The Number Mysteries: A Mathematical Odyssey through Everyday Life
du Sautoy, Marcus
One THE CURIOUS INCIDENT OF THE NEVER-ENDING PRIMES
p. 29
The Chinese were probably one of the first cultures to single out the primes as important numbers. They believed that each number had its own gender—even numbers were female, and odd numbers were male. They realized that some odd numbers were rather special. For example, if you have 15 stones, there is a way to arrange them into a nice-looking rectangle, in three rows of five. But if you have 17 stones, you can’t make a neat array: all you can do is line them up in a straight line. For the Chinese, the primes were therefore the really macho numbers. The odd numbers that aren’t prime, though they were male, were somehow rather effeminate. This ancient Chinese perspective homed in on the essential property of being prime, because the number of stones in a pile is prime if there is no way to arrange them into a nice rectangle.
Applied Combinatorics
Prelude
第6页
This book seeks to develop facility at combinatorial reasoning, which is the basis for analyzing a wide range of problems in computer science and discrete applied mathematics.
Preface
第1页
Combinatorial reasoning underlies all analysis of computer systems. It plays similar role in discrete operations research problems and in finite probability.
This book teaches students in the mathematical sciences how to reason and model combinatorially. It seeks to develop proficiency in basic discrete math problem solving in the way that a calculus textbook develops proficiency in basic analysis problems solving.
The three principal aspects of combinatorial reasoning emphasized in this book are: the systematic analysis of different possibilities, the exploration of the logical structure of a problem(e.g.,finding manageable sub pieces or first solving the problem with three objects instead of n), and ingenuity.
Becoming a Better Programmer
chapter 14 Software Development is
第131页
Too much modern software is like my Alphabetti custard: it’s the wrong thing, written the wrong way.
chapter 24 Live to Love to Learn
第221页
The Knowledge Portfolio
Be aware of the risk/reward balance of the items in your portfolio.
Purposefully manage your knowledge portfolio.
Complexity
第233页
What Is Network Thinking?
Network thinking means focusing on relationships between entities rather than the entities themselves.
Prime Numbers
prime number theorem and the prime counting function
