2019-07-31

妄人妄语

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The equations that evolved in the minds of geniuses are the sonnets of God. Newton’s law of universal gravitation was superseded by Einstein equations. Dirac equation was as prophetic as Maxwell–Heaviside equations.

	\(\displaystyle \huge F=G{\frac {m_{1}m_{2}}{r^{2}}}\)
	\(\displaystyle \huge R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu } = 8 \pi T_{\mu \nu } \)
	\( \displaystyle \huge \begin{align} \nabla \cdot \mathbf {E} &={\frac {\rho }{\varepsilon _{0}}} \\ \nabla \cdot \mathbf {B} &=0 \\ \nabla \times \mathbf {E} &=-{\frac {\partial \mathbf {B} }{\partial t}} \\ \nabla \times \mathbf {B} &=\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right) \end{align} \)
	\( \displaystyle \huge (i\hbar \gamma ^{\mu }\partial _{\mu }-mc)\psi =0 \)

Many men have had intuitions well ahead of their time … Gibbs rather than Einstein or Heisenberg or Planck to whom we must attribute the first great revolution of twentieth-century physics. (Norbert Wiener)

Science has no national borders. National boundaries are not evident when we view the Earth from space. (Carl Sagan)

There are two avenues by which opinions are received into the soul, which are its two principal powers: the understanding and the will. (Blaise Pascal)

Two things fill the mind with ever-increasing wonder and awe, the more often and the more intensely the mind of thought is drawn to them: the starry heavens above me and the moral law within me. — Immanuel Kant, Critique of Practical Reason (1788)

The world won’t care about your self-esteem. The world will expect you to accomplish something before you feel good about yourself. (Charlie Sykes)

whereas in thinking of God, a man must abstract Him from all substance, and not think of Him as a form of extension, but as an order of events. God was no Being, but pure Act — the principle of a transcendental world-order. — Heinrich Heine, Reisebilder

人念及上帝时，应该把祂抽离一切实在，而不将祂想象为占空间的形式，上帝不是存在，而是世易时移之力，一个形而上世界的秩序的原素。— 海涅，《游记》

For we are mistaken when we look forward to death; the major portion of death has already passed. Whatever years be behind us are in death’s hands. (Seneca the Younger)

Before you were born your parents weren’t as boring as they are now. They got that way paying bills, cleaning your room, and listening to you tell them how idealistic you are. … — Charles Sykes(author of DUMBING DOWN OUR KIDS)

Democracy is the worst form of government, except for all the others. ― Winston S. Churchill

Winston Churchill: The empires of the future are the empires of the mind.

Winston Churchill: The price of greatness is responsibility.

Winston Churchill: Where there is great power there is great responsibility.

责任体现伟大之价值（丘吉尔）

American Dream: that dream of a land in which life should be better and richer and fuller for everyone, with opportunity for each according to ability or achievement. - The Epic of America, James Truslow Adams

To die is only to be as we were before we were born; [on the fear of death] (William Hazlitt)

司马贺：实践和思考，是学生学到知识的途径，也是唯一的途径。

Anyone who is not shocked by quantum theory has not understood it. - Niels Bohr

Debugging sucks. Testing rocks!

A wealth of information creates a poverty of attention. — Herbert A. Simon

We are all of us more or less the slaves of opinion. - William Hazlitt

Newton was far from the crude thought that explanation of phenomena could be attained by abstraction. — Bernhard Riemann

The word hypothesis has now a somewhat different significance from that given it by Newton. We are now accustomed to understand by hypothesis all thoughts connected with the phenomena. — Bernhard Riemann

Natural science is the attempt to comprehend nature by precise concepts. — Bernhard Riemann

I have always depended on the kindness of strangers. — Blanche DuBois in play A Streetcar Named Desire

Nothing will work unless you do. — Maya Angelou

I started to realize that maybe Thomas Edison did a lot more to improve the world than Karl Marx and Neem Karoli Baba put together. — Steve Jobs

万般带不走，唯有业随身
Men are judged by what they do. A man must take the consequence of his own deeds.

There isn’t time, so brief is life, for bickerings, apologies, heartburnings, callings to account. There is only time for loving, and but an instant, so to speak, for that. — Mark Twain

Here is my secret. It is very simple: It is only with the heart that one can see rightly; what is essential is invisible to the eye.
But the eyes are blind. One must look with the heart…
— The Little Prince

I don’t know who my grandfather was; I am much more concerned to know what his grandson will be. - Abraham Lincoln

Aristotle: The totality is not, as it were, a mere heap, but the whole is something besides the parts. Euclid: The whole is greater than the part.

人的情绪有一种类似植物顶端优势的机制，一旦一种情绪盘踞心头，别的情绪在当下就被边缘化。调节情绪就像认识自我一样不容易。性格，它左右我这一生。情绪，它主宰我每一天。

居里夫人在给她女儿的信中写道：我的亲爱的女儿，愿你们每天都愉快地过着生活，不要等到日子过去了才找出它们的可爱之点，也不要把所有特别合意的希望都放在将来。

I know not with what weapons World War III will be fought, but World War IV will be fought with sticks and stones.
― Albert Einstein

第三次世界大戰將要使用的武器我並不知道，但是第四次世界大戰將會用木棍和石頭開戰。（愛因斯坦）

Internet的确产生了大众文化，但是我们不应该忽略，它也可能降低创造性，因为许多人在寻求信息，而自己却没有时间进行思考。 – Ilya Prigogine

前乌克兰党魁伽弗里洛夫是个业余数学家，20世纪60年代利用大权在握，设法让他给出的黎曼假设的错误证明发表出来，蓋爾范德为此差点心脏病发作，因为他认为这一黎曼假设错误证明的发表，对苏联的数学造成了很大的伤害。

‘It’s really hard to find maintainers…’ Linus Torvalds ponders the future of Linux

Science is the only news. — Stewart Brand

John Polkinghorne suggests that the mechanistic explanations of the world that have continued from Laplace to Richard Dawkins should be replaced by an understanding that most of nature is cloud-like rather than clock-like.

René Descartes is a philosopher who created analytic geometry, by contrast, Hegel is just a philosopher who was ignorant in mathematics, ‘Such things are nowhere more at home than among philosophers who are not mathematicians’,as Gauss put it. Schopenhauer is better than Hegel.

A Time for Leadership by Margaret Thatcher - China will not become a superpower to match the United States — at least, not while it is held back by the deadweight of socialism.

The tides caused by the moon are key to life forming on earth. As Maggie concluded “the moon is the mother of us all”

Your smartphone is making you stupid, antisocial and unhealthy. So why can’t you put it down?

In the fight against climate change, we pay now or our children pay later

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Our main business is not to see what lies dimly at a distance, but to do what lies clearly at hand.

今天的我是昨天的我选择的结果，明天的我将是今天的我选择的结果，我们就这样在一天天地选择着未来的自己，我的选择就是我的未来，我的习性就是我的选择。

大多数人，都是自己的热情和欲念的奴隶。

所谓的活着，不过是在消耗粮食污染环境。

豆瓣

“民科”和“民哲”不仅在思维、心理上高度相似，就像是同一个心理物种，而且甚至可以相互转化，成为相同的社会物种。

我是宇宙的一块拼图，体量上也许微不足道，但个性上可能独一无二。

伽罗瓦：扩域。黎曼：解析延拓。Cohen：力迫，扩张模型。朗兰兹：数学分支奇妙的内在联系

电视时代，沙发上的土豆。触屏时代，低头的屏奴。娱乐至死的时代。唯一能安慰我们之可悲的东西就是消遣，可是它也是我们可悲之中的最大的可悲。……若是没有它，我们就会陷于无聊，而这种无聊就会推动我们去寻找一种更牢靠的解脱办法了。可是消遣却使得我们开心，并使我们不知不觉地走到死亡。

乔治·华盛顿曾经说过：”如果被剥夺了言论自由，我们就会变得愚笨沉默，如绵羊般被拉向屠宰场。”

我们这些没本事的平庸之人，只能堕入“拿命换钱，拿钱续命”的有限循环螺旋式死亡之旅。

衰老是慢性病？

约西亚·威拉德·吉布斯：科学存在一种建设力，能从混沌中重建次序；科学存在一种分析力，能够区分真实与虚假；科学存在一种整合力，能看到一个真理而没哟忘记另一个真理；拥有这三种才能的，才是真正的科学家。天才敬天才，麦克斯韦和爱因斯坦都赞赏吉布斯。

Bitcoin has evolved to become better-suited to being an asset than being a means of exchange.

手游让我们娱乐至死

你的兴趣，是痴心迷恋的真兴趣？还是叶公好龙的假爱好？

Well-being and happiness never appeared to me as an absolute aim. I am even inclined to compare such moral aims to the ambitions of a pig.

希尔伯特：哥廷根随便找个学数学的小孩都比爱因斯坦更懂黎曼几何，但广义相对论只有爱因斯坦才能想得明白。
陈省身：我不久即看到爱因斯坦所遭遇的问题之极度困难与数学及物理学的不同。不知道他的物理如何？数学也就那样。丛和联络的概念都很简洁，爱因斯坦会喜欢的。

余英时：西方经典《赞愚》中就说，越是愚钝的人越有智慧，愚钝的人不是言词辩诘，而是对人生有某些很深的体验。越聪明的人越是糟糕，聪明等于是一个工具，就像科技一样… - 精神空虚的中国没有大师

许知远：利润，如今变成了连接这个国家的纽带，看看今天中国人信仰的庸俗哲学吧：一切都是利益驱动的，一切现象都可以通过利益来解释。 - 价值的真空

欧拉全集75卷，黎曼全集1卷，欧拉是高斯的祖父辈，欧拉比高斯大了整整七十岁，高斯是黎曼的伯父辈，高斯比黎曼大四十九岁。黎曼为了证明高斯关于素数密度可以用对数积分来估计的猜想，把欧拉乘积恒等式推广到复域，提出了zeta函数及其函数方程，为素数定理的证明理清了理路。

浮生若梦，人生几何？有首流行歌曲叫《光阴的故事》，有部情景喜剧叫《成长的烦恼》

耶稣会的教士都接受过良好的欧洲人文教育并经受过准军事训练，利玛窦、汤若望、南怀仁都是这一组织中出类拔萃的人才，他们身后有足够的支持，供他们以精密的仪器、华美的书籍、先进的技术赢得中国上流社会的青睐。

有时我觉得自己学习中文非常有益，陶冶性灵，但更多的时候，我觉得自己是中毒太深，不敢再把情绪放纵到诗与文里面，以免如梦如幻，贻误终身。

西奥多·罗斯福把这些喜欢揭露社会阴暗面的记者称为“扒粪者”（muckraker），但他肯定了这些揭露社会黑暗的记者们的价值，认为只有把丑恶暴露在阳光下，社会才可能改善。前后持续十几年的“扒粪运动”,有效地净化了美国的政治和商务生态，也给美国新闻行业建立了“调查”和“批评”的传统。

弗里曼·戴森认为博士制度令人讨厌, Freeman Dyson没有博士学位。
博士制度是在19世纪为了教育德国教授而发明的，它适合那些少数将要终身当教授的人，但现在却变成了为了工作而必须获得的某种会员卡，无论是成为教授还是从事其它职业。它迫使人们浪费许多年的时间和生命，去假装做一些他们其实完全不适合的研究。

民科的命操牛顿的心

工作撵跑三个魔鬼：无聊、堕落和贫穷。

浮生若梦，人生最大的痛苦是梦醒了发现自己无路可走。

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《道德经》中“修之于身，其德乃真”的本义，似乎不是在嘲笑那些伪君子言行不一“说一套、做一套”、仁义道德嘴把式，而是：尽管你认同那些更有利于使你“做最好的自己”的种种道理，你却不能够把它们身体力行，这些能够“德润身”的道理就只是些虚的道理，而坏习惯依旧像鬼魂一样形影不离地笼罩着你。

明-洪应明-菜根谭：迷则乐境成苦海，悟则苦海为乐境。John Milton, Paradise Lost: The mind is its own place, and in it self Can make a Heav’n of Hell, a Hell of Heav’n.

“饭吃七分饱”的富兰克林表达法：在我见到的人中，饿死的人寥寥无几，撑死的人倒有成千上万。 Eat not to Dulness. Drink not to Elevation.

相当一部分中国公民以经济为惟一生活目标。科名之心不可有，经济之才不可无。Money talks. In God we trust, the rest pay cash. Money changed hands.

有没有一个科学的价值观？什么样的价值观能让我不卑不亢？什么样的价值观能让我心平气和？什么样的价值观能让我泰然自若？内隐的价值观，挥之不去的内心指令，根深蒂固的核心假设，心理捷径累积成思维定势，生活的磨砺在内心深处的沉淀，价值的迷失是全盘的迷失，是注定要误入歧途、无可救药的迷失。

子不语怪、力、乱、神。知之为知之，不知为不知，是知也。子曰：未知生，焉知死？未能事人，焉能事鬼？敬鬼神而远之。～疑邻偷斧，疑心生暗鬼。～精气为物，游魂为变。～一念之善，吉神随之；一念之恶，厉鬼尾之。～暗示能作用于心理，人的知觉伴生错觉甚至幻觉，居心不良的人就有可能在其中搞鬼。

德裔美籍数学家 Hans Lewy（烈伟，1904～1988，R.柯朗（D.希尔伯特的学生）的学生）曾以数学家的口吻说道：“在许多方面分析（即逻辑严谨的微积分）比其它一些数学分支更接近于生活。”意思好像是说数学分析的问题不管是陈述、前提，还是结论、趋势都不是那么清晰明了，需要下功夫，如同生活的真面目。

E.T.BELL数学三清最得意的成就：① Αρχιμήδης：球的体积计算公式，被刻在其墓碑上和菲尔兹奖章背面；② Newton：用三棱镜发现光的色散，牛顿自称：“在我看来，如果那不是迄今对自然的演变所作的最重要的发现，也是最有趣的发现。” ③ Gauß：高斯-博内公式，高斯称其为Theorema Egregium。

19世纪并肩齐名的数学与物理的集大成者：黎曼（G.F.B.Riemann, 1826～1866），师承高斯、狄利克雷，黎曼几何、黎曼zeta函数等关键数学概念的首创者，堪称现代数学的摩西；麦克斯韦（James Clerk Maxwell，1831～1879），师承开尔文勋爵、斯托克斯爵士，巨著《电磁通论》指引人类步入电气时代的康庄大道。

德鲁克（管理学的Guru）针对知识社会的重要课题【自我管理】提了几个问题：1. 我的优势是什么？不要试图改变自己，这样做是不能成功的。相反，你需要用心改进做事的方式方法。不要采用你做不到或做得不好的方法来做事。2. 做事的方式方法？3. 价值观？4. 取舍？5. 我能贡献什么？6. 建设性的人际关系

科学精神希冀人类的理性能够逼近三位一体的神所走的无穷的道路，于是有了集合论中的悖论和量子力学中的佯谬。罗素理发师悖论：朴素集合论经不起推敲；巴拿赫-塔斯基悖论：选择公理会变魔术；波粒二象性佯谬：实相无相，微妙法门；测量佯谬：波包一测就塌缩。薛定谔的猫到底是活着还是死了？这很难说。

科学家的临终憾事：爱因斯坦：统一场论，规范场纲领已经形式上统一了电弱强这三种基本力，但引力很难纳入规范场纲领；海森堡：湍流，虽然以测不准原理而闻名于世，但海森堡的博士论文却是 -《关于流体运动的层流和湍流》；哈代：黎曼假设，我敢说没有一种微积分和伽罗瓦理论那样的新思想新方法，甭想证明RH。

美国很有创造性的微分几何学家 H.惠特尼对数学的研究与学习的方法论方面有相当独到的见解，他认为最好的学习方法是学以致用，没有自己的想法那不是真正的学习，他对研究过程的描述相当吻合认知心理学关于知识的心理表征、认知地图、图式、问题解决与创造性的观点。

在科学天才的头脑中进化着的科学方程，是造物主的十四行诗。牛顿以横绝一世的数学才能综合出让天文学家奉为圭皋的万有引力定律，爱因斯坦以最能体现理性思维张力的广义相对论引力场方程超越之；麦克斯韦与亥维赛德提炼出以对称之美著称的电磁场方程组，狄拉克以魔法般的相对论性量子电动力学方程组令人叹为观止。

《社会的管理》结尾写道：从机械的宇宙向生物的宇宙的转变，最终将需要一种新的哲学综合。康德也许会把它叫做Einsicht或叫做一种“纯粹洞察批判”。这是Peter F. Drucker的原话，还是选编者 Atsuo Ueda（笃男植田）添油加醋的话？

针对物理学家 Eugene Wigner（尤金·维格纳，1902—1995）“数学在物理学中不可思议的有效性”的说法，数学家 Israïl Moseevich Gelfand（I.M.盖尔范德，1913—2009）说了句饶舌的俏皮话：这世上惟一比“数学在物理学中不可思议的有效性”更加不可思议的是“数学在生物学中不可思议的无效性”。

科学精神的理念旨在寻求自洽、贯通、循序的理路，它的困难在于要借助最终展现为逻辑链路的思维去探究一个非线性的、无穷维自由度的、混沌的世界，它的稳妥、公信力与继往开来式发展使它永远存在着揭不开的谜团，它寻求的是“人同此心，心同此理”的道与理，而非“只可意会，不可言传”的感与悟。

只有科学的进步才像太阳的光辉一样普照芸芸众生。

工作是银，健康是金。1992年世界卫生组织在加拿大维多利亚召开的国际心脏健康会议上发表的关于保健的《维多利亚宣言》认为健康的四大基石是：1、均衡膳食，2、戒烟限酒，3、适量运动，4、心理平衡。

1976年愚人节，乔布斯和沃兹尼克、Ron Wayne一起创建了苹果公司，11天后Ron Wayne选择了退出，并把他名下如今价值220亿美元的10%股权以800美元卖给了乔布斯和沃兹尼克。有个CNN记者采访了与巨额财富失之交臂的Ron Wayne，Ron Wayne说道:”What can I say? You make a decision based on your understanding of the circumstances, and you live with it, But when you’re at a focal point of history, you don’t realize you’re at a focal point of history.”

2019-06-30

React Native Cheat Sheet

Accessing console logs

react-native log-ios
react-native log-android

xcrun simctl list devices

react-native run-ios –simulator “iPhone 8”
react-native run-android

Sometimes you may need to reset or clear the React Native packager’s cache. To do so, you can pass the –reset-cache flag to the start script:

npm start – –reset-cache
yarn start – –reset-cache

2019-06-07

人人都能用英语 - 李笑来

怎样才能学好英语？
一个字：“用”。

digested from © 人人都能用英语

不要再 “学” 英语，你就该 “用” 英语！

你英文发音再难听，听的人也不会因此猝死；你英文语法错误再多，读的人也不会因此疼痛；别人对你说英文你没听懂或者给你看英文你没读懂，若非极端情况，你也不会因此就从此真的无颜见人。
只 “学” 不 “用”（这是大多数中国学生的写照）的下场就是在十几年之后依然在学依然无用（这是大多数中国学生的现实）。
只有不断地 “用”，才能真正地 “学”，因为所有技能的习得，都要靠试错（Trial and Error）。很多人宁愿 “学一辈子”，却坚持 “一辈子不用” 的原因就在于害怕犯错。儿时犯错往往招致惩罚，成年之后，就算没有来自他人的惩罚，还有因为犯错而导致自己自卑和尴尬，所以，很多人是 “不惜一切代价” 避免出错的。然而，要知道知识的习得过程离不开试错，没有试错，就不可能有全面而真实的进步。所以，要知道犯错是正常的，甚至是不可或缺的。做事的时候，出错是必然的，如果正在做事却一点错都没有，那不是做事──那是在做梦。

我是如何摆脱哑巴英语的？

我以前总以为学生的问题是 “不知道怎么说”（HOW）──我曾以为我自己的问题也是如此；现在我认为学生所面临的问题不仅仅是这个，更重要的是他们 “没什么话可说”（WHAT）。
后来我就养成了习惯，不再指望自己能够脱口而出，而是希望自己有备无患。每当我遇到一些要讲英文的正式场合，我一定会提前花时间先写 “逐字稿”，不会写的就去查查词典，查词典查不到的，就用 Google 搜索，连 Google 都搜不到，那就想想看有没有可替代的说法 …… 而后再花时间修订，熟悉，复述。如此这般之后，到了现场，基本上能够做到 “自如” 应对。
请允许我重复一遍：你的问题也许不在于你不会说，而在于你没什么话可说。

其实连哑巴英语都并不那么坏

说得形象点，说不出像哑巴，听不懂像聋子，读不懂像瞎子；而写不出呢？——大多数文化并不默认一个人写不了字是一种残疾。先不管能否写得出，只谈哑巴、聋子和瞎子。如果你必须成为这三种 “残疾” 中的一种，你会选哪个？我认真想过，我会选 “聋子” —— 因为不瞎不哑，起码还保留了输入和输出的渠道。我最怕成为 “瞎子” —— 因为那是最重要的输入手段，尤其是因为文字输入几乎占所有有效输入的绝大部分。某种意义上来说，学习一门外语最大的公用莫过于增加一个不一样的信息获得渠道。从这一点上来看，哑巴比聋子、瞎子强多了，而后两者之中，瞎子又远不如聋子。

发音很重要，但显然不是最重要的

中国学生往往不是不会说英文，也不是不愿说英文，更不是不能说英文，基本上都是不敢说英文。为什么不敢呢？很多原因。其中有一个是最普遍的，害怕自己的发音不标准。可是第二语言习得者发音不准不是很正常的事情么？就算是母语，我们都是花了很长时间才可以做到基本上说清楚的。当你能够用母语清楚地表达自己的时候，多大了？那凭什么一个人可以从一开始就能用第二语言做到清楚准确表达呢？并且还要 “发音标准”？
全世界所有的语言都是如此，每种语言都有各种各样的口音。英语也许是地球上口音最多的语言。在美国，南加利福尼亚和北加利福尼亚的口音就已经非常不同，大抵上相当于在中国山东人之间山西人讲中文的差异。纽约人和底特律人的发音当然也非常不一样。在伦敦，东部和南部的口音差异就已经非常明显。更不消说还有 “苏格兰口音”、“加拿大口音”、“澳大利亚口音”、“新西兰口音”、“印度口音”……
为将英语作为第二语言使用的人，完全不必因为自己的发音不标准、不好听、不清楚感到自卑，那其实是正常的、自然的、不可避免的。而语言使用，本质上以沟通为目的。要知道仅仅发音标准，并不意味着说就肯定可以有效沟通。有效沟通还需要用词、文法、逻辑、内容等等更多因素，而后面提到的所有这些因素，无一不比 “标准发音” 更重要。想像一下吧，联合国开会的时候，难道每个国家的发言人都用的是 “标准英音”？或者 “标准美音”？尽管每个国家的发言人都要用英文发言，但全都用掺杂自己特定口音，可是从未影响有效沟通。

多听多听再多听

最有效的方法其实是零成本的 —— 大幅度提高听觉输入量。
我常常建议自己的学生不要把自己的输入材料只限制于 “标准美音” 或者 “标准英音”；其实无所谓的，连颇具特色的 “黑人英语” 都可以听，甚至，越杂越好。我常常推荐的是CNN的广播，里面有各种各样腔调的英语，真的可以大开 “耳” 界。
有一个小技巧，听英语音频的时候不要两只耳朵全都戴上耳机 —— 只用一只耳朵戴耳机。因为自然语音输入和耳机输入是不一样的。在自然环境中，我们听到的语言语音从来都不是 “单独” 的 —— 总是伴随着各种各样的背景声音。戴着耳机的时候却基本上就只有 “纯粹的语音” 了，这对我们重建自己的语音过滤器来说并不是好事。只用一只耳朵戴耳机的另外一个好处是可以经常换着耳朵听，不至于损伤耳朵。
至少要坚持六个月，我个人建议每天的输入时间不要低于四个小时 —— 只要开始做，就会发现其实并不难，因为 “哪怕听不懂都无所谓”。听得多了，听得久了，早晚有一天想听不懂都不太容易。当然，即便是最初的时候，为了效果更佳，可以有意识地渐渐提高文本难度，并且最好配合精读。这期间几乎所有的人都会感觉没什么进步，但是，这种 “感觉” 是不靠谱的 —— 事实上，我们的感觉几乎总是极不靠谱。

一定要学会音标

中国学生学习音标还有另外一个苦恼。我们的课本里大多所使用的是 D.J.音标（英音），但这并不是唯一的音标体系。除了 D.J.之外，有些地方的教材使用的是 K.K.音标（美音）；牛津词典和剑桥词典尽管都声称自己使用的是 IPA 国际音标，但多多少少各不相同；而有些学生在准备 SAT 或者 GRE 的时候，根据学长的建议开始使用 Merriam-Webster 词典，结果发现里面是彻头彻尾另外一个体系的音标 —— 事实上，几乎市面上所有的词典都在使用各不相同的音标体系。原本就不太好学的东西却又有那么多的版本 —— 当然更加令人气馁。

跟读训练具体步骤

拿来跟读材料之后，第一步是精读文本。不认识的词全部都要查过，然后确定该单词在当前句子中的确切含义，而后抄写在文本边上。当然，今天我们还有 MS Word，可以很方便地在文本上添加 “批注”。而在 Word 里，还有一个内建的词典，非常好用 …… 如果每个单词全都查过，却依然搞不懂句意，那么往往应该是有词组存在，再逐一查过。
查每一个生词的时候，都要记录重音和元音长度（必要时把整个音标写下来，也可以使用简化标记法）。
第二步是反复听录音，做自然语流修正标记。
第三步了。反复跟读。刚开始可以录音放一句，自己跟几遍，细心纠正自己的前提是大声朗读。熟悉了之后再录音放一句就跟一句，再熟悉一点之后就 “异步” 朗读。所谓的异步朗读，就是 “慢一拍跟读”。听到录音说了一个词之后我们再开始，嘴里重复的是录音里刚刚说完的那个词，而耳朵里同时听到的是自己的声音和录音里所说的下一个词，然后循环往复，在录音说完一句话的时候，我们再说一个词也就正好结束。这种训练可以很微妙地提高我们的英语瞬间记忆力。再熟悉到一定程度的时候，就可以 “同步” 朗读了。
第四步是录音矫正。每隔一段时间，可以把自己的朗读声音录下来存好，过上一个星期之后再翻出来听。很多人事倍功半的原因是录下来之后马上就去听，但这样的话，基本上没有什么矫正余地 —— 因为录音之后和录音之前的你还没有任何变化呢。当时你就觉得那么说是对的所以才那么说的，仅仅两三分钟之后，你不可能有什么巨大的或者哪怕是足够的进步；于是，没有变化的你，其实根本听不出自己哪儿不对了，也没有能力为自己进行矫正。但是，你一直在练，每天都在练，一个星期之后，你的进步就算不是巨大也是足够，于是你可能就会很容易地听出若干出过去出错的地方。

朗读

朗读是语文教育的最古老、最普及、成本最低、效果最好的训练方式。
说来可惜，但大多数人确实并不重视朗读训练 —— 无论是母语还是外语的习得过程中都是如此。朗读训练既简单又有效，并且可以解决很多许多人花很多钱去各种各样的培训班解决不了的问题。
朗读训练是提高文字理解能力的最有效方式。
朗读训练会潜移默化地提高阅读理解速度
不必专门练习听力，朗读就够了
大量的朗读训练，可以使学生不必专门练习 “听力”。某种意义上，很多学生花费时间去专门练习听力其实非常荒谬。不聋不哑的正常人是没必要专门训练什么听力的，事实上也没办法专门练 —— 大家的耳朵构造是相同的，怎么练耳廓也不会增大，耳膜也不会变得更薄……
其实道理很简单，只要说得出，就能听得懂 —— 不管是哪一种语言。所以，只需要练说，而没必要专门练听。很多人所谓的 “听力不好” 其实是说得不好造成的，然而，他们舍本求末，就是不说，而后专门练听，这不是荒唐是什么？事实上，哪怕说得不好，也一样能够听懂。举例来说，我国有很多地区的人普通话说得并不标准，讲话掺杂着浓重的本地口音，甚至使用大量的本地特有词汇，但是，你遇到过他们之中的哪一个向你抱怨说中央电视台的新闻联播听不懂么？
听写训练几乎是最浪费时间最无效果的所谓方法了。
朗读训练可以提高语言文字记忆能力
朗读训练能够提高表达能力
朗读训练可以提高语言文字模式识别能力

词典

查词典与朗读一样是提高阅读理解能力的最直接有效的手段之一。
英语老师与学生最大的不同可能只有一个，英语老师必须查词典（如果还有别的话，就是语法书之类的参考书），而学生却有除了查词典之外的另外一个选择 ——参加各种各样的课程。英语课上老师做什么呢？其实大抵上只不过是把昨天晚上他查过的单词、词组，以及他通过查词典（以及其它的工具书）再动脑才搞明白的句子给学生们讲一遍。而学生呢？做在下面听。学生们倒是听了，然而，本质上却没有参与阅读理解的过程 —— 那个过程里应该有苦恼、迷失、无助、慌张（人人都讨厌这些）和恍然大悟（人人都想只要这个）；所以大多学生根本就没有动脑，于是顶多是以为自己搞懂了（事实上，没有之前的两个境界的铺垫，“蓦然回首” 根本看不见 “那人站在灯火阑珊处” 的。）
查词典并不难。但多一点点细心和耐心，却并不像想象得那么容易。
中国学生为什么总是疏于去查 “phrasal verb dictionary” 或者 “dictionary of idioms”
关于 Merriam-Webster 的权威性，基本上不容置疑。Merriam-Webster 的电子版，目前在网上可以找得到的有两种版本：2.5 版和 3.0 版。我个人认为 3.0 版并没有什么真正有意义的改进。2.5 版反倒相对更好用一些。
Merriam-Webster 电子版的 “真人发音”，是我个人认为目前可以找到的所有电子版词典中制作最为精良的 —— 发音最清晰（准确当然不用提），音量最稳定。
最后，Merriam-Webster 最牛的地方在于它有一个 “Spelling Help”。查找英文单词的时候，一个常见的窘境是，我们只知道某个单词的发音却不知道拼写，于是无法查到那个单词。然而，有了 “Spelling Help”，就非常方便了。
Collins Cobuild – Lexicon on CD-ROM我所推荐使用的是这部词典电子版的第三版，而非最新的第五版。

把 Word 打造成英语学习利器

对中国学生来说，MS Word 不仅仅是 “字处理工具”，更是一个非常强大的学习利器。
MS Word 2007 的 “鼠标取词” 功能
MS Word 2007 的 “词典面板”
MS Word 2007 的 “同近义辞典”（Thesaurus）
MS Word 2007 的 “英语助手”
为 MS Word 2007 设置单词朗读功能
用 Word 2007 为自己定制阅读文章词汇列表

2019-05-09

LeetCode - Algorithms - 215. Kth Largest Element in an Array

It seemed that every leetcode algorithms problem has mulple solutions, so do this one. I just verified solutions of other peer.

Java

Quickselect

class Solution {
	private void exch(int[] a, int i, int j) {
		int swap = a[i];
		a[i] = a[j];
		a[j] = swap;
	}
    
	private int partition(int[] a, int lo, int hi) {
		int pivot = lo + new Random().nextInt(hi - lo + 1);
		exch(a, hi, pivot);
		for (int i = lo; i < hi; i++) {
			if (a[i] > a[hi]) {
				exch(a, i, lo);
				lo++;
			}
		}
		exch(a, lo, hi);
		return lo;
	}
    
	private int quickselect(int[] a, int left, int right, int k) {
		if (left <= right) {
			int pivot = partition(a, left, right);
			if (pivot == k) {
				return a[k];
			}
			if (pivot > k) {
				return quickselect(a, left, pivot - 1, k);
			}
			return quickselect(a, pivot + 1, right, k);
		}
		return Integer.MIN_VALUE;
	}    
    
    public int findKthLargest(int[] nums, int k) {
        return quickselect(nums, 0, nums.length - 1, k - 1);
    }
}

Submission Detail

32 / 32 test cases passed.
Runtime: 1 ms, faster than 98.04% of Java online submissions for Kth Largest Element in an Array.
Memory Usage: 39.3 MB, less than 11.22% of Java online submissions for Kth Largest Element in an Array.

Solution 1: sorting

import java.util.Arrays;

class Solution {
    public int findKthLargest(int[] nums, int k) {
		Arrays.sort(nums);
		return nums[nums.length - k];        
    }
}

Submission Detail

32 / 32 test cases passed.
Runtime: 3 ms, faster than 84.58% of Java online submissions for Kth Largest Element in an Array.
Memory Usage: 35.7 MB, less than 98.57% of Java online submissions for Kth Largest Element in an Array.

Solution 2: min heap

import java.util.PriorityQueue;

class Solution {
    public int findKthLargest(int[] nums, int k) {
    	PriorityQueue<Integer> minHeap = new PriorityQueue<Integer>(k);
    	for(int i : nums) {
    		minHeap.offer(i);
    		if (minHeap.size()>k)
    			minHeap.poll();
    	}
    	return minHeap.peek();       
    }
}

Submission Detail

32 / 32 test cases passed.
Runtime: 8 ms, faster than 51.68% of Java online submissions for Kth Largest Element in an Array.
Memory Usage: 35.7 MB, less than 98.37% of Java online submissions for Kth Largest Element in an Array.

Solution 3: a similar method like quick sort

class Solution {
    public int findKthLargest(int[] nums, int k) {
		if (k < 1 || nums == null) {
			return 0;
		}
		return getKth(nums.length - k + 1, nums, 0, nums.length - 1);       
    }

	private int getKth(int k, int[] nums, int start, int end) {
		int pivot = nums[end];
		int left = start;
		int right = end;
		while (true) {
			while (nums[left] < pivot && left < right) {
				left++;
			}
			while (nums[right] >= pivot && right > left) {
				right--;
			}
			if (left == right) {
				break;
			}
			swap(nums, left, right);
		}
		swap(nums, left, end);
		if (k == left + 1) {
			return pivot;
		} else if (k < left + 1) {
			return getKth(k, nums, start, left - 1);
		} else {
			return getKth(k, nums, left + 1, end);
		}
	}

	private void swap(int[] arr, int i, int j) {
		int temp = arr[i];
		arr[i] = arr[j];
		arr[j] = temp;
	}
}

Submission Detail

32 / 32 test cases passed.
Runtime: 22 ms, faster than 37.24% of Java online submissions for Kth Largest Element in an Array.
Memory Usage: 37.5 MB, less than 94.43% of Java online submissions for Kth Largest Element in an Array.

LeetCode – Kth Largest Element in an Array (Java)
QuickSelect: The QuickSelect algorithm quickly finds the k-th smallest element of an unsorted array of n elements. It is a RandomizedAlgorithm, so we compute the worst-case expected running time.
Quickselect

2019-05-04

The Story of Maths - 4. To Infinity and Beyond - Subtitles

texts below are from © https://subsaga.com/bbc/documentaries/science/the-story-of-maths/4-to-infinity-and-beyond.html

1
00:00:17,280 –> 00:00:21,400
Mathematics is about solving problems

2
00:00:21,400 –> 00:00:26,240
and it’s the great unsolved problems that make maths really alive.

3
00:00:28,240 –> 00:00:29,440
In the summer of 1900,

4
00:00:29,440 –> 00:00:32,160
the International Congress of Mathematicians

5
00:00:32,160 –> 00:00:34,440
was held here in Paris in the Sorbonne.

6
00:00:34,440 –> 00:00:36,680
It was a pretty shambolic affair,

7
00:00:36,680 –> 00:00:39,280
not helped by the sultry August heat.

8
00:00:39,280 –> 00:00:43,000
But it will be remembered as one of the greatest congresses of all time

9
00:00:43,000 –> 00:00:47,360
thanks to a lecture given by the up-and-coming David Hilbert.

10
00:00:48,880 –> 00:00:51,600
Hilbert, a young German mathematician,

11
00:00:51,600 –> 00:00:56,360
boldly set out what he believed were the 23 most important problems

12
00:00:56,360 –> 00:00:58,560
for mathematicians to crack.

13
00:00:58,560 –> 00:01:04,200
He was trying to set the agenda for 20th-century maths and he succeeded.

14
00:01:04,200 –> 00:01:09,480
These Hilbert problems would define the mathematics of the modern age.

15
00:01:09,480 –> 00:01:15,360
Of those who tried to crack Hilbert’s challenges, some would experience immense triumphs,

16
00:01:15,360 –> 00:01:18,920
whilst others would be plunged into infinite despair.

17
00:01:30,120 –> 00:01:33,120
The first problem on Hilbert’s list emerged from here,

18
00:01:33,120 –> 00:01:36,320
Halle, in East Germany.

19
00:01:36,320 –> 00:01:41,200
It was where the great mathematician Georg Cantor spent all his adult life.

20
00:01:41,200 –> 00:01:45,280
And where he became the first person to really understand the meaning

21
00:01:45,280 –> 00:01:50,160
of infinity and give it mathematical precision.

22
00:01:50,160 –> 00:01:52,480
The statue in the town square, however,

23
00:01:52,480 –> 00:01:57,360
honours Halle’s other famous son, the composer George Handel.

24
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To discover more about Cantor, I had to take a tram way out of town.

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For 50 years, Halle was part of Communist East Germany

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and the Communists loved celebrating their scientists.

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So much so, they put Cantor on the side of a large cube that they commissioned.

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But, being communists, they didn’t put the cube

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in the middle of town. They put it out amongst the people.

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When I eventually found the estate, I started to fear

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that maybe I had got the location wrong.

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This looks the most unlikely venue for a statue to a mathematician.

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Excuse me?

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00:02:42,840 –> 00:02:43,960
Ein Frage.

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00:02:43,960 –> 00:02:47,560

Can you help me a minute?
Wie bitte?
Do you speak English?
No!
No?

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Ich suche ein Wurfel.

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Ein Wurfel, ja?

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00:02:51,600 –> 00:02:52,880
Is that right? A “Wurfel”?

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00:02:52,880 –> 00:02:55,160
A cube? Yeah? Like that?

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00:02:55,160 –> 00:02:58,680
Mit ein Bild der Mathematiker?

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00:02:58,680 –> 00:03:01,160
Yeah? Go round there?

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Die Name ist Cantor.

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Somewhere over here. Ah! There it is!

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It’s much bigger than I thought.

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I thought it was going to be something like this sort of size.

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Aha, here we are. On the dark side of the cube.

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here’s the man himself, Cantor.

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Cantor’s one of my big heroes actually.

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00:03:18,880 –> 00:03:23,040
I think if I had to choose some theorems that I wish I’d proved,

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I think the couple that Cantor proved

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would be up there in my top ten.

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‘This is because before Cantor,

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00:03:30,200 –> 00:03:33,200
‘no-one had really understood infinity.’

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It was a tricky, slippery concept that didn’t seem to go anywhere.

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But Cantor showed that infinity could be perfectly understandable.

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Indeed, there wasn’t just one infinity,

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but infinitely many infinities.

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First Cantor took the numbers 1, 2, 3, 4 and so on.

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Then he thought about comparing them with a much smaller set…

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something like 10, 20, 30, 40…

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What he showed is that these two infinite sets of numbers

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actually have the same size because we can pair them up -

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1 with 10, 2 with 20, 3 with 30 and so on.

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So these are the same sizes of infinity.

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But what about the fractions?

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After all, there are infinitely many fractions between any two whole numbers.

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Surely the infinity of fractions is much bigger

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than the infinity of whole numbers.

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Well, what Cantor did was to find a way to pair up

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all of the whole numbers with an infinite load of fractions.

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And this is how he did it.

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He started by arranging all the fractions in an infinite grid.

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The first row contained the whole numbers, fractions with one on the bottom.

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In the second row came the halves, fractions with two on the bottom.

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And so on. Every fraction appears somewhere in this grid.

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Where’s two thirds? Second column, third row.

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Now imagine a line snaking back and forward diagonally through the fractions.

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By pulling this line straight, we can match up every fraction with one of the whole numbers.

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This means the fractions are the same sort of infinity

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as the whole numbers.

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So perhaps all infinities have the same size.

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Well, here comes the really exciting bit

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because Cantor now considers the set of all infinite decimal numbers.

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And here he proves that they give us a bigger infinity because

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however you tried to list all the infinite decimals, Cantor produced

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a clever argument to show how to construct a new decimal number

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that was missing from your list.

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Suddenly, the idea of infinity opens up.

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There are different infinities, some bigger than others.

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It’s a really exciting moment.

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For me, this is like the first humans understanding how to count.

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But now we’re counting in a different way. We are counting infinities.

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A door has opened and an entirely new mathematics lay before us.

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00:06:19,320 –> 00:06:21,480
But it never helped Cantor much.

95
00:06:21,480 –> 00:06:25,240
I was in the cemetery in Halle where he is buried

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and where I had arranged to meet Professor Joe Dauben.

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He was keen to make the connections between Cantor’s maths and his life.

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00:06:33,720 –> 00:06:36,280
He suffered from manic depression.

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One of the first big breakdowns he has is in 1884

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but then around the turn of the century

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these recurrences of the mental illness

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become more and more frequent.

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A lot of people have tried to say that his mental illness

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was triggered by the incredible abstract mathematics he dealt with.

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Well, he was certainly struggling, so there may have been a connection.

