【Mathematics / 數學(xué)】
What is it like to have an understanding of very advanced mathematics?
懂得很高深的數學(xué)，是什么感覺(jué)？
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.org/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(lái)很難的問(wèn)題。但你不會(huì )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(gè)問(wèn)題是否可以由幾個(gè)強大的、通用的目標“模型”（比如說(shuō)，連續方程、幾何和代數的一致性、線(xiàn)性代數、通過(guò)某些定律將無(wú)限維問(wèn)題轉化為有限）結合其他你在特定的領(lǐng)域了解到的事實(shí)來(lái)解答。人們用來(lái)解決問(wèn)題的基本方法和技巧，似乎令人驚訝地有限——看看http://www.tricki.org/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(huì )在得到嚴密證明之前相信某個(gè)結論是正確的（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(jué)判斷如果X是錯的，就會(huì )與其他的你知道是對的的東西產(chǎn)生矛盾，所以你會(huì )傾向于認為X是對的來(lái)構成概念空間的和諧?？赡芎芏鄷r(shí)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(huì )感覺(jué)輕松，即使你覺(jué)得對于你所學(xué)的問(wèn)題沒(méi)有深層次的理解。事實(shí)上，當你有深層次的理解時(shí)，就意味著(zhù)你已經(jīng)解決了這個(gè)問(wèn)題，該做點(diǎn)別的事情了。這會(huì )使你一生中浪費在對自己取得的成就沾沾自喜的時(shí)間大大減少。對于任何研究人員來(lái)說(shuō)，一個(gè)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(shuō)明中仍然會(huì )多次涉及這一點(diǎn)。
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(gè)問(wèn)題的直覺(jué)往往是創(chuàng )造性并且經(jīng)過(guò)很好的組織，所以你幾乎不會(huì )浪費時(shí)間在無(wú)目標的迷惑中。舉個(gè)例子，當被問(wèn)及一個(gè)關(guān)于高維空間的問(wèn)題（比如，一個(gè)五個(gè)維度的物體作確定的旋轉時(shí)，空間中是否存在一個(gè)“不動(dòng)點(diǎn)”，它的位置不隨物體的旋轉而變化。）時(shí)，你不會(huì )花費很多時(shí)間竭力在常見(jiàn)的二維和三維空間想象這樣的現象，因為這種運動(dòng)不會(huì )有顯然的模擬在這兩個(gè)維度中。（對于很多初學(xué)數學(xué)的學(xué)生來(lái)說(shuō)，他們對數學(xué)的沮喪很大程度來(lái)自于違背了這條準則，他們不知道其實(shí)他們不應該去想象一個(gè)在低維度中并沒(méi)有適當模型的高維問(wèn)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(zhù)去認識一個(gè)新事物的時(shí)候，你會(huì )自然的關(guān)注一些你會(huì )輕易想起來(lái)簡(jiǎn)單模型，在此基礎上你借助自己的直覺(jué)將之改造成更為明確的概念。比如，你可能會(huì )想象與你關(guān)注問(wèn)題類(lèi)似的在二或三維空間的旋轉運動(dòng)，進(jìn)而考察它是否擁有你所希望的特性。接著(zhù)你會(huì )關(guān)注例子中關(guān)鍵本質(zhì)并嘗試將其轉化為符號語(yǔ)言。經(jīng)常性的，你在符號化演算中所依賴(lài)的關(guān)系并不會(huì )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(lái)越高級的數學(xué)時(shí)，你所考慮的模型其實(shí)都是很多簡(jiǎn)單模型組合來(lái)的，你現在認為的“簡(jiǎn)單情形”當初可是花了你兩年時(shí)間才拿下的！但是對于你的任何階段，你都不會(huì )試圖依仗“神的光芒”來(lái)解決難題，你會(huì )自己動(dòng)手將之簡(jiǎn)化為你熟悉的問(wèn)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.wordpress.com/career-advice/does-one-have-to-be-agenius- to-do-maths/).
