按：這篇筆記是系列筆記的第五篇，第一部分有4節(jié)，每節(jié)對(duì)應(yīng)1-2篇筆記。
筆記的方式，是引用一段個(gè)人覺得比較有亮點(diǎn)的英文原文，再給一段簡(jiǎn)化的中文說明，不采用中文版的翻譯，不自行做直接翻譯，只說明要點(diǎn)。因?yàn)椴豢赡艽蠖未蠖蔚厝ヒ茫厝粫?huì)有語境的丟失，會(huì)做一些補(bǔ)充說明，以“按：”開始。對(duì)中文版翻譯進(jìn)行更正或調(diào)整的說明，以“注：”開始。偶爾也會(huì)插入自己的議論，以“評(píng)：”開始。
前四篇筆記為：
I.4 The General Goals of Mathematical Research （數(shù)學(xué)研究的一般目的）
1 Solving Equations （解方程）
Mathematics is full of objects and structures (of a mathematical kind), but they do not simply sit there for our contemplation: we also like to do things to them.
數(shù)學(xué)里充滿了對(duì)象和結(jié)構(gòu)，但是它們并不是靜靜地呆在那里供我們沉思：我們也想要對(duì)它們實(shí)施各種操作。
Transformations like these give rise to a never-ending source of interesting problems. If we have defined some mathematical process, then a rather obvious mathematical project is to invent techniques for carrying it out. This leads to what one might call direct questions about the process. However, there is also a deeper set of inverse questions, which take the following form. Suppose you are told what process has been carried out and what answer it has produced. Can you then work out what the mathematical object was that the process was applied to?
像這樣的變換會(huì)產(chǎn)生無窮無盡的有趣問題。如果我們定義了某個(gè)數(shù)學(xué)過程，那么創(chuàng)造一種方法來執(zhí)行它，就是一個(gè)特別顯然的數(shù)學(xué)課題。這會(huì)引出可稱之為該數(shù)學(xué)過程的直接問題。然而，存在一組更深層次的逆問題，具備如下形式：假定你已被告知怎樣的數(shù)學(xué)過程被執(zhí)行了，以及它所產(chǎn)生的答案，你能否找到該數(shù)學(xué)過程是作用于什么數(shù)學(xué)對(duì)象之上的？
Because x and y can be very much more general objects than numbers, the notion of solving equations is itself very general, and for that reason it is central to mathematics.
因?yàn)閤和y這些未知項(xiàng)可以是比數(shù)一般得多的對(duì)象，解方程式的概念本身也就是非常一般的，因此它成為了數(shù)學(xué)的中心問題之一。
We have just discussed the generalization of linear equations from one variable to several variables. Another direction in which one can generalize them is to think of linear functions as polynomials of degree 1 and consider functions of higher degree.
我們剛才討論了線性方程從一元到多元的推廣。另外一個(gè)推廣它們的方向是將線性函數(shù)看作一次多項(xiàng)式，從而考慮更高次數(shù)的函數(shù)。
What matters about an equation is often the existence and properties of solutions and not so much whether one can find a formula for them.
對(duì)于一個(gè)方程最關(guān)鍵的通常是解的存在性與其性質(zhì)，而不那么在于能否找到解的公式。
In many instances the explicit solubility of an equation is a relative notion.
在許多情況下，解的顯式的可解性是一個(gè)相對(duì)的概念。（比如根號(hào)2這樣的數(shù)的定義就是它是x^2=2的解，允許根式出現(xiàn)在解的公式中，本身就放寬了公式的定義來使得相當(dāng)多的方程變得顯式可解了。）
The answer to the question of whether a particular equation has a solution varies according to where the solution is allowed to be.
一個(gè)特定的方程是否有解，取決于允許解被允許在哪里求解。（這里的哪里指的是數(shù)域或者數(shù)學(xué)對(duì)象的集合。）
......a typical Diophantine equation, the name given to an equation if one is looking for integer solutions.
(上面的例子是)一個(gè)典型的丟番圖方程。如果要找的是方程的整數(shù)解，就可以給該方程這個(gè)名字。
What does this tell us about Diophantine equations? We can no longer dream of a final theory that will encompass them all, so instead we are forced to restrict our attention to individual equations or special classes of equations, continually developing different methods for solving them.
