【Mathematics / 數(shù)學】

What is it like to have an understanding of very advanced mathematics?

懂得很高深的數(shù)學，是什么感覺？

Anon User

You can answer many seemingly difficult questions quickly.

But you are not very impressed by what can look like magic, because you know the trick. The trick is that your brain can quickly decide if question is answerable by one of a few powerful general purpose "machines" (e.g., continuity arguments, the correspondences between geometric and algebraic objects, linear algebra, ways to reduce the infinite to the finite through various forms of compactness) combined with specific facts you have learned about your area. The number of fundamental ideas and techniques that people use to solve problems is, perhaps surprisingly, pretty small — see http://www./tricki/map for a partial list, maintained by Timothy Gowers.

      你可以很快回答很多表面上看起來很難的問題。但你不會對看上去很神奇的東西印象深刻，因為你知道其中的奧妙。奧妙就在于你的大腦可以迅速判斷出這個問題是否可以由幾個強大的、通用的目標“模型”（比如說，連續(xù)方程、幾何和代數(shù)的一致性、線性代數(shù)、通過某些定律將無限維問題轉(zhuǎn)化為有限）結(jié)合其他你在特定的領域了解到的事實來解答。人們用來解決問題的基本方法和技巧，似乎令人驚訝地有限——看看http://www./tricki/map，所列的就是其中的一部分，該網(wǎng)站是Timothy Gowers維護的。

You are often confident that something is true long before you have an airtight proof for it (this happens especially often in geometry). The main reason is that you have a large catalogue of connections between concepts, and you can quickly intuit that if X were to be false, that would create tensions with other things you know to be true, so you are inclined to believe X is probably true to maintain the harmony of the conceptual space. It's not so much that you can imagine the situation perfectly, but you can quickly imagine many other things that are logically connected to it.

      你經(jīng)常會在得到嚴密證明之前相信某個結(jié)論是正確的（尤其是在幾何中）。主要原因在于，你已經(jīng)建立了一大堆互相關(guān)聯(lián)的概念，你可以憑直覺判斷如果X是錯的，就會與其他的你知道是對的的東西產(chǎn)生矛盾，所以你會傾向于認為X是對的來構(gòu)成概念空間的和諧?？赡芎芏鄷r候你不能遇到完全符合的情況，但你可以快速想到其他邏輯上相關(guān)的東西。

You are comfortable with feeling like you have no deep understanding of the problem you are studying. Indeed, when you do have a deep understanding, you have solved the problem, and it is time to do something else. This makes the total time you spend in life reveling in your mastery of something quite brief. One of the main skills of research scientists of any type is knowing how to work comfortably and productively in a state of confusion. More on this in the next few bullets.

      你完全會感覺輕松，即使你覺得對于你所學的問題沒有深層次的理解。事實上，當你有深層次的理解時，就意味著你已經(jīng)解決了這個問題，該做點別的事情了。這會使你一生中浪費在對自己取得的成就沾沾自喜的時間大大減少。對于任何研究人員來說，一個重要的技能就是知道如何在迷惑狀態(tài)下保持輕松和高效地工作。在后面的說明中仍然會多次涉及這一點。

Your intuitive thinking about a problem is productive and usefully structured, wasting little time on being aimlessly puzzled. For example, when answering a question about a high-dimensional space (e.g., whether a certain kind of rotation of a five-dimensional object has a "fixed point" that does not move during the rotation), you do not spend much time straining to visualize those things that do not have obvious analogues in two and three dimensions. (Violating this principle is a huge source of frustration for beginning maths students who don't know that they shouldn't be straining to visualize things for which they don't seem to have the visualizing machinery.) Instead . . .

       你對于某個問題的直覺往往是創(chuàng)造性并且經(jīng)過很好的組織，所以你幾乎不會浪費時間在無目標的迷惑中。舉個例子，當被問及一個關(guān)于高維空間的問題（比如，一個五個維度的物體作確定的旋轉(zhuǎn)時，空間中是否存在一個“不動點”，它的位置不隨物體的旋轉(zhuǎn)而變化。）時，你不會花費很多時間竭力在常見的二維和三維空間想象這樣的現(xiàn)象，因為這種運動不會有顯然的模擬在這兩個維度中。（對于很多初學數(shù)學的學生來說，他們對數(shù)學的沮喪很大程度來自于違背了這條準則，他們不知道其實他們不應該去想象一個在低維度中并沒有適當模型的高維問題模型。）相反，

When trying to understand a new thing, you automatically focus on very simple examples that are easy to think about, and then you leverage intuition about the examples into more impressive insights. For example, you might imagine two- and three-dimensional rotations that are analogous to the one you really care about, and think about whether they clearly do or don't have the desired property. Then you think about what was important to the examples and try to distill those ideas into symbols. Often, you see that the key idea in the symbolic manipulations doesn't depend on anything about two or three dimensions, and you know how to answer your hard question.

