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懂得很高深的数学,是什么感觉?

2014-01-08  参谋指挥...

Mathematics / 数学

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

懂得很高深的数学,是什么感觉?

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来很难的问题。但你不会对看上去很神奇的东西印象深刻,因为你知道其中的奥妙。奥妙就在于你的大脑可以迅速判断出这个问题是否可以由几个强大的、通用的目标“模型”(比如说,连续方程、几何和代数的一致性、线性代数、通过某些定律将无限维问题转化为有限)结合其他你在特定的领域了解到的事实来解答。人们用来解决问题的基本方法和技巧,似乎令人惊讶地有限——看看http://www.tricki.org/tricki/map,所列的就是其中的一部分,该网站是Timothy Gowers维护的。

 

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.

      你经常会在得到严密证明之前相信某个结论是正确的(尤其是在几何中)。主要原因在于,你已经建立了一大堆互相关联的概念,你可以凭直觉判断如果X是错的,就会与其他的你知道是对的的东西产生矛盾,所以你会倾向于认为X是对的来构成概念空间的和谐。可能很多时候你不能遇到完全符合的情况,但你可以快速想到其他逻辑上相关的东西。

 

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.

      你完全会感觉轻松,即使你觉得对于你所学的问题没有深层次的理解。事实上,当你有深层次的理解时,就意味着你已经解决了这个问题,该做点别的事情了。这会使你一生中浪费在对自己取得的成就沾沾自喜的时间大大减少。对于任何研究人员来说,一个重要的技能就是知道如何在迷惑状态下保持轻松和高效地工作。在后面的说明中仍然会多次涉及这一点。

 

 

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 . . .

       你对于某个问题的直觉往往是创造性并且经过很好的组织,所以你几乎不会浪费时间在无目标的迷惑中。举个例子,当被问及一个关于高维空间的问题(比如,一个五个维度的物体作确定的旋转时,空间中是否存在一个“不动点”,它的位置不随物体的旋转而变化。)时,你不会花费很多时间竭力在常见的二维和三维空间想象这样的现象,因为这种运动不会有显然的模拟在这两个维度中。(对于很多初学数学的学生来说,他们对数学的沮丧很大程度来自于违背了这条准则,他们不知道其实他们不应该去想象一个在低维度中并没有适当模型的高维问题模型。)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.

      当你试着去认识一个新事物的时候,你会自然的关注一些你会轻易想起来简单模型,在此基础上你借助自己的直觉将之改造成更为明确的概念。比如,你可能会想象与你关注问题类似的在二或三维空间的旋转运动,进而考察它是否拥有你所希望的特性。接着你会关注例子中关键本质并尝试将其转化为符号语言。经常性的,你在符号化演算中所依赖的关系并不会局限于二或三维空间中,并且你知道怎样解决你碰到的难题。

 

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.

       当你接触到越来越高级的数学时,你所考虑的模型其实都是很多简单模型组合来的,你现在认为的“简单情形”当初可是花了你两年时间才拿下的!但是对于你的任何阶段,你都不会试图依仗“神的光芒”来解决难题,你会自己动手将之简化为你熟悉的问题。

 

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,对数学不是很擅长的人对于数学家们最大的误解是数学家们运用了什么神奇的技能使得他们可以一下子解决难题。实际上,一个人只能提前想到有限的几步,穷尽自己对于与此相关问题的简单模型的经验,试着得到部分结论,或者尝试去类比自己理解的其他结论。这与你在大学的数学课程中或者比赛中解决问题的思路是一样的。当你学到更高级数学时只是你积累的数学模型更多了,你的思维因为锻炼而更加迅捷了,与此同时你有有更多例子去参考,因此你会想出利于解决问题的更好猜想。有时,在这个过程中,一个灵感降临,但若没有之前的纠结阶段,这你是想都别想的。

 

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.

          事实上,这儿总结的感受与很多对数学的认真对待的尚处在本科阶段的学生的很类似,当你接触到更多数学的时候,这些感受和那些经历过同样基础阶段后的后期过程中产生的非常像。

 

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.

       你思考问题越来越抽象,且抽象程度不断加强。你昨天研究的对象变成了今天你能想起的一个模型或者构成它的一部分,举例来说在微积分中你研究函数和曲线,在泛函分析或代数几何中你研究由函数或曲线作为点构成的空间——也就是说,你通过抽象将所有函数都变成了空间中的一个由其他函数经过同样抽象简化得到的点所包围的点。运用这种抽象的技巧,你可以将非常复杂的问题以简单的形式理解——复杂到,如果具体描述,可能需要几页纸才能讲明白。这样的抽象简化会使你得以通过你有限的脑容量和演算能力解决巨复杂的问题。

 

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.