我印象中，對數學(xué)不是很擅長(cháng)的人對于數學(xué)家們最大的誤解是數學(xué)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(shí)際上，一個(gè)人只能提前想到有限的幾步，窮盡自己對于與此相關(guān)問(wèn)題的簡(jiǎn)單模型的經(jīng)驗，試著(zhù)得到部分結論，或者嘗試去類(lèi)比自己理解的其他結論。這與你在大學(xué)的數學(xué)課程中或者比賽中解決問(wèn)題的思路是一樣的。當你學(xué)到更高級數學(xué)時(shí)只是你積累的數學(xué)模型更多了，你的思維因為鍛煉而更加迅捷了，與此同時(shí)你有有更多例子去參考，因此你會(huì )想出利于解決問(wèn)題的更好猜想。有時(shí)，在這個(gè)過(guò)程中，一個(gè)靈感降臨，但若沒(méi)有之前的糾結階段，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(shí)上，這兒總結的感受與很多對數學(xué)的認真對待的尚處在本科階段的學(xué)生的很類(lèi)似，當你接觸到更多數學(xué)的時(shí)候，這些感受和那些經(jīng)歷過(guò)同樣基礎階段后的后期過(guò)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(wèn)題越來(lái)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(gè)模型或者構成它的一部分，舉例來(lái)說(shuō)在微積分中你研究函數和曲線(xiàn)，在泛函分析或代數幾何中你研究由函數或曲線(xiàn)作為點(diǎn)構成的空間——也就是說(shuō)，你通過(guò)抽象將所有函數都變成了空間中的一個(gè)由其他函數經(jīng)過(guò)同樣抽象簡(jiǎn)化得到的點(diǎn)所包圍的點(diǎn)。運用這種抽象的技巧，你可以將非常復雜的問(wèn)題以簡(jiǎn)單的形式理解——復雜到，如果具體描述，可能需要幾頁(yè)紙才能講明白。這樣的抽象簡(jiǎn)化會(huì )使你得以通過(guò)你有限的腦容量和演算能力解決巨復雜的問(wèn)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(lǐng)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底與你已知的數學(xué)知識。你通常對于你學(xué)大部分理論和技巧的能力會(huì )表現的很自信。我的一個(gè)理論物理學(xué)家朋友喜歡半開(kāi)玩笑說(shuō)，任何一本內容晦澀難懂（比如定量化學(xué)、廣義相對論、證券定價(jià)、經(jīng)典認識論）的書(shū)都應該在標題中注明“僅限數學(xué)家讀”的字樣。這些書(shū)往往都很簡(jiǎn)練，因為這些領(lǐng)域的許多關(guān)鍵概念數學(xué)家們都已經(jīng)深入理解并掌握了。許多時(shí)候，那些部分都可以表述的更加簡(jiǎn)潔美妙如果那些描述建立在數學(xué)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(xué)習另外一個(gè)特定領(lǐng)域的特定原理依然會(huì )存在難度，比如說(shuō)，物理和經(jīng)濟學(xué)的直覺(jué)似乎不止依賴(lài)于通過(guò)數學(xué)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(lǐng)域不是很實(shí)用的你所積累起來(lái)的數學(xué)習慣，作為數學(xué)家所練就的這些技巧會(huì )讓你在學(xué)習其他領(lǐng)域的知識時(shí)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形式似乎非常不同的關(guān)于問(wèn)題的描述形式間。比如，很多問(wèn)題和概念似乎更有代數意義（更接近數的本質(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換，在適當的時(shí)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(shí)上，數學(xué)中的一些很厲害的概念（例如，雙重性，伽羅瓦理論，代數幾何等）對于外部世界的運動(dòng)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(chēng)性的認識（比如嚴格的八邊形的運動(dòng)）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(wú)能為力。一旦你了解了萬(wàn)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(diǎn)展開(kāi)來(lái)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(guò)分倚賴(lài)，除非很有必要，否則你總會(huì )忽略掉那些冗雜的計算和一步一步的論證過(guò)程。數學(xué)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。數學(xué)家們經(jīng)常會(huì )花費數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(huì )導出其他的結論，而不是去糾結于一連串的演算。事實(shí)上，你會(huì )傾向于選擇那些由純粹的洞察力激發(fā)出的問(wèn)題，而不是靠列舉出各種可能性就能解決的毫無(wú)啟發(fā)性可言的問(wèn)題。（不過(guò)，詳盡的計算卻是在深入了解問(wèn)題的初級階段必需的一部分，這跟領(lǐng)域有關(guān)，有些時(shí)候計算就在最終的完美的證明中扮演著(zhù)重要的角色。）
In A Mathematician's Apology, (http://www.math.ualberta.ca/~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(shū)（此書(shū)是我所知道的描述數學(xué)家是怎樣的書(shū)中最富有詩(shī)意的一本）中，G.H. Hardy 寫(xiě)道：
“在這些定理（包括證明過(guò)程）中，存在著(zhù)很大的不可預知性，同時(shí)伴有必然性和簡(jiǎn)潔性。相比那些很難理解的結論，證明過(guò)程中所用的論據是如此的簡(jiǎn)單，令人感覺(jué)驚奇,但同樣推導出了結論。（推導過(guò)程中）并沒(méi)有冗余的細節——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(lǐng)會(huì )這種境界的前提是你需要對技術(shù)很精通。我們不希望在數學(xué)證明中看到很多種可能性的證明，事實(shí)上，對各種情況的羅列，算是數學(xué)論證方面最無(wú)趣的方式之一。一個(gè)數學(xué)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?！?/span>
“（棋類(lèi)問(wèn)題）屬于數學(xué)中特殊的問(wèn)題，對這類(lèi)問(wèn)題的解自有其價(jià)值，但這正是那種“通過(guò)羅列出各種可能進(jìn)行證明的問(wèn)題”（或羅列出至少不算偏離很大的情況），而這則遭受主流數學(xué)家們鄙視?！?/font>
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.wikipedia.org/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.cam.ac.uk/~wtg10/2cultures.pdf.)
