當代最偉大的數學家：英語精讀講義

You don't usually see crowds gathering for a math professor,

通常你不會看到人群為一位數學教授而聚集，

but Terence Tao is anything but ordinary. 慣用語

但 Terence Tao（陶哲軒）絕不普通。

By the age of two,

在兩歲時，

he taught himself to read and count.

他就自學了閱讀和數數。

By nine,

到了九歲，

he was studying university-level math.

他已經在學習大學水準的數學了。

And by 20,

而到二十歲，

he graduated with a PhD from Princeton.

他就獲得了 Princeton 的博士學位。

Most math geniuses are considered a little strange.

多數數學天才都被認為有些奇怪。

Isaac Newton was intense and obsessive.

Isaac Newton 情緒暴躁且偏執。

John Nash was paranoid with delusions.

John Nash 患有妄想症。

Grigori Perelman withdrew almost completely from public life.

Grigori Perelman 幾乎完全退出了公眾視線。

My videos on them are linked in the description.

關於他們的影片連結都在資訊欄裡。

On the other hand, 表達脈絡

另一方面，

Tao is super normal.

Tao 卻非常正常。

That's a credit to his parents. 片語搭配

這要歸功於他的父母。

His mom,

他的媽媽，

a high school math teacher from Hong Kong,

一位來自香港的高中數學老師，

and his father,

還有他的爸爸，

a pediatrician,

一位小兒科醫生，

raised him and his two younger brothers in Adelaide, Australia.

在澳洲的阿德雷德撫養他和他的兩個弟弟。

They didn't push him relentlessly,

他們沒有冷酷地逼迫他，

ensuring he would develop emotionally and socially,

而是確保他在情感和社交上得到發展，

even though they knew he was special.

即便他們知道他很特別。

When he was two,

兩歲時，

he taught children twice his age how to count using wooden blocks. 文法

他就用積木教比他大一倍的孩子如何數數。

As a child,

在孩童時期，

he was assessed to have an IQ over 160,

他被評估為智商超過160，

and he scored 760 on the math portion of the SAT 文化背景

他在 SAT 的數學部分考了760分，

when he was just eight years old.

而那時他才僅僅八歲。

He was hooked on math 慣用語

他迷上了數學，

when he realized it could help explain the world,

當他意識到數學可以幫助解釋這個世界，

like how satellites can guide millions of drivers

比如衛星是如何引導數百萬駕駛，

to within a few meters 片語搭配

精確到幾公尺之內，

using precise geometry and orbits.

所用的就是精密的幾何學和軌道知識。

What sets Tao apart is how he thinks. 講解

讓 Tao 脫穎而出的是他的思考方式。

When he struggled with a problem involving rotating waves,

當他在解一道關於旋轉波的難題時，

this is how he came up with the solution. 片語動詞

他是這樣想出解決辦法的。

You know, I tried to visualize

你知道，我試圖想像

being the wave that the equation was

自己就是那個方程式

uh was modeling.

呃，正在模擬的波。

And I remember having my eyes closed on the floor.

我記得我閉著眼睛躺在地板上。

I was rolling around

我在地上滾來滾去，

trying to imagine how different parts of the wave evolved.

試圖想像波的不同部分是如何演變的。

Um and uh in fact,

嗯，呃，事實上，

I was staying at my aunt's place at the time

當時我正住在我姑姑家，

and she saw me rolling on the floor,

她看到我在地板上打滾，

and I couldn't explain what I was doing.

而我也無法解釋我在做什麼。

But I did actually understand 文法

但我確實明白了，

um I found a way to transform the solution

嗯，我找到了一種轉換解法的方法，

to make it easier to solve. 文法

讓它變得更容易解決。

And it actually helped quite a lot to...

這真的有很大的幫助……

so physically physically do that.

像那樣進行身體上的動作。

For the longest time,

在很長一段時間裡，

he didn't think being a mathematician was a real career

他並不認為當數學家是一份真正的職業，

and figured he'd end up as a shopkeeper balancing the books. 講解

並以為自己最終會成為一名算帳的店主。

But when his undergraduate advisor stepped in, 片語動詞

但是當他的大學指導教授介入後，

everything changed.

