It's the Second World War. A German prison camp. And this man, Archie Cochrane, is a prisoner of war and a doctor, and he has a problem. The problem is that the men under his care are suffering from an excruciating and debilitating condition that Archie doesn't really understand. The symptoms are this horrible swelling up of fluids under the skin. But he doesn't know whether it's an infection, whether it's to do with malnutrition. He doesn't know how to cure it. And he's operating in a hostile environment. And people do terrible things in wars. The German camp guards, they've got bored. They've taken to just firing into the prison camp at random for fun. On one particular occasion, one of the guards threw a grenade into the prisoners' lavatory while it was full of prisoners. He said he heard suspicious laughter. And Archie Cochrane, as the camp doctor, was one of the first men in to clear up the mess. And one more thing: Archie was suffering from this illness himself.
這是在第二次世界大戰時 德國的一個戰俘營 而這位軍人 是已故英國臨床流行病學者亞契.柯克倫(Archie Cochrane) 既是戰俘也是醫生, 他當時有個難題 多名接受他治療的病人 正飽受令人費解之病痛 這種病使患者感到劇烈疼痛, 渾身無力 亞契苦思不得其解 他們的症狀 是皮下嚴重水腫 然而他無法確定水腫是由病菌所感染, 還是營養不良所引起 那時,他遍尋不著良方 更何況那是在惡劣的環境之下 兵馬倥傯戰亂之中 德軍軍營守衛如果閒著無聊 就對戰俘營裡恣意掃射 打發時間並尋開心。 某次 一名警衛扔手榴彈 到人滿為患的犯人廁所內。 守衛的理由是,他聽到可疑的笑聲。 亞契.柯克倫 ,這名營區的醫生, 是首批進去收拾善後, 處理慘狀的人之一。 另外, 亞契也身染該病。
So the situation seemed pretty desperate. But Archie Cochrane was a resourceful person. He'd already smuggled vitamin C into the camp, and now he managed to get hold of supplies of marmite on the black market. Now some of you will be wondering what marmite is. Marmite is a breakfast spread beloved of the British. It looks like crude oil. It tastes ... zesty. And importantly, it's a rich source of vitamin B12. So Archie splits the men under his care as best he can into two equal groups. He gives half of them vitamin C. He gives half of them vitamin B12. He very carefully and meticulously notes his results in an exercise book. And after just a few days, it becomes clear that whatever is causing this illness, marmite is the cure.
這千鈞一髮的情況下, 足智多謀的亞契 想辦法取得資源 悄悄地走私維他命C進去營區 而且當時 他還設法換取 黑市裡的麴精 在座的各位,也許正在猜想麴精是什麼? 其實那是英式早餐的一種麵包醬 外型看似未提煉的原油,黝黑色。 嘗起來 棒透了。 更重要的是, 麵包裡富含 維他命B12。 亞契將這群他苦心照料的病人 分成兩組。 一組餵以維他命C。 另一組則以維他命B12。 醫生鉅細靡遺地 在筆記簿上記下結果。 數日之後, 一切撥雲見日。 無論病因為何, 麵包醬就是解藥。
So Cochrane then goes to the Germans who are running the prison camp. Now you've got to imagine at the moment -- forget this photo, imagine this guy with this long ginger beard and this shock of red hair. He hasn't been able to shave -- a sort of Billy Connolly figure. Cochrane, he starts ranting at these Germans in this Scottish accent -- in fluent German, by the way, but in a Scottish accent -- and explains to them how German culture was the culture that gave Schiller and Goethe to the world. And he can't understand how this barbarism can be tolerated, and he vents his frustrations. And then he goes back to his quarters, breaks down and weeps because he's convinced that the situation is hopeless. But a young German doctor picks up Archie Cochrane's exercise book and says to his colleagues, "This evidence is incontrovertible. If we don't supply vitamins to the prisoners, it's a war crime." And the next morning, supplies of vitamin B12 are delivered to the camp, and the prisoners begin to recover.
