按：我是在同事的朋友圈發現《普林斯頓數學指引》的，看了介紹感覺會很有啟發，隨即就下了英文電子版，然後又向同事借來了中文版的第一卷。恰逢幾個整段時間，看到100多頁，然後又中斷了1個月，又用了幾個夜晚從頭再讀並同時在英文電子版上標註筆記，終於咀嚼完了提綱挈領的第一部分（此時也已經粗讀完了第二部分），所以在此做個筆記，記錄感想。

中文版的譯者雖然是我自《重溫微積分》就接觸過的齊民友教授，也能看得出翻譯的態度是認真的，但依然存在很多不流暢、不準確的翻譯，甚至有多處因理解不到位對原文做的「更正」，筆記中會部分指出。中文版對於閱讀的加速作用還是很明顯的，但常常需要對照英文版確認，這點依然還是很不理想。網上的英文版除了沒有目錄之外，也比中文版少了部分段落，但不多。我在網上沒能找到譯作的官方勘誤表，只有原編者在他的博客上發布了一些勘誤。

筆記的方式，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，不採用中文版的翻譯，不自行做直接翻譯，只說明要點。因為不可能大段大段地去引用，必然會有語境的丟失，會做一些補充說明，以「按：」開始。對中文版翻譯進行更正或調整的說明，以「註：」開始。偶爾也會插入自己的議論，以「評：」開始。

這篇筆記是系列筆記的第一篇，第一部分有4節，對應4則筆記。

下一則筆記是《普林斯頓數學指引》讀書筆記——I.2 數學的語言和語法

I.1 What Is Mathematics About? （數學是關於什麼的）

Algebra, Geometry, and Analysis（代數、幾何與分析）

按：為了回答「數學是什麼」這個棘手的問題，作者從一個粗略的近似分類開始，即先把數學粗分成代數、幾何和分析這三個領域，再在這個基礎上細分修正。

Algebra versus Geometry（代數與幾何的區別與聯繫）

The objects of geometry, and the processes that they undergo, have a much more visual character than the equations of algebra.

幾何比代數天生具備更多可視的特性，無論是幾何研究的對象，還是它們經歷的過程。

The methods used to solve geometrical problems very often involve a great deal of symbolic manipulation, although good powers of visualization may be needed to find and use these methods and pictures will typically underlie what is going on.

解決幾何問題的方法常常需要很多符號操作（這點和代數是相似的），但要找到這些方法，需要很強的可視化能力，而且問題的內在機理背後也蘊含著圖形。

註：中文版將「pictures will typically underlie what is going on」譯作「在它的下面，典型地有圖形在」，非常生硬。這裡嘗試把「whats going on」當作「how it works」意會成了「內在機理」，不一定準確，請大家體會原文。

As for algebra, is it 「mere」 symbolic manipulation? Not at all: very often one solves an algebraic problem by finding a way to visualize it.

代數也不僅僅是符號的操作，常常會需要可視化的手法來幫助找到解法。

評：符號操作和圖形化，是代數和幾何各自的突出特點，但它們的研究領域和方法你中有我、我中有你，沒有清晰的界限。

Mathematicians vary widely in their ability and willingness to follow an argument like that one. If you cannot quite visualize it well enough to see that it is definitely correct, then you may prefer an algebraic approach.

It is often possible to translate a piece of mathematics from algebra into geometry or vice versa. Nevertheless, there is a definite difference between algebraic and geometric methods of thinking——one more symbolic and one more pictorial——and this can have a profound influence on the subjects that mathematicians choose to pursue.

按：上文對一個代數和幾何都可以解決的問題給出了一個非常典型的幾何式的解法。

數學家們使用上述幾何化的論述方式的能力和意願是很不一樣的。如果你難以建立對上述思路直觀的、可視化的一種把握，並且看出來其顯然是對的，那你一定傾向於代數的思路。

一個代數問題往往可以變換為一個幾何問題來解決，反之亦然。但是，代數化和幾何化的思考方式是涇渭分明的，這對於數學家選擇什麼樣的數學主題去追尋有非常顯著的影響。

評：一下子擊中了我的軟肋。我在數學上能明顯感覺到的自己的瓶頸，就是不容易在腦海中建立一個立體的可視化的圖景，即使我已經對問題有了清晰的理性了解。我在初中的時候還不能明白並有意識地去鍛煉自己來突破這個瓶頸，於是就非常傾向於解析幾何的思路來把幾何問題轉化為代數問題，而不喜歡去憑藉（我缺乏的）圖像直覺來畫輔助線、感覺圖形中的各種距離和角度關係。這個問題到了高中的立體幾何變得更為突出，但我依然沒有察覺，迷戀上了微分幾何，沉浸在更多的符號里。與此同時，因為自己的大腦不能提供出色的可視化，所以特別喜歡用編程的方式（從而依然是符號化的領域）來進行可視化，從LOGO到Mathematica到Dot再到D3再到WebGL。

Algebra versus Analysis（代數與分析的區別和聯繫）

As a first approximation, one might say that a branch of mathematics belongs to analysis if it involves limiting processes, whereas it belongs to algebra if you can get to the answer after just a finite sequence of steps.

