent systems are useless um because anything at all follows from a contradiction so if you've got a system that can prove a contradiction you can prove absolutely anything you know the existence of God I mean anything you can just prove anything from an inconsistent system so girdle's second incompleteness there I'm saying that within a system you can never prove the consistency of that system um really uh was a was a blow to formalism it really sounds rather shocking I mean to me I mean how how did the VES or for that matter the rest of the world take to this well the very interesting thing
立刻就有了现实意义，因为希尔伯特说过需要做的事，就是证明算术的一致性。第二不完备性定理，它是第一定理的一个推论，说的是在一个形式系统里你无法证明的东西之一，就是这个系统自身的一致性。再说一遍什么叫一致性——一个系统如果对某个命题 P，既能证明 P 又能证明非 P，那它就是不一致的，这就是一个不一致的系统。不一致的系统是没用的，因为从矛盾出发什么都能推出来。所以如果你有一个能证明矛盾的系统，你就能证明任何东西，比如上帝的存在，我是说任何东西，从一个不一致的系统里你什么都能证明出来。所以哥德尔的第二不完备性定理说，在一个系统内部你永远无法证明这个系统的一致性，这对形式主义真的是一记重击。这听上去确实相当令人震惊，我是说对我来说是这样。那么维也纳学派，或者说世界上其他人，是怎么接受这件事的呢？很有意思的一点
06柯尼斯堡的咕哝与冯·诺依曼的领悟
19:51
is that when so he had formed these great this he had these huge beliefs placed ISM he believed that mathematics was descriptive of a trans empirical realm he wanted a mathematical proof that he felt was going to prove that prove something mathemat metam mathematically and audacious uh ambition that he may or may not have carried through on but he certainly did something big and um he announced it at a conference in coxburg very softly mumbled it on the last day he said it might it might indeed be possible uh that there are true propositions of the order of goldbach's conjecture and foras Last Theorem to very important unsolved problems and in that day theer miles Last Theorem has now been proved by Dr Wilds um but that there might very well be that for every formal system there are true propositions of that that OT that can't be proved it kind of mumbled it that was the announcement of this incredible uh Discovery and nobody paid him any um mind at all and in fact when the um proceedings of that conference was
是，他当时已经形成了这些重大的信念，他是个柏拉图主义者，他相信数学描述的是一个超越经验的领域。他想要一个数学证明，他觉得这个证明能在元数学的层面上证明某种东西，这是个大胆的雄心，他也许实现了，也许没有，但他确实做成了一件大事。他是在柯尼斯堡的一次会议上宣布的，在最后一天，非常轻声地咕哝了一句。他说，很可能确实存在像哥德巴赫猜想、费马大定理这一类的真命题——这是两个非常重要的未解难题，在当时；费马大定理现在已经被怀尔斯博士证明了——但很可能对每一个形式系统来说，都存在该系统中无法被证明的真命题。他就那么咕哝了一句，这就是这个惊人发现的宣布方式，而根本没有人理会他。事实上，当那次会议的论文集
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written up in ER kentus uh a uh an organ of logical positivism published in Berlin it wasn't even mentioned no nobody even mentioned this but there was a a great mathematician uh y John Fon noyman who was there at that conference um he was standing in for David Hilbert he was presenting Hilbert's formalism and he came over to this young shy mathematician afterwards and he said wait a minute did you just say what I thought you said and he spoke to um G enough so that uh um he thought there's really something here he went back to Princeton he was at the institute for advanced study started to think some more wrote to girle and said it seems to be that if a consequence of your uh first theorem is that you can't prove the consistency of arithmetic and um girle wrote back and said oh yes you've got it right I have the very formal proof here a very rigorous proof here and it was F Lyman who spoke up Inc completeness theorem to everybody um and uh he was the one who disseminated did it Gregory you you also have a
发表在《认识》上时——那是逻辑实证主义的一份刊物，在柏林出版——这件事甚至都没被提到，没有，根本没人提。但当时会场上有一位伟大的数学家，约翰·冯·诺依曼。他是代表大卫·希尔伯特出席的，他在会上讲的是希尔伯特的形式主义。会后他走到这位年轻害羞的数学家跟前，说，等一下，你刚才说的是我以为你说的那个意思吗？他跟哥德尔谈了一会儿，觉得这里头确实有东西。他回到普林斯顿，他当时在高等研究院，开始进一步思考，然后写信给哥德尔说，在我看来，你第一定理的一个推论就是无法证明算术的一致性。哥德尔回信说，是的，你说对了，我这里已经有一个非常形式化、非常严格的证明了。是冯·诺依曼把不完备性定理讲给所有人听的，是他把它传播开来的。格雷戈里，你也有一个关于
07Chaitin：Ω数与数学的无穷复杂性
22:31
incomplete Ander can you tell us right you know from the horse's mouth what what what it's all about sure Paul um you know 80 years later we still don't know what the hell GLE proved the problem you see you see the the normal idea is mathematics gives absolute certainty it's pure thought if there's any place of their certainly it's in the world of pure thought and in 1931 GLE pulls the rug out from under that so what the hell is mathematics and it's 80 years and we're still arguing so the Fallout has not stopped the earthquake is still there are still Tremors now my own effort to understand what's going on I've been using ideas that you refer to in your introduction of complexity ideas that come from biology in a way inspired by biology but I've been using them in mathematical logic and so with a notion of complexity or information content uh I can show that the world of pure mathematics has Infinity complexity or contains an infinite amount of information but any mathematical Theory only has a finite amount of complexity so so that makes in completen seem natural uh another way to look at it is I have a number called Omega which comes from Turing halting problem but it's sort of like the DNA for pure mathematics and this number is a concrete example showing a
不完备性的角度，你能不能亲口给我们讲讲，这到底是怎么回事？当然可以，保罗。你知道，八十年过去了，我们还是不知道哥德尔到底证明了什么。问题在于，通常的想法是数学给出绝对的确定性，它是纯粹的思维，如果说哪里有确定性，那一定是在纯粹思维的世界里。而 1931 年哥德尔把这块地毯从底下抽走了。那数学到底是什么？八十年了我们还在争论，所以余波并没有停止，地震还在继续，还有余震。我自己为了理解这一切，一直在使用你在介绍里提到的那些复杂性的概念，这些概念来自生物学，某种意义上是受生物学启发的，但我把它们用在数理逻辑里。用复杂性或者说信息含量这个概念，我可以证明纯数学的世界具有无穷的复杂性，或者说包含无穷多的信息，而任何一个数学理论都只有有限的复杂性，这样一来不完备性就显得很自然了。另一种看待它的方式是，我有一个叫 Omega 的数，它来自图灵的停机问题，但它有点像纯数学的 DNA。这个数是一个具体的例子，展示了纯数学中
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place in pure mathematics where there's infinite irreducible complexity the Omega number if you write it in binary each bit is a complete surprise it looks completely accidental and and this is a very concrete example showing that pure mathematics contains infinite irreducible complexity so these are mathematical facts that seem to happen for no reason and it's an infinite number of them so so so what this what I'm starting to think now is that do you mean by that you can't predict I mean you can't predict what's coming next if you you can't predict what's coming next you can't prove what's coming next it looks like God is tossing a coin in mathematical truth now as a platonist I think each bit is determined nobody is playing dice with mathematical truth but to us it looks that way with the with any of the mathematical tools that we will ever have so this is a place in pure mathematics where things look accidental where they really Escape our our beyond our powers now the most what I'm beginning to think is that this is opening a door to biology I'm beginning to think that what GLE and Turing in 1936 came up with was the first step in the direction this is Alan Turing Allen Turing 1936 right but what they really
存在无穷、不可约复杂性的地方。Omega 这个数，如果你把它写成二进制，每一位都是一个完全的意外，看上去完全是偶然的。这是一个非常具体的例子，说明纯数学包含无穷的不可约复杂性。所以这些数学事实似乎是毫无理由地发生的，而且有无穷多个。所以我现在开始想的是——你的意思是说无法预测？我是说你无法预测下一位是什么。如果你无法预测下一位，你也就无法证明下一位是什么。看起来上帝在数学真理里掷硬币。当然作为一个柏拉图主义者，我认为每一位都是确定的，没有人在拿数学真理掷骰子，但对我们来说，用我们所能拥有的任何数学工具，它看起来就是那样。所以这是纯数学中一个事情看起来纯属偶然的地方，一个真正超出我们能力范围的地方。而我现在开始越来越觉得，这打开了一扇通向生物学的门。我开始认为，哥德尔和图灵在 1936 年提出的东西是朝这个方向迈出的第一步——是艾伦·图灵，艾伦·图灵，1936 年，对。但他们真正
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what the door was opening taking pure mathematics in a New Direction which I feel is the direction of biology actually I think history may say this in 50 or 100 years because you see if you look at the uh Omega number what it shows is that pure mathematics contains infinite irreducible complexity now normally people think that uh the the field of science where there's complexity where is biology biology is the domain of the complex there are no simple laws right but mathematics and and physics are supposed to be simple governed by beautiful simple equations so so what GLE and Turing open the door to is the the fact that in fact pure mathematics has infinite complexity and in a way is even worse than biology because biology has very large complexity but it's only finite so pure math is even is even worse and based on this hint I've been trying to to uh to study uh the fundamental ideas of biology in an effort to come up with a theoretical biology which is very hard in fact impossible so I'm trying to do a theoretical metabiology which is instead of trying to have a math a theoretical mathematics for real biology which is just too messy and complicated it's hopeless I'm trying to see if you can come up with a mathematical toy model that captures some of the essential
打开的那扇门，是把纯数学带向一个新方向，我觉得那就是生物学的方向。我其实觉得五十年或一百年后历史可能会这么说。因为你看，如果你看 Omega 这个数，它表明纯数学包含无穷的不可约复杂性。而通常人们认为，有复杂性的科学领域是生物学，生物学是复杂性的领域，那里没有简单的定律，对吧？而数学和物理学本应是简单的，由优美简洁的方程支配。所以哥德尔和图灵打开的门，其实揭示了纯数学具有无穷的复杂性，某种意义上它甚至比生物学还糟，因为生物学的复杂性虽然非常大，但毕竟是有限的，所以纯数学更糟。基于这个线索，我一直在试图研究生物学的基本思想，想搞出一套理论生物学，这非常难，事实上是不可能的。所以我试着做的是一种理论元生物学，也就是说，与其为真实的生物学建立一套理论数学——那太混乱太复杂了，是没希望的——我想看看能不能构造一个数学玩具模型，抓住其中一些本质的
08柏拉图主义、形式主义与逻辑主义
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ideas of biology and it's a software model and the basic idea is very simple I'll summarize it in two phrases uh well you said it DNA is digital software that's the basic idea so I'm running with that idea as a mathematician and the two phrases that uh Define this metab biological uh effort that uh I think is promising but is just beginning is to to model to look at biology with uh through the evolution of mutating software is one of the key phrases in metabology and the other key phrases is random walks in software space so uh I'm going to just wave this mysterious words in front of you and say come back in 50 years and maybe there'll be a field of mathematics there well I might I might come back in 15 minutes of that actually but we'll um we'll see where how much progress we've made in that time but both of you um mentioned the platonist and I think it might be interesting to get a sense of realism and non-realism in mathematics would somebody like to pick up on that explain what it is well I'm happy to talk about platonism versus formalism that she mentioned and so on and what are these things so Plato is you know uh Alfred North Whitehead once wrote that all of Western philosophy is a series of footnotes to Plato um and uh so he was a very smart person yeah um but uh you
……生物学的思想，它是一个软件模型。基本想法非常简单，我可以用两句话来概括。嗯，你刚才也说了，DNA 就是数字化的软件，这就是基本思想。所以我作为一个数学家，就顺着这个思路往下走。界定这个「元生物学」（metabiology）方向的有两个短语——我认为这个方向很有前景，但才刚刚起步——一个是：通过不断变异的软件的演化来建模、来看待生物学，这是元生物学的关键短语之一；另一个关键短语是：软件空间中的随机游走。所以我就先在你们面前晃一晃这些神秘的词，然后说：五十年后再回来，也许那时候就会有这么一个数学领域了。嗯，其实我可能十五分钟后就会再回到这个话题，但我们……我们看看在这段时间里能取得多少进展。不过你们两位都提到了柏拉图主义者，我想也许可以让大家对数学中的实在论和非实在论有个概念，有谁愿意接着这个话题，解释一下这是怎么回事？好，我很乐意谈谈她提到的柏拉图主义与形式主义之类的，谈谈这些究竟是什么。柏拉图——你知道，阿尔弗雷德·诺思·怀特海曾经写过，全部西方哲学不过是对柏拉图的一系列脚注。嗯，所以他是个非常聪明的人。是啊。不过，呃，你……
27:58
know maybe it's best what he thought is maybe best explained by his allegory of the cave he said that we humans you know are like prisoners in a cave somebody's throwing Shadows on the wall of the Cave the prisoners are forced to see the shadows and that's what they think reality is about so Plato basically said mathematics exist in some world out there uh this is a platonic world of mathematical forms so when I draw a circle on a piece of paper that's not the real Circle the real circle is in that world of mathematics this is just a crude approximation for that Circle uh most working mathematicians are platonists in their hearts namely they think that mathematics has this existence out there and we're merely discovering the truths of mathematics uh the school that Rebecca described before of the formalist um she actually used this this analogy of Chess basically says no no it's all just you know I write down some rules and or some set of axioms and I can derive all of mathematics from that it's like I give you the rules of Chess we play chess I change the rules we play a different game basically the idea is I write down one set of axom I get some form of mathematics I change the axom I get another mathematics and the best example of this was the ukian geometry that we
……知道，也许他的想法最好还是用他的洞穴寓言来解释。他说，我们人类就像是洞穴里的囚徒，有人把影子投在洞壁上，囚徒们只能看到这些影子，他们就以为那就是实在。所以柏拉图基本上是说，数学存在于外面的某个世界里，这是一个由数学形式构成的柏拉图世界。所以当我在纸上画一个圆时，那并不是真正的圆，真正的圆在那个数学世界里，我画的只是对那个圆的粗糙近似。大多数做研究的数学家在内心里都是柏拉图主义者，也就是说，他们认为数学有那样一种独立于我们的存在，而我们只是在发现数学的真理。而丽贝卡刚才描述的那个形式主义学派——她当时用了国际象棋这个类比——基本上是说：不不不，这一切只不过是，我写下一些规则、或者一组公理，然后我就能从中推导出全部数学。就好像我把国际象棋的规则给你，我们下国际象棋；我改一改规则，我们下的就是另一种棋了。基本思想就是：我写下一组公理，得到某种形式的数学；我换一组公理，就得到另一种数学。而这方面最好的例子，就是我们大家……
29:25