因此,柯克倫便去見德軍營長, 在座的各位,試想在此時此刻-- 先把這張照片丟至九霄雲外,只要想像 一下巴剪不斷理還亂的鬍鬚,和一頭震懾人的紅髮 許久未能整理儀容,猶如比利康諾利的柯克倫 朝著這些德國人咆哮 帶著蘇格蘭口音 卻極其流利的德文 數落對方,坐擁孕育世界偉人席勒和歌德 的偉大文化 竟能容忍這些野蠻行徑。 他完全無法理解這些德國人。 他發完這頓牢騷之後, 踱步回房。 情緒失控,落下男兒淚。 畢竟他認為情況陷入絕望,束手無策。 此時,一名年輕的德軍軍醫 拾起亞契.科克倫的筆記簿 對同僚說: 「事實擺在眼前, 倘若我們再不提供戰俘維他命, 會有違人道。」 於是,翌日清晨, 營區便提供維他命B12補給 而戰俘便逐漸好轉。
Now I'm not telling you this story because I think Archie Cochrane is a dude, although Archie Cochrane is a dude. I'm not even telling you the story because I think we should be running more carefully controlled randomized trials in all aspects of public policy, although I think that would also be completely awesome. I'm telling you this story because Archie Cochrane, all his life, fought against a terrible affliction, and he realized it was debilitating to individuals and it was corrosive to societies. And he had a name for it. He called it the God complex. Now I can describe the symptoms of the God complex very, very easily. So the symptoms of the complex are, no matter how complicated the problem, you have an absolutely overwhelming belief that you are infallibly right in your solution.
我敘述這個故事的原因 並非因為我視亞契.柯克倫為堂堂漢子 雖然他的確也是。 講這個故事的理由是 我們都該更 小心去掌控公共政策的所有層面 下的隨機試驗。 即便,個人淺見,隨機試驗本身其實很棒。 向各位闡述故事是為了 解釋亞契. 柯克倫, 終其一生與苦難對抗 而他明瞭,苦難對個體的蠶食 對整個社會的鯨吞。 他將此命名為: 上帝情結。 接下來,為各位簡述上帝情結的症狀。 這些症狀就像是 無論遭遇的問題有多錯綜複雜 你總是有無與倫比的信心 總而言之,你認為你的解決之道正確且毋庸置疑
Now Archie was a doctor, so he hung around with doctors a lot. And doctors suffer from the God complex a lot. Now I'm an economist, I'm not a doctor, but I see the God complex around me all the time in my fellow economists. I see it in our business leaders. I see it in the politicians we vote for -- people who, in the face of an incredibly complicated world, are nevertheless absolutely convinced that they understand the way that the world works. And you know, with the future billions that we've been hearing about, the world is simply far too complex to understand in that way.
身為一名醫生, 亞契與醫生為伍 醫生很容易有上帝情結。 而我是一名經濟學家,並非醫生, 在我眼裡,患有上帝情結的經濟學者, 比比皆是。 舉凡企業頭子、 甚至是我們票選出的政治人物。 這些人,身處在複雜程度難以置信的世界裡, 卻百分百相信 世界的運轉,自己瞭若指掌 在座各位也曉得,我們將與未來的數十億人口共存, 如果還是用老法子去思考, 是難以理解這複雜的世界。
Well let me give you an example. Imagine for a moment that, instead of Tim Harford in front of you, there was Hans Rosling presenting his graphs. You know Hans: the Mick Jagger of TED. (Laughter) And he'd be showing you these amazing statistics, these amazing animations. And they are brilliant; it's wonderful work. But a typical Hans Rosling graph: think for a moment, not what it shows, but think instead about what it leaves out. So it'll show you GDP per capita, population, longevity, that's about it. So three pieces of data for each country -- three pieces of data. Three pieces of data is nothing. I mean, have a look at this graph.
讓我為各位舉個例子。 花幾秒想像一下 站在台上的人不是我, 而是漢斯.羅史齡,在展示圖表。 各位眼中的漢斯: 就好比TED集團的米克.傑格。 (笑聲) 他會秀出這些驚人的數據, 讚為觀止的動畫。 這挺棒的! 一個典型的漢斯.羅史齡會用的圖表: 試想圖表除了本身, 還喻含的意義。 各位將會看到國內生產毛額、 人口、壽命, 大概就是這些東西。 若每個國家都 列舉三種資訊 才三種並沒有什麼。 且讓我們看一下該表。
This is produced by the physicist Cesar Hidalgo. He's at MIT. Now you won't be able to understand a word of it, but this is what it looks like. Cesar has trolled the database of over 5,000 different products, and he's used techniques of network analysis to interrogate this database and to graph relationships between the different products. And it's wonderful, wonderful work. You show all these interconnections, all these interrelations. And I think it'll be profoundly useful in understanding how it is that economies grow. Brilliant work. Cesar and I tried to write a piece for The New York Times Magazine explaining how this works. And what we learned is Cesar's work is far too good to explain in The New York Times Magazine.