有人可能會認為分析就是涉及到極限過程的數學分支，而如果能夠通過有限步得到答案的問題則屬於代數。

However, here again the first approximation is so crude as to be misleading, and for a similar reason: if one looks more closely one finds that it is not so much branches of mathematics that should be classified into analysis or algebra, but mathematical techniques.

其實被區分為分析或代數的，不應該是數學問題的分支，而是數學技藝。

評：這點和在討論幾何與代數的區別時簡直如出一轍。可見數學問題本身是不應該被分類的，一個問題可以有來自多個數學領域的解法，或者解法會需要多個數學領域結合。所以能分類的，也只是數學技藝/技術。

註：mathematical techniques中文版譯作「數學技巧」，容易引起誤解，無論是技術、技藝、方法，都更準確一些。

There are two features......that are typical of analysis. First, although the statement we wished to prove was about a limiting process, and was therefore 「infinitary,」 the actual work that we needed to do to prove it was entirely finite. Second, the nature of that work was to find sufficient conditions for a certain fairly simple inequality to be true.

分析有兩個非常典型的特點。首先，雖然我們想要證明的命題涉及到極限的過程，因此是無限的，但為了證明它所要做的工作，卻完全是有限的。其次，這些證明工作的本質，是去找到相當簡單的特定不等式成立的充分條件。

The ... argument is somewhat long, but each step consists in proving a rather simple inequality——this is the sense in which the proof is typical of analysis.

上述的論證雖然比較長，但是每一步僅僅包含證明一個非常簡單的不等式，正是在這樣一個意義上，這個證明是典型的分析證明。

Succinctly, algebraists like equalities and analysts like inequalities.

代數學家喜歡等式，分析學家喜歡不等式。

The Main Branches of Mathematics（數學的主要分支）

Algebra（代數）

按：這小節的介紹比較一般，所以只有下面這段被摘錄，不代表全小節的中心思想就是下面這段。下同。

(There is) a contrast that appears in many branches of mathematics, namely the distinction between general, abstract statements and particular, concrete ones. One algebraist might be thinking about groups, say, in order to understand a particular rather complicated group of symmetries, while another might be interested in the general theory of groups on the grounds that they are a fundamental class of mathematical objects.

在數學的許多分支中，都有這樣一種對比，即一般的、抽象的命題和特殊的、具體的命題之間的區別。一個代數學家考察群，可能是為了理解一個特定的、複雜的對稱群，而另外一個代數學家，則可能是因為群是數學對象的一個基本類別，從而對其一般理論感興趣。

註：中文版將contrast譯作「對立」，不妥。此處譯為「對比」，感覺也未達意，請大家體會原文。

Number Theory（數論）

Most number theorists are not directly trying to solve equations in integers; instead they are trying to understand structures that were originally developed to study such equations but which then took on a life of their own and became objects of study in their own right.

絕大多數的數論學家，並不直接嘗試用整數解方程，而是去努力理解各種最初是為了解整數方程而發展起來的數學結構。

As a rough rule of thumb, the study of equations in integers leads to algebraic number theory and the study of prime numbers leads to analytic number theory.

粗糙地說，研究方程的整數解，引導到代數數論，而解析數論的根源是素數的研究。

Geometry（幾何）

Within the study of manifolds, one can attempt a further classification, according to when two manifolds are regarded as 「genuinely distinct.」

在研究流形時，可以依據「什麼條件下可以把兩個流形看成是真正不同」而做進一步的分類。

......relative distances are not important to topologists, since one can change them by suitable continuous stretches. A differential topologist asks for the deformations to be 「smooth」 (which means 「sufficiently differentiable」).

對於拓撲學家，相對距離是不重要的，因為距離可以被適當的連續拉伸來改變。一個微分拓撲學家，則還會要求變形是「光滑的」（即「充分可微」）。

At the other, more 「geometrical,」 end of the spectrum are mathematicians who are much more concerned by the precise nature of the distances between points on a manifold (a concept that would not make sense to a topologist) and in auxiliary structures that one can associate with a manifold. See Riemannian Metrics [I.3 §6.10] and Ricci Flow [III.80]

在數學研究領域裡的更為「幾何」的另一端，則有這樣的數學家，他們對流形上的點之間的距離的精確的本性，更為感興趣，更加關心可以與流形相關聯的輔助結構，參見黎曼度量和里奇流。

註：「auxiliary structures that one can associate with a manifold」中文版和此處的譯法都不太好，請體會原文。

Algebraic Geometry（代數幾何）

Algebraic geometers also study manifolds, but with the important difference that their manifolds are defined using polynomials.