丟番圖方程告訴了我們什么呢？我們?cè)僖膊荒軌?mèng)想有一個(gè)囊括所有這種方程（解法）的最終理論，相反，我們被迫聚焦于個(gè)別或特定類別的方程，并且持續(xù)地對(duì)它們發(fā)展不同的解法。
For many reasons, differential equations represent a jump in sophistication. One is that the unknowns are functions, which are much more complicated objects than numbers or n-dimensional points. (For example, the first equation above asks what function x of t has the property that if you differentiate it twice then you get -k^2 times the original function.) A second is that the basic operations one performs on functions include differentiation and integration, which are considerably less “basic” than addition and multiplication. A third is that differential equations that can be solved in “closed form,” that is, by means of a formula for the unknown function f , are the exception rather than the rule, even when the equations are natural and important.
有許多理由說明求解微分方程在精巧性上是一個(gè)飛躍。其中一個(gè)是此時(shí)未知項(xiàng)是函數(shù)，它相比數(shù)或者n維空間的點(diǎn)來說是復(fù)雜得多的對(duì)象（例如，上面的第一個(gè)方程要求，關(guān)于t的函數(shù)x在微分兩次之后還原為原來的函數(shù)，但乘上一個(gè)-k^2的因子）。第二個(gè)理由是現(xiàn)在施加于函數(shù)上的運(yùn)算包括了微分和積分，它們遠(yuǎn)不如加法和乘法那么“基本”。第三個(gè)理由是可以“封閉形式”求解的微分方程（即通過未知函數(shù)f的公式來表達(dá)）只是例外而非常規(guī)，即使方程非常自然和重要。
As with polynomial equations, this can depend on what you count as an allowable solution. Sometimes we are in the position we were in with the equation x^2 = 2: it is not too hard to prove that solutions exist and all that is left to do is name them.
就像多項(xiàng)式方程一樣，（微分方程的可解性）可能取決于把什么當(dāng)作被允許的解。有時(shí)候我們就像回到了面對(duì)方程x的平方等于2時(shí)的局面：證明解的存在并不難，只需要給它取一個(gè)名字就行了。
Sometimes there are ways of proving that solutions exist even if they cannot be easily specified. Then one may ask not for precise formulas, but for general descriptions. For example, if the equation has a time dependence (as, for instance, the heat equation and wave equations have), one can ask whether solutions tend to decay over time, or blow up, or remain roughly the same. These more qualitative questions concern what is known as asymptotic behavior, and there are techniques for answering some of them even when a solution is not given by a tidy formula.
有時(shí)候存在多種方法證明解是存在的，哪怕這些解不能被輕易指定。這時(shí)，人們會(huì)不要求得到精確的公式，只希望得到一般性的描述。例如，如果這個(gè)方程有對(duì)時(shí)間的依賴性（比如熱的方程和波的方程就都有），人們就會(huì)問解是否隨著時(shí)間而衰減，或者爆發(fā)，或者大體上不變。這些更加定性的問題，關(guān)注的是所謂的漸進(jìn)性態(tài)，有一些技巧來回答部分此類問題，即使此時(shí)解無法由干凈利落的公式給出。
2 Classifying（分類）
If one is trying to understand a new mathematical structure, such as a group [I.3 §2.1] or a manifold [I.3 §6.9], one of the first tasks is to come up with a good supply of examples. Sometimes examples are very easy to find, in which case there may be a bewildering array of them that cannot be put into any sort of order. Often, however, the conditions that an example must satisfy are quite stringent, and then it may be possible to come up with something like an infinite list that includes every single one.