第183页
De la Vallée Poussin also showed that Gauss’s estimate Li(x) is a better approximation to π (x) than x/(log x - a) no matter what value is assigned to the constant a (and also that the best value for a is 1).
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And yet—Pythagoras claimed that the universe was made of numbers and Leopold Kronecker (1823–1891) claimed that “God made the integers and all the rest is the work of man.” Who knows? Perhaps the world is even more cunningly constructed out of the prime numbers!
当代数学大师-沃尔夫数学奖得主及其建树与见解-第四版
A.塞尔伯格( Selberg, Atle)
第1876页
当有人问他，如果作为非专业人士问你“黎曼假设告诉了我们关于素数的哪些东西？” 他明快地答道：“它告诉我们素数分布得相当好，以最大可能分布得均衡，分布得适宜。当然，我们不能期望一个完全均衡的分布。”
当有人问他，“你是否认为黎曼假设是正确的？” 他答道：“如果在我们的宇宙中还有什么东西是正确的，那必定就是黎曼假设，若找不出正确的理由，那纯粹的美学理由就够了。
谈谈方法
第三部分
第739页
一定要经过长期训练，反复思考，才能熟练地从这个角度去看万事万物。我相信，那些古代哲学家之所以能够摆脱命运的干扰，漠视痛苦和贫困，安乐赛过神仙，其秘密主要就在于此。因为他们不断地考察自然给他们划定的界限，终于大彻大悟，确信除了自己的思想之外，没有一样东西可以由他们做主，确信只要认清这一点就可以心无挂碍，不为外物所动；他们对自己的思想作出了绝对的支配，因此也就有理由认为自己又富又强，逍遥安乐，胜过所有的别人，别人不懂这种哲学，不管得到自然和命运多大优待，还是不能支配一切、事事如愿以偿的。
第734页
我的第三条准则是：永远只求克服自己，不求克服命运，只求改变自己的愿望，不求改变世间的秩序。总之，要始终相信：除了我们自己的思想以外，没有一样事情可以完全由我们做主。
七堂极简物理课
尾声 我们
第578页
我们的爱与真诚与生俱来，我们天生就渴望懂得更多，渴望不断学习。我们对世界的认知在不断增长。在知识的边界，我们的求知欲在燃烧。我们渴望探索空间纹理的细微之处，探索宇宙的起源，时间的本质，黑洞的现象，以及我们思维的运行。
现在，在人类已知事物的最前沿，我们将要航行于未知的海洋，世界的奥秘与美丽熠熠生辉，让我们目眩神迷。
第564页
诚如古罗马哲学家卢克莱修（Titus Lucretius Carus）所言：“我们对生命的胃口是贪得无厌的，我们对生命的渴求是永不满足的。”（《物性论》卷三，第1084行）
终极理论之梦
第一章 序幕
第1页
在我看来，好的科学哲学是对历史和科学发现的迷人解说。但是，我们不应指望靠它来指导今天的科学家如何去工作，或告诉他们将要发现什么。
从异教徒到基督徒
第五章 澄清佛教的迷雾
二、 罪与业
第1777页
如果宗教是意味着超脱凡世的，我反对它。如果宗教是意味着我们必须从这个现世、知觉的生活中走出，且有多快就多快地“逃避”它，像一只老鼠放弃快要下沉的船一样，我是和它对立的。我认为一个人必须有中国人的共有意识，勇敢地接受现世的生活，且像禅宗的信徒一样和它和平共处。而我强烈地觉得宗教（任何宗教）一天固执于一个来世，趋向于否定现世，且从上帝所赐给我们的这般丰富且有知觉的生命中逃避，我们将因此种做法而妨碍宗教（任何宗教）与近代青年的意识接触。我们将是上帝真正不知感恩的儿女，甚至不值得禅宗的信徒称我们为堂兄弟。
第六章 理性在宗教
二、现在的姿态
第1850页
在所谓宗教信仰与意见的繁茂丛林中，一切谬见，弗朗西斯·培根的“四个假象”，都被介绍了：一切偏见（种族的假象），例如上帝必然是一种人性的存在，一个神与人同性的上帝的观念；一切与个人的或国家的成见相符的信仰（洞穴的假象），例如做一个基督徒和做一个白人，事实上有同样意义的流行习惯；一切言辞的虚构及混乱（市场的假象）；一切以人造的哲学系统为根据的不合理的教条（舞台的假象），例如加尔文的“完全堕落”的教义。
三、可理解的止境
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柏拉图说得对，我们所能看见及知道的，只是一个影子的世界。我们感官的知觉，只能给我们一个现象世界的图，这是理性所能告诉我们的一切：在现象的背后是本性，是物体本身，而我们永远不能凭我们心的推理来知道绝对的真理。多么可怜！这是对人类缺陷的悲哀的宣判：它是以官觉的知识为根据，自然的东西的存在是知识而已，我们所认为存在的不过是知觉，且可能是一种幻觉，我们的体质注定我们要隔着一张幕来看东西，而且永远不能和绝对真理面对面。尽我们想做的来做，某些东西仍常留在后面，即那些可知世界的剩余区。这是对人类智力的侮辱，悲哀地宣判人的心智已至绝境。对此，人自觉无力反抗。佛曾宣讲它，柏拉图曾说明它，一群献身于对机械与攻击人类知识定律经历世纪之久的哲学家，伤心地承认它，而新近的科学也证实了它。
第1929页