all learn in school it was based on a set of 10 axom one of which say that from if you have a line and a point outside the line you can pass a single parallel line to that line that was thought to be truth from God until the beginning of the 19th century when three mathematicians independently showed that actually you can dispose of that axum all together and you can develop new mathematics one that can be described on a Surface curved like a s one on the surface like a sphere and so on in which this axom doesn't hold true at all and those are equally good descriptions of space you know and everything so basically formally say write down the axioms give me what there is what girle did was that he pulled the rug underneath this entire premise because basically you know to summarize in one sentence what Rebecca said he basically showed that no formal system no system of axioms you're going to write are going to ever capture all the truths of mathematics this is a simple way of saying this you know in physics I'm a physicist we people here may have heard the term Theory of Everything we strive
……在学校里都学过的欧几里得几何。它建立在一组十条公理之上，其中一条说，如果有一条直线和直线外的一个点，那么过这个点只能作一条与该直线平行的线。直到十九世纪初，这一直被认为是来自上帝的真理，后来有三位数学家各自独立地证明，其实你完全可以把那条公理整个抛弃掉，从而发展出新的数学——一种可以描述像马鞍那样弯曲的曲面上的几何，一种像球面那样的曲面上的几何，等等，在这些几何里那条公理根本不成立。而这些对空间的描述同样是很好的描述，你知道，一切都说得通。所以形式主义基本上就是说：把公理写下来，看看能得到什么。而哥德尔所做的，是把这整个前提脚下的地毯抽走了。因为，基本上，用一句话来概括丽贝卡刚才说的：他证明了，没有任何形式系统、没有任何你能写下的公理系统，能够囊括数学的全部真理。这是一种简单的说法。你知道，在物理学里——我是个物理学家——在座各位可能听说过「万物理论」这个词，我们努力……
30:47
to get this Theory of Everything in physics what G's theorem shows that there is no theorem of everything in mathematics namely you cannot you know write down some set of axom and from that derive all of mathematics so these were the the tensions between these particular group groups on top of that there was this third Group which is that of the logicist which is people like beran Russell golob FR and other said no no no no no it all actually comes out of logic I can write down some axioms of logic for example I tell you either the butler killed the millionaire or his daughter killed him and then I tell you the daughter didn't kill him then you have to conclude that the butler did it and this conclusion is not dependent on how old is the butler what's the length of the nose of the daughter or any such thing so there are some axioms of logic and gotl fr said that he could prove all of arithmetic and by that I mean even 1+ one equals two from some axioms of logic and that looked EX extremely good I mean you know this was almost everybody was happy because you know the formalist said okay it's all a game but maybe there is a mother of all games uh the platonist said oh if it all comes from some axum of logic surely those are
……在物理学中得到这个「万物理论」。而哥德尔定理表明，在数学里不存在「万物定理」，也就是说，你不可能写下某一组公理，然后从中推导出全部数学。这些就是这几个阵营之间的张力所在。除此之外，还有第三个群体，就是逻辑主义者，比如伯特兰·罗素、戈特洛布·弗雷格等人，他们说：不不不不不，这一切其实都出自逻辑。我可以写下一些逻辑公理，比如我告诉你，要么是管家杀了那个百万富翁，要么是他女儿杀了他；然后我再告诉你，女儿没有杀他，那你就必须得出结论：是管家干的。而这个结论并不取决于管家多大年纪、女儿的鼻子有多长，或者任何诸如此类的事情。所以存在一些逻辑公理，而戈特洛布·弗雷格说，他能从一些逻辑公理出发证明全部算术——我指的甚至包括一加一等于二。这看上去极其漂亮，我是说，几乎所有人都很高兴，因为形式主义者说：好吧，这一切都是一场游戏，但也许存在一个「万游戏之母」；柏拉图主义者则说：哦，如果这一切都出自某些逻辑公理，那这些公理肯定就在……
32:15
somewhere there in that world of mathematics you know and so on so all of this looked very good except that when FR was ready to send his fat book to the print bertr and Russell sent him a paradox which is a little bit like what is sometimes called The Barbers Paradox you know there's this Barber who has this sign I shave all and only the men of the village who don't shave themselves and this all sounds perfectly logical because the PE the men who shave themselves don't need the services of the barber and the barber shaves everybody else until you stop and ask okay but who shaves The Barber and if the barber shaves himself he's one of those men that shave themselves and therefore he doesn't shave himself if if the barber doesn't shave himself then he's one of those that don't shave himself and then he has to shave himself so the the Paradox that that Russell sent to uh to uh FR was not quite this Paradox but it had a similar flavor to it and so it basically showed that our logical system can be fallible um so the there was this crisis you mentioned shock you know in in goodles theorem and so on there was a sense that the whole foundations of mathematics became rather shaky and this is what what for me as a physicist makes the puzzle even bigger
……那个数学世界里的某个地方，等等。所以这一切看起来都非常好，只不过，就在弗雷格正准备把他那本厚厚的书送去付印时，伯特兰·罗素给他寄来了一个悖论，这个悖论有点像人们有时说的「理发师悖论」。你知道，有这么一个理发师，他挂着一块牌子说：我给村里所有不给自己刮胡子的男人刮胡子，而且只给他们刮。这听上去完全合乎逻辑，因为自己刮胡子的男人不需要理发师的服务，而理发师给其他所有人刮胡子——直到你停下来问一句：好，那谁给理发师刮胡子呢？如果理发师给自己刮胡子，那他就属于那些自己刮胡子的人，因此他就不该给自己刮；如果理发师不给自己刮胡子，那他就属于那些不给自己刮胡子的人，那他就得给自己刮。罗素寄给弗雷格的那个悖论并不完全是这个悖论，但味道很相似。所以它基本上表明，我们的逻辑系统是可能出错的。所以就出现了那场危机——你刚才提到了震动，在哥德尔定理等等这些事情上——当时有一种感觉，就是整个数学基础都变得相当不牢靠。而这一点，对我这个物理学家来说，让这个谜题变得更大了……
09Minsky：公众对量子与演化的误解
33:48
in the sense that still while these foundations were shaking of mathematics mathematics became this extraordinar powerful tool in explaining the universe around us so how can you explain the whole universe with something that has such shaky tools I spoke a little bit long but I hope this well I'm going to pick you up on that in a moment that very point but does anybody else want to comment more on this realism non-realism clayon ISM business uh I have a whole bunch of remarks about but they're a little tangential because um I'm going to talk a little bit about the limit of public [Music] understanding because all sorts of remarkable things are happening in science and there are a few directions in which the public has been misinformed in a a strange way uh and the result has been some strange beliefs U so let me mention one um this is a little different but you've all heard of the uncertainty principle and the mysteries of quantum mechanics and and uh many philosophers who know a little physics have explained to the
……因为，就在数学的这些基础摇摇欲坠的同时，数学却成了解释我们周围这个宇宙的一种极其强大的工具。那么，你怎么能用一个工具如此不牢靠的东西去解释整个宇宙呢？我说得有点长了，但我希望这讲清楚了。好，我待会儿就要就这一点来追问你。不过还有别人想就实在论、非实在论、柏拉图主义这些事再说几句吗？呃，我有一大堆想法，但都有点跑题，因为呃，我想稍微谈一谈公众理解的局限【音乐】，因为科学里正在发生各种了不起的事情，而在某些方向上，公众以一种很奇怪的方式被误导了，结果就产生了一些奇怪的信念。那我就提一个吧。嗯，这个有点不太一样，但你们都听说过不确定性原理，听说过量子力学的种种神秘之处，还有很多懂一点物理的哲学家向……解释说
35:13
public that the World of Newton in which things were determined and mechanical and predictable was really a nice beginning in physics but the real physics is very different and things are un certain and uh there's a famous uh Heisenberg Principle that you can't know the position and velocity of a particle both at the same time uh but only the uh the more certain you are of one then the less certain you are of the others and that's perfectly true but the numbers are very strange because these Quantum effects in most cases are very small so uh here's the joke U suppose we take the World of Newton and talk about uh the kinds of things that uh Dr chayton is interested in uh Evolution and genetics and so forth and think of a little molecule of DNA and as you everybody knows that's the code which determines a lot about how uh biology works well if you take the World of Newton and look at the solar system that's sort of like an atom there's a big star and little planets going around it and then we know from about 1900 that
公众被告知：牛顿的世界——那个一切都是决定论的、机械的、可预测的世界——在物理学里只是个不错的开端，而真正的物理学非常不一样，事物是不确定的，有个著名的海森堡原理，说你没法同时知道一个粒子的位置和速度，你对其中一个越确定，对另一个就越不确定。这完全是对的，但那些数值很奇怪，因为这些量子效应在大多数情况下都非常小。所以笑话来了：假设我们回到牛顿的世界，来谈谈 Chaitin 博士感兴趣的那类东西，进化、遗传学之类的，想一想一个小小的 DNA 分子，大家都知道那是决定生物学如何运作的密码。那么，如果你在牛顿的世界里看太阳系，它有点像一个原子，中间是一颗大恒星，小行星绕着它转，而我们从大约 1900 年起就知道
36:41
atoms are the same sort of thing there's a nucleus and little electrons going around it but the joke is that things like solar systems aren't stable so in the World of Newton uh for example some of uh I think susman and Jack wisdom at MIT some years ago showed that probably in the solar system that we live in Jupiter is so large and has such a big gravitational field and Pluto is so far uh away from other things that in a few billion years probably Jupiter will throw Pluto out so it's not clear how long the solar system lasts wouldn't be such a big disaster it's not a planet anymore that's right but on the other hand if you think of a molecule of DNA at room temperature that's stable for a billion years and so this is why evolution is possible because of quantum mechanics not in spite of it so that's just one in other words the public has been told that in modern physics things are much vaguer and uncertain and it's exactly wrong in the Newtonian world you couldn't have the kinds of stable
原子也是同一类东西，有个原子核，小电子绕着它转。但好笑的是，太阳系这种东西并不稳定。所以在牛顿的世界里，比如说，我记得 MIT 的 Sussman 和 Jack Wisdom 几年前就证明过，在我们所处的太阳系里，木星实在太大、引力场太强，而冥王星又离其他天体太远，可能再过几十亿年木星就会把冥王星甩出去，所以太阳系能维持多久并不清楚。——反正它也不算行星了，也不算多大的灾难。——没错。但另一方面，你想想一个 DNA 分子，在室温下它能稳定十亿年。所以进化之所以可能，恰恰是因为量子力学，而不是尽管有量子力学。这只是其中一个例子。换句话说，公众被告知在现代物理学里事情更加模糊和不确定，而这恰恰说反了：在牛顿式的世界里，你根本不可能有那种稳定的
37:58
structures well there about five more but uh here's another one we all learn about Evolution and uh there are some people who say well how could random changes and the kinds of variations that uh jadon's talking about lead to anything so magnificent as uh people with all their intelligence and Mathematics and philosophy uh how could that have happened when we're just making random changes and selecting them well have you ever thought about it there's a problem with Evolution as it's explained with darwinian Evolution I'm not saying it isn't true it's absolutely true but there is a little trouble with it namely the way Evolution works is you make little changes in organism uh generally random although not always and uh then some of them live and some of them die because the ones uh with the wrong kind of toes fall out of the trees and and yet it you know all sorts of bad things happen what does evolution remember it remembers the things that worked but it forgets what killed all the others there's no record of that and so uh when it comes to intelligence it may be that you couldn't evolve a brain with darwinian Evolution alone but I'm not being mysterian here I'm just saying uh the answer is
结构。这样的例子我还有五个左右，不过这里再说一个：我们都学过进化论，有些人会说，随机的变化、Chaitin 说的那种变异，怎么可能带来像人这样宏伟的东西，带来智能、数学和哲学？我们只是在做随机改变然后做选择，这怎么可能发生呢？你想过没有，达尔文式进化论的解释其实是有个问题的。我不是说它不对，它绝对是对的，但确实有点小麻烦，也就是说：进化的运作方式是你对生物体做些小改动，一般是随机的，虽然不总是，然后有些活下来了，有些死掉了，因为那些长错脚趾的从树上摔下去了，各种糟糕的事都会发生。那么进化记住了什么？它记住了那些管用的东西，却忘了那些害死其他个体的东西，那些都没有记录。所以说到智能，也许光靠达尔文式进化你演化不出一个大脑。但我不是在搞什么神秘主义，我想说答案是
39:26
language culture Society maybe some chimpanzee like creature gets a few words or a few symbols and what what do we tell our children well if you look at the fairy tales uh I think the first great psychologist really was Sigman Freud who had this idea that the mind is many parts and very complicated and one of the things Freud talked about is early experiences having big effects well look at the Legends that populate all cultures almost all the Legends uh like The Three Little Pigs and so forth are about mistakes you can make and they're usually fatal and the answer is uh a human child would never survive without a culture because it wouldn't live long enough to learn all the Fatal mistakes that are Comming so I could go on but what I'm saying is that we when we talk about the limits of understanding I think we need need to know much more about how human understanding itself evolved and develops and my feeling is that we're just at uh one is always making fun of Ancient Ancient people and their foolish Legends and I think we're just coming
语言、文化、社会。也许某种类似黑猩猩的生物学会了几个词、几个符号。那我们又是怎么教孩子的呢？看看童话故事。我觉得第一位真正伟大的心理学家其实是弗洛伊德，他提出心智是由许多部分构成的、非常复杂，而且他谈到过早期经验会产生巨大影响。你看看遍布各个文化的那些传说，几乎所有的传说，比如《三只小猪》之类的，都是关于你可能犯的错误，而且通常是致命的错误。答案就是：人类的孩子没有文化根本活不下来，因为他活不到足够长的时间去把所有那些致命错误一一学会。我还能继续讲，但我想说的是，当我们谈论理解的界限时，我认为我们需要更多地了解人类的理解本身是如何演化和发展起来的。我的感觉是，我们不过刚刚——人们总爱嘲笑古人和他们愚蠢的传说，而我觉得我们只是刚刚走
10数学为何「不可思议地有效」
40:43
out of one of maybe 20 different levels of ignorance going into the next one and we're learning more and more every year well one thing I I always feel ignorant about I must say or find difficult is um what's called the unreasonable effectiveness of mathematics in explaining the world I never quite know why that should be the case can somebody help me sure yeah so uh the phrase by the way was coined by uh Nobel or Eugene vigner the unreasonable effectiveness of mathematics in the physical sciences uh there is by the way a paper entitled The unreasonable ineffectiveness of mathematics in biology um but uh but uh in the physical world so so what what is meant by this unreasonable Effectiveness so let me give you just a couple of very quick example so there was this astronomer Johannes Kepler and he made observations 400 years ago and they were not very accurate they were accurate for his time but not very accurate they were about accurate to within about 4% yet from those relatively scanty observations Sir Isaac Newton managed to write a mathematic iCal law of gravity