我指的是這個物理學家所作的表。 凱薩.荷德果 畢業於MIT(麻省理工)。 目前各位可能一點也看不懂這個表。 但看起來像是這樣子。 凱薩把逾5000多筆的產品的資料庫 提取分析。 利用網路分析的技術 藉此整合資料庫, 並將不同結果間的關係視覺圖像化。 創出這件傑出的作品, 所有的互動、交互關聯一目了然。 欲瞭解經濟如何成長, 看這個圖表便能略知一二。 真是傑作。 凱薩和我試圖替紐時雜誌撰寫一篇文章, 藉此解釋運作方式。 從中,我們習得的是 僅一篇的篇幅,不夠解釋 凱薩的傑作。
Five thousand products -- that's still nothing. Five thousand products -- imagine counting every product category in Cesar Hidalgo's data. Imagine you had one second per product category. In about the length of this session, you would have counted all 5,000. Now imagine doing the same thing for every different type of product on sale in Walmart. There are 100,000 there. It would take you all day. Now imagine trying to count every different specific product and service on sale in a major economy such as Tokyo, London or New York. It's even more difficult in Edinburgh because you have to count all the whisky and the tartan. If you wanted to count every product and service on offer in New York -- there are 10 billion of them -- it would take you 317 years. This is how complex the economy we've created is. And I'm just counting toasters here. I'm not trying to solve the Middle East problem. The complexity here is unbelievable. And just a piece of context -- the societies in which our brains evolved had about 300 products and services. You could count them in five minutes.
5000筆結果 不費吹灰之力。 5000筆結果耶! 試想算出每個 在資料庫裡的分類 一秒鐘 能算出的分類結果。 同個時間範圍內, 各位會得到5000筆。 如果,如法炮製到 沃爾瑪超市內特價的各種商品 那裡有十萬種商品,數完要花上一整天的時間。 試想, 在一個主要的經濟體內,像是在東京、倫敦或紐約 計算在打折的 每種不同的產品和服務。 算愛丁堡內的甚至會難上加難, 因為還得加算威士忌和格子呢。 假設想算出每種 在紐約所提供的不同的產品和服務-- 答案將會是100億個 各位得花317年才算得出來 這就是人類所孕育出,無比複雜的經濟。 而我不過就是在這裡算吐司麵包。 並不想解決中東問題。 這裡的複雜性令人難以置信。 只要一小塊人腦 所創造出的社會 就擁有逾300樣的產品與服務。 卻只消五分鐘便算完了。
So this is the complexity of the world that surrounds us. This perhaps is why we find the God complex so tempting. We tend to retreat and say, "We can draw a picture, we can post some graphs, we get it, we understand how this works." And we don't. We never do. Now I'm not trying to deliver a nihilistic message here. I'm not trying to say we can't solve complicated problems in a complicated world. We clearly can. But the way we solve them is with humility -- to abandon the God complex and to actually use a problem-solving technique that works. And we have a problem-solving technique that works. Now you show me a successful complex system, and I will show you a system that has evolved through trial and error.
這就是我們周遭複雜的世界 也許 這就是為何上帝情結誘人。 我們嘗試要撤退並聲明「我們可以畫圖, 我們可以張貼圖表, 我們全盤瞭解世界的遊戲規則。」 然而,事實並非如此。 我們的能力從未所及。 我並無要散播虛無主義的用意 也不是暗指我們無法解決 複雜世界裡的複雜問題。 顯然,我們能。 但是用的是 謙虛-- 放棄上帝情結 找出真正解決之道 我們的確擁有有效的解決之道 假設在座的各位 拿給我一個成功且複雜的系統 那我會還給你們 一個經過試驗和錯誤失敗中重生的系統
Here's an example. This baby was produced through trial and error. I realize that's an ambiguous statement. Maybe I should clarify it. This baby is a human body: it evolved. What is evolution? Over millions of years, variation and selection, variation and selection -- trial and error, trial and error. And it's not just biological systems that produce miracles through trial and error. You could use it in an industrial context.