      當你試著去認識一個新事物的時候，你會自然的關(guān)注一些你會輕易想起來簡單模型，在此基礎上你借助自己的直覺將之改造成更為明確的概念。比如，你可能會想象與你關(guān)注問題類似的在二或三維空間的旋轉(zhuǎn)運動，進而考察它是否擁有你所希望的特性。接著你會關(guān)注例子中關(guān)鍵本質(zhì)并嘗試將其轉(zhuǎn)化為符號語言。經(jīng)常性的，你在符號化演算中所依賴的關(guān)系并不會局限于二或三維空間中，并且你知道怎樣解決你碰到的難題。

As you get more mathematically advanced, the examples you consider easy are actually complex insights built up from many easier examples; the "simple case" you think about now took you two years to become comfortable with. But at any given stage, you do not strain to obtain a magical illumination about something intractable; you work to reduce it to the things that feel friendly.

       當你接觸到越來越高級的數(shù)學時，你所考慮的模型其實都是很多簡單模型組合來的，你現(xiàn)在認為的“簡單情形”當初可是花了你兩年時間才拿下的！但是對于你的任何階段，你都不會試圖依仗“神的光芒”來解決難題，你會自己動手將之簡化為你熟悉的問題。

To me, the biggest misconception that non-mathematicians have about how mathematicians think is that there is some mysterious mental faculty that is used to crack a problem all at once. In reality, one can ever think only a few moves ahead, trying out possible attacks from one's arsenal on simple examples relating to the problem, trying to establish partial results, or looking to make analogies with other ideas one understands. This is the same way that one solves problems in one's first real maths courses in university and in competitions. What happens as you get more advanced is simply that the arsenal grows larger, the thinking gets somewhat faster due to practice, and you have more examples to try, perhaps making better guesses about what is likely to yield progress. Sometimes, during this process, a sudden insight comes, but it would not be possible without the painstaking groundwork (http://terrytao./career-advice/does-one-have-to-be-agenius- to-do-maths/).

        我印象中，對數(shù)學不是很擅長的人對于數(shù)學家們最大的誤解是數(shù)學家們運用了什么神奇的技能使得他們可以一下子解決難題。實際上，一個人只能提前想到有限的幾步，窮盡自己對于與此相關(guān)問題的簡單模型的經(jīng)驗，試著得到部分結(jié)論，或者嘗試去類比自己理解的其他結(jié)論。這與你在大學的數(shù)學課程中或者比賽中解決問題的思路是一樣的。當你學到更高級數(shù)學時只是你積累的數(shù)學模型更多了，你的思維因為鍛煉而更加迅捷了，與此同時你有有更多例子去參考，因此你會想出利于解決問題的更好猜想。有時，在這個過程中，一個靈感降臨，但若沒有之前的糾結(jié)階段，這你是想都別想的。

Indeed, most of the bullet points here summarize feelings familiar to many serious students of mathematics who are in the middle of their undergraduate careers; as you learn more mathematics, these experiences apply to "bigger" things but have the same fundamental flavor.

          事實上，這兒總結(jié)的感受與很多對數(shù)學的認真對待的尚處在本科階段的學生的很類似，當你接觸到更多數(shù)學的時候，這些感受和那些經(jīng)歷過同樣基礎階段后的后期過程中產(chǎn)生的非常像。

You go up in abstraction, "higher and higher." The main object of study yesterday becomes just an example or a tiny part of what you are considering today. For example, in calculus classes you think about functions or curves. In functional analysis or algebraic geometry, you think of spaces whose points are functions or curves — that is, you "zoom out" so that every function is just a point in a space, surrounded by many other "nearby" functions. Using this kind of zooming out technique, you can say very complex things in short sentences — things that, if unpacked and said at the zoomed-in level, would take up pages. Abstracting and compressing in this way allows you to consider extremely complicated issues while using your limited memory and processing power.