         很多其他领域特殊的抽象或理论的部分都变得可行因为它们最终都归根结底与你已知的数学知识。你通常对于你学大部分理论和技巧的能力会表现的很自信。我的一个理论物理学家朋友喜欢半开玩笑说,任何一本内容晦涩难懂(比如定量化学、广义相对论、证券定价、经典认识论)的书都应该在标题中注明“仅限数学家读”的字样。这些书往往都很简练,因为这些领域的许多关键概念数学家们都已经深入理解并掌握了。许多时候,那些部分都可以表述的更加简洁美妙如果那些描述建立在数学知识和抽象概念上。

 

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.

        学习另外一个特定领域的特定原理依然会存在难度,比如说,物理和经济学的直觉似乎不止依赖于通过数学训练所获得的智力上的技巧,但是只要你愿意保持谦逊并且不断修正那些在其他领域不是很实用的你所积累起来的数学习惯,作为数学家所练就的这些技巧会让你在学习其他领域的知识时总能找到捷径。

 

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现形式似乎非常不同的关于问题的描述形式间。比如,很多问题和概念似乎更有代数意义(更接近数的本质),而另外的更有几何意义(更接近形的本质)。你在他们之间自由转换,在适当的时候运用更有帮助的形式。

 

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.

        事实上,数学中的一些很厉害的概念(例如,双重性,伽罗瓦理论,代数几何等)对于外部世界的运动提供了惊人的预测。比如,伽罗瓦理论使得我们可以利用我们对于形状对称性的认识(比如严格的八边形的运动)去认识为什么你能解决任何封闭形式的四次多项式,却对于五次多项式无能为力。一旦你了解了万物之间的联系,你就能轻而易举的在容易卡壳的地方解脱。下面的两条将就这点展开来讲。)

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.)

       由于对自己熟悉工具的过分倚赖,除非很有必要,否则你总会忽略掉那些冗杂的计算和一步一步的论证过程。数学家们锻炼出了一种既有深度又不失优雅的思维方式,这种思维方式虽然不是绝对的,总是脱离于机械演算。数学家们经常会花费数天去弄明白为什么一些非常深奥但却为人们所理解的形式会导出其他的结论,而不是去纠结于一连串的演算。事实上,你会倾向于选择那些由纯粹的洞察力激发出的问题,而不是靠列举出各种可能性就能解决的毫无启发性可言的问题。(不过,详尽的计算却是在深入了解问题的初级阶段必需的一部分,这跟领域有关,有些时候计算就在最终的完美的证明中扮演着重要的角色。)

 

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 这本书(此书是我所知道的描述数学家是怎样的书中最富有诗意的一本)中,G.H. Hardy 写道:

      “在这些定理(包括证明过程)中,存在着很大的不可预知性,同时伴有必然性和简洁性。相比那些很难理解的结论,证明过程中所用的论据是如此的简单,令人感觉惊奇,但同样推导出了结论。(推导过程中)并没有冗余的细节——对每种情况一行描述就已足够,对于很多更为复杂定理的证明也是这样,完全领会这种境界的前提是你需要对技术很精通。我们不希望在数学证明中看到很多种可能性的证明,事实上,对各种情况的罗列,算是数学论证方面最无趣的方式之一。一个数学证明应该像夜空中轮廓清楚的星座,而非银河系中零散分布的星团。”

       “(棋类问题)属于数学中特殊的问题,对这类问题的解自有其价值,但这正是那种“通过罗列出各种可能进行证明的问题”(或罗列出至少不算偏离很大的情况),而这则遭受主流数学家们鄙视。”

 

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.)

     你培养出这样的偏好:相对于针对特定问题的解法,你更喜欢能涵盖更多问题的更通用性的概念。数学家们并不真正在意针对特定问题的解答,甚至是那些悬而未解的定理,比如费马大定理仅仅只是一个逗引,它的存在只是在提醒我们需要发明出更先进的工具并且理解更新的东西以去证明它。最宝贵的是我们在推进它的过程中获得的知识,而非“它被证明了”这个结果。数学家追求的成就是在不同领域的概念间发现联系。因此,许多数学家并不关注他们的研究成果中实用性或者计算结果的寓意(而这却往往成为超抽象方法的缺点);相反,他们想简单的找到更通用、强大的联系。Timothy Gowers关于这点有一些很有意思的评论,以及数学社区中的一些对于该种观点的反对意见。

 

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.)

     理解一些抽象的东西或者证明某些东西是正确的越来越变成一件类似构筑某种东西的任务。你会想:“首先我设定这个基础,然后我将用这些熟悉的模块构建这样一个框架,只留下主体部分等待后面填补,接着我要检验证明过程……”所有这些步骤都具有数学上的类似性,并且具有一定模式的结构化的步骤会使得你可以在不感到迷茫和沮丧的前提下花费好几天时间思考一些你不明白的一些东西(我不得不说,对于所谓“不感到迷茫和沮丧”,有时候这种感觉是不可避免的,关键是把其控制在一个可忍受的程度)。

 

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 )

 

      安德鲁 威尔士,证明了费马大定理的人,使用过一个“探险者”隐喻:“也许我可以将我研究数学的经历完美的诠释为一段穿越漆黑的未经探索的宅子的经历。你进入宅子的第一间屋子,它一片漆黑,你被四周的家具羁绊,但最终你了解到了各间家具的位置。最后,六个多月后,你找到了灯的开关,你打开灯,突然间四周一片光明,你能清楚的看到你在何处。接着,你进入下一间屋子并且又度过六个月的黑暗的日子。所以,每一步进展,甚至有的时候它是瞬息万变的,有时候是一两天时间,它们的形成离不开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.stanford.edu/~vakil/potentialstudents.html.)     

