陶的数学之路

数学是一个多层面的学科,我们的经验和品味会随着时间和阅历的改变而改变。

当我上小学时,我被数学中形式计算的抽象美所吸引,并且醉心于利用简单规则得到非平凡答案的超常能力。

当我是一个中学生时,我把数学竞赛当做一种运动,通过采取巧妙的设计数学谜题并且寻找正确的“技巧”来解决这些问题,乐此不疲。

当我是一个本科生时,第一次瞥见丰富深刻而又令人着迷的理论以及存在于现代核心数学的结构,这让我心存敬畏。

当我是一个研究生时,我体会到了拥有自己的研究计划的自豪感,以及通过原创的讨论来重新解决之前公开的问题的满足感。

直到现在作为职业数学研究开始学术生涯,我开始观察蕴含于现代数学理论背后的直觉与动机,当意识到非常复杂艰深的结果常常由非常简单寻常的例子来指引,常常让我欣喜不已。掌握这些原理并且突然发现一大批数学是怎样启示与阐释,这真是一个非凡的“原来如此”经历。

数学中仍然有许多值得去发现的东西,直到最近,我已经掌握了足够多的数学领域,并且开始朝着数学作为一个整体的学科而努力,同时联系其他科学等学科。

Mathematics is a multifaceted subject, and our experience and appreciation of it changes with time and experience.

As a primary school student, I was drawn to mathematics by the abstract beauty of formal manipulation, and the remarkable ability to repeatedly use simple rules to achieve non-trivial answers.

As a high school student, competing in mathematics competitions, I enjoyed mathematics as a sport, taking cleverly designed mathematical puzzle problems and searching for the right “trick” that would unlock each one.

As an undergraduate, I was awed by my first glimpses of the rich, deep, and fascinating theories and structures which lie at the core of modern mathematics today.

As a graduate student, I learnt the pride of having one’s own research project, and the unique satisfaction that comes from creating an original argument that resolved previously open question.

Upon starting my career as a professional research mathematician, I began to see the intuition and motivation that lay behind the theories and problems of modern mathematics, and was delighted when realizing how even very complex and deep results are often at heart be guided by very simple, even common-sensical, principles. The “Aha!” experience of grasping one of these principles, and suddenly seeing how it illuminates and informs a large body of mathematics, is a truly remarkable one.

And there are yet more aspects of mathematics to discover; it is only recently for me that I have grasped enough fields of mathematics to begin to get a sense of the endeavour of modern mathematics as a unified subject, and how it connects to the sciences and other disciplines.

我并没有任何魔力 。我看一个问题,看到它和我之前看到的的很类似,我认为之前的想法也许可以用来解决现在的问题。如果不能,那么你会认为一个小的技巧会使得问题改观但仍然不完全正确。

我与数学玩耍,直到我搞清楚发生了什么。大多数人面对一个数学问题,并且试图直接解决这个问题。即使他们已经完全解决了问题,但是也许并非真正明白其中的道理。

在我写出全部细节之前,我会先提出一个策略。一旦你有一个策略,一个非常复杂的问题就可以分解成许多小的问题。我从未满足刚刚解决的问题。我想看看如果改变一些会发生什么,仍然凑效吗?

如果你足够有经验,你就会理解的越深刻。不久,当相似的事情伴随而来的时候,你就会知道行不行。这与你是否聪明敏捷无关。

这有一点像爬山:如果你非常强壮而且很快,并且有很多绳索,这会帮助你,但是你需要设计出好的路线是你到达目的地。

快速计算并且知道许多事实,这类似于一个攀岩者拥有力气、速度以及好的工具。你仍然需要一个计划-这会是很困难的部分-使得你看到更加广阔的图景。

 

 

I don’t have any magical ability. I look at a problem, and if it looks something like one I’ve done before; I think maybe the idea that worked before will work here. If nothing’s working out; then you think of a small trick that makes it a little better but still is not quite right.

I play with the problem, and after a while, I figure out what’s going on. Most people, faced with a math problem, will try to solve the problem directly. Even if they get it, they might not understand exactly what they did.

Before I work out any details, I work on the strategy. Once you have a strategy, a very complicated problem can split up into a lot of mini-problems. I’ve never really been satisfied with just solving the problem. I want to see what happens if I make some changes; will it still work?

If you experiment enough, you get a deeper understanding. After a while, when something similar comes along, you get an idea of what works and what doesn’t work. It’s not about being smart or even fast.

It’s like climbing a cliff: If you’re very strong and quick and have a lot of rope, it helps, but you need to devise a good route to get up there.

Doing calculations quickly and knowing a lot of facts are like a rock climber with strength, quickness and good tools. You still need a plan — that’s the hard part — and you have to see the bigger picture.

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