舉個例子。 經過測試,這個寶寶出生了。 我的說法有點模稜兩可 也許我該修飾一下 這是進化過的人類嬰兒 何謂進化? 歷經數百萬年的 演進天擇, 測試和 從錯誤中修正。 這不僅是從試驗和錯誤中 產下的奇蹟。 若放到工業的框架下
So let's say you wanted to make detergent. Let's say you're Unilever and you want to make detergent in a factory near Liverpool. How do you do it? Well you have this great big tank full of liquid detergent. You pump it at a high pressure through a nozzle. You create a spray of detergent. Then the spray dries. It turns into powder. It falls to the floor. You scoop it up. You put it in cardboard boxes. You sell it at a supermarket. You make lots of money. How do you design that nozzle? It turns out to be very important. Now if you ascribe to the God complex, what you do is you find yourself a little God. You find yourself a mathematician; you find yourself a physicist -- somebody who understands the dynamics of this fluid. And he will, or she will, calculate the optimal design of the nozzle. Now Unilever did this and it didn't work -- too complicated. Even this problem, too complicated.
假設各位要製造清潔劑 設想我們是聯合利華股份有限公司 想在利物浦附近的工廠生產清潔劑 該如何下手? 目前有滿滿的一大桶清潔液 先利用高壓將清潔液從管口抽出。 然後把清潔液灑成霧狀 待噴霧乾凅,便轉成粉末。 落到地面。 用勺子取出,放置到紙盒。 拿到超市販售。 錢財滾滾來。 高壓管該如何設計? 這個很重要。 倘若當下,各位患了上帝情結的大頭症 你們可能誤以為,自己跟上帝也相去不遠。 自以為是數學家、物理學家, 或是理解液體動學的某人 然後就會 計算出最佳的噴管設計 不過,聯合利華也這麼做,卻沒效 因為太複雜了。 就連這樣的問題,都複雜。
But the geneticist Professor Steve Jones describes how Unilever actually did solve this problem -- trial and error, variation and selection. You take a nozzle and you create 10 random variations on the nozzle. You try out all 10; you keep the one that works best. You create 10 variations on that one. You try out all 10. You keep the one that works best. You try out 10 variations on that one. You see how this works, right? And after 45 generations, you have this incredible nozzle. It looks a bit like a chess piece -- functions absolutely brilliantly. We have no idea why it works, no idea at all. And the moment you step back from the God complex -- let's just try to have a bunch of stuff; let's have a systematic way of determining what's working and what's not -- you can solve your problem.
基因學者史帝芬.瓊斯 描述聯合利華是如何解決問題 經由試驗和錯誤, 變異和選擇。 選個管子 10種隨機變異製造的管子 10種都試試看,然後留下最好用的那只。 以那只為基礎,繼續做出10種。 全都測試。然後留下最棒的。 再從屏雀中選的那一只,繼續測試。 各位眼見為憑。 在45輪的測試後, 擁有了這只無敵的噴管 外型看似西洋棋 運作完美 我們對於為何它運作無礙 毫無頭緒 打破頭腦也想不出來 此時,各位回到上帝情結的大頭症 讓我們來測試這一堆 系統化精密計算過的管子 一樣可以解決問題
Now this process of trial and error is actually far more common in successful institutions than we care to recognize. And we've heard a lot about how economies function. The U.S. economy is still the world's greatest economy. How did it become the world's greatest economy? I could give you all kinds of facts and figures about the U.S. economy, but I think the most salient one is this: ten percent of American businesses disappear every year. That is a huge failure rate. It's far higher than the failure rate of, say, Americans. Ten percent of Americans don't disappear every year. Which leads us to conclude American businesses fail faster than Americans, and therefore American businesses are evolving faster than Americans. And eventually, they'll have evolved to such a high peak of perfection that they will make us all their pets -- (Laughter) if, of course, they haven't already done so. I sometimes wonder. But it's this process of trial and error that explains this great divergence, this incredible performance of Western economies. It didn't come because you put some incredibly smart person in charge. It's come through trial and error.
