英文阅读｜数学家们的想法是怎么来的（可能与你想象的不一样）

写在前面

不知道大家有没有遇到这个问题：单词背也背了（有时背得还很辛苦），但就是用不出来（口语&写作中）

其实解决方式也不难：

1.结合原文和大量例句看别人是怎么用的（这个过程叫“输入”）

p.s. 例句不要只看1个，否则自己还是学不到在【不同的语境中】该词的用法，这也是我会一个单词配好几个例句的原因

2.自己造句（这个过程叫“输出”），通过造句，你就“逼”自己不得不思考“我可以把这个词用在哪呢？”“用的时候要注意什么？”等等问题，你在【主动学习】，那以后在口语或写作中，你当然就可以自然而然地用出来了，不需要在“用词”上烦恼了，只需要focus在你口语/写作的“内容”上，这才是核心，所以，记得完成页面底部的“趁热打铁”部分，要学就学透了，要学就追求高质量！

原 文 泛 读

人们的想法是从哪里来的呢？

《纽约客》的这篇文章通过一个数学家的自述告诉我们数学家们的ideas是怎么来的，他们可不是钻破脑袋想出来的哦～

全文以叙事为主，用词纯熟老道，整篇文章的画面感很强，“文青”气息也很有，不愧为“文青必读”杂志

[文中下划线的部分是必备，其他句子和段落选背]

THE MYTH AND MAGIC OF GENERATING NEW IDEAS

A mathematician on how to get the mind into motion.

Where do ideas come from? That’s a big question. Here’s a smaller one: Where do mathematical ideas come from? I’ve wondered about this from the time I first contemplated being a mathematician until long after I officially became one.

My earliest memory of anything like a mathematical idea comes from a childhood walk with my dad. We left the house and made our way toward downtown Metuchen, the tiny town in central Jersey where I grew up, to a little luncheonette called the Corner Confectionery. I can still picture it: the rack of newspapers, magazines, and comic books; the ice-cream treats in the back corner; the long counter with stools, where I used to sit and spin until I was told to stop. It was about a mile-long walk, reserved for special occasions. On that bright fall morning, we strolled up Spring Street—a beautiful street lined with huge oak trees—and talked about fractions, though I wouldn’t have known to call them that. We were puzzling over—or, rather, I was puzzling over—how to fairly divide a pie (probably one of the Corner Confectionery’s apple pies). My dad, a mathematical physicist, a man with an active mind, but one of few words, was a gentle guide, letting me think through things on my own.

We took our time walking, and we also took our time thinking and talking about the basic properties of numbers. In my head, it was easy to cut the pie in half, and then in half again, and again: two, four, or eight pieces. But, somewhere near Main Street, I got stuck on how to reliably create three, five, or six pieces. I started thinking about making bigger numbers out of smaller numbers. This leisurely walk through the neighborhood soon led me to the exciting idea that twelve was a great number. Twelve could be divided by one, two, three, four, and six. That’s a lot of numbers! If I had a pie cut into twelve pieces, it would be easy to divvy up dessert for many different-sized groups of friends. By the time we crossed the railroad tracks and arrived at the door of the Confectionery, I thought that I had made a remarkable discovery: Everyone! Stop! We need to think about the world in terms of twelves!

Ten or so years later, when I was a college freshman, I would learn that I had stumbled upon an instance of what is called an abundant number, a phenomenon first studied by the ancient Greeks. An abundant number is smaller than the sum of its divisors: in my case, the sum of one, two, three, four, and six (twelve’s divisors) is sixteen. That morning with my dad, I didn’t have a name for this phenomenon, but I was happy nonetheless, and maybe even happier because I was ignorant of the larger picture. It was my own surprising little discovery, born of walking and puzzling. Magic all around.

As odd as it might sound, I’ve never been particularly confident of my mathematical abilities. I don’t mean the arithmetic part, the part that people usually associate with being a mathematician. (“Hey, let Dan calculate the tip! Ha ha!”) It’s true that I’m probably better than average at mental math, but that’s not really what makes a mathematician a mathematician. My job is to come up with ideas. Sometimes we mathematicians call the things we think about and work with “objects,” which doesn’t mean triangles, spheres, or other shapes. Mathematical objects are big ideas about algebra, geometry, and logic, about the properties and definitions of numbers.

