皮波迪先生对《Godel, Escher, Bach》的笔记(5)

皮波迪先生 (择善固执,止于至善)

在读 Godel, Escher, Bach

Godel, Escher, Bach
  • 书名: Godel, Escher, Bach
  • 作者: Douglas R Hofstadter
  • 副标题: An Eternal Golden Braid
  • 页数: 824
  • 出版社: Penguin
  • 出版年: 2000-03-30
  • Introduction: A musico-logical offering
    P3 Frederick The Great, King of Prussia, came to power in 1740. ... The celebrated mathematician Leonhard Euler spent twenty-five years there

    Potsdam where Fredrick’s courts located.

    Many other mathematicians and scientists came, as well as philosophers - including Voltaire and La Mettrie, who wrote some of their most influential works while there.
    But music was Frederick’s real love. He was an avid flutist and composer.
    This Bach’s compositions were somewhat notorious. Some called them “turgid and confused”, while others claimed they were incomparable masterpieces. But no one disputed Bach’s ability to improvise on the organ.
    P4 Carl Philipp Emanuel Bach was the Capellmeister at the court of King Frederick. ... It was Frederick’s custom to have evening concerts of chamber music in this court. Often he himself would be the soloist in concerto for flute. ... At the cembalo is C. P. E. Bach.

    It is very probably that those BWV 1030~1035 flute sonata series were composed by C. P. E. Bach instead of J. S. Bach.

    P7 In the copy which Bach sent to King Frederick, on the page preceding the first sheet of music, was the following inscription:
    Regis Iusfu Cantio Et Reliqua Canonica Arte Refolula.
    Here Bach is punning on the word "canonic", since it means not only "with canons" but also "in the best possible way". The initials of this inscription are
    --- an Italian word, meaning "to seek".


    To write a decent fugue of even two voices based on it would not be easy for the average musician!
    P17 All consistent axiomatic formulations of number theory include undecidable propositions.
    P19 Gödel showed that provability is a weaker notion than truth, no matter what axiomatic system is involved.……no fixed system, no matter how complicated, could represent the complexity of the whole numbers: 0, 1, 2, 3, ……
    P19 Aristotle codified syllogisms, and Euclid codified geometry;
    P23 If paradoxes could pop up so easily in set theory --- a theory whose basic concept, that of a set, is surely very intuitively appealing --- then might they not also exist in other branches of mathematics?……In fact, this very question --- "Are mathematics and logic distincet, or separate?" --- was the source of much controversy.
    P25 In an 1842 memoir, she wrote that the A.E. 「Analytical Engine proposed by Babbage」 "might act upon other things besides number". While Babbage dreamt of creating a chess or tic-tac-toe automaton, she suggested that his Engine, with pitches and harmonies coded into its spinning cylinders, "might compose elaborate and scientific pieces of music of any degree of complexity or extent." In nearly the same breath, however, she cautions that "The Analytical Engine has no pretensions whatever to originate anything. It can do whatever we know how to order it to perform." Though she well understood the power of artificial computation, Lady Lovelace was skeptical about the artificial creation of intelligence.
    P26 This is what Artificial Intelligence (AI) research is all about. And the strange flavor of AI work is that people try to put together long sets of rules in strict formalisms which tell inflexible machines how to be flexible.


    P27 The flexibility of intelligence comes from the enormous number of different rules, and levels of rules.
    P27 Without doubt, Strange Loops involving rules that change themselves, directly or indirectly, are at the core of intelligence.

    Lisp language.

    P27 In the year 1754, four years after the death of J. S. Bach, the Leipzig theologian Johann Michael Schmidt wrote, in a treatise on music and the soul, the following noteworthy passage:
    Not many years ago it was reported from France that a man had made a statue that could play various pieces on the Fleuttraversiere, placed the flute to its lips and took it down again, rolled its eyes, etc.
    2017-11-25 16:44:04 回应
  • Chapter 1: The MU-puzzle

    Formal systems

    P33 Requirement of Formality: The major point, which almost doesn't need stating, is that you must not do anything which is outside the rules. We might call this restriction the “Requirement of Formality”.