但是經由測試和錯誤 實際上成功率比較高, 高過我們所認為的機率。 我們時有所聞經濟如何運作的方式。 美國的經濟體仍是世界的龍頭。 如何才能成為世界最大的經濟體呢? 我可以提供各位五花八門的事實和數據 只要關於美國,應有盡有。 然而,個人所見,其中最重要的一點是; 每年, 美國有10%的商業實體消失。 這個數據意味著驚人的失敗率。 比起美國的人口數,高的太多。 美國人口不會每年遞減10%。 總結一句, 比起美國人,美國的企業太容易被淘汰, 因此美國的企業經營進化神速。 以致於,經營模式演化至完美極致 使我們都成為他們的搖擺狗。 (笑聲) 如果,他們還沒這麼做的話。 我不時在想 試驗和錯誤 解釋中間的巨大差異 西方經濟表現出色, 並非靠一群絕頂聰明之人掌權。 靠得是透過不斷試驗和從錯誤中學習。
Now I've been sort of banging on about this for the last couple of months, and people sometimes say to me, "Well Tim, it's kind of obvious. Obviously trial and error is very important. Obviously experimentation is very important. Now why are you just wandering around saying this obvious thing?"
過去的幾個月以來, 我反覆思量這個概念。 周遭的人們會對我說:「提姆, 道理顯而易見,不需一直反覆述說。 所有人都懂試驗和錯誤是非常重要。 每個人也都理解,實驗的精神很重要。 所以,為何要四處跑著宣傳這些常識呢?」
So I say, okay, fine. You think it's obvious? I will admit it's obvious when schools start teaching children that there are some problems that don't have a correct answer. Stop giving them lists of questions every single one of which has an answer. And there's an authority figure in the corner behind the teacher's desk who knows all the answers. And if you can't find the answers, you must be lazy or stupid. When schools stop doing that all the time, I will admit that, yes, it's obvious that trial and error is a good thing. When a politician stands up campaigning for elected office and says, "I want to fix our health system. I want to fix our education system. I have no idea how to do it. I have half a dozen ideas. We're going to test them out. They'll probably all fail. Then we'll test some other ideas out. We'll find some that work. We'll build on those. We'll get rid of the ones that don't." -- when a politician campaigns on that platform, and more importantly, when voters like you and me are willing to vote for that kind of politician, then I will admit that it is obvious that trial and error works, and that -- thank you.
於是,我回答: 「我懂你的意思。 你認為這是常識? 我承認道理很簡單, 當學校開始 教學生 有些問題並沒有所謂的正解時。 別再給他們一成串的問題, 而每個問題都有個固定答案。 更別提,權威的主流思想 暗藏在老師的桌下,這些自識萬能的老師。 學生如果找不到答案, 就會被視為天性懶惰或是天資駑鈍。 學校一旦停止這般的教育時, 我會承認, 試驗和錯誤會有所助益。 如果,當一名政客起身 替選情登高一呼: 「我承諾當選後,會整頓醫療系統。 還有教育組織。 不過我毫無頭續。 雖然腦子裡有半打的主意。 即便這些點子全都將陣亡,但是我們還是會一個一個測試。 然後,我們會繼續想出其他方法。 其中有些會有效。我們便可奠基於此。 剔除那些無效的方式。」 如同你我,選民聽到這般的牛肉, 從政客口中說出, 會願意投他一票。 接下來, 我願意承認,試驗和錯誤這個常識有效。感謝。
(Applause)
(掌聲)
Until then, until then I'm going to keep banging on about trial and error and why we should abandon the God complex. Because it's so hard to admit our own fallibility. It's so uncomfortable. And Archie Cochrane understood this as well as anybody. There's this one trial he ran many years after World War II. He wanted to test out the question of, where is it that patients should recover from heart attacks? Should they recover in a specialized cardiac unit in hospital, or should they recover at home? All the cardiac doctors tried to shut him down. They had the God complex in spades. They knew that their hospitals were the right place for patients, and they knew it was very unethical to run any kind of trial or experiment.
在那時以前, 我將會繼續傳播我的思想, 為何我們應禁絕上帝情結的大頭症。 因為,承認我們會失敗, 是件困難事。 還會令人渾身不對勁。 和各位一樣,亞契.科克倫體會到這點。 二次大戰結束的數年後, 他另外又做了個測試。 他想測出 心臟病發的患者 該在何處較能 從病痛中康復? 如果醫院安排他們在特別的心臟治療區,會康復嗎? 還是該讓他們在家靜養? 全部的心臟科醫生都不支持這項作法。 他們都有大頭症。 認為醫院一定是最佳場所。 他們也認為進行任何的測試 是有違人道。
Nevertheless, Archie managed to get permission to do this. He ran his trial. And after the trial had been running for a little while, he gathered together all his colleagues around his table, and he said, "Well, gentlemen, we have some preliminary results. They're not statistically significant. But we have something. And it turns out that you're right and I'm wrong. It is dangerous for patients to recover from heart attacks at home. They should be in hospital." And there's this uproar, and all the doctors start pounding the table and saying, "We always said you were unethical, Archie. You're killing people with your clinical trials. You need to shut it down now. Shut it down at once." And there's this huge hubbub. Archie lets it die down. And then he says, "Well that's very interesting, gentlemen, because when I gave you the table of results, I swapped the two columns around. It turns out your hospitals are killing people, and they should be at home. Would you like to close down the trial now, or should we wait until we have robust results?" Tumbleweed rolls through the meeting room.
