《Godel, Escher, Bach》的笔记-Two-part invention & Chapter 2 Meaning and form in mathematics
- 章节名：Two-part invention & Chapter 2 Meaning and form in mathematics
- 2017-12-22 09:02:42
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 mathematicsP47 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.
- Trial and error.
- 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.
皮波迪先生对本书的所有笔记 · · · · · ·
Formal systems P33 Requirement of Formality: The major point, which almost doesn't need...
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...
Two-part invention & Chapter 2 Meaning and form in mathematics
Primes vs. composites P64 Rules for typographical operations reading and recognizing an...
说明 · · · · · ·