《Godel, Escher, Bach》的笔记-Chapter 3 Figure and ground
- 章节名：Chapter 3 Figure and ground
- 2017-12-31 11:45:13
Primes vs. compositesP64 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 compositenessP65 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 primesP67 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 groundP67 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 musicP70 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 setsP72 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 saysa “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 groundP73 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.
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Chapter 3 Figure and ground
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