你培養出這樣的偏好：相對于針對特定問(wèn)題的解法，你更喜歡能涵蓋更多問(wèn)題的更通用性的概念。數學(xué)家們并不真正在意針對特定問(wèn)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(gè)逗引，它的存在只是在提醒我們需要發(fā)明出更先進(jìn)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(jìn)它的過(guò)程中獲得的知識，而非“它被證明了”這個(gè)結果。數學(xué)家追求的成就是在不同領(lǐng)域的概念間發(fā)現聯(lián)系。因此，許多數學(xué)家并不關(guān)注他們的研究成果中實(shí)用性或者計算結果的寓意（而這卻往往成為超抽象方法的缺點(diǎn)）；相反，他們想簡(jiǎn)單的找到更通用、強大的聯(lián)系。Timothy Gowers關(guān)于這點(diǎn)有一些很有意思的評論，以及數學(xué)社區中的一些對于該種觀(guān)點(diǎn)的反對意見(jiàn)。
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(lái)越變成一件類(lèi)似構筑某種東西的任務(wù)。你會(huì )想：“首先我設定這個(gè)基礎，然后我將用這些熟悉的模塊構建這樣一個(gè)框架，只留下主體部分等待后面填補，接著(zhù)我要檢驗證明過(guò)程……”所有這些步驟都具有數學(xué)上的類(lèi)似性，并且具有一定模式的結構化的步驟會(huì )使得你可以在不感到迷茫和沮喪的前提下花費好幾天時(shí)間思考一些你不明白的一些東西（我不得不說(shuō)，對于所謂“不感到迷茫和沮喪”，有時(shí)候這種感覺(jué)是不可避免的，關(guān)鍵是把其控制在一個(gè)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.pbs.org/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(guò)一個(gè)“探險者”隱喻：“也許我可以將我研究數學(xué)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(jìn)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(gè)多月后，你找到了燈的開(kāi)關(guān)，你打開(kāi)燈，突然間四周一片光明，你能清楚的看到你在何處。接著(zhù)，你進(jìn)入下一間屋子并且又度過(guò)六個(gè)月的黑暗的日子。所以，每一步進(jìn)展，甚至有的時(shí)候它是瞬息萬(wàn)變的，有時(shí)候是一兩天時(shí)間，它們的形成離不開(kāi)那些你在黑暗中磕碰的日子，是由那些日子所促成的高潮?！?/font>
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.stanford.edu/~vakil/potentialstudents.html.)