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00:06:57,280 –> 00:07:01,920
Yeah, I mean I must say, when you start to contemplate the infinite…

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I am pretty happy with the bottom end of the infinite,

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00:07:05,080 –> 00:07:07,240
but as you build it up more and more,

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I must say I start to feel a bit unnerved

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about what’s going on here and where is it going.

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For much of Cantor’s life, the only place it was going was here -

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the university’s sanatorium.

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There was no treatment then for manic depression

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or indeed for the paranoia that often accompanied Cantor’s attacks.

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Yet the clinic was a good place to be -

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comfortable, quiet and peaceful.

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And Cantor often found his time here gave him the mental strength

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to resume his exploration of the infinite.

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Other mathematicians would be bothered by the paradoxes

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that Cantor’s work had created.

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Curiously, this was one thing Cantor was not worried by.

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He was never as upset about the paradox of the infinite

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as everybody else was because Cantor believed that

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there are certain things that I have been able to show,

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we can establish with complete mathematical certainty

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and then the absolute infinite which is only in God.

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He can understand all of this and there’s still that final paradox

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that is not given to us to understand, but God does.

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But there was one problem that Cantor couldn’t leave

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in the hands of the Almighty,

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a problem he wrestled with for the rest of his life.

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It became known as the continuum hypothesis.

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Is there an infinity sitting between the smaller infinity

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of all the whole numbers and the larger infinity of the decimals?

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Cantor’s work didn’t go down well with many of his contemporaries

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but there was one mathematician from France who spoke up for him,

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arguing that Cantor’s new mathematics of infinity

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was “beautiful, if pathological”.

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Fortunately this mathematician was the most famous and respected mathematician of his day.

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When Bertrand Russell was asked by a French politician who he thought

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the greatest man France had produced, he replied without hesitation, “Poincare”.

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The politician was surprised that he’d chosen

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the prime minister Raymond Poincare above the likes of Napoleon, Balzac.

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Russell replied, “I don’t mean Raymond Poincare but his cousin,

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“the mathematician, Henri Poincare.”

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Henri Poincare spent most of his life in Paris,

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a city that he loved even with its uncertain climate.

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In the last decades of the 19th century, Paris was a centre

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for world mathematics and Poincare became its leading light.

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Algebra, geometry, analysis, he was good at everything.

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His work would lead to all kinds of applications,

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from finding your way around on the underground,

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to new ways of predicting the weather.

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Poincare was very strict about his working day.

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Two hours of work in the morning

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and two hours in the early evening.

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Between these periods,

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he would let his subconscious carry on working on the problem.

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He records one moment when he had a flash of inspiration which occurred

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almost out of nowhere, just as he was getting on a bus.

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And one such flash of inspiration led to an early success.

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In 1885, King Oscar II of Sweden and Norway

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offered a prize of 2,500 crowns for anyone who could establish mathematically once and for all

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whether the solar system would continue turning like clockwork,

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or might suddenly fly apart.

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If the solar system has two planets then Newton had already proved that their orbits would be stable.

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The two bodies just travel in ellipsis round each other.

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But as soon as soon as you add three bodies like the earth, moon and sun,

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the question of whether their orbits were stable or not stumped even the great Newton.

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The problem is that now you have some 18 different variables,

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like the exact coordinates of each body

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and their velocity in each direction.

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So the equations become very difficult to solve.

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But Poincare made significant headway in sorting them out.

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Poincare simplified the problem by making successive approximations to the orbits which he believed

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wouldn’t affect the final outcome significantly.

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Although he couldn’t solve the problem in its entirety,

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his ideas were sophisticated enough to win him the prize.

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He developed this great sort of arsenal of techniques,

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mathematical techniques

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in order to try and solve it

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and in fact, the prize that he won was essentially

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more for the techniques than for solving the problem.

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But when Poincare’s paper was being prepared for publication

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by the King’s scientific advisor, Mittag-Leffler,

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one of the editors found a problem.

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Poincare realised he’d made a mistake.

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Contrary to what he had originally thought, even a small change in the

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initial conditions could end up producing vastly different orbits.

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His simplification just didn’t work.

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But the result was even more important.

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The orbits Poincare had discovered indirectly led to what we now know as chaos theory.

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Understanding the mathematical rules of chaos explain why a butterfly’s wings

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could create tiny changes in the atmosphere

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that ultimately might cause

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a tornado or a hurricane to appear on the other side of the world.

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So this big subject of the 20th century, chaos,

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actually came out of a mistake that Poincare made

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and he spotted at the last minute.

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Yes! So the essay had actually been published in its original form,

201
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and was ready to go out and Mittag-Leffler had sent copies out to various people,

202
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and it was to his horror when Poincare wrote to him to say, “Stop!”

203
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Oh, my God. This is every mathematician’s worst nightmare.

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Absolutely. “Pull it!”

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Hold the presses!

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Owning up to his mistake, if anything,

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enhanced Poincare’s reputation.

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00:13:12,920 –> 00:13:15,800
He continued to produce a wide range of original work

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throughout his life.

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00:13:16,960 –> 00:13:20,000
Not just specialist stuff either.

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He also wrote popular books, extolling the importance of maths.

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00:13:24,480 –> 00:13:28,560
Here we go. Here’s a section on the future of mathematics.

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00:13:30,080 –> 00:13:34,240
It starts, “If we wish to foresee the future of mathematics,

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“our proper course is to study the history and present the condition of the science.”

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So, I think Poincare might have approved of my journey to uncover the story of maths.

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He certainly would have approved of the next destination.

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Because to discover perhaps Poincare’s most important contribution to modern mathematics,

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I had to go looking for a bridge.

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Seven bridges in fact.

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The Seven bridges of Konigsberg.

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Today the city is known as Kaliningrad, a little outpost

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of Russia on the Baltic Sea surrounded by Poland and Lithuania.

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Until 1945, however, when it was ceded to the Soviet Union,

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it was the great Prussian City of Konigsberg.

225
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Much of the old town sadly has been demolished.

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There is now no sign at all of two of the original seven bridges

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and several have changed out of all recognition.

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This is one of the original bridges.

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It may seem like an unlikely setting for the beginning of a mathematical story, but bear with me.

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It started as an 18th-century puzzle.

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Is there a route around the city which crosses each of these seven bridges only once?

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Finding the solution is much more difficult than it looks.

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It was eventually solved by the great mathematician Leonhard Euler,

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who in 1735 proved that it wasn’t possible.

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There could not be a route that didn’t cross at least one bridge twice.

236
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He solved the problem by making a conceptual leap.

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He realised, you don’t really care what the distances are between the bridges.

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What really matters is how the bridges are connected together.

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This is a problem of a new sort of geometry of position - a problem of topology.

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Many of us use topology every day.

241
00:15:40,960 –> 00:15:43,440
Virtually all metro maps the world over

242
00:15:43,440 –> 00:15:46,040
are drawn on topological principles.

243
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You don’t care how far the stations are from each other

244
00:15:49,400 –> 00:15:51,200
but how they are connected.

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There isn’t a metro in Kaliningrad,

246
00:15:53,840 –> 00:15:58,560
but there is in the nearest other Russian city, St Petersburg.

247
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The topology is pretty easy on this map.

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It’s the Russian I am having difficulty with.

249
00:16:03,200 –> 00:16:06,360

Can you tell me which…?
What’s the problem?

250
00:16:06,360 –> 00:16:09,760
I want to know what station this one was.

251
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I had it the wrong way round even!

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00:16:14,640 –> 00:16:18,280
Although topology had its origins in the bridges of Konigsberg,

253
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it was in the hands of Poincare that the subject evolved

254
00:16:22,520 –> 00:16:26,120
into a powerful new way of looking at shape.

255
00:16:26,120 –> 00:16:29,840
Some people refer to topology as bendy geometry

256
00:16:29,840 –> 00:16:34,640
because in topology, two shapes are the same if you can bend or morph

257
00:16:34,640 –> 00:16:37,240
one into another without cutting it.

258
00:16:37,240 –> 00:16:42,360
So for example if I take a football or rugby ball, topologically they

259
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are the same because one can be morphed into the other.

260
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Similarly a bagel and a tea-cup are the same because one can be morphed into the other.

261
00:16:51,960 –> 00:16:58,720
Even very complicated shapes can be unwrapped to become much simpler from a topological point of view.

262
00:16:58,720 –> 00:17:02,760
But there is no way to continuously deform a bagel to morph it into a ball.

263
00:17:02,760 –> 00:17:06,560
The hole in the middle makes these shapes topologically different.

264
00:17:06,560 –> 00:17:11,800
Poincare knew all the possible two-dimensional topological surfaces.

265
00:17:11,800 –> 00:17:15,560
But in 1904 he came up with a topological problem

266
00:17:15,560 –> 00:17:17,480
he just couldn’t solve.

267
00:17:17,480 –> 00:17:21,320
If you’ve got a flat two-dimensional universe then Poincare worked out

268
00:17:21,320 –> 00:17:24,520
all the possible shapes he could wrap that universe up into.

269
00:17:24,520 –> 00:17:29,600
It could be a ball or a bagel with one hole, two holes or more holes in.

270
00:17:29,600 –> 00:17:35,200
But we live in a three-dimensional universe so what are the possible shapes that our universe can be?

271
00:17:35,200 –> 00:17:39,240
That question became known as the Poincare Conjecture.

272
00:17:39,240 –> 00:17:43,960
It was finally solved in 2002 here in St Petersburg

273
00:17:43,960 –> 00:17:47,560
by a Russian mathematician called Grisha Perelman.

274
00:17:47,560 –> 00:17:51,240
His proof is very difficult to understand, even for mathematicians.

275
00:17:51,240 –> 00:17:57,960
Perelman solved the problem by linking it to a completely different area of mathematics.

276
00:17:57,960 –> 00:18:03,800
To understand the shapes, he looked instead at the dynamics of the way things can flow over the shape

277
00:18:03,800 –> 00:18:06,880
which led to a description of all the possible ways

278
00:18:06,880 –> 00:18:11,320
that three dimensional space can be wrapped up in higher dimensions.

279
00:18:11,320 –> 00:18:16,000
I wondered if the man himself could help in unravelling the intricacies of his proof,

280
00:18:16,000 –> 00:18:23,400
but I’d been told that finding Perelman is almost as difficult as understanding the solution.

281
00:18:23,400 –> 00:18:26,040
The classic stereotype of a mathematician

282
00:18:26,040 –> 00:18:29,800
is a mad eccentric scientist, but I think that’s a little bit unfair.

283
00:18:29,800 –> 00:18:33,040
Most of my colleagues are normal. Well, reasonably.

284
00:18:33,040 –> 00:18:35,120
But when it comes to Perelman,

285
00:18:35,120 –> 00:18:37,920
there is no doubt he is a very strange character.

286
00:18:37,920 –> 00:18:40,880
He’s received prizes and offers of professorships

287
00:18:40,880 –> 00:18:43,560
from distinguished universities in the West

288
00:18:43,560 –> 00:18:46,280
but he’s turned them all down.

289
00:18:46,280 –> 00:18:49,800
Recently he seems to have given up mathematics completely

290
00:18:49,800 –> 00:18:52,000
and retreated to live as a semi-recluse

291
00:18:52,000 –> 00:18:54,720
in this very modest housing estate with his mum.

292
00:18:54,720 –> 00:19:01,320
He refuses to talk to the media but I thought he might just talk to me as a fellow mathematician.

293
00:19:01,320 –> 00:19:03,480
I was wrong.

294
00:19:03,480 –> 00:19:07,320
Well, it’s interesting. I think he’s actually turned off his buzzer.

295
00:19:07,320 –> 00:19:09,600
Probably too many media have been buzzing it.

296
00:19:09,600 –> 00:19:12,920
I tried a neighbour and that rang but his doesn’t ring at all.

297
00:19:12,920 –> 00:19:18,560
I think his papers, his mathematics speaks for itself in a way.

298
00:19:18,560 –> 00:19:21,080
I don’t really need to meet the mathematician

299
00:19:21,080 –> 00:19:23,560
and in this age of Big Brother and Big Money,

300
00:19:23,560 –> 00:19:26,840
I think there’s something noble about the fact he gets his kick

301
00:19:26,840 –> 00:19:29,520
out of proving theorems and not winning prizes.

302
00:19:32,960 –> 00:19:36,000
One mathematician would certainly have applauded.

303
00:19:36,000 –> 00:19:40,440
For solving any of his 23 problems, David Hilbert offered no prize

304
00:19:40,440 –> 00:19:45,760
or reward beyond the admiration of other mathematicians.

305
00:19:45,760 –> 00:19:49,360
When he sketched out the problems in Paris in 1900,

306
00:19:49,360 –> 00:19:52,360
Hilbert himself was already a mathematical star.

307
00:19:52,360 –> 00:19:56,320
And it was in Gottingen in northern Germany that he really shone.

308
00:19:59,440 –> 00:20:05,560
He was by far the most charismatic mathematician of his age.

309
00:20:05,560 –> 00:20:09,960
It’s clear that everyone who knew him thought he was absolutely wonderful.

310
00:20:12,880 –> 00:20:17,600
He studied number theory and brought everything together that was there

311
00:20:17,600 –> 00:20:20,760
and then within a year or so he left that completely

312
00:20:20,760 –> 00:20:24,320
and revolutionised the theory of integral equation.

313
00:20:24,320 –> 00:20:26,880
It’s always change and always something new,

314
00:20:26,880 –> 00:20:29,840
and there’s hardly anybody who is…

315
00:20:29,840 –> 00:20:34,800
who was so flexible and so varied in his approach as Hilbert was.

316
00:20:34,800 –> 00:20:41,800
His work is still talked about today and his name has become attached to many mathematical terms.

317
00:20:41,800 –> 00:20:46,160
Mathematicians still use the Hilbert Space, the Hilbert Classification,

318
00:20:46,160 –> 00:20:51,120
the Hilbert Inequality and several Hilbert theorems.

319
00:20:51,120 –> 00:20:54,800
But it was his early work on equations that marked him out

320
00:20:54,800 –> 00:20:57,520
as a mathematician thinking in new ways.

321
00:20:57,520 –> 00:21:01,480
Hilbert showed that although there are infinitely many equations,

322
00:21:01,480 –> 00:21:04,800
there are ways to divide them up so that they are built

323
00:21:04,800 –> 00:21:08,160
out of just a finite set, like a set of building blocks.

324
00:21:08,160 –> 00:21:13,880
The most striking element of Hilbert’s proof was that he couldn’t actually construct this finite set.

325
00:21:13,880 –> 00:21:17,440
He just proved it must exist.

326
00:21:17,440 –> 00:21:20,760
Somebody criticised this as theology and not mathematics

327
00:21:20,760 –> 00:21:22,400
but they’d missed the point.

328
00:21:22,400 –> 00:21:26,280
What Hilbert was doing here was creating a new style of mathematics,

329
00:21:26,280 –> 00:21:28,840
a more abstract approach to the subject.

330
00:21:28,840 –> 00:21:31,280
You could still prove something existed,

331
00:21:31,280 –> 00:21:34,240
even though you couldn’t construct it explicitly.

332
00:21:34,240 –> 00:21:37,960
It’s like saying, “I know there has to be a way to get

333
00:21:37,960 –> 00:21:42,360
“from Gottingen to St Petersburg even though I can’t tell you

334
00:21:42,360 –> 00:21:44,440
“how to actually get there.”

335
00:21:44,440 –> 00:21:49,120
As well as challenging mathematical orthodoxies, Hilbert was also happy

336
00:21:49,120 –> 00:21:54,840
to knock the formal hierarchies that existed in the university system in Germany at the time.

337
00:21:54,840 –> 00:22:01,000
Other professors were quite shocked to see Hilbert out bicycling and drinking with his students.

338
00:22:01,000 –> 00:22:03,440

He liked very much parties.
Yeah?

339
00:22:03,440 –> 00:22:07,240

Yes.
Party animal. That’s my kind of mathematician.

340
00:22:07,240 –> 00:22:13,360
He liked very much dancing with young women. He liked very much to flirt.

341
00:22:13,360 –> 00:22:17,880
Really? Most mathematicians I know are not the biggest of flirts.

342
00:22:17,880 –> 00:22:22,000
‘Yet this lifestyle went hand in hand with an absolute commitment to maths.’

343
00:22:22,000 –> 00:22:26,200
Hilbert was of course somebody who thought

344
00:22:26,200 –> 00:22:30,240
that everybody who has a mathematical skill,

345
00:22:30,240 –> 00:22:36,400
a penguin, a woman, a man, or black, white or yellow,

346
00:22:36,400 –> 00:22:40,280
it doesn’t matter, he should do mathematics

347
00:22:40,280 –> 00:22:42,360
and he should be admired for his.

348
00:22:42,360 –> 00:22:46,200
The mathematics speaks for itself in a way.

349
00:22:46,200 –> 00:22:49,720

It doesn’t matter…
When you’re a penguin.

350
00:22:49,720 –> 00:22:54,360
Yeah, if you can prove the Riemann hypothesis, we really don’t mind.

351
00:22:54,360 –> 00:22:58,280

Yes, mathematics was for him a universal language.
Yes.

352
00:22:58,280 –> 00:23:02,080
Hilbert believed that this language was powerful enough

353
00:23:02,080 –> 00:23:04,360
to unlock all the truths of mathematics,

354
00:23:04,360 –> 00:23:07,640
a belief he expounded in a radio interview he gave

355
00:23:07,640 –> 00:23:11,400
on the future of mathematics on the 8th September 1930.

356
00:23:16,080 –> 00:23:20,280
In it, he had no doubt that all his 23 problems would soon be solved

357
00:23:20,280 –> 00:23:23,720
and that mathematics would finally be put

358
00:23:23,720 –> 00:23:26,840
on unshakeable logical foundations.

359
00:23:26,840 –> 00:23:30,160
There are absolutely no unsolvable problems, he declared,

360
00:23:30,160 –> 00:23:32,520
a belief that’s been held by mathematicians

361
00:23:32,520 –> 00:23:34,480
since the time of the Ancient Greeks.

362
00:23:34,480 –> 00:23:40,040
He ended with this clarion call, “We must know, we will know.”

363
00:23:40,040 –> 00:23:44,640
‘Wir mussen wissen, wir werden wissen.’

364
00:23:45,960 –> 00:23:48,480
Unfortunately for him, the very day before

365
00:23:48,480 –> 00:23:52,320
in a scientific lecture that was not considered worthy of broadcast,

366
00:23:52,320 –> 00:23:55,520
another mathematician would shatter Hilbert’s dream

367
00:23:55,520 –> 00:23:59,480
and put uncertainty at the heart of mathematics.

368
00:23:59,480 –> 00:24:02,400
The man responsible for destroying Hilbert’s belief

369
00:24:02,400 –> 00:24:05,520
was an Austrian mathematician, Kurt Godel.

370
00:24:10,400 –> 00:24:12,440
And it all started here - Vienna.

371
00:24:12,440 –> 00:24:15,360
Even his admirers, and there are a great many,

372
00:24:15,360 –> 00:24:19,920
admit that Kurt Godel was a little odd.

373
00:24:19,920 –> 00:24:23,840
As a child, he was bright, sickly and very strange.

374
00:24:23,840 –> 00:24:25,880
He couldn’t stop asking questions.

375
00:24:25,880 –> 00:24:30,720
So much so, that his family called him Herr Warum - Mr Why.

376
00:24:30,720 –> 00:24:35,160
Godel lived in Vienna in the 1920s and 1930s,

377
00:24:35,160 –> 00:24:38,000
between the fall of the Austro-Hungarian Empire

378
00:24:38,000 –> 00:24:39,960
and its annexation by the Nazis.

379
00:24:39,960 –> 00:24:45,520
It was a strange, chaotic and exciting time to be in the city.

380
00:24:45,520 –> 00:24:48,160
Godel studied mathematics at Vienna University

381
00:24:48,160 –> 00:24:50,600
but he spent most of his time in the cafes,

382
00:24:50,600 –> 00:24:52,960
the internet chat rooms of their time,

383
00:24:52,960 –> 00:24:55,920
where amongst games of backgammon and billiards,

384
00:24:55,920 –> 00:24:59,040
the real intellectual excitement was taking place.

385
00:24:59,040 –> 00:25:02,320
Particularly amongst a highly influential group

386
00:25:02,320 –> 00:25:05,920
of philosophers and scientists called the Vienna Circle.

387
00:25:05,920 –> 00:25:10,080
In their discussions, Kurt Godel would come up with an idea

388
00:25:10,080 –> 00:25:13,000
that would revolutionise mathematics.

389
00:25:13,000 –> 00:25:15,960
He’d set himself a difficult mathematical test.

390
00:25:15,960 –> 00:25:18,760
He wanted to solve Hilbert’s second problem

391
00:25:18,760 –> 00:25:22,000
and find a logical foundation for all mathematics.

392
00:25:22,000 –> 00:25:25,520
But what he came up with surprised even him.

393
00:25:25,520 –> 00:25:28,960
All his efforts in mathematical logic not only couldn’t provide

394
00:25:28,960 –> 00:25:33,840
the guarantee Hilbert wanted, instead he proved the opposite.

395
00:25:33,840 –> 00:25:35,440
Got it.

396
00:25:35,440 –> 00:25:38,800
It’s called the Incompleteness Theorem.

397
00:25:38,800 –> 00:25:42,360
Godel proved that within any logical system for mathematics

398
00:25:42,360 –> 00:25:46,200
there will be statements about numbers which are true

399
00:25:46,200 –> 00:25:48,200
but which you cannot prove.

400
00:25:48,200 –> 00:25:53,000
He starts with the statement, “This statement cannot be proved.”

401
00:25:53,000 –> 00:25:55,480
This is not a mathematical statement yet.

402
00:25:55,480 –> 00:25:58,360
But by using a clever code based on prime numbers,

403
00:25:58,360 –> 00:26:03,480
Godel could change this statement into a pure statement of arithmetic.

404
00:26:03,480 –> 00:26:08,640
Now, such statements must be either true or false.

405
00:26:08,640 –> 00:26:13,320
Hold on to your logical hats as we explore the possibilities.

406
00:26:13,320 –> 00:26:17,960
If the statement is false, that means the statement could be proved,

407
00:26:17,960 –> 00:26:21,320
which means it would be true, and that’s a contradiction.

408
00:26:21,320 –> 00:26:23,880
So that means, the statement must be true.

409
00:26:23,880 –> 00:26:28,320
In other words, here is a mathematical statement that is true

410
00:26:28,320 –> 00:26:30,840
but can’t be proved.

411
00:26:30,840 –> 00:26:32,440
Blast.

412
00:26:32,440 –> 00:26:35,520
Godel’s proof led to a crisis in mathematics.

413
00:26:35,520 –> 00:26:39,320
What if the problem you were working on, the Goldbach conjecture, say,

414
00:26:39,320 –> 00:26:43,600
or the Riemann hypothesis, would turn out to be true but unprovable?

415
00:26:43,600 –> 00:26:46,720
It led to a crisis for Godel too.

416
00:26:46,720 –> 00:26:50,400
In the autumn of 1934, he suffered the first of what became

417
00:26:50,400 –> 00:26:55,520
a series of breakdowns and spent time in a sanatorium.

418
00:26:55,520 –> 00:26:58,960
He was saved by the love of a good woman.

419
00:26:58,960 –> 00:27:02,880
Adele Nimbursky was a dancer in a local night club.

420
00:27:02,880 –> 00:27:06,200
She kept Godel alive.

421
00:27:06,200 –> 00:27:10,040
One day, she and Godel were walking down these very steps.

422
00:27:10,040 –> 00:27:13,120
Suddenly he was attacked by Nazi thugs.

423
00:27:13,120 –> 00:27:17,360
Godel himself wasn’t Jewish, but many of his friends in the Vienna Circle were.

424
00:27:17,360 –> 00:27:19,840
Adele came to his rescue.

425
00:27:19,840 –> 00:27:24,400
But it was only a temporary reprieve for Godel and for maths.

426
00:27:24,400 –> 00:27:29,680
All across Austria and Germany, mathematics was about to die.

427
00:27:33,680 –> 00:27:36,240
In the new German empire in the late 1930s

428
00:27:36,240 –> 00:27:39,760
there weren’t colourful balloons flying over the universities,

429
00:27:39,760 –> 00:27:41,600
but swastikas.

430
00:27:41,600 –> 00:27:46,280
The Nazis passed a law allowing the removal of any civil servant

431
00:27:46,280 –> 00:27:47,680
who wasn’t Aryan.

432
00:27:47,680 –> 00:27:51,200
Academics were civil servants in Germany then and now.

433
00:27:53,520 –> 00:27:56,200
Mathematicians suffered more than most.

434
00:27:56,200 –> 00:27:59,600
144 in Germany would lose their jobs.

435
00:27:59,600 –> 00:28:04,040
14 were driven to suicide or died in concentration camps.

436
00:28:07,680 –> 00:28:10,600
But one brilliant mathematician stayed on.

437
00:28:10,600 –> 00:28:12,400
David Hilbert helped arrange

438
00:28:12,400 –> 00:28:15,000
for some of his brightest students to flee.

439
00:28:15,000 –> 00:28:17,640
And he spoke out for a while about the dismissal

440
00:28:17,640 –> 00:28:19,200
of his Jewish colleagues.

441
00:28:19,200 –> 00:28:23,400
But soon, he too became silent.

442
00:28:26,720 –> 00:28:29,240
It’s not clear why he didn’t flee himself

443
00:28:29,240 –> 00:28:31,320
or at least protest a little more.

444
00:28:31,320 –> 00:28:33,600
He did fall ill towards the end of his life

445
00:28:33,600 –> 00:28:35,800
so maybe he just didn’t have the energy.

446
00:28:35,800 –> 00:28:38,440
All around him, mathematicians and scientists

447
00:28:38,440 –> 00:28:42,160
were fleeing the Nazi regime until it was only Hilbert left

448
00:28:42,160 –> 00:28:47,480
to witness the destruction of one of the greatest mathematical centres of all time.

449
00:28:50,000 –> 00:28:53,640
David Hilbert died in 1943.

450
00:28:53,640 –> 00:28:56,360
Only ten people attended the funeral

451
00:28:56,360 –> 00:28:59,600
of the most famous mathematician of his time.

452
00:28:59,600 –> 00:29:01,880
The dominance of Europe,

453
00:29:01,880 –> 00:29:05,680
the centre for world maths for 500 years, was over.

454
00:29:05,680 –> 00:29:12,000
It was time for the mathematical baton to be handed to the New World.

455
00:29:13,840 –> 00:29:17,120
Time in fact for this place.

456
00:29:17,120 –> 00:29:22,040
The Institute for Advanced Study had been set up in Princeton in 1930.

457
00:29:22,040 –> 00:29:24,880
The idea was to reproduce the collegiate atmosphere

458
00:29:24,880 –> 00:29:28,880
of the old European universities in rural New Jersey.

459
00:29:28,880 –> 00:29:32,200
But to do this, it needed to attract the very best,

460
00:29:32,200 –> 00:29:34,280
and it didn’t need to look far.

461
00:29:34,280 –> 00:29:37,480
Many of the brightest European mathematicians

462
00:29:37,480 –> 00:29:39,920
were fleeing the Nazis for America.

463
00:29:39,920 –> 00:29:42,520
People like Hermann Weyl, whose research

464
00:29:42,520 –> 00:29:45,680
would have major significance for theoretical physics.

465
00:29:45,680 –> 00:29:48,280
And John Von Neumann, who developed game theory

466
00:29:48,280 –> 00:29:50,840
and was one of the pioneers of computer science.

467
00:29:50,840 –> 00:29:55,400
The Institute quickly became the perfect place

468
00:29:55,400 –> 00:29:59,440
to create another Gottingen in the woods.

469
00:29:59,440 –> 00:30:04,760
One mathematician in particular made the place a home from home.

470
00:30:04,760 –> 00:30:06,320
Every morning Kurt Godel,

471
00:30:06,320 –> 00:30:09,360
dressed in a white linen suit and wearing a fedora,

472
00:30:09,360 –> 00:30:13,040
would walk from his home along Mercer Street to the Institute.

473
00:30:13,040 –> 00:30:16,520
On his way, he would stop here at number 112,

474
00:30:16,520 –> 00:30:22,640
to pick up his closest friend, another European exile, Albert Einstein.

475
00:30:22,640 –> 00:30:26,960
But not even relaxed, affluent Princeton could help Godel

476
00:30:26,960 –> 00:30:29,040
finally escape his demons.

477
00:30:29,040 –> 00:30:31,640
Einstein was always full of laughter.

478
00:30:31,640 –> 00:30:35,520
He described Princeton as a banishment to paradise.

479
00:30:35,520 –> 00:30:40,080
But the much younger Godel became increasingly solemn and pessimistic.

480
00:30:43,160 –> 00:30:46,400
Slowly this pessimism turned into paranoia.

481
00:30:46,400 –> 00:30:50,520
He spent less and less time with his fellow mathematicians in Princeton.

482
00:30:50,520 –> 00:30:54,200
Instead, he preferred to come here to the beach, next to the ocean,

483
00:30:54,200 –> 00:30:59,240
walk alone, thinking about the works of the great German mathematician, Leibniz.

484
00:31:01,400 –> 00:31:05,320
But as Godel was withdrawing into his own interior world,

485
00:31:05,320 –> 00:31:09,320
his influence on American mathematics paradoxically

486
00:31:09,320 –> 00:31:12,000
was growing stronger and stronger.

487
00:31:12,000 –> 00:31:16,160
And a young mathematician from just along the New Jersey coast

488
00:31:16,160 –> 00:31:19,840
eagerly took on some of the challenges he posed.

489
00:31:19,840 –> 00:31:23,760

One, two, three, four, five, six, seven, eight, nine, ten

490
00:31:23,760 –> 00:31:25,880

Times a day I could love you…

491
00:31:25,880 –> 00:31:27,040
In 1950s America,

492
00:31:27,040 –> 00:31:31,440
the majority of youngsters weren’t preoccupied with mathematics.

493
00:31:31,440 –> 00:31:35,160
Most went for a more relaxed, hedonistic lifestyle

494
00:31:35,160 –> 00:31:38,840
in this newly affluent land of ice-cream and doughnuts.

495
00:31:38,840 –> 00:31:42,560
But one teenager didn’t indulge in the normal pursuits

496
00:31:42,560 –> 00:31:45,640
of American adolescence but chose instead

497
00:31:45,640 –> 00:31:49,200
to grapple with some of the major problems in mathematics.

498
00:31:49,200 –> 00:31:50,680
From a very early age,

499
00:31:50,680 –> 00:31:55,080
Paul Cohen was winning mathematical competitions and prizes.

500
00:31:55,080 –> 00:31:58,960
But he found it difficult at first to discover a field in mathematics

501
00:31:58,960 –> 00:32:01,280
where he could really make his mark…

502
00:32:01,280 –> 00:32:05,720
Until he read about Cantor‘s continuum hypothesis.

503
00:32:05,720 –> 00:32:09,280
That’s the one problem, as I had learned back in Halle,

504
00:32:09,280 –> 00:32:11,760
that Cantor just couldn’t resolve.

505
00:32:11,760 –> 00:32:15,400
It asks whether there is or there isn’t an infinite set of numbers

506
00:32:15,400 –> 00:32:18,080
bigger than the set of all whole numbers

507
00:32:18,080 –> 00:32:20,960
but smaller than the set of all decimals.

508
00:32:20,960 –> 00:32:24,280
It sounds straightforward, but it had foiled all attempts

509
00:32:24,280 –> 00:32:29,160
to solve it since Hilbert made it his first problem way back in 1900.

510
00:32:29,160 –> 00:32:31,480
With the arrogance of youth,

511
00:32:31,480 –> 00:32:36,040
the 22-year-old Paul Cohen decided that he could do it.

512
00:32:36,040 –> 00:32:40,720
Cohen came back a year later with the extraordinary discovery

513
00:32:40,720 –> 00:32:43,200
that both answers could be true.

514
00:32:43,200 –> 00:32:47,160
There was one mathematics where the continuum hypothesis

515
00:32:47,160 –> 00:32:49,080
could be assumed to be true.

516
00:32:49,080 –> 00:32:51,800
There wasn’t a set between the whole numbers

517
00:32:51,800 –> 00:32:53,440
and the infinite decimals.

518
00:32:55,160 –> 00:32:59,200
But there was an equally consistent mathematics

519
00:32:59,200 –> 00:33:03,440
where the continuum hypothesis could be assumed to be false.

520
00:33:03,440 –> 00:33:08,280
Here, there was a set between the whole numbers and the infinite decimals.

521
00:33:08,280 –> 00:33:11,480
It was an incredibly daring solution.

522
00:33:11,480 –> 00:33:13,840
Cohen’s proof seemed true,

523
00:33:13,840 –> 00:33:19,160
but his method was so new that nobody was absolutely sure.

524
00:33:19,160 –> 00:33:22,720
There was only one person whose opinion everybody trusted.

525
00:33:22,720 –> 00:33:26,640
There was a lot of scepticism and he had to come and make a trip here,

526
00:33:26,640 –> 00:33:29,320
to the Institute right here, to visit Godel.

527
00:33:29,320 –> 00:33:32,720
And it was only after Godel gave his stamp of approval

528
00:33:32,720 –> 00:33:34,240
in quite an unusual way,

529
00:33:34,240 –> 00:33:37,880
He said, “Give me your paper”, and then on Monday he put it back

530
00:33:37,880 –> 00:33:40,360
in the box and said, “Yes, it’s correct.”

531
00:33:40,360 –> 00:33:42,040
Then everything changed.

532
00:33:43,240 –> 00:33:46,200
Today mathematicians insert a statement

533
00:33:46,200 –> 00:33:50,840
that says whether the result depends on the continuum hypothesis.

534
00:33:50,840 –> 00:33:54,880
We’ve built up two different mathematical worlds

535
00:33:54,880 –> 00:33:57,320
in which one answer is yes and the other is no.

536
00:33:57,320 –> 00:34:01,440
Paul Cohen really has rocked the mathematical universe.

537
00:34:01,440 –> 00:34:05,680
It gave him fame, riches, and prizes galore.

538
00:34:07,680 –> 00:34:12,880
He didn’t publish much after his early success in the ‘60s.

539
00:34:12,880 –> 00:34:15,040
But he was absolutely dynamite.

540
00:34:15,040 –> 00:34:18,840
I can’t imagine anyone better to learn from, and he was very eager

541
00:34:18,840 –> 00:34:23,840
to learn, to teach you anything he knew or even things he didn’t know.

542
00:34:23,840 –> 00:34:27,640
With the confidence that came from solving Hilbert’s first problem,

543
00:34:27,640 –> 00:34:30,320
Cohen settled down in the mid 1960s

544
00:34:30,320 –> 00:34:34,440
to have a go at the most important Hilbert problem of them all -

545
00:34:34,440 –> 00:34:36,960
the eighth, the Riemann hypothesis.

546
00:34:36,960 –> 00:34:43,000
When he died in California in 2007, 40 years later, he was still trying.

547
00:34:43,000 –> 00:34:46,200
But like many famous mathematicians before him,

548
00:34:46,200 –> 00:34:48,280
Riemann had defeated even him.

549
00:34:48,280 –> 00:34:52,440
But his approach has inspired others to make progress towards a proof,

550
00:34:52,440 –> 00:34:55,560
including one of his most famous students, Peter Sarnak.

551
00:34:55,560 –> 00:34:59,440
I think, overall, absolutely loved the guy.

552
00:34:59,440 –> 00:35:01,840
He was my inspiration.

553
00:35:01,840 –> 00:35:04,600
I’m really glad I worked with him.

554
00:35:04,600 –> 00:35:06,800
He put me on the right track.

555
00:35:09,960 –> 00:35:14,240
Paul Cohen is a good example of the success of the great American Dream.

556
00:35:14,240 –> 00:35:16,800
The second generation Jewish immigrant

557
00:35:16,800 –> 00:35:18,960
becomes top American professor.

558
00:35:18,960 –> 00:35:23,640
But I wouldn’t say that his mathematics was a particularly American product.

559
00:35:23,640 –> 00:35:25,720
Cohen was so fired up by this problem

560
00:35:25,720 –> 00:35:29,680
that he probably would have cracked it whatever the surroundings.

561
00:35:31,200 –> 00:35:33,680
Paul Cohen had it relatively easy.

562
00:35:33,680 –> 00:35:36,640
But another great American mathematician of the 1960s

563
00:35:36,640 –> 00:35:40,320
faced a much tougher struggle to make an impact.

564
00:35:40,320 –> 00:35:43,440
Not least because she was female.

565
00:35:43,440 –> 00:35:48,240
In the story of maths, nearly all the truly great mathematicians have been men.

566
00:35:48,240 –> 00:35:51,560
But there have been a few exceptions.

567
00:35:51,560 –> 00:35:54,000
There was the Russian Sofia Kovalevskaya

568
00:35:54,000 –> 00:35:58,920
who became the first female professor of mathematics in Stockholm in 1889,

569
00:35:58,920 –> 00:36:03,400
and won a very prestigious French mathematical prize.

570
00:36:03,400 –> 00:36:07,080
And then Emmy Noether, a talented algebraist who fled from the Nazis

571
00:36:07,080 –> 00:36:10,600
to America but then died before she fully realised her potential.

572
00:36:10,600 –> 00:36:15,920
Then there is the woman who I am crossing America to find out about.

573
00:36:15,920 –> 00:36:19,680
Julia Robinson, the first woman ever to be elected president

574
00:36:19,680 –> 00:36:22,080
of the American Mathematical Society.

575
00:36:31,440 –> 00:36:34,840
She was born in St Louis in 1919,

576
00:36:34,840 –> 00:36:38,160
but her mother died when she was two.

577
00:36:38,160 –> 00:36:42,360
She and her sister Constance moved to live with their grandmother

578
00:36:42,360 –> 00:36:45,720
in a small community in the desert near Phoenix, Arizona.

579
00:36:47,720 –> 00:36:49,800
Julia Robinson grew up around here.

580
00:36:49,800 –> 00:36:53,440
I’ve got a photo which shows her cottage in the 1930s,

581
00:36:53,440 –> 00:36:55,480
with nothing much around it.

582
00:36:55,480 –> 00:36:58,160
The mountains pretty much match those over there

583
00:36:58,160 –> 00:37:00,640
so I think she might have lived down there.

584
00:37:01,600 –> 00:37:04,160
Julia grew up a shy, sickly girl,

585
00:37:04,160 –> 00:37:09,440
who, when she was seven, spent a year in bed because of scarlet fever.

586
00:37:09,440 –> 00:37:12,240
Ill-health persisted throughout her childhood.

587
00:37:12,240 –> 00:37:15,120
She was told she wouldn’t live past 40.

588
00:37:15,120 –> 00:37:20,400
But right from the start, she had an innate mathematical ability.

589
00:37:20,400 –> 00:37:25,240
Under the shade of the native Arizona cactus, she whiled away the time

590
00:37:25,240 –> 00:37:28,720
playing endless counting games with stone pebbles.

591
00:37:28,720 –> 00:37:31,960
This early searching for patterns would give her a feel

592
00:37:31,960 –> 00:37:35,320
and love of numbers that would last for the rest of her life.

593
00:37:35,320 –> 00:37:39,160
But despite showing an early brilliance, she had to continually

594
00:37:39,160 –> 00:37:44,080
fight at school and college to simply be allowed to keep doing maths.

595
00:37:44,080 –> 00:37:47,920
As a teenager, she was the only girl in the maths class

596
00:37:47,920 –> 00:37:50,600
but had very little encouragement.

597
00:37:50,600 –> 00:37:55,480
The young Julia sought intellectual stimulation elsewhere.

598
00:37:55,480 –> 00:37:59,440
Julia loved listening to a radio show called the University Explorer

599
00:37:59,440 –> 00:38:02,440
and the 53rd episode was all about mathematics.

600
00:38:02,440 –> 00:38:04,960
The broadcaster described how he discovered

601
00:38:04,960 –> 00:38:08,560
despite their esoteric language and their seclusive nature,

602
00:38:08,560 –> 00:38:12,320
mathematicians are the most interesting and inspiring creatures.

603
00:38:12,320 –> 00:38:16,240
For the first time, Julia had found out that there were mathematicians,

604
00:38:16,240 –> 00:38:17,920
not just mathematics teachers.

605
00:38:17,920 –> 00:38:20,440
There was a world of mathematics out there,

606
00:38:20,440 –> 00:38:22,240
and she wanted to be part of it.

607
00:38:26,080 –> 00:38:29,680
The doors to that world opened here at the University of California,

608
00:38:29,680 –> 00:38:31,960
at Berkeley near San Francisco.

609
00:38:33,760 –> 00:38:38,680
She was absolutely obsessed that she wanted to go to Berkeley.

610
00:38:38,680 –> 00:38:44,200
She wanted to go away to some place where there were mathematicians.

611
00:38:44,200 –> 00:38:46,720
Berkeley certainly had mathematicians,

612
00:38:46,720 –> 00:38:50,320
including a number theorist called Raphael Robinson.

613
00:38:50,320 –> 00:38:53,400
In their frequent walks around the campus

614
00:38:53,400 –> 00:38:59,960
they found they had more than just a passion for mathematics. They married in 1952.

615
00:38:59,960 –> 00:39:03,200
Julia got her PhD and settled down

616
00:39:03,200 –> 00:39:05,720
to what would turn into her lifetime’s work -

617
00:39:05,720 –> 00:39:07,280
Hilbert’s tenth problem.

618
00:39:07,280 –> 00:39:10,000
She thought about it all the time.

