按：這篇筆記是系列筆記的第二篇洋访，第一部分有4節(jié)，每節(jié)對應1-2則筆記斑司。
筆記的方式渗饮，是引用一段個人覺得比較有亮點的英文原文，再給一段簡化的中文說明宿刮，不采用中文版的翻譯互站，不自行做直接翻譯，只說明要點糙置。因為不可能大段大段地去引用云茸，必然會有語境的丟失，會做一些補充說明谤饭，以“按：”開始。對中文版翻譯進行更正或調(diào)整的說明，以“注：”開始揉抵。偶爾也會插入自己的議論亡容，以“評：”開始。
上一篇筆記為《普林斯頓數(shù)學指引》讀書筆記——I.1 數(shù)學是關于什么的 冤今。
下一篇筆記為《普林斯頓數(shù)學指引》讀書筆記——I.3 一些基本的數(shù)學定義（上）
I.2 The Language and Grammar of Mathematics（數(shù)學的語言和語法）
本節(jié)寫得特別出彩闺兢，完全可以當作一篇自包含的essay來讀。學過數(shù)理邏輯的同學對本節(jié)介紹的內(nèi)容并不陌生戏罢，然而本節(jié)精彩之處正是清晰闡述了數(shù)學家們選擇從自然語言走向數(shù)學語言的內(nèi)在邏輯屋谭。
這里先做一個簡表，將本節(jié)介紹的符號和含義對應列出龟糕，后面不再贅述桐磁，而專注于更有洞見的內(nèi)容。
| 符號 | 含義 | 例子 |
|---|---|---|
| 屬于 | ||
| 不屬于 | ||
| 映射 |
|
|
| 映射 |
|
|
| 等價于 | ||
| 蘊含 | ||
| 且 | ||
| 或 | ||
| 對于所有…… | ||
| 存在…… |
Introduction（引言）
The main reason for using mathematical grammar is that the statements of mathematics are supposed to be completely precise, and it is not possible to achieve complete precision unless the language one uses is free of many of the vaguenesses and ambiguities of ordinary speech. Mathematical sentences can also be highly complex: if the parts that made them up were not clear and simple, then the unclarities would rapidly accumulate and render the sentences unintelligible.
使用數(shù)學語法主要是因為我擂，數(shù)學命題理應是完全精確的。要做到完全精確缓艳，使用的語言就必須擺脫充斥于日常語言中的含糊和歧義校摩。數(shù)學命題也可能是高度復雜的：如果構成這些命題的各個部分并不簡明清晰，這些不清晰的地方就會很快地積累起來阶淘，使得整個表述無法理解衙吩。
注：上面這段點睛的話，中文版有多處錯譯（如“are supposed to be”溪窒、“is free of”分井、“ordinary speech”、“not clear and simple”等）霉猛，容易產(chǎn)生誤解尺锚，大家可對照看下。
Four Basic Concepts（四大基本概念）
Sets are also very useful if one is trying to do metamathematics, that is, to prove statements not about mathematical objects but about the process of mathematical reasoning itself.
研究元數(shù)學的時候惜浅，集合時常是很有用的瘫辩。元數(shù)學的目標是去證明一類特別的命題，這些命題并非關于數(shù)學對象坛悉，而是關于數(shù)學推理過程自身的伐厌。
Sets allow one to reduce greatly the number of parts of speech that one needs, turning almost all of them into nouns.
集合通過把幾乎所有的詞都變成名詞，使我們能夠大為減少所需的詞類的數(shù)量裸影。
Over and over again, throughout mathematics, it is useful to think of a mathematical phenomenon, which may be complex and very un-thinglike, as a single object.
貫穿數(shù)學諸多領域挣轨，把一個數(shù)學現(xiàn)象（哪怕它可能很復雜，很不像是一個東西）看成一個單一對象轩猩，一次又一次地被證明是非常有用的卷扮。
Many algebraic structures are most naturally thought of as sets of functions. (See, for example, the discussion of groups and symmetry in [I.3 §2.1]. See also Hilbert Spaces [III.37], Function Spaces [III.29], and Vector Spaces [I.3 §2.3].)
許多代數(shù)結構荡澎，看成函數(shù)的集合最為自然。參見[I.3 §2.1]關于群和對稱的討論晤锹，又可見希爾伯特空間摩幔、函數(shù)空間以及向量空間。
To specify a function, therefore, one must be careful to specify two sets as well: the domain, which is the set of objects to be transformed, and the range, which is the set of objects they are allowed to be transformed into.
因此要確定一個函數(shù)鞭铆，就必須同時仔細地確定兩個集合：一個是定義域（domain）或衡，即要被變換的對象的集合，另一個是值域（range）车遂，即被允許變換成的對象的集合封断。
“Most” functions, though not most functions that one actually uses, are so arbitrary that they cannot be defined. (Such functions may not be useful as individual objects, but they are needed so that the set of all functions from one set to another has an interesting mathematical structure.)
