理解高級數(shù)學(xué)是怎樣的體驗？

1. 你可以很快回答很多看起來很難的問題。但是你不會被這些看起來很神奇的東西所驚訝到，因為你知道其中的訣竅。

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. 在你想到一個無懈可擊的證明之前，你經(jīng)常會相信某個結(jié)果是對的（這經(jīng)常發(fā)生在幾何領(lǐng)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. 你會覺得你對正在研究的問題并沒有深刻的理解，但這并不影響你的研究。

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. 當(dāng)試著學(xué)習(xí)新東西時，你自動會關(guān)注哪些非常簡單的例子，之后才被例子中的直觀感覺引入到更深刻的內(nèi)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. 你學(xué)得會越來越抽象，“越來越高級”。昨天的主要對象今天已經(jīng)變成了一個例子或者非常細(xì)節(jié)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. 你在表達(dá)一個問題時，可以自由地用不同的看起來相差很遠(yuǎn)的觀點來說明。

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. 理解一個抽象的東西或者證明某件事是對的就好像是建造一座房子。

你會這樣想：“首先我需要打基礎(chǔ)，接著我要用熟悉的東西給出框架，但是細(xì)節(jié)留到以后去填，接著我需要測試...”

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. 在聽討論班或讀文章時，你不會被熟悉的東西卡住，許多東西可以當(dāng)作黑匣子來處理。

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. 你在抽象新的問題時，你非常擅長提出自己的定義和猜想。

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. 你會變得謙遜，因為你認(rèn)識到數(shù)學(xué)并不能解決所有問題，你會對暫時解決不了的問題泰然處之。

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.

原文見http://www./Mathematics/What-is-it-like-to-understand-advanced-mathematics