然而,亞契排除萬難 進行他的實驗。 實驗進行沒多久 他集合所有的同僚 到他的桌前 宣佈:「各位, 一些初步的結果已經出來了, 雖然數據上並不顯著, 但是 結果是我錯了。 在家靜養 對病人風險太高。 他們該待在醫院。」 此時,一陣喧囂,所有的醫生都開始搥桌 怒罵:「我們老早就警告過你了,亞契。 你這麼做只是拿病人的性命開玩笑,你得立刻停止這一切。 馬上!」 接著又是一陣叫喊嘈雜。 亞契試著平息眾怒。 接著他對他們說:各位, 我會給你們一桌的報告, 交換兩個專欄的版面,因為結果饒富趣味, 你們的醫院事實上荼毒生靈, 病患其實該待在家中。 所以,各位想立刻停止實驗, 還是想靜待有力證據出爐呢? 風吹但草不動。 會議室一片靜默。
But Cochrane would do that kind of thing. And the reason he would do that kind of thing is because he understood it feels so much better to stand there and say, "Here in my own little world, I am a god, I understand everything. I do not want to have my opinions challenged. I do not want to have my conclusions tested." It feels so much more comfortable simply to lay down the law. Cochrane understood that uncertainty, that fallibility, that being challenged, they hurt. And you sometimes need to be shocked out of that. Now I'm not going to pretend that this is easy. It isn't easy. It's incredibly painful.
柯克倫會做這樣的事情 原因是 他瞭解 他會因此感覺良好 並且能站出來並宣佈 :「此地,我為王。 我是上帝,瞭解萬物, 不讓任何人挑戰我。 不讓任何人檢視我的研究結果。」 這種感覺無以倫比。 不過就是清楚地表明遊戲規則。 柯克倫懂 不確定性、錯誤性 會被挑戰,他們會失去江城。 有時,會震驚。 我不會假裝,這一切看似容易。 其實一點也不。 極其痛苦。
And since I started talking about this subject and researching this subject, I've been really haunted by something a Japanese mathematician said on the subject. So shortly after the war, this young man, Yutaka Taniyama, developed this amazing conjecture called the Taniyama-Shimura Conjecture. It turned out to be absolutely instrumental many decades later in proving Fermat's Last Theorem. In fact, it turns out it's equivalent to proving Fermat's Last Theorem. You prove one, you prove the other. But it was always a conjecture. Taniyama tried and tried and tried and he could never prove that it was true. And shortly before his 30th birthday in 1958, Yutaka Taniyama killed himself. His friend, Goro Shimura -- who worked on the mathematics with him -- many decades later, reflected on Taniyama's life. He said, "He was not a very careful person as a mathematician. He made a lot of mistakes. But he made mistakes in a good direction. I tried to emulate him, but I realized it is very difficult to make good mistakes."
自從我討論這個主題, 研究它之後, 有個不知名的東西一直縈繞著我, 是日本的一名數學家說的話 戰後不久後, 谷山豐(Yutaka Taniyama)那時還是個年輕人, 他發明出這個 稱為「谷山- 志村猜想」的理論。 日後,該猜想地位重要。 數十年後, 費馬大定理被證明。 事實上, 谷山-志村猜想和費馬大定理並駕齊驅。 證明一者,亦證出另一者。 但猜想永遠是猜想。 谷山試了又試, 從未能夠證明正確性。 1958年,在他30歲生日沒多久後, 谷山便自我了斷。 身兼同事與友人的志村五郎(Goro Shimura ) 和他一起研究數學, 在數十年後, 回憶起谷山的一生, 他說:「他不是個小心翼翼的數學家。 身為數學家 他犯了很多錯誤。 但都導向正確的道路。 我曾試著迎頭趕上, 但終究 我理解到,這極其困難。 特別是犯良性錯誤。」
Thank you.
謝謝各位的聆聽。
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