絕大多數(shù)函數(shù)（雖然里面大部分函數(shù)我們很少實際使用）都非常的任意，以至于他們難以被干凈地定義舶担。這些函數(shù)坡疼，雖然作為個別的對象不一定有用，但是需要有它們柄沮，從一個集合到另外一個集合的所有的函數(shù)的集合才會具備有趣的數(shù)學結構回梧。
To use “<” in a sentence, one should precede it by a noun and follow it by a noun. For the resulting grammatically correct sentence to make sense, the nouns should refer to numbers (or perhaps to more general objects that can be put in order). A mathematical “object” that behaves like this is called a relation, though it might be more accurate to call it a potential relationship. “Equals” and “is an element of” are two other examples of relations.
如果要在一個句子里面使用符號<，它的前后就要各放一個名詞祖搓。這樣得到的句子語法上雖然正確狱意，但要有意義宴霸，這兩個名詞都應該代表數(shù)稿黍，或者具有次序關系的更一般的數(shù)學對象。一個具備上述這種行為的數(shù)學對象稱為一個關系肋拔，雖然稱它為潛在的關系更為準確一些镐作〔亟悖“等于”和“屬于”，是關系的另外兩個例子该贾。
There are many situations in mathematics where one wishes to regard different objects as “essentially the same,” and to help us make this idea precise there is a very important class of relations known as equivalence relations.
在數(shù)學里有許多時候羔杨，我們希望把不同的對象看成是“本質(zhì)上相同”的。為了把握這種思想杨蛋，有一類非常重要的關系兜材，稱為“等價關系”。
What exactly is it that these two relations have in common? The answer is that they both take a set (in the first case the set of all geometrical shapes, and in the second the set of all whole numbers) and split it into parts, called equivalence classes, where each part consists of objects that one wishes to regard as essentially the same.
這兩個關系究竟有什么共同之處呢逞力？答案是曙寡，它們都取一個集合（前者是所有幾何圖形的集合，后者是所有整數(shù)的集合）寇荧，并把它劃分成幾個部分举庶，其中每個部分各自都由希望被看作“本質(zhì)上相同”的對象組成，稱為一個等價類揩抡。
One of the main uses of equivalence relations is to make precise the notion of quotient [I.3 §3.3] constructions.
等價關系的主要用途之一是使得商這個概念的構造變得精確户侥。
These last two operations raise another issue: unless the set A is chosen carefully, they may not always be defined. For example, if one restricts one’s attention to the positive integers, then the expression 3 ? 5 has no meaning. There are two conventions one could imagine adopting in response to this. One might decide not to insist that a binary operation should be defined for every pair of elements of A, and to regard it as a desirable extra property of an operation if it is defined everywhere. But the convention actually in force is that binary operations do have to be defined everywhere, so that “minus,” though a perfectly good binary operation on the set of all integers, is not a binary operation on the set of all positive integers.
后幾種運算（減法镀琉、除法和升冪）提出來另外一個問題：除非集合A選擇得很仔細，這幾種運算并非總能被定義添祸。例如滚粟，如果限制只關注正整數(shù)寻仗，則表達式3-5就沒有意義刃泌。
不難想到兩種解決上述問題的約定。第一種約定決定不再堅持“一個二元運算要對每一對元素都有定義”署尤，從而將一個二元運算處處有定義看成是一個額外的耙替、令人喜歡的特性。
然而曹体，真正實施的約定是第二種約定：一個二元運算必須處處有定義俗扇。因此，減法雖然在所有整數(shù)的集合上是良好定義的二元運算箕别，在所有正整數(shù)的集合上則不是一個二元運算铜幽。
注：此處中文版將第一種約定的兩個子句當作兩個約定來處理，不符合原義串稀，尤其是考慮這樣譯兩個約定其實就是一個意思除抛，所以此處做了修改。
These basic properties of binary operations are fundamental to the structures of abstract algebra. See four important algebraic structures [I.3 §2] for further details.
二元運算的這些基本性質(zhì)母截，是各個抽象代數(shù)結構的基礎到忽，詳見四個重要的代數(shù)結構 [I.3 §2] （即群、域清寇、向量空間和環(huán)）喘漏。
一點初等邏輯（Some Elementary Logic）
In English the word “implies” suggests some sort of connection between P and Q, that P in some way causes Q or is at least relevant to it. If P causes Q then certainly P cannot be true without Q being true, but all a mathematician cares about is this logical consequence and not whether there is any reason for it. Thus, if you want to prove that P ? Q, all you have to do is rule out the possibility that P could be true and Q false at the same time.
在日常的英語里，“implies”（蘊含）一詞暗示在P和Q之間有某種聯(lián)系华烟，即P以某種方式導致了Q翩迈，至少與這個過程有關聯(lián)。如果確實是P（唯一地）導致了Q盔夜，當然P非真時Q也無法為真负饲。然而數(shù)學家關心的只是邏輯推論關系，而不是背后的理由比吭。所以如果需要證明P ? Q绽族，唯一要做的就是排除P為真而Q非真這種情況（而無需排除P為非真時Q為真這種情況）。
Words like “all,” “some,” “any,” “every,” and “nothing” are called quantifiers, and in the English language they are highly prone to this kind of ambiguity. Mathematicians therefore make do with just two quantifiers, and the rules for their use are much stricter. They tend to come at the beginning of sentences, and can be read as “for all” (or “for every”) and “there exists” (or “for some”).