       你思考問題越來越抽象，且抽象程度不斷加強。你昨天研究的對象變成了今天你能想起的一個模型或者構(gòu)成它的一部分，舉例來說在微積分中你研究函數(shù)和曲線，在泛函分析或代數(shù)幾何中你研究由函數(shù)或曲線作為點構(gòu)成的空間——也就是說，你通過抽象將所有函數(shù)都變成了空間中的一個由其他函數(shù)經(jīng)過同樣抽象簡化得到的點所包圍的點。運用這種抽象的技巧，你可以將非常復雜的問題以簡單的形式理解——復雜到，如果具體描述，可能需要幾頁紙才能講明白。這樣的抽象簡化會使你得以通過你有限的腦容量和演算能力解決巨復雜的問題。

The particularly "abstract" or "technical" parts of many other subjects seem quite accessible because they boil down to maths you already know. You generally feel confident about your ability to learn most quantitative ideas and techniques. A theoretical physicist friend likes to say, only partly in jest, that there should be books titled "______ for Mathematicians," where ______ is something generally believed to be difficult (quantum chemistry, general relativity, securities pricing, formal epistemology). Those books would be short and pithy, because many key concepts in those subjects are ones that mathematicians are well equipped to understand. Often, those parts can be explained more briefly and elegantly than they usually are if the explanation can assume a knowledge of maths and a facility with abstraction.

         很多其他領域特殊的抽象或理論的部分都變得可行因為它們最終都歸根結(jié)底與你已知的數(shù)學知識。你通常對于你學大部分理論和技巧的能力會表現(xiàn)的很自信。我的一個理論物理學家朋友喜歡半開玩笑說，任何一本內(nèi)容晦澀難懂（比如定量化學、廣義相對論、證券定價、經(jīng)典認識論）的書都應該在標題中注明“僅限數(shù)學家讀”的字樣。這些書往往都很簡練，因為這些領域的許多關(guān)鍵概念數(shù)學家們都已經(jīng)深入理解并掌握了。許多時候，那些部分都可以表述的更加簡潔美妙如果那些描述建立在數(shù)學知識和抽象概念上。

Learning the domain-specific elements of a different field can still be hard — for instance, physical intuition and economic intuition seem to rely on tricks of the brain that are not learned through mathematical training alone. But the quantitative and logical techniques you sharpen as a mathematician allow you to take many shortcuts that make learning other fields easier, as long as you are willing to be humble and modify those mathematical habits that are not useful in the new field.

        學習另外一個特定領域的特定原理依然會存在難度，比如說，物理和經(jīng)濟學的直覺似乎不止依賴于通過數(shù)學訓練所獲得的智力上的技巧，但是只要你愿意保持謙遜并且不斷修正那些在其他領域不是很實用的你所積累起來的數(shù)學習慣，作為數(shù)學家所練就的這些技巧會讓你在學習其他領域的知識時總能找到捷徑。

You move easily between multiple seemingly very different ways of representing a problem. For example, most problems and concepts have more algebraic representations (closer in spirit to an algorithm) and more geometric ones (closer in spirit to a picture). You go back and forth between them naturally, using whichever one is more helpful at the moment.

       你可以游刃有余地穿梭于表現(xiàn)形式似乎非常不同的關(guān)于問題的描述形式間。比如，很多問題和概念似乎更有代數(shù)意義（更接近數(shù)的本質(zhì)），而另外的更有幾何意義（更接近形的本質(zhì)）。你在他們之間自由轉(zhuǎn)換，在適當?shù)臅r候運用更有幫助的形式。

Indeed, some of the most powerful ideas in mathematics (e.g., duality, Galois theory, algebraic geometry), provide "dictionaries" for moving between "worlds" in ways that, exante, are very surprising. For example, Galois theory allows us to use our understanding of symmetries of shapes (e.g., rigid motions of an octagon) to understand why you can solve any fourth-degree polynomial equation in closed form, but not any fifth-degree polynomial equation. Once you know these threads between different parts of the universe, you can use them like wormholes to extricate yourself from a place where you would otherwise be stuck. The next two bullets expand on this.