    Even jumping out of the system to think about the meaning.

    Theorems, axioms, rules

    P35 Such strings, producible by the rules, are called theorems.
    I gave you a theorem for free at the beginning, namely MI. Such a “free” theorem is called an axiom

    Are the mathematical axioms obtained from God? At least they should be a priori.

    P36 A derivation of a theorem is an explicit, line-by-line demonstration of how to produce that theorem according to the rules of the formal system.

    Inside and outside the system

    P36 But what if you asked a friend to try to generate U? It would not surprise you if he came back after a while, complaining that he can't get rid of the initial M, and therefore it is a wild goose chase. Even if a person is not very bright, he still cannot help making some observations about what he is doing, and these observations give him good insight into the task – insight which the computer program, as we have described it, lacks.
    It is possible to program a machine to do a routine task in such as way that the machine will never notice even the most obvious facts about what it is doing; but it is inherent in human consciousness to notice some facts about the things one is doing.
    P37 a car will never pick up the idea, no matter how much or how well it is driven, that it is supposed to avoid other cars and obstacles on the road; and it will never learn even the most frequent traveled routes of its owner.


    The difference, then, is that it is possible for a machine to act unobservant; it is impossible for a human to act unobservant. Notice I am not saying that all machines are necessarily incapable of making sophisticated observations; just that some machines are. Nor am I saying that all people are always making sophisticated observations; people, in fact, are often very unobservant. But machines can be made to be totally unobservant; and people cannot. And in fact, most machines made so far are pretty close to being totally unobservant. Probably for this reason, the property of being unobservant seems to be the characteristic feature of machines, to most people. For example, if somebody says that some task is “mechanical”, it does not mean that people are incapable of doing the task; it implies, though, that only a machine could do it over and over without ever complaining, or feeling bored.

    Jumping out of the system

    P37 It is an inherent property of intelligence that it can jump out of the task which it is performing, and survey what it has done; it is always looking for, and often finding, patterns.


    There are cases where only a rare individual will have t he vision to perceive a system which governs many peoples' lives, a system which had never before even been recognized as a system; then such people often devote their lives to convincing other people that the system really is there, and that it ought to be exited from!

    They are philosophers!

    P38 It is very important when studying formal systems to distinguish working within the system from making statements or observations about the system.


    I am sure that every human being is capable to some extent of working inside a system and simultaneously thinking about what he is doing.


    Decision procedures

    P40 If there is a test for theoremhood, a test which does always terminate in a finite amount of time, then that test is called a decision procedure for the given formal system.
    P41 Certainly the rules of inference and the axioms of the MIU-system do characterize implicitly, those strings that are theorems. Even more implicitly, they characterize those strings that are not theorems. But implicit characterization is not enough, for many purposes. If someone claims to have a characterization of all theorems, but it takes him infinitely long to deduce that some particular string is not a theorem, you would probably tend to say that there is something lacking in that characterization – it is not quite concrete enough. And that is why discovering means, in effect, is that you can perform a test for theoremhood of a string, and that, even if the test is complicated, it is guaranteed to terminate. In principle, the test is just as easy, just as mechanical, just as finite, just as full of certitude, as checking whether the first letter of the string is M. A decision procedure is a “litmus test” for theoremhood!


    The set of axioms must be characterized by a decision procedure – there must be a litmus test for axiomhood.