諸如“all”（所有）衩藤、“some”（有些）吧慢、“any”（任一）、“every”（每一）赏表、“nothing”（沒有）检诗，這樣的一些單詞匈仗，都稱為量詞，而在日常的英語語言里逢慌，很容易產(chǎn)生歧義悠轩。所以數(shù)學家們，只需要用到兩個量詞來解決所有的問題攻泼，而且使用的規(guī)則也要嚴格得多火架。它們總是被放在句首，其中?可被讀作“for all”（對于所有……）或“for every”（對于每一個……）忙菠，?則讀作“there exists”（存在……）或“for some”（對于某些）何鸡。
Let us take A to be a set of positive integers and ask ourselves what the negation is of the sentence “Every number in the set A is odd.” Many people when asked this question will suggest, “Every number in the set A is even.” However, this is wrong: if one thinks carefully about what exactly would have to happen for the first sentence to be false, one realizes that all that is needed is that at least one number in A should be even. So in fact the negation of the sentence is, “There exists a number in A that is even.”
取一個由正整數(shù)構成的集合A，并且問“A中的每個數(shù)均為奇數(shù)”這句話的反面是什么牛欢？許多人被問到這個問題的時候都會說骡男，應該是“A中的每個數(shù)均為偶數(shù)”。然而這是錯的：如果我們仔細考慮了要使得該命題非真究竟需要什么傍睹，就會看到隔盛，需要的是A中至少有一個數(shù)是偶數(shù)，所以事實上拾稳，這句話的反面是“A中存在一個數(shù)為偶數(shù)”吮炕。
A variable such as
m, which denotes a specific object, is called a free variable. It sort of hovers there, free to take any value. A variable likeaandb, of the kind that does not denote a specific object, is called a bound variable, or sometimes a dummy variable. (The word “bound” is used mainly when the variable appears just after a quantifier.
代表一個特定對象的變元稱之“自由變元”，它可以自由地取任何值熊赖。并不代表某一特定對象的變元来屠，稱為“約束變元”，或者“啞變元”震鹉【愕眩“約束”一詞主要用在變元緊跟著量詞的情況下（其他情況一般叫“啞變元”）。
注：此處中文版將“variable”譯作“變項”并不為錯传趾，但并不自然迎膜，而且“項”一般是一個表達式了，“元”更基本一些浆兰，所以改為“變元”（當然磕仅，此處語境顯然不適合“變量”的譯法）。
Yet another indication that a variable is a dummy variable is when the sentence in which it occurs can be rewritten without it.
一個變元是啞變元還有一個標志簸呈，就是它們所在的句子原則上可以改寫成沒有它們的形式（按：即通過將它們展開為實際的值榕订，見原書例子）。
The language typically used is a careful compromise between fully colloquial English, which would indeed run the risk of being unacceptably imprecise, and fully formal symbolism, which would be a nightmare to read. The ideal is to write in as friendly and approachable a way as possible, while making sure that the reader (who, one assumes, has plenty of experience and training in how to read mathematics) can see easily how what one writes could be made more formal if it became important to do so. And sometimes it does become important: when an argument is difficult to grasp it may be that the only way to convince oneself that it is correct is to rewrite it more formally.
數(shù)學家們實際使用的典型語言蜕便，則是完全口語化的英語與完全形式化的符號語言之間的一種仔細推敲后的折衷產(chǎn)物劫恒。完全用前者會冒著引入無法接受的不精確的風險，完全用后者，人們讀起來則像在做噩夢一樣两嘴。理想的辦法是用一種盡可能對讀者友好丛楚、易于接受的形式，同時又確保讀者（假使他們擁有相當多的閱讀數(shù)學文章的經(jīng)驗和訓練）在認為必要的情況下憔辫，可以很容易看出如何將其進行更形式化的改寫趣些。這種必要的情況有時的確會出現(xiàn)：當一個論證很難掌握時，使人確信這個論證為正確的唯一辦法贰您，正是對其進行更形式化的改寫坏平。
注：上面這段話，中文版有幾處錯譯（如“run the risk of”枉圃、“making sure”等）功茴，容易產(chǎn)生誤解庐冯，大家可對照看下孽亲。
In practice, there are many different levels of formality, and mathematicians are adept at switching between them. It is this that makes it possible to feel completely confident in the correctness of a mathematical argument even when it is not presented in the manner of (18)—though it is also this that allows mistakes to slip through the net from time to time.
實踐中，形式化有多種不同的尺度展父，而數(shù)學家們很善于在其中轉(zhuǎn)換返劲。這使得一個數(shù)學論證即使沒有寫得完全形式化，數(shù)學家們也能對其正確性有完全的信心——雖然這種轉(zhuǎn)換也使得種種錯誤如同漏網(wǎng)之魚般時不時地鉆了進來栖茉。