It’s not at all obvious how to go about thinking up some new twist on these things—the transformation from test-taker to theorem poser and then theorem prover is difficult to articulate. My ideas have always felt contingent and magical to me. I don’t think I’m alone, at least as far as the magic goes. Henri Poincaré, the father of chaos theory and the co-discoverer of special relativity, is famous for a story that appears in his 1908 book “Science and Method,” about an insight being jarred loose while boarding a bus: “At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it.” The Irish mathematician Sir William Rowan Hamilton, who devoted many years to searching for a way to multiply numbers in higher dimensions, had a similar epiphany, in 1843, just as he was strolling by the Brougham Bridge, in Dublin, while on a walk with his wife. He was so delighted that he stopped and carved the defining algebraic equation into the bridge: i2=j2=k2=ijk=-1. One person’s graffiti is another person’s breakthrough.

These stories suggest that an initial period of concentration—conscious, directed attention—needs to be followed by some amount of unconscious processing. Mathematicians will often speak of the first phase of this process as “worrying” about a problem or idea. It’s a good word, because it evokes anxiety and upset while also conjuring an image of productivity: a dog worrying a bone, chewing at it to get to the marrow—the rich, meaty part of the problem that will lead to its solution. In this view of creative momentum, the key to solving a problem is to take a break from worrying, to move the problem to the back burner, to let the unwatched pot boil.

All problem solvers and problem inventors have had the experience of thinking, and then overthinking, themselves into a dead end. The question we’ve all encountered—and, inevitably, will encounter again—is how to get things moving and keep them moving. That is, how to get unstuck.

For me, the quest for a breakthrough often requires getting myself into literal motion; one small step for Poincaré but a whole sequence of steps for me. I’ll take a long hike, during which my mind has nothing to worry about except putting one foot in front of the other, or I’ll go for a long drive, so that my primary focus is on the road. Maybe it’s the endorphins, or maybe it’s refocussing my attention on some other activity which enables a new idea. Perhaps it is the momentary feeling of being untethered that gives the mind free rein—the space to have a good idea.

词 汇 卡 片

1. contemplate [kn.tm.plet] vi./vt. C2

to spend time considering a possible future action, or to consider one particular thing for a long time in a serious and quiet way

盘算；沉思，冥想

- [ + -ing verb ] I"m contemplating going abroad for a year.

我打算出国一年。

- They were contemplating a move to California.

他们打算搬到加利福尼亚去。

- It"s too awful/horrific/dangerous to contemplate.

这件事太糟糕／可怕／危险，简直不敢想。

2. luncheonette [ln.tnet] CN

a small restaurant serving simple, light meals

快餐店

3. picture vt. C1

to imagine sb/sth; to create an image of sb/sth in you mind

想象；设想；忆起

- I can still picture the house I grew up in.

我还能回忆起我童年时住的那栋房子。

- We found it hard to picture him as the father of teenage sons.

我们很难想像他居然是有几个十几岁儿子的父亲了。

- [VN -ing] When he did not come home she pictured him lying dead on the roadside somewhere.

他还没回家的时候，她想像着他已横尸路边了。

- [V wh-] I tried to picture what it would be like to live alone.

我努力设想一个人单独生活是什么情景。

4. treat CN

a special food that tastes good, especially one that you do not eat very often

（尤指不经常吃的）美味食物

- The cafe serves an assortment of gourmet treats.

这家小餐馆供应各种美味佳肴。

注：

介绍该词的一个Idiom:

a treat

very well; with good results

很好；效果不错

- To prevent red wine from staining, put some salt on it - it works a treat.

要想不让红葡萄酒留下污渍，在污渍处洒些盐——效果很不错。

- That soup went down a treat (= tasted very good)!

那汤很好喝！

- I polished that old desk of grandma"s and it came up a treat (= its appearance improved).

我把奶奶的那张旧书桌擦了擦，马上就焕然一新了。

5. stool [stul] CN

a seat without any support for the back or arms凳子

- a bar/kitchen/piano stool

吧台／厨房／钢琴凳

- a three-legged stool

3条腿的凳子

6. divvy [dv.i] sth up

to share something between a number of people

分享；分担，分摊

- They haven"t yet decided how to divvy up the proceeds from the sale.

他们还没决定好如何分配销售收入。

- There"s a lot of artistic expression to divvy up.

艺术表现形式真是多种多样啊。

7.an abundant number

an imperfect number that is less than the sum of all its divisors (as 12)

过剩数

百科知识：

在数论中，若一个正整数除了本身外之所有正约数之和比此数自身大，则称此数为过剩数。（又称作丰数或盈数）。例如12除了本身外的所有正约数有 1,2,3,4,6，而1+2+3+4+6=16，16>12，所以12可称为过剩数。

注：

其实如果你是在做阅读理解，没必要知道an abundant number的中文是什么啦，因为原文中的下一句有解释啊：An abundant number is smaller than the sum of its divisors

[英文阅读中，能与生词“打交道”是一种能力，如果全文一个生词都没有或者每个背景知识都是自己知道的，那么，要不这篇文章也没啥意思，要不就是你很强哦～]

8. It was my own surprising little discovery, born of walking and puzzling.这句话中的born of简直让Fred看哭，born of用得太棒了，其英文释义为existing as the result of something

- With a courage born of necessity, she seized the gun and ran at him.