代數幾何學家也研究流形，不過他們研究的流形，是由多項式來定義的。

Algebraic geometry is algebraic in the sense that it is 「all about polynomials」 but geometric in the sense that the set of solutions of a polynomial in several variables is a geometric object.

從「完全是關於多項式」的角度來看，代數幾何是代數的，但是，就多變數多項式的解的集合是一個幾何對象而言，它又是幾何的。

An important part of algebraic geometry is the study of singularities. Often the set of solutions to a system of polynomial equations is similar to a manifold, but has a few exceptional, singular points.

代數幾何的一個重要部分是對奇點的研究。一個多項式方程組的解的集合，通常類似於一個流形，但會有少數例外的奇點。

註：中文版將singularities誤譯為「奇異性」。

Analysis（分析）

Like algebra, analysis has some abstract structures that are central objects of study, such as Banach Spaces [III.64], Hilbert Spaces [III.37], C*-Algebras [IV.19 ∽3], and Von Neumann Algebras [IV.19 ∽2]. These are all infinite-dimensional vector spaces [I.3 ∽2.3], and the last two are 「algebras,」 which means that one can multiply their elements together as well as adding them and multiplying them by scalars. Because these structures are infinite dimensional, studying them involves limiting arguments, which is why they belong to analysis. However, the extra algebraic structure of C.-algebras and von Neumann algebras means that in those areas substantial use is made of algebraic tools as well. And as the word 「space」 suggests, geometry also has a very important role.

和代數一樣，分析也有其抽象的一面。例如，巴拿赫空間、希爾伯特空間、C*-代數、馮·諾依曼代數都處在研究的中心。這些都是無限維向量空間，後兩個還是代數，這意味著其中的元素，不但可以相加，可以與標量相乘，還可以彼此相乘。因為這些構造都是無限維的，研究它們的過程中要用到極限的論證，這就是為什麼它們可以被歸類到分析。然而C*-代數和馮·諾依曼代數里額外的代數結構也就意味著會大量地使用到代數工具。而「空間」一詞，又表示幾何也會起到重要的作用。

註：中文版對substantial use錯譯為「本質地應用」。

Dynamics [IV.15] is another significant branch of analysis. It is concerned with what happens when you take a simple process and do it over and over again.......The answer turns out to depend in a complicated way on the original number . The study of how it depends on is a question in dynamics.

動力學是分析的另外一個引人注目的分支，它研究的是，當我們將一個非常簡單的過程，反覆地進行下去，會發生什麼。……其結果，是這個（狀態的）序列以一種複雜的方式依賴於初始狀態z0。對這個序列如何依賴於初始狀態的研究，正是動力學中的一個問題。

Much of dynamics is concerned with the long-term behavior of solutions to these (partial differential equations).

動力學的相當大的一部分，就是關於偏微分方程的解的漸近性態的研究。

註：中文版在此處將原文的偏微分方程更正為常微分方程，理由不充分，存疑，此處不採納；中文版將the long-term behavior of solutions（直接意思是「解的長期行為」）翻譯為漸進性態，而漸進性態對應的英文術語是Asymptotic Behavior，不確定是否正確，待查證確認。

邏輯

The word 「logic」 is sometimes used as a shorthand for all branches of mathematics that are concerned with fundamental questions about mathematics itself, notably set theory [IV.1], category theory [III.8], model theory [IV.2], and logic in the narrower sense of 「rules of deduction.」

「邏輯」一詞常常用作對「關於數學自身的基本問題的所有數學分支」的一個簡稱，特別值得關注的包括集合理論、範疇理論、模型理論，而狹義的邏輯，指的則是「演繹的規則」。

Category theory is another subject that began as a study of the processes of mathematics and then became a mathematical subject in its own right. It differs from set theory in that its focus is less on mathematical objects themselves than on what is done to those objects——in particular, the maps that transform one to another.

範疇理論本來是來自對數學過程的研究，後來其自身也變成了一門數學學科。它與集合理論的不同在於，它較少關注數學對象本身，而是研究加諸這些對象之上的過程，尤其是從一個對象變換到另外一個對象的映射。

註：中文版將「another subject」誤譯作「另一個例子」。

A model for a collection of axioms is a mathematical structure for which those axioms, suitably interpreted, are true. For example, any concrete example of a group is a model for the axioms of group theory.