        事實上，數(shù)學中的一些很厲害的概念（例如，雙重性，伽羅瓦理論，代數(shù)幾何等）對于外部世界的運動提供了驚人的預測。比如，伽羅瓦理論使得我們可以利用我們對于形狀對稱性的認識（比如嚴格的八邊形的運動）去認識為什么你能解決任何封閉形式的四次多項式，卻對于五次多項式無能為力。一旦你了解了萬物之間的聯(lián)系，你就能輕而易舉的在容易卡殼的地方解脫。下面的兩條將就這點展開來講。）

Spoiled by the power of your best tools, you tend to shy away from messy calculations or long, case-by-case arguments unless they are absolutely unavoidable. Mathematicians develop a powerful attachment to elegance and depth, which are in tension with, if not directly opposed to, mechanical calculation. Mathematicians will often spend days figuring out why a result follows easily from some very deep and general pattern that is already well-understood, rather than from a string of calculations. Indeed, you tend to choose problems motivated by how likely it is that there will be some "clean" insight in them, as opposed to a detailed but ultimately unenlightening proof by exhaustively enumerating a bunch of possibilities. (Nevertheless, detailed calculation of an example is often a crucial part of beginning to see what is really going on in a problem; and, depending on the field, some calculation often plays an essential role even in the best proof of a result.)

       由于對自己熟悉工具的過分倚賴，除非很有必要，否則你總會忽略掉那些冗雜的計算和一步一步的論證過程。數(shù)學家們鍛煉出了一種既有深度又不失優(yōu)雅的思維方式，這種思維方式雖然不是絕對的，總是脫離于機械演算。數(shù)學家們經(jīng)常會花費數(shù)天去弄明白為什么一些非常深奧但卻為人們所理解的形式會導出其他的結(jié)論，而不是去糾結(jié)于一連串的演算。事實上，你會傾向于選擇那些由純粹的洞察力激發(fā)出的問題，而不是靠列舉出各種可能性就能解決的毫無啟發(fā)性可言的問題。（不過，詳盡的計算卻是在深入了解問題的初級階段必需的一部分，這跟領域有關(guān)，有些時候計算就在最終的完美的證明中扮演著重要的角色。）

In A Mathematician's Apology, (http://www.math./~mss/misc/A Mathematician's Apology.pdf，the most poetic book I know on what it is "like" to be a mathematician), G.H. Hardy wrote: 

"In both [these example] theorems (and in the theorems, of course, I include the proofs) there is a very high degree ofunexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions. There are no complications of detail — one line of attack is enough in each case; and this is true too of the proofs of many much more difficult theorems, the full appreciation of which demands quite a high degree of technical proficiency. We do not want many 'variations’ in the proof of a mathematical theorem: 'enumeration of cases’, indeed, is one of the duller forms of mathematical argument. A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way."

" . . . [A solution to a difficult chess problem] is quite genuine mathematics, and has its merits; but it is just that 'proof by enumeration of cases’ (and of cases which do not, at bottom, differ at all profoundly) which a real mathematician tends to despise."

       在 A Mathematician’s Apology 這本書（此書是我所知道的描述數(shù)學家是怎樣的書中最富有詩意的一本）中，G.H. Hardy 寫道：

      “在這些定理（包括證明過程）中，存在著很大的不可預知性，同時伴有必然性和簡潔性。相比那些很難理解的結(jié)論，證明過程中所用的論據(jù)是如此的簡單，令人感覺驚奇,但同樣推導出了結(jié)論。（推導過程中）并沒有冗余的細節(jié)——對每種情況一行描述就已足夠，對于很多更為復雜定理的證明也是這樣，完全領會這種境界的前提是你需要對技術(shù)很精通。我們不希望在數(shù)學證明中看到很多種可能性的證明，事實上，對各種情況的羅列，算是數(shù)學論證方面最無趣的方式之一。一個數(shù)學證明應該像夜空中輪廓清楚的星座，而非銀河系中零散分布的星團。”

       “（棋類問題）屬于數(shù)學中特殊的問題，對這類問題的解自有其價值，但這正是那種“通過羅列出各種可能進行證明的問題”（或羅列出至少不算偏離很大的情況），而這則遭受主流數(shù)學家們鄙視。”

You develop a strong aesthetic preference for powerful and general ideas that connect hundreds of difficult questions, as opposed to resolutions of particular puzzles. Mathematicians don't really care about "the answer" to any particular question; even the most sought-after theorems, like Fermat's Last Theorem (http://en./wiki/Fermat's_Last_Theorem ) are only tantalizing because their difficulty tells us that we have to develop very good tools and understand very new things to have a shot at proving them. It is what we get in the process, and not the answer per se, that is the valuable thing. The accomplishment a mathematician seeks is finding a new dictionary or wormhole between different parts of the conceptual universe. As a result, many mathematicians do not focus on deriving the practical or computational implications of their studies (which can be a drawback of the hyper-abstract approach!); instead, they simply want to find the most powerful and general connections. Timothy Gowers has some interesting comments on this issue, and disagreements within the mathematical community about it. (https://www.dpmms./~wtg10/2cultures.pdf.)