出大概二十个不同的无知层级中的一个，正走进下一个，而我们每年都在学到更多。有件事我得承认我一直觉得自己很无知，或者说觉得很难懂，就是所谓“数学在解释世界方面不可思议的有效性”。我一直不太明白为什么会是这样，有人能帮帮我吗？——当然。这个说法顺便说一句是诺贝尔奖得主尤金·维格纳提出的，“数学在自然科学中不可思议的有效性”。顺便一提，还有一篇论文叫《数学在生物学中不可思议的无效性》。但在物理世界里——那么这个“不可思议的有效性”是什么意思呢？我给你举几个很快的例子。有位天文学家叫约翰内斯·开普勒，他在 400 年前做了观测，那些观测并不算很精确，在他那个时代算精确，但并不算很精确，大概准到 4% 左右。然而就凭这些相对贫乏的观测，艾萨克·牛顿爵士竟然写出了一条引力的数学定律，
41:58
that already by the 1950s was shown to be accurate to better than one part in a million in fact in an experiment done in 2008 they showed that the law inverse Square Law of Newton holds down to a distance of 56 microns one micron is 1 millionth of a meter these are distances that Newton could have had no idea his mathematical law should hold true what is it that gives mathematics such Powers we have a theory of everything that's Electric and magnetic it's called Quantum electrodynamics in this Theory it's a highly mathematical Theory you know electrons that moov in atoms they they are like little magnets you can use this Theory to calculate the strength of this magnet we can calculate the strength to parts per trillion everyone now knows what the trillion is it's the size of our annual deficit you you you calculate this magnet to parts per TR in 2006 this magnetic strength of the electron was measured to parts per trillion and the two results agreed to within eight parts per trillion but Mario why are you surprised the world is built out of mathematics God is a mathematician so this is the most natural thing I I wrote a a TI a book with this title but with a question mark at the end because you're a physicist and I'm a mathematician
到了 1950 年代人们就已经证明它准确到优于百万分之一。事实上在 2008 年做的一个实验里，他们证明牛顿的平方反比定律一直到 56 微米的距离都成立，一微米是一米的百万分之一。这些距离是牛顿根本不可能想到他的数学定律还会适用的。是什么让数学有这样的威力？我们有一个关于电和磁的“万物理论”，叫量子电动力学，这是个高度数学化的理论。你知道，在原子中运动的电子就像一块小磁铁，你可以用这个理论算出这块磁铁的强度，我们能算到万亿分之一的精度。现在人人都知道万亿是多大了，就是我们年度赤字的规模。你把这块磁铁算到万亿分之几，而在 2006 年，电子的这个磁性强度被测量到了万亿分之一的精度，两个结果吻合到万亿分之八以内。——可是 Mario，你为什么会感到惊讶呢？世界就是由数学构成的，上帝是个数学家，所以这是最自然不过的事。——我写过一本书就叫这个书名，不过结尾加了个问号，因为你是物理学家，而我是数学家。
43:24
so so um in in so the question is indeed uh how come mathematics is as powerful as it is in particular there is this thing which I sometimes call the passive Effectiveness which is mathematicians sit down you know people like Marvin and like Gregory here they sit down and they develop branches of mathematics sometimes with absolutely no application whatsoever in mind in in fact in fact they are proud that there are no applications uh and yet sometimes decades sometimes centuries later those very precise branches of mathematics are found to be exactly what is needed to Theory you know this was the case with Einstein's general relativity this was the case with group Theory you know and so so uh so these things exactly provide this how come why is mathematics as powerful as that I gu Mario then Marvin I or you can do Marvin do you mind being interrupted no absolutely not well I think I have one uh funny answer to that which is that uh let's think what would happen if it weren't if the world didn't obey simple principles and the answer is um do you think the world exists people argue a lot of religions are based on the idea that well here's the world somebody must have made it of course
那么问题确实在于：数学为什么会有这么大的威力？特别是还有一种我有时称之为“被动有效性”的东西，就是数学家们坐下来——像 Marvin，像这位 Gregory 这样的人——他们坐下来发展数学的各个分支，有时脑子里完全没有任何应用；事实上，他们还为没有应用而自豪。可有时候几十年后、有时候几个世纪后，恰恰是这些精确的数学分支被发现正是某个理论所需要的。爱因斯坦的广义相对论就是这样，群论也是这样。所以这些例子恰恰摆出了这个问题：数学怎么会有这么强的威力？——我想先请 Mario，然后 Marvin……或者你来吧。Marvin，你介意被打断吗？——完全不介意。我想我对此有个挺好笑的回答，就是：我们来想想，如果不是这样会怎样，如果世界并不遵守简单的原理会怎样？答案是——你觉得世界存在吗？很多宗教都建立在这样的想法上：世界在这儿，那一定有人造了它。当然，
44:55
then you have to ask the question not not just how does the world work but uh how did the creator work but the real problem is that the word existence doesn't make any sense it's all right to say this table exists because that means this table is in the universe it doesn't make any sense to say the universe exists so my argument is one this is just a possible universe and now let's consider all the possible universes and suppose you had a universe in which energy weren't conserved uh then all of a sudden things would blow up and there wouldn't DNA wouldn't last a billion years at room temperature and so if you take all of the things that physicists say look at this amazing mathematical thing it's accurate to one part in 10 to the 16th then you say well what would happen if it weren't and if you look carefully you'd see that well then everything would explode or everything would disappear and in fact the current theory is that once in a while some little thing in fact does explode and that's a big bang and another possible Universe this multiverses is this so in other words maybe you don't have to answer this question because the answer is obvious in Worlds where there are no laws there is no DNA and no philosophers
接着你就得问，不只是世界如何运作，还得问造物主是如何运作的。但真正的问题在于，“存在”这个词根本讲不通。说“这张桌子存在”是可以的，因为那意味着这张桌子在宇宙之中；但说“宇宙存在”就毫无意义了。所以我的论点是：这只是一个可能的宇宙。现在我们来考虑所有可能的宇宙，假设你有个宇宙，其中能量不守恒，那么突然之间东西就会炸开，DNA 就不可能在室温下维持十亿年。所以，如果你把物理学家说的那些“看这个了不起的数学事实，它准确到 10 的 16 次方分之一”的东西拿来，然后问：如果不是这样会怎样？你仔细一看就会发现，那样的话一切都会爆炸，或者一切都会消失。而事实上，现在的理论认为，偶尔真的会有某个小东西爆炸，那就是一次大爆炸，那就是另一个可能的宇宙——这就是多重宇宙。换句话说，也许你根本不必回答这个问题，因为答案是显而易见的：在没有定律的世界里，没有 DNA，也没有哲学家来
46:14
to discuss [Music] it so so let me just say say a few a few things about I only read half your book and but you read the you read the right half you read the right half so let let me just say say a few a few words about this so uh because you know I wrote about the Multiverse and things like this so let me just say a few things so uh Marvin is absolutely right in the following uh statement that he makes he talks about a world in which there are no laws of physics or the laws of physics are very very different from what we know them and so on it is certainly the case that if there were no law of physics we might not have existed here and furthermore we wouldn't be talking about describing those laws of physics by mathematics I mean the fact that our universe is symmetric in the sense that for example an atom here behaves the same as an atom 12 billion light years from here this is what allows us to try to explain anything otherwise you know we wouldn't be now that however still does not answer the question why is our mathema matics so adequate for describing the laws that exist and how come that mathematics that are built with absolutely no application in mind turn out to be very descriptive
讨论它。【音乐】那我就说几句吧。——我只读了你那本书的一半。——但你读的是对的那一半，你读的是对的那一半。——那我就这个话题说几句，因为你知道我写过关于多重宇宙之类的东西。Marvin 有一点绝对是对的，就是他说到一个没有物理定律、或者物理定律和我们所知的非常非常不同的世界。确实，如果没有物理定律，我们可能压根就不会存在于此，更不会在这里谈论用数学去描述这些物理定律。我是说，我们的宇宙是对称的——比如这里的一个原子和 120 亿光年之外的一个原子行为是一样的——正是这一点让我们有可能去解释任何事情，否则我们根本做不到。然而这仍然没有回答那个问题：为什么我们的数学如此适合描述现存的这些定律？为什么那些完全不考虑任何应用而建立起来的数学，最后却极为贴切地描述了
47:40
of physical phenomena but the answer there is is actually it's complicated but it is not that difficult to understand it's a combination of many things Mario the world has to be built out of something why not out of mathematics that's the most beautiful thing we have what do you have against my reason for all of this the world is built out of pure mathematics it's the most beautiful the most fundamental thing that exists it has the most beautiful structure obviously God or the great computer programmer who made this simulation we're in would use that beautiful stuff I right what else what else would you make the world out of marshmallows so here's another possible explanation and that is that our son science starting in the 17th century what physics did was to isolate structure um and and how do we describe structure in the language of mathematics um and so our laws that we've been able to um formulate so far have all been structural um therefore mathematical but are not getting at everything so that there are certain problems for example my favorite being the hard problem of Consciousness right uh that we have not yet been able to tackle it it's not just that we can't tackle it you might be disagreeing with me here but um we don't know how to tackle
物理现象？不过那个答案其实……很复杂，但并不难理解，它是很多因素的组合。——Mario，世界总得由某种东西构成吧，为什么不能是由数学构成呢？那可是我们拥有的最美的东西啊。你有什么理由反对我这个说法？世界就是由纯数学构成的，那是存在的最美、最基本的东西，它有着最美的结构。显然，上帝，或者说造出我们身处的这个模拟的那位伟大程序员，会用这么美的材料。对吧？不然你还能用什么造世界，棉花糖吗？——这里还有另一种可能的解释，那就是：我们的科学从 17 世纪开始，物理学所做的是把结构分离出来。而我们用什么来描述结构呢？用数学的语言。所以我们迄今为止能够表述的那些定律，全都是结构性的，因而是数学的，但它们并没有触及一切。所以有些问题——我最喜欢的例子就是意识的难问题——我们至今无法处理。而且不只是我们处理不了，你在这一点上可能不同意我，但我们不知道该怎么去处理
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it um because we're using this kind of our notion that we've gotten from physics of what a law is which is a isol ating structure that we can that we can express mathematically but isn't exhaust it's an incompleteness it's another incompleteness theorem it's not exhausting the stuff of matter um and so that that would be another explanation for why we see laws of you physics are always mathematical those are the only ones we can get at we have the tools uh to get at that I'm giving up chairing I'm just let me try to weave together some of what the three of them said so Rebecca now touched um something that that is really very important which is uh one of the things is that these are the things that we have at our disposal and and Gregory touched about that too um so we try to use them the best we can and not only that we try to actually tailor the mathematics we use a little bit to the problem at hand for example if if I were to put Pebbles into this bottle and I
它，因为我们用的是从物理学里得来的那种“定律”的概念，即把结构分离出来、用数学表达出来，但那并不穷尽一切。这是一种不完备性，是另一个不完备性定理，它没有穷尽物质的全部内容。所以这也算是另一个解释：为什么我们看到的物理定律总是数学的？因为那是我们唯一够得着的，我们只有工具去够到那部分。——我要放弃主持人的角色了，我就……让我试着把他们三位说的东西串起来。Rebecca 刚才触及了一个非常重要的点：这些是我们手头能用的工具，Gregory 刚才也提到了这一点。所以我们尽可能地去用它们，而且不仅如此，我们还会针对手头的问题对所用的数学稍作剪裁。比如说，如果我往这个瓶子里放小石子，先放三颗，再放四颗，那我就会用算术来描述发生的事，对吧，3 + 4 = 7。但如果
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would put three Pebbles then four Pebbles then I would be using arithmetic to describe what happens right 3 + 4 equal 7 if on the other hand I pour water into this bottle I pour three times water then four times water I wouldn't describe arithmetic to I wouldn't say there is seven water in this bottle right so we tailored the mathematics that we use a little bit to the type of physics we we want to solve then we touched upon this thing that not everything can be described by mathematics and and Marvin here described the well I'll tell you why let me tell you why you know GLE would not like the way you gentlemen are arguing as Rebecca points out in her beautiful book GLE is a man from the Middle Ages and he was against imperical science he said pure mathematics is the only field Which con conserves medieval theology essentially which conserves idealism and platonism and he didn't believe in empirical science all truth should be a priori truth necessary truth right well I think right so you gentlemen you gentlemen have a very uh how do you say an attitude which is quite common now an empirical attitude
我往这个瓶子里倒水，倒三次水，再倒四次水，我就不会用算术来描述了，我不会说这瓶子里有“七个水”，对吧。所以我们会根据想解决的物理类型，对所用的数学稍作剪裁。接着我们谈到了并非一切都能被数学描述这件事，Marvin 在这里描述的是……——好，我来告诉你为什么。让我告诉你，哥德尔不会喜欢你们几位这样的论证方式。正如 Rebecca 在她那本很棒的书里指出的，哥德尔是个来自中世纪的人，他反对经验科学。他说纯数学是唯一能保存中世纪神学的领域，本质上是保存唯心论和柏拉图主义的领域。他不相信经验科学，一切真理都应该是先验真理、必然真理，对吧。——嗯，我觉得……——对，所以你们几位持有的是一种如今很常见的、怎么说呢，一种经验主义的态度，
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uh you know the word God can't be mentioned mathematical Beauty can't be mentioned but but ghetto belongs to the Middle Ages you know and he would be very uncomfortable so I'm trying to let let let me say something I tell you what ghet might have responded let me let me let me take the example that that that that Marvin used yeah but you know GLE lived a few tens of years ago we have advanced since then I'm talking about something that we know how to write equations for okay I want to describe what what every atom in the atmosphere is going to do so it sounds simple right I know the equations if I just have a big enough computer I'll be able ble to calculate that guess what not true and you know why because you will need as an input to that computer to put the position and the speed of each atom when you start the calculation now quantum mechanics tells us that I cannot tell precisely the position and the and the speed of a single atom let alone of all the atoms in the atmosphere yet the system is chaotic which means it is infinitely sensitive to initial conditions you change the position of one atom by a little bit and the final answer is completely different so there are problems where quantum mechanics and chaotic systems tell us that we will never be able to calculate and but we
