思維方式

費瑪最後定理 這本書第六章中有一段記錄了 Andrew Wiles 做研究的方式：

我習慣於到樓上我的書房去，著手嘗試尋找一些模式。 我設法做一些計算來解釋某一小段數學， 設法使它符合某些以前對某部分數學的泛泛的概念性理解，這有助於澄清我正在思考的具體問題。 有時候還得去書上查考。以便弄明白在那裡它是怎麼完成的。 有時它只是做一點補充計算，進行一點修放的問題。 而有時候我發覺以前所做的事情根本都是沒用的，於是我就必須找出一些全新的東西 -- 它從哪裡冒出來的？這件事有點神祕。 Andrew Wiles, p. 204

以下是原文：

I used to come up to my study, and start trying to find patterns. I tried doing calculations which explain some little piece of mathematics. I tried to fit it in with some previous broad conceptual understanding of some part of mathematics that would clarify the particular problem I was thinking about. Sometimes that would involve going and looking it up in a book to see how it's done there.Sometimes it was a question of modifying things a bit, doing a little extra calculation. And sometimes I realized that nothing that had ever been done before was any use at all.Then I just had to find something completely new -- it's a mystery where that comes from. Andrew Wiles, p. 207

我覺得值得一學。 接下來的一段提到了遇到瓶頸時，潛意識居然可以解決問題！

基本上說，它還是思維的結果。 你常常會寫下一些話來闡明你的想法，但並不一定如此。 特別是當你真的進入死胡同的時候。 當有一個真正的問題需要你去征服的時候，那種循規蹈矩的數學思維對你來說毫無用處。 導致那一類新的想法必須經過長時間的對那個問題的極其專注的思考， 不能有任何分心。這之後似乎有一段鬆弛期，在這期間潛意識出現，佔據你的腦海。 正是在這段期間，某種新見解冒出來了。 Andrew Wiles, p. 204

以下是原文：

Basically it's just a matter of thinking. Often you write something down to clarify your thoughts, but no necessarily. In particular when you've reached a real impasse, when there's a real problem that you want to overcome, then the routine kind of mathematical thinking is of no use to you. Leading up to that kind of new idea there has to be a long period of tremendous foucus on the problem without any distraction. You have to really think about nothing but that problem -- just concentrate on it. Then you stop. Afterwards there seems to be a kind of period of relaxation during which the subconscious appears to take over, and it's during that time that some new insight comes. Andrew Wiles, p. 208

我在 mathlearningcenter 查到一個關於潛意識的連結。