如果想理解一個(gè)數(shù)學(xué)結(jié)構(gòu)，例如群[I.3 §2.1] 或者一個(gè)流形[I.3 §6.9]，首要任務(wù)之一就是想出足夠多的例子。有時(shí)候例子找起來很容易，卻只是多得眼花繚亂而理不出任何秩序的一大堆。然而，時(shí)常這些例子必須要滿足的條件相當(dāng)嚴(yán)格，于是這時(shí)，想出的就像一個(gè)無限長(zhǎng)的、包含每一個(gè)具體例子的清單。
評(píng)：這一段無論是中文還是英文，都始終沒有理解作者的意思。為什么主體是分類，談的卻是舉例？如果談的是舉出典型的例子，并且圍繞著典型的例子進(jìn)行聚類分組，還可以理解，但這里并沒有表達(dá)這個(gè)意思。提出說存在容易找例子但不好分類組織的情況，但沒有給出解決方案，就直接說變成了一個(gè)無限長(zhǎng)的清單，是逃避了問題，還是這種情況每個(gè)例子就的確是一類，與其他例子沒有共性？再考慮到下面一段話提到對(duì)清單中每一個(gè)例子驗(yàn)證某一個(gè)命題成立，則該命題對(duì)于該數(shù)學(xué)結(jié)構(gòu)成立，卻沒有在這段話中交代如何確保分類是不漏不重（Mutually Exclusive Collectively Exhaustive，MECE）的。
Classifications are very useful because if we can classify a mathematical structure then we have a new way of proving results about that structure: instead of deducing a result from the axioms that the structure is required to satisfy, we can simply check that it holds for every example on the list, confident in the knowledge that we have thereby proved it in general. This is not always easier than the more abstract, axiomatic approach, but it certainly is sometimes. Indeed, there are several results proved using classifications that nobody knows how to prove in any other way. More generally, the more examples you know of a mathematical structure, the easier it is to think about that structure—testing hypotheses, finding counterexamples, and so on. If you know all the examples of the structure, then for some purposes your understanding is complete.
分類是非常有用的，因?yàn)槿绻軌驅(qū)σ环N數(shù)學(xué)結(jié)構(gòu)進(jìn)行分類，就有了一種證明有關(guān)該結(jié)構(gòu)的結(jié)果的新方法：不必從這個(gè)結(jié)構(gòu)所必須滿足的公理中來推出結(jié)果，而只需要驗(yàn)證這個(gè)結(jié)果是否對(duì)于這個(gè)（分類）清單中的每一個(gè)例子都成立，我們就可以很有信心地知道我們借此一般性地證明了該結(jié)果。
這樣做并不總是比更抽象的公理方法更容易，但有時(shí)確實(shí)要容易一些。事實(shí)上，有一些結(jié)果就是用分類才證明的，而且至今無人知曉如何用其他方式證明。更一般地說，對(duì)于一個(gè)數(shù)學(xué)結(jié)構(gòu)知道的例子越多，對(duì)這個(gè)結(jié)構(gòu)進(jìn)行思考就越容易——檢驗(yàn)假設(shè)、尋找反例等等。如果已經(jīng)知道了一個(gè)結(jié)構(gòu)所有的例子，則對(duì)于某些目的而言，我們完全懂得了這個(gè)結(jié)構(gòu)。
We therefore know that the regular polytopes in dimensions three and higher fall into three families—the n-dimensional versions of the tetrahedron, cube, and octahedron—together with five “exceptional” examples—the dodecahedron, the icosahedron, and the three four-dimensional polytopes just described. This situation is typical of many classification theorems. The exceptional examples, often called “sporadic,” tend to have a very high degree of symmetry—it is almost as if we have no right to expect this degree of symmetry to be possible, but just occasionally by a happy chance it is. The families and sporadic examples that occur in different classification results are often closely related, and this can be a sign of deep connections between areas that do not at first appear to be connected at all. Sometimes one does not try to classify all mathematical structures of a given kind, but instead identifies a certain class of “basic” structures out of which all the others can be built in a simple way.
