So it turns out that mathematics is a very powerful language. It has generated considerable insight in physics, in biology and economics, but not that much in the humanities and in history. I think there's a belief that it's just impossible, that you cannot quantify the doings of mankind, that you cannot measure history. But I don't think that's right. I want to show you a couple of examples why.
數學是十分強大的語言 可幫助人更深入探討物理學 生物學和經濟學 但對人文和歷史卻沒有太大幫助 我想大家都認為這是不可能的 因為我們無法量化人類的行為 也無法計量歷史 但對此,我抱持著不同的看法 我想舉幾個例子跟大家解釋原因
So my collaborator Erez and I were considering the following fact: that two kings separated by centuries will speak a very different language. That's a powerful historical force. So the king of England, Alfred the Great, will use a vocabulary and grammar that is quite different from the king of hip hop, Jay-Z. (Laughter) Now it's just the way it is. Language changes over time, and it's a powerful force.
我和合作夥伴 Erez 對於以下這件史實是這麼想的 相隔幾個世紀的兩位君王 會說截然不同的語言 這就是歷史的強大力量 因此古代英國國王艾佛烈大帝 使用的字彙和文法 跟當代嘻哈之王 Jay-Z 非常不一樣 (笑聲) 事實就是這樣 語言隨時間變化,而且還是一股強大的力量
So Erez and I wanted to know more about that. So we paid attention to a particular grammatical rule, past-tense conjugation. So you just add "ed" to a verb at the end to signify the past. "Today I walk. Yesterday I walked." But some verbs are irregular. "Yesterday I thought." Now what's interesting about that is irregular verbs between Alfred and Jay-Z have become more regular. Like the verb "to wed" that you see here has become regular.
Erez 和我想更深入了解這件事 我們因而注意到一條特別的文法規則:過去式動詞變化 也就是在動詞後加上「ed」來代表過去 今天我走 (walk),昨天我走了 (walked) 但是還有一些不規則動詞 昨天我想 (thought) 而這件事的有趣之處在於 從艾佛列大帝到 Jay-Z 的年代 不規則動詞是否變得更加規則? 如你所見,像動詞「結婚」就變得更加規則
So Erez and I followed the fate of over 100 irregular verbs through 12 centuries of English language, and we saw that there's actually a very simple mathematical pattern that captures this complex historical change, namely, if a verb is 100 times more frequent than another, it regularizes 10 times slower. That's a piece of history, but it comes in a mathematical wrapping.
因此 Erez 和我觀察了超過 100 個不規則動詞的演進變化 時間橫跨 12 世紀的英文 我們發現其中存在著一個很簡單的數學模式 足以描述這個複雜的歷史演變 當一動詞的常用程度比其他動詞高出 100 倍時 其規則化的速度就比其他動詞慢 10 倍 這就是一個可以用數學概括描述的歷史片段
Now in some cases math can even help explain, or propose explanations for, historical forces. So here Steve Pinker and I were considering the magnitude of wars during the last two centuries. There's actually a well-known regularity to them where the number of wars that are 100 times deadlier is 10 times smaller. So there are 30 wars that are about as deadly as the Six Days War, but there's only four wars that are 100 times deadlier -- like World War I. So what kind of historical mechanism can produce that? What's the origin of this?
而在其他狀況下,數學也有助於解釋歷史 或者為其提出假說 Steve Pinker 和我 思考兩個世紀以來戰爭規模的變化 其中其實存在著大家都很孰悉的規則性 死傷人數高達其它戰爭100 倍的戰爭數量 其實只有 10 分之 1 有 30 場戰爭其死亡人數與以阿六日戰爭相同 但死傷人數是其 100 倍的戰爭只有四場 例如第一次世界大戰 而這又是哪種歷史機制造成的呢? 其起因又是什麼?
So Steve and I, through mathematical analysis, propose that there's actually a very simple phenomenon at the root of this, which lies in our brains. This is a very well-known feature in which we perceive quantities in relative ways -- quantities like the intensity of light or the loudness of a sound. For instance, committing 10,000 soldiers to the next battle sounds like a lot. It's relatively enormous if you've already committed 1,000 soldiers previously. But it doesn't sound so much, it's not relatively enough, it won't make a difference if you've already committed 100,000 soldiers previously. So you see that because of the way we perceive quantities, as the war drags on, the number of soldiers committed to it and the casualties will increase not linearly -- like 10,000, 11,000, 12,000 -- but exponentially -- 10,000, later 20,000, later 40,000. And so that explains this pattern that we've seen before.
Steve 和我透過數學分析 提出了其實起因來自很簡單的現象 這種現象就存在大腦之中 是種大家都熟知的特性 也就是我們對「量」的觀感是相對的 如光線「強度」或音量「大小」 舉例來說,派 1 萬名士兵去打下一場仗 感覺好像很多 假使先前已派了 1 千名士兵的狀況下 感覺的確是如此 但是有時我們並不覺得人數有這麼多 因為「量」的對比不大,因此無法感受差異 假使先前其實已經派了 10 萬大軍 所以大家現在可以體會我們對「量」的觀感 隨著戰爭持續下去 派遣的士兵和死傷人數 將不會呈線性成長 趨勢不是 1 萬、1萬1、1 萬2 而會呈指數成長,從 1 萬到 2 萬到 4 萬 這解釋了我們先前看過的模式
So here mathematics is able to link a well-known feature of the individual mind with a long-term historical pattern that unfolds over centuries and across continents.
在此數學可以連結大眾熟知的思維特性 以及長期歷史模式 這種模式跨越幾個世紀橫跨幾大洲慢慢成形
So these types of examples, today there are just a few of them, but I think in the next decade they will become commonplace. The reason for that is that the historical record is becoming digitized at a very fast pace. So there's about 130 million books that have been written since the dawn of time. Companies like Google have digitized many of them -- above 20 million actually. And when the stuff of history is available in digital form, it makes it possible for a mathematical analysis to very quickly and very conveniently review trends in our history and our culture.
這種例子即便在今日仍然屈指可數 但我認為十年之後便將成為常態 原因是歷史記錄 以很快的速度數位化 有史以來 人類已經寫了13 億本書 Google 這類公司已經將其中的一大部分數位化 實際數量超過 2 億本 因此當歷史記錄被數位化後 就能拿來做數學分析 讓我們能很快又便利的 檢視歷史和文化趨勢
So I think in the next decade, the sciences and the humanities will come closer together to be able to answer deep questions about mankind. And I think that mathematics will be a very powerful language to do that. It will be able to reveal new trends in our history, sometimes to explain them, and maybe even in the future to predict what's going to happen.
因此我認為接下來十年 科學和人文將更緊密結合 而且能夠回答一些與人類相關的深層問題 我更認為數學這種強大語言將能做到此點 數學將能揭開歷史的新趨勢 有時加以解釋 最後甚至也有可能可以預測未來
Thank you very much.
謝謝各位
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