619
00:39:10,000 –> 00:39:14,120
She said to me she just wouldn’t wanna die without knowing that answer

620
00:39:14,120 –> 00:39:16,240
and it had become an obsession.

621
00:39:17,280 –> 00:39:21,200
Julia’s obsession has been shared with many other mathematicians

622
00:39:21,200 –> 00:39:24,560
since Hilbert had first posed it back in 1900.

623
00:39:24,560 –> 00:39:28,400
His tenth problem asked if there was some universal method

624
00:39:28,400 –> 00:39:34,200
that could tell whether any equation had whole number solutions or not.

625
00:39:34,200 –> 00:39:36,520
Nobody had been able to solve it.

626
00:39:36,520 –> 00:39:39,520
In fact, the growing belief was

627
00:39:39,520 –> 00:39:42,440
that no such universal method was possible.

628
00:39:42,440 –> 00:39:44,520
How on earth could you prove that,

629
00:39:44,520 –> 00:39:48,400
however ingenious you were, you’d never come up with a method?

630
00:39:50,080 –> 00:39:51,800
With the help of colleagues,

631
00:39:51,800 –> 00:39:55,640
Julia developed what became known as the Robinson hypothesis.

632
00:39:55,640 –> 00:39:58,920
This argued that to show no such method existed,

633
00:39:58,920 –> 00:40:03,280
all you had to do was to cook up one equation whose solutions

634
00:40:03,280 –> 00:40:06,040
were a very specific set of numbers.

635
00:40:06,040 –> 00:40:09,280
The set of numbers needed to grow exponentially,

636
00:40:09,280 –> 00:40:13,960
like taking powers of two, yet still be captured by the equations

637
00:40:13,960 –> 00:40:16,520
at the heart of Hilbert’s problem.

638
00:40:16,520 –> 00:40:21,600
Try as she might, Robinson just couldn’t find this set.

639
00:40:21,600 –> 00:40:25,880
For the tenth problem to be finally solved,

640
00:40:25,880 –> 00:40:28,880
there needed to be some fresh inspiration.

641
00:40:28,880 –> 00:40:34,280
That came from 5,000 miles away - St Petersburg in Russia.

642
00:40:34,280 –> 00:40:37,840
Ever since the great Leonhard Euler set up shop here

643
00:40:37,840 –> 00:40:39,040
in the 18th century,

644
00:40:39,040 –> 00:40:42,960
the city has been famous for its mathematics and mathematicians.

645
00:40:42,960 –> 00:40:44,760
Here in the Steklov Institute,

646
00:40:44,760 –> 00:40:47,480
some of the world’s brightest mathematicians

647
00:40:47,480 –> 00:40:50,160
have set out their theorems and conjectures.

648
00:40:50,160 –> 00:40:54,320
This morning, one of them is giving a rare seminar.

649
00:40:57,120 –> 00:41:00,040
It’s tough going even if you speak Russian,

650
00:41:00,040 –> 00:41:02,080
which unfortunately I don’t.

651
00:41:02,080 –> 00:41:06,320
But we do get a break in the middle to recover before the final hour.

652
00:41:06,320 –> 00:41:08,320
There is a kind of rule in seminars.

653
00:41:08,320 –> 00:41:12,880
The first third is for everyone, the second third for the experts

654
00:41:12,880 –> 00:41:16,080
and the last third is just for the lecturer.

655
00:41:16,080 –> 00:41:19,080
I think that’s what we’re going to get next.

656
00:41:19,080 –> 00:41:22,800
The lecturer is Yuri Matiyasevich and he is explaining

657
00:41:22,800 –> 00:41:26,520
his latest work on - what else? - the Riemann hypothesis.

658
00:41:28,720 –> 00:41:33,160
As a bright young graduate student in 1965, Yuri’s tutor

659
00:41:33,160 –> 00:41:36,000
suggested he have a go at another Hilbert problem,

660
00:41:36,000 –> 00:41:39,000
the one that had in fact preoccupied Julia Robinson.

661
00:41:39,000 –> 00:41:40,280
Hilbert’s tenth.

662
00:41:43,160 –> 00:41:45,080
It was the height of the Cold War.

663
00:41:45,080 –> 00:41:48,440
Perhaps Matiyasevich could succeed for Russia

664
00:41:48,440 –> 00:41:52,080
where Julia and her fellow American mathematicians had failed.

665
00:41:52,080 –> 00:41:55,000

At first I did not like their approach.
Oh, right.

666
00:41:55,000 –> 00:41:59,640
The statement looked to me rather strange and artificial

667
00:41:59,640 –> 00:42:03,520
but after some time I understood it was quite natural,

668
00:42:03,520 –> 00:42:07,200
and then I understood that she had a new brilliant idea

669
00:42:07,200 –> 00:42:10,000
and I just started to further develop it.

670
00:42:11,520 –> 00:42:17,000
In January 1970, he found the vital last piece in the jigsaw.

671
00:42:17,000 –> 00:42:21,880
He saw how to capture the famous Fibonacci sequence of numbers

672
00:42:21,880 –> 00:42:26,040
using the equations that were at the heart of Hilbert’s problem.

673
00:42:26,040 –> 00:42:28,920
Building on the work of Julia and her colleagues,

674
00:42:28,920 –> 00:42:30,720
he had solved the tenth.

675
00:42:30,720 –> 00:42:34,240
He was just 22 years old.

676
00:42:34,240 –> 00:42:37,920
The first person he wanted to tell was the woman he owed so much to.

677
00:42:39,800 –> 00:42:41,720
I got no answer

678
00:42:41,720 –> 00:42:44,600
and I believed they were lost in the mail.

679
00:42:44,600 –> 00:42:47,720
It was quite natural because it was Soviet time.

680
00:42:47,720 –> 00:42:50,800
But back in California, Julia had heard rumours

681
00:42:50,800 –> 00:42:54,840
through the mathematical grapevine that the problem had been solved.

682
00:42:54,840 –> 00:42:57,120
And she contacted Yuri herself.

683
00:42:58,120 –> 00:43:01,480
She said, I just had to wait for you to grow up

684
00:43:01,480 –> 00:43:06,160
to get the answer, because she had started work in 1948.

685
00:43:06,160 –> 00:43:07,960
When Yuri was just a baby.

686
00:43:07,960 –> 00:43:11,240
Then he responds by thanking her

687
00:43:11,240 –> 00:43:16,160
and saying that the credit is as much hers as it is his.

688
00:43:18,240 –> 00:43:20,520
YURI: I met Julia one year later.

689
00:43:20,520 –> 00:43:25,080
It was in Bucharest. I suggested after the conference in Bucharest

690
00:43:25,080 –> 00:43:30,120
Julia and her husband Raphael came to see me here in Leningrad.

691
00:43:30,120 –> 00:43:35,400
Together, Julia and Yuri worked on several other mathematical problems

692
00:43:35,400 –> 00:43:39,160
until shortly before Julia died in 1985.

693
00:43:39,160 –> 00:43:41,960
She was just 55 years old.

694
00:43:41,960 –> 00:43:45,640
She was able to find the new ways.

695
00:43:45,640 –> 00:43:49,640
Many mathematicians just combine previous known methods

696
00:43:49,640 –> 00:43:55,560
to solve new problems and she had really new ideas.

697
00:43:55,560 –> 00:43:59,160
Although Julia Robinson showed there was no universal method

698
00:43:59,160 –> 00:44:01,560
to solve all equations in whole numbers,

699
00:44:01,560 –> 00:44:05,840
mathematicians were still interested in finding methods

700
00:44:05,840 –> 00:44:08,760
to solve special classes of equations.

701
00:44:08,760 –> 00:44:11,320
It would be in France in the early 19th century,

702
00:44:11,320 –> 00:44:13,560
in one of the most extraordinary stories

703
00:44:13,560 –> 00:44:17,120
in the history of mathematics, that methods were developed

704
00:44:17,120 –> 00:44:20,240
to understand why certain equations could be solved

705
00:44:20,240 –> 00:44:21,760
while others couldn’t.

706
00:44:27,840 –> 00:44:32,520
It’s early morning in Paris on the 29th May 1832.

707
00:44:32,520 –> 00:44:37,120
Evariste Galois is about to fight for his very life.

708
00:44:37,120 –> 00:44:40,680
It is the reign of the reactionary Bourbon King, Charles X,

709
00:44:40,680 –> 00:44:43,960
and Galois, like many angry young men in Paris then,

710
00:44:43,960 –> 00:44:46,680
is a republican revolutionary.

711
00:44:46,680 –> 00:44:52,000
Unlike the rest of his comrades though, he has another passion - mathematics.

712
00:44:53,560 –> 00:44:56,480
He had just spent four months in jail.

713
00:44:56,480 –> 00:45:00,160
Then, in a mysterious saga of unrequited love,

714
00:45:00,160 –> 00:45:02,280
he is challenged to a duel.

715
00:45:02,280 –> 00:45:04,280
He’d been up the whole previous night

716
00:45:04,280 –> 00:45:07,360
refining a new language for mathematics he’d developed.

717
00:45:07,360 –> 00:45:14,160
Galois believed that mathematics shouldn’t be the study of number and shape, but the study of structure.

718
00:45:14,160 –> 00:45:17,240
Perhaps he was still pre-occupied with his maths.

719
00:45:17,240 –> 00:45:18,800
GUNSHOT

720
00:45:18,800 –> 00:45:21,680
There was only one shot fired that morning.

721
00:45:21,680 –> 00:45:27,280
Galois died the next day, just 20 years old.

722
00:45:27,280 –> 00:45:30,320
It was one of mathematics greatest losses.

723
00:45:30,320 –> 00:45:33,080
Only by the beginning of the 20th century

724
00:45:33,080 –> 00:45:37,640
would Galois be fully appreciated and his ideas fully realised.

725
00:45:42,400 –> 00:45:46,520
Galois had discovered new techniques to be able to tell

726
00:45:46,520 –> 00:45:49,920
whether certain equations could have solutions or not.

727
00:45:49,920 –> 00:45:54,000
The symmetry of certain geometric objects seemed to be the key.

728
00:45:54,000 –> 00:45:58,520
His idea of using geometry to analyse equations

729
00:45:58,520 –> 00:46:03,880
would be picked up in the 1920s by another Parisian mathematician, Andre Weil.

730
00:46:03,880 –> 00:46:09,520
I was very much interested and so far as school was concerned

731
00:46:09,520 –> 00:46:13,720
quite successful in all possible branches.

732
00:46:13,720 –> 00:46:17,480
And he was. After studying in Germany as well as France,

733
00:46:17,480 –> 00:46:21,000
Andre settled down at this apartment in Paris

734
00:46:21,000 –> 00:46:25,760
which he shared with his more-famous sister, the writer Simone Weil.

735
00:46:25,760 –> 00:46:31,040
But when the Second World War broke out, he found himself in very different circumstances.

736
00:46:31,040 –> 00:46:37,040
He dodged the draft by fleeing to Finland where he was almost executed for being a Russian spy.

737
00:46:37,040 –> 00:46:42,720
On his return to France he was put in prison in Rouen to await trial for desertion.

738
00:46:42,720 –> 00:46:45,320
At the trial, the judge gave him a choice.

739
00:46:45,320 –> 00:46:49,120
Five more years in prison or serve in a combat unit.

740
00:46:49,120 –> 00:46:52,240
He chose to join the French army, a lucky choice

741
00:46:52,240 –> 00:46:56,120
because just before the Germans invaded a few months later,

742
00:46:56,120 –> 00:46:58,280
all the prisoners were killed.

743
00:46:58,280 –> 00:47:05,400
Weil only spent a few months in prison, but this time was crucial for his mathematics.

744
00:47:05,400 –> 00:47:11,000
Because here he built on the ideas of Galois and first developed algebraic geometry

745
00:47:11,000 –> 00:47:15,720
a whole new language for understanding solutions to equations.

746
00:47:15,720 –> 00:47:18,720
Galois had shown how new mathematical structures

747
00:47:18,720 –> 00:47:22,600
can be used to reveal the secrets behind equations.

748
00:47:22,600 –> 00:47:24,640
Weil’s work led him to theorems

749
00:47:24,640 –> 00:47:28,800
that connected number theory, algebra, geometry and topology

750
00:47:28,800 –> 00:47:33,720
and are one of the greatest achievements of modern mathematics.

751
00:47:33,720 –> 00:47:36,760
Without Andre Weil, we would never have heard

752
00:47:36,760 –> 00:47:41,400
of the strangest man in the history of maths, Nicolas Bourbaki.

753
00:47:43,720 –> 00:47:50,400
There are no photos of Bourbaki in existence but we do know he was born in this cafe in the Latin Quarter

754
00:47:50,400 –> 00:47:54,520
in 1934 when it was a proper cafe, the cafe Capoulade,

755
00:47:54,520 –> 00:47:58,000
and not the fast food joint it has now become.

756
00:47:58,000 –> 00:48:03,200
Just down the road, I met up with Bourbaki expert David Aubin.

757
00:48:03,200 –> 00:48:06,400
When I was a graduate student I got quite frightened

758
00:48:06,400 –> 00:48:08,120
when I used to go into the library

759
00:48:08,120 –> 00:48:10,960
because this guy Bourbaki had written so many books.

760
00:48:10,960 –> 00:48:14,400
Something like 30 or 40 altogether.

761
00:48:14,400 –> 00:48:19,680
In analysis, in geometry, in topology, it was all new foundations.

762
00:48:19,680 –> 00:48:23,360
Virtually everyone studying Maths seriously anywhere in the world

763
00:48:23,360 –> 00:48:28,200
in the 1950s, ‘60s and ‘70s would have read Nicolas Bourbaki.

764
00:48:28,200 –> 00:48:31,160
He applied for membership of the American Maths Society, I heard.

765
00:48:31,160 –> 00:48:33,360
At which point he was denied membership

766
00:48:33,360 –> 00:48:36,320

on the grounds that he didn’t exist.
Oh!

767
00:48:36,320 –> 00:48:38,160
The Americans were right.

768
00:48:38,160 –> 00:48:41,880
Nicolas Bourbaki does not exist at all. And never has.

769
00:48:41,880 –> 00:48:46,200
Bourbaki is in fact the nom de plume for a group of French mathematicians

770
00:48:46,200 –> 00:48:49,880
led by Andre Weil who decided to write a coherent account

771
00:48:49,880 –> 00:48:52,480
of the mathematics of the 20th century.

772
00:48:52,480 –> 00:48:57,200
Most of the time mathematicians like to have their own names on theorems.

773
00:48:57,200 –> 00:48:59,600
But for the Bourbaki group,

774
00:48:59,600 –> 00:49:03,440
the aims of the project overrode any desire for personal glory.

775
00:49:03,440 –> 00:49:07,120
After the Second World War, the Bourbaki baton was handed down

776
00:49:07,120 –> 00:49:10,080
to the next generation of French mathematicians.

777
00:49:10,080 –> 00:49:15,400
And their most brilliant member was Alexandre Grothendieck.

778
00:49:15,400 –> 00:49:17,000
Here at the IHES in Paris,

779
00:49:17,000 –> 00:49:21,520
the French equivalent of Princeton’s Institute for Advanced Study,

780
00:49:21,520 –> 00:49:27,160
Grothendieck held court at his famous seminars in the 1950s and 1960s.

781
00:49:29,920 –> 00:49:33,600
He had this incredible charisma.

782
00:49:33,600 –> 00:49:40,240
He had this amazing ability to see a young person and somehow know

783
00:49:40,240 –> 00:49:46,280
what kind of contribution this person could make to this incredible vision

784
00:49:46,280 –> 00:49:48,920
he had of how mathematics could be.

785
00:49:48,920 –> 00:49:54,520
And this vision enabled him to get across some very difficult ideas indeed.

786
00:49:54,520 –> 00:49:58,240
He says, “Suppose you want to open a walnut.

787
00:49:58,240 –> 00:50:02,200
“So the standard thing is you take a nutcracker and you just break it open.”

788
00:50:02,200 –> 00:50:04,800
And he says his approach is more like,

789
00:50:04,800 –> 00:50:08,120
you take this walnut and you put it out in the snow

790
00:50:08,120 –> 00:50:10,160
and you leave it there for a few months

791
00:50:10,160 –> 00:50:13,760
and then when you come back to it, it just opens.

792
00:50:13,760 –> 00:50:15,760
Grothendieck is a Structuralist.

793
00:50:15,760 –> 00:50:19,720
What he’s interested in are the hidden structures

794
00:50:19,720 –> 00:50:22,120
underneath all mathematics.

795
00:50:22,120 –> 00:50:27,560
Only when you get down to the very basic architecture and think in very general terms

796
00:50:27,560 –> 00:50:31,160
will the patterns in mathematics become clear.

797
00:50:31,160 –> 00:50:37,120
Grothendieck produced a new powerful language to see structures in a new way.

798
00:50:37,120 –> 00:50:39,720
It was like living in a world of black and white

799
00:50:39,720 –> 00:50:42,960
and suddenly having the language to see the world in colour.

800
00:50:42,960 –> 00:50:46,640
It’s a language that mathematicians have been using ever since

801
00:50:46,640 –> 00:50:51,640
to solve problems in number theory, geometry, even fundamental physics.

802
00:50:53,160 –> 00:50:56,440
But in the late 1960s, Grothendieck decided

803
00:50:56,440 –> 00:51:01,640
to turn his back on mathematics after he discovered politics.

804
00:51:01,640 –> 00:51:06,320
He believed that the threat of nuclear war and the questions

805
00:51:06,320 –> 00:51:12,440
of nuclear disarmament were more important than mathematics

806
00:51:12,440 –> 00:51:17,480
and that people who continue to do mathematics

807
00:51:17,480 –> 00:51:21,240
rather than confront this threat of nuclear war

808
00:51:21,240 –> 00:51:22,920
were doing harm in the world.

809
00:51:26,440 –> 00:51:29,040
Grothendieck decided to leave Paris

810
00:51:29,040 –> 00:51:32,040
and move back to the south of France where he grew up.

811
00:51:32,040 –> 00:51:36,680
Bursts of radical politics followed and then a nervous breakdown.

812
00:51:36,680 –> 00:51:40,720
He moved to the Pyrenees and became a recluse.

813
00:51:40,720 –> 00:51:45,600
He’s now lost all contact with his old friends and mathematical colleagues.

814
00:51:46,600 –> 00:51:51,040
Nevertheless, the legacy of his achievements means that Grothendieck stands

815
00:51:51,040 –> 00:51:57,440
alongside Cantor, Godel and Hilbert as someone who has transformed the mathematical landscape.

816
00:51:59,200 –> 00:52:03,800
He changed the whole subject in a really fundamental way. It will never go back.

817
00:52:03,800 –> 00:52:08,800
Certainly, he’s THE dominant figure of the 20th century.

818
00:52:16,200 –> 00:52:18,280
I’ve come back to England, though,

819
00:52:18,280 –> 00:52:22,440
thinking again about another seminal figure of the 20th century.

820
00:52:22,440 –> 00:52:26,640
The person that started it all off, David Hilbert.

821
00:52:26,640 –> 00:52:32,400
Of the 23 problems Hilbert set mathematicians in the year 1900,

822
00:52:32,400 –> 00:52:34,880
most have now been solved.

823
00:52:34,880 –> 00:52:37,160
However there is one great exception.

824
00:52:37,160 –> 00:52:40,360
The Riemann hypothesis, the eighth on Hilbert’s list.

825
00:52:40,360 –> 00:52:43,160
That is still the holy grail of mathematics.

826
00:52:44,960 –> 00:52:50,200
Hilbert’s lecture inspired a generation to pursue their mathematical dreams.

827
00:52:50,200 –> 00:52:55,120
This morning, in the town where I grew up, I hope to inspire another generation.

828
00:52:55,120 –> 00:52:57,280
CHEERING AND APPLAUSE

829
00:53:01,680 –> 00:53:04,120
Thank you. Hello. My name’s Marcus du Sautoy

830
00:53:04,120 –> 00:53:05,960
and I’m a Professor of Mathematics

831
00:53:05,960 –> 00:53:08,120
up the road at the University of Oxford.

832
00:53:08,120 –> 00:53:10,320
It was actually in this school here,

833
00:53:10,320 –> 00:53:14,520
in fact this classroom is where I discovered my love for mathematics.

834
00:53:14,520 –> 00:53:17,120
‘This love of mathematics that I first acquired

835
00:53:17,120 –> 00:53:20,400
‘here in my old comprehensive school still drives me now.

836
00:53:20,400 –> 00:53:22,280
‘It’s a love of solving problems.

837
00:53:22,280 –> 00:53:25,680
‘There are so many problems I could tell them about,

838
00:53:25,680 –> 00:53:27,720
‘but I’ve chosen my favourite.’

839
00:53:27,720 –> 00:53:30,840
I think that a mathematician is a pattern searcher

840
00:53:30,840 –> 00:53:33,960
and that’s really what mathematicians try and do.

841
00:53:33,960 –> 00:53:37,080
We try and understand the patterns and the structure

842
00:53:37,080 –> 00:53:40,440
and the logic to explain the way the world around us works.

843
00:53:40,440 –> 00:53:43,480
And this is really at the heart of the Riemann hypothesis.

844
00:53:43,480 –> 00:53:48,360
The task is - is there any pattern in these numbers which can help me say

845
00:53:48,360 –> 00:53:50,440
where the next number will be?

846
00:53:50,440 –> 00:53:52,760
What’s the next one after 31? How can I tell?

847
00:53:52,760 –> 00:53:55,760
‘These numbers are, of course, prime numbers -

848
00:53:55,760 –> 00:53:58,200
‘the building blocks of mathematics.’

849
00:53:58,200 –> 00:54:01,520
‘The Riemann hypothesis, a conjecture about the distribution

850
00:54:01,520 –> 00:54:04,720
‘of the primes, goes to the very heart of our subject.’

851
00:54:04,720 –> 00:54:07,560
Why on earth is anybody interested in these primes?

852
00:54:07,560 –> 00:54:11,040
Why is the army interested in primes, why are spies interested?

853
00:54:11,040 –> 00:54:14,800

Isn’t it to encrypt stuff?
Exactly.

854
00:54:14,800 –> 00:54:18,280
I study this stuff cos I think it’s all really beautiful and elegant

855
00:54:18,280 –> 00:54:20,200
but actually, there’s a lot of people

856
00:54:20,200 –> 00:54:24,440
who are interested in these numbers because of their very practical use.

857
00:54:24,440 –> 00:54:28,720
‘The bizarre thing is that the more abstract and difficult mathematics becomes,

858
00:54:28,720 –> 00:54:32,480
‘the more it seems to have applications in the real world.

859
00:54:32,480 –> 00:54:36,560
‘Mathematics now pervades every aspect of our lives.

860
00:54:36,560 –> 00:54:41,560
‘Every time we switch on the television, plug in a computer, pay with a credit card.

861
00:54:41,560 –> 00:54:46,160
‘There’s now a million dollars for anyone who can solve the Riemann hypothesis.

862
00:54:46,160 –> 00:54:48,600
‘But there’s more at stake than that.’

863
00:54:48,600 –> 00:54:51,800
Anybody who proves this theorem will be remembered forever.

864
00:54:51,800 –> 00:54:55,640
They’ll be on that board ahead of any of those other mathematicians.

865
00:54:55,640 –> 00:54:59,600
‘That’s because the Riemann hypothesis is a corner-stone of maths.

866
00:54:59,600 –> 00:55:02,800
‘Thousands of theorems depend on it being true.

867
00:55:02,800 –> 00:55:06,000
‘Very few mathematicians think that it isn’t true.

868
00:55:06,000 –> 00:55:10,640
‘But mathematics is about proof and until we can prove it

869
00:55:10,640 –> 00:55:12,840
‘there will still be doubt.’

870
00:55:12,840 –> 00:55:17,160
Maths has grown out of this passion to get rid of doubt.

871
00:55:17,160 –> 00:55:20,760
This is what I have learned in my journey through the history of mathematics.

872
00:55:20,760 –> 00:55:25,080
Mathematicians like Archimedes and al-Khwarizmi, Gauss and Grothendieck

873
00:55:25,080 –> 00:55:30,520
were driven to understand the precise way numbers and space work.

874
00:55:30,520 –> 00:55:33,200
Maths in action, that one.

875
00:55:33,200 –> 00:55:35,440
It’s beautiful. Really nice.

876
00:55:35,440 –> 00:55:39,200
Using the language of mathematics, they have told us stories

877
00:55:39,200 –> 00:55:43,760
that remain as true today as they were when they were first told.

878
00:55:43,760 –> 00:55:48,760
In the Mediterranean, I discovered the origins of geometry.

879
00:55:48,760 –> 00:55:51,840
Mathematicians and philosophers flocked to Alexandria

880
00:55:51,840 –> 00:55:55,240
driven by a thirst for knowledge and the pursuit of excellence.

881
00:55:55,240 –> 00:55:59,080
In India, I learned about another discovery

882
00:55:59,080 –> 00:56:02,880
that it would be impossible to imagine modern life without.

883
00:56:02,880 –> 00:56:07,240
So here we are in one of the true holy sites of the mathematical world.

884
00:56:07,240 –> 00:56:10,080
Up here are some numbers,

885
00:56:10,080 –> 00:56:12,680
and here’s the new number.

886
00:56:12,680 –> 00:56:14,320
Its zero.

887
00:56:14,320 –> 00:56:19,600
In the Middle East, I was amazed at al-Khwarizmi‘s invention of algebra.

888
00:56:19,600 –> 00:56:22,480
He developed systematic ways to analyse problems

889
00:56:22,480 –> 00:56:26,160
so that the solutions would work whatever numbers you took.

890
00:56:26,160 –> 00:56:28,080
In the Golden Age of Mathematics,

891
00:56:28,080 –> 00:56:31,600
in Europe in the 18th and 19th centuries, I found how maths

892
00:56:31,600 –> 00:56:35,760
discovered new ways for analysing bodies in motion and new geometries

893
00:56:35,760 –> 00:56:40,520
that helped us understand the very strange shape of space.

894
00:56:40,520 –> 00:56:43,840
It is with Riemann’s work that we finally have

895
00:56:43,840 –> 00:56:49,280
the mathematical glasses to be able to explore such worlds of the mind.

896
00:56:49,280 –> 00:56:53,480
And now my journey into the abstract world of 20th-century mathematics

897
00:56:53,480 –> 00:56:56,600
has revealed that maths is the true language

898
00:56:56,600 –> 00:56:58,800
the universe is written in,

899
00:56:58,800 –> 00:57:02,120
the key to understanding the world around us.

900
00:57:02,120 –> 00:57:05,840
Mathematicians aren’t motivated by money and material gain

901
00:57:05,840 –> 00:57:09,160
or even by practical applications of their work.

902
00:57:09,160 –> 00:57:13,400
For us, it is the glory of solving one of the great unsolved problems

903
00:57:13,400 –> 00:57:18,560
that have outwitted previous generations of mathematicians.

904
00:57:18,560 –> 00:57:21,920
Hilbert was right. It’s the unsolved problems of mathematics

905
00:57:21,920 –> 00:57:23,720
that make it a living subject,

906
00:57:23,720 –> 00:57:27,160
which obsess each new generation of mathematicians.

907
00:57:27,160 –> 00:57:30,960
Despite all the things we’ve discovered over the last seven millennia,

908
00:57:30,960 –> 00:57:33,600
there are still many things we don’t understand.

909
00:57:33,600 –> 00:57:39,960
And its Hilbert’s call of, “We must know, we will know”, which drives mathematics.

910
00:57:42,240 –> 00:57:45,440
You can learn more about The Story Of Maths

911
00:57:45,440 –> 00:57:48,480
with the Open University at…

912
00:58:00,600 –> 00:58:03,640
Subtitled by Red Bee Media Ltd

913
00:58:03,640 –> 00:58:06,680
E-mail subtitling@bbc.co.uk

Subtitles by © Red Bee Media Ltd

2019-05-04

The Story of Maths - 3. The Frontiers of Space - Subtitles

texts below are from © https://subsaga.com/bbc/documentaries/science/the-story-of-maths/3-the-frontiers-of-space.html

1
00:00:23,920 –> 00:00:27,000
I’m walking in the mountains of the moon.

2
00:00:29,080 –> 00:00:33,640
I’m on the trail of the Renaissance artist, Piero della Francesca,

3
00:00:33,640 –> 00:00:38,080
so I’ve come to the town in northern Italy which Piero made his own.

4
00:00:38,080 –> 00:00:41,240
There it is, Urbino.

5
00:00:41,240 –> 00:00:45,040
I’ve come here to see some of Piero’s finest works,

6
00:00:45,040 –> 00:00:50,320
masterpieces of art, but also masterpieces of mathematics.

7
00:00:52,080 –> 00:00:55,520
The artists and architects of the early Renaissance brought back

8
00:00:55,520 –> 00:01:00,200
the use of perspective, a technique that had been lost for 1,000 years,

9
00:01:00,200 –> 00:01:03,120
but using it properly turned out to be a lot

10
00:01:03,120 –> 00:01:05,160
more difficult than they’d imagined.

11
00:01:05,160 –> 00:01:10,640
Piero was the first major painter to fully understand perspective.

12
00:01:10,640 –> 00:01:15,720
That’s because he was a mathematician as well as an artist.

13
00:01:15,720 –> 00:01:17,880
I came here to see his masterpiece,

14
00:01:17,880 –> 00:01:21,440
The Flagellation of Christ, but there was a problem.

15
00:01:21,440 –> 00:01:24,000
I’ve just been to see The Flagellation, and it’s an

16
00:01:24,000 –> 00:01:27,320
absolutely stunning picture, but unfortunately, for various

17
00:01:27,320 –> 00:01:30,680
kind of Italian reasons, we’re not allowed to go and film in there.

18
00:01:30,680 –> 00:01:34,160
But this is a maths programme, after all, and not an arts programme,

19
00:01:34,160 –> 00:01:38,440
so I’ve used a bit of mathematics to bring this picture alive.

20
00:01:38,440 –> 00:01:43,840
We can’t go to the picture, but we can make the picture come to us.

21
00:01:43,840 –> 00:01:46,080
The problem of perspective is how

22
00:01:46,080 –> 00:01:51,160
to represent the three-dimensional world on a two-dimensional canvas.

23
00:01:51,160 –> 00:01:54,600
To give a sense of depth, a sense of the third dimension,

24
00:01:54,600 –> 00:01:57,720
Piero used mathematics.

25
00:01:57,720 –> 00:02:00,440
How big is he going to paint Christ,

26
00:02:00,440 –> 00:02:03,400
if this group of men here were a certain distance away

27
00:02:03,400 –> 00:02:05,520
from these men in the foreground?

28
00:02:07,040 –> 00:02:11,440
Get it wrong and the illusion of perspective is shattered.

29
00:02:11,440 –> 00:02:14,600
It’s far from obvious how a three-dimensional world

30
00:02:14,600 –> 00:02:19,160
can be accurately represented on a two-dimensional surface.

31
00:02:19,160 –> 00:02:23,000
Look at how the parallel lines in the three-dimensional world

32
00:02:23,000 –> 00:02:26,800
are no longer parallel in the two-dimensional canvas, but meet

33
00:02:26,800 –> 00:02:28,680
at a vanishing point.

34
00:02:30,520 –> 00:02:33,600
And this is what the tiles in the picture really look like.

35
00:02:39,200 –> 00:02:41,200
What is emerging here is a new

36
00:02:41,200 –> 00:02:45,320
mathematical language which allows us to map one thing into another.

37
00:02:45,320 –> 00:02:49,120
The power of perspective unleashed a new way to see the world,

38
00:02:49,120 –> 00:02:53,280
a perspective that would cause a mathematical revolution.

39
00:02:55,400 –> 00:02:59,640
Piero’s work was the beginning of a new way to understand geometry,

40
00:02:59,640 –> 00:03:02,080
but it would take another 200 years

41
00:03:02,080 –> 00:03:05,800
before other mathematicians would continue where he left off.

42
00:03:13,840 –> 00:03:16,440
Our journey has come north.

43
00:03:16,440 –> 00:03:19,480
By the 17th century, Europe had taken over

44
00:03:19,480 –> 00:03:23,960
from the Middle East as the world’s powerhouse of mathematical ideas.

45
00:03:23,960 –> 00:03:26,000
Great strides had been made in the geometry

46
00:03:26,000 –> 00:03:27,640
of objects fixed in time and space.

47
00:03:27,640 –> 00:03:30,800
In France, Germany, Holland and Britain,

48
00:03:30,800 –> 00:03:35,320
the race was now on to understand the mathematics of objects in motion

49
00:03:35,320 –> 00:03:38,600
and the pursuit of this new mathematics started here in this

50
00:03:38,600 –> 00:03:42,240
village in the centre of France.

51
00:03:42,240 –> 00:03:45,720
Only the French would name a village after a mathematician.

52
00:03:45,720 –> 00:03:47,400
Imagine in England a town called

53
00:03:47,400 –> 00:03:50,800
Newton or Ball or Cayley. I don’t think so!

54
00:03:50,800 –> 00:03:54,880
But in France, they really value their mathematicians.

55
00:03:54,880 –> 00:03:57,400
This is the village of Descartes in the Loire Valley.

56
00:03:57,400 –> 00:04:00,080
It was renamed after the famous philosopher

57
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and mathematician 200 years ago.

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Descartes himself was born here in 1596, a sickly child who lost

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his mother when very young, so he was allowed to stay in bed every

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morning until 11.00am, a practice he tried to continue all his life.

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To do mathematics, sometimes you just need to remove

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all distractions, to float off into a world of shapes and patterns.

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Descartes thought that the bed was the best place to achieve

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this meditative state.

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I think I know what he means.

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The house where Descartes undertook his bedtime meditations

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is now a museum dedicated to all things Cartesian.

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Come with me.

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Its exhibition pieces arranged, by curator Sylvie Garnier, show how

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his philosophical, scientific and mathematical ideas all fit together.

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It also features less familiar aspects

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of Descartes’ life and career.

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So he decided to be a soldier…in the army,

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in the Protestant Army

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and too in the Catholic Army, not a problem for him

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because no patriotism.

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Sylvie is putting it very nicely,

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but Descartes was in fact a mercenary.

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He fought for the German Protestants, the French Catholics

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and anyone else who would pay him.

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Very early one autumn morning in 1628, he was in the Bavarian Army

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camped out on a cold river bank.

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Inspiration very often strikes in very strange places.

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The story is told how Descartes couldn’t sleep one night,

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maybe because he was getting up so late

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or perhaps he was celebrating St Martin’s Eve

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and had just drunk too much.

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Problems were tumbling around in his mind.

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He was thinking about his favourite subject, philosophy.

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He was finding it very frustrating.

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How can you actually know anything at all?!

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Then he slips into a dream…

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and in the dream he understood that the key was to build philosophy

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on the indisputable facts of mathematics.

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Numbers, he realised, could brush away the cobwebs of uncertainty.

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He wanted to publish all his radical ideas, but he was worried how they’d

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be received in Catholic France, so he packed his bags and left.

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Descartes found a home here in Holland.

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He’d been one of the champions of the new scientific revolution

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which rejected the dominant view that the sun went around the earth,

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an opinion that got scientists like Galileo

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into deep trouble with the Vatican.

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Descartes reckoned that here amongst the Protestant Dutch

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he would be safe, especially

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at the old university town of Leiden

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where they valued maths and science.

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I’ve come to Leiden too.

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Unfortunately, I’m late!

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Hello. Yeah, I’m sorry.

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I got a puncture. It took me a bit of time, yeah, yeah.

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Henk Bos is one of Europe’s most eminent Cartesian scholars.

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He’s not surprised the French scholar ended up in Leiden.

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He came to talk with people and some people were open to his ideas.

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This was not only mathematic. It was also a mechanics specially.

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He merged algebra and geometry.

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Right.
So you could have formulas and figures and go back and forth.

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So a sort of dictionary between the two?
Yeah, yeah.

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This dictionary, which was finally published here in Holland in 1637,

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included mainly controversial

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philosophical ideas, but the most radical thoughts

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were in the appendix, a proposal to link algebra and geometry.

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Each point in two dimensions can be described by two numbers,

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one giving the horizontal location, the second number giving the point’s

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vertical location.

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As the point moves around a circle, these coordinates change,

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but we can write down an equation that identifies the changing value

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of these numbers at any point in the figure.

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Suddenly, geometry has turned into algebra.

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Using this transformation

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from geometry into numbers, you could tell, for example,

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if the curve on this bridge was part of a circle or not.

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You didn’t need to use your eyes.

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Instead, the equations of the curve would reveal its secrets,

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but it wouldn’t stop there.

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Descartes had unlocked the possibility of navigating geometries

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of higher dimensions, worlds our eyes will never see but are central

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to modern technology and physics.

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There’s no doubt that Descartes was one of the giants of mathematics.

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Unfortunately, though, he wasn’t the nicest of men.

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I think he was not an easy person, so…

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And he could be… he was very much concerned about

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his image. He was entirely

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self-convinced that he was right, also when he was wrong and his first

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reaction would be that the other one was stupid that hadn’t understood it.

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Descartes may not have been the most congenial person,

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but there’s no doubt that his insight into the connection

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between algebra and geometry transformed mathematics forever.

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For his mathematical revolution to work, though, he needed one other

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vital ingredient.

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To find that, I had to say goodbye to Henk and Leiden and go to church.

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CHORAL SINGING

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I’m not a believer myself, but there’s little doubt

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that many mathematicians from the time of Descartes

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had strong religious convictions.

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Maybe it’s just a coincidence,

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but perhaps it’s because mathematics and religion are both building ideas

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upon an undisputed set of axioms - one plus one equals two. God exists.

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I think I know which set of axioms I’ve got my faith in.

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In the 17th century,

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there was a Parisian monk who went to the same school as Descartes.

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He loved mathematics as much as he loved God.

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Indeed, he saw maths and science as evidence of the existence of God,

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Marin Mersenne was a first-class mathematician.

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One of his discoveries in prime numbers is still named after him.

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But he’s also celebrated for his correspondence.

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From his monastery in Paris, Mersenne acted like some kind of

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17th century internet hub, receiving ideas and then sending them on.

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It’s not so different now.

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We sit like mathematical monks thinking about our ideas, then

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sending a message to a colleague and hoping for some reply.

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There was a spirit of mathematical communication in 17th century Europe

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which had not been seen since the Greeks.

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Mersenne urged people to read Descartes’ new work on geometry.

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He also did something just as important.

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He publicised some new findings on the properties of numbers

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by an unknown amateur who would end up rivalling Descartes as the

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greatest mathematician of his time, Pierre de Fermat.

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Here in Beaumont-de-Lomagne

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near Toulouse, residents and visitors have come

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out to celebrate the life and work of the village’s most famous son.

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But I’m not too sure what these gladiators are doing here!

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And the appearance of this camel came as a bit of a surprise too.

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The man himself would have hardly approved of

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the ideas of using fun and games to advance an interest in mathematics.

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Unlike the aristocratic Descartes, Fermat wouldn’t have considered it

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worthless or common to create a festival of mathematics.

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Maths in action, that one.

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It’s beautiful, really nice, yeah.

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Fermat’s greatest contribution to mathematics was to virtually invent

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modern number theory.

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He devised a wide range of conjectures

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and theorems about numbers including his famous Last Theorem,

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the proof of which would puzzle mathematicians for over 350 years,

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but it’s little help to me now.

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Getting it apart is the easy bit.

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It’s putting it together, isn’t it, that’s the difficult bit.

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How many bits have I got? I’ve got six bits.

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I think what I need to do is put some symmetry into this.

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I’m afraid he’s going to tell me how to do it and I don’t want to see.

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I hate being told how to do a problem. I don’t want to look.

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And he’s laughing at me now because I can’t do it.

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That’s very unfair!

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Here we go.

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Can I put them together?

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I got it!

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Now that’s the buzz of doing mathematics when

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the thing clicks together and suddenly you see the right answer.

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Remarkably, Fermat only tackled mathematics in his spare time.

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By day he was a magistrate.

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Battling with mathematical problems was his hobby and his passion.

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The wonderful thing about mathematics is

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you can do it anywhere.

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You don’t have to have a laboratory.

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You don’t even really need a library.

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Fermat used to do much of his work while sitting at the kitchen table

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or praying in his local church or up here on his roof.

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He may have looked like an amateur,

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but he took his mathematics very seriously indeed.

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Fermat managed to find several new patterns in numbers

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that had defeated mathematicians for centuries.

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One of my favourite theorems of Fermat

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is all to do with prime numbers.

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If you’ve got a prime number which when you divide it by four

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leaves remainder one, then Fermat showed you could

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always rewrite this number as two square numbers added together.

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For example, I’ve got 13 cloves of garlic here,

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a prime number which has remainder one when I divide it by four.

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Fermat proved you can rewrite this number as two square numbers added

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together, so 13 can be rewritten

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as three squared plus two squared, or four plus nine.

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The amazing thing is that Fermat proved this will work however big

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the prime number is. Provided it has remainder one on division by four,

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you can always rewrite that number

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as two square numbers added together.

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Ah, my God!

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What I love about this sort of day is the playfulness of mathematics

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and Fermat certainly enjoyed playing around with numbers. He loved

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looking for patterns in numbers and then the puzzle side of mathematics,

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he wanted to prove that these patterns would be there forever.

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But as well as being the basis for fun and games in the years to come,

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Fermat’s mathematics would have some very serious applications.

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One of his theorems, his Little Theorem, is

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the basis of the codes that protect our credit cards on the internet.