一切都改變了。

His advisor encouraged him to focus on mathematical research

他的指導教授鼓勵他專注於數學研究，

and to study abroad.

並出國留學。

When Tao applied to Princeton,

當 Tao 申請 Princeton 時，

his application included a letter of recommendation

他的申請資料中包含了一封推薦信，

from the great Hungarian mathematician Paul Erdős.

來自偉大的匈牙利數學家 Paul Erdős。

I am sure he will develop into a first-rate mathematician

「我確信他將成長為一名一流的數學家，

and perhaps into a really great one.

甚至可能成為一位真正偉大的數學家。

I recommend him in the highest possible terms. 片語搭配

我以最高評價向您推薦他。」

Tao would become famous for tackling problems involving prime numbers.

Tao 後來因為攻克涉及質數的難題而聲名大噪。

One of the reasons he won the Fields Medal, 文化背景

這也是他贏得 Fields Medal 的原因之一，

the Nobel Prize of math, in 2006.

該獎項在2006年被譽為數學界的諾貝爾獎。

Prime numbers are divisible only by one and themselves. 用詞

質數只能被一和它本身整除。

And as numbers grow larger,

隨著數字越來越大，

primes become rarer

質數變得越來越稀少，

with ever-increasing gaps between them.

它們之間的間隔也不斷增加。

For centuries,

幾個世紀以來，

mathematicians wondered,

數學家們都在想，

do primes behave randomly,

質數的分布是隨機的嗎，

or is there hidden order beneath the surface? 慣用語

還是表面之下隱藏著秩序？

In 2004,

在2004年，

Tao and British mathematician Ben Green proved

Tao 和英國數學家 Ben Green 證明了，

that no matter how sparse prime numbers become, 文法

無論質數變得多麼稀疏，

you can always find long sequences of evenly spaced primes, 用詞

你總能找到由等距質數組成的長序列，

called arithmetic progressions,

也就是等差數列，

if you look far enough.

只要你找得足夠遠。

3, 5, and 7

3、5和7

are prime numbers spaced evenly by two.

是間距均勻為2的質數。

That's an arithmetic progression of length three.

那是一個長度為3的等差數列。

There are longer ones.

還有更長的。

7, 37, 67, and 97

7、37、67和97

are all prime,

都是質數，

each separated by 30,

每個間隔為30，

with an arithmetic progression of length four.

組成了一個長度為4的等差數列。

However far out you go, 文法

無論你探索得多遠，

sequences like this exist for whatever length you desire.

無論你想要多長的序列，這樣的序列都存在。

They didn't prove it by hunting for patterns one by one. 慣用語

他們並不是透過逐一尋找規律來證明的。

They used powerful tools from different branches of math

他們使用了來自不同數學分支的強大工具，

to show that the structure of the primes

來證明質數的結構

forces these progressions to exist. 文法

必然會導致這些數列的存在。

Tao compared the idea to counting grains of sand in a box. 片語搭配

Tao 將這個想法比喻為數盒子裡的沙粒。

You don't count each grain one by one.

你不會一粒一粒地去數。

You weigh the box,

你會稱一下盒子的重量，

subtract the container,

減去容器的重量，

and divide by the average weight of a single grain

然後除以單粒沙子的平均重量，

to get an accurate count of how many are likely to be inside.

從而得出裡面可能有多少沙粒的準確數字。

Unlike many mathematicians who spend a lifetime specializing in one area,

與許多一生只專注於一個領域的數學家不同，

Tao moves freely across math fields,

Tao 在數學各個領域中游刃有餘，

often working collaboratively.

並經常與人合作。

One day in 2004,

2004年的一天，

Stanford mathematician Emmanuel Candès

Stanford 的數學家 Emmanuel Candès

brought Tao a problem he'd run into 片語動詞

把他在與放射科醫師合作時

while working with radiologists.

遇到的一個問題帶給了 Tao。

They were trying to make MRI scans faster,

他們試圖讓 MRI 掃描更快，

but the math just wasn't working.