把柏拉图洞穴的比喻放在现代科学中来看，是否适用和正确是超乎我们所能估计的。爱丁丝说：“真正了解到物质科学所谈及的是一个影子世界，是近代最有意义的进步之一。”而杰恩斯追求以太的量子及波长的时候说：“人们已开始觉得这个宇宙看来像一个伟大的思想，多过像一架伟大的机器。”量子的确成为物理学上的困惑。量子让我们首次看到物质与能力渡过不可见的边界的地方，使我们确认对于物质的老概念已不再适用。当我们对物质做进一步的探究，到了把次原子的极小量充以一百万伏特的电，我们简直是失去了它。这是今天舆论的客观趋势。
第1829页
其实人想及上帝时，必须把他抽离一切实在，而不将他想象为一个占据空间的形式，只是一种事物的秩序。上帝不是存在，纯是动力——一个形而上学世界秩序的元素。
现实不似你所见
11. 无穷的终结
第1950页
《数沙者》像是在开玩笑，但意义深远。凭借比启蒙运动早大约一千年的想象力，阿基米德对某种认识做出了反抗，这种认识坚持认为存在一些人类思想本质上无法触及的奥秘。他没有宣称确切地知道宇宙的维度，或者沙子的具体数目。他主张的不是知识的完备性，正好相反，他十分清楚他估算的近似性和暂时性。他谈到宇宙真实的大小有哪些可能，但没有做出明确的选择。重要的不是假设我们通晓一切，而是相反：意识到昨天的无知可能被今天阐明，今天的无知可以被明天照亮。
4. 量子
第1187页
在量子力学描述的世界中，实在只存在于物理系统之间的关联之中。并不是事物进入关联，而是关联是“事物”的基础。量子力学的世界不是物体的世界，它是事件的世界。事物通过基本事件的发生而建立，就像哲学家尼尔森·古德曼（Nelson Goodman）在20世纪50年代写出的美妙语句那样：“物体是一个不变的过程”。
黎曼猜想漫谈
未竟的探索
第183页
在所有高难度的数学猜想中，若以它们跟其他数学命题之间的关系，乃至与物理学那样的自然科学领域之间的关系（这些关系在很大程度上决定了一个数学猜想的重要性）而论，黎曼猜想可以说是无与伦比的。
与费马猜想或哥德巴赫猜想那种连中学生都能看懂题意的数学猜想不同，理解黎曼猜想是有一定“门槛”的，因为仅仅理解其表述就需要有一些复分析方面的知识。
《黎曼猜想漫谈》读后感（代序）
第10页
一旦这些RH解决了，人类就站在一个不知比现在高多少的数学平台上，看到更远得多的风景。— 王元
从存在到演化
第3章 量子力学
第918页
算符的引入从根本上改变了我们对自然的描述，因此把它说成“量子革命”是很恰当的。
一个算符的本征函数用来描述系统的状态，其中该算符所代表的物理量具有确定的值（即本征值）。因此用物理学的术语来说，不可对易性意味着，不可能存在这样的状态，在其中，比如说坐标q和动量p同时具有确定的值。这就是著名的海森伯测不准关系的内容。
第四章 热力学
第61页
总是和耗散结构相连的三个方面：用化学方程所表达的功能；不稳定性所产生的时空结构；以及触发这个不稳定性的涨落。
The Riemann Hypothesis
Prime Time
第13页
Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate. — Leonhard Euler
The Millennium Problems
The Riemann Hypothesis - Bernhard Riemann
第25页
While for many mathematicians “intuitive work” can be hit-or-miss, Riemann’s mathematical intuitions were incredibly acute, and his results generally turned out to be correct.
ZERO The Gauntlet is Thrown
第14页
Perhaps the best way to read this book is to think of the seven problems as the commonplace mathematics of the twenty-fifth century.
心理学与生活
chapter 12 Emotion, Stress, And Health — A toast to your health
第430页
- Never say bad things about yourself.
在成年时期，社会关系和个人的成就占重要地位。
社会亲密感是成年期心理健康的先决条件，重要的不是社会交往的数量，而是质量。亲密感是一种对他人承担性爱、情感及道德承诺的能力。它要求坦率、勇气、伦理感，并且往往要牺牲一些个人偏好。
再生力的本质：良好的适应，依赖于成为一名“关注他人”的人－即成为一名关心他人、富有同情心并拥有良好的社会关系的人士。
编程珠玑(第二版)
第1章 开篇 – 1.7 进阶阅读
第9页
程序员的主要问题不一定是技术上的，更可能是心理上的；因为他试图解决一个错误的问题，所以他不能取得进步。通过打破概念上的障碍，转而解决一个更简单的问题，这样我们最终解决他的问题了。
James L.Adams所著的《Conceptual Blocking》研究了这种跳跃，通常它可以说是一种通向创造性思维的令人愉悦的激励。Adams将概念性障碍定义为“妨碍问题解决者正确认识问题或获得解答的心理屏障”。
第二版的尾声
第167页
Tom Duff：尽可能盗用别人的代码。
库非常伟大，任何时候都尽可能使用它们。
Thinking, Fast and Slow
Part 4. choices — 32 keeping score — Regret
第346页
Regret is one of the counterfactual emotions that are triggered by the availability of alternatives to reality.