你知道，“上帝”这个词不能提，“数学之美”也不能提。但哥德尔属于中世纪，你知道吗，他会非常不自在的。——那我说点什么吧。——我来告诉你哥德尔可能会怎么回应。——让我用一下 Marvin 刚才举的那个例子。——不过你知道，哥德尔是几十年前的人了，我们从那以后已经进步了。——我说的是一个我们知道怎么写出方程的东西。好，我想描述大气中每一个原子接下来会做什么。听起来很简单，对吧？我知道那些方程，只要有一台足够大的计算机，我就能算出来。结果呢？做不到。你知道为什么吗？因为你得给那台计算机输入初始条件，也就是开始计算时每个原子的位置和速度。而量子力学告诉我们，我连一个原子的位置和速度都无法精确给出，更别说大气中所有的原子了。而且这个系统是混沌的，也就是说它对初始条件无限敏感，你把一个原子的位置稍微改一点点，最终结果就完全不同。所以确实存在一些问题，量子力学和混沌系统告诉我们我们永远算不出来。但我们
11意识是一个谜还是26个问题
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are able to try to address fundamental problems in physics and there I agree with Gregory completely mathematics is our most beautiful tool to do this that's what I do for a living I use mathematics to try to explain the universe well I I would like to go back to this Consciousness because I think that's a very strange situation because the word Consciousness it's an it's an old word and I think it's a social word uh and if you look in my book uh I have something like 26 different meanings for the same word and the reason Consciousness is considered a mystery is because the philosophers who talk about it don't recognize that they're using it as a trash basket for for 26 or I forget the number actually of different problems about psychology that they don't understand if you look at early Freud he's saying that Consciousness is the collection of activities that result when there are conf conflicts between Basic Instincts and higher level constraints between what the individual wants and what a culture wants and uh Freud's model is very complicated it doesn't just have a
确实有能力去处理物理学中的根本问题，在这一点上我完全同意 Gregory：数学是我们做这件事最美的工具，这就是我赖以谋生的东西，我用数学去试着解释宇宙。——我想回到意识这个话题，因为我觉得那是个非常奇怪的状况。“意识”这个词是个很老的词，我认为它是个社会性的词。你翻我那本书，我在里面列了这个词大概 26 种不同的含义。意识之所以被当成一个谜，是因为谈论它的哲学家们没有意识到，他们是把这个词当成一个垃圾桶在用，用来装 26 个——具体数字我忘了——他们弄不懂的、关于心理学的不同问题。你看早期的弗洛伊德，他说意识是一系列活动的集合，是当基本本能和更高层次的约束之间发生冲突时、当个体的欲求和文化的要求之间发生冲突时所产生的活动。弗洛伊德的模型非常复杂，它不只有一个
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conscious he does use the word and he has a preconscious and he has sensors and and facilitators that act at the gates but the mystery of Consciousness that uh it's very popular nowadays people like Steven Hornet and other philosophers uh make a big uh fuss about it but as far as I can see they never look at that word and see that this is just all the Mysteries that all the things about higher level thinking that they don't understand giving a single word to it of course makes it seem like a big question a very hard question but if you break it into 26 easy questions which in chapter four of the emotion machine well then you've got a lot of things to work on and there's no General mystery at all as far as I can see well here's how philosophers um often describe it for example um Tom Nagel and what is it like to be a bat for something to be conscious means that there is something that it's like for that thing to be that thing what do you mean fact of the matter there are facts of the matter so right now is there a something it's like
意识——他确实用了这个词——他还有前意识，还有检查者和促进者在各个关口起作用。但如今很流行的那个“意识之谜”，像 Steven Hornet 之类的哲学家把它渲染得很了不得，可在我看来，他们从来没有正视过那个词，没有看到它其实就是把所有那些谜、所有那些他们不理解的高层思维现象，统统塞进一个词里。给它一个单一的名字，当然会让它显得像一个大问题、一个非常难的问题。但如果你把它拆成 26 个容易的问题——就像《情感机器》第四章里那样——那你就有一大堆可以着手研究的东西了，在我看来根本不存在什么普遍的谜。——哲学家们通常是这么描述它的，比如托马斯·内格尔的《成为一只蝙蝠是什么感觉》：一个东西是有意识的，意味着“成为那个东西”是有某种感觉的。——你这是什么意思？——是有事实的，这里有事实层面的东西。所以此刻，是不是有某种“感觉起来是怎样”的东西，
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or is there a of comp things so for right now for example I'm I'm conscious I'm not sure about anybody else but I am and you could describe me um and as a novelist of course I do this all the time I describe what is it like for that character to be in the world what is the Char what is the world like for that character uh what does it feel like um so there are Sensations there are memories there are emotions there so there 36 things what's the problem um so the thing is that could you in getting a description of that person and purely a another way of putting all of these facts that they're facts about one's subjectivity your brain has 20,000 genes do you want it to be 10 words or 50 here's the question the question is what's a description the description of a purely physical description as we have it now right a purely objective description would you be able to get out of of that the description of all that it's like to be that person at that moment or even any of them any of if it has 30,000 equations why not it's just hard from equation you could get out that right now um I am what the right now have to do with it well because Consciousness is something that takes place in the right it's not something it's a word it's an experience it is it
还是说只有一堆东西？——比如说此刻，我是有意识的，别人我不确定，但我是有意识的。你可以描述我，而作为小说家我当然一直在做这件事，我描述那个角色置身世界中是什么感觉，世界对那个角色而言是什么样的，感受起来是怎样的。——所以有各种感觉、有记忆、有情绪，所以有 36 样东西，问题在哪儿？——问题在于，在得到对那个人的描述时……换个说法，所有这些事实都是关于一个人的主观性的事实。——你的大脑有两万个基因，你是想用 10 个词还是 50 个词来说它？——问题是这样的：一个描述，一个纯粹物理的描述，就像我们现在所拥有的那种，一个纯粹客观的描述，你能不能从中推出“成为那个人在那一刻是什么感觉”的全部描述，或者哪怕其中任何一部分？——如果它有三万个方程，为什么不行？只是从方程里推出来比较难罢了。——你能从中推出，此刻我……——“此刻”跟这有什么关系？——因为意识是发生在……——那不是一个东西，那是一个词。——那是一种体验，那是
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is fact of what is like to be a living person it it there's no it let me how do I put this how do I put it without yes Marvin Marvin with all due respect you can't solve a problem by declaring that it's meaningless I know this is a popular tactic I said it's I said it's 26 problems no it didn't say it didn't say it's meaningless what Marvin is saying is that when you say Consciousness you don't mean a single thing of there are there are lots of problems involved in what one tries to bury under the concept of Consciousness and if you actually try to address please correct me if I'm paraphrasing you not in the right way if you actually try to address individual problems which you will phrase correctly and pose correctly you may or may not be able to find physical solutions to each one of those problems let me give one example uh if I might be wrong about this but I was reading Galileo once a long time ago and he had an expression it was written a long time yeah he he he
关于身为一个活人是什么感觉的事实。——根本没有那么个“它”。——让我……我该怎么说呢，我怎么说才好呢——Marvin，恕我直言，你不能靠宣布一个问题没有意义来解决它。我知道这是个很流行的策略。——我说的是它是 26 个问题。——不，他没有说、他没有说它没有意义。Marvin 的意思是，当你说“意识”时，你指的并不是单一的一个东西，人们试图埋进“意识”这个概念之下的问题有很多个。如果你真的去处理——如果我转述得不对请纠正我——如果你真的去逐个处理那些被正确表述、正确提出的具体问题，你或许能、或许不能为其中每一个找到物理层面的解答。——我举个例子，我可能记错了，不过我很久以前读过伽利略，他有个说法……——那是很久以前写的了。——对，他有个
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had an expression Vis Viva or the life of motion or whatever it means and here's the story he's talking about this thing and it's 100 years before Newton and he's got momentum which is MV and energy which is MV squ and he's using the same word that's all I'm saying the reason people are puzzled about Consciousness is it's the difference between Galileo and Newton and they should be using 26 words but they stick to it and they say it it what's it like there's no it there's 26 things that's probably more than 26 would be my guess yeah there are a trillion neurons right you know what we're talking about here I mean the fact that we're describing this in purely objective terms and purely physical terms uh a a material system and I'm not a you know I'm not a dualist I think that in fact we are our brains and that our our Consciousness is a function of our brains right using it all the time I don't use is a function of our brains how is that where's the it uh I I don't I I don't Consciousness is it it's it it's a brace basket that all
说法叫“活力”（vis viva），或者说“运动之生命”，随便它是什么意思。故事是这样的：他在谈这个东西，那是牛顿之前一百年，他手里既有动量 MV，又有能量 MV²，而他用的是同一个词。我要说的就是这个：人们对意识感到困惑，原因就在于这是伽利略和牛顿之间的差别，他们本该用 26 个词，可他们死抱着一个词，说“是有那么个感觉”——根本没有那个“它”，有的是 26 样东西。——我猜大概不止 26 样吧。——是啊，有上万亿个神经元呢。——你知道我们在谈的是什么吗？我是说，我们是在用纯粹客观的、纯粹物理的词汇来描述这件事，一个物质系统。而我不是——你知道，我不是二元论者，我认为我们事实上就是我们的大脑，我们的意识是大脑的一个功能。——对，我一直在用它。——“是大脑的一个功能”，这怎么可能？那个“它”在哪儿？我不……我不……意识是……那是个垃圾桶，装着所有
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everything that is true of me is true of my physical body um I would say except the relationships and N the things but everything that is intrinsically true of me true of my physical body so it's not I'm not I'm not proposing cartisian dualism here are two separate things but that if you were to describe as we can describe it now using biological physical uh descriptions everything about my physical body we couldn't get out of that description other facts that are true of me most people would look at a macas and say it's doing this what he's saying is that what what you say may be true that okay maybe we don't know now because we don't even know all the functions that we need exactly science is not even wrong how we would say exactly that a complete agreement there right but but but it's not as if we know that in principle that's not possible exactly exactly does it help us at all to think about um the problems in building a machine that has Consciousness just to think about what the issues are there does that help us that's why I wrote that chapter because
凡是关于我为真的事，都是关于我这具身体为真的。——我会说，除了各种关系和别的一些东西之外——但凡是内在地关于我为真的，都是关于我这具身体为真的。所以我不是在主张笛卡尔式的二元论、不是在说有两个分离的东西。但如果你用我们现在能用的生物学的、物理的描述，把关于我这具身体的一切都描述出来，我们还是没法从那个描述里推出其他一些关于我为真的事实。——大多数人看到一台机器都会说“它在做这个”。——他的意思是，你说的可能是对的，就是说，好吧，也许我们现在不知道，因为我们连需要哪些功能都还不清楚。——正是如此。——在科学上我们会说这甚至还谈不上对错。——正是这个意思，这一点我们完全一致。——对，但问题是，我们并不知道那在原则上就是不可能的。——正是，正是。——那么，去思考如何造出一台有意识的机器，思考其中涉及哪些问题，对我们有帮助吗？——我写那一章就是因为
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it seemed to me uh there are a lot of function you want a mind to have and when I started uh cataloging them and trying to get students to program them I found that there were a whole lot of them and that the philosophers who talk about it and like Chalmer and the hide the hard problem of Consciousness they are trying to do the opposite they're saying they talk about qualia what is the quality of seeing red instead of seeing this as associations with blood and with you know painting and with a hundred different things as though there's a redness but we know that some women have two Reds there isn't any red some people have no red and how do we know this about these women is it be the description of their of this because they have four kinds of cones and when you measure the right how do we know but we we ask them and we're asking them what is the quality of this experience and how are these Reds different you always have to go to the subject they're not novelists you're a novelist so when you describe quality it takes a whole book when a philosopher describes it they just say
在我看来，你希望心智具备的功能有很多。当我开始给它们编目录、并试着让学生去把它们编成程序时，我发现这样的功能有一大堆。而那些谈论意识的哲学家，比如 Chalmers 和“意识的难问题”，他们在做的恰恰相反：他们谈“感受质”，谈看到红色的那种质感是什么，而不是把它看成与血、与绘画、与上百样别的东西的联想，好像存在一种“红性”似的。可我们知道有些女性有两种红，根本不存在唯一的红，也有些人一种红都没有。而我们是怎么知道这些女性的情况的呢？——是通过对她们那个……的描述吗，因为她们有四种视锥细胞？——对，我们怎么知道的？我们去问她们。——而我们问她们的是：这个体验的质感是什么？这些红有什么不同？你总得回到主体身上。——她们又不是小说家。你是小说家，所以你描述质感要用一整本书；而哲学家描述它时，就只说
12自由意志与哥德尔式定理的野心
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oh there's a there's a qualium well if if if we were to do justice to conscious experience it would always take at least a novel uh at every moment sure well since we've all agreed on Consciousness now um what about Free Will anything anything to say about free so uh no takers no takers wouldn't wouldn't it wouldn't it be horrible if there were such a thing well Einstein would always quote schopenhauer right he said you can do what you like but you can't like what you like so like that and John Horton Conway just uh last year uh proved the theorem where basically he showed that uh if we have free will then so do electrons more or less so I mean a free will meaning things are not determined uh that our actions are not determined or our states are not determined that that sense of of free will because um philosophers often redefine free will so that it is uh compatible with determinism it's known as compatibilism uh and that in fact it demands determinism Free Will demands determinism uh if in fact our actions are are caused are determined by our own
“哦，这里有个感受质”。——嗯，如果我们要对意识体验做到公正的描述，那每一个瞬间都至少得写一部小说。——当然。好，既然我们现在都在意识问题上达成一致了，那自由意志呢？关于自由意志有什么要说的吗？——没人接？没人接？——如果真有那种东西，那不是很可怕吗？——爱因斯坦总是引用叔本华，对吧，他说：你可以做你想做的，但你不能想你所想的。就是那个意思。——还有约翰·何顿·康威去年证明了一个定理，基本上他证明了：如果我们有自由意志，那么电子也有，大致如此。我是说，这里的自由意志是指事情不是被决定的，我们的行为不是被决定的，或者我们的状态不是被决定的，是那个意义上的自由意志。因为哲学家常常重新定义自由意志，让它和决定论相容，这叫相容论；而且事实上它还要求决定论——自由意志要求决定论，只要我们的行为确实是由我们自己的