    2017-12-01 23:07:58 回应
  • 全书生词表(持续更新)

    abolition 废除, 废止

    ad infinitum 无限地;永久地;无止境地

    afford vt. 给予,提供;买得起

    alight 1. vi. 下来;飞落2. adj. 烧着的;点亮着的

    all along 自始至终,一直

    allegory n. 寓言

    allegro 1. n. 急速的乐章;快板2. adj. 快速的

    allusion n. 暗示;提及

    anticipate 预料

    apologist 辩护士,辩解者

    apparent adj. 显然的;表面上的

    arrest 1. vt. 吸引;逮捕;阻止2. n. 逮捕;监禁

    assent 1. vi. 赞成;同意2. n. 赞成;同意

    at a loss 亏本地;困惑不解

    austere adj. 简朴的;严峻的;苦行的;无装饰的

    automaton 自动机器, 机器人

    avid adj. 渴望的,贪婪的;热心的

    Babbage 巴贝奇

    banish vt. 放逐;驱逐

    bastion n. 棱堡;堡垒

    before long 不久以后

    befuddle vt. 使迷惑;使昏沉

    bizarre adj. 奇异的(指态度,容貌,款式等)

    bland 1. adj. 乏味的;冷漠的;温和的2. vt. 使…变得淡而无味;除掉…的特性

    blatant adj. 炫耀的;喧嚣的;俗丽的;公然的

    boggle 1. vi. 犹豫,退缩;惊恐2. vt. 搞糟,弄坏;使……惊奇;使……困惑

    bore hole 炮眼;钻孔

    brand 1. vt. 打烙印于;印…商标于;铭刻于,铭记2. n. 商标,牌子;烙印

    breach 1. n. 违背,违反;缺口2. vt. 打破;违反,破坏

    brink n. (峭壁的)边缘

    by all means 尽一切办法;一定,务必

    Carrollian adj. 具有卡罗尔风格的

    catalyze vt. [化]催化;刺激,促进

    caution 1. n. 小心,谨慎;警告,警示2. vt. 警告

    certitude n. 确信;确实

    champion 1. n. 冠军;拥护者;战士2. vt. 拥护;支持

    chorale n. 赞美诗

    codify vt. 编纂;将...编成法典;编成法典

    commonplace 平凡的, 陈腐的

    concoct vt. 捏造;混合而制;调合;图谋

    conjure 1. vt. 念咒召唤;用魔法变出;想象2. vi. 以念咒召唤神灵;施魔法,变魔术

    consummate 完善

    converse 1. adj. 相反的,逆向的;颠倒的2. vi. 交谈,谈话;认识

    crux n. 关键;难题;十字架形,坩埚

    culprit n. 犯人,罪犯;被控犯罪的人

    cushioning 缓冲

    deductive 演绎

    defy 1. vt. 藐视;公然反抗;挑衅;使落空2. n. 挑战;对抗

    deity n. 神;神性

    demolish vt. 毁坏;推翻;破坏;拆除;驳倒

    designate 指定的

    dilettante 1. n. 业余爱好者;一知半解者2. adj. 业余艺术爱好的;浅薄的

    discord 1. n. 不和;不调和;嘈杂声2. vi. 不一致;刺耳

    dishearten 使沮丧, 使泄气

    disorient 迷惑

    disparate 不同

    dread 1. n. 恐惧;可怕的人(或物)2. vi. 惧怕;担心

    dreamily adv. 梦似地,朦胧地;爱梦想地

    dysphoria 烦躁不安

    eccentric 怪人

    eerie adj. 可怕的;怪异的

    Elea n. 埃里亚(意大利南部的地名)