素数论
第5章 重要的猜想
第105页
素数首先就是一些谜。
第2章 Riemann zeta函数
第41页
19世纪上半叶，Gauss根据当时已有的素数表，猜测对数积分li(x)给出了素数稀疏分布律的一个极好的近似，从而将Legendre的直觉精确化。
法文版前言
素数表显示了素数的混沌特性，而其表面的无序性最终却与一些，比如说源于物理现象的经典随机模型相吻合。…为什么像素数这样高度确定的序列能够包含着那样的随机性。
确定性的终结
引言 一种新的理性？
第3页
James Clerk Maxwell就谈到＂一种新型的知识＂会克服决定论的偏见。
问题解决心理学
第三章 问题表征：＂顿悟＂的案例
第63页
Reproductive thinking是问题的structurally blind,其表现是没有真正理解问题后面的结构。
与“众”不同的心理学
第十章 人类认知的死穴
第235页
很多人头脑里不具备概率推理的基本定律，或者即使有也不够用。
第一章 充满活力的心理学
第19页
公众更倾向于用哲学的思辩、神学的预言或是民间的常识来解释世界。
博大精深的素数
第四章 素数是如何分布的？
第162页
许多数学家都认为素数分布问题具有极大的美学动力。
机遇与混沌
11. 混沌：一个新的范式
第65页
物理学家杨振宁（C.N.Yang）邀请我去做了一个报告。之后，他取笑了我的＂有争议的湍流思想＂—在当时这是一个恰如其分的评价。
学会提问
第1章 提出正确问题的益处
第5页
如果一个读者始终依赖于海绵式思维，那么他将始终相信其最后接收到的信息。不管对个人还是社会，成为别人思想的傀儡都是一件可怕的事。
海绵式思维强调知识的获得，而淘金式思维强调与知识积极的互动。
认知心理学
第11章 问题解决与创造性
第312页
专家不仅仅拥有丰富的知识，而且知识的组织性更好，这样才使得他们能更有效地利用他们的知识。而且，专家的图式不仅包括更多某个问题领域的陈述性知识，而且与该领域相关的程序性策略也比新手更多。
薄伽梵歌论
附：薄伽梵歌 1957年海外初版序言
第465页
然舍是则无以立，不得已必落于言筌，则曰至真，即至善而尽美，曰太极，即全智而遍能；在印度教辄曰超上大梵，曰彼一，人格化而为薄伽梵，薄伽梵者，称谓之至尊，佛乘固尝以此尊称如来者也。欧西文字，辄译曰：天主，上帝，皆是也。
旁观者
第3章 怀恩师
第70页
不要从别人的错误中学习，看看别人是怎么做对的。
数学的语言
序曲 何谓数学
第3页
数学是研究模式的科学（science of patterns）。
我们内心的冲突
序言
第2页
神经症是由文化因素引起的。
神经症产生于人际关系的紊乱失调。
第1页
精神失调者…而是饱受内心冲突折磨的人。
第十章 人格衰竭
第111页
表明冲突存在的三种大的紊乱失调：其一是遇事都犹豫不决。精力被分散的第二个典型症状是一种普遍性的办事无效率，即一个人由于内心有冲突不能发挥最好的能力造成的。第三个典型的紊乱症状是普遍性怠惰，主动性和行动能力瘫痪，奋斗的方向不明确，对自我的严重疏离。
乔布斯产品圣经
179 发明比思想更能改变世界
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那时我开始认为，也许爱迪生给世界带来的改变，比马克思和印度高僧尼姆•卡洛里•巴巴加起来的还要多。
项目百态
模式 33 扑克之夜
第80页
互相熟悉可以使彼此互相信任，也可以使彼此更有耐心。
除了在一起玩耍，我们从来没有在其他事情上面感觉到如此地有活力，如此放开地表现自己，以及如此地专注痴迷。—Charles E.Schaefer
混沌
第4章 生命的盛衰
第73页
在世界上压倒一切的是非线性。
如果有更多的人明白简单的非线性系统并不必然具有简单的动力学性质，那我们大家的日子都会好过得多。
数学的故事
第237页
Sophus Lie的新颖想法：Galois理论在微分方程上应该有一个相应的理论。
第352页
Jack Wisdom和Jacques Laskar：1、发现太阳系的动力学特性是混沌的。2、这俩位天文学家还证明了：月球潮汐能使地球保持稳定，否则的话，就会导致混沌运动，而致使天气迅速变化，从暖期一下子变到冰川时代，又从冰川时代迅速变回暖期。因此，混沌理论证明，如果没有月球，地球将会是一个非常不宜居的地方。
完美的证明
第110页
Everyone is a bastard, everyone is bad, with the possible exception of Jesus Christ. Einstein is bad too, because he did not leave America after the nuclear bomb was detonated over his objections.In the end, through the general interconnectedness of events, a person becomes, in some way or another, to a greater or lesser extent, party to everything that happens in the world, and if he can exert any influence whatsoever on any event, then he becomes responsible for it.