     你培養(yǎng)出這樣的偏好：相對于針對特定問題的解法，你更喜歡能涵蓋更多問題的更通用性的概念。數(shù)學家們并不真正在意針對特定問題的解答，甚至是那些懸而未解的定理，比如費馬大定理僅僅只是一個逗引，它的存在只是在提醒我們需要發(fā)明出更先進的工具并且理解更新的東西以去證明它。最寶貴的是我們在推進它的過程中獲得的知識，而非“它被證明了”這個結(jié)果。數(shù)學家追求的成就是在不同領域的概念間發(fā)現(xiàn)聯(lián)系。因此，許多數(shù)學家并不關(guān)注他們的研究成果中實用性或者計算結(jié)果的寓意（而這卻往往成為超抽象方法的缺點）；相反，他們想簡單的找到更通用、強大的聯(lián)系。Timothy Gowers關(guān)于這點有一些很有意思的評論，以及數(shù)學社區(qū)中的一些對于該種觀點的反對意見。

Understanding something abstract or proving that something is true becomes a task a lot like building something. You think: "First I will lay this foundation, then I will build this framework using these familiar pieces, but leave the walls to fill in later, then I will test the beams . . . " All these steps have mathematical analogues, and structuring things in a modular way allows you to spend several days thinking about something you do not understand without feeling lost or frustrated. (I should say, "without feeling unbearably lost and frustrated some amount of these feelings is inevitable, but the key is to reduce them to a tolerable degree.)

     理解一些抽象的東西或者證明某些東西是正確的越來越變成一件類似構(gòu)筑某種東西的任務。你會想：“首先我設定這個基礎，然后我將用這些熟悉的模塊構(gòu)建這樣一個框架，只留下主體部分等待后面填補，接著我要檢驗證明過程……”所有這些步驟都具有數(shù)學上的類似性，并且具有一定模式的結(jié)構(gòu)化的步驟會使得你可以在不感到迷茫和沮喪的前提下花費好幾天時間思考一些你不明白的一些東西（我不得不說，對于所謂“不感到迷茫和沮喪”，有時候這種感覺是不可避免的，關(guān)鍵是把其控制在一個可忍受的程度）。

Andrew Wiles, who proved Fermat's Last Theorem, used an "exploring" metaphor:

 "Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it's completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it's all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they're momentary, sometimes over a period of a day or two, they are the culmination of — and couldn't exist without — the many months of stumbling around in the dark that proceed them." （http://www./wgbh/nova/physics/andrew-wiles-fermat.html )

      安德魯 威爾士，證明了費馬大定理的人，使用過一個“探險者”隱喻：“也許我可以將我研究數(shù)學的經(jīng)歷完美的詮釋為一段穿越漆黑的未經(jīng)探索的宅子的經(jīng)歷。你進入宅子的第一間屋子，它一片漆黑，你被四周的家具羈絆，但最終你了解到了各間家具的位置。最后，六個多月后，你找到了燈的開關(guān)，你打開燈，突然間四周一片光明，你能清楚的看到你在何處。接著，你進入下一間屋子并且又度過六個月的黑暗的日子。所以，每一步進展，甚至有的時候它是瞬息萬變的，有時候是一兩天時間，它們的形成離不開那些你在黑暗中磕碰的日子，是由那些日子所促成的高潮。”

In listening to a seminar or while reading a paper, you don't get stuck as much as you used to in youth because you are good at modularizing a conceptual space, taking certain calculations or arguments you don't understand as "black boxes," and considering their implications anyway. You can sometimes make statements you know are true and have good intuition for, without understanding all the details. You can often detect where the delicate or interesting part of something is based on only a very high-level explanation. (I first saw these phenomena highlighted by Ravi Vakil, who offers insightful advice on being a mathematics student: (http://math./~vakil/potentialstudents.html.)     