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Technology we now rely on today all comes from the scribblings

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of a 17th-century mathematician.

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But the usefulness of Fermat’s mathematics is nothing compared to

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that of our next great mathematician and he comes not from France at all,

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but from its great rival.

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In the 17th century, Britain was emerging as a world power.

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Its expansion and ambitions required new methods of measurement

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and computation and that gave a great boost to mathematics.

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The university towns of Oxford and Cambridge

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were churning out mathematicians who were in great demand

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and the greatest of them was Isaac Newton.

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I’m here in Grantham, where Isaac Newton grew up,

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and they’re very proud of him here.

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They have a wonderful statue to him.

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They’ve even got

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the Isaac Newton Shopping Centre, with a nice apple logo up there.

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There’s a school that he went to with a nice blue plaque

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and there’s a museum over here in the Town Hall, although, actually,

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one of the other famous residents here, Margaret Thatcher,

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has got as big a display as Isaac Newton.

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In fact, the Thatcher cups have

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sold out and there’s loads of Newton ones still left,

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so I thought I would support mathematics by buying a Newton cup.

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And Newton’s maths does need support.

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Newton’s very famous here. Do you know what he’s famous for?
No.

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No, I don’t.
Discovering gravity.
Gravity?
Gravity, yes.
Gravity?

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Apple tree and all that, gravity.
‘That pretty much summed it up.

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‘If people know about Newton’s work at all, it is his physics,

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‘his laws of gravity in motion, not his mathematics.’

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I’m in a rush!
You’re in a rush. OK.

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Acceleration, you see? One of Newton’s laws!

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Eight miles south of Grantham,

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in the village of Woolsthorpe, where Newton was born,

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I met up with someone who does share my passion for his mathematics.

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This is the house.

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Wow, beautiful. ‘Jackie Stedall is a Newton fan and more than willing

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‘to show me around the house where Newton was brought up.’

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So here is the…

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you might call it the dining room. I’m sure they didn’t call it that,

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but the room where they ate, next to the kitchen.

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Of course, there would have been a huge fire in there.

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Yes! Gosh, I wish it was there now!

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His father was an illiterate farmer,

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but he died shortly before Newton was born.

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Otherwise, the young Isaac’s fate might have been very different.

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And here’s his room.

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Oh, lovely, wow.

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They present it really nicely.
Yes.

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It’s got a real feel of going back in time.
It does, yes.

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I can see he’s as scruffy as I am. Look at the state of that bed.

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That’s how, I think, I left my bed this morning.

295
00:18:13,480 –> 00:18:18,160
Newton hated his stepfather, but it was this man who ensured

296
00:18:18,160 –> 00:18:21,240
he became a mathematician rather than a sheep farmer.

297
00:18:21,240 –> 00:18:23,480
I don’t think he was particularly remarkable as a child.

298
00:18:23,480 –> 00:18:26,800

OK.
So there’s hope for all those kids out there.
Yes, yes.

299
00:18:26,800 –> 00:18:28,400
I think he had a sort of average school report.

300
00:18:28,400 –> 00:18:32,280
He had very few close friends. I don’t feel he’s someone

301
00:18:32,280 –> 00:18:33,880
I particularly would have wanted to meet,

302
00:18:33,880 –> 00:18:37,760
but I do love his mathematics. It’s wonderful.

303
00:18:37,760 –> 00:18:40,320
Newton came back to Lincolnshire from Cambridge

304
00:18:40,320 –> 00:18:46,600
during the Great Plague of 1665 when he was just 22 years old.

305
00:18:46,600 –> 00:18:50,880
In two miraculous years here, he developed a new theory of light,

306
00:18:50,880 –> 00:18:52,400
discovered gravitation

307
00:18:52,400 –> 00:18:57,960
and scribbled out a revolutionary approach to maths, the calculus.

308
00:18:57,960 –> 00:18:59,880
It works like this.

309
00:18:59,880 –> 00:19:03,920
I’m going to accelerate this car from 0 to 60 as quickly as I can.

310
00:19:03,920 –> 00:19:07,520
The speedometer is showing me that the speed’s changing all the time,

311
00:19:07,520 –> 00:19:09,360
but this is only an average speed.

312
00:19:09,360 –> 00:19:11,480
How can I tell precisely what my speed is

313
00:19:11,480 –> 00:19:15,400
at any particular instant? Well, here’s how.

314
00:19:15,400 –> 00:19:20,320
As the car races along the road, we can draw a graph above the road

315
00:19:20,320 –> 00:19:23,560
where the height above each point in the road records how long it took

316
00:19:23,560 –> 00:19:26,400
the car to get to that point.

317
00:19:26,400 –> 00:19:28,840
I can calculate the average speed between

318
00:19:28,840 –> 00:19:33,240
two points, A and B, on my journey by recording the distance travelled

319
00:19:33,240 –> 00:19:37,760
and dividing by the time it took to get between these two points,

320
00:19:37,760 –> 00:19:42,000
but what about the precise speed at the first point, A?

321
00:19:43,520 –> 00:19:48,200
If I move point B closer and closer to the first point, I take a smaller

322
00:19:48,200 –> 00:19:51,440
and smaller window of time and the speed gets closer

323
00:19:51,440 –> 00:19:55,240
and closer to the true value, but eventually, it looks like

324
00:19:55,240 –> 00:19:59,320
I have to calculate 0 divided by 0.

325
00:19:59,320 –> 00:20:03,920
The calculus allows us to make sense of this calculation.

326
00:20:03,920 –> 00:20:08,320
It enables us to work out the exact speed and also the precise distance

327
00:20:08,320 –> 00:20:11,280
travelled at any moment in time.

328
00:20:11,280 –> 00:20:15,080
I mean, it does make sense, the things we take for granted so much,

329
00:20:15,080 –> 00:20:16,720
things like… if I drop this apple…

330
00:20:16,720 –> 00:20:18,280
Its distance is changing and its

331
00:20:18,280 –> 00:20:20,920
speed is changing and calculus can deal with all of that.

332
00:20:20,920 –> 00:20:22,480
Which is quite in contrast to the Greeks.

333
00:20:22,480 –> 00:20:25,120
It was a very static geometry.

334
00:20:25,120 –> 00:20:27,000

Yes, it is.
And here we see…

335
00:20:27,000 –> 00:20:29,880
so the calculus is used by

336
00:20:29,880 –> 00:20:33,200
every engineer, physicist, because it can describe the moving world.

337
00:20:33,200 –> 00:20:36,720
Yes, and it’s the only way really you can deal with the mathematics of

338
00:20:36,720 –> 00:20:38,480
motion or with change.

339
00:20:38,480 –> 00:20:40,080
There’s a lot of mathematics in this apple!

340
00:20:42,360 –> 00:20:46,040
Newton’s calculus enables us to really understand

341
00:20:46,040 –> 00:20:50,600
the changing world, the orbits of planets, the motions of fluids.

342
00:20:50,600 –> 00:20:54,200
Through the power of the calculus, we have a way of describing, with

343
00:20:54,200 –> 00:20:58,840
mathematical precision, the complex, ever-changing natural world.

344
00:21:04,800 –> 00:21:09,080
But it would take 200 years to realise its full potential.

345
00:21:09,080 –> 00:21:12,640
Newton himself decided not to publish, but just to circulate

346
00:21:12,640 –> 00:21:14,960
his thoughts among friends.

347
00:21:14,960 –> 00:21:17,240
His reputation, though, gradually spread.

348
00:21:17,240 –> 00:21:21,480
He became a professor, an MP, and then Warden of the Royal Mint

349
00:21:21,480 –> 00:21:23,640
here in the City of London.

350
00:21:25,600 –> 00:21:28,760
On his regular trips to the Royal Society from the Royal Mint,

351
00:21:28,760 –> 00:21:33,120
he preferred to think about theology and alchemy rather than mathematics.

352
00:21:33,120 –> 00:21:35,440
Developing the calculus just got crowded out

353
00:21:35,440 –> 00:21:39,720
by all his other interests until he heard about a rival…

354
00:21:41,800 –> 00:21:46,080
a rival who was also a member of the Royal Society and who came up

355
00:21:46,080 –> 00:21:48,800
with exactly the same idea as him,

356
00:21:48,800 –> 00:21:50,960
Gottfried Leibniz.

357
00:21:50,960 –> 00:21:54,240
Every word Leibniz wrote has been preserved and catalogued

358
00:21:54,240 –> 00:21:57,800
in his hometown of Hanover in northern Germany.

359
00:21:57,800 –> 00:22:01,040
His actual manuscripts are kept under lock and key,

360
00:22:01,040 –> 00:22:04,360
particularly the manuscript which shows how Leibniz

361
00:22:04,360 –> 00:22:09,720
also discovered the miracle of calculus, shortly after Newton.

362
00:22:09,720 –> 00:22:11,520
What age was he when he wrote…

363
00:22:11,520 –> 00:22:16,720
He was 29 years old and that’s the time, within two months, he developed

364
00:22:16,720 –> 00:22:19,640

differential calculus and integral calculus.
In two months?

365
00:22:19,640 –> 00:22:21,600

Yeah.
Fast and furious, when it comes, er…

366
00:22:21,600 –> 00:22:23,240
Yeah.

367
00:22:23,240 –> 00:22:26,440
There is a little scrap of paper over here. What’s that one?

368
00:22:26,440 –> 00:22:29,840

A letter or…
That’s a small manuscript of Leibniz’s notes.

369
00:22:32,560 –> 00:22:37,280
“Sometimes it happens that in the morning lying in the bed,

370
00:22:37,280 –> 00:22:40,960
“I have so many ideas that it takes the whole morning and sometimes

371
00:22:40,960 –> 00:22:45,760
“even longer to note all these ideas and bring them to paper.”

372
00:22:45,760 –> 00:22:47,280
I suppose, that’s beautiful.

373
00:22:47,280 –> 00:22:51,480
I suppose that he liked to lie in the bed in the morning.

374
00:22:51,480 –> 00:22:53,400

A true mathematician.
Yeah.

375
00:22:53,400 –> 00:22:55,680
He spends his time thinking in bed.

376
00:22:55,680 –> 00:22:58,640
I see you’ve got some paintings down here.

377
00:22:58,640 –> 00:23:00,280
A painting.

378
00:23:00,280 –> 00:23:02,360
This is what he looked like. Right.

379
00:23:03,880 –> 00:23:07,280
Even though he didn’t become quite the 17th century celebrity

380
00:23:07,280 –> 00:23:10,560
that Newton did, it wasn’t such a bad life.

381
00:23:10,560 –> 00:23:12,520
Leibniz worked for the Royal Family

382
00:23:12,520 –> 00:23:16,600
of Hanover and travelled around Europe representing their interests.

383
00:23:16,600 –> 00:23:19,040
This gave him plenty of time to indulge in

384
00:23:19,040 –> 00:23:23,400
his favourite intellectual pastimes, which were wide, even for the time.

385
00:23:23,400 –> 00:23:26,960
He devised a plan for reunifying the Protestant and Roman Catholic

386
00:23:26,960 –> 00:23:32,000
churches, a proposal for France to conquer Egypt and contributions to

387
00:23:32,000 –> 00:23:36,280
philosophy and logic which are still highly rated today.

388
00:23:36,280 –> 00:23:39,880

He wrote all these letters?
Yeah.
That’s absolutely extraordinary.

389
00:23:39,880 –> 00:23:43,080
He must have cloned himself. I can’t believe there was just one Leibniz!

390
00:23:43,080 –> 00:23:46,040
‘But Leibniz was not just man of words.

391
00:23:46,040 –> 00:23:47,640
‘He was also one of the first people

392
00:23:47,640 –> 00:23:49,480
‘to invent practical calculating machines

393
00:23:49,480 –> 00:23:54,520
‘that worked on the binary system, true forerunners of the computer.

394
00:23:54,520 –> 00:23:58,680
‘300 years later, the engineering department at Leibniz University

395
00:23:58,680 –> 00:24:02,880
‘in Hanover have put them together following Leibniz’s blueprint.’

396
00:24:02,880 –> 00:24:04,760
I love all the ball bearings, so these are going to be all

397
00:24:04,760 –> 00:24:06,680
of our zeros and ones. So a ball bearing is a one.

398
00:24:06,680 –> 00:24:10,720
Only zero and one. Now we represent a number 127.

399
00:24:10,720 –> 00:24:15,960

In binary, it means that we have the first seven digits in one.
Yeah.

400
00:24:15,960 –> 00:24:18,880

And now I give the number one.
OK.

401
00:24:18,880 –> 00:24:24,360
Now we add 127 plus one - is 128, which is two, power eight.

402
00:24:24,360 –> 00:24:28,000

Oh, OK. So there’s going to be lots of action.
Would you show this here?

403
00:24:28,000 –> 00:24:30,480
This is the money shot.

404
00:24:30,480 –> 00:24:33,560
So we’re going to add one. Oops. Here we go. They’re all carrying.

405
00:24:33,560 –> 00:24:36,520
So this 128 is two power eight.

406
00:24:36,520 –> 00:24:42,360
Excellent, so 127 in binary is 1, 1, 1, 1, 1, 1, 1, which is

407
00:24:42,360 –> 00:24:44,320
all the ball bearings here.

408
00:24:44,320 –> 00:24:46,320
To add one it all gets

409
00:24:46,320 –> 00:24:50,920
carried, this goes to 0, 0, 0, 0, and we have a power of two here.

410
00:24:50,920 –> 00:24:53,080
So this mechanism gets rid of all the ball bearings that you

411
00:24:53,080 –> 00:24:56,680

don’t need. It’s like pinball, mathematical pinball.
Exactly.

412
00:24:56,680 –> 00:24:58,200
I love this machine!

413
00:25:03,680 –> 00:25:08,120
After a hard day’s work, Leibniz often came here,

414
00:25:08,120 –> 00:25:10,080
the famous gardens of Herrenhausen,

415
00:25:10,080 –> 00:25:14,800
now in the middle of Hanover, but then on the outskirts of the city.

416
00:25:14,800 –> 00:25:17,400
There’s something about mathematics and walking.

417
00:25:17,400 –> 00:25:21,040
I don’t know, you’ve been working at your desk all day, all morning

418
00:25:21,040 –> 00:25:22,640
on some problem and your head’s all

419
00:25:22,640 –> 00:25:25,040
fuzzy, and you just need to come and have a walk.

420
00:25:25,040 –> 00:25:27,760
You let your subconscious mind kind of take over and sometimes

421
00:25:27,760 –> 00:25:31,880
you get your breakthrough just looking at the trees or whatever.

422
00:25:31,880 –> 00:25:35,160
I’ve had some of my best ideas whilst walking in my local park,

423
00:25:35,160 –> 00:25:39,120
so I’m hoping to get a little bit of inspiration here on Leibniz’s

424
00:25:39,120 –> 00:25:40,760
local stomping ground.

425
00:25:44,240 –> 00:25:47,120
I didn’t get the chance to purge my mind of mathematical challenges

426
00:25:47,120 –> 00:25:49,240
because in the years since Leibniz lived here,

427
00:25:49,240 –> 00:25:50,440
someone has built a maze.

428
00:25:50,440 –> 00:25:53,520
Well, there is a mathematical formula for getting out of a maze,

429
00:25:53,520 –> 00:25:57,200
which is if you put your left hand on the side of the maze and just

430
00:25:57,200 –> 00:26:00,760
keep it there, keep on winding round, you eventually get out.

431
00:26:00,760 –> 00:26:03,760
That’s the theory, at least. Let’s see whether it works!

432
00:26:11,080 –> 00:26:13,600
Leibniz had no such distractions.

433
00:26:13,600 –> 00:26:17,320
Within five years, he’d worked out the details of the calculus,

434
00:26:17,320 –> 00:26:19,160
seemingly independent from Newton,

435
00:26:19,160 –> 00:26:21,680
although he knew about Newton’s work,

436
00:26:21,680 –> 00:26:26,200
but unlike Newton, Leibniz was quite happy to make his work known

437
00:26:26,200 –> 00:26:29,440
and so mathematicians across Europe heard about the calculus first

438
00:26:29,440 –> 00:26:35,680
from him and not from Newton, and that’s when all the trouble started.

439
00:26:35,680 –> 00:26:39,200
Throughout mathematical history, there have been lots of priority

440
00:26:39,200 –> 00:26:40,800
disputes and arguments.

441
00:26:40,800 –> 00:26:43,800
It may seem a little bit petty and schoolboyish.

442
00:26:43,800 –> 00:26:46,600
We really want our name to be on that theorem.

443
00:26:46,600 –> 00:26:49,800
This is our one chance for a little bit of immortality because that

444
00:26:49,800 –> 00:26:54,120
theorem’s going to last forever and that’s why we dedicate so much time

445
00:26:54,120 –> 00:26:55,920
to trying to crack these things.

446
00:26:55,920 –> 00:26:57,800
Somehow we can’t believe that somebody else

447
00:26:57,800 –> 00:27:00,000
has got it at the same time as us.

448
00:27:00,000 –> 00:27:03,040
These are our theorems, our babies, our children and we

449
00:27:03,040 –> 00:27:06,000
don’t want to share the credit.

450
00:27:06,000 –> 00:27:08,440
Back in London, Newton certainly didn’t want

451
00:27:08,440 –> 00:27:13,040
to share credit with Leibniz, who he thought of as a Hanoverian upstart.

452
00:27:13,040 –> 00:27:16,160
After years of acrimony and accusation, the Royal Society

453
00:27:16,160 –> 00:27:21,120
in London was asked to adjudicate between the rival claims.

454
00:27:21,120 –> 00:27:23,080
The Royal Society gave Newton credit

455
00:27:23,080 –> 00:27:25,240
for the first discovery of the calculus

456
00:27:25,240 –> 00:27:28,880
and Leibniz credit for the first publication,

457
00:27:28,880 –> 00:27:33,400
but in their final judgment, they accused Leibniz of plagiarism.

458
00:27:33,400 –> 00:27:36,640
However, that might have had something to do with the fact that

459
00:27:36,640 –> 00:27:41,920
the report was written by their President, one Sir Isaac Newton.

460
00:27:44,040 –> 00:27:46,440
Leibniz was incredibly hurt.

461
00:27:46,440 –> 00:27:50,400
He admired Newton and never really recovered.

462
00:27:50,400 –> 00:27:52,440
He died in 1716.

463
00:27:52,440 –> 00:27:56,200
Newton lived on another 11 years and was buried in the grandeur of

464
00:27:56,200 –> 00:27:58,240
Westminster Abbey.

465
00:27:58,240 –> 00:28:00,360
Leibniz’s memorial, by contrast,

466
00:28:00,360 –> 00:28:02,520
is here in this small church in Hanover.

467
00:28:02,520 –> 00:28:06,040
The irony is that it’s Leibniz’s mathematics which

468
00:28:06,040 –> 00:28:08,800
eventually triumphs, not Newton’s.

469
00:28:11,040 –> 00:28:13,720
I’m a big Leibniz fan.

470
00:28:13,720 –> 00:28:16,920
Quite often revolutions in mathematics are about producing the

471
00:28:16,920 –> 00:28:19,680
right language to capture a new vision and that’s what

472
00:28:19,680 –> 00:28:21,520
Leibniz was so good at.

473
00:28:21,520 –> 00:28:25,280
Leibniz’s notation, his way of writing the calculus,

474
00:28:25,280 –> 00:28:27,360
captured its true spirit.

475
00:28:27,360 –> 00:28:29,960
It’s still the one we use in maths today.

476
00:28:29,960 –> 00:28:34,320
Newton’s notation was, for many mathematicians, clumsy and difficult

477
00:28:34,320 –> 00:28:38,600
to use and so while British mathematics loses its way a little,

478
00:28:38,600 –> 00:28:43,360
the story of maths switches to the very heart of Europe, Basel.

479
00:28:48,560 –> 00:28:52,280
In its heyday in the 18th century, the free city of Basel in

480
00:28:52,280 –> 00:28:56,840
Switzerland was the commercial hub of the entire Western world.

481
00:28:56,840 –> 00:28:59,640
Around this maelstrom of trade, there developed a tradition of

482
00:28:59,640 –> 00:29:03,520
learning, particularly learning which connected with commerce

483
00:29:03,520 –> 00:29:06,400
and one family summed all this up.

484
00:29:06,400 –> 00:29:11,160
It’s kind of curious - artists often have children who are artists.

485
00:29:11,160 –> 00:29:15,480
Musicians, their children are often musicians, but us mathematicians,

486
00:29:15,480 –> 00:29:17,680
our children don’t tend to be mathematicians.

487
00:29:17,680 –> 00:29:19,720
I’m not sure why it is.

488
00:29:19,720 –> 00:29:23,000
At least that’s my view, although others dispute it.

489
00:29:23,000 –> 00:29:25,000
What no-one disagrees with

490
00:29:25,000 –> 00:29:30,080
is there is one great dynasty of mathematicians, the Bernoullis.

491
00:29:30,080 –> 00:29:33,760
In the 18th and 19th centuries they produced half a dozen

492
00:29:33,760 –> 00:29:37,040
outstanding mathematicians, any of which we would have been

493
00:29:37,040 –> 00:29:41,800
proud to have had in Britain, and they all came from Basel.

494
00:29:41,800 –> 00:29:44,960
You might have great minds like Newton and Leibniz who make

495
00:29:44,960 –> 00:29:48,440
these fundamental breakthroughs, but you also need the disciples

496
00:29:48,440 –> 00:29:51,680
who take that message, clarify it, realise its implications,

497
00:29:51,680 –> 00:29:55,480
then spread it wide. The family were originally merchants,

498
00:29:55,480 –> 00:29:57,440
and this is one of their houses.

499
00:29:57,440 –> 00:30:00,360
It’s now part of the University of Basel

500
00:30:00,360 –> 00:30:03,440
and it’s been completely refurbished, apart from one room,

501
00:30:03,440 –> 00:30:07,360
which has been kept very much as the family would have used it.

502
00:30:07,360 –> 00:30:09,720
Dr Fritz Nagel, keeper of the Bernoulli Archive,

503
00:30:09,720 –> 00:30:12,480
has promised to show it to me.

504
00:30:12,480 –> 00:30:15,120

If we can find it.
No, we’re on the wrong floor.

505
00:30:15,120 –> 00:30:17,440
Wrong floor, OK. Right!

506
00:30:17,440 –> 00:30:19,560
Oh, look.

507
00:30:19,560 –> 00:30:21,440
Can we take an apple?

508
00:30:21,440 –> 00:30:24,000
‘No, wrong mathematician.

509
00:30:24,000 –> 00:30:26,480
‘Eventually, we got there.’

510
00:30:26,480 –> 00:30:28,840
This is where the Bernoullis would have done

511
00:30:28,840 –> 00:30:30,600
some of their mathematics.

512
00:30:30,600 –> 00:30:33,680
‘I was really just being polite.

513
00:30:33,680 –> 00:30:36,400
‘The only thing of interest was an old stove.’

514
00:30:36,400 –> 00:30:40,200
Now, of the Bernoullis, which is your favourite?

515
00:30:40,200 –> 00:30:44,080
My favourite Bernoulli is Johann I.

516
00:30:44,080 –> 00:30:49,640
He is the most smart mathematician.

517
00:30:49,640 –> 00:30:54,160
Perhaps his brother Jakob was the mathematician

518
00:30:54,160 –> 00:30:57,160
with the deeper insight into problems,

519
00:30:57,160 –> 00:30:59,800
but Johann found elegant solutions.

520
00:30:59,800 –> 00:31:03,920
The brothers didn’t like each other much, but both worshipped Leibniz.

521
00:31:03,920 –> 00:31:06,560
They corresponded with him, stood up for him

522
00:31:06,560 –> 00:31:10,960
against Newton’s allies, and spread his calculus throughout Europe.

523
00:31:10,960 –> 00:31:15,440
Leibnitz was very happy to have found two gifted mathematicians

524
00:31:15,440 –> 00:31:20,640
outside of his personal circle of friends who mastered his calculus

525
00:31:20,640 –> 00:31:23,680
and could distribute it in the scientific community.

526
00:31:23,680 –> 00:31:28,320

That was very important for Leibniz.
And important for maths, too.

527
00:31:28,320 –> 00:31:32,440
Without the Bernoullis, it would have taken much longer for calculus

528
00:31:32,440 –> 00:31:36,200
to become what it is today, a cornerstone of mathematics.

529
00:31:36,200 –> 00:31:38,760
At least, that is Dr Nagel’s contention.

530
00:31:38,760 –> 00:31:41,240
And he is a great Bernoulli fan.

531
00:31:41,240 –> 00:31:44,520
He has arranged for me to meet Professor Daniel Bernoulli,

532
00:31:44,520 –> 00:31:46,960
the latest member of the family,

533
00:31:46,960 –> 00:31:49,680
whose famous name ensures he gets some odd e-mails.

534
00:31:49,680 –> 00:31:51,320
Another one of which I got was,

535
00:31:51,320 –> 00:31:54,440
“Professor Bernoulli, can you give me a hand with calculus?”

536
00:31:54,440 –> 00:31:58,560
To find a Bernoulli, you expect them to be able to do calculus.

537
00:31:58,560 –> 00:32:02,640
‘But this Daniel Bernoulli is a professor of geology.

538
00:32:02,640 –> 00:32:05,880
‘The maths gene seems to have truly died out.

539
00:32:05,880 –> 00:32:07,880
‘And during our very hearty dinner,

540
00:32:07,880 –> 00:32:11,200
‘I found myself wandering back to maths.’

541
00:32:11,200 –> 00:32:14,400
It is a bit unfair on the Bernoullis to describe them simply

542
00:32:14,400 –> 00:32:16,040
as disciples of Leibniz.

543
00:32:16,040 –> 00:32:18,960
One of their many great contributions to mathematics

544
00:32:18,960 –> 00:32:23,800
was to develop the calculus to solve a classic problem of the day.

545
00:32:23,800 –> 00:32:26,360
Imagine a ball rolling down a ramp.

546
00:32:26,360 –> 00:32:29,320
The task is to design a ramp that will get the ball

547
00:32:29,320 –> 00:32:32,440
from the top to the bottom in the fastest time possible.

548
00:32:32,440 –> 00:32:36,080
You might think that a straight ramp would be quickest.

549
00:32:36,080 –> 00:32:37,920
Or possibly a curved one like this

550
00:32:37,920 –> 00:32:40,720
that gives the ball plenty of downward momentum.

551
00:32:40,720 –> 00:32:42,880
In fact, it’s neither of these.

552
00:32:42,880 –> 00:32:45,960
Calculus shows that it is what we call a cycloid,

553
00:32:45,960 –> 00:32:49,640
the path traced by a point on the rim of a moving bicycle wheel.

554
00:32:49,640 –> 00:32:53,360
This application of the calculus by the Bernoullis, which became known

555
00:32:53,360 –> 00:32:55,520
as the calculus of variation,

556
00:32:55,520 –> 00:32:58,600
has become one of the most powerful aspects of the mathematics

557
00:32:58,600 –> 00:33:01,560
of Leibniz and Newton. Investors use it to maximise profits.

558
00:33:01,560 –> 00:33:05,240
Engineers exploit it to minimise energy use.

559
00:33:05,240 –> 00:33:08,560
Designers apply it to optimise construction.

560
00:33:08,560 –> 00:33:10,680
It has now become one of the linchpins

561
00:33:10,680 –> 00:33:12,840
of our modern technological world.

562
00:33:12,840 –> 00:33:17,160
Meanwhile, things were getting more interesting in the restaurant.

563
00:33:17,160 –> 00:33:18,760
Here is my second surprise.

564
00:33:18,760 –> 00:33:22,000
Let me introduce Mr Leonhard Euler.

565
00:33:22,000 –> 00:33:23,720
Daniel Bernoulli.

566
00:33:23,720 –> 00:33:27,920
‘Leonhard Euler, one of the most famous names in mathematics.

567
00:33:27,920 –> 00:33:29,600
‘This Leonhard is a descendant

568
00:33:29,600 –> 00:33:34,080
‘of the original Leonhard Euler, star pupil of Johann Bernoulli.’

569
00:33:34,080 –> 00:33:36,640
I am the ninth generation,

570
00:33:36,640 –> 00:33:39,840
the fourth Leonhard in our family

571
00:33:39,840 –> 00:33:42,440
after Leonard Euler I, the mathematician.

572
00:33:42,440 –> 00:33:44,840
OK. And yourself, are you a mathematician?

573
00:33:44,840 –> 00:33:47,840
Actually, I am a business analyst.

574
00:33:47,840 –> 00:33:51,920
I can’t study mathematics with my name.

575
00:33:51,920 –> 00:33:55,320
Everyone will expect you to prove that the Riemann hypothesis!

576
00:33:55,320 –> 00:33:58,600
Perhaps it’s just as well that Leonhard decided

577
00:33:58,600 –> 00:34:02,240
not to follow in the footsteps of his illustrious ancestor.

578
00:34:02,240 –> 00:34:04,600
He’d have had a lot to live up to.

579
00:34:13,000 –> 00:34:15,000
I am going in a boat across the Rhine,

580
00:34:15,000 –> 00:34:17,560
and I’m feeling a little bit worse for wear.

581
00:34:17,560 –> 00:34:21,120
Last night’s dinner with Mr Euler and Professor Bernoulli

582
00:34:21,120 –> 00:34:25,480
degenerated into toasts to all the theorems the Bernoullis and Eulers

583
00:34:25,480 –> 00:34:28,600
have proved, and by God, they have proved quite a lot of them!

584
00:34:28,600 –> 00:34:30,880
Never again.

585
00:34:30,880 –> 00:34:34,800
I was getting disapproving glances from my fellow passengers as well.

586
00:34:34,800 –> 00:34:37,360
Luckily, it was only a short trip.

587
00:34:37,360 –> 00:34:41,960
Not like the trip that Euler took in 1728 to start a new life.

588
00:34:41,960 –> 00:34:45,240
Euler may have been the prodigy of Johann Bernoulli,

589
00:34:45,240 –> 00:34:47,800
but there was no room for him in the city.

590
00:34:47,800 –> 00:34:49,520
If your name wasn’t Bernoulli,

591
00:34:49,520 –> 00:34:53,240
there was little chance of getting a job in mathematics here in Basel.

592
00:34:53,240 –> 00:34:55,600
But Daniel, the son of Johann Bernoulli,

593
00:34:55,600 –> 00:34:57,120
was a great friend of Euler

594
00:34:57,120 –> 00:35:00,360
and managed to get him a job at his university.

595
00:35:00,360 –> 00:35:03,280
But to get there would take seven weeks,

596
00:35:03,280 –> 00:35:05,800
because Daniel’s university was in Russia.

597
00:35:08,280 –> 00:35:11,720
It wasn’t an intellectual powerhouse like Berlin or Paris,

598
00:35:11,720 –> 00:35:17,320
but St Petersburg was by no means unsophisticated in the 18th century.

599
00:35:17,320 –> 00:35:21,440
Peter the Great had created a city very much in the European style.

600
00:35:21,440 –> 00:35:26,080
And every fashionable city at the time had a scientific academy.

601
00:35:27,840 –> 00:35:30,040
Peter’s Academy is now a museum.

602
00:35:30,040 –> 00:35:34,320
It includes several rooms full of the kind of grotesque curiosities

603
00:35:34,320 –> 00:35:38,000
that are usually kept out of the public display in the West.

604
00:35:38,000 –> 00:35:39,960
But in the 1730s,

605
00:35:39,960 –> 00:35:44,400
this building was a centre for ground-breaking research.

606
00:35:44,400 –> 00:35:46,880
It is where Euler found his intellectual home.

607
00:35:50,280 –> 00:35:57,000

I am sure that there could never be a more contented man than me…

608
00:35:58,000 –> 00:36:00,840
Many of the ideas that were bubbling away at the time -

609
00:36:00,840 –> 00:36:02,480
calculus of variation,

610
00:36:02,480 –> 00:36:06,560
Fermat’s theory of numbers - crystallised in Euler’s hands.

611
00:36:06,560 –> 00:36:09,560
But he was also creating incredibly modern mathematics,

612
00:36:09,560 –> 00:36:12,040
topology and analysis.

613
00:36:12,040 –> 00:36:15,240
Much of the notation that I use today as a mathematician

614
00:36:15,240 –> 00:36:19,240
was created by Euler, numbers like e and i.

615
00:36:19,240 –> 00:36:23,000
Euler also popularised the use of the symbol pi.

616
00:36:23,000 –> 00:36:25,200
He even combined these numbers together

617
00:36:25,200 –> 00:36:28,120
in one of the most beautiful formulas of mathematics,

618
00:36:28,120 –> 00:36:32,920
e to the power of i times pi is equal to -1.

619
00:36:32,920 –> 00:36:36,600
An amazing feat of mathematical alchemy.

620
00:36:36,600 –> 00:36:39,960
His life, in fact, is full of mathematical magic.

621
00:36:39,960 –> 00:36:43,560
Euler applied his skills to an immense range of topics,

622
00:36:43,560 –> 00:36:46,440
from prime numbers to optics to astronomy.

623
00:36:46,440 –> 00:36:49,840
He devised a new system of weights and measures, wrote a textbook

624
00:36:49,840 –> 00:36:54,520
on mechanics, and even found time to develop a new theory of music.

625
00:36:59,360 –> 00:37:01,440
I think of him as the Mozart of maths.

626
00:37:01,440 –> 00:37:04,800
And that view is shared by the mathematician Nikolai Vavilov,

627
00:37:04,800 –> 00:37:07,360
who met me at the house that was given to Euler

628
00:37:07,360 –> 00:37:10,040
by Catherine the Great.

629
00:37:10,040 –> 00:37:14,360
Euler lived here from ‘66 to ‘83, which means from the year

630
00:37:14,360 –> 00:37:17,640
he came back to St Petersburg to the year he died.

631
00:37:17,640 –> 00:37:22,720
And he was a member of the Russian Academy of Sciences,

632
00:37:22,720 –> 00:37:24,760
and their greatest mathematician.

633
00:37:24,760 –> 00:37:27,360
That is exactly what it says.

634
00:37:27,360 –> 00:37:29,360

What is it now?
It is a school.

635
00:37:29,360 –> 00:37:30,920
Shall we go in and see?

636
00:37:30,920 –> 00:37:33,760
OK.

637
00:37:33,760 –> 00:37:38,920
‘I’m not sure Nikolai entirely approved. But nothing ventured…’

638
00:37:38,920 –> 00:37:41,320
Perhaps we should talk to the head teacher.

639
00:37:46,200 –> 00:37:48,320
The head didn’t mind at all.

640
00:37:48,320 –> 00:37:50,680
I rather got the impression that she was used

641
00:37:50,680 –> 00:37:53,200
to people dropping in to talk about Euler.

642
00:37:53,200 –> 00:37:57,040
She even had a couple of very able pupils suspiciously close to hand.

643
00:37:57,040 –> 00:38:02,240
These two young ladies are ready to tell a few words about the scientist

644
00:38:02,240 –> 00:38:04,400
and about this very building.

645
00:38:04,400 –> 00:38:06,200
They certainly knew their stuff.

646
00:38:06,200 –> 00:38:09,880
They had undertaken an entire classroom project on Euler,

647
00:38:09,880 –> 00:38:13,160
his long life, happy marriage and 13 children.

648
00:38:13,160 –> 00:38:16,160
And then his tragedies - only five of his children

649
00:38:16,160 –> 00:38:17,720
survived to adulthood.

650
00:38:17,720 –> 00:38:21,200
His first wife, who he adored, died young.

651
00:38:21,200 –> 00:38:23,640
He started losing most of his eyesight.

652
00:38:26,720 –> 00:38:31,480
So for the last years of his life, he still continued to work, actually.

653
00:38:31,480 –> 00:38:34,560
He continued his mathematical research.

654
00:38:34,560 –> 00:38:36,480
I read a quote that said now with his blindness,

655
00:38:36,480 –> 00:38:38,640
he hasn’t got any distractions,

656
00:38:38,640 –> 00:38:42,480
he can finally get on with his mathematics. A positive attitude.

657
00:38:42,480 –> 00:38:46,200
It was a totally unexpected and charming visit.

658
00:38:46,200 –> 00:38:49,200
Although I couldn’t resist sneaking back and correcting

659
00:38:49,200 –> 00:38:53,640
one of the equations on the board when everyone else had left.

660
00:38:54,960 –> 00:38:59,960
To demonstrate one of my favourite Euler theorems, I needed a drink.

661
00:38:59,960 –> 00:39:02,920
It concerns calculating infinite sums,

662
00:39:02,920 –> 00:39:06,280
the discovery that shot Euler to the top of the mathematical pops

663
00:39:06,280 –> 00:39:08,840
when it was announced in 1735.

664
00:39:11,120 –> 00:39:15,680
Take one shot glass full of vodka and add it to this tall glass here.

665
00:39:17,960 –> 00:39:22,400
Next, take a glass which is a quarter full, or a half squared,

666
00:39:22,400 –> 00:39:24,120
and add it to the first glass.

667
00:39:25,880 –> 00:39:30,240
Next, take a glass which is a ninth full, or a third squared,

668
00:39:30,240 –> 00:39:31,920
and add that one.

669
00:39:31,920 –> 00:39:36,880
Now, if I keep on adding infinitely many glasses where each one

670
00:39:36,880 –> 00:39:43,200
is a fraction squared, how much will be in this tall glass here?

671
00:39:43,200 –> 00:39:45,080
It was called the Basel problem

672
00:39:45,080 –> 00:39:47,760
after the Bernoullis tried and failed to solve it.

673
00:39:47,760 –> 00:39:52,600
Daniel Bernoulli knew that you would not get an infinite amount of vodka.

674
00:39:52,600 –> 00:39:57,280
He estimated that the total would come to about one and three fifths.

675
00:39:57,280 –> 00:39:59,280
But then Euler came along.

676
00:39:59,280 –> 00:40:03,520
Daniel was close, but mathematics is about precision.

677
00:40:03,520 –> 00:40:06,640
Euler calculated that the total height of the vodka

678
00:40:06,640 –> 00:40:10,960
would be exactly pi squared divided by six.

679
00:40:13,040 –> 00:40:15,160
It was a complete surprise.

680
00:40:15,160 –> 00:40:17,800
What on earth did adding squares of fractions

681
00:40:17,800 –> 00:40:20,520
have to do with the special number pi?

682
00:40:20,520 –> 00:40:23,600
But Euler’s analysis showed that they were two sides

683
00:40:23,600 –> 00:40:25,240
of the same equation.

684
00:40:25,240 –> 00:40:29,280
One plus a quarter plus a ninth plus a sixteenth

685
00:40:29,280 –> 00:40:34,560
and so on to infinity is equal to pi squared over six.

686
00:40:34,560 –> 00:40:38,080
That’s still quite a lot of vodka, but here goes.

687
00:40:43,280 –> 00:40:46,440
Euler would certainly be a hard act to follow.

688
00:40:46,440 –> 00:40:49,560
Mathematicians from two countries would try.

689
00:40:49,560 –> 00:40:53,680
Both France and Germany were caught up in the age of revolution

690
00:40:53,680 –> 00:40:56,960
that was sweeping Europe in the late 18th century.

691
00:40:56,960 –> 00:40:59,760
Both desperately needed mathematicians.

692
00:40:59,760 –> 00:41:04,600
But they went about supporting mathematics rather differently.

693
00:41:04,600 –> 00:41:05,960
Here in France,

694
00:41:05,960 –> 00:41:09,560
the Revolution emphasised the usefulness of mathematics.

695
00:41:09,560 –> 00:41:12,280
Napoleon recognised that if you were going to have

696
00:41:12,280 –> 00:41:14,920
the best military machine, the best weaponry,

697
00:41:14,920 –> 00:41:17,720
then you needed the best mathematicians.

698
00:41:17,720 –> 00:41:21,120
Napoleon’s reforms gave mathematics a big boost.

699
00:41:21,120 –> 00:41:24,400
But this was a mathematics that was going to serve society.

700
00:41:25,920 –> 00:41:30,000
Here in the German states, the great educationalist Wilhelm von Humboldt

701
00:41:30,000 –> 00:41:33,840
was also committed to mathematics, but a mathematics that was detached

702
00:41:33,840 –> 00:41:36,360
from the demands of the State and the military.

703
00:41:36,360 –> 00:41:42,200
Von Humboldt’s educational reforms valued mathematics for its own sake.

704
00:41:42,200 –> 00:41:46,080
In France, they got wonderful mathematicians, like Joseph Fourier,

705
00:41:46,080 –> 00:41:49,280
whose work on sound waves we still benefit from today.

706
00:41:49,280 –> 00:41:53,360
MP3 technology is based on Fourier analysis.

707
00:41:53,360 –> 00:41:56,680
But in Germany, they got, at least in my opinion,

708
00:41:56,680 –> 00:41:58,680
the greatest mathematician ever.

709
00:42:01,960 –> 00:42:03,920
Quaint and quiet,

710
00:42:03,920 –> 00:42:08,080
the university town of Gottingen may seem like a bit of a backwater.

711
00:42:08,080 –> 00:42:12,000
But this little town has been home to some of the giants of maths,

712
00:42:12,000 –> 00:42:14,320
including the man who’s often described

713
00:42:14,320 –> 00:42:19,360
as the Prince of Mathematics, Carl Friedrich Gauss.

714
00:42:19,360 –> 00:42:23,240
Few non-mathematicians, however, seem to know anything about him.

715
00:42:23,240 –> 00:42:25,040
Not in Paris.