    epitome n. 缩影;摘要;象征

    erudite 1. adj. 博学的;有学问的2. n. 饱学之士

    escalation 扩大, 增加

    esoteric adj. 秘传的;限于圈内人的;难懂的

    evasive adj. 逃避的;托辞的;推托的

    exhilarate 兴奋

    exorcise vt. 驱邪;除怪

    extempore 即席的, 当场的

    extemporise vt. 即兴创作,即席演奏

    fancy 1. n. 想像力;爱好;幻想2. adj. 想象的;奇特的;精选的;昂贵的

    fleet 1. adj. 快速的,敏捷的2. n. 舰队;小河;港湾

    flirt 1. vi. 调情;玩弄;轻率地对待;摆动2. vt. 挥动;忽然弹出

    footrace n. 赛跑;竞走

    foremost 1. adj. 最先的;最重要的2. adv. 首先;居于首位地

    forthwith 立刻

    fortuitous 偶然

    fugal adj. [音]赋格曲的

    genie n. 鬼;妖怪

    Grecian 1. adj. 希腊的;希腊式的2. n. 希腊学家;希腊语

    gulf 1. n. 海湾;深渊;漩涡;分歧2. vt. 吞没

    havoc 1. n. 大破坏;浩劫;蹂躏2. vt. 严重破坏

    head on 迎面地

    Heterological 异源

    highbrow 1. adj. 自炫博学的;知识分子的;不切实际的2. n. 卖弄知识的人;知识分子

    hither and thither 到处

    illuminate 照明,阐释,说明

    illusory adj. 错觉的;产生幻觉的;幻影的;虚假的

    immodest adj. 不谦虚的;不庄重的

    impart vt. 给予(尤指抽象事物),传授;告知,透露

    incarnate 体现

    incidental 附带的

    incidentally 顺便

    indubitable adj. 不容置疑的;明确的

    ineluctable adj. 不可避免的;无法逃避的

    interim 1. adj. 临时的,暂时的;中间的;间歇的2. n. 过渡时期,中间时期;暂定

    intricate 复杂的,难懂的

    irreparable adj. 不能挽回的;不能修补的

    jeopardise 危害

    keen 1. adj. 敏锐的,敏捷的;热心的;锐利的;渴望的;强烈的2. n. 痛哭,挽歌

    knob 1. n. 把手;瘤;球形突出物2. vi. 鼓起

    koan n. [宗]心印;[宗]以心传心

    leaps and bounds v. 跳跃(leap的过去分词)

    lichens 1. n. 地衣类;地衣(lichen的复数)2. v. 使长满地衣(lichen的三单形式)

    liken vt. 把…比作;比拟

    literally 从字面上

    lithograph 版画

    lithography 光刻技术

    litmus n. [化]石蕊

    lodging n. 寄宿处;寄宿;出租的房间、住房

    loom 1. n. 织布机;若隐若现的景象2. vi. 可怕地出现;朦胧地出现;隐约可见

    mammoth 1. n. 长毛象;猛犸象;庞然大物2. adj. 猛犸似的;巨大的,庞大的

    memoir n. 自传;实录;回忆录;研究报告

    metaphorical 隐喻性的, 比喻性的

    methodical 有条不紊的,井然的

    mock 1. vt. 嘲弄;模仿;使…失望;使…无效2. vi. 嘲弄,嘲笑

    mockery n. 嘲弄;笑柄;徒劳无功;拙劣可笑的模仿或歪曲

    musing 1. adj. 沉思的;瞑想的2. v. 沉思;凝望(muse的ing形式)

    none other than 不是别的而正是…

    nonplussed 不知所措

    noteworthy 值得注意的

    notion n. 概念;见解;打算

    nuisance n. 麻烦事;损害;讨厌的人;讨厌的东西

    numeral 1. n. 数字2. adj. 表示数字的;数字的

    obligato 1. n. 助奏;助唱(等于obbligato)2. adj. 不能省略的

    obliterate 涂去, 擦去, 删除

    offhand 1. adj. 即时的;随便的;无准备的;即席的2. adv. 即席地;随便地;即时地

    opus n. 作品

    Organ Grinder 柯蒂氏器官;螺旋器

    originate 发起

    outrageous 不像话

    outwit vt. 瞒骗;以智取胜

    overkill 1. vt. 过度地杀伤2. n. 过度的杀伤威力

    overtake 1. vt. 赶上;压倒;突然来袭2. vi. 超车

    pamphlet n. 小册子

    parlance n. 说法;用语;语调;发言

    pathological 病理学的

    patriarch 家长

    pentasyllable n. 五音节

    pest n. 害虫;有害之物;讨厌的人

    plodder n. 沉重行走的人;辛勤工作的人

    plodding 1. adj. 单调乏味的;沉重缓慢的2. v. 沉重地走;辛勤工作(plod的ing形式)