      在參加研究小組或者讀論文的時候你不會像初期那樣經(jīng)常被卡住，因為你已經(jīng)非常擅長模塊化概念空間，將那些你不明白的推定或論據(jù)以“黑盒子”表示，再以任何方式去考慮它們的推論。你有時對你認為是對的東西有很好的直覺并可以做出陳述，而不必了解其所有細節(jié)。你經(jīng)?？梢园l(fā)現(xiàn)某些運用很高級概念的東西的巧妙或者有意思之處。（我首次看到這個現(xiàn)象被Ravi Vakil強調(diào)，他給數(shù)學系學生提出了很有見識的建議。）

You are good at generating your own definitions and your own questions in thinking about some new kind of abstraction. One of the things one learns fairly late in a typical mathematical education (often only at the stage of starting to do research) is how to make good, useful definitions. Something I've reliably heard from people who know parts of mathematics well but never went on to be professional mathematicians (i.e., write articles about new mathematics for a living) is that they were good at proving difficult propositions that were stated in a textbook exercise, but would be lost if presented with a mathematical structure and asked to find and prove some interesting facts about it. Concretely, the ability to do this amounts to being good at making definitions and, using the newly defined concepts, formulating precise results that other mathematicians find intriguing or enlightening.

     你會很善于在思考一些新的抽象概念的時候產(chǎn)生你自己對其的定義并且經(jīng)常提出自己獨到的問題。在正統(tǒng)的數(shù)學教育中一個人學的非?？亢蟮臇|西（經(jīng)常僅在開始做研究的時候）是怎樣作出好的、有用的限定。我非?？煽康膹哪切┲啦糠謹?shù)學但卻永遠不會成為數(shù)學家的人那里聽說到他們非常擅長證明那些在課本練習中陳述的問題，但當面對數(shù)學的結(jié)構(gòu)并被要求證明關(guān)于其的一些有趣的事實時往往會迷失。實際上，要做好這件事需要擅長作出假定，運用新定義的概念，簡要陳述其他數(shù)學家認為有趣或有啟發(fā)性的準確的結(jié)果的能力。

This kind of challenge is like being given a world and asked to find events in it that come together to form a good detective story. Unlike a more standard detective, you have to figure out what the "crime" (interesting question) might be; you'll have to generate your own "clues" by building up deductively from the basic axioms. To do these things, you use analogies with other detective stories (mathematical theories) that you know and a taste for what is surprising or deep. How this process works is perhaps the most difficult aspect of mathematical work to describe precisely but also the thing that I would guess is the strongest thing that mathematicians have in common.

     這種挑戰(zhàn)就如同給你一個世界，要求你去找出可以組合在一起形成一個絕妙偵探小說的各個事件。和標準的偵探不同的是，你需要弄明白所謂的“案件”（有意思的問題）可能是什么；你需要從基本公理中產(chǎn)生自己的“線索”。為了做這些事，你需要和你了解的其他偵探故事（數(shù)學理論）類比以及利用自己對于如何更驚奇或者更深入的品味。這個過程如何起作用也許就是如何更準確描述數(shù)學工作最難的一方面同時我想也是所有數(shù)學家所一定共有的東西。

You are easily annoyed by imprecision in talking about the quantitative or logical. This is mostly because you are trained to quickly think about counterexamples that make an imprecise claim seem obviously false.

     你會容易被討論數(shù)學量或邏輯方面的不嚴謹而惹怒。這很大程度上是因為你受過訓練，能很快的想出可以證明不嚴密的聲明是明顯錯位的案例。

On the other hand, you are very comfortable with intentional imprecision or "hand-waving" in areas you know, because you know how to fill in the details. Terence Tao is very eloquent about this here (http://terrytao./career-advice/therea??s-more-tomathematics-than-rigour-and-proofs/ )

     另一方面，你對于有意識地不嚴密的表述或者在你熟悉領域的領域“空洞”的話卻會感覺都很舒服，因為你知道該怎么去充實其中的細節(jié)。Terence Tao在這方面很有說法，參看…

"[After learning to think rigorously, comes the] 'post-rigorous' stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. (For instance, in this stage one would be able to quickly and accurately perform computations

in vector calculus by using analogies with scalar calculus, or informal and semi-rigorous use of infinitesimals, big-O notation, and so forth, and be able to convert all such calculations into a rigorous argument whenever required.) The emphasis is now on applications, intuition, and the 'big picture.' This stage usually occupies the late graduate years and beyond."