一組公理的模型，是（在適當詮釋後）使得公理成立的數學結構。例如，任何一個群的具體例子，就是群論公理的一個模型。

組合數學（Combinatorics）

A first definition is that combinatorics is about counting things.

第一種定義：組合數學是關於如何對事物進行計數的。

Combinatorics is sometimes called 「discrete mathematics」 because it is concerned with 「discrete」 as opposed to 「continuous」 structures.

第二種定義：組合數學，有時又被稱之為離散數學，因為它考慮的是離散的結構，而不是連續的結構。

A third definition is that combinatorics is concerned with mathematical structures that have 「few constraints.」

第三種定義：組合數學處理的是具有極少量約束的數學結構。

註：中文版將constraints譯為限制，而不使用數學文獻中常見的約束，不妥。

The first question counts as number theory, since it concerns a very specific sequence—the sequence of squares—and one would expect to use properties of this special set of numbers in order to determine the answer, which turns out to be yes......The second question concerns a far less structured sequence. All we know about an is its rough size—it is fairly close to —but we know nothing about its more detailed properties...The second problem belongs to combinatorics. The answer is not known. If the answer turns out to be yes, then it will show that, in a sense, the number theory in the first problem was an illusion and that all that really mattered was the rough rate of growth of the sequence of squares.

按：上文中舉了兩個例子來說明數論和組合數學的區別。兩個問題都是問是否存在一個可以用1000種不同方法寫成兩數之和的正整數。區別是第一個問題里的「兩數」，都是平方數，第二個問題里的「兩數」，則都來自一個正整數序列，裡面每一項的值都位於和之間。

第一個問題可以算作數論問題，因為它考察的是一個非常特定的序列——完全平方數序列，而且會希望用特定的數的集合的性質來回答。這個問題的答案是肯定的，的確存在這樣一個正整數。

第二個問題，則是關於一個結構化程度低得多的序列。我們只知道其粗略的大小（接近），但對其更精確的性質一無所知。……第二個問題屬於組合數學，答案如何尚不得而知。如果答案是肯定的，則它在一定意義上表明，數論對第一個問題的回答與解釋只是一個幻象，真正起作用的，其實是完全平方序列粗略的增長率。

註：中文版將「a far less structured sequence」譯為「構造要少得多的對象」，對理解並無幫助，且丟失了「序列」的信息量，不妥。

Theoretical Computer Science（理論計算機科學）

Theoretical computer science is concerned with efficiency of computation, meaning the amounts of various resources, such as time and computer memory, needed to perform given computational tasks. There are mathematical models of computation that allow one to study questions about computational efficiency in great generality without having to worry about precise details of how algorithms are implemented.

理論計算機科學關注的是計算的效率的問題，如完成一定的計算任務所需的各種資源量，比如時間和計算機內存。計算的數學模型，使得數學家可以以很大的通用性來研究計算效率的問題，而無需考慮演算法如何具體執行。

Probability（概率論）

It may happen that there is a 「critical probability」 p with the following property: if the probability of infection after contact of a certain kind is above p then an epidemic may very well result, whereas if it is below p then the disease will almost certainly die out. A dramatic difference in behavior like this is called a phase transition. (See probabilistic models of critical phenomena [IV.26] for further discussion.)

（對於疾病的傳播，）可能有一個具備如下屬性的「臨界概率」p存在：如果接觸後感染的概率高於p，就很可能發生傳染，而小於p時疾病幾乎一定會自行消失。這樣性態上的劇變被稱為相變，參見臨界現象的概率模型。

Mathematical Physics（數學物理）

There is still a big cultural difference between the two subjects: mathematicians are far more interested in finding rigorous proofs, whereas physicists, who use mathematics as a tool, are usually happy with a convincing argument for the truth of a mathematical statement, even if that argument is not actually a proof. The result is that physicists, operating under less stringent constraints, often discover fascinating mathematical phenomena long before mathematicians do.

數學和物理學，仍有著巨大的文化上的差異：數學家們，對於尋找嚴格證明的興趣要大得多，而物理學家們則是把數學作為一種工具，對於一個數學命題是否為真，只要有了令人信服的論證，哪怕這種論證還不真正就是一個證明，物理學家也就滿足了。結果是，物理學家們是在不太嚴苛的限制下工作的，所以他們發現迷人的數學現象，常常比數學家早不少。

The articles Vertex Operator Algebras [IV.13], Mirror Symmetry [IV.14], General Relativity and the Einstein Equations [IV.17], and Operator Algebras [IV.19] describe some fascinating examples of how mathematics and physics have enriched each other.

頂點運算元代數、鏡面對稱、廣義相對論和愛因斯坦場方程、運算元代數，這些條目描述了一些數學和物理豐富彼此的精彩例子。