但數學計算卻總是行不通。

That conversation led them to launch an entirely new field,

那次對話促使他們開創了一個全新的領域，

compressed sensing. 用詞

壓縮感知。

To understand compressed sensing,

要理解壓縮感知，

think about an image.

想像一張圖片。

If you delete pixels,

如果你刪除像素，

the information is gone and the image is ruined.

資訊就會遺失，圖片也就毀了。

But compressed sensing doesn't see pixels, 表達脈絡

但壓縮感知看到的不是像素，

it sees patterns.

它看到的是模式。

If an image has sufficient underlying structure,

如果一張圖片有足夠的底層結構，

you don't need to record every pixel to recover it.

你就不需要記錄每一個像素來還原它。

Even if some of the information is discarded, 文法

即使丟棄了一些資訊，

you can still reconstruct the original image.

你仍然可以重建原始圖像。

Here's a simple analogy.

這裡有一個簡單的類比。

You can probably read this without much trouble.

你大概可以毫不費力地讀出這句話。

Even if lots of letters are missing,

即使遺失了很多字母，

you can still read it because English has structure.

你仍然能讀懂，因為英語有其結構。

Their work led to faster MRI scans

他們的工作不僅讓 MRI 掃描變得更快，

and also attracted the military,

還吸引了軍方的注意，

where the goal is often to see an area using many simple sensors

軍方的目標通常是使用許多簡單的感測器來觀察一個區域，

rather than relying on extremely expensive ones. 文法

而不是依賴極其昂貴的感測器。

Work like this made him quickly stand out. 片語動詞

像這樣的工作讓他迅速脫穎而出。

He became a tenured professor at UCLA when he was 24, 文化背景

他在24歲時成為 UCLA 的終身教授，

teaching students who were almost the same age.

教授的學生幾乎和他同年紀。

He later married Laura,

後來他與 Laura 結婚，

a former student of his,

她曾是他的學生，

who went on to become an engineer at NASA's Jet Propulsion Laboratory. 片語動詞

後來成為 NASA Jet Propulsion Laboratory 的工程師。

Together,

在一起後，

they have two children,

他們有兩個孩子，

and he's passed his spark of curiosity on to them. 片語動詞

他也把自己的好奇心傳遞給了他們。

When his son was four,

當他的兒子四歲時，

Tao would ask questions like, 文法

Tao 會問一些這樣的問題，

"If there are 10 cookies and five people in the living room,

「如果客廳裡有10塊餅乾和5個人，

how many does each person get?"

每個人能分到多少塊？」

His son also showed a strong inclination toward languages,

他的兒子也表現出了強烈的語言傾向，

unlike Tao,

這與 Tao 不同，

who struggled with English and Latin in school.

Tao 在學校時英語和拉丁語成績並不好。

He preferred mathematics,

他更喜歡數學，

where answers were concrete,

因為數學的答案是具體的，

true or false,

非真即假，

right or wrong.

非對即錯。

At one point,

有一次，

a hedge fund approached him,

一家避險基金聯絡了他，

eager to put his extraordinary mathematical mind to work in finance, 慣用語

渴望將他非凡的數學頭腦用於金融領域，

a job that would have been extremely lucrative, 文法

那份工作絕對利潤豐厚，

but he turned it down for the pure joy of staying in academia. 片語動詞

但他拒絕了，只因為享受留在學術界的那份純粹的快樂。

And while he's certainly not hurting for money, 慣用語

雖然他確實不缺錢，

his UCLA professorship pays handsomely,

他在 UCLA 的教授職位薪水豐厚，

with public records showing he earned 784,000 in 2024.

公開紀錄顯示他在2024年的收入為78.4萬美元。

He's been more interested in the questions that fascinate him 文法

但他一直對自己感興趣的問題更為痴迷，

than in the dollars they might generate.

而不是它們可能帶來的金錢。

One of those fascinating questions is whether AI can solve

其中一個引人入勝的問題是，AI 是否能解決

long-standing open problems in pure mathematics. 用詞

純數學領域長期懸而未決的公開問題。

Tao has been tracking how AI tools

Tao 一直在追蹤 AI 工具

like advanced versions of ChatGPT

比如高級版本的 ChatGPT

tackle unsolved puzzles by Paul Erdős,

如何處理 Paul Erdős 留下的未解難題，

even maintaining a public database of these attempts.