因此我們知道在3維和更高維的情況下，正多胞形分為三個(gè)族類，即n維版本的正四面體、正六面體、正八面體，再加上五個(gè)特例——三維的正12面體和正20面體，還有剛才描述的三個(gè)4維多胞體（按：這里不方便展開介紹，可參見 List of regular polytopes and compounds）。這種情形在許多分類定理中非常典型。這些特例，常稱為“零星的”，通常傾向于具有非常高的對(duì)稱性——這種程度的對(duì)稱性我們不敢期望其可能，卻奇跡般地存在。這些出現(xiàn)在不同分類結(jié)果中的族類以及特例，時(shí)常具備相互間緊密的聯(lián)系，這正是乍一看毫無關(guān)聯(lián)的領(lǐng)域間有深刻聯(lián)系的一個(gè)信號(hào)。有時(shí)我們并不打算把某一類型的數(shù)學(xué)結(jié)構(gòu)全部加以分類，而是從中識(shí)別出某一類基本的結(jié)構(gòu)，使得其他結(jié)構(gòu)全可以由它們簡(jiǎn)單地構(gòu)造出來。
Why should nonequivalence be harder to prove than equivalence? The answer is that in order to show that two objects are equivalent, all one has to do is find a single transformation that demonstrates this equivalence. However, to show that two objects are not equivalent, one must somehow consider all possible transformations and show that not one of them works. How can one rule out the existence of some wildly complicated continuous deformation that is impossible to visualize but happens, remarkably, to turn a sphere into a torus?
為什么不等價(jià)比等價(jià)要難證明呢？答案在于，要證明兩個(gè)對(duì)象等價(jià)，只要找到一個(gè)變換來演示其等價(jià)性，而要證明兩個(gè)對(duì)象不等價(jià)，就要考慮一切可能的變換，而且證明沒有一個(gè)有效。我們?nèi)绾尾拍芘懦嬖谝粋€(gè)極為復(fù)雜的、無法可視化的連續(xù)變換把一個(gè)球面變成一個(gè)環(huán)面的可能性呢？
The Euler number is an example of an invariant. This means a function φ, the domain of which is the set of all objects of the kind one is studying, with the property that if X and Y are equivalent objects, then φ(X) = φ(Y). To show that X is not equivalent to Y, it is enough to find an invariant φ for which φ(X) and φ(Y) are different. Sometimes the values φ takes are numbers (as with the Euler number), but often they will be more complicated objects such as polynomials or groups.
歐拉示性數(shù)是不變式的一個(gè)例子。不變式即是一個(gè)函數(shù)φ，它的定義域是要研究的那一類的全部對(duì)象的集合，而且具有如下的屬性，如果兩個(gè)對(duì)象X和Y等價(jià)，則φ(X) = φ(Y)。為了證明X和Y不等價(jià)，只需要找到一個(gè)不變式使得這兩個(gè)不變式不相等即可，有時(shí)候這個(gè)不變式的值φ是一個(gè)數(shù)，但往往會(huì)是更復(fù)雜的數(shù)學(xué)結(jié)構(gòu)，例如多項(xiàng)式或者群。
3 Generalizing （推廣）
When an important mathematical definition is formulated, or theorem proved, that is rarely the end of the story. However clear a piece of mathematics may seem, it is nearly always possible to understand it better, and one of the most common ways of doing so is to present it as a special case of something more general.
當(dāng)一個(gè)重要的數(shù)學(xué)定義得以提出，或者一個(gè)重要的數(shù)學(xué)定理得以證明，往往并不是故事的結(jié)束。不論一項(xiàng)數(shù)學(xué)工作看似多么清楚明顯，總存在更好地理解它的可能，而最常見的達(dá)成這個(gè)目標(biāo)的方式，就是將其重述為一個(gè)更一般的概念的特例。
Why did it help to generalize the problem in this way? One might think that it would be harder to prove a result if one assumed less. However, that is often not true. The less you assume, the fewer options you have when trying to use your assumptions, and that can speed up the search for a proof. Had we not generalized the problem above, we would have had too many options.
為什么用這種方式把問題推廣會(huì)有利于問題的解決？人們可能以為，假設(shè)更少時(shí)，證明一個(gè)結(jié)果更難 ，然而實(shí)際情況往往并非如此。假設(shè)得更少，嘗試使用這些假設(shè)時(shí)需要作出的選擇就更少，這可以加速對(duì)證明（路徑）的搜尋。如果我們沒有把問題一般化，我們會(huì)擁有過多的選擇（需要去嘗試）。
There is no clear distinction between weakening hypotheses and strengthening conclusions, since if we are asked to prove a statement of the form P ? Q, we can always reformulate it as ?Q ??P. Then, if we weaken P we are weakening the hypotheses of P ? Q but strengthening the conclusion of ?Q??P.