    preach 1. vt. 说教;讲道;传道;鼓吹;反复灌输2. vi. 鼓吹;说教;讲道;宣扬

    pretension 预紧

    principia 数学原理

    prolix adj. 冗长的;说话啰嗦的

    prophecy n. 预言;预言能力;[宗]预言书

    pun 1. vi. 说双关语;说俏皮话2. n. 双关语;俏皮话

    purport 1. vt. 声称;意指;意图;打算2. n. 意义,主旨;意图

    quirk n. 怪癖;急转;借口

    quirky adj. 诡诈的;古怪的;离奇的

    rage 1. n. 愤怒;狂暴,肆虐;情绪激动2. vi. 大怒,发怒;流行,风行

    recession 不景气,后退

    recherché 循环

    redeem vt. 赎回;补偿;挽回;恢复;履行;兑换

    relate 1. vt. 叙述;使…有联系2. vi. 与…有某种联系;认同;符合;涉及

    relish 1. n. 滋味;风味;食欲;开胃小菜;含义2. vt. 喜爱;品味;给…加佐料

    repercussion n. 反响;弹回;反射;[医]浮动诊胎法

    repertoire n. 全部节目;计算机指令系统;(美)某人或机器的全部技能

    resign 1. vt. 辞职;放弃;使听从;委托2. vi. 辞职

    retrograde 退化

    ricercar n. 寻觅曲

    rid vt. 使去掉;使摆脱

    ritual 1. n. 仪式;惯例;礼制2. adj. 仪式的;礼节性的;例行的

    rod n. 棒;枝条;惩罚;权力

    round out 完成

    savor 1. vt. 使有风味;加调味品于;尽情享受2. n. 滋味;气味;食欲

    select 1. vt. 挑选;选拔2. adj. 精选的;挑选出来的;极好的

    serenade 1. n. 小夜曲2. vt. 为…唱小夜曲

    simplistic 简单

    skirt 1. n. 裙子2. vt. 绕过,回避;位于…边缘

    stereotype 铅板

    stratification 分层

    stratify 1. vt. 成层;分层;使形成阶层2. vi. 分层;成层;阶层化

    stupendous 惊人的, 巨大的

    submissive 服从的

    suspense n. 悬念;焦虑;悬疑;悬而不决

    swerve 1. vi. 突然转向;转弯;背离2. vt. 使突然转向;使转弯;使背离

    syllogism n. 推论;三段论

    synopsis n. 概要,大纲

    take stock of 观察;估量

    teaser n. 戏弄者;强求者

    theologian 神学专家

    therein adv. 在那里;在其中

    thinly adv. 稀疏地;薄;瘦;细

    To the Hilt 切题, 切中要害

    transplant 移植

    treatise n. 论文;论述;专著

    turgid adj. 浮夸的;肿胀的;浮肿的

    unobservant 眛

    utterance 发声

    vehement adj. 热烈的;激烈的,猛烈的

    veneration n. 尊敬;崇拜

    vitriolic 硫酸

    wallow 1. vi. 打滚;颠簸;沉迷2. n. 打滚;堕落;泥坑

    walrus n. 海象

    will 1. n. 意志;意图;心愿;情感;遗嘱2. vt. 用意志力使;遗赠;决心要

    wiseacre n. 自以为聪明者

    wordplay n. 字句的争论;双关语

    2017-12-02 12:04:46 回应
  • Two-part invention & Chapter 2 Meaning and form in mathematics

    Two-part invention

    P45 「 以上无穷推理的过程相当于把人看作了没有意识和逻辑从而无法跳出这个无穷级的规则系统而在其外对其进行审视。而这便等同于不承认先验知识的存在性,也不认可其合理性。」

    Chapter 2 Meaning and form in mathematics

    P47 If --p---q- turns out to be a theorem, then so will --p----q--.
    the statement establishes a causal connection between the theoremhood of two strings, but without asserting theoremhood for either one on its own.


