skid懂得很高深的数学，是什么感觉？
【cd romMathematics / 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 Ur
You can answer many emingly difficult questions quickly.
单眼皮眼线膏的画法>person怎么读But you are not very impresd by what can look like magic, becau you know the trick. The trick is that your brain can quickly decide if question is answerable by one of a few powerful general purpo "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 u to solve
problems is, perhaps surprisingly, pretty small — e /tricki/map for a partial list, maintained by Timothy Gowers.全国大学英语四六级
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/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 fal, 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 situati
on 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 el. This makes the total time you spend in life reveling in your mastery of something quite brief. One of the main skills of rearch 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.
cio是什么意思
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。
cucYour intuitive thinking about a problem is productive and ufully 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 tho 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 em 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 tho ideas into symbols. Often, you e 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 ca" 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.规格英语
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 mathe
sgp是哪个国家的缩写maticians think is that there is some mysterious mental faculty that is ud 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 arnal 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 cours in university and in competitions. What happens as you get more advanced is simply that the arnal grows larger, the thinking gets somewhat faster due to practice, and you have more examples to try, perhaps making better guess 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 (/career-advice/does-one-have-to-be-agenius- to-do-maths/).