甚至還維護了一個公開的嘗試資料庫。

Recently,

最近，

ChatGPT solved Erdős's problem number 281,

ChatGPT 解決了 Erdős 的第281號問題，

generating a proof that has held up under expert review. 片語動詞

並生成了一個經得起專家審查的證明。

Tao called it perhaps the most unambiguous

Tao 稱這或許是最明確無誤的

instance of AI independently solving an open problem.

AI 獨立解決公開問題的案例。

Today,

如今，

Terence Tao is one of the most recognizable mathematicians in the world.

Terence Tao 是世界上最著名的數學家之一。

He is so sought after that his website includes a simple notice, 文法

他非常受歡迎，以至於他的網站上包含了一條簡單的通知，

"I am declining almost all interviews at this time." 語氣意圖

「目前我拒絕幾乎所有的採訪。」

I reached out as well, 片語動詞

我也試著聯絡了他，

knowing full well the answer might be no. 片語搭配

雖然我很清楚答案可能是「不」。

If I do hear back, 片語動詞

如果我確實收到了回覆，

I'll leave an update in a pinned comment.

我會在置頂留言裡更新。

Despite the attention,

儘管備受矚目，

the Mozart of math has remained remarkably humble. 慣用語

這位數學界的 Mozart 卻保持著非凡的謙遜。

That attitude reflects the Chinese philosophy he grew up with,

這種態度反映了他從小接觸的中國哲學，

which emphasizes humility.

強調謙虛。

As his father put it,

正如他父親所說，

"Even if you are very successful,

「即使你非常成功，

you tend to say,

你也會傾向於說，

"No, I'm just pretty average."

『不，我只是挺普通的。』」

In Terence Tao's case,

但在 Terence Tao 身上，

that couldn't be further from the truth. 慣用語

事實並非如此。

Tao wants to teach more non-mathematicians how to think mathematically,

Tao 想要教導更多非數學家如何用數學方式思考，

saying, "I believe you can teach this to almost anybody."

他說：「我相信你可以把這門學問教給幾乎任何人。」

And he's right,

而且他是對的，

if you're taught the right way.

前提是你使用了正確的教學方法。

That's one of the reasons I love using Brilliant.

這就是我喜歡使用 Brilliant 的原因之一。

It's a completely different way of learning.

這是一種完全不同的學習方式。

Instead of passively sitting through lectures, 講解

不是被動地坐在那裡聽課，

you're actively solving problems step by step,

而是主動地一步一步去解決問題，

learning intuitively.

進行直觀地學習。

I've been playing around with Brilliant's exponential functions. 片語動詞

我一直在把玩 Brilliant 的指數函數課程。

What's great is how interactive it is.

最棒的是它的互動性。

It just clicks way more naturally. 慣用語

它讓你理解起來更加自然流暢。

Brilliant has a range of courses in algebra and calculus

Brilliant 提供了一系列代數和微積分課程，

that meet you at your skill level 慣用語

能匹配你的技能水準，

while letting you challenge yourself and grow.

同時也能讓你挑戰自我、不斷成長。

They also have a whole host of courses in coding, 片語搭配

他們還有大量程式設計課程，

science,

科學，

and AI,

以及 AI 領域的課程，

all built around learning by doing. 講解

全都是圍繞「邊做邊學」的理念建構的。

You can start learning for free at brilliant.org/newsthink

你可以在 brilliant.org/newsthink 免費開始學習，

or scan the QR code on your screen.

或者掃描螢幕上的行動條碼。

Brilliant's also giving you 20% off their annual premium subscription

Brilliant 還提供年度高級訂閱的20%折扣，

if you use my custom link in the description,

只要你使用資訊欄裡我的專屬連結，

giving you unlimited access to thousands of lessons.

你就能無限次訪問數千堂課程。

Thanks for watching.

感謝收看。

For Newsthink,

我是 Newsthink 的

I'm Cindy Pom.

Cindy Pom。