    P47 let us give the name well-formed string to any string which begins with a hyphen-group, then has one p, then has a second hyphen-group, then a q, and then a final hyphen-group.

    A well-formed string may not be a theorem.

    P48 Any formal system which tells you how to make longer theorems from shorter ones, but never the reverse, has got to have a decision procedure for its theorems.
    P48 In this way, you “reduce” the problem to determining whether any of several new but shorter strings is a theorem.

    The number of procedures for this operation is finite, because the string to be checked has shorter and shorter length as the operation continues.

    P49 My answer would be that we have perceived an isomorphism between pq-theorems and additions.

    This mapping, isomorphism, is fascinating.

    P50 It is cause for joy when a mathematician discovers an isomorphism between two structures which he knows. It is often a “bolt from the blue”, and a source of wonderment. The perception of an isomorphism between two known structures is a significant advance in knowledge – and I claim that it is such perceptions of isomorphism which create meanings in the minds of people.


    P50 This symbol-word correspondence has a name: interpretation.

    Interpretation is an association on purpose.

    P50 You may make several tentative stabs in the dark before finding a good set of words to associate with the symbols. It is very similar to attempts to crack a code, or to decipher inscriptions in an unknown language like Linear B of Crete: the only way to proceed is by trial and error, based on educated guesses. When you hit a good choice, a “meaningful” choice, all of a sudden things just feel right, and work speeds up enormously. Pretty soon everything falls into place. The excitement of such an experience is captured in The Decipherment of Linear B by John Chadwick.


    1. Trial and error.
    2. Educated guess.
    P50 Mathematicians (and more recently, linguists, philosophers, and some others) are the only users of formal systems, and they invariably have an interpretation in mind for the formal systems which they use and publish. The idea of these people is to set up a formal system whose theorems reflect some portion of reality isomorphically. In such a case, the choice of symbols is a highly motivated one, as is the choice of typographical rules of production.
    P52 But wishing doesn't change the fact it isn't.
    P52 In a formal system, the meaning must remain passive.

    Interpretation is passive.

    P53 However, reality and the formal system are independent. Nobody need be aware that there is an isomorphism between the two. Each side stands by itself – one plus one equals two, whether or not we know that -p-q-- is a theorem; and -p-q-- is still a theorem whether or not we connect it with addition.

    Of course, the formal system can be mapped to another part of reality.

    P53 You might wonder whether making this formal system, or any formal system, sheds new light on truths in the domain of its interpretation? Have we learned any new additions by producing pq-theorems? Certainly not; but we have learned something about the nature of addition as a process – namely, that it is easily mimicked by a typographical rule governing meaningless symbols.
    P56 The hard-edged rules that govern “ideal” numbers constitute arithmetic, and their more advanced consequences constitute number theory.
    P59 Although Euclid's proof is a proof that all numbers have a certain property, it avoids treating each of the infinitely many cases separately.

    The power of generalization.

    2017-12-31 10:51:51 回应
  • Chapter 3 Figure and ground

    Primes vs. composites

    P64 Rules for typographical operations
    1. reading and recognizing any of a finite set of symbols;
    2. writing down any symbol belong to that set;
    3. copying any of those symbols from one place to another;
    4. erasing any of these symbols;
    5. checking to see whether one symbol is the same as another;
    6. keeping and using a list of previously generated theorems.
    P64 Definition of formal system: compound some of these operations 「typographical operations」to make a formal system.

    Capturing compositeness

    P65 Characterize composite numbers: If x-ty-qz is a theorem, then Cz is a theorem.

    x-ty-qz means (X+1)×(Y+1)=Z, where the capitalized letters X, Y and Z represent the number of hyphens in the strings x, y and z respectively. The conclusion Cz is a predicate about the fact that z is a composite number.