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desires which are themselves determined uh but if it's caused by a desire uh then it's free and that's a useful notion of of uh Freedom it also ties in with our notion of responsibility um so if uh I go in and hold up a bank uh because somebody has threatened to kill my children if I don't um you could say that my action was not free or maybe it was free because I wanted my children to live but in any case I mean the action of trying to save my children this was free but holding up the bank was not free because it was not my desire to hold up the bank and so I should not be held morally responsible for it basically we want a notion of Freedom that will help us out with a notion of moral responsibility so if you want to go from one large question to another I think we should get the Templeton Foundation to have another festival for this well anyway I I'm making my way towards God but we'll get there in the end but I think this this the nature of discussion indicates also the magnitude of gto's achievement as uh to return to a theme that Rebecca touched on because you see we're all these are words as a mathematician I can fantasize that someone will prove a theorem that we have free will or that we don't or that someone will prove a theorem that we have Consciousness or we
欲望引起、被欲望决定的，而这些欲望本身又是被决定的。但如果它是由欲望引起的，那它就是自由的，这是一个有用的自由概念，它也和我们的责任概念挂钩。所以，假如我去抢银行，是因为有人威胁说我不去就杀我的孩子，你可以说我的行为不是自由的；或者也可以说它是自由的，因为我想让我的孩子活下去。但无论如何，我是说，试图救我孩子这个行为是自由的，而抢银行不是自由的，因为抢银行并不是我的欲望，所以我不该为此承担道德责任。基本上，我们想要的是一个能帮我们处理道德责任概念的自由概念。——如果你要从一个大问题跳到另一个大问题，那我觉得我们该让邓普顿基金会再办一个这样的节了。——不管怎样，我正朝着“上帝”这个话题走，我们最后会走到那儿的。不过我觉得，这场讨论的性质也显示出哥德尔成就的量级，回到 Rebecca 刚才提到的那个主题。因为你看，这些都只是词语。作为数学家，我可以幻想有人会证明一个定理说我们有自由意志，或者说我们没有；或者有人证明一个定理说我们有意识，或者说我们
13哥德尔之后，纯数学到底是什么
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don't but we're a long way from that but girle proved a theorem with such deep philosophical consequences that 80 years later we're still discussing where does this leave mathematics where does this leave certainty where does this leave pure reason yeah so so as a pure mathematician I think that's Splendid and I would like us to do the same with other deep philosophical questions except I have the clue how but does it upset you as a mathematician yes it does I it upsets me that that it can't be done that's why that's why I'd like to see a theorem proving that darwinian Evolution works or that it doesn't because it's so basic and there ought to be a nice little theorem showing that it works otherwise we're not doing our job it's really it was only I mean mathem mathematicians usually prove theorems that only have to do with mathematics it's very it's not it's not pure mathematic you know a rigorous look at G GLE and Turing are really the only two that did this that proved mathematical theorems that spill out into philosophy well let's keep it up we have there good example let's not stop there so I'm not sure why why Gregory is upset uh to be honest because the fact that because the G's incompleteness theorems showed us
没有。但我们离那还远得很。可是哥德尔证明了一个哲学后果如此深刻的定理，以至于八十年后我们还在讨论：这让数学处于什么位置？这让确定性处于什么位置？这让纯粹理性处于什么位置？作为一个纯数学家，我觉得那太了不起了，我也希望我们能对其他深刻的哲学问题做同样的事，只不过我完全不知道该怎么做。——但作为数学家，这让你不安吗？——是的，让我不安。让我不安的是这件事做不到。所以我才想看到一个定理，证明达尔文式进化是行得通的，或者证明它行不通。因为这太基本了，本该有个漂亮的小定理证明它管用，否则就是我们没尽到本分。——数学家通常只证明和数学有关的定理，那不是纯粹的、你知道，严格的……——哥德尔和图灵大概是仅有的两个做到这一点的人，他们证明的数学定理外溢到了哲学里。——那我们继续保持下去吧，我们有很好的先例，别就停在那儿。——我其实不太明白 Gregory 为什么感到不安，说实话。因为哥德尔的不完备性定理向我们展示的是
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limitation of mathematics within mathematics uh Good's incompleteness theorems did not stop phys physicists from continuing to apply mathematics to the explain the universe in fact uh just but did it make them feel uncomfortable make at all can I disagree with so why is that why does it not make them feel uncomfortable well it's because it it demonstrated more the shortcomings of formal system exactly not of mathematics not mathematics as a whole necessarily or formal systems it's like you you may remember that's supposedly what mathematics is according to the current uh fashion and that's what he showed was wrong you remember but then what is pure mathematics and nobody knows you you remember that it's not a formal system it's not a for in school uh you know ukl and the people who in PL were crazy about proving everything with a straight edge and a compass anything you couldn't do with a straight edge and a compass was not considered right you know and so on until you know others came and said wait a second why do I need these chains on me I can do things otherwise you know and so on so in the formal system let me go back to the barber that I mentioned before so there is this Paradox right about the barber
数学在数学内部的局限。哥德尔的不完备性定理并没有阻止物理学家继续用数学去解释宇宙。事实上————可它有没有让他们觉得不自在？——一点儿也不。——我能不能不同意……——那为什么不会让他们不自在呢？——因为它更多是揭示了形式系统的缺陷。——正是。——而不是数学的缺陷，不一定是整个数学的缺陷。——或者说是形式系统的缺陷。你也许还记得，按照当时流行的看法，数学被认为就是形式系统，而哥德尔证明的正是这一点是错的。你记得吧？但那纯数学到底是什么呢？没人知道。你记得的，它不是一个形式系统，不是一个形——在学校里，你知道，欧几里得和那些人痴迷于用直尺和圆规证明一切，凡是直尺圆规做不出来的就被认为不正当，如此等等，直到后来别人来了说，等一下，我干嘛要戴着这些镣铐，我可以用别的方式做嘛，等等。所以在形式系统里，让我回到我之前提到的那个理发师，有那个悖论对吧，关于理发师的
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not bothered because you're a physicist we mathematicians some of us were bothered then we solve the problem the way usually do if you can't solve a problem convince yourself that it's not important forget about it but wait wait a second I mean good good the problem that g posed is is still is still on the table what is pure mathematics Hilbert said pure mathematics is a formal acatic system which I'm going to discover and that's it and now we know exactly what it means something is trueth if I can prove it according to these rules GTO show that won't work you're right so now where does that leave pure mathematic the answer is nobody knows I'll tell you where it I I'll tell you what you don't care because you're a physicist no no I'm number one I'm a theoretical physicist which means I use mathematics every day now but but here listen we know that we have a problem with the foundation of quantum mechanics we have serious problems with the foundation of quantum mechanics you know all these things about how does Observer and observer determine the results you know and so on the famous if a tree falls in the middle of the forest I I there is another version of it that I like better which is if a man talks in the middle of a forest and
——你不困扰，因为你是物理学家；我们数学家里有些人当时是很困扰的。后来我们用惯常的办法解决了问题：如果你解决不了一个问题，就说服自己它不重要，然后忘掉它。——但等一下，等一下，我是说，哥德尔提出的那个问题现在仍然摆在桌面上：纯数学到底是什么？希尔伯特说纯数学就是一个形式公理系统，我要把它找出来，就这样，然后我们就确切知道它是什么意思了：如果我能按这些规则证明某个东西，它就是真的。哥德尔表明这行不通。——你说得对。——那现在纯数学处于什么位置？答案是没人知道。——我来告诉你它——我来告诉你你为什么不在乎，因为你是物理学家。——不不，第一，我是理论物理学家，也就是说我每天都在用数学。但你听我说，我们知道量子力学的基础是有问题的，量子力学的基础有严重的问题，你知道那些说法，观察者如何、观察者决定结果之类的，等等。那个著名的“如果一棵树在森林中央倒下”——我更喜欢它的另一个版本，就是：如果一个男人在森林中央说话，而
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there is no woman around is he still wrong so uh so uh yes I knew you would say that I knew you would say that no so we know that there is a problem with the foundation of quantum mechanics there actually serious problem with the has that stopped physics from going on the answer is no have gle's theorems stopped mathematics from going onit a second physics go on as technology but who cares about technology the fundamental conceptual issues of the foundations of quantum mechanics are really terrible and if you think if you're interested in thinking and understanding the world this is a serious problem now if you're interested only in making transistors and making a billion dollars you don't care but if you want to understand how the World functions it's important that problem you're too practical I don't make was not a practical guy GLE was not interested in making money I don't I don't have the first notion of how to make a transistor I tried to understand what dark energy is or if we live in a Multiverse no I mean okay you're forgiven
「如果周围没有女人在，他还算不算错？」——是的，我就知道你会这么说，我就知道你会这么说。不，我们确实知道量子力学的基础有问题，而且是很严重的问题。但这阻止物理学继续往前走了吗？答案是没有。哥德尔的那些定理阻止数学继续往前走了吗？——等一下，物理学是作为技术在往前走，可谁在乎技术呢？量子力学基础中那些根本性的概念问题真的很糟糕。如果你感兴趣的是思考、是理解这个世界，那这就是个严肃的问题。当然，如果你只想造晶体管、赚十亿美元，那你不在乎。但如果你想理解世界是怎么运作的，那个问题就很重要。——你太讲实用了。——我可不是个讲实用的人，哥德尔也对赚钱没兴趣。我连怎么造一个晶体管都毫无概念，我想弄明白暗能量是什么、我们是不是生活在多重宇宙里。——好吧，那原谅你了。
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my my point my point is that G's theor theorems in spite of the fact that you know certainly they are of huge import they eventually not stopped the progress of mathematics either and that's an interesting question how come that's an interesting question in itself which I would like to understand better and I don't understand but girdle himself said I mean that he was showing the limitations of formal systems to capture everything that we know mathematically but he wasn't showing the um he wasn't showing the limitations of mathematical knowledge I mean he was in in his uh uh paper what is uh Canter's Continuum hypothesis he talks about this and he says what he's shown is that mathematical intuition can't be eliminated in it can't be eliminated from for formal system so the question comes back to the mystery of mathematical intuition which is partly The Mystery of the hard problem of Consciousness well yeah I mean ghetto himself has some lovely philosophical essays arguing that his theorem does not limit the power of mathematicians to solve any problem however if you look at these papers they are full of words they look like philosophy papers and therefore they're open to discussion endlessly exactly the greatness of gle's
我的意思是，尽管哥德尔的那些定理无疑意义重大，但它们最终也并没有让数学的进展停下来。这本身就是个有意思的问题：为什么会这样？这是我很想更好地理解、但目前并不理解的问题。不过哥德尔自己说过，他展示的是形式系统在捕捉我们全部数学知识方面的局限，而不是数学知识本身的局限。在他那篇《什么是康托尔的连续统假设？》里他谈到过这一点，他说他所证明的是：数学直觉是无法被消除的，无法从形式系统中被消除掉。所以问题又回到了数学直觉之谜，而这在某种程度上就是意识难题之谜的一部分。——是啊，哥德尔自己写过一些很漂亮的哲学随笔，论证他的定理并没有限制数学家解决任何问题的能力。然而你去看那些文章，它们全是文字，看起来就像哲学论文，因此可以被无休止地争论下去。——正是如此。哥德尔第一篇论文的伟大之处在于
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first paper is that it deals with a fundamental philosophical issue and it settles it via a mathematical proof GLE himself was not happy with these essays that he wrote at the end of his life some of them he never allowed to be published because he never found any proofs it was they were all plausible arguments they were all words and so what we need is in my opinion is to duplicate GLE himself I think was trying to find a proof of some of these results and he never allowed these papers to be published because they were just words and he was looking for a mathema IAL proof so I think that we still need to understand these questions better sure but when you say that the first paper was so fantastic because he got a philosophical you said it I got it from your book this your profound Insight I agree with got your book all right I was going to ask you but what is that philosophical Insight that comes out of the 1931 paper well that mathematics doesn't give absolute certainty because it's not a formal system that's one way to put it I don't think girl would agree with that I think that he would say that mathematics does give absolute certainty even though it's not a formal system okay so this is 80 years
它处理的是一个根本性的哲学问题，而且是用一个数学证明把它解决掉的。哥德尔本人对他晚年写的那些随笔并不满意，其中有些他从来不允许发表，因为他始终没找到证明——那些全都只是看起来合理的论证，全都只是文字。所以在我看来，我们需要做的是重复哥德尔本人的努力：我想他一直在试图为其中一些结论找到证明，而他不允许这些文章发表，就是因为它们只是文字，他要的是数学证明。所以我认为，我们仍然需要更好地理解这些问题。——当然。不过你说第一篇论文之所以了不起，是因为他把一个哲学问题……——是你说的。——我是从你书里读到的，这是你深刻的洞见，我很赞同。——你拿到我的书了，好吧。——我本来想问你：从1931年那篇论文里得出的那个哲学洞见到底是什么？——就是数学并不提供绝对确定性，因为它不是一个形式系统。这是一种说法。——我不认为哥德尔会同意这个说法。我想他会说，数学确实提供绝对确定性，尽管它不是一个形式系统。——好吧，这都过了八十年了，
14水母思想实验：数学是发现还是发明
73:39
later and we're still arguing over what GLE did or did not do could I ask another could I ask another question if um if I I think that Michael AA may have said this I can't remember if I was a jellyfish at the bottom of the ocean was that yeah I can I can repeat this story repeat the story and then when you've done that um tell me that if you were a jellyfish in instead of feel I am would you have a different concept of the physics a different concept so here so here is it's not physics really it is the question is was our mathematics in some sense inevitable so our mathematics started by the B iians and the Greeks and the Egyptians and it started in particular with two Fields arithmetic and geometry and in arithmetic the most basic concept you can think of are natural numbers 1 2 3 4 5 6 right so you might have thought that there is no way that any kind of civilizations wherever they are they would have to come up with the natural number so those are inevitable so then comes the question question suppose that intelligence resided not in humans but in some
我们还在争论哥德尔到底做了什么、没做什么。——我能再问一个问题吗？我想这话可能是迈克尔·阿蒂亚说过的，我记不清了——「如果我是海底的一只水母」……——对，这个故事我可以再讲一遍。——你先讲这个故事，讲完以后再告诉我：如果你是一只水母，而不是你现在感觉自己所是的东西，你会不会有一套不同的物理学概念？——好，那么，这其实不太算物理学的问题，问题是：我们的数学在某种意义上是不是必然的？我们的数学是从巴比伦人、希腊人和埃及人那里开始的，尤其是从两个领域开始的：算术和几何。在算术里，你能想到的最基本的概念就是自然数：1、2、3、4、5、6，对吧。所以你可能会以为，任何一种文明，不管它在哪里，都不可能不发展出自然数——它们是必然的。那么问题就来了：假设智能并不寓于人类，而是寓于某种