      在参加研究小组或者读论文的时候你不会像初期那样经常被卡住,因为你已经非常擅长模块化概念空间,将那些你不明白的推定或论据以“黑盒子”表示,再以任何方式去考虑它们的推论。你有时对你认为是对的东西有很好的直觉并可以做出陈述,而不必了解其所有细节。你经常可以发现某些运用很高级概念的东西的巧妙或者有意思之处。(我首次看到这个现象被Ravi Vakil强调,他给数学系学生提出了很有见识的建议。)

 

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.

     你会很善于在思考一些新的抽象概念的时候产生你自己对其的定义并且经常提出自己独到的问题。在正统的数学教育中一个人学的非常靠后的东西(经常仅在开始做研究的时候)是怎样作出好的、有用的限定。我非常可靠的从那些知道部分数学但却永远不会成为数学家的人那里听说到他们非常擅长证明那些在课本练习中陈述的问题,但当面对数学的结构并被要求证明关于其的一些有趣的事实时往往会迷失。实际上,要做好这件事需要擅长作出假定,运用新定义的概念,简要陈述其他数学家认为有趣或有启发性的准确的结果的能力。

 

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.

     这种挑战就如同给你一个世界,要求你去找出可以组合在一起形成一个绝妙侦探小说的各个事件。和标准的侦探不同的是,你需要弄明白所谓的“案件”(有意思的问题)可能是什么;你需要从基本公理中产生自己的“线索”。为了做这些事,你需要和你了解的其他侦探故事(数学理论)类比以及利用自己对于如何更惊奇或者更深入的品味。这个过程如何起作用也许就是如何更准确描述数学工作最难的一方面同时我想也是所有数学家所一定共有的东西。

 

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.

     你会容易被讨论数学量或逻辑方面的不严谨而惹怒。这很大程度上是因为你受过训练,能很快的想出可以证明不严密的声明是明显错位的案例。

 

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,你对于有意识地不严密的表述或者在你熟悉领域的领域“空洞”的话却会感觉都很舒服,因为你知道该怎么去充实其中的细节。Terence Tao在这方面很有说法,参看…

 

"[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."

    “(在学习如何严密思考以后,接着是)'过严密’阶段,在这个阶段中你往往会对你所选领域的严密的基础习以为常,并已经作好准备重新审视并且完善你在此方面的以前觉得严密的直觉了,但是此时直觉是通过严密的理论所支撑的。(比如说,在这个阶段你可以通过类比标量计算非常快速和准确的进行矢量运算,或者非正式或半严密的使用无穷小,无穷大记号以及其他,你能将所有计算转换为需要的严密的形式)现在强调的是运用,直觉以及所谓的'蓝图’。这个阶段经常持续到后面的研究生阶段以及更远。”

 

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.

     特别地,以前花数个小时才正确理解的点子首次(“对任意小的总可以找到一个小使得条件成立。”//啊哈,很熟悉!有木有。。。)成为你以后可以不做过多思考就可使用一个基本元素。

 

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!

    在搁笔(//这里其实应该是Ctrl+S和Alt+F4)之前,提醒以下事实是非常必要的:数学家们并不对大多数其他人面对的限制免疫。他们并不是所谓的智力上的超级英雄。比如,他们也经常排斥新的观点,也会对不是他们自己的思考方式(即使是跟数学有关)感到不舒服。他们也会对智力竞赛持抵触情绪,拒绝其他人或者在争论中显得偏狭。以上,我试着总结出如何进行数学式的思考、感觉以及工作,无意关注数学家们的个人缺点或者不同数学领域的争论。这些东西值得他们写出自己的详尽答案!

 

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.

    你对自己的知识很谦逊因为你意识到了数学的无力,并且你对于在很多问题上你并无想法的事实处之泰然。我们只对非常有限的数学问题有合理的明确的答案。任意一个数学家随便就能很好的解决的数学问题显然就更少了。经过两到三年的标准大学课程,一个出色的数学研究生可以毫不费力的写出数以百计的可以使即使最好的数学家也不敢冒险给出试探性答案的数学问题。(理论计算机学家Richard Lipton列举出了一些潜在的很深的无知的例子如下。)这使得被很多问题困住的情况显得习以为常;那种你粗略的知道哪些问题是易于处理的以及哪些是目前我们无能为力的的感觉是粗陋的,但这也会使你变得不那么自卑,因为你深知你对于解决这种问题的强有力的工具是熟悉的。

 

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