    P65 I am defending this new rule by giving you some “Intelligent mode” justification for it. That is because you are a human being, and want to know why there is such as rule. If you were operating exclusively in the “Mechanical mode”, you would not need any justification, since M-mode workers just follow the rules mechanically and happily, never questioning them!

    Or the M-mode workers will never work happily but merely seem being happy. In addition, humans with intelligence are always liable to ask for reasons and jump out of the system.

    P65 The Requirement of Formality, which in Chapter I probably seemed puzzling (because it seemed so obvious), here becomes tricky, and crucial.

    Illegally characterizing primes

    P67 The reason for hesitating is that the holes are only negatively defined – they are the things that are left out of a list which is positively defined.

    Figure and ground

    P67 A message written in such an alphabet is shown below. At first it looks like a collection of somewhat random blobs, but if you step back a ways and stare at it for a while, all of a sudden, you will see seven letters appear in this . . .

    The answer is: MAIL BOX. This mode or sense seems to be activated in my mind suddenly without control.

    Figure and ground in music

    P70 it is surprising when we find, in the lower lines of a piece of music, recognizable melodies. This does not happen too often in post-baroque music. Usually the harmonies are not thought of as foreground. But in baroque music – in Bach above all – the distinct lines, whether high or low or in between, all act as “figures”. In this sense, pieces by Bach can be called “recursive”.

    Here, the word “recursive” means, in artistic domain, both the foreground and background in a figure are deliberately designed and have meanings.

    Recursively enumerable sets vs. recursive sets

    P72 There exist formal systems whose negative space (set of nontheorems) is not the positive space (set of theorems) of any formal system.


    P72 There exist recursively enumerable sets which are not recursive.

    In the above terminology, recursively enumerable (abbreviated as “r.e.”) is the mathematical counterpart to our artistic notion of “cursively drawable”. Recursive is the counterpart of “recursive” in the artistic domain, i.e. both the foreground and background in a figure are deliberately designed and have meanings. Therefore, the book then says

    a “recursive set” is like a figure whose ground is also a figure – not only is it r.e., but its complement is also r.e.
    P72 A typographical decision procedure is a method which tells theorems from nontheorems.
    P73 It is important to understand that if the members of F were always generated in order of increasing size, then we could always characterize G.
    P73 We can agree that all the numbers in set F have some common “form” – but can the same be said about numbers in set G? It is a strange question. When we are dealing with an infinite set to start with – the natural numbers – the holes created by removing some subset may be very hard to define in any explicit way. And so it may be that they are not connected by any common attribute or “form”.


    01/03/18 综合以上摘录可以发现,侯先生只能定性、形象地描述的东西被康先生用理性阐述清楚了。

    There exist formal systems whose negative space (set of nontheorems) is not the positive space (set of theorems) of any formal system.
    There exist recursively enumerable sets which are not recursive.
    We can agree that all the numbers in set F have some common “form” – but can the same be said about numbers in set G? It is a strange question. When we are dealing with an infinite set to start with – the natural numbers – the holes created by removing some subset may be very hard to define in any explicit way. And so it may be that they are not connected by any common attribute or “form”.



    Primes as figure rather than ground

    P73 Axiom schema: xyDNDx where x and y are hyphen-strings.

    此公理意为:所有较大的数都无法整除相对较小的数。即,if X > Y, then xDNDy.

    P74 RULE: If xDNDy is a theorem, then so is xDNDxy.

    该规则的意思是如果X无法整除Y,则无论给Y增加多少倍的X,X仍旧无法将其整除。即,mod(Y, X)==mod(NX+Y, X).

    P74 we can't be so vague in formal systems as to say “et cetera”. We must spell things out.
    P74 It is this “monotonicity” or unidirectionality – this absence of cross-play between lengthening and shortening, increasing and decreasing – that allows primality to be captured.

    2017-12-31 11:46:29 回应