《Godel, Escher, Bach》的笔记Chapter 3 Figure and ground
皮波迪先生 (择善固执，止于至善)
 章节名：Chapter 3 Figure and ground
 20171231 11:45:13
Primes vs. composites
P64 Rules for typographical operations
 reading and recognizing any of a finite set of symbols;
 writing down any symbol belong to that set;
 copying any of those symbols from one place to another;
 erasing any of these symbols;
 checking to see whether one symbol is the same as another;
 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 xtyqz is a theorem, then Cz is a theorem.xtyqz 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 Mmode workers just follow the rules mechanically and happily, never questioning them!Or the Mmode 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 postbaroque 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 hyphenstrings.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 crossplay between lengthening and shortening, increasing and decreasing – that allows primality to be captured.
皮波迪先生对本书的所有笔记 · · · · · ·

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...

Twopart invention & Chapter 2 Meaning and form in mathematics
Twopart 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...

Chapter 3 Figure and ground
说明 · · · · · ·
表示其中内容是对原文的摘抄