在參加研究小組或者讀論文的時(shí)候你不會(huì )像初期那樣經(jīng)常被卡住，因為你已經(jīng)非常擅長(cháng)模塊化概念空間，將那些你不明白的推定或論據以“黑盒子”表示，再以任何方式去考慮它們的推論。你有時(shí)對你認為是對的東西有很好的直覺(jué)并可以做出陳述，而不必了解其所有細節。你經(jīng)?？梢园l(fā)現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(gè)現象被Ravi Vakil強調，他給數學(xué)系學(xué)生提出了很有見(jiàn)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(huì )很善于在思考一些新的抽象概念的時(shí)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(wèn)題。在正統的數學(xué)教育中一個(gè)人學(xué)的非?？亢蟮臇|西（經(jīng)常僅在開(kāi)始做研究的時(shí)候）是怎樣作出好的、有用的限定。我非?？煽康膹哪切┲啦糠謹祵W(xué)但卻永遠不會(huì )成為數學(xué)家的人那里聽(tīng)說(shuō)到他們非常擅長(cháng)證明那些在課本練習中陳述的問(wèn)題，但當面對數學(xué)的結構并被要求證明關(guān)于其的一些有趣的事實(shí)時(shí)往往會(huì )迷失。實(shí)際上，要做好這件事需要擅長(cháng)作出假定，運用新定義的概念，簡(jiǎn)要陳述其他數學(xué)家認為有趣或有啟發(fā)性的準確的結果的能力。
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.
這種挑戰就如同給你一個(gè)世界，要求你去找出可以組合在一起形成一個(gè)絕妙偵探小說(shuō)的各個(gè)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(wèn)題）可能是什么；你需要從基本公理中產(chǎn)生自己的“線(xiàn)索”。為了做這些事，你需要和你了解的其他偵探故事（數學(xué)理論）類(lèi)比以及利用自己對于如何更驚奇或者更深入的品味。這個(gè)過(guò)程如何起作用也許就是如何更準確描述數學(xué)工作最難的一方面同時(shí)我想也是所有數學(xué)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(huì )容易被討論數學(xué)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(guò)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.wordpress.com/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(lǐng)域的領(lǐng)域“空洞”的話(huà)卻會(huì )感覺(jué)都很舒服，因為你知道該怎么去充實(shí)其中的細節。Terence Tao在這方面很有說(shuō)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(xué)習如何嚴密思考以后，接著(zhù)是)'過(guò)嚴密’階段，在這個(gè)階段中你往往會(huì )對你所選領(lǐng)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(jué)得嚴密的直覺(jué)了，但是此時(shí)直覺(jué)是通過(guò)嚴密的理論所支撐的。（比如說(shuō)，在這個(gè)階段你可以通過(guò)類(lèi)比標量計算非?？焖俸蜏蚀_的進(jìn)行矢量運算，或者非正式或半嚴密的使用無(wú)窮小，無(wú)窮大記號以及其他，你能將所有計算轉換為需要的嚴密的形式）現在強調的是運用，直覺(jué)以及所謂的'藍圖’。這個(gè)階段經(jīng)常持續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，以前花數個(gè)小時(shí)才正確理解的點(diǎn)子首次（“對任意小的總可以找到一個(gè)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(guò)多思考就可使用一個(gè)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(shí)應該是Ctrl+S和Alt+F4）之前，提醒以下事實(shí)是非常必要的：數學(xué)家們并不對大多數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(guān)點(diǎn)，也會(huì )對不是他們自己的思考方式（即使是跟數學(xué)有關(guān)）感到不舒服。他們也會(huì )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(zhù)總結出如何進(jìn)行數學(xué)式的思考、感覺(jué)以及工作，無(wú)意關(guān)注數學(xué)家們的個(gè)人缺點(diǎn)或者不同數學(xué)領(lǐng)域的爭論。這些東西值得他們寫(xiě)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.wordpress.com/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學(xué)的無(wú)力，并且你對于在很多問(wèn)題上你并無(wú)想法的事實(shí)處之泰然。我們只對非常有限的數學(xué)問(wèn)題有合理的明確的答案。任意一個(gè)數學(xué)家隨便就能很好的解決的數學(xué)問(wèn)題顯然就更少了。經(jīng)過(guò)兩到三年的標準大學(xué)課程，一個(gè)出色的數學(xué)研究生可以毫不費力的寫(xiě)出數以百計的可以使即使最好的數學(xué)家也不敢冒險給出試探性答案的數學(xué)問(wèn)題。（理論計算機學(xué)家Richard Lipton列舉出了一些潛在的很深的無(wú)知的例子如下。）這使得被很多問(wèn)題困住的情況顯得習以為常；那種你粗略的知道哪些問(wèn)題是易于處理的以及哪些是目前我們無(wú)能為力的的感覺(jué)是粗陋的，但這也會(huì )使你變得不那么自卑，因為你深知你對于解決這種問(wèn)題的強有力的工具是熟悉的。
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