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理解高级数学是怎样的体验？ 1. 你可以很快回答很多看起来很难的问题。但是你不会被这些看起来很神奇的东西所惊讶到，因为你知道其中的诀窍。 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. 2. 在你想到一个无懈可击的证明之前，你经常会相信某个结果是对的（这经常发生在几何领域）。 The main reason is that you have a large catalog of connections between concepts, and you can quickly intuit that if Xwere to be false, that would create tensions with other things you know to be true, so you are inclined to believeXis 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. 3. 你会觉得你对正在研究的问题并没有深刻的理解，但这并不影响你的研究。 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 omfortably and productively in a state of confusion. 4. 当试着学习新东西时，你自动会关注哪些非常简单的例子，之后才被例子中的直观感觉引入到更深刻的内容中去。 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. 5. 你学得会越来越抽象，“越来越高级”。昨天的主要对象今天已经变成了一个例子或者非常细节的部分。 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 makes it possible to consider extremely complicated issues with one's (very) limited memory and processing power. 6. 你在表达一个问题时，可以自由地用不同的看起来相差很远的观点来说明。 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. 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,ex ante, are very surprising. For example, Galois the-ory 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 equa-tion. 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. 7. 理解一个抽象的东西或者证明某件事是对的就好像是建造一座房子。 你会这样想：“首先我需要打基础，接着我要用熟悉的东西给出框架，但是细节留到以后去填，接着我需要测试...” 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. 8. 在听讨论班或读文章时，你不会被熟悉的东西卡住，许多东西可以当作黑匣子来处理。 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. 9. 你在抽象新的问题时，你非常擅长提出自己的定义和猜想。 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. You have to figure out who the characters should be (the concepts and objects you define) and what the interesting mystery might be. 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 diffcult aspect of mathematical work to describe precisely but also the thing that I would guess is the strongest thing that mathematicians have in common. 10. 你会变得谦逊，因为你认识到数学并不能解决所有问题，你会对暂时解决不了的问题泰然处之。 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. 原文见http://www./Mathematics/What-is-it-like-to-understand-advanced-mathematics |
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