Alexandrov：
每个人都是混账、坏蛋，只有耶稣基督除外。爱因斯坦也是个坏蛋，因为在原子弹被引爆之后，他并没有离开美国，尽管他曾极力反对使用原子弹。
最终，通过事物之间的普遍联系，一个人会以这样或那样的方式，在深浅不等的程度上与世界上发生的所有事物都扯上干系。如果一个人可以对任何一个事件施加任何影响，那么他就要为此承担责任。
第二章 如何培养数学家
第17页
佩雷尔曼的母亲名叫Lubov，是犹太人，20世纪60年代的数学专业研究生，当了名数学老师。
The Music of the Primes
chapter one : who wants to be a millionaire?
第1页
Somebody allegedly asked Hilbert, ‘If you were to be received like Barbarossa, after five hundred years, what would you do?’ His reply:’I would ask, “Has someone proved the Riemann Hypothesis?”’
第10页
The dependence of so many results on Riemann’s challenge is why mathematicians refer to it as a hypothesis rather than a conjecture.
the atoms of arithmetic
第55页
As we shall see, Riemann’s Hypothesis can be interpreted as an example of a general philosophy among mathematicians that, given a choice between an ugly world and an aesthetic one, Nature always choose the latter.
Riemann’s Imaginary Mathematical Looking-Glass
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His contemporaries were to see nothing of him while he waded through Cauchy’s output. Several weeks later Riemann resurfaced, declaring that ‘this is a new mathematics’. What had captured Cauchy and Riemann’s imagination was the emerging power of imaginary numbers.
chapter five The Mathematical RelayRace: Realising Riemann’s Revolution
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Euclid in Alexandria. Euler in St. Petersburg. The Göttingen trio — Gauss, Dirichlet, Riemann. The problem of prime numbers had been passed on like a baton from one generation to another. The new perspectives of each generation provided impetus for a fresh surge along the track. Each wave of mathematicians left its characteristic mark on the primes, a reflection of their era’s particular cultural outlook on the mathematical world.
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As Hilbert wrote in 1897, he wanted to implement ‘Riemann’s principle according to which proofs should be impelled by thought alone and not by computation’.
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But Gauss had made a second conjecture: that his guess would always overestimate the number of primes…
But in 1912 Littlewood proved that as you counted higher you would eventually come to regions of numbers where Gauss’s guess switch from overestimating to underestimating the number of primes.
Littlewood also showed that Riemann’s refinement might look more accurate as we count through the first million numbers, but in the farther reaches of universe of numbers Gauss’s guess would sometimes give the better prediction.
Indeed, to this day no one has actually counted far enough to arrive at a region of numbers where Gauss’s guess underestimates the primes. It is only through Littlewood’s theoretical analysis and the power of mathematical proof that we can be sure that somewhere along the line Gauss’s original prediction is false.
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From now on, proof was everything. Nothing could be trusted without conclusive evidence.
chapter six Ramanujan, the Mathematical Mystic
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Mathematics seems to bring out the cranks.
chapter seven Mathematical Exodus: From Göttingen to Princeton
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Selberg spoke about his view on the Riemann Hypothesis. Although he had made a major contribution on the way to a proof. he stressed that there was still very little to support its truth. ‘I think the reason that we were tempted to believe the Riemann Hypothesis then was essentially that it is the most beautiful and simple distribution that we can have. You have this symmetry down the line. It would lead also to the most natural distribution of primes. You think that at least something should be right in this universe.’