716
00:42:25,040 –> 00:42:27,000
Qui s’appelle Carl Friedrich Gauss?

717
00:42:27,000 –> 00:42:28,880

Non.
Non?

718
00:42:28,880 –> 00:42:30,480
‘Not in Oxford.’

719
00:42:30,480 –> 00:42:34,440

I’ve heard the name but I couldn’t tell you.
No idea.
No idea?
No.

720
00:42:34,440 –> 00:42:37,480
‘And I’m afraid to say, not even in modern Germany.’

721
00:42:37,480 –> 00:42:39,400

Nein.
Nein? OK.

722
00:42:39,400 –> 00:42:41,040

I don’t know.
You don’t know?

723
00:42:41,040 –> 00:42:44,600
But in Gottingen, everyone knows who Gauss is.

724
00:42:44,600 –> 00:42:47,040
He’s the local hero.

725
00:42:47,040 –> 00:42:49,440
His father was a stonemason

726
00:42:49,440 –> 00:42:52,560
and it’s likely that Gauss would have become one, too.

727
00:42:52,560 –> 00:42:55,720
But his singular talent was recognised by his mother,

728
00:42:55,720 –> 00:42:57,560
and she helped ensure

729
00:42:57,560 –> 00:43:01,320
that he received the best possible education.

730
00:43:01,320 –> 00:43:05,080
Every few years in the news, you hear about a new prodigy

731
00:43:05,080 –> 00:43:08,240
who’s passed their A-levels at ten, gone to university at 12,

732
00:43:08,240 –> 00:43:10,240
but nobody compares to Gauss.

733
00:43:10,240 –> 00:43:13,680
Already at the age of 12, he was criticising Euclid’s geometry.

734
00:43:13,680 –> 00:43:16,960
At 15, he discovered a new pattern in prime numbers

735
00:43:16,960 –> 00:43:20,240
which had eluded mathematicians for 2,000 years.

736
00:43:20,240 –> 00:43:24,000
And at 19, he discovered the construction of a 17-sided figure

737
00:43:24,000 –> 00:43:26,880
which nobody had known before this time.

738
00:43:30,200 –> 00:43:34,160
His early successes encouraged Gauss to keep a diary.

739
00:43:34,160 –> 00:43:36,120
Here at the University of Gottingen,

740
00:43:36,120 –> 00:43:40,000
you can still read it if you can understand Latin.

741
00:43:40,000 –> 00:43:42,120
Fortunately, I had help.

742
00:43:44,200 –> 00:43:46,960
The first entry is in 1796.

743
00:43:46,960 –> 00:43:49,600

Is it possible to lift it up?
Yes, but be careful.

744
00:43:49,600 –> 00:43:54,160
It’s really one of the most valuable things that this library possesses.

745
00:43:54,160 –> 00:43:56,680

Yes, I can believe that.
He writes beautifully.

746
00:43:56,680 –> 00:43:59,120
It is aesthetically very pleasing,

747
00:43:59,120 –> 00:44:02,560
even if people don’t understand what it is.

748
00:44:02,560 –> 00:44:05,320
I’m going to put this down. It’s very delicate.

749
00:44:05,320 –> 00:44:08,520
The diary proves that some of Gauss’ ideas

750
00:44:08,520 –> 00:44:10,240
were 100 years ahead of their time.

751
00:44:10,240 –> 00:44:15,520
Here are some sines and integrals. Very different sort of mathematics.

752
00:44:15,520 –> 00:44:20,400
Yes, this was the first intimations of the theory

753
00:44:20,400 –> 00:44:25,040
of elliptic functions, which was one of his other great developments.

754
00:44:25,040 –> 00:44:28,600
And here you see something that is basically

755
00:44:28,600 –> 00:44:30,720
the Riemann zeta function appearing.

756
00:44:30,720 –> 00:44:34,200
Wow, gosh! That’s very impressive.

757
00:44:34,200 –> 00:44:38,880
The zeta function has become a vital element in our present understanding

758
00:44:38,880 –> 00:44:43,600
of the distribution of the building blocks of all numbers, the primes.

759
00:44:43,600 –> 00:44:47,280
There is somewhere in the diary here where he says,

760
00:44:47,280 –> 00:44:49,280
“I have made this wonderful discovery

761
00:44:49,280 –> 00:44:52,000
“and incidentally, a son was born today.”

762
00:44:52,000 –> 00:44:53,640
We see his priorities!

763
00:44:53,640 –> 00:44:55,560
Yes, indeed!

764
00:44:55,560 –> 00:44:58,600
I think I know a few mathematicians like that, too.

765
00:45:00,320 –> 00:45:03,800
My priorities, though, for the rest of the afternoon were clear.

766
00:45:03,800 –> 00:45:05,560
I needed another walk.

767
00:45:05,560 –> 00:45:08,960
Fortunately, Gottingen is surrounded by good woodland trails.

768
00:45:08,960 –> 00:45:10,920
It was a perfect setting for me

769
00:45:10,920 –> 00:45:13,440
to think more about Gauss’ discoveries.

770
00:45:22,400 –> 00:45:26,280
Gauss’ mathematics has touched many parts of the mathematical world,

771
00:45:26,280 –> 00:45:31,320
but I’m going to just choose one of them, a fun one - imaginary numbers.

772
00:45:31,320 –> 00:45:34,920
In the 16th and 17th century, European mathematicians

773
00:45:34,920 –> 00:45:40,120
imagined the square root of minus one and gave it the symbol i.

774
00:45:40,120 –> 00:45:42,760
They didn’t like it much, but it solved equations

775
00:45:42,760 –> 00:45:45,240
that couldn’t be solved any other way.

776
00:45:46,320 –> 00:45:49,760
Imaginary numbers have helped us to understand radio waves,

777
00:45:49,760 –> 00:45:52,000
to build bridges and aeroplanes.

778
00:45:52,000 –> 00:45:54,240
They’re even the key to quantum physics,

779
00:45:54,240 –> 00:45:56,560
the science of the sub-atomic world.

780
00:45:56,560 –> 00:46:01,400
They’ve provided a map to see how things really are.

781
00:46:01,400 –> 00:46:05,560
But back in the early 19th century, they had no map, no picture

782
00:46:05,560 –> 00:46:08,560
of how imaginary numbers connected with real numbers.

783
00:46:08,560 –> 00:46:10,760
Where is this new number?

784
00:46:10,760 –> 00:46:14,240
There’s no room on the number line for the square root of minus one.

785
00:46:14,240 –> 00:46:16,320
I’ve got the positive numbers running out here,

786
00:46:16,320 –> 00:46:17,880
the negative numbers here.

787
00:46:17,880 –> 00:46:21,600
The great step is to create a new direction of numbers,

788
00:46:21,600 –> 00:46:23,560
perpendicular to the number line,

789
00:46:23,560 –> 00:46:26,720
and that’s where the square root of minus one is.

790
00:46:28,880 –> 00:46:32,600
Gauss was not the first to come up with this two-dimensional picture

791
00:46:32,600 –> 00:46:36,720
of numbers, but he was the first person to explain it all clearly.

792
00:46:36,720 –> 00:46:38,760
He gave people a picture to understand

793
00:46:38,760 –> 00:46:40,920
how imaginary numbers worked.

794
00:46:40,920 –> 00:46:43,080
And once they’d developed this picture,

795
00:46:43,080 –> 00:46:46,200
their immense potential could really be unleashed.

796
00:46:46,200 –> 00:46:49,680
Guten Morgen. Ein Kaffee, bitte.

797
00:46:49,680 –> 00:46:53,120
His maths led to a claim and financial security for Gauss.

798
00:46:53,120 –> 00:46:56,360
He could have gone anywhere, but he was happy enough

799
00:46:56,360 –> 00:47:01,680
to settle down and spend the rest of his life in sleepy Gottingen.

800
00:47:01,680 –> 00:47:03,920
Unfortunately, as his fame developed,

801
00:47:03,920 –> 00:47:06,080
so his character deteriorated.

802
00:47:06,080 –> 00:47:08,440
A naturally conservative, shy man,

803
00:47:08,440 –> 00:47:12,760
he became increasingly distrustful and grumpy.

804
00:47:12,760 –> 00:47:16,600
Many young mathematicians across Europe regarded Gauss as a god

805
00:47:16,600 –> 00:47:18,720
and they would send in their theorems,

806
00:47:18,720 –> 00:47:20,720
their conjectures, even some proofs.

807
00:47:20,720 –> 00:47:23,560
But most of the time, he wouldn’t respond, and even when he did,

808
00:47:23,560 –> 00:47:26,480
it was generally to say either that they’d got it wrong

809
00:47:26,480 –> 00:47:28,480
or he’d proved it already.

810
00:47:28,480 –> 00:47:32,600
His dismissal or lack of interest in the work of lesser mortals

811
00:47:32,600 –> 00:47:35,360
sometimes discouraged some very talented mathematicians

812
00:47:35,360 –> 00:47:38,120
from pursuing their ideas.

813
00:47:38,120 –> 00:47:40,240
But occasionally, Gauss also failed

814
00:47:40,240 –> 00:47:45,040
to follow up on his own insights, including one very important insight

815
00:47:45,040 –> 00:47:48,240
that might have transformed the mathematics of his time.

816
00:47:50,400 –> 00:47:53,640
15 kilometres outside Gottingen stands what is known today

817
00:47:53,640 –> 00:47:55,640
as the Gauss Tower.

818
00:47:55,640 –> 00:47:57,960
Wow, that is stunning.

819
00:47:57,960 –> 00:48:01,640
It is really a fantastic view here, yes.

820
00:48:01,640 –> 00:48:05,040
Gauss took on many projects for the Hanoverian government,

821
00:48:05,040 –> 00:48:09,320
including the first proper survey of all the lands of Hanover.

822
00:48:09,320 –> 00:48:12,560
Was this Gauss’ choice to do this surveying?

823
00:48:12,560 –> 00:48:16,120
For a mathematician, it sounds like the last thing I’d want to do.

824
00:48:16,120 –> 00:48:17,320
He wanted to do it.

825
00:48:17,320 –> 00:48:23,280
The major point in doing this was to discover the shape of the earth.

826
00:48:23,280 –> 00:48:25,280
But he also started speculating

827
00:48:25,280 –> 00:48:29,880
about something even more revolutionary - the shape of space.

828
00:48:29,880 –> 00:48:34,720
So he’s thinking there may not be anything flat in the universe?

829
00:48:34,720 –> 00:48:37,280
Yes. And if we were living in a curved universe,

830
00:48:37,280 –> 00:48:40,480
there wouldn’t be anything flat.

831
00:48:40,480 –> 00:48:44,680
This led Gauss to question one of the central tenets of mathematics -

832
00:48:44,680 –> 00:48:47,280
Euclid’s geometry.

833
00:48:47,280 –> 00:48:50,160
He realised that this geometry, far from universal,

834
00:48:50,160 –> 00:48:52,960
depended on the idea of space as flat.

835
00:48:52,960 –> 00:48:56,160
It just didn’t apply to a universe that was curved.

836
00:48:56,160 –> 00:48:59,520
But in the early 19th century, Euclid’s geometry

837
00:48:59,520 –> 00:49:03,320
was seen as God-given and Gauss didn’t want any trouble.

838
00:49:03,320 –> 00:49:05,640
So he never published anything.

839
00:49:05,640 –> 00:49:09,200
Another mathematician, though, had no such fears.

840
00:49:11,960 –> 00:49:16,000
In mathematics, it’s often helpful to be part of a community

841
00:49:16,000 –> 00:49:19,320
where you can talk to and bounce ideas off others.

842
00:49:19,320 –> 00:49:22,160
But inside such a mathematical community,

843
00:49:22,160 –> 00:49:25,400
it can sometimes be difficult to come up with that one idea

844
00:49:25,400 –> 00:49:28,760
that completely challenges the status quo,

845
00:49:28,760 –> 00:49:33,560
and then the breakthrough often comes from somewhere else.

846
00:49:33,560 –> 00:49:36,840
Mathematics can be done in some pretty weird places.

847
00:49:36,840 –> 00:49:38,440
I’m in Transylvania,

848
00:49:38,440 –> 00:49:42,040
which is fairly appropriate, cos I’m in search of a lone wolf.

849
00:49:42,040 –> 00:49:45,200
Janos Bolyai spent much of his life

850
00:49:45,200 –> 00:49:49,520
hundreds of miles away from the mathematical centres of excellence.

851
00:49:49,520 –> 00:49:53,600
This is the only portrait of him that I was able to find.

852
00:49:53,600 –> 00:49:56,600
Unfortunately, it isn’t actually him.

853
00:49:56,600 –> 00:50:00,040
It’s one that the Communist Party in Romania started circulating

854
00:50:00,040 –> 00:50:04,000
when people got interested in his theories in the 1960s.

855
00:50:04,000 –> 00:50:06,480
They couldn’t find a picture of Janos.

856
00:50:06,480 –> 00:50:09,520
So they substituted a picture of somebody else instead.

857
00:50:11,800 –> 00:50:15,520
Born in 1802, Janos was the son of Farkas Bolyai,

858
00:50:15,520 –> 00:50:17,120
who was a maths teacher.

859
00:50:17,120 –> 00:50:20,400
He realised his son was a mathematical prodigy,

860
00:50:20,400 –> 00:50:23,720
so he wrote to his old friend Carl Friedrich Gauss,

861
00:50:23,720 –> 00:50:25,640
asking him to tutor the boy.

862
00:50:25,640 –> 00:50:28,880
Sadly, Gauss declined.

863
00:50:28,880 –> 00:50:31,760
So instead of becoming a professional mathematician,

864
00:50:31,760 –> 00:50:33,920
Janos joined the Army.

865
00:50:33,920 –> 00:50:37,080
But mathematics remained his first love.

866
00:50:40,680 –> 00:50:44,320
Maybe there’s something about the air here because Bolyai carried on

867
00:50:44,320 –> 00:50:46,720
doing his mathematics in his spare time.

868
00:50:46,720 –> 00:50:50,360
He started to explore what he called imaginary geometries,

869
00:50:50,360 –> 00:50:55,040
where the angles in triangles add up to less than 180.

870
00:50:55,040 –> 00:50:58,240
The amazing thing is that these imaginary geometries

871
00:50:58,240 –> 00:51:00,720
make perfect mathematical sense.

872
00:51:04,520 –> 00:51:09,280
Bolyai’s new geometry has become known as hyperbolic geometry.

873
00:51:09,280 –> 00:51:12,800
The best way to imagine it is a kind of mirror image of a sphere

874
00:51:12,800 –> 00:51:15,440
where lines curve back on each other.

875
00:51:15,440 –> 00:51:18,320
It’s difficult to represent it since we are so used

876
00:51:18,320 –> 00:51:21,680
to living in space which appears to be straight and flat.

877
00:51:23,800 –> 00:51:25,480
In his hometown of Targu Mures,

878
00:51:25,480 –> 00:51:29,600
I went looking for more about Bolyai’s revolutionary mathematics.

879
00:51:29,600 –> 00:51:33,040
His memory is certainly revered here.

880
00:51:33,040 –> 00:51:36,760
The museum contains a collection of Bolyai-related artefacts,

881
00:51:36,760 –> 00:51:40,520
some of which might be considered distinctly Transylvanian.

882
00:51:40,520 –> 00:51:42,480
It’s still got some hair on it.

883
00:51:42,480 –> 00:51:45,160
It’s kind of a little bit gruesome.

884
00:51:45,160 –> 00:51:46,760
But the object I like most here

885
00:51:46,760 –> 00:51:50,400
is a beautiful model of Bolyai’s geometry.

886
00:51:50,400 –> 00:51:54,000
You got the shortest distance between here and here

887
00:51:54,000 –> 00:51:56,760
if you stick on this surface. It’s not a straight line,

888
00:51:56,760 –> 00:51:59,160
but this curved line which of bends into the triangle.

889
00:51:59,160 –> 00:52:03,760
Here is a surface where the shortest distances which define the triangle

890
00:52:03,760 –> 00:52:06,040
add up to less than 180.

891
00:52:06,040 –> 00:52:09,440
Bolyai published his work in 1831.

892
00:52:09,440 –> 00:52:12,360
His father sent his old friend Gauss a copy.

893
00:52:12,360 –> 00:52:16,280
Gauss wrote back straightaway giving his approval,

894
00:52:16,280 –> 00:52:19,440
but Gauss refused to praise the young Bolyai,

895
00:52:19,440 –> 00:52:22,560
because he said the person he should be praising was himself.

896
00:52:22,560 –> 00:52:26,200
He had worked it all out a decade or so before.

897
00:52:26,200 –> 00:52:29,760
Actually, there is a letter from Gauss

898
00:52:29,760 –> 00:52:32,200
to another friend of his where he says,

899
00:52:32,200 –> 00:52:34,840
“I regard this young geometer boy

900
00:52:34,840 –> 00:52:37,960
“as a genius of the first order.”

901
00:52:37,960 –> 00:52:41,560
But Gauss never thought to tell Bolyai that.

902
00:52:41,560 –> 00:52:44,520
And young Janos was completely disheartened.

903
00:52:44,520 –> 00:52:47,040
Another body blow soon followed.

904
00:52:47,040 –> 00:52:49,880
Somebody else had developed exactly the same idea,

905
00:52:49,880 –> 00:52:52,000
but had published two years before him -

906
00:52:52,000 –> 00:52:55,080
the Russian mathematician Nicholas Lobachevsky.

907
00:52:57,560 –> 00:53:00,080
It was all downhill for Bolyai after that.

908
00:53:00,080 –> 00:53:04,080
With no recognition or career, he didn’t publish anything else.

909
00:53:04,080 –> 00:53:06,960
Eventually, he went a little crazy.

910
00:53:08,440 –> 00:53:13,160
In 1860, Janos Bolyai died in obscurity.

911
00:53:15,280 –> 00:53:19,040
Gauss, by contrast, was lionised after his death.

912
00:53:19,040 –> 00:53:22,560
A university, the units used to measure magnetic induction,

913
00:53:22,560 –> 00:53:25,520
even a crater on the moon would be named after him.

914
00:53:28,760 –> 00:53:31,600
During his lifetime, Gauss lent his support

915
00:53:31,600 –> 00:53:33,960
to very few mathematicians.

916
00:53:33,960 –> 00:53:38,840
But one exception was another of Gottingen’s mathematical giants -

917
00:53:38,840 –> 00:53:41,840
Bernhard Riemann.

918
00:53:48,280 –> 00:53:49,800
His father was a minister

919
00:53:49,800 –> 00:53:54,080
and he would remain a sincere Christian all his life.

920
00:53:54,080 –> 00:53:58,280
But Riemann grew up a shy boy who suffered from consumption.

921
00:53:58,280 –> 00:54:00,640
His family was large and poor and the only thing

922
00:54:00,640 –> 00:54:04,560
the young boy had going for him was an excellence at maths.

923
00:54:04,560 –> 00:54:07,720
That was his salvation.

924
00:54:07,720 –> 00:54:11,240
Many mathematicians like Riemann had very difficult childhoods,

925
00:54:11,240 –> 00:54:14,960
were quite unsociable. Their lives seemed to be falling apart.

926
00:54:14,960 –> 00:54:18,800
It was mathematics that gave them a sense of security.

927
00:54:21,920 –> 00:54:24,800
Riemann spent much of his early life in the town of Luneburg

928
00:54:24,800 –> 00:54:26,840
in northern Germany.

929
00:54:26,840 –> 00:54:30,440
This was his local school, built as a direct result

930
00:54:30,440 –> 00:54:34,280
of Humboldt’s educational reforms in the early 19th century.

931
00:54:34,280 –> 00:54:37,040
Riemann was one of its first pupils.

932
00:54:37,040 –> 00:54:41,360
The head teacher saw a way of bringing out the shy boy.

933
00:54:41,360 –> 00:54:44,320
He was given the freedom of the school’s library.

934
00:54:44,320 –> 00:54:46,880
It opened up a whole new world to him.

935
00:54:46,880 –> 00:54:48,680
One of the books he found in there

936
00:54:48,680 –> 00:54:51,480
was a book by the French mathematician Legendre,

937
00:54:51,480 –> 00:54:53,000
all about number theory.

938
00:54:53,000 –> 00:54:55,680
His teacher asked him how he was getting on with it.

939
00:54:55,680 –> 00:55:01,360
He replied, “I have understood all 859 pages of this wonderful book.”

940
00:55:01,360 –> 00:55:04,520
It was a strategy that obviously suited Riemann

941
00:55:04,520 –> 00:55:07,080
because he became a brilliant mathematician.

942
00:55:07,080 –> 00:55:12,280
One of his most famous contributions to mathematics was a lecture in 1852

943
00:55:12,280 –> 00:55:16,400
on the foundations of geometry. In the lecture,

944
00:55:16,400 –> 00:55:20,120
Riemann first described what geometry actually was

945
00:55:20,120 –> 00:55:22,160
and its relationship with the world.

946
00:55:22,160 –> 00:55:25,240
He then sketched out what geometry could be -

947
00:55:25,240 –> 00:55:28,240
a mathematics of many different kinds of space,

948
00:55:28,240 –> 00:55:31,240
only one of which would be the flat Euclidian space

949
00:55:31,240 –> 00:55:32,880
in which we appear to live.

950
00:55:32,880 –> 00:55:36,080
He was just 26 years old.

951
00:55:36,080 –> 00:55:40,560
Was it received well? Did people recognise the revolution?

952
00:55:40,560 –> 00:55:42,840
There was no way that people could actually

953
00:55:42,840 –> 00:55:45,040
make these ideas concrete.

954
00:55:45,040 –> 00:55:50,640
That only occurred 50, 60 years after this, with Einstein.

955
00:55:50,640 –> 00:55:53,400
So this is the beginning, really, of the revolution

956
00:55:53,400 –> 00:55:56,960

which ends with Einstein’s relativity.
Exactly.

957
00:55:56,960 –> 00:56:01,640
Riemann’s mathematics changed how we see the world.

958
00:56:01,640 –> 00:56:04,400
Suddenly, higher dimensional geometry appeared.

959
00:56:04,400 –> 00:56:06,640
The potential was there from Descartes,

960
00:56:06,640 –> 00:56:11,120
but it was Riemann’s imagination that made it happen.

961
00:56:11,120 –> 00:56:15,160
He began without putting any restriction

962
00:56:15,160 –> 00:56:18,680
on the dimensions whatsoever. This was something quite new,

963
00:56:18,680 –> 00:56:21,320
his way of thinking about things.

964
00:56:21,320 –> 00:56:24,800
Someone like Bolyai was really thinking about new geometries,

965
00:56:24,800 –> 00:56:26,920
but new two-dimensional geometries.

966
00:56:26,920 –> 00:56:30,160
New two-dimensional geometries. Riemann then broke away

967
00:56:30,160 –> 00:56:35,240
from all the limitations of two or three dimensions

968
00:56:35,240 –> 00:56:37,880
and began to think in in higher dimensions.

969
00:56:37,880 –> 00:56:39,400
And this was quite new.

970
00:56:39,400 –> 00:56:41,960
Multi-dimensional space is at the heart

971
00:56:41,960 –> 00:56:44,520
of so much mathematics done today.

972
00:56:44,520 –> 00:56:48,080
In geometry, number theory, and several other branches of maths,

973
00:56:48,080 –> 00:56:51,800
Riemann’s ideas still perplex and amaze.

974
00:56:52,760 –> 00:56:55,920
He died, though, in 1866.

975
00:56:55,920 –> 00:56:59,480
He was only 39 years old.

976
00:56:59,480 –> 00:57:02,960
Today, the results of Riemann’s mathematics are everywhere.

977
00:57:02,960 –> 00:57:07,520
Hyperspace is no longer science fiction, but science fact.

978
00:57:07,520 –> 00:57:11,280
In Paris, they have even tried to visualise what shapes

979
00:57:11,280 –> 00:57:13,880
in higher dimensions might look like.

980
00:57:15,680 –> 00:57:18,640
Just as the Renaissance artist Piero would have drawn a square

981
00:57:18,640 –> 00:57:22,880
inside a square to represent a cube on the two-dimensional canvas,

982
00:57:22,880 –> 00:57:27,360
the architect here at La Defense has built a cube inside a cube

983
00:57:27,360 –> 00:57:31,720
to represent a shadow of the four-dimensional hypercube.

984
00:57:31,720 –> 00:57:34,640
It is with Riemann’s work that we finally have

985
00:57:34,640 –> 00:57:37,120
the mathematical glasses to be able to explore

986
00:57:37,120 –> 00:57:39,360
such worlds of the mind.

987
00:57:42,480 –> 00:57:44,920
It’s taken a while to make these glasses fit,

988
00:57:44,920 –> 00:57:47,320
but without this golden age of mathematics,

989
00:57:47,320 –> 00:57:50,480
from Descartes to Riemann, there would be no calculus,

990
00:57:50,480 –> 00:57:55,240
no quantum physics, no relativity, none of the technology we use today.

991
00:57:55,240 –> 00:57:57,440
But even more important than that,

992
00:57:57,440 –> 00:58:00,800
their mathematics blew away the cobwebs

993
00:58:00,800 –> 00:58:04,520
and allowed us to see the world as it really is -

994
00:58:04,520 –> 00:58:07,680
a world much stranger than we ever thought.

995
00:58:11,080 –> 00:58:13,400
You can learn more about the story of maths

996
00:58:13,400 –> 00:58:16,000
at the Open University at:

997
00:58:26,680 –> 00:58:29,440
Subtitles by Red Bee Media Ltd

998
00:58:29,440 –> 00:58:33,320
Email subtitling@bbc.co.uk

Subtitles by © Red Bee Media Ltd

2019-05-04

The Story of Maths - 2. The Genius of the East - Subtitles

texts below are from © https://subsaga.com/bbc/documentaries/science/the-story-of-maths/2-the-genius-of-the-east.html

1
00:00:10,000 –> 00:00:15,520
From measuring time to understanding our position in the universe,

2
00:00:15,520 –> 00:00:19,960
from mapping the Earth to navigating the seas,

3
00:00:19,960 –> 00:00:24,240
from man’s earliest inventions to today’s advanced technologies,

4
00:00:24,240 –> 00:00:28,680
mathematics has been the pivot on which human life depends.

5
00:00:34,680 –> 00:00:37,040
The first steps of man’s mathematical journey

6
00:00:37,040 –> 00:00:42,000
were taken by the ancient cultures of Egypt, Mesopotamia and Greece -

7
00:00:42,000 –> 00:00:49,240
cultures which created the basic language of number and calculation.

8
00:00:49,240 –> 00:00:51,680
But when ancient Greece fell into decline,

9
00:00:51,680 –> 00:00:54,160
mathematical progress juddered to a halt.

10
00:00:58,040 –> 00:01:00,120
But that was in the West.

11
00:01:00,120 –> 00:01:04,360
In the East, mathematics would reach dynamic new heights.

12
00:01:08,320 –> 00:01:11,240
But in the West, much of this mathematical heritage

13
00:01:11,240 –> 00:01:14,600
has been conveniently forgotten or shaded from view.

14
00:01:14,600 –> 00:01:18,200
Due credit has not been given to the great mathematical breakthroughs

15
00:01:18,200 –> 00:01:21,280
that ultimately changed the world we live in.

16
00:01:21,280 –> 00:01:24,800
This is the untold story of the mathematics of the East

17
00:01:24,800 –> 00:01:29,000
that would transform the West and give birth to the modern world.

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00:02:04,040 –> 00:02:08,800
The Great Wall of China stretches for thousands of miles.

19
00:02:08,800 –> 00:02:12,960
Nearly 2,000 years in the making, this vast, defensive wall

20
00:02:12,960 –> 00:02:17,800
was begun in 220BC to protect China’s growing empire.

21
00:02:20,520 –> 00:02:23,840
The Great Wall of China is an amazing feat of engineering

22
00:02:23,840 –> 00:02:26,600
built over rough and high countryside.

23
00:02:26,600 –> 00:02:28,720
As soon as they started building,

24
00:02:28,720 –> 00:02:31,600
the ancient Chinese realised they had to make calculations

25
00:02:31,600 –> 00:02:36,080
about distances, angles of elevation and amounts of material.

26
00:02:36,080 –> 00:02:38,560
So perhaps it isn’t surprising that this inspired

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00:02:38,560 –> 00:02:43,320
some very clever mathematics to help build Imperial China.

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00:02:43,320 –> 00:02:46,480
At the heart of ancient Chinese mathematics

29
00:02:46,480 –> 00:02:49,440
was an incredibly simple number system

30
00:02:49,440 –> 00:02:53,320
which laid the foundations for the way we count in the West today.

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00:02:57,760 –> 00:03:03,480
When a mathematician wanted to do a sum, he would use small bamboo rods.

32
00:03:03,480 –> 00:03:07,920
These rods were arranged to represent the numbers one to nine.

33
00:03:14,080 –> 00:03:16,760
They were then placed in columns,

34
00:03:16,760 –> 00:03:20,840
each column representing units, tens,

35
00:03:20,840 –> 00:03:23,000
hundreds, thousands and so on.

36
00:03:25,000 –> 00:03:28,440
So the number 924 was represented by putting

37
00:03:28,440 –> 00:03:33,400
the symbol 4 in the units column, the symbol 2 in the tens column

38
00:03:33,400 –> 00:03:36,600
and the symbol 9 in the hundreds column.

39
00:03:43,280 –> 00:03:46,440
This is what we call a decimal place-value system,

40
00:03:46,440 –> 00:03:49,520
and it’s very similar to the one we use today.

41
00:03:49,520 –> 00:03:52,720
We too use numbers from one to nine, and we use their position

42
00:03:52,720 –> 00:03:57,040
to indicate whether it’s units, tens, hundreds or thousands.

43
00:03:57,040 –> 00:04:00,960
But the power of these rods is that it makes calculations very quick.

44
00:04:00,960 –> 00:04:04,240
In fact, the way the ancient Chinese did their calculations

45
00:04:04,240 –> 00:04:06,720
is very similar to the way we learn today in school.

46
00:04:12,080 –> 00:04:14,000
Not only were the ancient Chinese

47
00:04:14,000 –> 00:04:17,480
the first to use a decimal place-value system,

48
00:04:17,480 –> 00:04:21,840
but they did so over 1,000 years before we adopted it in the West.

49
00:04:21,840 –> 00:04:25,760
But they only used it when calculating with the rods.

50
00:04:25,760 –> 00:04:28,120
When writing the numbers down,

51
00:04:28,120 –> 00:04:31,160
the ancient Chinese didn’t use the place-value system.

52
00:04:33,760 –> 00:04:37,120
Instead, they used a far more laborious method,

53
00:04:37,120 –> 00:04:42,520
in which special symbols stood for tens, hundreds, thousands and so on.

54
00:04:44,040 –> 00:04:46,920
So the number 924 would be written out

55
00:04:46,920 –> 00:04:50,880
as nine hundreds, two tens and four.

56
00:04:50,880 –> 00:04:53,080
Not quite so efficient.

57
00:04:54,600 –> 00:04:56,120
The problem was

58
00:04:56,120 –> 00:04:59,200
that the ancient Chinese didn’t have a concept of zero.

59
00:04:59,200 –> 00:05:02,920
They didn’t have a symbol for zero. It just didn’t exist as a number.

60
00:05:02,920 –> 00:05:04,760
Using the counting rods,

61
00:05:04,760 –> 00:05:08,600
they would use a blank space where today we would write a zero.

62
00:05:08,600 –> 00:05:12,120
The problem came with trying to write down this number, which is why

63
00:05:12,120 –> 00:05:15,840
they had to create these new symbols for tens, hundreds and thousands.

64
00:05:15,840 –> 00:05:20,000
Without a zero, the written number was extremely limited.

65
00:05:23,960 –> 00:05:26,080
But the absence of zero didn’t stop

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00:05:26,080 –> 00:05:29,920
the ancient Chinese from making giant mathematical steps.

67
00:05:29,920 –> 00:05:32,440
In fact, there was a widespread fascination

68
00:05:32,440 –> 00:05:34,920
with number in ancient China.

69
00:05:34,920 –> 00:05:39,040
According to legend, the first sovereign of China,

70
00:05:39,040 –> 00:05:41,560
the Yellow Emperor, had one of his deities

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00:05:41,560 –> 00:05:44,800
create mathematics in 2800BC,

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00:05:44,800 –> 00:05:49,720
believing that number held cosmic significance. And to this day,

73
00:05:49,720 –> 00:05:53,880
the Chinese still believe in the mystical power of numbers.

74
00:05:56,960 –> 00:06:01,440
Odd numbers are seen as male, even numbers, female.

75
00:06:01,440 –> 00:06:04,880
The number four is to be avoided at all costs.

76
00:06:04,880 –> 00:06:07,320
The number eight brings good fortune.

77
00:06:08,440 –> 00:06:11,920
And the ancient Chinese were drawn to patterns in numbers,

78
00:06:11,920 –> 00:06:15,440
developing their own rather early version of sudoku.

79
00:06:17,520 –> 00:06:20,160
It was called the magic square.

80
00:06:24,080 –> 00:06:28,440
Legend has it that thousands of years ago, Emperor Yu was visited

81
00:06:28,440 –> 00:06:32,640
by a sacred turtle that came out of the depths of the Yellow River.

82
00:06:32,640 –> 00:06:34,480
On its back were numbers

83
00:06:34,480 –> 00:06:37,600
arranged into a magic square, a little like this.

84
00:06:46,160 –> 00:06:47,960
In this square,

85
00:06:47,960 –> 00:06:51,640
which was regarded as having great religious significance,

86
00:06:51,640 –> 00:06:55,880
all the numbers in each line - horizontal, vertical and diagonal -

87
00:06:55,880 –> 00:06:59,320
all add up to the same number - 15.

88
00:07:02,280 –> 00:07:05,320
Now, the magic square may be no more than a fun puzzle,

89
00:07:05,320 –> 00:07:07,960
but it shows the ancient Chinese fascination

90
00:07:07,960 –> 00:07:10,880
with mathematical patterns, and it wasn’t too long

91
00:07:10,880 –> 00:07:13,880
before they were creating even bigger magic squares

92
00:07:13,880 –> 00:07:18,000
with even greater magical and mathematical powers.

93
00:07:25,400 –> 00:07:28,160
But mathematics also played

94
00:07:28,160 –> 00:07:33,320
a vital role in the running of the emperor’s court.

95
00:07:33,320 –> 00:07:35,880
The calendar and the movement of the planets

96
00:07:35,880 –> 00:07:38,960
were of the utmost importance to the emperor,

97
00:07:38,960 –> 00:07:43,760
influencing all his decisions, even down to the way his day was planned,

98
00:07:43,760 –> 00:07:47,440
so astronomers became prized members of the imperial court,

99
00:07:47,440 –> 00:07:50,960
and astronomers were always mathematicians.

100
00:07:55,720 –> 00:07:58,920
Everything in the emperor’s life was governed by the calendar,

101
00:07:58,920 –> 00:08:02,280
and he ran his affairs with mathematical precision.

102
00:08:02,280 –> 00:08:05,480
The emperor even got his mathematical advisors

103
00:08:05,480 –> 00:08:08,120
to come up with a system to help him sleep his way

104
00:08:08,120 –> 00:08:11,480
through the vast number of women he had in his harem.

105
00:08:11,480 –> 00:08:14,680
Never one to miss a trick, the mathematical advisors decided

106
00:08:14,680 –> 00:08:19,480
to base the harem on a mathematical idea called a geometric progression.

107
00:08:19,480 –> 00:08:23,360
Maths has never had such a fun purpose!

108
00:08:23,360 –> 00:08:26,440
Legend has it that in the space of 15 nights,

109
00:08:26,440 –> 00:08:30,200
the emperor had to sleep with 121 women…

110
00:08:36,120 –> 00:08:38,400
..the empress,

111
00:08:38,400 –> 00:08:40,680
three senior consorts,

112
00:08:40,680 –> 00:08:43,120
nine wives,

113
00:08:43,120 –> 00:08:45,200
27 concubines

114
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and 81 slaves.

115
00:08:49,280 –> 00:08:52,200
The mathematicians would soon have realised

116
00:08:52,200 –> 00:08:56,040
that this was a geometric progression - a series of numbers

117
00:08:56,040 –> 00:08:58,560
in which you get from one number to the next

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00:08:58,560 –> 00:09:03,080
by multiplying the same number each time - in this case, three.

119
00:09:04,280 –> 00:09:08,520
Each group of women is three times as large as the previous group,

120
00:09:08,520 –> 00:09:12,840
so the mathematicians could quickly draw up a rota to ensure that,

121
00:09:12,840 –> 00:09:14,840
in the space of 15 nights,

122
00:09:14,840 –> 00:09:18,200
the emperor slept with every woman in the harem.

123
00:09:19,720 –> 00:09:23,520
The first night was reserved for the empress.

124
00:09:23,520 –> 00:09:26,360
The next was for the three senior consorts.

125
00:09:26,360 –> 00:09:29,040
The nine wives came next,

126
00:09:29,040 –> 00:09:35,200
and then the 27 concubines were chosen in rotation, nine each night.

127
00:09:35,200 –> 00:09:38,880
And then finally, over a period of nine nights,

128
00:09:38,880 –> 00:09:42,040
the 81 slaves were dealt with in groups of nine.

129
00:09:47,920 –> 00:09:50,800
Being the emperor certainly required stamina,

130
00:09:50,800 –> 00:09:52,560
a bit like being a mathematician,

131
00:09:52,560 –> 00:09:54,600
but the object is clear -

132
00:09:54,600 –> 00:09:58,440
to procure the best possible imperial succession.

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00:09:58,440 –> 00:10:00,440
The rota ensured that the emperor

134
00:10:00,440 –> 00:10:04,240
slept with the ladies of highest rank closest to the full moon,

135
00:10:04,240 –> 00:10:06,640
when their yin, their female force,

136
00:10:06,640 –> 00:10:11,200
would be at its highest and be able to match his yang, or male force.

137
00:10:16,440 –> 00:10:20,560
The emperor’s court wasn’t alone in its dependence on mathematics.

138
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It was central to the running of the state.

139
00:10:23,360 –> 00:10:28,320
Ancient China was a vast and growing empire with a strict legal code,

140
00:10:28,320 –> 00:10:30,000
widespread taxation

141
00:10:30,000 –> 00:10:33,640
and a standardised system of weights, measures and money.

142
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The empire needed

143
00:10:37,360 –> 00:10:41,560
a highly trained civil service, competent in mathematics.

144
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And to educate these civil servants was a mathematical textbook,

145
00:10:47,640 –> 00:10:51,800
probably written in around 200BC - the Nine Chapters.

146
00:10:54,360 –> 00:10:58,200
The book is a compilation of 246 problems

147
00:10:58,200 –> 00:11:02,920
in practical areas such as trade, payment of wages and taxes.

148
00:11:05,760 –> 00:11:08,800
And at the heart of these problems lies

149
00:11:08,800 –> 00:11:13,400
one of the central themes of mathematics, how to solve equations.

150
00:11:16,160 –> 00:11:19,160
Equations are a little bit like cryptic crosswords.

151
00:11:19,160 –> 00:11:21,560
You’re given a certain amount of information

152
00:11:21,560 –> 00:11:24,440
about some unknown numbers, and from that information

153
00:11:24,440 –> 00:11:27,320
you’ve got to deduce what the unknown numbers are.

154
00:11:27,320 –> 00:11:29,600
For example, with my weights and scales,

155
00:11:29,600 –> 00:11:31,640
I’ve found out that one plum…

156
00:11:32,760 –> 00:11:35,920
..together with three peaches

157
00:11:35,920 –> 00:11:39,480
weighs a total of 15g.

158
00:11:41,240 –> 00:11:42,640
But…

159
00:11:43,840 –> 00:11:45,760
..two plums

160
00:11:45,760 –> 00:11:48,440
together with one peach

161
00:11:48,440 –> 00:11:50,720
weighs a total of 10g.

162
00:11:50,720 –> 00:11:55,040
From this information, I can deduce what a single plum weighs

163
00:11:55,040 –> 00:11:59,120
and a single peach weighs, and this is how I do it.

164
00:12:00,640 –> 00:12:02,640
If I take the first set of scales,

165
00:12:02,640 –> 00:12:05,280
one plum and three peaches weighing 15g,

166
00:12:05,280 –> 00:12:11,560
and double it, I get two plums and six peaches weighing 30g.

167
00:12:14,320 –> 00:12:18,160
If I take this and subtract from it the second set of scales -

168
00:12:18,160 –> 00:12:20,880
that’s two plums and a peach weighing 10g -

169
00:12:20,880 –> 00:12:23,640
I’m left with an interesting result -

170
00:12:23,640 –> 00:12:25,960
no plums.

171
00:12:25,960 –> 00:12:28,120
Having eliminated the plums,

172
00:12:28,120 –> 00:12:31,360
I’ve discovered that five peaches weighs 20g,

173
00:12:31,360 –> 00:12:34,320
so a single peach weighs 4g,

174
00:12:34,320 –> 00:12:39,640
and from this I can deduce that the plum weighs 3g.

175
00:12:39,640 –> 00:12:42,680
The ancient Chinese went on to apply similar methods

176
00:12:42,680 –> 00:12:45,560
to larger and larger numbers of unknowns,

177
00:12:45,560 –> 00:12:50,680
using it to solve increasingly complicated equations.

178
00:12:50,680 –> 00:12:52,560
What’s extraordinary is

179
00:12:52,560 –> 00:12:55,280
that this particular system of solving equations

180
00:12:55,280 –> 00:12:59,160
didn’t appear in the West until the beginning of the 19th century.

