Godel, Escher, Bach

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.

Theorem即为可以从正面予以"attack"（表征、确定、给出）的，而nontheorem则是通过排除法或者取theorem集合的补集来间接得到的。

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”.

能够有确定的规则来生成定理——无论它有多么复杂，但只要是确定的和有限步数的——就相当于我们发现了其中（定理集合）的共性和规律。而显然地，这些定理的补集nontheorems是否仍旧包含确定的生成规则，以及呈现其共性的规律、形式就不得而知了。

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”.

康德在《纯粹理性批判》中提到：

如果我只是指出直观的客体不是怎样，却不能说在它里面究竟包含着什么，这毕竟不是真正的知识；在这种情况下，我根本没有表象一个客体对我的纯粹知性概念来说的可能性，因为我不能给予与它相应的直观，而是只能说我们的直观对它无效。但这里最重要的是，就连一个范畴也不能被运用于这样一种某物；例如实体的概念，也就是说关于作为主词、但却绝不能纯然作为谓词能够实存的某物的概念，对于它，如果不是经验性的直观给予我运用的事例，我就根本不知道是否有某个与这种思想规定相应的事物存在。不过，更多的东西留待后面再说。

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.

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.

即描述的只是两个陈述之间的关系，而不是陈述本身。这一点非常像是康德的观点：人类的理性只能认识事物呈现给我们的表象，而并非客体本身。

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.

对两个看似毫无关联的系统或结构进行类比、映射，然后再去证明其中所有的定理。这便是人类思维的能力，而机器难以获得。

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.

在黑暗中的尝试与探索。其中遵循的步骤是：

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.

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

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.

现在有了人工智能技术，自动驾驶的问题就得以解决了。当然，这里作者说得也没有错：无论人工智能算法能够把汽车开得多么好，它本身从心智上来说是无法理解这算法的——它自己只不过是“机械化”程序的执行者。

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.

人工智能算法是否也可以跳出当前正在执行的任务，“回过头来”对其加以审视，并发现其中的模式呢？这种监督与审视正在执行的任务——即算法——的功能就是算法的算法。这样看来，人工智能算法在一定程度上可以说是真的拥有“智能”的。当然，这种“智能”毕竟也是部分的和有限的，因为算法毕竟无法跳出算法的算法从而形成算法的算法的算法。或者即便可以做到这一点，但也终究不能无休止地跳出——因为一个合理的、现实的算法应该是在有限的时间内能够终止的。

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.

跳出系统看问题。

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.

身在曹营心在汉。

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!

若依据康德的观点，这里说的characterization显然是属于分析的而不是综合的。然而，我们也没有必要因为它是分析的就觉得悲观，毕竟characterization本身绝不是一件轻松或微不足道的工作。人类能够做到水晶般清晰和透彻地、具体地（concrete）表征一个系统，尽管是分析地，就已实属了不起了。

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