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Some misinterpreted his comments, thinking that Selberg was casting doubt on the validity of the Riemann Hypothesis. Yet he was not as pessimistic as Littlewood who believed the lack of evidence meant the Hypothesis was false. ‘I have always been a strong believer in the Riemann Hypothesis. I would never bet against it. But at that stage I maintained that we didn’t really have any results either numerical or theoretical that pointed very strongly to its truth. What the results pointed to was that it was mostly true.’
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Hardy once dismissively declared, ‘Probability is not a notion of pure mathematics but of philosophy or physics.’
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Erdös liked this probabilistic interpretation of the Riemann Hypothesis.
chapter eight Machines of the Mind
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Hilbert responded with a view shared by many mathematicians: ‘It is a completely misunderstanding of our science to construct differences according to peoples and races and the reasons for which this has been done are very shabby ones. Mathematics knows no races…for mathematics, the whole cultural world is a single country.’
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Julia Robinson: ‘We can conceive of a chemistry that is different from ours, or a biology, but we cannot conceive a different mathematics of numbers. What is proved about numbers will be a fact in any universe.’
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Cohen has been one of the few mathematicians to admit that he was actively working on this notoriously difficult problem. So far, though, it has held firm against his attack.
Chapter Nine The Computer Age: From the Mind to the Desktop
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The Mertens Conjecture is very closely related to the Riemann Hypothesis, and its disproof showed mathematicians that if the Riemann Hypothesis were true, it was only just true.
chapter 10 - Cracking Numbers and Codes
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If Gauss were alive today, he would be a hacker. — Peter Sarnak
Chapter 11 - From Orderly Zeros to Quantum Chaos
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Gödel himself had voiced such concerns in relation to the Riemann Hypothesis: perhaps the axioms that formed the foundation of the mathematical edifice were not broad enough to carry the required proof, in which case you might continue building upwards and never find a connection to the Hypothesis.
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As Littlewood had advised Montgomery during his time in Cambridge, ‘Don’t be afraid to work on hard problems because you might solve something interesting along the way.’
Atle Selberg had become something of a latter-day Gauss.
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Chowla, who dragged Montgomery over and introduce to Dyson, is the only person ever to have bullied Selberg into writing a joint paper.
Chapter 12 - The Missing Piece of the Jigsaw
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One epic in particular was to become a close companion throughout André Weil’s life: the Bhagavad-Gita, the song of God from the Mahabharata.
Chapter 12 - The Missing Piece of the Jigsaw
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Many of the mathematicians who have contributed to our understanding of the primes have been rewarded with long lives. Having proved the prime number theorem in 1896, Jacques Hadamard and Charles de la Vallée-Poussin both lived into their nineties. People had begun to believe that their having proved the Prime Number Theorem had made them immortal. The belief in a connection between longevity and the primes has been further fulled by Atle Selberg and Paul Erdös, whose alternative elementary proof the Prime Number Theorem in the 1940s saw both of them live into their eighties. Mathematicians joke about a new conjecture: anyone who prove the Riemann Hypothesis will indeed become immortal.
Dr.Riemann’s Zeros
Proofs and refutations
There was a mathematician in the Soviet Union … called Nikolai Gavrilov, who in the early 1960s was a big Communist boss in one of the Ukrainian cities. He was an ameteur mathematician, and he got hold of the Riemann Hypothesis and thought he had a proof, and being a man of power he organized for his proof to be published. The fortunate thing is that the proof did get published. You should know that at that time the Soviets were competing, especially with the United States, in all fields - the cosmos, high technology, and so on - and they had fine number theorists and mathematicians in general. Especially one Gelfond and one Yuri Linnik. Now, when these men got hold of a false proof of the Riemann Hypothesis, it is said, Gelfond had a heart attack and Linnik was choking and gasping for breath. He almost died from shock. They were so frustrated because much damage had been done to Soviet mathematics by this false proof of the Riemann Hypothesis appearing.