    “(在學習如何嚴密思考以后，接著是)'過嚴密’階段，在這個階段中你往往會對你所選領域的嚴密的基礎習以為常，并已經(jīng)作好準備重新審視并且完善你在此方面的以前覺得嚴密的直覺了，但是此時直覺是通過嚴密的理論所支撐的。（比如說，在這個階段你可以通過類比標量計算非?？焖俸蜏蚀_的進行矢量運算，或者非正式或半嚴密的使用無窮小，無窮大記號以及其他，你能將所有計算轉(zhuǎn)換為需要的嚴密的形式）現(xiàn)在強調(diào)的是運用，直覺以及所謂的'藍圖’。這個階段經(jīng)常持續(xù)到后面的研究生階段以及更遠?！?/font>

In particular, an idea that took hours to understand correctly the first time ("for any arbitrarily small epsilon I can find a small delta so that this statement is true") becomes such a basic element of your later thinking that you don't give it conscious thought.

     特別地，以前花數(shù)個小時才正確理解的點子首次（“對任意小的總可以找到一個小使得條件成立?！?/啊哈，很熟悉！有木有。。。）成為你以后可以不做過多思考就可使用一個基本元素。

Before wrapping up, it is worth mentioning that mathematicians are not immune to the limitations faced by most others. They are not typically intellectual superheroes. For instance, they often become resistant to new ideas and uncomfortable with ways of thinking (even about mathematics) that are not their own. They can be defensive about intellectual turf, dismissive of others, or petty in their disputes. Above, I have tried to summarize how the mathematical way of thinking feels and works at its best, without focusing on personality flaws of mathematicians or on the politics of various mathematical fields. These issues are worthy of their own long answers!

    在擱筆（//這里其實應該是Ctrl+S和Alt+F4）之前，提醒以下事實是非常必要的：數(shù)學家們并不對大多數(shù)其他人面對的限制免疫。他們并不是所謂的智力上的超級英雄。比如，他們也經(jīng)常排斥新的觀點，也會對不是他們自己的思考方式（即使是跟數(shù)學有關(guān)）感到不舒服。他們也會對智力競賽持抵觸情緒，拒絕其他人或者在爭論中顯得偏狹。以上，我試著總結(jié)出如何進行數(shù)學式的思考、感覺以及工作，無意關(guān)注數(shù)學家們的個人缺點或者不同數(shù)學領域的爭論。這些東西值得他們寫出自己的詳盡答案！

You are humble about your knowledge because you are aware of how weak maths is, and you are comfortable with the fact that you can say nothing intelligent about most problems. There are only very few mathematical questions to which we have reasonably insightful answers. There are even fewer questions, obviously, to which any given mathematician can give a good answer. After two or three years of a standard university curriculum, a good maths undergraduate can effortlessly write down hundreds of mathematical questions to which the very best mathematicians could not venture even a tentative answer. (The theoretical computer scientist Richard Lipton lists some examples of potentially "deep" ignorance here: http://rjlipton./2009/12/26/mathematicalembarrassments/.) This makes it more comfortable to be stumped by most problems; a sense that you know roughly what questions are tractable and which are currently far beyond our abilities is humbling, but also frees you from being very intimidated, because you do know you are familiar with the most powerful apparatus we have for dealing with these kinds of problems.

    你對自己的知識很謙遜因為你意識到了數(shù)學的無力，并且你對于在很多問題上你并無想法的事實處之泰然。我們只對非常有限的數(shù)學問題有合理的明確的答案。任意一個數(shù)學家隨便就能很好的解決的數(shù)學問題顯然就更少了。經(jīng)過兩到三年的標準大學課程，一個出色的數(shù)學研究生可以毫不費力的寫出數(shù)以百計的可以使即使最好的數(shù)學家也不敢冒險給出試探性答案的數(shù)學問題。（理論計算機學家Richard Lipton列舉出了一些潛在的很深的無知的例子如下。）這使得被很多問題困住的情況顯得習以為常；那種你粗略的知道哪些問題是易于處理的以及哪些是目前我們無能為力的的感覺是粗陋的，但這也會使你變得不那么自卑，因為你深知你對于解決這種問題的強有力的工具是熟悉的。