181
00:12:59,160 –> 00:13:03,520
In 1809, while analysing a rock called Pallas in the asteroid belt,

182
00:13:03,520 –> 00:13:05,360
Carl Friedrich Gauss,

183
00:13:05,360 –> 00:13:08,160
who would become known as the prince of mathematics,

184
00:13:08,160 –> 00:13:09,840
rediscovered this method

185
00:13:09,840 –> 00:13:13,440
which had been formulated in ancient China centuries earlier.

186
00:13:13,440 –> 00:13:17,400
Once again, ancient China streets ahead of Europe.

187
00:13:21,240 –> 00:13:23,800
But the Chinese were to go on to solve

188
00:13:23,800 –> 00:13:27,800
even more complicated equations involving far larger numbers.

189
00:13:27,800 –> 00:13:31,040
In what’s become known as the Chinese remainder theorem,

190
00:13:31,040 –> 00:13:35,600
the Chinese came up with a new kind of problem.

191
00:13:35,600 –> 00:13:38,440
In this, we know the number that’s left

192
00:13:38,440 –> 00:13:42,280
when the equation’s unknown number is divided by a given number -

193
00:13:42,280 –> 00:13:44,320
say, three, five or seven.

194
00:13:46,160 –> 00:13:50,360
Of course, this is a fairly abstract mathematical problem,

195
00:13:50,360 –> 00:13:54,520
but the ancient Chinese still couched it in practical terms.

196
00:13:56,600 –> 00:13:59,600
So a woman in the market has a tray of eggs,

197
00:13:59,600 –> 00:14:02,440
but she doesn’t know how many eggs she’s got.

198
00:14:02,440 –> 00:14:05,800
What she does know is that if she arranges them in threes,

199
00:14:05,800 –> 00:14:08,960
she has one egg left over.

200
00:14:08,960 –> 00:14:13,360
If she arranges them in fives, she gets two eggs left over.

201
00:14:13,360 –> 00:14:15,840
But if she arranged them in rows of seven,

202
00:14:15,840 –> 00:14:18,640
she found she had three eggs left over.

203
00:14:18,640 –> 00:14:22,320
The ancient Chinese found a systematic way to calculate

204
00:14:22,320 –> 00:14:26,320
that the smallest number of eggs she could have had in the tray is 52.

205
00:14:26,320 –> 00:14:29,400
But the more amazing thing is that you can capture

206
00:14:29,400 –> 00:14:31,240
such a large number, like 52,

207
00:14:31,240 –> 00:14:34,760
by using these small numbers like three, five and seven.

208
00:14:34,760 –> 00:14:36,760
This way of looking at numbers

209
00:14:36,760 –> 00:14:40,520
would become a dominant theme over the last two centuries.

210
00:14:49,160 –> 00:14:53,880
By the 6th century AD, the Chinese remainder theorem was being used

211
00:14:53,880 –> 00:14:57,360
in ancient Chinese astronomy to measure planetary movement.

212
00:14:57,360 –> 00:15:00,760
But today it still has practical uses.

213
00:15:00,760 –> 00:15:05,600
Internet cryptography encodes numbers using mathematics

214
00:15:05,600 –> 00:15:09,800
that has its origins in the Chinese remainder theorem.

215
00:15:17,600 –> 00:15:19,400
By the 13th century,

216
00:15:19,400 –> 00:15:22,480
mathematics was long established on the curriculum,

217
00:15:22,480 –> 00:15:26,480
with over 30 mathematics schools scattered across the country.

218
00:15:26,480 –> 00:15:30,440
The golden age of Chinese maths had arrived.

219
00:15:32,200 –> 00:15:36,360
And its most important mathematician was called Qin Jiushao.

220
00:15:38,560 –> 00:15:43,480
Legend has it that Qin Jiushao was something of a scoundrel.

221
00:15:43,480 –> 00:15:47,280
He was a fantastically corrupt imperial administrator

222
00:15:47,280 –> 00:15:50,840
who crisscrossed China, lurching from one post to another.

223
00:15:50,840 –> 00:15:54,360
Repeatedly sacked for embezzling government money,

224
00:15:54,360 –> 00:15:57,280
he poisoned anyone who got in his way.

225
00:15:59,600 –> 00:16:02,240
Qin Jiushao was reputedly described as

226
00:16:02,240 –> 00:16:04,680
as violent as a tiger or a wolf

227
00:16:04,680 –> 00:16:07,760
and as poisonous as a scorpion or a viper

228
00:16:07,760 –> 00:16:10,880
so, not surprisingly, he made a fierce warrior.

229
00:16:10,880 –> 00:16:13,800
For ten years, he fought against the invading Mongols,

230
00:16:13,800 –> 00:16:17,240
but for much of that time he was complaining that his military life

231
00:16:17,240 –> 00:16:19,520
took him away from his true passion.

232
00:16:19,520 –> 00:16:22,520
No, not corruption, but mathematics.

233
00:16:34,080 –> 00:16:36,640
Qin started trying to solve equations

234
00:16:36,640 –> 00:16:39,920
that grew out of trying to measure the world around us.

235
00:16:39,920 –> 00:16:41,800
Quadratic equations involve numbers

236
00:16:41,800 –> 00:16:46,600
that are squared, or to the power of two - say, five times five.

237
00:16:47,960 –> 00:16:49,640
The ancient Mesopotamians

238
00:16:49,640 –> 00:16:52,680
had already realised that these equations

239
00:16:52,680 –> 00:16:55,400
were perfect for measuring flat, two-dimensional shapes,

240
00:16:55,400 –> 00:16:57,160
like Tiananmen Square.

241
00:17:00,120 –> 00:17:02,320
But Qin was interested

242
00:17:02,320 –> 00:17:06,280
in more complicated equations - cubic equations.

243
00:17:08,000 –> 00:17:10,800
These involve numbers which are cubed,

244
00:17:10,800 –> 00:17:15,520
or to the power of three - say, five times five times five,

245
00:17:15,520 –> 00:17:19,320
and they were perfect for capturing three-dimensional shapes,

246
00:17:19,320 –> 00:17:21,720
like Chairman Mao’s mausoleum.

247
00:17:23,240 –> 00:17:26,000
Qin found a way of solving cubic equations,

248
00:17:26,000 –> 00:17:28,560
and this is how it worked.

249
00:17:32,400 –> 00:17:34,440
Say Qin wants to know

250
00:17:34,440 –> 00:17:37,600
the exact dimensions of Chairman Mao’s mausoleum.

251
00:17:39,800 –> 00:17:42,080
He knows the volume of the building

252
00:17:42,080 –> 00:17:45,320
and the relationships between the dimensions.

253
00:17:47,000 –> 00:17:49,320
In order to get his answer,

254
00:17:49,320 –> 00:17:53,880
Qin uses what he knows to produce a cubic equation.

255
00:17:53,880 –> 00:17:57,800
He then makes an educated guess at the dimensions.

256
00:17:57,800 –> 00:18:01,520
Although he’s captured a good proportion of the mausoleum,

257
00:18:01,520 –> 00:18:03,600
there are still bits left over.

258
00:18:05,080 –> 00:18:09,040
Qin takes these bits and creates a new cubic equation.

259
00:18:09,040 –> 00:18:11,120
He can now refine his first guess

260
00:18:11,120 –> 00:18:15,200
by trying to find a solution to this new cubic equation, and so on.

261
00:18:18,320 –> 00:18:21,960
Each time he does this, the pieces he’s left with

262
00:18:21,960 –> 00:18:26,440
get smaller and smaller and his guesses get better and better.

263
00:18:28,120 –> 00:18:31,640
What’s striking is that Qin’s method for solving equations

264
00:18:31,640 –> 00:18:34,880
wasn’t discovered in the West until the 17th century,

265
00:18:34,880 –> 00:18:39,360
when Isaac Newton came up with a very similar approximation method.

266
00:18:39,360 –> 00:18:41,840
The power of this technique is

267
00:18:41,840 –> 00:18:46,000
that it can be applied to even more complicated equations.

268
00:18:46,000 –> 00:18:49,720
Qin even used his techniques to solve an equation

269
00:18:49,720 –> 00:18:51,960
involving numbers up to the power of ten.

270
00:18:51,960 –> 00:18:56,000
This was extraordinary stuff - highly complex mathematics.

271
00:18:58,400 –> 00:19:00,800
Qin may have been years ahead of his time,

272
00:19:00,800 –> 00:19:03,120
but there was a problem with his technique.

273
00:19:03,120 –> 00:19:05,960
It only gave him an approximate solution.

274
00:19:05,960 –> 00:19:09,880
That might be good enough for an engineer - not for a mathematician.

275
00:19:09,880 –> 00:19:13,440
Mathematics is an exact science. We like things to be precise,

276
00:19:13,440 –> 00:19:16,320
and Qin just couldn’t come up with a formula

277
00:19:16,320 –> 00:19:19,840
to give him an exact solution to these complicated equations.

278
00:19:27,840 –> 00:19:30,280
China had made great mathematical leaps,

279
00:19:30,280 –> 00:19:34,240
but the next great mathematical breakthroughs were to happen

280
00:19:34,240 –> 00:19:37,040
in a country lying to the southwest of China -

281
00:19:37,040 –> 00:19:40,360
a country that had a rich mathematical tradition

282
00:19:40,360 –> 00:19:43,360
that would change the face of maths for ever.

283
00:20:13,840 –> 00:20:18,560
India’s first great mathematical gift lay in the world of number.

284
00:20:18,560 –> 00:20:22,640
Like the Chinese, the Indians had discovered the mathematical benefits

285
00:20:22,640 –> 00:20:24,560
of the decimal place-value system

286
00:20:24,560 –> 00:20:28,520
and were using it by the middle of the 3rd century AD.

287
00:20:30,600 –> 00:20:34,200
It’s been suggested that the Indians learned the system

288
00:20:34,200 –> 00:20:38,800
from Chinese merchants travelling in India with their counting rods,

289
00:20:38,800 –> 00:20:42,640
or they may well just have stumbled across it themselves.

290
00:20:42,640 –> 00:20:46,120
It’s all such a long time ago that it’s shrouded in mystery.

291
00:20:48,320 –> 00:20:51,840
We may never know how the Indians came up with their number system,

292
00:20:51,840 –> 00:20:54,880
but we do know that they refined and perfected it,

293
00:20:54,880 –> 00:20:58,800
creating the ancestors for the nine numerals used across the world now.

294
00:20:58,800 –> 00:21:01,480
Many rank the Indian system of counting

295
00:21:01,480 –> 00:21:05,040
as one of the greatest intellectual innovations of all time,

296
00:21:05,040 –> 00:21:09,200
developing into the closest thing we could call a universal language.

297
00:21:27,120 –> 00:21:29,600
But there was one number missing,

298
00:21:29,600 –> 00:21:33,440
and it was the Indians who would introduce it to the world.

299
00:21:39,960 –> 00:21:44,400
The earliest known recording of this number dates from the 9th century,

300
00:21:44,400 –> 00:21:48,080
though it was probably in practical use for centuries before.

301
00:21:49,720 –> 00:21:53,560
This strange new numeral is engraved on the wall

302
00:21:53,560 –> 00:21:57,360
of small temple in the fort of Gwalior in central India.

303
00:22:01,480 –> 00:22:05,400
So here we are in one of the holy sites of the mathematical world,

304
00:22:05,400 –> 00:22:08,840
and what I’m looking for is in this inscription on the wall.

305
00:22:09,800 –> 00:22:12,600
Up here are some numbers, and…

306
00:22:12,600 –> 00:22:14,880
here’s the new number.

307
00:22:14,880 –> 00:22:16,880
It’s zero.

308
00:22:21,600 –> 00:22:25,720
It’s astonishing to think that before the Indians invented it,

309
00:22:25,720 –> 00:22:28,120
there was no number zero.

310
00:22:28,120 –> 00:22:31,280
To the ancient Greeks, it simply hadn’t existed.

311
00:22:31,280 –> 00:22:35,520
To the Egyptians, the Mesopotamians and, as we’ve seen, the Chinese,

312
00:22:35,520 –> 00:22:39,720
zero had been in use but as a placeholder, an empty space

313
00:22:39,720 –> 00:22:42,040
to show a zero inside a number.

314
00:22:45,320 –> 00:22:48,400
The Indians transformed zero from a mere placeholder

315
00:22:48,400 –> 00:22:51,320
into a number that made sense in its own right -

316
00:22:51,320 –> 00:22:54,280
a number for calculation, for investigation.

317
00:22:54,280 –> 00:22:58,480
This brilliant conceptual leap would revolutionise mathematics.

318
00:23:02,400 –> 00:23:06,760
Now, with just ten digits - zero to nine - it was suddenly possible

319
00:23:06,760 –> 00:23:09,760
to capture astronomically large numbers

320
00:23:09,760 –> 00:23:12,040
in an incredibly efficient way.

321
00:23:15,040 –> 00:23:18,360
But why did the Indians make this imaginative leap?

322
00:23:18,360 –> 00:23:20,560
Well, we’ll never know for sure,

323
00:23:20,560 –> 00:23:24,520
but it’s possible that the idea and symbol that the Indians use for zero

324
00:23:24,520 –> 00:23:27,720
came from calculations they did with stones in the sand.

325
00:23:27,720 –> 00:23:31,040
When stones were removed from the calculation,

326
00:23:31,040 –> 00:23:33,800
a small, round hole was left in its place,

327
00:23:33,800 –> 00:23:37,160
representing the movement from something to nothing.

328
00:23:39,800 –> 00:23:44,120
But perhaps there is also a cultural reason for the invention of zero.

329
00:23:44,120 –> 00:23:47,680
HORNS BLOW AND DRUMS BANG

330
00:23:47,680 –> 00:23:50,600
METALLIC BEATING

331
00:23:53,040 –> 00:23:57,520
For the ancient Indians, the concepts of nothingness and eternity

332
00:23:57,520 –> 00:24:00,440
lay at the very heart of their belief system.

333
00:24:04,920 –> 00:24:07,360
BELL CLANGS AND SILENCE FALLS

334
00:24:09,880 –> 00:24:13,880
In the religions of India, the universe was born from nothingness,

335
00:24:13,880 –> 00:24:17,000
and nothingness is the ultimate goal of humanity.

336
00:24:17,000 –> 00:24:18,840
So it’s perhaps not surprising

337
00:24:18,840 –> 00:24:22,680
that a culture that so enthusiastically embraced the void

338
00:24:22,680 –> 00:24:25,880
should be happy with the notion of zero.

339
00:24:25,880 –> 00:24:30,080
The Indians even used the word for the philosophical idea of the void,

340
00:24:30,080 –> 00:24:33,920
shunya, to represent the new mathematical term “zero”.

341
00:24:47,280 –> 00:24:52,680
In the 7th century, the brilliant Indian mathematician Brahmagupta

342
00:24:52,680 –> 00:24:55,680
proved some of the essential properties of zero.

343
00:25:01,480 –> 00:25:04,320
Brahmagupta’s rules about calculating with zero

344
00:25:04,320 –> 00:25:08,280
are taught in schools all over the world to this day.

345
00:25:09,240 –> 00:25:12,240
One plus zero equals one.

346
00:25:13,280 –> 00:25:16,640
One minus zero equals one.

347
00:25:16,640 –> 00:25:19,920
One times zero is equal to zero.

348
00:25:24,120 –> 00:25:28,680
But Brahmagupta came a cropper when he tried to do one divided by zero.

349
00:25:28,680 –> 00:25:31,880
After all, what number times zero equals one?

350
00:25:31,880 –> 00:25:35,760
It would require a new mathematical concept, that of infinity,

351
00:25:35,760 –> 00:25:38,000
to make sense of dividing by zero,

352
00:25:38,000 –> 00:25:41,920
and the breakthrough was made by a 12th-century Indian mathematician

353
00:25:41,920 –> 00:25:45,040
called Bhaskara II, and it works like this.

354
00:25:45,040 –> 00:25:51,200
If I take a fruit and I divide it into halves, I get two pieces,

355
00:25:51,200 –> 00:25:54,080
so one divided by a half is two.

356
00:25:54,080 –> 00:25:57,480
If I divide it into thirds, I get three pieces.

357
00:25:57,480 –> 00:26:00,920
So when I divide it into smaller and smaller fractions,

358
00:26:00,920 –> 00:26:04,640
I get more and more pieces, so ultimately,

359
00:26:04,640 –> 00:26:06,600
when I divide by a piece

360
00:26:06,600 –> 00:26:10,400
which is of zero size, I’ll have infinitely many pieces.

361
00:26:10,400 –> 00:26:14,560
So for Bhaskara, one divided by zero is infinity.

362
00:26:22,880 –> 00:26:26,680
But the Indians would go further in their calculations with zero.

363
00:26:27,840 –> 00:26:31,920
For example, if you take three from three and get zero,

364
00:26:31,920 –> 00:26:35,240
what happens when you take four from three?

365
00:26:35,240 –> 00:26:37,480
It looks like you have nothing,

366
00:26:37,480 –> 00:26:39,720
but the Indians recognised that this

367
00:26:39,720 –> 00:26:43,720
was a new sort of nothing - negative numbers.

368
00:26:43,720 –> 00:26:47,440
The Indians called them “debts”, because they solved equations like,

369
00:26:47,440 –> 00:26:51,040
“If I have three batches of material and take four away,

370
00:26:51,040 –> 00:26:53,200
“how many have I left?”

371
00:26:56,880 –> 00:26:58,840
This may seem odd and impractical,

372
00:26:58,840 –> 00:27:01,400
but that was the beauty of Indian mathematics.

373
00:27:01,400 –> 00:27:04,680
Their ability to come up with negative numbers and zero

374
00:27:04,680 –> 00:27:08,080
was because they thought of numbers as abstract entities.

375
00:27:08,080 –> 00:27:11,360
They weren’t just for counting and measuring pieces of cloth.

376
00:27:11,360 –> 00:27:15,000
They had a life of their own, floating free of the real world.

377
00:27:15,000 –> 00:27:19,000
This led to an explosion of mathematical ideas.

378
00:27:30,880 –> 00:27:34,560
The Indians’ abstract approach to mathematics soon revealed

379
00:27:34,560 –> 00:27:38,440
a new side to the problem of how to solve quadratic equations.

380
00:27:38,440 –> 00:27:42,000
That is equations including numbers to the power of two.

381
00:27:43,520 –> 00:27:47,520
Brahmagupta’s understanding of negative numbers allowed him to see

382
00:27:47,520 –> 00:27:50,720
that quadratic equations always have two solutions,

383
00:27:50,720 –> 00:27:52,600
one of which could be negative.

384
00:27:55,120 –> 00:27:57,040
Brahmagupta went even further,

385
00:27:57,040 –> 00:28:00,000
solving quadratic equations with two unknowns,

386
00:28:00,000 –> 00:28:04,040
a question which wouldn’t be considered in the West until 1657,

387
00:28:04,040 –> 00:28:05,920
when French mathematician Fermat

388
00:28:05,920 –> 00:28:08,600
challenged his colleagues with the same problem.

389
00:28:08,600 –> 00:28:11,760
Little did he know that they’d been beaten to a solution

390
00:28:11,760 –> 00:28:14,680
by Brahmagupta 1,000 years earlier.

391
00:28:20,000 –> 00:28:24,640
Brahmagupta was beginning to find abstract ways of solving equations,

392
00:28:24,640 –> 00:28:27,800
but astonishingly, he was also developing

393
00:28:27,800 –> 00:28:31,120
a new mathematical language to express that abstraction.

394
00:28:32,440 –> 00:28:36,640
Brahmagupta was experimenting with ways of writing his equations down,

395
00:28:36,640 –> 00:28:40,120
using the initials of the names of different colours

396
00:28:40,120 –> 00:28:42,680
to represent unknowns in his equations.

397
00:28:44,640 –> 00:28:47,400
A new mathematical language was coming to life,

398
00:28:47,400 –> 00:28:49,840
which would ultimately lead to the x’s and y’s

399
00:28:49,840 –> 00:28:52,880
which fill today’s mathematical journals.

400
00:29:07,160 –> 00:29:10,840
But it wasn’t just new notation that was being developed.

401
00:29:13,200 –> 00:29:15,840
Indian mathematicians were responsible for making

402
00:29:15,840 –> 00:29:19,560
fundamental new discoveries in the theory of trigonometry.

403
00:29:22,400 –> 00:29:26,640
The power of trigonometry is that it acts like a dictionary,

404
00:29:26,640 –> 00:29:29,880
translating geometry into numbers and back.

405
00:29:29,880 –> 00:29:33,120
Although first developed by the ancient Greeks,

406
00:29:33,120 –> 00:29:35,720
it was in the hands of the Indian mathematicians

407
00:29:35,720 –> 00:29:37,760
that the subject truly flourished.

408
00:29:37,760 –> 00:29:42,280
At its heart lies the study of right-angled triangles.

409
00:29:44,520 –> 00:29:48,000
In trigonometry, you can use this angle here

410
00:29:48,000 –> 00:29:52,240
to find the ratios of the opposite side to the longest side.

411
00:29:52,240 –> 00:29:55,000
There’s a function called the sine function

412
00:29:55,000 –> 00:29:58,040
which, when you input the angle, outputs the ratio.

413
00:29:58,040 –> 00:30:01,720
So for example in this triangle, the angle is about 30 degrees,

414
00:30:01,720 –> 00:30:05,720
so the output of the sine function is a ratio of one to two,

415
00:30:05,720 –> 00:30:10,320
telling me that this side is half the length of the longest side.

416
00:30:12,800 –> 00:30:16,800
The sine function enables you to calculate distances

417
00:30:16,800 –> 00:30:21,080
when you’re not able to make an accurate measurement.

418
00:30:21,080 –> 00:30:25,160
To this day, it’s used in architecture and engineering.

419
00:30:25,160 –> 00:30:28,000
The Indians used it to survey the land around them,

420
00:30:28,000 –> 00:30:32,840
navigate the seas and, ultimately, chart the depths of space itself.

421
00:30:34,800 –> 00:30:37,760
It was central to the work of observatories,

422
00:30:37,760 –> 00:30:39,600
like this one in Delhi,

423
00:30:39,600 –> 00:30:42,480
where astronomers would study the stars.

424
00:30:42,480 –> 00:30:45,000
The Indian astronomers could use trigonometry

425
00:30:45,000 –> 00:30:48,120
to work out the relative distance between Earth and the moon

426
00:30:48,120 –> 00:30:49,560
and Earth and the sun.

427
00:30:49,560 –> 00:30:53,360
You can only make the calculation when the moon is half full,

428
00:30:53,360 –> 00:30:56,560
because that’s when it’s directly opposite the sun,

429
00:30:56,560 –> 00:31:01,080
so that the sun, moon and Earth create a right-angled triangle.

430
00:31:02,640 –> 00:31:04,480
Now, the Indians could measure

431
00:31:04,480 –> 00:31:07,800
that the angle between the sun and the observatory

432
00:31:07,800 –> 00:31:09,640
was one-seventh of a degree.

433
00:31:10,880 –> 00:31:14,160
The sine function of one-seventh of a degree

434
00:31:14,160 –> 00:31:18,080
gives me the ratio of 400:1.

435
00:31:18,080 –> 00:31:23,240
This means the sun is 400 times further from Earth than the moon is.

436
00:31:23,240 –> 00:31:25,120
So using trigonometry,

437
00:31:25,120 –> 00:31:28,400
the Indian mathematicians could explore the solar system

438
00:31:28,400 –> 00:31:31,440
without ever having to leave the surface of the Earth.

439
00:31:39,000 –> 00:31:42,600
The ancient Greeks had been the first to explore the sine function,

440
00:31:42,600 –> 00:31:46,960
listing precise values for some angles,

441
00:31:46,960 –> 00:31:50,600
but they couldn’t calculate the sines of every angle.

442
00:31:50,600 –> 00:31:55,120
The Indians were to go much further, setting themselves a mammoth task.

443
00:31:55,120 –> 00:31:57,200
The search was on to find a way

444
00:31:57,200 –> 00:32:01,200
to calculate the sine function of any angle you might be given.

445
00:32:17,920 –> 00:32:21,440
The breakthrough in the search for the sine function of every angle

446
00:32:21,440 –> 00:32:24,480
would be made here in Kerala in south India.

447
00:32:24,480 –> 00:32:27,560
In the 15th century, this part of the country

448
00:32:27,560 –> 00:32:31,360
became home to one of the most brilliant schools of mathematicians

449
00:32:31,360 –> 00:32:33,160
to have ever worked.

450
00:32:35,280 –> 00:32:38,560
Their leader was called Madhava, and he was to make

451
00:32:38,560 –> 00:32:42,320
some extraordinary mathematical discoveries.

452
00:32:45,120 –> 00:32:49,080
The key to Madhava’s success was the concept of the infinite.

453
00:32:49,080 –> 00:32:52,680
Madhava discovered that you could add up infinitely many things

454
00:32:52,680 –> 00:32:54,520
with dramatic effects.

455
00:32:54,520 –> 00:32:57,840
Previous cultures had been nervous of these infinite sums,

456
00:32:57,840 –> 00:33:00,320
but Madhava was happy to play with them.

457
00:33:00,320 –> 00:33:02,880
For example, here’s how one can be made up

458
00:33:02,880 –> 00:33:05,320
by adding infinitely many fractions.

459
00:33:06,840 –> 00:33:11,200
I’m heading from zero to one on my boat,

460
00:33:11,200 –> 00:33:15,440
but I can split my journey up into infinitely many fractions.

461
00:33:15,440 –> 00:33:18,200
So I can get to a half,

462
00:33:18,200 –> 00:33:21,920
then I can sail on a quarter,

463
00:33:21,920 –> 00:33:24,920
then an eighth, then a sixteenth, and so on.

464
00:33:24,920 –> 00:33:29,320
The smaller the fractions I move, the nearer to one I get,

465
00:33:29,320 –> 00:33:33,720
but I’ll only get there once I’ve added up infinitely many fractions.

466
00:33:36,040 –> 00:33:38,160
Physically and philosophically,

467
00:33:38,160 –> 00:33:41,640
it seems rather a challenge to add up infinitely many things,

468
00:33:41,640 –> 00:33:45,680
but the power of mathematics is to make sense of the impossible.

469
00:33:45,680 –> 00:33:47,240
By producing a language

470
00:33:47,240 –> 00:33:49,600
to articulate and manipulate the infinite,

471
00:33:49,600 –> 00:33:52,480
you can prove that after infinitely many steps

472
00:33:52,480 –> 00:33:54,440
you’ll reach your destination.

473
00:33:57,640 –> 00:34:01,880
Such infinite sums are called infinite series, and Madhava

474
00:34:01,880 –> 00:34:04,520
was doing a lot of research into the connections

475
00:34:04,520 –> 00:34:07,560
between these series and trigonometry.

476
00:34:08,560 –> 00:34:12,200
First, he realised that he could use the same principle

477
00:34:12,200 –> 00:34:14,840
of adding up infinitely many fractions to capture

478
00:34:14,840 –> 00:34:19,360
one of the most important numbers in mathematics - pi.

479
00:34:20,880 –> 00:34:25,680
Pi is the ratio of the circle’s circumference to its diameter.

480
00:34:25,680 –> 00:34:29,880
It’s a number that appears in all sorts of mathematics,

481
00:34:29,880 –> 00:34:32,360
but is especially useful for engineers,

482
00:34:32,360 –> 00:34:36,600
because any measurements involving curves soon require pi.

483
00:34:38,200 –> 00:34:42,800
So for centuries, mathematicians searched for a precise value for pi.

484
00:34:48,320 –> 00:34:52,320
It was in 6th-century India that the mathematician Aryabhata

485
00:34:52,320 –> 00:34:57,160
gave a very accurate approximation for pi - namely 3.1416.

486
00:34:57,160 –> 00:34:58,840
He went on to use this

487
00:34:58,840 –> 00:35:02,000
to make a measurement of the circumference of the Earth,

488
00:35:02,000 –> 00:35:05,480
and he got it as 24,835 miles,

489
00:35:05,480 –> 00:35:09,800
which, amazingly, is only 70 miles away from its true value.

490
00:35:09,800 –> 00:35:12,360
But it was in Kerala in the 15th century

491
00:35:12,360 –> 00:35:15,240
that Madhava realised he could use infinity

492
00:35:15,240 –> 00:35:17,680
to get an exact formula for pi.

493
00:35:21,200 –> 00:35:24,800
By successively adding and subtracting different fractions,

494
00:35:24,800 –> 00:35:28,320
Madhava could hone in on an exact formula for pi.

495
00:35:30,640 –> 00:35:34,160
First, he moved four steps up the number line.

496
00:35:34,160 –> 00:35:36,520
That took him way past pi.

497
00:35:38,040 –> 00:35:41,080
So next he took four-thirds of a step,

498
00:35:41,080 –> 00:35:44,400
or one-and-one-third steps, back.

499
00:35:44,400 –> 00:35:46,560
Now he’d come too far the other way.

500
00:35:47,800 –> 00:35:51,520
So he headed forward four-fifths of a step.

501
00:35:51,520 –> 00:35:56,320
Each time, he alternated between four divided by the next odd number.

502
00:36:03,040 –> 00:36:06,160
He zigzagged up and down the number line,

503
00:36:06,160 –> 00:36:08,640
getting closer and closer to pi.

504
00:36:08,640 –> 00:36:12,000
He discovered that if you went through all the odd numbers,

505
00:36:12,000 –> 00:36:15,520
infinitely many of them, you would hit pi exactly.

506
00:36:19,920 –> 00:36:22,640
I was taught at university that this formula for pi

507
00:36:22,640 –> 00:36:26,480
was discovered by the 17th-century German mathematician Leibniz,

508
00:36:26,480 –> 00:36:29,880
but amazingly, it was actually discovered here in Kerala

509
00:36:29,880 –> 00:36:31,760
two centuries earlier by Madhava.

510
00:36:31,760 –> 00:36:34,360
He went on to use the same sort of mathematics

511
00:36:34,360 –> 00:36:36,280
to get infinite-series expressions

512
00:36:36,280 –> 00:36:38,640
for the sine formula in trigonometry.

513
00:36:38,640 –> 00:36:42,080
And the wonderful thing is that you can use these formulas now

514
00:36:42,080 –> 00:36:46,040
to calculate the sine of any angle to any degree of accuracy.

515
00:36:56,760 –> 00:37:00,520
It seems incredible that the Indians made these discoveries

516
00:37:00,520 –> 00:37:03,400
centuries before Western mathematicians.

517
00:37:06,160 –> 00:37:10,760
And it says a lot about our attitude in the West to non-Western cultures

518
00:37:10,760 –> 00:37:14,720
that we nearly always claim their discoveries as our own.

519
00:37:14,720 –> 00:37:18,760
What is clear is the West has been very slow to give due credit

520
00:37:18,760 –> 00:37:22,320
to the major breakthroughs made in non-Western mathematics.

521
00:37:22,320 –> 00:37:25,520
Madhava wasn’t the only mathematician to suffer this way.

522
00:37:25,520 –> 00:37:28,600
As the West came into contact more and more with the East

523
00:37:28,600 –> 00:37:30,480
during the 18th and 19th centuries,

524
00:37:30,480 –> 00:37:33,120
there was a widespread dismissal and denigration

525
00:37:33,120 –> 00:37:35,200
of the cultures they were colonising.

526
00:37:35,200 –> 00:37:38,000
The natives, it was assumed, couldn’t have anything

527
00:37:38,000 –> 00:37:40,240
of intellectual worth to offer the West.

528
00:37:40,240 –> 00:37:43,160
It’s only now, at the beginning of the 21st century,

529
00:37:43,160 –> 00:37:45,880
that history is being rewritten.

530
00:37:45,880 –> 00:37:49,880
But Eastern mathematics was to have a major impact in Europe,

531
00:37:49,880 –> 00:37:53,040
thanks to the development of one of the major powers

532
00:37:53,040 –> 00:37:54,720
of the medieval world.

533
00:38:17,440 –> 00:38:20,960
In the 7th century, a new empire began to spread

534
00:38:20,960 –> 00:38:23,200
across the Middle East.

535
00:38:23,200 –> 00:38:25,680
The teachings of the Prophet Mohammed

536
00:38:25,680 –> 00:38:28,560
inspired a vast and powerful Islamic empire

537
00:38:28,560 –> 00:38:30,920
which soon stretched from India in the east

538
00:38:30,920 –> 00:38:35,160
to here in Morocco in the west.

539
00:38:41,960 –> 00:38:46,480
And at the heart of this empire lay a vibrant intellectual culture.

540
00:38:51,400 –> 00:38:56,160
A great library and centre of learning was established in Baghdad.

541
00:38:56,160 –> 00:38:59,640
Called the House of Wisdom, its teaching spread

542
00:38:59,640 –> 00:39:01,840
throughout the Islamic empire,

543
00:39:01,840 –> 00:39:05,080
reaching schools like this one here in Fez.

544
00:39:05,080 –> 00:39:08,360
Subjects studied included astronomy, medicine,

545
00:39:08,360 –> 00:39:10,240
chemistry, zoology

546
00:39:10,240 –> 00:39:11,920
and mathematics.

547
00:39:13,480 –> 00:39:18,160
The Muslim scholars collected and translated many ancient texts,

548
00:39:18,160 –> 00:39:20,600
effectively saving them for posterity.

549
00:39:20,600 –> 00:39:23,880
In fact, without their intervention, we may never have known

550
00:39:23,880 –> 00:39:27,480
about the ancient cultures of Egypt, Babylon, Greece and India.

551
00:39:27,480 –> 00:39:30,440
But the scholars at the House of Wisdom weren’t content

552
00:39:30,440 –> 00:39:33,360
simply with translating other people’s mathematics.

553
00:39:33,360 –> 00:39:36,080
They wanted to create a mathematics of their own,

554
00:39:36,080 –> 00:39:37,920
to push the subject forward.

555
00:39:42,080 –> 00:39:46,080
Such intellectual curiosity was actively encouraged

556
00:39:46,080 –> 00:39:49,320
in the early centuries of the Islamic empire.

557
00:39:51,320 –> 00:39:54,880
The Koran asserted the importance of knowledge.

558
00:39:54,880 –> 00:39:58,640
Learning was nothing less than a requirement of God.

559
00:40:01,720 –> 00:40:05,400
In fact, the needs of Islam demanded mathematical skill.

560
00:40:05,400 –> 00:40:07,920
The devout needed to calculate the time of prayer

561
00:40:07,920 –> 00:40:10,640
and the direction of Mecca to pray towards,

562
00:40:10,640 –> 00:40:13,640
and the prohibition of depicting the human form

563
00:40:13,640 –> 00:40:15,520
meant that they had to use

564
00:40:15,520 –> 00:40:18,520
much more geometric patterns to cover their buildings.

565
00:40:18,520 –> 00:40:22,080
The Muslim artists discovered all the different sorts of symmetry

566
00:40:22,080 –> 00:40:26,320
that you can depict on a two-dimensional wall.

567
00:40:34,000 –> 00:40:37,040
The director of the House of Wisdom in Baghdad

568
00:40:37,040 –> 00:40:40,400
was a Persian scholar called Muhammad Al-Khwarizmi.

569
00:40:43,520 –> 00:40:48,440
Al-Khwarizmi was an exceptional mathematician who was responsible

570
00:40:48,440 –> 00:40:52,680
for introducing two key mathematical concepts to the West.

571
00:40:52,680 –> 00:40:55,680
Al-Khwarizmi recognised the incredible potential

572
00:40:55,680 –> 00:40:57,520
that the Hindu numerals had

573
00:40:57,520 –> 00:41:00,480
to revolutionise mathematics and science.

574
00:41:00,480 –> 00:41:03,040
His work explaining the power of these numbers

575
00:41:03,040 –> 00:41:06,000
to speed up calculations and do things effectively

576
00:41:06,000 –> 00:41:09,400
was so influential that it wasn’t long before they were adopted

577
00:41:09,400 –> 00:41:13,240
as the numbers of choice amongst the mathematicians of the Islamic world.

578
00:41:13,240 –> 00:41:16,000
In fact, these numbers have now become known

579
00:41:16,000 –> 00:41:18,320
as the Hindu-Arabic numerals.

580
00:41:18,320 –> 00:41:21,360
These numbers - one to nine and zero -

581
00:41:21,360 –> 00:41:25,160
are the ones we use today all over the world.

582
00:41:29,680 –> 00:41:34,640
But Al-Khwarizmi was to create a whole new mathematical language.

583
00:41:36,280 –> 00:41:38,240
It was called algebra

584
00:41:38,240 –> 00:41:42,760
and was named after the title of his book Al-jabr W’al-muqabala,

585
00:41:42,760 –> 00:41:46,120
or Calculation By Restoration Or Reduction.

586
00:41:50,960 –> 00:41:56,080
Algebra is the grammar that underlies the way that numbers work.

587
00:41:56,080 –> 00:41:58,480
It’s a language that explains the patterns

588
00:41:58,480 –> 00:42:01,640
that lie behind the behaviour of numbers.

589
00:42:01,640 –> 00:42:05,560
It’s a bit like a code for running a computer program.

590
00:42:05,560 –> 00:42:09,240
The code will work whatever the numbers you feed in to the program.

591
00:42:11,040 –> 00:42:14,680
For example, mathematicians might have discovered

592
00:42:14,680 –> 00:42:16,960
that if you take a number and square it,

593
00:42:16,960 –> 00:42:19,240
that’s always one more than if you’d taken

594
00:42:19,240 –> 00:42:22,240
the numbers either side and multiplied those together.

595
00:42:22,240 –> 00:42:25,440
For example, five times five is 25,

596
00:42:25,440 –> 00:42:29,360
which is one more than four times six - 24.

597
00:42:29,360 –> 00:42:33,160
Six times six is always one more than five times seven and so on.

598
00:42:33,160 –> 00:42:34,880
But how can you be sure

599
00:42:34,880 –> 00:42:38,080
that this is going to work whatever numbers you take?

600
00:42:38,080 –> 00:42:41,040
To explain the pattern underlying these calculations,

601
00:42:41,040 –> 00:42:43,320
let’s use the dyeing holes in this tannery.

602
00:42:51,280 –> 00:42:56,520
If we take a square of 25 holes, running five by five,

603
00:42:56,520 –> 00:43:00,760
and take one row of five away and add it to the bottom,

604
00:43:00,760 –> 00:43:03,640
we get six by four with one left over.

605
00:43:05,880 –> 00:43:09,440
But however many holes there are on the side of the square,

606
00:43:09,440 –> 00:43:12,320
we can always move one row of holes down in a similar way

607
00:43:12,320 –> 00:43:16,240
to be left with a rectangle of holes with one left over.

608
00:43:18,880 –> 00:43:20,960
Algebra was a huge breakthrough.

609
00:43:20,960 –> 00:43:22,680
Here was a new language

610
00:43:22,680 –> 00:43:25,720
to be able to analyse the way that numbers worked.

611
00:43:25,720 –> 00:43:27,880
Previously, the Indians and the Chinese

612
00:43:27,880 –> 00:43:30,120
had considered very specific problems,

613
00:43:30,120 –> 00:43:33,600
but Al-Khwarizmi went from the specific to the general.

614
00:43:33,600 –> 00:43:37,200
He developed systematic ways to be able to analyse problems

615
00:43:37,200 –> 00:43:40,800
so that the solutions would work whatever the numbers that you took.

616
00:43:40,800 –> 00:43:44,560
This language is used across the mathematical world today.

617
00:43:46,080 –> 00:43:50,800
Al-Khwarizmi’s great breakthrough came when he applied algebra

618
00:43:50,800 –> 00:43:52,480
to quadratic equations -

619
00:43:52,480 –> 00:43:55,560
that is equations including numbers to the power of two.

620
00:43:55,560 –> 00:43:58,360
The ancient Mesopotamians had devised

621
00:43:58,360 –> 00:44:02,120
a cunning method to solve particular quadratic equations,

622
00:44:02,120 –> 00:44:06,240
but it was Al-Khwarizmi’s abstract language of algebra

623
00:44:06,240 –> 00:44:10,000
that could finally express why this method always worked.

624
00:44:11,600 –> 00:44:14,200
This was a great conceptual leap

625
00:44:14,200 –> 00:44:17,920
and would ultimately lead to a formula that could be used to solve

626
00:44:17,920 –> 00:44:22,160
any quadratic equation, whatever the numbers involved.

627
00:44:30,480 –> 00:44:32,440
The next mathematical Holy Grail

628
00:44:32,440 –> 00:44:37,040
was to find a general method that could solve all cubic equations -

629
00:44:37,040 –> 00:44:40,640
equations including numbers to the power of three.

630
00:44:57,920 –> 00:45:00,640
It was an 11th-century Persian mathematician

631
00:45:00,640 –> 00:45:04,000
who took up the challenge of cracking the problem of the cubic.

632
00:45:08,440 –> 00:45:11,960
His name was Omar Khayyam, and he travelled widely

633
00:45:11,960 –> 00:45:15,600
across the Middle East, calculating as he went.

634
00:45:17,520 –> 00:45:21,440
But he was famous for another, very different, reason.

635
00:45:21,440 –> 00:45:24,080
Khayyam was a celebrated poet,

636
00:45:24,080 –> 00:45:28,040
author of the great epic poem the Rubaiyat.

637
00:45:30,920 –> 00:45:35,120
It may seem a bit odd that a poet was also a master mathematician.