弱化結(jié)論與強(qiáng)化結(jié)論，并不是那么涇渭分明，因?yàn)槿绻覀儽灰笞C明形如P ? Q的命題，我們總可以將其重新表述為 ?Q ??P。于是，如果我們?nèi)趸薖，我們是在弱化 P ? Q 的假設(shè)，卻同時(shí)也是在強(qiáng)化?Q??P的結(jié)論。
The abstract concept of a group helps one to see Fermat’s little theorem in a completely new way: it can be viewed as a special case of a more general result, but a result that cannot even be stated until one has developed some new, abstract concepts. This process of abstraction has many benefits. Most obviously, it provides us with a more general theorem, one that has many other interesting particular cases. Once we see this, then we can prove the general result once and for all rather than having to prove each case separately. A related benefit is that it enables us to see connections between results that may originally have seemed quite different. And finding surprising connections between different areas of mathematics almost always leads to significant advances in the subject.
群的抽象概念可以幫助人們以全新的視角看待費(fèi)馬小定理：它可以被看作更為一般的結(jié)果的一個(gè)特例，但這個(gè)更一般的結(jié)果在某種新的抽象概念被發(fā)展出來之前，甚至無法被陳述。這種抽象的過程具備許多益處。最明顯的是，它為我們提供了更為一般的定理，該定理具備許多其他有趣的特定應(yīng)用。一旦我們看到了這一點(diǎn)，我們就可以一勞永逸地證明一般性的結(jié)果，而不必分別證明每種情況。
The word “abstract” is often used to refer to a part of mathematics where it is more common to use characteristic properties of an object than it is to argue directly from a definition of the object itself (though, as the example of √ 2 shows, this distinction can be somewhat hazy). The ultimate in abstraction is to explore the consequences of a system of axioms, such as those for a group or a vector space. However, sometimes, in order to reason about such algebraic structures, it is very helpful to classify them, and the result of classification is to make them more concrete again.
“抽象”這個(gè)詞常常被用來指代更常使用對(duì)象特征性質(zhì)而非直接從對(duì)象自身的定義出發(fā)論證的那一部分?jǐn)?shù)學(xué)（雖然 √ 2 的例子說明，這種區(qū)別可能會(huì)有些模糊）。抽象的最終目的是探索一系列公理的后果，例如一個(gè)群或者一個(gè)向量空間的公理。然而，有時(shí)候，為了就這樣的代數(shù)結(jié)構(gòu)進(jìn)行推理，對(duì)它們進(jìn)行分類會(huì)很有幫助，而分類的結(jié)果卻又把它們變得更為具體了。
one reformulates part of geometry in terms of a certain algebraic structure and then generalizes the algebra.
人們用某個(gè)特定的代數(shù)結(jié)構(gòu)來重新表示幾何的一部分，然后推廣該代數(shù)。
A process that has generated many of the most important problems and results in mathematics, particularly over the last century or so: the process of generalization from one variable to several variables.
從一元推廣到多元的過程，在數(shù)學(xué)中產(chǎn)生了相當(dāng)多最重要的問題和結(jié)果，尤其是在過去的一個(gè)世紀(jì)里。
This geometrical interpretation is important, and goes a long way toward explaining why extensions of definitions and theorems from one variable to several variables are so interesting. If we generalize a piece of algebra from one variable to several variables, we can also think of what we are doing as generalizing from a one-dimensional setting to a higher-dimensional setting. This idea leads to many links between algebra and geometry, allowing techniques from one area to be used to great effect in the other.
這種幾何解釋是重要的，而且相當(dāng)有助于解釋為何定義和定理從一元到多元的擴(kuò)展如此有趣。如果我們將一項(xiàng)代數(shù)工作從一元推廣到多元，我們也可以將這個(gè)過程視為從1維的設(shè)定推廣到高維的設(shè)定。這種思想引出了代數(shù)與幾何之間的許多關(guān)聯(lián)，使得一個(gè)領(lǐng)域的數(shù)學(xué)技藝能夠運(yùn)用到另一個(gè)領(lǐng)域并發(fā)揮極大作用。