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isolated jellyfish living at the bottom of the Pacific Ocean all that this jellyfish can feel is the temperature of the water the pressure of the water the motion of the water would this jellyfish have come up with the natural numbers and it can't count there's no fish for it to count there is nothing to count this is why I said isolated isolated isolated so so it's not not sure that it would have come up with the with that with that system what about pellets of food what isn't it eating something yeah so it may be it may be just taking in water and taking whatever goes with it get EIN has a remark a remark on this that I feel is very deep Einstein has uh an essay or two where he says even the positive integers are not a priori they you think they're a necessary tool of thought but they are actually a free creation of the human mind invented by us to organize our sense perceptions and this is a quasi empirical view of mathematics that we invent mathematics and that even the most fundamental concepts in pure mathematics are inventions and this is a
孤零零生活在太平洋海底的水母身上。这只水母能感觉到的只有水的温度、水的压力、水的流动。它会发展出自然数吗？——它也没法数数啊，它没有鱼可数，什么可数的东西都没有。——所以我才说「孤立的」，孤立的。——所以说，它是不是会发展出那样一套体系，并不确定。——那食物颗粒呢？它总得吃点什么吧？——是啊，也许它就是把水吸进来，顺带把水里的东西一起吃进去。——爱因斯坦对这个有一段评论，我觉得非常深刻。爱因斯坦在一两篇文章里说过：即便是正整数也不是先验的，你以为它们是思维的必然工具，但其实它们是人类心智的自由创造，是我们为了整理感官知觉而发明出来的。这是一种准经验主义的数学观：数学是我们发明的，甚至纯数学中最根本的概念也是发明。而这也是
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view that GLE discusses because the two of them talked all the time in those late essays yeah but the question I call it a quas empirical view following lcos of the found of pure mathematics so I was trying to go one step further you said it wasn't a question of physics because the reason I was asking the question was because mathematics to us can make so much sense of physics if I was a jellyfish would I create a different physics or or a different mathematics because I would have a different math so the question is for example why you you may ask and that's a deep question why did the Ang ient Babylonians start with arithmetic and geometry and I'll give you what why I think no that's why they chose a number particular number for their base but the question is why at all to start with arithmetic and geometry and I'll tell you what I think this is this is a personal opinion I think that this has a lot to do with our perception system our perception system and and and Gregory just alluded to that our perception system we are very very good at seeing the boundaries of things I can see that this is the end of Rebecca and then there is some background and so on we're very good at this this had to do with the fact that we started counting things we are very
哥德尔讨论过的观点，因为那些晚年的随笔时期他们俩一直在交谈。——是的，不过按照拉卡托斯的说法，我把这叫作对纯数学基础的准经验主义观点。——我想再往前推一步。你说这不是物理学的问题，但我之所以问这个问题，是因为数学能帮我们把物理讲得那么通。如果我是一只水母，我会不会创造出一套不同的物理学？或者一套不同的数学，因为我会有不同的数学？——所以问题是，比如说你可以问——这是个很深的问题——为什么古巴比伦人是从算术和几何开始的？——我可以告诉你我认为的原因。——不，那是他们为什么选了某个特定的数作底数。问题是，为什么一开始是从算术和几何入手的。我告诉你我的想法，这是个人看法：我认为这跟我们的感知系统有很大关系。我们的感知系统——格雷戈里刚才也提到了——我们非常非常擅长看出事物的边界。我能看出这里是丽贝卡的边缘，那边是背景，等等，我们对此非常在行。这跟我们开始数东西有关。我们还非常
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very good at seeing straight lines and curves we can tell that something is a circle and not an ellipse again because of the way our perception system works this is I think why we started with geometry maybe if we saw everything in the infrared let's say where every all the lines would have been blurred you know and so on maybe we would have come up with a different concept because we could have started with a different concept where absolute truth has has has gone it's completely gone nowadays but but G believed in absolute truth and I still think it's a it's a good thing to think about even if it's not fashionable it may come back you know when you mean a good thing to think about a good thing to take seriously or just a good thing to Wonder upon all of the above I don't quite see the difference yeah so so Marvin for example can tell you that we could have started mathematics with these things which we call Cellular automata which are like little computer programs you start with a certain small set of rules you put one White pebble next to it you put a black Pebble you know and so on you can show that you can derive much of our mathematics and physics from such small
非常擅长看直线和曲线，我们能分辨出某个东西是圆而不是椭圆，这同样是因为我们感知系统的工作方式。我想这就是为什么我们从几何开始。也许，如果我们看到的一切都是红外的，所有线条都是模糊的，那我们可能会发展出不同的概念，因为我们会从不同的概念出发。——「绝对真理」这个东西如今已经彻底不见了，完全没有了，但哥德尔是相信绝对真理的，我仍然觉得这是件值得思考的事，哪怕它现在不时髦，说不定以后还会回来。——你说「值得思考」，是指值得认真对待，还是仅仅值得琢磨琢磨？——以上皆是，我看不出有什么区别。——是啊，比如马文就会告诉你，我们本可以从所谓的细胞自动机出发来建立数学——那些像小计算机程序一样的东西：你从一小组规则开始，放一颗白石子，旁边放一颗黑石子，如此等等；可以证明，你能从这样的小
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computer programs in experiment that Danny babro and I did once in the early days uh of computers uh namely we uh wrote a program that listed all possible touring machines starting with the very simplest ones and uh in the first few thousand of them most of them did nothing and some of them did little circles like counting up to three and starting over and a few of them counted up started counting the integers and uh in the first uh I think 50 or 100,000 uh none of the machines did anything interesting except the ones that counted interesting interesting so in other words arithmetic is among the simplest things that can happen that's a beautiful experiment you know this raises an issue which is the traditional plate in this view is God is a mathematician the world is built out of pure math there is a more modern version of this which Mar which Marvin was alluding to directly which is that God is a computer programmer the world is built out of software yeah that's sort of the current uh Reincarnation of that older view in some sense this doesn't
程序里推导出我们大部分的数学和物理。——早年计算机刚出现的时候，我和丹尼·博布罗做过一个实验：我们写了一个程序，把所有可能的图灵机从最简单的开始列出来。在最初的几千台里，大多数什么也不做，有一些会做点小循环，比如数到三然后重新开始，还有少数几台开始数整数。在最初的——我想是五万还是十万台里——除了那些会计数的机器之外，没有一台机器做出任何有意思的事。——有意思，有意思。——换句话说，算术是可能发生的最简单的事情之一。——这个实验真漂亮。——你知道，这就引出一个问题：传统的柏拉图式观点是「上帝是数学家」，世界是由纯数学构成的；而它有一个更现代的版本，也就是马文刚才直接暗示的：上帝是个程序员，世界是由软件构成的。——是啊，这可以说是那个古老观点在当下的转世。——在某种意义上，这在我看来
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seem to me so very interesting that is that what had to be true of our cognitive faculties in order for us to discover mathematics or if our cognitive faculties were different we could have discovered these structures as well or maybe you know our if our cognitive faculties were so very different mathematics couldn't be discovered we would not be able to access it doesn't seem to me to address the question of all is does there exist something independent there to be discovered whether we can you know it's a lucky fluke that we evolved intelligences that could discover this I mean to me it's amazing that we know anything at all um that uh that we the the the the products of blind processes of evolution know anything can sit up here talking about gdle incompleteness theorems or or relativity Theory or quantum mechanics um it seems to me extraordinary so um the fact that if we had different cognitive abilities we wouldn't be able to access the the few things that we do access does doesn't seem to me to get at
并没有那么有意思。也就是说，我们的认知能力必须具备什么条件，我们才能发现数学？如果我们的认知能力不同，我们是不是也能发现这些结构？或者，如果我们的认知能力差别太大，数学根本就发现不了，我们无法触及它？在我看来，这些都没有触及那个真正的问题：那里是否存在某种独立于我们、等着被发现的东西？至于我们能不能发现它——我们演化出了能够发现这些东西的智能，这只是一个幸运的偶然。对我来说，我们居然能知道任何东西，这本身就很惊人：我们这些盲目演化过程的产物，居然能知道些什么，居然能坐在这里谈论哥德尔不完备定理、相对论或者量子力学，在我看来这非同寻常。所以说，「如果我们有不同的认知能力，就无法触及我们现在所触及的这一点点东西」——在我看来这并没有触及
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the heart of the objectivity of what we're probably most planets don't don't have time because it took about half a billion years uh when did life start fairly early life started very early 3.7 billion years okay but planets like this with the sun we only have three more billion years before the sun goes Nova no it doesn't go Nova becomes a red giant yeah uh and so uh it might be that uh most planets don't have mathematicians it's too bad so well there's a pressing question let's say we get signals from some other star so there are these intelligent jellyfish right on some other solar system and obviously we will want to communicate but we don't have much in common with these intelligent jellyfish so what can we talk about gum maybe GLE theum maybe uh uh prime number the prime number theorem number Theory maybe logic maybe philosophy or maybe nothing you know maybe we have nothing in common we can just send them uh cartoons or so yeah I have to Pi up our TV TV programs uh GLE had had very little um faith in our
我们所谈论之物的客观性的核心。——很可能大多数行星都没有足够的时间，因为这大概花了五亿年。——生命是什么时候开始的？——相当早，生命开始得非常早，三十七亿年前。——好，但像这样有一颗太阳的行星，我们在太阳变成新星之前只剩三十亿年了。——它不会变成新星，是变成红巨星。——对。所以可能大多数行星上都没有数学家，太可惜了。——那么有个很迫切的问题：假设我们从别的恒星那里收到了信号，那边某个星系里真有一群聪明的水母，显然我们会想跟它们交流，可我们跟这些聪明的水母没多少共同点，那我们能聊什么呢？——也许聊哥德尔定理，也许聊素数定理、数论，也许聊逻辑，也许聊哲学，也可能什么都聊不了——也许我们根本没有共同点，只能给它们发点漫画之类的。——是啊，还有我们的电视节目。——哥德尔对我们
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ability to be able to communicate with one another with words um and that's why he wanted to although he had these deep philosophical convictions um he never breathed a word of them to these people who were discussing them he wanted a proof a proof is was it and there is this one story I love so much of he he was coming back from um a meeting of the Vienna Circle and he said to another mathematician minger um uh well I think minger said to him tonight we out Vicken Stein all those Vicken steinan we kept completely silent because Vicken Stein ends his tractatus by saying um you know about what we can't speak we must be silent you know a lovely tautology and um uh GLE responded back he said the more I think about language the less uh I believe that we can talk about anything of interest to one another that we can understand each other through language at all I often have that feeling in talking but hopefully but hopefully we haven't shown that to be the case this there's a whole film it's called Lost in
用语言相互交流的能力抱有非常小的信心。所以尽管他有那些深刻的哲学信念，他从来没有对那些正在讨论这些问题的人透露过一个字，他要的是证明，只有证明才算数。有个故事我特别喜欢：他有一次从维也纳学派的聚会回来，对另一位数学家门格尔说——其实是门格尔对他说的——「今晚我们比维特根斯坦还维特根斯坦」，因为我们全程一言不发。维特根斯坦的《逻辑哲学论》结尾就是那句「对于不可言说之物，必须保持沉默」，一句可爱的同义反复。而哥德尔回应说：我越是思考语言，就越不相信我们能够就任何有意思的事情彼此交谈，越不相信我们能通过语言相互理解。——我在谈话中也常有这种感觉。——但愿——但愿我们今天没有印证这一点。——这里有一整部电影，就叫《Lost in
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Translation which is so so I I I want to just come back to you you asked the question and and Rebecca just touched on it is so are we discovering mathematics or are we inventing them and I I'll give you this is a personal opinion and mathematicians discuss this to death and Gregory will say I'm a physicist but yes okay so no pler says we remember mathematics remember meno yes so we don't discover it or invent it so so so here is the thing when you POS the question like this was mathematics discovered or invented I think you have already committed an error and the error is that when you pose it like this people immediately assume that the answer has to be that it is discovered or it is invented and it it cannot be something different now the way I think about this is that actually mathematics is an intricate combinations of inventions and discoveries and let me just say very broadly very broadly we invent the concepts and then we discover the relations among the concepts I'll give you a specific example there is no square root of minus one there is no number that is the square root of minus one okay humans at some point had to invent that concept they invented the concept once they invented the concept they discovered that there is all kinds of
Translation》（迷失东京）。——我想回到你刚才提的那个问题，丽贝卡也刚刚触及了：我们究竟是在发现数学，还是在发明数学？我给个个人看法——数学家们对这个已经争论到死了，而格雷戈里会说……——我是物理学家。——对，好吧。——柏拉图说我们是在「回忆」数学，记得《美诺篇》吗？所以我们既不是发现它也不是发明它。——是这样：当你像这样提出问题——数学是被发现的还是被发明的——我认为你已经犯了一个错误。错误在于，你这么一问，人们立刻就假定答案必须是「发现」或者「发明」，不可能是别的什么东西。而我的看法是，数学其实是发明与发现的复杂交织。非常粗略地说：我们发明概念，然后我们发现这些概念之间的关系。举个具体的例子：负一的平方根是不存在的，没有哪个数是负一的平方根。人类在某个时刻必须发明出这个概念。而一旦发明了这个概念，他们就发现，围绕它可以做出各种各样的