638
00:45:35,120 –> 00:45:38,560
After all, the combination doesn’t immediately spring to mind.

639
00:45:38,560 –> 00:45:42,200
But there’s quite a lot of similarity between the disciplines.

640
00:45:42,200 –> 00:45:45,560
Poetry, with its rhyming structure and rhythmic patterns,

641
00:45:45,560 –> 00:45:49,520
resonates strongly with constructing a logical mathematical proof.

642
00:45:53,000 –> 00:45:55,320
Khayyam’s major mathematical work

643
00:45:55,320 –> 00:46:02,040
was devoted to finding the general method to solve all cubic equations.

644
00:46:02,040 –> 00:46:04,600
Rather than looking at particular examples,

645
00:46:04,600 –> 00:46:08,640
Khayyam carried out a systematic analysis of the problem,

646
00:46:08,640 –> 00:46:11,920
true to the algebraic spirit of Al-Khwarizmi.

647
00:46:13,760 –> 00:46:16,280
Khayyam’s analysis revealed for the first time

648
00:46:16,280 –> 00:46:19,480
that there were several different sorts of cubic equation.

649
00:46:19,480 –> 00:46:21,560
But he was still very influenced

650
00:46:21,560 –> 00:46:24,320
by the geometric heritage of the Greeks.

651
00:46:24,320 –> 00:46:27,080
He couldn’t separate the algebra from the geometry.

652
00:46:27,080 –> 00:46:30,440
In fact, he wouldn’t even consider equations in higher degrees,

653
00:46:30,440 –> 00:46:33,840
because they described objects in more than three dimensions,

654
00:46:33,840 –> 00:46:35,640
something he saw as impossible.

655
00:46:35,640 –> 00:46:37,520
Although the geometry allowed him

656
00:46:37,520 –> 00:46:40,120
to analyse these cubic equations to some extent,

657
00:46:40,120 –> 00:46:43,280
he still couldn’t come up with a purely algebraic solution.

658
00:46:45,800 –> 00:46:51,400
It would be another 500 years before mathematicians could make the leap

659
00:46:51,400 –> 00:46:54,720
and find a general solution to the cubic equation.

660
00:46:56,240 –> 00:47:01,400
And that leap would finally be made in the West - in Italy.

661
00:47:15,400 –> 00:47:18,880
During the centuries in which China, India and the Islamic empire

662
00:47:18,880 –> 00:47:20,520
had been in the ascendant,

663
00:47:20,520 –> 00:47:24,760
Europe had fallen under the shadow of the Dark Ages.

664
00:47:26,280 –> 00:47:30,560
All intellectual life, including the study of mathematics, had stagnated.

665
00:47:35,760 –> 00:47:41,400
But by the 13th century, things were beginning to change.

666
00:47:41,400 –> 00:47:46,680
Led by Italy, Europe was starting to explore and trade with the East.

667
00:47:46,680 –> 00:47:51,120
With that contact came the spread of Eastern knowledge to the West.

668
00:47:51,120 –> 00:47:53,120
It was the son of a customs official

669
00:47:53,120 –> 00:47:56,640
that would become Europe’s first great medieval mathematician.

670
00:47:56,640 –> 00:48:00,240
As a child, he travelled around North Africa with his father,

671
00:48:00,240 –> 00:48:03,440
where he learnt about the developments of Arabic mathematics

672
00:48:03,440 –> 00:48:06,720
and especially the benefits of the Hindu-Arabic numerals.

673
00:48:06,720 –> 00:48:08,760
When he got home to Italy he wrote a book

674
00:48:08,760 –> 00:48:10,640
that would be hugely influential

675
00:48:10,640 –> 00:48:13,240
in the development of Western mathematics.

676
00:48:29,320 –> 00:48:31,800
That mathematician was Leonardo of Pisa,

677
00:48:31,800 –> 00:48:34,440
better known as Fibonacci,

678
00:48:35,480 –> 00:48:37,920
and in his Book Of Calculating,

679
00:48:37,920 –> 00:48:40,720
Fibonacci promoted the new number system,

680
00:48:40,720 –> 00:48:44,080
demonstrating how simple it was compared to the Roman numerals

681
00:48:44,080 –> 00:48:47,560
that were in use across Europe.

682
00:48:47,560 –> 00:48:52,640
Calculations were far easier, a fact that had huge consequences

683
00:48:52,640 –> 00:48:55,080
for anyone dealing with numbers -

684
00:48:55,080 –> 00:48:59,920
pretty much everyone, from mathematicians to merchants.

685
00:48:59,920 –> 00:49:02,640
But there was widespread suspicion of these new numbers.

686
00:49:02,640 –> 00:49:06,320
Old habits die hard, and the authorities just didn’t trust them.

687
00:49:06,320 –> 00:49:09,200
Some believed that they would be more open to fraud -

688
00:49:09,200 –> 00:49:11,040
that you could tamper with them.

689
00:49:11,040 –> 00:49:14,520
Others believed that they’d be so easy to use for calculations

690
00:49:14,520 –> 00:49:17,800
that it would empower the masses, taking authority away

691
00:49:17,800 –> 00:49:21,800
from the intelligentsia who knew how to use the old sort of numbers.

692
00:49:27,200 –> 00:49:31,200
The city of Florence even banned them in 1299,

693
00:49:31,200 –> 00:49:34,400
but over time, common sense prevailed,

694
00:49:34,400 –> 00:49:37,200
the new system spread throughout Europe,

695
00:49:37,200 –> 00:49:40,960
and the old Roman system slowly became defunct.

696
00:49:40,960 –> 00:49:46,440
At last, the Hindu-Arabic numerals, zero to nine, had triumphed.

697
00:49:48,360 –> 00:49:51,720
Today Fibonacci is best known for the discovery of some numbers,

698
00:49:51,720 –> 00:49:55,200
now called the Fibonacci sequence, that arose when he was trying

699
00:49:55,200 –> 00:49:58,240
to solve a riddle about the mating habits of rabbits.

700
00:49:58,240 –> 00:50:01,040
Suppose a farmer has a pair of rabbits.

701
00:50:01,040 –> 00:50:03,520
Rabbits take two months to reach maturity,

702
00:50:03,520 –> 00:50:07,240
and after that they give birth to another pair of rabbits each month.

703
00:50:07,240 –> 00:50:09,080
So the problem was how to determine

704
00:50:09,080 –> 00:50:12,560
how many pairs of rabbits there will be in any given month.

705
00:50:14,800 –> 00:50:20,000
Well, during the first month you have one pair of rabbits,

706
00:50:20,000 –> 00:50:24,200
and since they haven’t matured, they can’t reproduce.

707
00:50:24,200 –> 00:50:28,400
During the second month, there is still only one pair.

708
00:50:28,400 –> 00:50:32,000
But at the beginning of the third month, the first pair

709
00:50:32,000 –> 00:50:36,400
reproduces for the first time, so there are two pairs of rabbits.

710
00:50:36,400 –> 00:50:38,720
At the beginning of the fourth month,

711
00:50:38,720 –> 00:50:40,800
the first pair reproduces again,

712
00:50:40,800 –> 00:50:45,160
but the second pair is not mature enough, so there are three pairs.

713
00:50:46,840 –> 00:50:50,000
In the fifth month, the first pair reproduces

714
00:50:50,000 –> 00:50:53,480
and the second pair reproduces for the first time,

715
00:50:53,480 –> 00:50:58,200
but the third pair is still too young, so there are five pairs.

716
00:50:58,200 –> 00:51:00,120
The mating ritual continues,

717
00:51:00,120 –> 00:51:02,240
but what you soon realise is

718
00:51:02,240 –> 00:51:05,760
the number of pairs of rabbits you have in any given month

719
00:51:05,760 –> 00:51:09,400
is the sum of the pairs of rabbits that you have had

720
00:51:09,400 –> 00:51:13,120
in each of the two previous months, so the sequence goes…

721
00:51:13,120 –> 00:51:17,280
1…1…2…3…

722
00:51:17,280 –> 00:51:21,120
5…8…13…

723
00:51:21,120 –> 00:51:26,640
21…34…55…and so on.

724
00:51:26,640 –> 00:51:29,680
The Fibonacci numbers are nature’s favourite numbers.

725
00:51:29,680 –> 00:51:31,600
It’s not just rabbits that use them.

726
00:51:31,600 –> 00:51:35,880
The number of petals on a flower is invariably a Fibonacci number.

727
00:51:35,880 –> 00:51:39,960
They run up and down pineapples if you count the segments.

728
00:51:39,960 –> 00:51:42,960
Even snails use them to grow their shells.

729
00:51:42,960 –> 00:51:46,920
Wherever you find growth in nature, you find the Fibonacci numbers.

730
00:51:51,560 –> 00:51:54,880
But the next major breakthrough in European mathematics

731
00:51:54,880 –> 00:51:58,800
wouldn’t happen until the early 16th century.

732
00:51:58,800 –> 00:52:00,800
It would involve

733
00:52:00,800 –> 00:52:04,240
finding the general method that would solve all cubic equations,

734
00:52:04,240 –> 00:52:08,880
and it would happen here in the Italian city of Bologna.

735
00:52:10,600 –> 00:52:14,040
The University of Bologna was the crucible

736
00:52:14,040 –> 00:52:17,560
of European mathematical thought at the beginning of the 16th century.

737
00:52:20,880 –> 00:52:24,720
Pupils from all over Europe flocked here and developed

738
00:52:24,720 –> 00:52:29,440
a new form of spectator sport - the mathematical competition.

739
00:52:31,120 –> 00:52:34,480
Large audiences would gather to watch mathematicians

740
00:52:34,480 –> 00:52:39,680
challenge each other with numbers, a kind of intellectual fencing match.

741
00:52:39,680 –> 00:52:42,920
But even in this questioning atmosphere

742
00:52:42,920 –> 00:52:46,400
it was believed that some problems were just unsolvable.

743
00:52:46,400 –> 00:52:51,120
It was generally assumed that finding a general method

744
00:52:51,120 –> 00:52:54,760
to solve all cubic equations was impossible.

745
00:52:54,760 –> 00:52:58,560
But one scholar was to prove everyone wrong.

746
00:53:01,200 –> 00:53:03,040
His name was Tartaglia,

747
00:53:03,040 –> 00:53:05,040
but he certainly didn’t look

748
00:53:05,040 –> 00:53:08,000
the heroic architect of a new mathematics.

749
00:53:08,000 –> 00:53:11,160
At the age of 12, he’d been slashed across the face

750
00:53:11,160 –> 00:53:13,720
with a sabre by a rampaging French army.

751
00:53:13,720 –> 00:53:16,320
The result was a terrible facial scar

752
00:53:16,320 –> 00:53:19,000
and a devastating speech impediment.

753
00:53:19,000 –> 00:53:22,880
In fact, Tartaglia was the nickname he’d been given as a child

754
00:53:22,880 –> 00:53:24,800
and means “the stammerer”.

755
00:53:30,040 –> 00:53:33,360
Shunned by his schoolmates,

756
00:53:33,360 –> 00:53:37,960
Tartaglia lost himself in mathematics, and it wasn’t long

757
00:53:37,960 –> 00:53:43,080
before he’d found the formula to solve one type of cubic equation.

758
00:53:43,080 –> 00:53:45,040
But Tartaglia soon discovered

759
00:53:45,040 –> 00:53:48,640
that he wasn’t the only one to believe he’d cracked the cubic.

760
00:53:48,640 –> 00:53:51,840
A young Italian called Fior was boasting

761
00:53:51,840 –> 00:53:57,000
that he too held the secret formula for solving cubic equations.

762
00:53:57,000 –> 00:53:59,840
When news broke about the discoveries

763
00:53:59,840 –> 00:54:02,440
made by the two mathematicians,

764
00:54:02,440 –> 00:54:06,360
a competition was arranged to pit them against each other.

765
00:54:06,360 –> 00:54:10,320
The intellectual fencing match of the century was about to begin.

766
00:54:17,840 –> 00:54:19,880
The trouble was that Tartaglia

767
00:54:19,880 –> 00:54:22,800
only knew how to solve one sort of cubic equation,

768
00:54:22,800 –> 00:54:24,800
and Fior was ready to challenge him

769
00:54:24,800 –> 00:54:27,120
with questions about a different sort.

770
00:54:27,120 –> 00:54:29,480
But just a few days before the contest,

771
00:54:29,480 –> 00:54:32,520
Tartaglia worked out how to solve this different sort,

772
00:54:32,520 –> 00:54:35,880
and with this new weapon in his arsenal he thrashed his opponent,

773
00:54:35,880 –> 00:54:38,240
solving all the questions in under two hours.

774
00:54:41,840 –> 00:54:44,080
Tartaglia went on

775
00:54:44,080 –> 00:54:48,000
to find the formula to solve all types of cubic equations.

776
00:54:48,000 –> 00:54:51,040
News soon spread, and a mathematician in Milan

777
00:54:51,040 –> 00:54:54,800
called Cardano became so desperate to find the solution

778
00:54:54,800 –> 00:54:59,360
that he persuaded a reluctant Tartaglia to reveal the secret,

779
00:54:59,360 –> 00:55:01,200
but on one condition -

780
00:55:01,200 –> 00:55:05,000
that Cardano keep the secret and never publish.

781
00:55:07,760 –> 00:55:09,600
But Cardano couldn’t resist

782
00:55:09,600 –> 00:55:14,000
discussing Tartaglia’s solution with his brilliant student, Ferrari.

783
00:55:14,000 –> 00:55:16,880
As Ferrari got to grips with Tartaglia’s work,

784
00:55:16,880 –> 00:55:19,120
he realised that he could use it to solve

785
00:55:19,120 –> 00:55:22,720
the more complicated quartic equation, an amazing achievement.

786
00:55:22,720 –> 00:55:25,920
Cardano couldn’t deny his student his just rewards,

787
00:55:25,920 –> 00:55:29,280
and he broke his vow of secrecy, publishing Tartaglia’s work

788
00:55:29,280 –> 00:55:32,720
together with Ferrari’s brilliant solution of the quartic.

789
00:55:35,120 –> 00:55:39,080
Poor Tartaglia never recovered and died penniless,

790
00:55:39,080 –> 00:55:42,600
and to this day, the formula that solves the cubic equation

791
00:55:42,600 –> 00:55:45,240
is known as Cardano’s formula.

792
00:55:54,040 –> 00:55:57,520
Tartaglia may not have won glory in his lifetime,

793
00:55:57,520 –> 00:56:01,200
but his mathematics managed to solve a problem that had bewildered

794
00:56:01,200 –> 00:56:05,720
the great mathematicians of China, India and the Arab world.

795
00:56:07,920 –> 00:56:11,440
It was the first great mathematical breakthrough

796
00:56:11,440 –> 00:56:13,440
to happen in modern Europe.

797
00:56:17,520 –> 00:56:20,880
The Europeans now had in their hands the new language of algebra,

798
00:56:20,880 –> 00:56:24,120
the powerful techniques of the Hindu-Arabic numerals

799
00:56:24,120 –> 00:56:27,120
and the beginnings of the mastery of the infinite.

800
00:56:27,120 –> 00:56:28,920
It was time for the Western world

801
00:56:28,920 –> 00:56:31,400
to start writing its own mathematical stories

802
00:56:31,400 –> 00:56:33,040
in the language of the East.

803
00:56:33,040 –> 00:56:35,920
The mathematical revolution was about to begin.

804
00:56:39,600 –> 00:56:43,560
You can learn more about The Story Of Maths with the Open University

805
00:56:43,560 –> 00:56:45,800
at open2.net.

2019-05-04

The Story of Maths - 1. The Language of the Universe - Subtitles

texts below are from © https://subsaga.com/bbc/documentaries/science/the-story-of-maths/1-the-language-of-the-universe.html

1
00:00:03,400 –> 00:00:07,040
Throughout history, humankind has struggled

2
00:00:07,040 –> 00:00:11,640
to understand the fundamental workings of the material world.

3
00:00:11,640 –> 00:00:16,440
We’ve endeavoured to discover the rules and patterns that determine the qualities

4
00:00:16,440 –> 00:00:22,080
of the objects that surround us, and their complex relationship to us and each other.

5
00:00:23,360 –> 00:00:28,040
Over thousands of years, societies all over the world have found that one discipline

6
00:00:28,040 –> 00:00:31,560
above all others yields certain knowledge

7
00:00:31,560 –> 00:00:35,000
about the underlying realities of the physical world.

8
00:00:35,000 –> 00:00:38,200
That discipline is mathematics.

9
00:00:38,200 –> 00:00:41,760
I’m Marcus Du Sautoy, and I’m a mathematician.

10
00:00:41,760 –> 00:00:46,440
I see myself as a pattern searcher, hunting down the hidden structures

11
00:00:46,440 –> 00:00:51,480
that lie behind the apparent chaos and complexity of the world around us.

12
00:00:52,920 –> 00:00:58,200
In my search for pattern and order, I draw upon the work of the great mathematicians

13
00:00:58,200 –> 00:01:02,480
who’ve gone before me, people belonging to cultures across the globe,

14
00:01:02,480 –> 00:01:06,920
whose innovations created the language the universe is written in.

15
00:01:06,920 –> 00:01:12,640
I want to take you on a journey through time and space, and track the growth of mathematics

16
00:01:12,640 –> 00:01:16,920
from its awakening to the sophisticated subject we know today.

17
00:01:18,040 –> 00:01:21,120
Using computer generated imagery, we will explore

18
00:01:21,120 –> 00:01:24,920
the trailblazing discoveries that allowed the earliest civilisations

19
00:01:24,920 –> 00:01:28,640
to understand the world mathematical.

20
00:01:28,640 –> 00:01:31,600
This is the story of maths.

21
00:01:51,120 –> 00:01:55,080
Our world is made up of patterns and sequences.

22
00:01:55,080 –> 00:01:57,400
They’re all around us.

23
00:01:57,400 –> 00:01:59,880
Day becomes night.

24
00:01:59,880 –> 00:02:04,880
Animals travel across the earth in ever-changing formations.

25
00:02:04,880 –> 00:02:08,840
Landscapes are constantly altering.

26
00:02:08,840 –> 00:02:12,680
One of the reasons mathematics began was because we needed to find a way

27
00:02:12,680 –> 00:02:15,680
of making sense of these natural patterns.

28
00:02:18,680 –> 00:02:23,200
The most basic concepts of maths - space and quantity -

29
00:02:23,200 –> 00:02:27,200
are hard-wired into our brains.

30
00:02:27,200 –> 00:02:30,200
Even animals have a sense of distance and number,

31
00:02:30,200 –> 00:02:36,120
assessing when their pack is outnumbered, and whether to fight or fly,

32
00:02:36,120 –> 00:02:40,840
calculating whether their prey is within striking distance.

33
00:02:40,840 –> 00:02:46,120
Understanding maths is the difference between life and death.

34
00:02:47,240 –> 00:02:50,160
But it was man who took these basic concepts

35
00:02:50,160 –> 00:02:52,760
and started to build upon these foundations.

36
00:02:52,760 –> 00:02:55,880
At some point, humans started to spot patterns,

37
00:02:55,880 –> 00:02:59,680
to make connections, to count and to order the world around them.

38
00:02:59,680 –> 00:03:04,720
With this, a whole new mathematical universe began to emerge.

39
00:03:11,200 –> 00:03:12,960
This is the River Nile.

40
00:03:12,960 –> 00:03:15,800
It’s been the lifeline of Egypt for millennia.

41
00:03:17,200 –> 00:03:20,120
I’ve come here because it’s where some of the first signs

42
00:03:20,120 –> 00:03:23,560
of mathematics as we know it today emerged.

43
00:03:25,440 –> 00:03:30,800
People abandoned nomadic life and began settling here as early as 6000BC.

44
00:03:30,800 –> 00:03:34,880
The conditions were perfect for farming.

45
00:03:38,160 –> 00:03:44,120
The most important event for Egyptian agriculture each year was the flooding of the Nile.

46
00:03:44,120 –> 00:03:49,720
So this was used as a marker to start each new year.

47
00:03:49,720 –> 00:03:54,000
Egyptians did record what was going on over periods of time,

48
00:03:54,000 –> 00:03:56,480
so in order to establish a calendar like this,

49
00:03:56,480 –> 00:03:59,960
you need to count how many days, for example,

50
00:03:59,960 –> 00:04:02,680
happened in-between lunar phases,

51
00:04:02,680 –> 00:04:08,880
or how many days happened in-between two floodings of the Nile.

52
00:04:10,440 –> 00:04:14,200
Recording the patterns for the seasons was essential,

53
00:04:14,200 –> 00:04:18,040
not only to their management of the land, but also their religion.

54
00:04:18,040 –> 00:04:21,200
The ancient Egyptians who settled on the Nile banks

55
00:04:21,200 –> 00:04:25,600
believed it was the river god, Hapy, who flooded the river each year.

56
00:04:25,600 –> 00:04:28,440
And in return for the life-giving water,

57
00:04:28,440 –> 00:04:32,840
the citizens offered a portion of the yield as a thanksgiving.

58
00:04:34,040 –> 00:04:38,920
As settlements grew larger, it became necessary to find ways to administer them.

59
00:04:38,920 –> 00:04:43,200
Areas of land needed to be calculated, crop yields predicted,

60
00:04:43,200 –> 00:04:45,520
taxes charged and collated.

61
00:04:45,520 –> 00:04:49,280
In short, people needed to count and measure.

62
00:04:50,800 –> 00:04:53,800
The Egyptians used their bodies to measure the world,

63
00:04:53,800 –> 00:04:56,840
and it’s how their units of measurements evolved.

64
00:04:56,840 –> 00:04:59,240
A palm was the width of a hand,

65
00:04:59,240 –> 00:05:03,880
a cubit an arm length from elbow to fingertips.

66
00:05:03,880 –> 00:05:07,320
Land cubits, strips of land measuring a cubit by 100,

67
00:05:07,320 –> 00:05:10,680
were used by the pharaoh’s surveyors to calculate areas.

68
00:05:13,880 –> 00:05:17,120
There’s a very strong link between bureaucracy

69
00:05:17,120 –> 00:05:20,360
and the development of mathematics in ancient Egypt.

70
00:05:20,360 –> 00:05:23,320
And we can see this link right from the beginning,

71
00:05:23,320 –> 00:05:25,640
from the invention of the number system,

72
00:05:25,640 –> 00:05:28,440
throughout Egyptian history, really.

73
00:05:28,440 –> 00:05:30,920
For the Old Kingdom, the only evidence we have

74
00:05:30,920 –> 00:05:34,960
are metrological systems, that is measurements for areas, for length.

75
00:05:34,960 –> 00:05:41,520
This points to a bureaucratic need to develop such things.

76
00:05:41,520 –> 00:05:46,760
It was vital to know the area of a farmer’s land so he could be taxed accordingly.

77
00:05:46,760 –> 00:05:51,680
Or if the Nile robbed him of part of his land, so he could request a rebate.

78
00:05:51,680 –> 00:05:54,640
It meant that the pharaoh’s surveyors were often calculating

79
00:05:54,640 –> 00:05:58,200
the area of irregular parcels of land.

80
00:05:58,200 –> 00:06:00,880
It was the need to solve such practical problems

81
00:06:00,880 –> 00:06:05,080
that made them the earliest mathematical innovators.

82
00:06:09,760 –> 00:06:13,760
The Egyptians needed some way to record the results of their calculations.

83
00:06:15,960 –> 00:06:20,560
Amongst all the hieroglyphs that cover the tourist souvenirs scattered around Cairo,

84
00:06:20,560 –> 00:06:25,760
I was on the hunt for those that recorded some of the first numbers in history.

85
00:06:25,760 –> 00:06:29,520
They were difficult to track down.

86
00:06:30,680 –> 00:06:33,480
But I did find them in the end.

87
00:06:36,560 –> 00:06:41,840
The Egyptians were using a decimal system, motivated by the 10 fingers on our hands.

88
00:06:41,840 –> 00:06:44,400
The sign for one was a stroke,

89
00:06:44,400 –> 00:06:50,280
10, a heel bone, 100, a coil of rope, and 1,000, a Lotus plant.

90
00:06:50,280 –> 00:06:52,560
How much is this T-shirt?

91
00:06:52,560 –> 00:06:54,280
Er, 25.

92
00:06:54,280 –> 00:07:00,080

25!
Yes!
So that would be 2 knee bones and 5 strokes.

93
00:07:00,080 –> 00:07:03,440

So you’re not gonna charge me anything up here?
Here, one million!

94
00:07:03,440 –> 00:07:05,480

One million?
My God!

95
00:07:05,480 –> 00:07:07,880
This one million.

96
00:07:07,880 –> 00:07:09,920
One million, yeah, that’s pretty big!

97
00:07:11,280 –> 00:07:16,760
The hieroglyphs are beautiful, but the Egyptian number system was fundamentally flawed.

98
00:07:18,360 –> 00:07:21,880
They had no concept of a place value,

99
00:07:21,880 –> 00:07:24,360
so one stroke could only represent one unit,

100
00:07:24,360 –> 00:07:26,080
not 100 or 1,000.

101
00:07:26,080 –> 00:07:29,120
Although you can write a million with just one character,

102
00:07:29,120 –> 00:07:33,400
rather than the seven that we use, if you want to write a million minus one,

103
00:07:33,400 –> 00:07:36,840
then the poor old Egyptian scribe has got to write nine strokes,

104
00:07:36,840 –> 00:07:40,000
nine heel bones, nine coils of rope, and so on,

105
00:07:40,000 –> 00:07:42,560
a total of 54 characters.

106
00:07:44,960 –> 00:07:50,160
Despite the drawback of this number system, the Egyptians were brilliant problem solvers.

107
00:07:52,160 –> 00:07:56,160
We know this because of the few records that have survived.

108
00:07:56,160 –> 00:07:59,160
The Egyptian scribes used sheets of papyrus

109
00:07:59,160 –> 00:08:02,560
to record their mathematical discoveries.

110
00:08:02,560 –> 00:08:06,360
This delicate material made from reeds decayed over time

111
00:08:06,360 –> 00:08:09,640
and many secrets perished with it.

112
00:08:09,640 –> 00:08:13,760
But there’s one revealing document that has survived.

113
00:08:13,760 –> 00:08:17,600
The Rhind Mathematical Papyrus is the most important document

114
00:08:17,600 –> 00:08:20,400
we have today for Egyptian mathematics.

115
00:08:20,400 –> 00:08:24,600
We get a good overview of what types of problems

116
00:08:24,600 –> 00:08:28,560
the Egyptians would have dealt with in their mathematics.

117
00:08:28,560 –> 00:08:34,040
We also get explicitly stated how multiplications and divisions were carried out.

118
00:08:35,760 –> 00:08:40,040
The papyri show how to multiply two large numbers together.

119
00:08:40,040 –> 00:08:44,560
But to illustrate the method, let’s take two smaller numbers.

120
00:08:44,560 –> 00:08:47,120
Let’s do three times six.

121
00:08:47,120 –> 00:08:50,720
The scribe would take the first number, three, and put it in one column.

122
00:08:53,000 –> 00:08:56,200
In the second column, he would place the number one.

123
00:08:56,200 –> 00:09:00,960
Then he would double the numbers in each column, so three becomes six…

124
00:09:04,520 –> 00:09:06,600
..and six would become 12.

125
00:09:11,000 –> 00:09:14,720
And then in the second column, one would become two,

126
00:09:14,720 –> 00:09:16,280
and two becomes four.

127
00:09:18,960 –> 00:09:21,400
Now, here’s the really clever bit.

128
00:09:21,400 –> 00:09:24,400
The scribe wants to multiply three by six.

129
00:09:24,400 –> 00:09:27,920
So he takes the powers of two in the second column,

130
00:09:27,920 –> 00:09:31,640
which add up to six. That’s two plus four.

131
00:09:31,640 –> 00:09:34,560
Then he moves back to the first column, and just takes

132
00:09:34,560 –> 00:09:37,480
those rows corresponding to the two and the four.

133
00:09:37,480 –> 00:09:39,120
So that’s six and the 12.

134
00:09:39,120 –> 00:09:43,960
He adds those together to get the answer of 18.

135
00:09:43,960 –> 00:09:47,800
But for me, the most striking thing about this method

136
00:09:47,800 –> 00:09:51,760
is that the scribe has effectively written that second number in binary.

137
00:09:51,760 –> 00:09:56,760
Six is one lot of four, one lot of two, and no units.

138
00:09:56,760 –> 00:09:59,360
Which is 1-1-0.

139
00:09:59,360 –> 00:10:03,640
The Egyptians have understood the power of binary over 3,000 years

140
00:10:03,640 –> 00:10:07,600
before the mathematician and philosopher Leibniz would reveal their potential.

141
00:10:07,600 –> 00:10:11,920
Today, the whole technological world depends on the same principles

142
00:10:11,920 –> 00:10:14,800
that were used in ancient Egypt.

143
00:10:16,600 –> 00:10:22,200
The Rhind Papyrus was recorded by a scribe called Ahmes around 1650BC.

144
00:10:22,200 –> 00:10:27,080
Its problems are concerned with finding solutions to everyday situations.

145
00:10:27,080 –> 00:10:30,200
Several of the problems mention bread and beer,

146
00:10:30,200 –> 00:10:33,960
which isn’t surprising as Egyptian workers were paid in food and drink.

147
00:10:33,960 –> 00:10:37,400
One is concerned with how to divide nine loaves

148
00:10:37,400 –> 00:10:41,880
equally between 10 people, without a fight breaking out.

149
00:10:41,880 –> 00:10:44,000
I’ve got nine loaves of bread here.

150
00:10:44,000 –> 00:10:47,800
I’m gonna take five of them and cut them into halves.

151
00:10:48,840 –> 00:10:51,560
Of course, nine people could shave a 10th off their loaf

152
00:10:51,560 –> 00:10:54,960
and give the pile of crumbs to the 10th person.

153
00:10:54,960 –> 00:10:58,800
But the Egyptians developed a far more elegant solution -

154
00:10:58,800 –> 00:11:02,480
take the next four and divide those into thirds.

155
00:11:04,040 –> 00:11:07,560
But two of the thirds I am now going to cut into fifths,

156
00:11:07,560 –> 00:11:10,000
so each piece will be one fifteenth.

157
00:11:12,760 –> 00:11:17,240
Each person then gets one half

158
00:11:17,240 –> 00:11:19,360
and one third

159
00:11:19,360 –> 00:11:23,080
and one fifteenth.

160
00:11:23,080 –> 00:11:26,000
It is through such seemingly practical problems

161
00:11:26,000 –> 00:11:29,560
that we start to see a more abstract mathematics developing.

162
00:11:29,560 –> 00:11:32,280
Suddenly, new numbers are on the scene - fractions -

163
00:11:32,280 –> 00:11:37,680
and it isn’t too long before the Egyptians are exploring the mathematics of these numbers.

164
00:11:39,640 –> 00:11:45,080
Fractions are clearly of practical importance to anyone dividing quantities for trade in the market.

165
00:11:45,080 –> 00:11:51,880
To log these transactions, the Egyptians developed notation which recorded these new numbers.

166
00:11:53,400 –> 00:11:56,560
One of the earliest representations of these fractions

167
00:11:56,560 –> 00:12:00,240
came from a hieroglyph which had great mystical significance.

168
00:12:00,240 –> 00:12:03,880
It’s called the Eye of Horus.

169
00:12:03,880 –> 00:12:09,000
Horus was an Old Kingdom god, depicted as half man, half falcon.

170
00:12:10,920 –> 00:12:15,760
According to legend, Horus’ father was killed by his other son, Seth.

171
00:12:15,760 –> 00:12:18,960
Horus was determined to avenge the murder.

172
00:12:18,960 –> 00:12:21,640
During one particularly fierce battle,

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Seth ripped out Horus’ eye, tore it up and scattered it over Egypt.

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But the gods were looking favourably on Horus.

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They gathered up the scattered pieces and reassembled the eye.

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Each part of the eye represented a different fraction.

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Each one, half the fraction before.

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Although the original eye represented a whole unit,

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the reassembled eye is 1/64 short.

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Although the Egyptians stopped at 1/64,

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implicit in this picture

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is the possibility of adding more fractions,

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halving them each time, the sum getting closer and closer to one,

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but never quite reaching it.

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This is the first hint of something called a geometric series,

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and it appears at a number of points in the Rhind Papyrus.

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But the concept of infinite series would remain hidden

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until the mathematicians of Asia discovered it centuries later.

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Having worked out a system of numbers, including these new fractions,

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it was time for the Egyptians to apply their knowledge

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to understanding shapes that they encountered day to day.

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These shapes were rarely regular squares or rectangles,

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and in the Rhind Papyrus, we find the area of a more organic form, the circle.

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What is astounding in the calculation

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of the area of the circle is its exactness, really.

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How they would have found their method is open to speculation,

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because the texts we have

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do not show us the methods how they were found.

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This calculation is particularly striking because it depends

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on seeing how the shape of the circle

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can be approximated by shapes that the Egyptians already understood.

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The Rhind Papyrus states that a circular field

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with a diameter of nine units

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is close in area to a square with sides of eight.

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But how would this relationship have been discovered?

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My favourite theory sees the answer in the ancient game of mancala.

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Mancala boards were found carved on the roofs of temples.

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Each player starts with an equal number of stones,

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and the object of the game is to move them round the board,

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capturing your opponent’s counters on the way.

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As the players sat around waiting to make their next move,

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perhaps one of them realised that sometimes the balls fill the circular holes

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of the mancala board in a rather nice way.

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He might have gone on to experiment with trying to make larger circles.

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Perhaps he noticed that 64 stones, the square of 8,

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can be used to make a circle with diameter nine stones.

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By rearranging the stones, the circle has been approximated by a square.

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And because the area of a circle is pi times the radius squared,

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the Egyptian calculation gives us the first accurate value for pi.

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The area of the circle is 64. Divide this by the radius squared,

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in this case 4.5 squared, and you get a value for pi.

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So 64 divided by 4.5 squared is 3.16,

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just a little under two hundredths away from its true value.

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But the really brilliant thing is, the Egyptians

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are using these smaller shapes to capture the larger shape.

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But there’s one imposing and majestic symbol of Egyptian

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mathematics we haven’t attempted to unravel yet -

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the pyramid.

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I’ve seen so many pictures that I couldn’t believe I’d be impressed by them.

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But meeting them face to face, you understand why they’re called

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one of the Seven Wonders of the Ancient World.

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They’re simply breathtaking.

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And how much more impressive they must have been in their day,

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when the sides were as smooth as glass, reflecting the desert sun.

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To me it looks like there might be mirror pyramids hiding underneath the desert,

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which would complete the shapes to make perfectly symmetrical octahedrons.

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Sometimes, in the shimmer of the desert heat, you can almost see these shapes.

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It’s the hint of symmetry hidden inside these shapes that makes them so impressive for a mathematician.

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The pyramids are just a little short to create these perfect shapes,

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but some have suggested another important mathematical concept

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might be hidden inside the proportions of the Great Pyramid - the golden ratio.

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Two lengths are in the golden ratio, if the relationship of the longest

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to the shortest is the same as the sum of the two to the longest side.

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Such a ratio has been associated with the perfect proportions one finds

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all over the natural world, as well as in the work of artists,

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architects and designers for millennia.

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Whether the architects of the pyramids were conscious of this important mathematical idea,

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or were instinctively drawn to it because of its satisfying aesthetic properties, we’ll never know.

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For me, the most impressive thing about the pyramids is the mathematical brilliance

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that went into making them, including the first inkling

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of one of the great theorems of the ancient world, Pythagoras’ theorem.

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In order to get perfect right-angled corners on their buildings

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and pyramids, the Egyptians would have used a rope with knots tied in it.

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At some point, the Egyptians realised that if they took a triangle with sides

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marked with three knots, four knots and five knots, it guaranteed them a perfect right-angle.

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This is because three squared, plus four squared, is equal to five squared.

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So we’ve got a perfect Pythagorean triangle.

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In fact any triangle whose sides satisfy this relationship will give me an 90-degree angle.

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But I’m pretty sure that the Egyptians hadn’t got

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this sweeping generalisation of their 3, 4, 5 triangle.

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We would not expect to find the general proof

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because this is not the style of Egyptian mathematics.

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Every problem was solved using concrete numbers and then

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if a verification would be carried out at the end, it would use the result

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and these concrete, given numbers,

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there’s no general proof within the Egyptian mathematical texts.

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It would be some 2,000 years before the Greeks and Pythagoras

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would prove that all right-angled triangles shared certain properties.

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This wasn’t the only mathematical idea that the Egyptians would anticipate.

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In a 4,000-year-old document called the Moscow papyrus, we find a formula for the volume

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of a pyramid with its peak sliced off, which shows the first hint of calculus at work.

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For a culture like Egypt that is famous for its pyramids, you would expect problems like this

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to have been a regular feature within the mathematical texts.

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The calculation of the volume of a truncated pyramid is one of the most

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advanced bits, according to our modern standards of mathematics,

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that we have from ancient Egypt.

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The architects and engineers would certainly have wanted such a formula

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to calculate the amount of materials required to build it.

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But it’s a mark of the sophistication

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of Egyptian mathematics that they were able to produce such a beautiful method.

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To understand how they derived their formula, start with a pyramid

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built such that the highest point sits directly over one corner.

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Three of these can be put together to make a rectangular box,

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so the volume of the skewed pyramid is a third the volume of the box.

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That is, the height, times the length, times the width, divided by three.

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Now comes an argument which shows the very first hints of the calculus at work,

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thousands of years before Gottfried Leibniz and Isaac Newton would come up with the theory.

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Suppose you could cut the pyramid into slices, you could then slide

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the layers across to make the more symmetrical pyramid you see in Giza.

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However, the volume of the pyramid has not changed, despite the rearrangement of the layers.

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So the same formula works.

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00:20:55,360 –> 00:20:58,880
The Egyptians were amazing innovators,

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and their ability to generate new mathematics was staggering.

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For me, they revealed the power of geometry and numbers, and made the first moves

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towards some of the exciting mathematical discoveries to come.

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But there was another civilisation that had mathematics to rival that of Egypt.

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And we know much more about their achievements.

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This is Damascus, over 5,000 years old,

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and still vibrant and bustling today.

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It used to be the most important point on the trade routes, linking old Mesopotamia with Egypt.

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The Babylonians controlled much of modern-day Iraq, Iran and Syria, from 1800BC.

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In order to expand and run their empire, they became masters of managing and manipulating numbers.

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We have law codes for instance that tell us

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about the way society is ordered.

305
00:21:56,200 –> 00:22:00,120
The people we know most about are the scribes, the professionally literate

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00:22:00,120 –> 00:22:05,280
and numerate people who kept the records for the wealthy families and for the temples and palaces.

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00:22:05,280 –> 00:22:10,320
Scribe schools existed from around 2500BC.

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00:22:10,320 –> 00:22:17,240
Aspiring scribes were sent there as children, and learned how to read, write and work with numbers.

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Scribe records were kept on clay tablets,

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which allowed the Babylonians to manage and advance their empire.

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00:22:24,200 –> 00:22:31,000
However, many of the tablets we have today aren’t official documents, but children’s exercises.

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It’s these unlikely relics that give us a rare insight into how the Babylonians approached mathematics.

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00:22:37,640 –> 00:22:42,440
So, this is a geometrical textbook from about the 18th century BC.

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I hope you can see that there are lots of pictures on it.

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And underneath each picture is a text that sets a problem about the picture.

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So for instance this one here says, I drew a square, 60 units long,

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00:22:55,800 –> 00:23:01,200
and inside it, I drew four circles - what are their areas?

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This little tablet here was written 1,000 years at least later than the tablet here,

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00:23:07,240 –> 00:23:10,120
but has a very interesting relationship.

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00:23:10,120 –> 00:23:12,520
It also has four circles on,

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00:23:12,520 –> 00:23:17,280
in a square, roughly drawn, but this isn’t a textbook, it’s a school exercise.

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00:23:17,280 –> 00:23:21,400
The adult scribe who’s teaching the student is being given this

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00:23:21,400 –> 00:23:25,320
as an example of completed homework or something like that.

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Like the Egyptians, the Babylonians appeared interested

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00:23:29,560 –> 00:23:32,920
in solving practical problems to do with measuring and weighing.

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00:23:32,920 –> 00:23:37,400
The Babylonian solutions to these problems are written like mathematical recipes.

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A scribe would simply follow and record a set of instructions to get a result.

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Here’s an example of the kind of problem they’d solve.

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I’ve got a bundle of cinnamon sticks here, but I’m not gonna weigh them.

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Instead, I’m gonna take four times their weight and add them to the scales.

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00:23:58,040 –> 00:24:04,640
Now I’m gonna add 20 gin. Gin was the ancient Babylonian measure of weight.

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I’m gonna take half of everything here and then add it again…

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00:24:07,960 –> 00:24:10,280
That’s two bundles, and ten gin.

334
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Everything on this side is equal to one mana. One mana was 60 gin.

335
00:24:16,320 –> 00:24:20,280
And here, we have one of the first mathematical equations in history,

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00:24:20,280 –> 00:24:23,160
everything on this side is equal to one mana.

337
00:24:23,160 –> 00:24:26,200
But how much does the bundle of cinnamon sticks weigh?

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00:24:26,200 –> 00:24:29,480
Without any algebraic language, they were able to manipulate

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00:24:29,480 –> 00:24:35,200
the quantities to be able to prove that the cinnamon sticks weighed five gin.