有了必要的勇气，她持枪向他走去。

[这句话让我想到了《致命女人》]

9. twist [twst] CN

a change in the way in which something happens

变化；转折；改变

- The story took a surprise twist today with media reports that the doctor had resigned.

媒体报道该医生已经辞职，从而使事情在今天有了惊人的转折。

- The incident was the latest twist in the continuing saga of fraud and high scandal in banks and stock brokerages.

此事件是银行和股票经纪公司持续不断的诈骗和高层丑闻的最新发展。

- But for a cruel twist of fate/fortune, he could now be running his own business.

要不是时运不济，他现在可能正在经营自己的公司了。

- There"s an unexpected twist in/to the plottowards the end of the film.

电影结尾部分的情节出人意料。

注：

这句话破折号后的内容即是twi

10. articulate [ɑtk.j.lt] vt.

to express in words

清楚地表达

- I found myself unable to articulate my feelings.

我感到无法用语言来表达我的感情。

- Many people are opposed to the new law, but have had no opportunity to articulate their opposition.

很多人不赞成这项新法律，但是却没有机会去表达他们的反对意见。

11. contingent [kntn.dnt] adj.

depending on something that may happen in the future依情况而变的，视条件而定的

- Further investment is contingent upon the company"s profit performances.

进一步的投资取决于公司的利润情况。

- Outdoor activities are, as ever, contingent on the weather.

室外活动从来都只能依天气而定。

- Our success is contingent upon your support.

我们的成功有赖于你的支持。

注：

原文的My ideas have always felt contingent and magical to me. 大白话理解一下就是：我的ideas总是来得莫名奇妙

12. 本文的jar可不是罐子的意思，而是用作及物动词震动，猛烈地摇动的意思(to hit or shake something forcefully)

- The earthquake jarred the buildings loose.

地震将房屋给震垮了。

注：

看到原文中的an insight being jarred loose后，Fred再次哭泣，用得太形象了吧，想象一下：一个人的想法都被“囚禁”在一个罐子里，突然有一瞬间这些想法都一下子被晃出来了（也就是本段后面的epiphany（顿悟））

[《纽约客》的风格就是很“文青”，看文章时大脑中会很有画面感]

13.epiphany [pf.n.i] CN/UN

a moment when you suddenly feel that you understand, or suddenly become conscious of, something that is very important to you

（对重要事物的）顿悟，突然明白

14.conjure [kn.dr] vt.

to make something appear or happen in a way which is not expected

使（某物）突然出现（发生）

- He has conjured victories from worse situations than this.

他曾在比这更糟的情况下出人意料地取得过胜利。

15. marrow [mr.] UN

soft tissue containing a lot of fat in the centre of a bone

髓，骨髓

16. Poincaré

儒勒·昂利·庞加莱

背景知识：

儒勒·昂利·庞加莱，通常称为昂利·庞加莱，法国最伟大的数学家之一，理论科学家和科学哲学家。庞加莱被公认是19世纪后和20世纪初的领袖数学家，是继高斯之后对于数学及其应用具有全面知识的最后数学家。 他对数学，数学物理，和天体力学做出了很多创造性的基础性的贡献。他提出的庞加莱猜想是数学中最著名的问题之一。

17. endorphin [end.fn] CN

a chemical naturally, released in the brain to reduce pain, that in large amounts can make you feel relaxed or full of energy内啡呔

18.tether [te.r] vt. 这个词本来指“拴住（畜生）”，所以本文的untethered就是“（畜生）未被拴住的”，这样明显语境不通，结合上下文我们知道，文本该词指“注意力不是只集中于某一件事上的”

语法/长难句分析

Sometimes we mathematicians call the things we think about and work with “objects,” which doesn’t mean triangles, spheres, or other shapes.

自己先试着分析：

1. 这句话的主干是什么？

2. 其他成分是什么？

主干：

Sometimes we mathematicians call the things “objects”

[我觉得有同学主干抓不对]

其他成分：

1. we think about and work with是定语从句修饰things，关系代词which/that省去了，因为其在定语从句中作介词about和with的宾语

2. , which...是非限制性定语从句修饰"things"，which指代“things”，在从句中作主语

[有同学也许会问："things"是复数呀，后面怎么用的doesn"t呢？答：这里将"things"看作一个整体]

趁 热 打 铁

任选上面18个词中的1个造句，下方评论告诉我哦，会一一回复哒～

分享是一种动力 点赞是一种鼓励