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mathematics you can do with that you know and so on and this so you invented the concept then you discovered lots of things prime numbers which Gregory mentioned wait wait he will disagree in a second yeah can I just give you a perspective from the other side of the wall you know there are lots of we all have to survive in this competitive world so you have to publish a lot a lot of the math papers that are published frankly look invented artificial you know they are giving answers to questions which really were better left unposed right yes but when you find a piece of mathematics that is really basic that is really fundamental you sometimes maybe it's an illusion you get the feeling that there was a certain inevitability to it that you're just discovering something that existed there in the pl I with you that's your feeling but most mathematics I agree it's right now is not reach this exalted take even take even the prime numbers which is something that is of course favorite of everybody's the Indian mathematicians Chinese mathematicians never actually invented the concept of prime numbers it does not mean they did know that prime numbers exist but as a concept they did not declare it as such and when you look
数学，等等。所以你先发明概念，然后发现大量的东西。比如格雷戈里提到的素数……——等等等等，他马上就要反对了。——是啊，我能不能从墙那边给个视角？我们都得在这个竞争激烈的世界里活下去，所以你得大量发表论文。老实说，已发表的数学论文里有很多看起来是发明出来的、人造的，它们是在回答一些其实最好别提出来的问题。——对，没错。——但是当你碰到一块真正基础、真正根本的数学时，你有时会有一种感觉——也许这是错觉——觉得它带着某种必然性，觉得你只是发现了本来就存在于那里的东西。——这一点我同意你，那是你的感受，但我同意，眼下大多数数学并没有达到那种崇高的层次。——就拿素数来说吧，这当然是大家都最喜欢的东西。印度数学家、中国数学家其实从来没有发明出「素数」这个概念。这不是说他们不知道素数的存在，而是作为一个概念，他们没有把它明确地确立起来。而当你去看
15收尾：三种「理解的极限」
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at their mathematics it doesn't appear there because they did not invent the concept now of course everybody knew that there are prime numbers there but the fact that ukl proved that there is an infinite number of them wrote a mathematical proof and so on they became a very very fundamental part of mathematics now Gregory is going to say yeah we discovered so many things there about them you know and so on they feel like a discovery yes most of them are a discovery once you decided that the concept is there right we're nearly at the end I think I want no more than one minute from each of you one no more than one minute from each of you on what you think is the major limit of our understanding starting with Gregory the major limit of our understanding um what what I what I do think is that we don't understand the goodle incompleteness yet it's it's still a um a challenge to us um the major limit of understanding complexity perhaps some things are too complicated for us um they it requires too much amount of too much information to hold
他们的数学时，素数并没有出现在里面，因为他们没有发明这个概念。当然，人人都知道那里有素数，但正因为欧几里得证明了素数有无穷多个、写下了数学证明等等，素数才成为数学中非常非常基础的一部分。现在格雷戈里要说了：是啊，我们围绕素数发现了那么多东西，它们感觉就像是被发现的。是的，其中大多数确实是发现——但前提是你已经决定了那个概念存在于那里。——好，我们快到尾声了。我想请每位不超过一分钟——每位不超过一分钟——谈谈你认为我们理解力的主要极限是什么。先从格雷戈里开始。——我们理解力的主要极限……我确实认为我们还没有理解哥德尔不完备性，它对我们仍然是个挑战。理解力的主要极限也许是复杂性：有些东西对我们来说太复杂了，需要太多的信息量才能装进
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in the human in the human mind uh in biology you start to stretch to reach those limits and um and um um so I think complexity is a is a is a very important limit Rebecca oh where just want begin um I there are just any philosophical problem measures the limits of our understanding I think my field philosophy is what it is is uh all of the problems that we are smart enough to ask some of us um and I mean it's it's really the hard thing is to even understand what the problems are uh just smart enough to ask and not smart enough to answer so this just go to philosophy and you will find uh all of the limits of our understanding all I think that right now what I see as the most major limit to our understanding is the fact that we cannot quite know what is truly fundamental and what may be just accidental Kepler thought that the number of planets in the solar system was a fundamental thing which he needed to explain from first principles today we know that's not the case it may be that some of the things that we now perceive as fundamental are not really
人的头脑里。在生物学中，你就开始逼近这些极限了。所以我认为复杂性是一个非常重要的极限。——丽贝卡？——哦，从哪儿说起呢……我觉得任何一个哲学问题都在丈量我们理解力的极限。我的领域——哲学——之所以是它现在这个样子，就是因为它包含了所有那些我们（其中一些人）聪明到足以提出的问题。而真正难的是甚至弄清楚这些问题究竟是什么。我们刚好聪明到能提出它们，却又不够聪明到能回答它们。所以你只要走进哲学，就会找到我们理解力的所有极限。——我认为，目前我看到的我们理解力最主要的极限，是我们无法真正分辨什么是真正根本的、什么可能只是偶然的。开普勒曾认为太阳系中行星的数目是一件根本性的事情，是他需要从第一性原理出发去解释的；今天我们知道并非如此。也许我们今天视为根本的某些东西其实并不
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such and they are accidental perhaps in a Multiverse you know that we happen to live in a universe that has such things and so on I think that the fact that we don't know this distinction is a major limit to our understanding Marvin well I have the a diff slightly different answer which I would hope that chayon and uh Rebecca would agree with with well chayon will uh it seems to me that the most important Discovery since girdle was the discovery by Chon solomonov and Cole mov of a concept called algorithmic probability which is the fundamental new theory of how to make predictions given a collection of experien and uh this is a beautiful Theory everybody should learn it and but it's got one problem which is that you can't actually calculate what this theory predicts because it's too hard and uh requires an infinite amount of work uh however it should be possible to make practical approximations to the chaon korov Solomon of theory uh that will make better predictions than anything we
根本，而是偶然的——比如在多重宇宙中，我们碰巧生活在一个具有这些性质的宇宙里，等等。我认为，我们不知道这个区分，正是我们理解力的一个主要极限。——马文？——我的答案略有不同，但我希望蔡廷和丽贝卡会同意——好吧，至少蔡廷会同意。在我看来，自哥德尔以来最重要的发现，是蔡廷、索罗门诺夫和柯尔莫哥洛夫发现的一个叫作「算法概率」的概念，它是关于「给定一堆经验之后如何做出预测」的根本性新理论。这是个很漂亮的理论，人人都该学一学。但它有一个问题：你实际上算不出这个理论预测的是什么，因为太难了，需要无穷多的工作量。不过，应该有可能对蔡廷–柯尔莫哥洛夫–索罗门诺夫理论做出实用的近似，从而做出比我们今天所有方法
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have today and everybody should learn all about that and spend the rest of their lives working on it I think this evening we've leared one thing at least that there's very little limit to the understanding of our panelists thank you very [Applause] much I think we go ladiesent e e e
都更好的预测。人人都该好好学这个，并把余生投入其中。——我想今晚我们至少学到了一件事：我们几位嘉宾的理解力几乎没有什么极限。非常感谢。（掌声）我想我们就……各位……
视频总结 · 一句话概括与核心要点
一句话概括
世界科学节的这场圆桌以哥德尔不完备定理为轴心，由数学家 Gregory Chaitin、哲学家 Rebecca Goldstein、天体物理学家 Mario Livio 和 AI 先驱 Marvin Minsky 探讨数学确定性的边界，进而延伸到数学为何"不合理地有效"、意识、自由意志、数学是发明还是发现等问题，最终各自给出对"人类理解之极限"的判断。
核心要点
- 哥德尔在 24 岁时击碎了希尔伯特纲领。 1920 年希尔伯特号召证明数学的一致性；1931 年哥德尔提出两条不完备定理：（1）任何足以表达算术的形式系统中都存在不可判定命题——它和它的否定都无法在系统内证明；（2）一个系统无法在自身内部证明自身的一致性。而不一致的系统毫无用处，因为从矛盾可以推出任何命题。
- 这一发现最初几乎无人理会，是冯·诺依曼将其传播开来。 哥德尔在柯尼斯堡会议最后一天低声宣布结果，会议纪要（发表于逻辑实证主义刊物《Erkenntnis》）甚至没有提及；只有代替希尔伯特出席的冯·诺依曼追问后意识到其重要性，回到普林斯顿后写信指出"推论是算术一致性不可证明"，哥德尔回复称已有严格证明。
- 哥德尔是身处维也纳学派中的沉默异见者。 维也纳学派信奉逻辑实证主义与形式主义，把"无意义"作为判罪术语，奉维特根斯坦为神；哥德尔却是柏拉图主义者，相信数学描述超经验的实在。他完全不同意周围人，但在拿到证明之前一言不发。Goldstein 称他是"中世纪的人"，相信先验必然真理、不信经验科学。
- Chaitin 用算法信息论把不完备性"自然化"：纯数学含有无限不可约复杂性。 他的 Omega 数（源自图灵停机问题）二进制展开的每一位都像"上帝掷硬币"，无法预测也无法证明；而任何数学理论只有有限复杂度，因此不完备是必然。他进一步认为哥德尔与图灵 1936 年的工作实际上是把数学推向生物学方向——生物复杂度虽大但有限，纯数学却更糟。他正在建立"元生物学"，核心是"DNA 是数字软件""变异软件的演化""软件空间中的随机游走"。
- 三大数学基础流派均遭重创。 形式主义（数学如国际象棋，改公理即改游戏，非欧几何是典型例证）被哥德尔证明无法穷尽数学真理——"数学没有万有理论"；逻辑主义（弗雷格、罗素，主张从逻辑公理推出全部算术）在弗雷格著作付印前收到罗素悖论（理发师悖论变体）而崩塌；柏拉图主义虽为多数工作数学家的心之所向，却无法解释我们如何获得对超验领域的知识。
- "数学不合理的有效性"有惊人数据支撑。 开普勒的观测精度只有约 4%，牛顿据此写出的引力定律到 1950 年代精确到百万分之一，2008 年实验证实平方反比律在 56 微米尺度仍成立；量子电动力学计算电子磁矩与 2006 年实验测量吻合到万亿分之八。Minsky 的回应是人择式解释：能量不守恒的宇宙里 DNA 无法稳定十亿年，也就没有哲学家来讨论此问题；Goldstein 则提出物理学只从世界中"分离出结构"，因此规律必然是数学的——这本身是另一种不完备。
- Minsky 指出公众对量子力学的误解恰好颠倒。 牛顿力学的太阳系并不稳定（Sussman 与 Wisdom 的计算表明木星最终可能把冥王星甩出），而室温下的 DNA 分子却能稳定十亿年——演化之所以可能是"因为"量子力学而非"尽管"有量子力学。他还指出达尔文演化只记住成功、忘记失败，单靠它可能演化不出大脑，需要语言与文化（童话大多在讲致命错误）来传递教训。
- 意识问题引发最激烈交锋。 Minsky 认为"意识"是装了 26 个不同问题的"垃圾篮"，一旦拆分就无神秘可言，如同伽利略用同一词"活力"混淆动量 mv 与能量 mv²；Goldstein 以 Nagel 的"作为蝙蝠是什么感觉"反驳，认为纯客观物理描述无法导出主观事实，且不能靠宣布问题无意义来解决问题。Livio 调和：目前不能，不代表原则上不能。
- 数学是"发明概念、发现关系"的交织。 Livio 以 √-1 为例：概念是发明的，其后的数学结构是发现的；印度和中国数学家从未把素数作为概念提出。Chaitin 补充：大量论文里的数学看起来是人为编造的，但真正基础的数学给人"不可避免"的发现感。Einstein 甚至认为正整数也不是先验的，而是人类为组织感知而自由创造的。Livio 的"海底孤立水母"思想实验暗示自然数并非必然；Minsky 与 Bobrow 早年枚举图灵机的实验则显示，前几万台机器中唯一"有趣"的行为就是计数——算术是最简单可能发生的事之一。
结论与值得注意的细节
- 四人对"理解的最大极限"的收尾回答各异：Chaitin——我们至今仍不理解哥德尔不完备性，且复杂性本身可能超出人脑容量（生物学已触及此边界）；Goldstein——每一个哲学问题都是理解极限的刻度，我们"聪明到能提问，却不够聪明去回答"；Livio——我们无法区分什么是真正基本的、什么只是偶然的（开普勒曾以为行星数目需要第一性原理解释，多重宇宙下许多"常数"可能只是偶然）；Minsky——哥德尔之后最重要的发现是 Solomonoff、Kolmogorov、Chaitin 的算法概率论，它是做预测的根本理论，只是不可计算，应致力于其实用近似。
- Chaitin 与 Livio 的分歧值得玩味：Chaitin 认为"纯数学到底是什么"至今无人知晓，且这令他作为数学家感到不安，他希望有人能像哥德尔那样用定理（而非哲学论文）证明达尔文演化是否可行；Livio 则指出，正如量子力学基础问题未阻止物理学前进，哥德尔定理也未阻止数学前进——因为它揭示的是形式系统的局限，而非数学本身的局限。哥德尔本人在《康托尔连续统假设》一文中也持此立场：定理证明了数学直觉不可从形式系统中消除。
- 哥德尔晚年撰写的哲学论文因"只有文字没有证明"而拒绝发表；他对语言极度不信任，曾对 Menger 说"我越思考语言，越不相信人们能通过语言谈论任何有趣的东西"。这与他的人生结局相呼应：因偏执地害怕被下毒而绝食，72 岁时在普林斯顿医院去世，体重仅 65 磅；爱因斯坦晚年称去研究院只是为了享受与哥德尔一同走回家的特权。
- 自由意志话题上，Chaitin 提到 Conway 去年证明的"自由意志定理"（若人有自由意志则电子也有），Goldstein 介绍相容论——自由意志要求决定论，其价值在于支撑道德责任概念（被威胁而抢银行的例子）。
- Livio 补充了一个常被忽略的计算极限：即使方程已知，大气中每个原子的演化也不可计算——量子力学禁止精确初始条件，而系统混沌意味着对初始条件无限敏感。
核心句型 · 10
1. X only makes sense in terms of Y
“Theories of the ultimate structure of matter only make sense in terms of mathematics”
表示「只有借助 Y 才能理解 X」。in terms of 引出解释框架，适合学术写作中说明某概念的依赖关系。仿写：This policy only makes sense in terms of long-term costs.
2. pull the rug out from under X
“In 1931 GLE pulls the rug out from under that”
习语，字面「从脚下抽走地毯」，指突然摧毁某观点或计划的基础。本场被 Chaitin 和 Livio 各用一次。适用于描述颠覆性发现。
3. not because of X but in spite of X（反转版：because of, not in spite of）
“This is why evolution is possible because of quantum mechanics not in spite of it”
用 because of / in spite of 的对举制造反直觉效果：常识以为 X 是阻碍，实则 X 是原因。仿写：She succeeded because of her doubts, not in spite of them.
4. It's not that …, it's that …
“It's not just that we can't tackle it … we don't know how to tackle it”
先否定一种浅层解释，再给出更深层的解释，是学术讨论中澄清分歧层次的常用结构。
5. Where does this leave X?
“Where does this leave mathematics where does this leave certainty where does this leave pure reason”
「这让 X 处于什么境地？」用于追问某一结论对既有概念的后果，排比连用可加强追问力度。
6. the likes of us / the likes of X
“How do the likes of us have this kind of certainty”
「像我们这样的人」，带自谦或轻微贬义。用于强调某种能力/待遇与主体身份不相称。
7. If you can't solve a problem, convince yourself that it's not important
“If you can't solve a problem convince yourself that it's not important forget about it”
祈使句叠加的讽刺性「处方」结构，用来讥讽回避问题的做法。仿写时可换动词：If you can't win, redefine winning.
8. when you pose the question like this, you have already committed an error
“When you POS the question like this … I think you have already committed an error”
质疑问题本身预设的表达方式：先复述问法，再指出「问法本身就错了」。适合驳斥二元对立式提问。
9. we invent X and then we discover Y
“We invent the concepts and then we discover the relations among the concepts”
用一对动词的先后顺序压缩一个哲学立场，句式简洁有力，适合给折中观点下定义。
10. smart enough to X and not smart enough to Y
“Just smart enough to ask and not smart enough to answer”
enough to 的对比结构，刻画「刚好够……却不够……」的尴尬处境，常用于自嘲式总结。
词汇精讲 · 154 · 按出现顺序
induction
/ɪnˈdʌkʃən/
n.
0:05
归纳法（从个别观察推出一般规律）
pondered
/ˈpɑːndərd/
v.
0:05
深思、反复思索
underpins
/ˌʌndərˈpɪnz/
v.
1:25
支撑、构成……的基础
Cornerstone
/ˈkɔːrnərstoʊn/
n.
1:25
基石、根本
axioms
/ˈæksiəmz/
n.
2:42
公理
self-contained
/ˌself kənˈteɪnd/
adj.
2:42
自足的、自洽的
provably
/ˈpruːvəbli/
adv.
2:42
可证明地
tenuous
/ˈtenjuəs/
adj.
2:42
脆弱的、站不住脚的
emancipation
/ɪˌmænsɪˈpeɪʃən/
n.
2:42
解放
softspoken
/ˌsɔːftˈspoʊkən/
adj.
4:18
说话轻声细语的
culmination
/ˌkʌlmɪˈneɪʃən/
n.
4:18
顶点、最高成就
hypochondriac
/ˌhaɪpəˈkɑːndriæk/
n.
4:18
疑病症患者
bordering on
phr.
4:18
近乎、接近于（某种状态）
reticent
/ˈretɪsənt/
adj.
5:52
寡言的、缄默的
withdrawn
/wɪðˈdrɔːn/
adj.
5:52
孤僻的、离群的
sway
/sweɪ/
v.
5:52
动摇、左右（某人的想法）
Forefront
/ˈfɔːrfrʌnt/
n.
7:32
最前沿（at the forefront of）
rationalism
/ˈræʃənəlɪzəm/
n.
8:49
理性主义（哲学）
fervent
/ˈfɜːrvənt/
adj.
10:13
热烈的（此处原意为 ferment，思想激荡）
in sync with
phr.
11:43
与……同步、协调一致
logical positivism
n.
11:43
逻辑实证主义
Criterion
/kraɪˈtɪriən/
n.
11:43
标准、准则
proposition
/ˌprɑːpəˈzɪʃən/
n.
11:43
命题
damnation
/dæmˈneɪʃən/
n.
11:43
谴责、贬斥
reverentially
/ˌrevəˈrenʃəli/
adv.
13:03
虔诚地、敬畏地
infallibility
/ɪnˌfæləˈbɪləti/
n.
13:03
不可错性、绝对可靠
formalism
/ˈfɔːrməlɪzəm/
n.
13:03
形式主义（数学哲学流派）
keeping his counsel
phr.
14:25
守口如瓶、不表态
empirical
/ɪmˈpɪrɪkəl/
adj.
14:25
经验的、基于观察的
cranium
/ˈkreɪniəm/
n.
14:25
头盖骨，引申为脑袋
a priori
/ˌeɪ praɪˈɔːraɪ/
adj./adv.
14:25
先验的（不依赖经验）
stipulated
/ˈstɪpjuleɪtɪd/
v.
15:45
规定、约定
rigorously
/ˈrɪɡərəsli/
adv.
15:45
严格地、严密地
spills out Beyond
phr.
15:45
溢出到……之外
coraly
/ˈkɔːrəleri/
n.
15:45
推论（corollary 的误拼）
dispel
/dɪˈspel/
v.
17:11
驱散、消除（疑虑、神秘感）
undecidable
/ˌʌndɪˈsaɪdəbəl/
adj.
17:11
不可判定的
negation
/nɪˈɡeɪʃən/
n.
17:11
否定（命题）
consistency
/kənˈsɪstənsi/
n.
18:31
（逻辑）一致性、无矛盾性
contradiction
/ˌkɑːntrəˈdɪkʃən/
n.
18:31
矛盾
trans empirical
adj.
19:51
超越经验的
audacious
/ɔːˈdeɪʃəs/
adj.
19:51
大胆的、无畏的
mumbled
/ˈmʌmbəld/
v.
19:51
咕哝、含糊地说
paid him any um mind
phr.
19:51
理会他（pay sb. mind）
standing in for
phr.
21:16
代替、顶替某人
disseminated
/dɪˈsemɪneɪtɪd/
v.
21:16
传播、散布
from the horse's mouth
phr.
22:31
来自当事人之口、第一手消息
pulls the rug out from under
phr.
22:31
釜底抽薪、突然拆台
Fallout
/ˈfɔːlaʊt/
n.
22:31
余波、后果
Tremors
/ˈtremərz/
n.
22:31
余震、震颤
halting problem
n.
22:31
停机问题（图灵）
irreducible
/ˌɪrɪˈduːsəbəl/
adj.
23:54
不可约的、不可简化的
playing dice with
phr.
23:54
拿……掷骰子（爱因斯坦名言变体）
toy model
n.
25:08
玩具模型（简化的理论模型）
mutating
/ˈmjuːteɪtɪŋ/
adj.
26:30
不断变异的
random walks
n.
26:30
随机游走
allegory
/ˈæləɡɔːri/
n.
27:58
寓言、讽喻
crude approximation
n.
27:58
粗糙的近似
dispose of
phr.
29:25
抛弃、去掉
premise
/ˈpremɪs/
n.
29:25
前提
tensions
/ˈtenʃənz/
n.
30:47
张力、对立关系
logicist
/ˈlɑːdʒɪsɪst/
n.
30:47
逻辑主义者
fallible
/ˈfæləbəl/
adj.
32:15
可能出错的
shaky
/ˈʃeɪki/
adj.
32:15
不稳固的、摇摇欲坠的
tangential
/tænˈdʒenʃəl/
adj.
33:48
离题的、旁枝末节的
misinformed
/ˌmɪsɪnˈfɔːrmd/
adj.
33:48
被误导的、得到错误信息的
uncertainty principle
n.
35:13
不确定性原理
gravitational field
n.
36:41
引力场
in spite of
phr.
36:41
尽管（与 because of 对比）
magnificent
/mæɡˈnɪfɪsənt/
adj.
37:58
宏伟的、壮丽的
mysterian
/mɪˈstɪriən/
n.
37:58
神秘主义者（认为某些问题原则上不可解）
populate
/ˈpɑːpjuleɪt/
v.
39:26
遍布于、充斥
coined
/kɔɪnd/
v.
40:43
创造（新词）
scanty
/ˈskænti/
adj.
40:43
贫乏的、不足的
inverse Square Law
n.
41:58
平方反比定律
Quantum electrodynamics
n.
41:58
量子电动力学
parts per trillion
phr.
41:58
万亿分之几（精度单位）
general relativity
n.
43:24
广义相对论
group Theory
n.
43:24
群论
conserved
/kənˈsɜːrvd/
v.
44:55
（物理量）守恒
multiverses
/ˈmʌltivɜːrsɪz/
n.
44:55
多重宇宙
symmetric
/sɪˈmetrɪk/
adj.
46:14
对称的
adequate
/ˈædɪkwət/
adj.
46:14
适合的、足够的
marshmallows
/ˈmɑːrʃmeloʊz/
n.
47:40
棉花糖（此处为讽刺反问）
isolate
/ˈaɪsəleɪt/
v.
47:40
分离出、单独抽取
tackle
/ˈtækəl/
v.
47:40
着手处理（难题）
exhausting
/ɪɡˈzɔːstɪŋ/
v.
49:15
穷尽（此处非「累」义）
weave together
phr.
49:15
把……编织/串联在一起
at our disposal
phr.
49:15
可供我们支配使用
tailor
/ˈteɪlər/
v.
49:15
量身定制、剪裁
idealism
/aɪˈdiːəlɪzəm/
n.
50:31
唯心论、观念论
chaotic
/keɪˈɑːtɪk/
adj.
51:44
混沌的（对初始条件敏感）
initial conditions
n.
51:44
初始条件
trash basket
n.
53:13
垃圾桶（比喻含混的概念容器）
Basic Instincts
n.
53:13
基本本能
preconscious
/ˌpriːˈkɑːnʃəs/
n.
54:39
前意识（弗洛伊德）
make a big uh fuss about
phr.
54:39
大惊小怪、小题大做
subjectivity
/ˌsʌbdʒekˈtɪvəti/
n.
55:55
主观性
with all due respect
phr.
57:23
恕我直言（礼貌反驳）
paraphrasing
/ˈpærəfreɪzɪŋ/
v.
57:23
转述、改述
momentum
/moʊˈmentəm/
n.
58:38
动量
dualist
/ˈduːəlɪst/
n.
58:38
二元论者
cartisian dualism
n.
60:01
笛卡尔二元论（Cartesian）
intrinsically
/ɪnˈtrɪnzɪkli/
adv.
60:01
内在地、本质上
not even wrong
phr.
60:01
连错都算不上（泡利名言，指无法检验）
cataloging
/ˈkætəlɔːɡɪŋ/
v.
61:23
编目、分类列举
qualia
/ˈkwɑːliə/
n.
61:23
感受质（主观体验的质感）
cones
/koʊnz/
n.
61:23
视锥细胞
do justice to
phr.
62:34
公正地对待、充分展现
compatibilism
/kəmˌpætəˈbɪlɪzəm/
n.
62:34
相容论（自由意志与决定论相容）
determinism
/dɪˈtɜːrmɪnɪzəm/
n.
62:34
决定论
hold up a bank
phr.
64:12
抢银行
held morally responsible
phr.
64:12
被追究道德责任
magnitude
/ˈmæɡnɪtuːd/
n.
64:12
量级、重大程度
Splendid
/ˈsplendɪd/
adj.
65:36
极好的、辉煌的
shortcomings
/ˈʃɔːrtkʌmɪŋz/
n.
67:02
缺陷、不足
straight edge and a compass
n.
67:02
直尺与圆规（尺规作图）
acatic
/ˌæksiəˈmætɪk/
adj.
68:19
公理化的（axiomatic 的误拼）
on the table
phr.
68:19
（问题）仍摆在桌面上、待讨论
transistors
/trænˈzɪstərz/
n.
69:33
晶体管
dark energy
n.
69:33
暗能量
of huge import
phr.
70:56
意义重大（import 作「重要性」）
Continuum hypothesis
n.
70:56
连续统假设
intuition
/ˌɪntuˈɪʃən/
n.
70:56
直觉
plausible
/ˈplɔːzəbəl/
adj.
72:23
看似合理的（但未证明）
duplicate
/ˈduːplɪkeɪt/
v.
72:23
复制、重复（他人的做法）
inevitable
/ɪnˈevɪtəbəl/
adj.
73:39
必然的、不可避免的
natural numbers
n.
73:39
自然数
positive integers
n.
75:00
正整数
quasi empirical
adj.
75:00
准经验主义的
alluded to
phr.
76:17
暗指、间接提到
perception system
n.
76:17
感知系统
ellipse
/ɪˈlɪps/
n.
77:38
椭圆
infrared
/ˌɪnfrəˈred/
n./adj.
77:38
红外线（的）
Cellular automata
n.
77:38
细胞自动机
Reincarnation
/ˌriːɪnkɑːrˈneɪʃən/
n.
78:53
转世、化身
cognitive faculties
n.
80:12
认知能力
lucky fluke
n.
80:12
侥幸、纯属偶然的好运
blind processes
n.
80:12
盲目的（无目的的）过程
objectivity
/ˌɑːbdʒekˈtɪvəti/
n.
81:26
客观性
red giant
n.
81:26
红巨星
prime number theorem
n.
81:26
素数定理
never breathed a word
phr.
82:45
从未透露一个字
tautology
/tɔːˈtɑːlədʒi/
n.
82:45
同义反复、恒真式
discuss this to death
phr.
83:57
把……讨论到烂
intricate
/ˈɪntrɪkət/
adj.
83:57
错综复杂的
square root
n.
83:57
平方根
better left unposed
phr.
85:23
（问题）最好别提出来
inevitability
/ɪnˌevɪtəˈbɪləti/
n.
85:23
必然性
exalted
/ɪɡˈzɔːltɪd/
adj.
85:23
崇高的、高层次的
stretch
/stretʃ/
v.
88:02
（能力）被拉伸到极限
first principles
n.
88:02
第一性原理
algorithmic probability
n.
89:31
算法概率
approximations
/əˌprɑːksɪˈmeɪʃənz/
n.
89:31
近似
理解自测 · 11 题
1. 哥德尔第一和第二不完备性定理分别说了什么？
第一定理：任何足以表达算术的形式系统中，都存在一个可表述但不可判定的命题——它和它的否定都无法在该系统内被证明；用更有争议的说法，即存在「真而不可证」的命题。第二定理是第一定理的推论：系统无法在自身内部证明自己的一致性。Goldstein 在「两条不完备性定理」一节中解释，一致性指不能同时证明 P 和非 P，而不一致系统「什么都能证」，因此毫无用处；第二定理正好击中希尔伯特要求证明算术一致性的纲领。
2. 哥德尔是如何宣布这一发现的？当时反应如何？
1930年在柯尼斯堡会议的最后一天，哥德尔非常轻声地「咕哝」了一句：可能存在哥德巴赫猜想、费马大定理那一类的真命题，在任何形式系统中都无法被证明。会场几乎无人理会，会议纪要发表在《认识》杂志时甚至没有提及。唯一当场领悟的是代表希尔伯特出席的冯·诺依曼，他会后找到哥德尔追问，回普林斯顿后独立推出「无法证明算术一致性」的推论并写信给哥德尔，哥德尔回信说已有严格证明。冯·诺依曼此后成为不完备性定理的传播者。
3. Minsky 为什么说「演化之所以可能是因为量子力学，而不是尽管有量子力学」？
Minsky 指出公众被误导为「现代物理比牛顿世界更模糊不定」。他用两个对比说明相反：牛顿式的太阳系并不稳定——Sussman 与 Wisdom 的计算显示木星可能在几十亿年内把冥王星甩出去；而一个 DNA 分子在室温下却能稳定十亿年。原子若像经典太阳系那样运作会迅速坍缩，正是量子化的能级让分子结构稳定，演化才有可依托的载体。所以量子力学提供的是稳定性而非模糊性，这是在「公众理解的局限」一节中的第一个例子。
4. Livio 说的「被动有效性」是什么意思？举了哪些例子？
「被动有效性」指数学家在完全没有应用意图下发展出的纯数学分支，几十年甚至几百年后被发现恰好是某个物理理论所需的语言。Livio 举了广义相对论（依赖此前的黎曼几何）和群论（成为粒子物理对称性的基础）。他把这与「主动」使用数学做工具区分开，认为前者才是 Wigner「不可思议的有效性」最难解释的部分。这一节还给出量化证据：牛顿定律从开普勒4%精度的数据推出，却在56微米尺度和百万分之一精度上仍成立；QED 对电子磁矩的计算与测量在万亿分之八内吻合。
5. Chaitin 用「信息量」重新解释不完备性的逻辑是什么？这和 Ω 数有什么关系？
Chaitin 的推理链：任何数学理论（公理系统）只包含有限的信息，而纯数学世界包含无穷多信息，因此必然有事实无法从有限公理推出——不完备性从「悖论」变成了「容量不足」的自然结果。Ω 数是具体例证：它是随机程序停机的概率，二进制展开中每一位都是「完全的意外」，无法压缩，任何公理系统最多只能确定有限位。他称之为纯数学中「无理由发生的事实」，并借此把纯数学与「复杂性的领域」生物学连接起来，引出元生物学的设想。
6. Minsky 与 Goldstein 关于「意识」的争论，分歧究竟在哪一层？
Minsky 主张「意识」是一个装了二十多个不同问题的「垃圾桶」词，像伽利略混用 vis viva 指动量与动能一样，把它拆成具体的功能问题后就没有「普遍的谜」。Goldstein 则援引 Nagel 的「成为一只蝙蝠是什么感觉」：问题不在于功能清单，而在于纯客观的物理描述能否推出主观感受的事实。Livio 调解指出 Minsky 并非说问题无意义，而是主张分而治之。两人最终在一点上一致：我们目前既不知道如何从物理描述推出主观事实，也不知道这在原则上是否不可能——即科学上「甚至还谈不上对错」。
7. Livio 和 Chaitin 在「哥德尔之后纯数学是什么」上为何争执？各自的核心理由是什么？
Livio 作为物理学家认为哥德尔只揭示了形式系统的局限而非数学本身的局限，正如量子力学基础问题未解也没阻止物理学前进；他类比尺规作图的限制被打破后几何反而更丰富。Chaitin 反驳：希尔伯特说「纯数学就是形式公理系统」，哥德尔证明这是错的，那纯数学到底是什么至今无人能答；物理学「作为技术」前进不等于「作为理解」前进，如果目标是理解世界而不是造晶体管赚钱，基础问题就不能被搁置。这场争论展示了实用主义与基础主义两种对待未解基础问题的态度。
8. 「水母思想实验」想说明什么？Minsky 的图灵机实验是如何回应它的？
Livio 转述 Atiyah 的假设：若智能存在于只能感知温度、压力和水流的孤立水母身上，它未必会发明自然数，因为没有可数的离散对象。这质疑了自然数的「必然性」，支持爱因斯坦「正整数是人类为整理感官经验的自由创造」的准经验主义观点。Livio 进一步把算术与几何的起源归因于人类视觉擅长分辨边界和直线曲线。Minsky 以早年与 Bobrow 枚举最简图灵机的实验回应：前数万台机器中除了「计数」之外没有任何有趣行为，因此「算术是能发生的最简单的事」——这为自然数的必然性提供了一个不依赖人类感知的计算论证。
9. Livio 说「问数学是发现还是发明本身就是错误」，他的替代方案是什么？Chaitin 如何补充？
Livio 认为二选一的问法预设了答案只能是其中之一。他的方案是「我们发明概念，然后发现概念之间的关系」：例如 √-1 不存在，是人类发明了这一概念，之后才发现围绕它能展开大量数学；素数在印度和中国数学中没有作为概念被明确确立，直到欧几里得证明其无穷多才成为基础。Chaitin 从「墙内」补充：大量论文回答的是「本不该问的问题」，明显是发明；只有真正基础的数学才带有「必然性」的发现感。两人实际上在「发现仅限于核心概念」上趋于一致。
10. 如果有人反驳 Minsky 的人择原理式解释「没有定律的世界里没有哲学家」，说它并没有解释数学为何如此精确地适用，Minsky 或 Livio 会如何回应？
这正是 Livio 当场提出的反驳。Minsky 的论证只说明：在能量不守恒、没有对称性的宇宙中不会有稳定的 DNA 和观察者，所以我们必然处在一个「有可描述规律」的宇宙。Livio 承认这一点，但指出它没有回答「为什么恰好是我们这套数学、尤其是毫无应用意图发明出来的数学如此贴切」。Goldstein 补充了一个可能的回应：物理学从17世纪起只「分离出结构」，而数学正是描述结构的语言，所以定律必然是数学的——但代价是承认物理定律并不穷尽实在，这是「另一个不完备性定理」。
11. 把 Chaitin「想证明达尔文演化行得通的定理」的野心放到今天的 AI 研究情境下，还成立吗？
Chaitin 的野心是像哥德尔那样用定理解决哲学级问题，他希望用「变异软件在程序空间中随机游走」的玩具模型证明演化能产生复杂性。Minsky 结尾提到的算法概率（Solomonoff–Kolmogorov–Chaitin）与这一方向同源：以程序长度做先验的归纳理论虽不可计算，但其近似可能成为最佳预测器。放到今天，「压缩即智能」「随机变异加选择能否产生开放式复杂性」仍是 AI 与人工生命领域的活跃问题，人们通过大规模实验而非定理逼近答案。Chaitin 的方向仍成立，但他自己也承认真实生物学「太混乱」，只能做元生物学；这与 Goldstein「聪明到能问、不够聪明到能答」的判断相呼应。
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