340
00:24:35,200 –> 00:24:40,560
In my mind, it’s this kind of problem which gives mathematics a bit of a bad name.

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00:24:40,560 –> 00:24:45,040
You can blame those ancient Babylonians for all those tortuous problems you had at school.

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00:24:45,040 –> 00:24:50,200
But the ancient Babylonian scribes excelled at this kind of problem.

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Intriguingly, they weren’t using powers of 10, like the Egyptians, they were using powers of 60.

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The Babylonians invented their number system, like the Egyptians, by using their fingers.

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00:25:05,320 –> 00:25:08,520
But instead of counting through the 10 fingers on their hand,

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00:25:08,520 –> 00:25:11,480
Babylonians found a more intriguing way to count body parts.

347
00:25:11,480 –> 00:25:14,000
They used the 12 knuckles on one hand,

348
00:25:14,000 –> 00:25:16,400
and the five fingers on the other to be able to count

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00:25:16,400 –> 00:25:20,520
12 times 5, ie 60 different numbers.

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00:25:20,520 –> 00:25:25,000
So for example, this number would have been 2 lots of 12, 24,

351
00:25:25,000 –> 00:25:29,120
and then, 1, 2, 3, 4, 5, to make 29.

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00:25:32,200 –> 00:25:35,920
The number 60 had another powerful property.

353
00:25:35,920 –> 00:25:39,360
It can be perfectly divided in a multitude of ways.

354
00:25:39,360 –> 00:25:41,360
Here are 60 beans.

355
00:25:41,360 –> 00:25:44,800
I can arrange them in 2 rows of 30.

356
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3 rows of 20.

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4 rows of 15.

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5 rows of 12.

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00:25:56,160 –> 00:25:59,320
Or 6 rows of 10.

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00:25:59,320 –> 00:26:04,560
The divisibility of 60 makes it a perfect base in which to do arithmetic.

361
00:26:04,560 –> 00:26:11,000
The base 60 system was so successful, we still use elements of it today.

362
00:26:11,000 –> 00:26:15,080
Every time we want to tell the time, we recognise units of 60 -

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00:26:15,080 –> 00:26:19,040
60 seconds in a minute, 60 minutes in an hour.

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00:26:19,040 –> 00:26:24,800
But the most important feature of the Babylonians’ number system was that it recognised place value.

365
00:26:24,800 –> 00:26:30,200
Just as our decimal numbers count how many lots of tens, hundreds and thousands you’re recording,

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00:26:30,200 –> 00:26:34,320
the position of each Babylonian number records the power of 60.

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00:26:41,360 –> 00:26:44,440
Instead of inventing new symbols for bigger and bigger numbers,

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00:26:44,440 –> 00:26:50,440
they would write 1-1-1, so this number would be 3,661.

369
00:26:54,000 –> 00:26:59,680
The catalyst for this discovery was the Babylonians’ desire to chart the course of the night sky.

370
00:27:07,400 –> 00:27:10,840
The Babylonians’ calendar was based on the cycles of the moon.

371
00:27:10,840 –> 00:27:15,200
They needed a way of recording astronomically large numbers.

372
00:27:15,200 –> 00:27:19,560
Month by month, year by year, these cycles were recorded.

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00:27:19,560 –> 00:27:25,720
From about 800BC, there were complete lists of lunar eclipses.

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00:27:25,720 –> 00:27:30,480
The Babylonian system of measurement was quite sophisticated at that time.

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00:27:30,480 –> 00:27:32,840
They had a system of angular measurement,

376
00:27:32,840 –> 00:27:36,960
360 degrees in a full circle, each degree was divided

377
00:27:36,960 –> 00:27:41,920
into 60 minutes, a minute was further divided into 60 seconds.

378
00:27:41,920 –> 00:27:48,560
So they had a regular system for measurement, and it was in perfect harmony with their number system,

379
00:27:48,560 –> 00:27:52,200
so it’s well suited not only for observation but also for calculation.

380
00:27:52,200 –> 00:27:56,360
But in order to calculate and cope with these large numbers,

381
00:27:56,360 –> 00:28:00,720
the Babylonians needed to invent a new symbol.

382
00:28:00,720 –> 00:28:03,760
And in so doing, they prepared the ground for one of the great

383
00:28:03,760 –> 00:28:06,880
breakthroughs in the history of mathematics - zero.

384
00:28:06,880 –> 00:28:11,240
In the early days, the Babylonians, in order to mark an empty place in

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00:28:11,240 –> 00:28:14,640
the middle of a number, would simply leave a blank space.

386
00:28:14,640 –> 00:28:19,960
So they needed a way of representing nothing in the middle of a number.

387
00:28:19,960 –> 00:28:25,360
So they used a sign, as a sort of breathing marker, a punctuation mark,

388
00:28:25,360 –> 00:28:28,480
and it comes to mean zero in the middle of a number.

389
00:28:28,480 –> 00:28:31,680
This was the first time zero, in any form,

390
00:28:31,680 –> 00:28:35,440
had appeared in the mathematical universe.

391
00:28:35,440 –> 00:28:42,000
But it would be over a 1,000 years before this little place holder would become a number in its own right.

392
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Having established such a sophisticated system of numbers,

393
00:28:53,920 –> 00:28:59,720
they harnessed it to tame the arid and inhospitable land that ran through Mesopotamia.

394
00:29:02,080 –> 00:29:06,400
Babylonian engineers and surveyors found ingenious ways of

395
00:29:06,400 –> 00:29:10,400
accessing water, and channelling it to the crop fields.

396
00:29:10,400 –> 00:29:15,760
Yet again, they used mathematics to come up with solutions.

397
00:29:15,760 –> 00:29:19,200
The Orontes valley in Syria is still an agricultural hub,

398
00:29:19,200 –> 00:29:26,320
and the old methods of irrigation are being exploited today, just as they were thousands of years ago.

399
00:29:26,320 –> 00:29:29,160
Many of the problems in Babylonian mathematics

400
00:29:29,160 –> 00:29:34,360
are concerned with measuring land, and it’s here we see for the first time

401
00:29:34,360 –> 00:29:39,920
the use of quadratic equations, one of the greatest legacies of Babylonian mathematics.

402
00:29:39,920 –> 00:29:43,560
Quadratic equations involve things where the unknown quantity

403
00:29:43,560 –> 00:29:46,920
you’re trying to identify is multiplied by itself.

404
00:29:46,920 –> 00:29:49,880
We call this squaring because it gives the area of a square,

405
00:29:49,880 –> 00:29:53,040
and it’s in the context of calculating the area of land

406
00:29:53,040 –> 00:29:55,960
that these quadratic equations naturally arise.

407
00:30:01,320 –> 00:30:03,280
Here’s a typical problem.

408
00:30:03,280 –> 00:30:06,160
If a field has an area of 55 units

409
00:30:06,160 –> 00:30:10,640
and one side is six units longer than the other,

410
00:30:10,640 –> 00:30:12,560
how long is the shorter side?

411
00:30:14,200 –> 00:30:18,640
The Babylonian solution was to reconfigure the field as a square.

412
00:30:18,640 –> 00:30:21,920
Cut three units off the end

413
00:30:21,920 –> 00:30:24,760
and move this round.

414
00:30:24,760 –> 00:30:29,920
Now, there’s a three-by-three piece missing, so let’s add this in.

415
00:30:29,920 –> 00:30:34,640
The area of the field has increased by nine units.

416
00:30:34,640 –> 00:30:38,040
This makes the new area 64.

417
00:30:38,040 –> 00:30:41,880
So the sides of the square are eight units.

418
00:30:41,880 –> 00:30:45,320
The problem-solver knows that they’ve added three to this side.

419
00:30:45,320 –> 00:30:49,520
So, the original length must be five.

420
00:30:50,520 –> 00:30:55,600
It may not look like it, but this is one of the first quadratic equations in history.

421
00:30:57,400 –> 00:31:02,400
In modern mathematics, I would use the symbolic language of algebra to solve this problem.

422
00:31:02,400 –> 00:31:07,400
The amazing feat of the Babylonians is that they were using these geometric games to find the value,

423
00:31:07,400 –> 00:31:10,200
without any recourse to symbols or formulas.

424
00:31:10,200 –> 00:31:13,920
The Babylonians were enjoying problem-solving for its own sake.

425
00:31:13,920 –> 00:31:17,960
They were falling in love with mathematics.

426
00:31:29,080 –> 00:31:34,080
The Babylonians’ fascination with numbers soon found a place in their leisure time, too.

427
00:31:34,080 –> 00:31:35,960
They were avid game-players.

428
00:31:35,960 –> 00:31:38,760
The Babylonians and their descendants have been playing

429
00:31:38,760 –> 00:31:43,160
a version of backgammon for over 5,000 years.

430
00:31:43,160 –> 00:31:45,840
The Babylonians played board games,

431
00:31:45,840 –> 00:31:52,200
from very posh board games in royal tombs to little bits of board games found in schools,

432
00:31:52,200 –> 00:31:56,280
to board games scratched on the entrances of palaces,

433
00:31:56,280 –> 00:32:00,520
so that the guardsmen must have played when they were bored,

434
00:32:00,520 –> 00:32:03,760
and they used dice to move their counters round.

435
00:32:04,880 –> 00:32:09,800
People who played games were using numbers in their leisure time to try and outwit their opponent,

436
00:32:09,800 –> 00:32:12,680
doing mental arithmetic very fast,

437
00:32:12,680 –> 00:32:17,280
and so they were calculating in their leisure time,

438
00:32:17,280 –> 00:32:21,000
without even thinking about it as being mathematical hard work.

439
00:32:23,320 –> 00:32:24,600
Now’s my chance.

440
00:32:24,600 –> 00:32:30,000
‘I hadn’t played backgammon for ages but I reckoned my maths would give me a fighting chance.’

441
00:32:30,000 –> 00:32:33,560

It’s up to you.
Six… I need to move something.

442
00:32:33,560 –> 00:32:36,560
‘But it wasn’t as easy as I thought.’

443
00:32:36,560 –> 00:32:38,680
Ah! What the hell was that?

444
00:32:38,680 –> 00:32:42,440

Yeah.
This is one, this is two.

445
00:32:42,440 –> 00:32:44,200
Now you’re in trouble.

446
00:32:44,200 –> 00:32:47,800

So I can’t move anything.
You cannot move these.

447
00:32:47,800 –> 00:32:49,200
Oh, gosh.

448
00:32:50,520 –> 00:32:52,320
There you go.

449
00:32:53,320 –> 00:32:54,960
Three and four.

450
00:32:54,960 –> 00:33:00,720
‘Just like the ancient Babylonians, my opponents were masters of tactical mathematics.’

451
00:33:00,720 –> 00:33:02,120
Yeah.

452
00:33:03,120 –> 00:33:05,840
Put it there. Good game.

453
00:33:07,120 –> 00:33:10,080
The Babylonians are recognised as one of the first cultures

454
00:33:10,080 –> 00:33:13,840
to use symmetrical mathematical shapes to make dice,

455
00:33:13,840 –> 00:33:17,440
but there is more heated debate about whether they might also

456
00:33:17,440 –> 00:33:20,920
have been the first to discover the secrets of another important shape.

457
00:33:20,920 –> 00:33:24,040
The right-angled triangle.

458
00:33:27,000 –> 00:33:32,360
We’ve already seen how the Egyptians use a 3-4-5 right-angled triangle.

459
00:33:32,360 –> 00:33:37,600
But what the Babylonians knew about this shape and others like it is much more sophisticated.

460
00:33:37,600 –> 00:33:42,120
This is the most famous and controversial ancient tablet we have.

461
00:33:42,120 –> 00:33:44,480
It’s called Plimpton 322.

462
00:33:45,480 –> 00:33:49,080
Many mathematicians are convinced it shows the Babylonians

463
00:33:49,080 –> 00:33:53,360
could well have known the principle regarding right-angled triangles,

464
00:33:53,360 –> 00:33:57,400
that the square on the diagonal is the sum of the squares on the sides,

465
00:33:57,400 –> 00:34:00,280
and known it centuries before the Greeks claimed it.

466
00:34:01,880 –> 00:34:06,320
This is a copy of arguably the most famous Babylonian tablet,

467
00:34:06,320 –> 00:34:08,040
which is Plimpton 322,

468
00:34:08,040 –> 00:34:12,680
and these numbers here reflect the width or height of a triangle,

469
00:34:12,680 –> 00:34:17,520
this being the diagonal, the other side would be over here,

470
00:34:17,520 –> 00:34:19,880
and the square of this column

471
00:34:19,880 –> 00:34:23,280
plus the square of the number in this column

472
00:34:23,280 –> 00:34:26,360
equals the square of the diagonal.

473
00:34:26,360 –> 00:34:31,120
They are arranged in an order of steadily decreasing angle,

474
00:34:31,120 –> 00:34:34,000
on a very uniform basis, showing that somebody

475
00:34:34,000 –> 00:34:38,600
had a lot of understanding of how the numbers fit together.

476
00:34:44,680 –> 00:34:50,800
Here were 15 perfect Pythagorean triangles, all of whose sides had whole-number lengths.

477
00:34:50,800 –> 00:34:56,160
It’s tempting to think that the Babylonians were the first custodians of Pythagoras’ theorem,

478
00:34:56,160 –> 00:35:01,200
and it’s a conclusion that generations of historians have been seduced by.

479
00:35:01,200 –> 00:35:03,960
But there could be a much simpler explanation

480
00:35:03,960 –> 00:35:07,760
for the sets of three numbers which fulfil Pythagoras’ theorem.

481
00:35:07,760 –> 00:35:12,800
It’s not a systematic explanation of Pythagorean triples, it’s simply

482
00:35:12,800 –> 00:35:17,640
a mathematics teacher doing some quite complicated calculations,

483
00:35:17,640 –> 00:35:21,160
but in order to produce some very simple numbers,

484
00:35:21,160 –> 00:35:26,120
in order to set his students problems about right-angled triangles,

485
00:35:26,120 –> 00:35:31,000
and in that sense it’s about Pythagorean triples only incidentally.

486
00:35:33,480 –> 00:35:39,040
The most valuable clues to what they understood could lie elsewhere.

487
00:35:39,040 –> 00:35:43,360
This small school exercise tablet is nearly 4,000 years old

488
00:35:43,360 –> 00:35:48,800
and reveals just what the Babylonians did know about right-angled triangles.

489
00:35:48,800 –> 00:35:54,360
It uses a principle of Pythagoras’ theorem to find the value of an astounding new number.

490
00:35:57,920 –> 00:36:05,000
Drawn along the diagonal is a really very good approximation to the square root of two,

491
00:36:05,000 –> 00:36:10,880
and so that shows us that it was known and used in school environments.

492
00:36:10,880 –> 00:36:12,880
Why’s this important?

493
00:36:12,880 –> 00:36:18,440
Because the square root of two is what we now call an irrational number,

494
00:36:18,440 –> 00:36:23,960
that is, if we write it out in decimals, or even in sexigesimal places,

495
00:36:23,960 –> 00:36:28,360
it doesn’t end, the numbers go on forever after the decimal point.

496
00:36:29,640 –> 00:36:33,640
The implications of this calculation are far-reaching.

497
00:36:33,640 –> 00:36:37,920
Firstly, it means the Babylonians knew something of Pythagoras’ theorem

498
00:36:37,920 –> 00:36:39,800
1,000 years before Pythagoras.

499
00:36:39,800 –> 00:36:45,560
Secondly, the fact that they can calculate this number to an accuracy of four decimal places

500
00:36:45,560 –> 00:36:50,600
shows an amazing arithmetic facility, as well as a passion for mathematical detail.

501
00:36:52,200 –> 00:36:56,440
The Babylonians’ mathematical dexterity was astounding,

502
00:36:56,440 –> 00:37:03,080
and for nearly 2,000 years they spearheaded intellectual progress in the ancient world.

503
00:37:03,080 –> 00:37:08,280
But when their imperial power began to wane, so did their intellectual vigour.

504
00:37:16,400 –> 00:37:23,280
By 330BC, the Greeks had advanced their imperial reach into old Mesopotamia.

505
00:37:25,200 –> 00:37:31,000
This is Palmyra in central Syria, a once-great city built by the Greeks.

506
00:37:33,800 –> 00:37:41,000
The mathematical expertise needed to build structures with such geometric perfection is impressive.

507
00:37:42,120 –> 00:37:48,320
Just like the Babylonians before them, the Greeks were passionate about mathematics.

508
00:37:50,520 –> 00:37:53,080
The Greeks were clever colonists.

509
00:37:53,080 –> 00:37:56,280
They took the best from the civilisations they invaded

510
00:37:56,280 –> 00:37:58,720
to advance their own power and influence,

511
00:37:58,720 –> 00:38:01,880
but they were soon making contributions themselves.

512
00:38:01,880 –> 00:38:07,080
In my opinion, their greatest innovation was to do with a shift in the mind.

513
00:38:07,080 –> 00:38:11,560
What they initiated would influence humanity for centuries.

514
00:38:11,560 –> 00:38:14,520
They gave us the power of proof.

515
00:38:14,520 –> 00:38:18,200
Somehow they decided that they had to have a deductive system

516
00:38:18,200 –> 00:38:19,640
for their mathematics

517
00:38:19,640 –> 00:38:21,800
and the typical deductive system

518
00:38:21,800 –> 00:38:25,720
was to begin with certain axioms, which you assume are true.

519
00:38:25,720 –> 00:38:29,080
It’s as if you assume a certain theorem is true without proving it.

520
00:38:29,080 –> 00:38:34,600
And then, using logical methods and very careful steps,

521
00:38:34,600 –> 00:38:37,480
from these axioms you prove theorems

522
00:38:37,480 –> 00:38:42,400
and from those theorems you prove more theorems, and it just snowballs.

523
00:38:43,520 –> 00:38:47,000
Proof is what gives mathematics its strength.

524
00:38:47,000 –> 00:38:51,360
It’s the power of proof which means that the discoveries of the Greeks

525
00:38:51,360 –> 00:38:55,480
are as true today as they were 2,000 years ago.

526
00:38:55,480 –> 00:39:01,120
I needed to head west into the heart of the old Greek empire to learn more.

527
00:39:08,720 –> 00:39:14,000
For me, Greek mathematics has always been heroic and romantic.

528
00:39:15,280 –> 00:39:20,240
I’m on my way to Samos, less than a mile from the Turkish coast.

529
00:39:20,240 –> 00:39:25,000
This place has become synonymous with the birth of Greek mathematics,

530
00:39:25,000 –> 00:39:27,920
and it’s down to the legend of one man.

531
00:39:31,000 –> 00:39:33,120
His name is Pythagoras.

532
00:39:33,120 –> 00:39:36,520
The legends that surround his life and work have contributed

533
00:39:36,520 –> 00:39:40,320
to the celebrity status he has gained over the last 2,000 years.

534
00:39:40,320 –> 00:39:44,960
He’s credited, rightly or wrongly, with beginning the transformation

535
00:39:44,960 –> 00:39:50,240
from mathematics as a tool for accounting to the analytic subject we recognise today.

536
00:39:54,160 –> 00:39:57,160
Pythagoras is a controversial figure.

537
00:39:57,160 –> 00:40:00,360
Because he left no mathematical writings, many have questioned

538
00:40:00,360 –> 00:40:04,920
whether he indeed solved any of the theorems attributed to him.

539
00:40:04,920 –> 00:40:07,960
He founded a school in Samos in the sixth century BC,

540
00:40:07,960 –> 00:40:13,440
but his teachings were considered suspect and the Pythagoreans a bizarre sect.

541
00:40:14,960 –> 00:40:19,720
There is good evidence that there were schools of Pythagoreans,

542
00:40:19,720 –> 00:40:22,360
and they may have looked more like sects

543
00:40:22,360 –> 00:40:25,920
than what we associate with philosophical schools,

544
00:40:25,920 –> 00:40:30,920
because they didn’t just share knowledge, they also shared a way of life.

545
00:40:30,920 –> 00:40:36,080
There may have been communal living and they all seemed to have been

546
00:40:36,080 –> 00:40:40,000
involved in the politics of their cities.

547
00:40:40,000 –> 00:40:45,440
One feature that makes them unusual in the ancient world is that they included women.

548
00:40:46,560 –> 00:40:52,280
But Pythagoras is synonymous with understanding something that eluded the Egyptians and the Babylonians -

549
00:40:52,280 –> 00:40:56,040
the properties of right-angled triangles.

550
00:40:56,040 –> 00:40:58,400
What’s known as Pythagoras’ theorem

551
00:40:58,400 –> 00:41:01,360
states that if you take any right-angled triangle,

552
00:41:01,360 –> 00:41:05,320
build squares on all the sides, then the area of the largest square

553
00:41:05,320 –> 00:41:09,320
is equal to the sum of the squares on the two smaller sides.

554
00:41:13,240 –> 00:41:16,680
It’s at this point for me that mathematics is born

555
00:41:16,680 –> 00:41:19,880
and a gulf opens up between the other sciences,

556
00:41:19,880 –> 00:41:24,600
and the proof is as simple as it is devastating in its implications.

557
00:41:24,600 –> 00:41:28,080
Place four copies of the right-angled triangle

558
00:41:28,080 –> 00:41:29,840
on top of this surface.

559
00:41:29,840 –> 00:41:31,720
The square that you now see

560
00:41:31,720 –> 00:41:35,440
has sides equal to the hypotenuse of the triangle.

561
00:41:35,440 –> 00:41:37,600
By sliding these triangles around,

562
00:41:37,600 –> 00:41:40,720
we see how we can break the area of the large square up

563
00:41:40,720 –> 00:41:43,160
into the sum of two smaller squares,

564
00:41:43,160 –> 00:41:47,280
whose sides are given by the two short sides of the triangle.

565
00:41:47,280 –> 00:41:52,040
In other words, the square on the hypotenuse is equal to the sum

566
00:41:52,040 –> 00:41:55,840
of the squares on the other sides. Pythagoras’ theorem.

567
00:41:58,040 –> 00:42:02,400
It illustrates one of the characteristic themes of Greek mathematics -

568
00:42:02,400 –> 00:42:07,600
the appeal to beautiful arguments in geometry rather than a reliance on number.

569
00:42:11,400 –> 00:42:16,000
Pythagoras may have fallen out of favour and many of the discoveries accredited to him

570
00:42:16,000 –> 00:42:21,840
have been contested recently, but there’s one mathematical theory that I’m loath to take away from him.

571
00:42:21,840 –> 00:42:25,840
It’s to do with music and the discovery of the harmonic series.

572
00:42:27,680 –> 00:42:31,480
The story goes that, walking past a blacksmith’s one day,

573
00:42:31,480 –> 00:42:33,800
Pythagoras heard anvils being struck,

574
00:42:33,800 –> 00:42:38,800
and noticed how the notes being produced sounded in perfect harmony.

575
00:42:38,800 –> 00:42:42,240
He believed that there must be some rational explanation

576
00:42:42,240 –> 00:42:46,080
to make sense of why the notes sounded so appealing.

577
00:42:46,080 –> 00:42:48,560
The answer was mathematics.

578
00:42:53,480 –> 00:42:58,120
Experimenting with a stringed instrument, Pythagoras discovered that the intervals between

579
00:42:58,120 –> 00:43:02,400
harmonious musical notes were always represented as whole-number ratios.

580
00:43:05,200 –> 00:43:08,160
And here’s how he might have constructed his theory.

581
00:43:10,720 –> 00:43:13,600
First, play a note on the open string.

582
00:43:13,600 –> 00:43:15,120
MAN PLAYS NOTE

583
00:43:15,120 –> 00:43:17,040
Next, take half the length.

584
00:43:18,960 –> 00:43:22,160
The note almost sounds the same as the first note.

585
00:43:22,160 –> 00:43:27,120
In fact it’s an octave higher, but the relationship is so strong, we give these notes the same name.

586
00:43:27,120 –> 00:43:28,960
Now take a third the length.

587
00:43:31,600 –> 00:43:35,640
We get another note which sounds harmonious next to the first two,

588
00:43:35,640 –> 00:43:41,240
but take a length of string which is not in a whole-number ratio and all we get is dissonance.

589
00:43:46,600 –> 00:43:51,000
According to legend, Pythagoras was so excited by this discovery

590
00:43:51,000 –> 00:43:54,440
that he concluded the whole universe was built from numbers.

591
00:43:54,440 –> 00:44:00,040
But he and his followers were in for a rather unsettling challenge to their world view

592
00:44:00,040 –> 00:44:05,120
and it came about as a result of the theorem which bears Pythagoras’ name.

593
00:44:07,120 –> 00:44:12,400
Legend has it, one of his followers, a mathematician called Hippasus,

594
00:44:12,400 –> 00:44:15,480
set out to find the length of the diagonal

595
00:44:15,480 –> 00:44:19,760
for a right-angled triangle with two sides measuring one unit.

596
00:44:19,760 –> 00:44:25,520
Pythagoras’ theorem implied that the length of the diagonal was a number whose square was two.

597
00:44:25,520 –> 00:44:29,560
The Pythagoreans assumed that the answer would be a fraction,

598
00:44:29,560 –> 00:44:36,000
but when Hippasus tried to express it in this way, no matter how he tried, he couldn’t capture it.

599
00:44:36,000 –> 00:44:38,600
Eventually he realised his mistake.

600
00:44:38,600 –> 00:44:43,320
It was the assumption that the value was a fraction at all which was wrong.

601
00:44:43,320 –> 00:44:49,440
The value of the square root of two was the number that the Babylonians etched into the Yale tablet.

602
00:44:49,440 –> 00:44:53,320
However, they didn’t recognise the special character of this number.

603
00:44:53,320 –> 00:44:55,040
But Hippasus did.

604
00:44:55,040 –> 00:44:57,560
It was an irrational number.

605
00:45:00,880 –> 00:45:04,800
The discovery of this new number, and others like it, is akin to an explorer

606
00:45:04,800 –> 00:45:09,240
discovering a new continent, or a naturalist finding a new species.

607
00:45:09,240 –> 00:45:13,520
But these irrational numbers didn’t fit the Pythagorean world view.

608
00:45:13,520 –> 00:45:19,120
Later Greek commentators tell the story of how Pythagoras swore his sect to secrecy,

609
00:45:19,120 –> 00:45:21,840
but Hippasus let slip the discovery

610
00:45:21,840 –> 00:45:25,600
and was promptly drowned for his attempts to broadcast their research.

611
00:45:27,080 –> 00:45:32,440
But these mathematical discoveries could not be easily suppressed.

612
00:45:32,440 –> 00:45:37,920
Schools of philosophy and science started to flourish all over Greece, building on these foundations.

613
00:45:37,920 –> 00:45:42,360
The most famous of these was the Academy.

614
00:45:42,360 –> 00:45:47,560
Plato founded this school in Athens in 387 BC.

615
00:45:47,560 –> 00:45:54,040
Although we think of him today as a philosopher, he was one of mathematics’ most important patrons.

616
00:45:54,040 –> 00:45:57,720
Plato was enraptured by the Pythagorean world view

617
00:45:57,720 –> 00:46:02,040
and considered mathematics the bedrock of knowledge.

618
00:46:02,040 –> 00:46:07,200
Some people would say that Plato is the most influential figure

619
00:46:07,200 –> 00:46:10,080
for our perception of Greek mathematics.

620
00:46:10,080 –> 00:46:15,120
He argued that mathematics is an important form of knowledge

621
00:46:15,120 –> 00:46:17,600
and does have a connection with reality.

622
00:46:17,600 –> 00:46:23,480
So by knowing mathematics, we know more about reality.

623
00:46:23,480 –> 00:46:29,240
In his dialogue Timaeus, Plato proposes the thesis that geometry is the key to unlocking

624
00:46:29,240 –> 00:46:33,480
the secrets of the universe, a view still held by scientists today.

625
00:46:33,480 –> 00:46:37,480
Indeed, the importance Plato attached to geometry is encapsulated

626
00:46:37,480 –> 00:46:43,960
in the sign that was mounted above the Academy, “Let no-one ignorant of geometry enter here.”

627
00:46:47,520 –> 00:46:53,720
Plato proposed that the universe could be crystallised into five regular symmetrical shapes.

628
00:46:53,720 –> 00:46:56,640
These shapes, which we now call the Platonic solids,

629
00:46:56,640 –> 00:46:59,600
were composed of regular polygons, assembled to create

630
00:46:59,600 –> 00:47:03,080
three-dimensional symmetrical objects.

631
00:47:03,080 –> 00:47:05,720
The tetrahedron represented fire.

632
00:47:05,720 –> 00:47:09,960
The icosahedron, made from 20 triangles, represented water.

633
00:47:09,960 –> 00:47:12,160
The stable cube was Earth.

634
00:47:12,160 –> 00:47:15,880
The eight-faced octahedron was air.

635
00:47:15,880 –> 00:47:19,440
And the fifth Platonic solid, the dodecahedron,

636
00:47:19,440 –> 00:47:22,280
made out of 12 pentagons, was reserved for the shape

637
00:47:22,280 –> 00:47:26,000
that captured Plato’s view of the universe.

638
00:47:29,600 –> 00:47:33,640
Plato’s theory would have a seismic influence and continued to inspire

639
00:47:33,640 –> 00:47:37,400
mathematicians and astronomers for over 1,500 years.

640
00:47:38,360 –> 00:47:41,120
In addition to the breakthroughs made in the Academy,

641
00:47:41,120 –> 00:47:45,040
mathematical triumphs were also emerging from the edge of the Greek empire,

642
00:47:45,040 –> 00:47:51,520
and owed as much to the mathematical heritage of the Egyptians as the Greeks.

643
00:47:51,520 –> 00:47:58,000
Alexandria became a hub of academic excellence under the rule of the Ptolemies in the 3rd century BC,

644
00:47:58,000 –> 00:48:04,320
and its famous library soon gained a reputation to rival Plato’s Academy.

645
00:48:04,320 –> 00:48:11,760
The kings of Alexandria were prepared to invest in the arts and culture,

646
00:48:11,760 –> 00:48:14,960
in technology, mathematics, grammar,

647
00:48:14,960 –> 00:48:19,680
because patronage for cultural pursuits

648
00:48:19,680 –> 00:48:27,000
was one way of showing that you were a more prestigious ruler,

649
00:48:27,000 –> 00:48:30,320
and had a better entitlement to greatness.

650
00:48:32,040 –> 00:48:35,360
The old library and its precious contents were destroyed

651
00:48:35,360 –> 00:48:38,240
But its spirit is alive in a new building.

652
00:48:40,240 –> 00:48:44,120
Today, the library remains a place of discovery and scholarship.

653
00:48:48,600 –> 00:48:51,920
Mathematicians and philosophers flocked to Alexandria,

654
00:48:51,920 –> 00:48:55,080
driven by their thirst for knowledge and the pursuit of excellence.

655
00:48:55,080 –> 00:48:59,040
The patrons of the library were the first professional scientists,

656
00:48:59,040 –> 00:49:02,600
individuals who were paid for their devotion to research.

657
00:49:02,600 –> 00:49:04,720
But of all those early pioneers,

658
00:49:04,720 –> 00:49:08,880
my hero is the enigmatic Greek mathematician Euclid.

659
00:49:12,560 –> 00:49:15,120
We know very little about Euclid’s life,

660
00:49:15,120 –> 00:49:19,360
but his greatest achievements were as a chronicler of mathematics.

661
00:49:19,360 –> 00:49:24,600
Around 300 BC, he wrote the most important text book of all time -

662
00:49:24,600 –> 00:49:27,080
The Elements. In The Elements,

663
00:49:27,080 –> 00:49:31,120
we find the culmination of the mathematical revolution

664
00:49:31,120 –> 00:49:32,960
which had taken place in Greece.

665
00:49:34,880 –> 00:49:39,240
It’s built on a series of mathematical assumptions, called axioms.

666
00:49:39,240 –> 00:49:44,000
For example, a line can be drawn between any two points.

667
00:49:44,000 –> 00:49:48,760
From these axioms, logical deductions are made and mathematical theorems established.

668
00:49:51,880 –> 00:49:56,360
The Elements contains formulas for calculating the volumes of cones

669
00:49:56,360 –> 00:49:59,400
and cylinders, proofs about geometric series,

670
00:49:59,400 –> 00:50:02,160
perfect numbers and primes.

671
00:50:02,160 –> 00:50:06,760
The climax of The Elements is a proof that there are only five Platonic solids.

672
00:50:09,560 –> 00:50:14,280
For me, this last theorem captures the power of mathematics.

673
00:50:14,280 –> 00:50:17,080
It’s one thing to build five symmetrical solids,

674
00:50:17,080 –> 00:50:22,600
quite another to come up with a watertight, logical argument for why there can’t be a sixth.

675
00:50:22,600 –> 00:50:26,600
The Elements unfolds like a wonderful, logical mystery novel.

676
00:50:26,600 –> 00:50:29,720
But this is a story which transcends time.

677
00:50:29,720 –> 00:50:33,560
Scientific theories get knocked down, from one generation to the next,

678
00:50:33,560 –> 00:50:39,920
but the theorems in The Elements are as true today as they were 2,000 years ago.

679
00:50:39,920 –> 00:50:43,480
When you stop and think about it, it’s really amazing.

680
00:50:43,480 –> 00:50:45,160
It’s the same theorems that we teach.

681
00:50:45,160 –> 00:50:49,960
We may teach them in a slightly different way, we may organise them differently,

682
00:50:49,960 –> 00:50:54,200
but it’s Euclidean geometry that is still valid,

683
00:50:54,200 –> 00:50:58,320
and even in higher mathematics, when you go to higher dimensional spaces,

684
00:50:58,320 –> 00:51:00,560
you’re still using Euclidean geometry.

685
00:51:02,080 –> 00:51:06,080
Alexandria must have been an inspiring place for the ancient scholars,

686
00:51:06,080 –> 00:51:12,360
and Euclid’s fame would have attracted even more eager, young intellectuals to the Egyptian port.

687
00:51:12,360 –> 00:51:18,680
One mathematician who particularly enjoyed the intellectual environment in Alexandria was Archimedes.

688
00:51:19,640 –> 00:51:23,200
He would become a mathematical visionary.

689
00:51:23,200 –> 00:51:28,080
The best Greek mathematicians, they were always pushing the limits,

690
00:51:28,080 –> 00:51:29,560
pushing the envelope.

691
00:51:29,560 –> 00:51:32,200
So, Archimedes…

692
00:51:32,200 –> 00:51:35,200
did what he could with polygons,

693
00:51:35,200 –> 00:51:37,520
with solids.

694
00:51:37,520 –> 00:51:40,360
He then moved on to centres of gravity.

695
00:51:40,360 –> 00:51:44,680
He then moved on to the spiral.

696
00:51:44,680 –> 00:51:50,800
This instinct to try and mathematise everything

697
00:51:50,800 –> 00:51:54,440
is something that I see as a legacy.

698
00:51:55,520 –> 00:52:00,280
One of Archimedes’ specialities was weapons of mass destruction.

699
00:52:00,280 –> 00:52:06,360
They were used against the Romans when they invaded his home of Syracuse in 212 BC.

700
00:52:06,360 –> 00:52:10,200
He also designed mirrors, which harnessed the power of the sun,

701
00:52:10,200 –> 00:52:12,760
to set the Roman ships on fire.

702
00:52:12,760 –> 00:52:17,520
But to Archimedes, these endeavours were mere amusements in geometry.

703
00:52:17,520 –> 00:52:20,280
He had loftier ambitions.

704
00:52:23,040 –> 00:52:29,560
Archimedes was enraptured by pure mathematics and believed in studying mathematics for its own sake

705
00:52:29,560 –> 00:52:33,800
and not for the ignoble trade of engineering or the sordid quest for profit.

706
00:52:33,800 –> 00:52:37,840
One of his finest investigations into pure mathematics

707
00:52:37,840 –> 00:52:41,840
was to produce formulas to calculate the areas of regular shapes.

708
00:52:43,760 –> 00:52:49,480
Archimedes’ method was to capture new shapes by using shapes he already understood.

709
00:52:49,480 –> 00:52:52,720
So, for example, to calculate the area of a circle,

710
00:52:52,720 –> 00:52:57,920
he would enclose it inside a triangle, and then by doubling the number of sides on the triangle,

711
00:52:57,920 –> 00:53:02,320
the enclosing shape would get closer and closer to the circle.

712
00:53:02,320 –> 00:53:04,360
Indeed, we sometimes call a circle

713
00:53:04,360 –> 00:53:07,360
a polygon with an infinite number of sides.

714
00:53:07,360 –> 00:53:11,200
But by estimating the area of the circle, Archimedes is, in fact,

715
00:53:11,200 –> 00:53:15,480
getting a value for pi, the most important number in mathematics.

716
00:53:16,480 –> 00:53:22,760
However, it was calculating the volumes of solid objects where Archimedes excelled.

717
00:53:22,760 –> 00:53:25,800
He found a way to calculate the volume of a sphere

718
00:53:25,800 –> 00:53:30,280
by slicing it up and approximating each slice as a cylinder.

719
00:53:30,280 –> 00:53:33,120
He then added up the volumes of the slices

720
00:53:33,120 –> 00:53:36,480
to get an approximate value for the sphere.

721
00:53:36,480 –> 00:53:39,440
But his act of genius was to see what happens

722
00:53:39,440 –> 00:53:42,280
if you make the slices thinner and thinner.

723
00:53:42,280 –> 00:53:47,040
In the limit, the approximation becomes an exact calculation.

724
00:53:51,080 –> 00:53:56,040
But it was Archimedes’ commitment to mathematics that would be his undoing.

725
00:53:58,120 –> 00:54:02,960
Archimedes was contemplating a problem about circles traced in the sand.

726
00:54:02,960 –> 00:54:05,600
When a Roman soldier accosted him,

727
00:54:05,600 –> 00:54:11,640
Archimedes was so engrossed in his problem that he insisted that he be allowed to finish his theorem.

728
00:54:11,640 –> 00:54:16,920
But the Roman soldier was not interested in Archimedes’ problem and killed him on the spot.

729
00:54:16,920 –> 00:54:21,800
Even in death, Archimedes’ devotion to mathematics was unwavering.

730
00:54:43,360 –> 00:54:46,480
By the middle of the 1st century BC,

731
00:54:46,480 –> 00:54:50,520
the Romans had tightened their grip on the old Greek empire.

732
00:54:50,520 –> 00:54:53,320
They were less smitten with the beauty of mathematics

733
00:54:53,320 –> 00:54:56,640
and were more concerned with its practical applications.

734
00:54:56,640 –> 00:55:02,520
This pragmatic attitude signalled the beginning of the end for the great library of Alexandria.

735
00:55:02,520 –> 00:55:06,760
But one mathematician was determined to keep the legacy of the Greeks alive.

736
00:55:06,760 –> 00:55:11,640
Hypatia was exceptional, a female mathematician,

737
00:55:11,640 –> 00:55:14,800
and a pagan in the piously Christian Roman empire.

738
00:55:16,680 –> 00:55:21,560
Hypatia was very prestigious and very influential in her time.

739
00:55:21,560 –> 00:55:27,440
She was a teacher with a lot of students, a lot of followers.

740
00:55:27,440 –> 00:55:31,680
She was politically influential in Alexandria.

741
00:55:31,680 –> 00:55:34,560
So it’s this combination of…

742
00:55:34,560 –> 00:55:40,840
high knowledge and high prestige that may have made her

743
00:55:40,840 –> 00:55:44,400
a figure of hatred for…

744
00:55:44,400 –> 00:55:46,080
the Christian mob.

745
00:55:51,760 –> 00:55:55,800
One morning during Lent, Hypatia was dragged off her chariot

746
00:55:55,800 –> 00:55:59,840
by a zealous Christian mob and taken to a church.

747
00:55:59,840 –> 00:56:03,560
There, she was tortured and brutally murdered.

748
00:56:06,280 –> 00:56:09,880
The dramatic circumstances of her life and death

749
00:56:09,880 –> 00:56:12,000
fascinated later generations.

750
00:56:12,000 –> 00:56:17,680
Sadly, her cult status eclipsed her mathematical achievements.

751
00:56:17,680 –> 00:56:20,720
She was, in fact, a brilliant teacher and theorist,

752
00:56:20,720 –> 00:56:26,440
and her death dealt a final blow to the Greek mathematical heritage of Alexandria.

753
00:56:33,800 –> 00:56:37,680
My travels have taken me on a fascinating journey to uncover

754
00:56:37,680 –> 00:56:42,880
the passion and innovation of the world’s earliest mathematicians.

755
00:56:42,880 –> 00:56:47,920
It’s the breakthroughs made by those early pioneers of Egypt, Babylon and Greece

756
00:56:47,920 –> 00:56:52,320
that are the foundations on which my subject is built today.

757
00:56:52,320 –> 00:56:55,760
But this is just the beginning of my mathematical odyssey.

758
00:56:55,760 –> 00:56:59,400
The next leg of my journey lies east, in the depths of Asia,

759
00:56:59,400 –> 00:57:02,560
where mathematicians scaled even greater heights

760
00:57:02,560 –> 00:57:04,800
in pursuit of knowledge.

761
00:57:04,800 –> 00:57:08,720
With this new era came a new language of algebra and numbers,

762
00:57:08,720 –> 00:57:12,920
better suited to telling the next chapter in the story of maths.

763
00:57:12,920 –> 00:57:16,600
You can learn more about the story of maths

764
00:57:16,600 –> 00:57:19,840
with the Open University at…

765
00:57:36,040 –> 00:57:39,080
Subtitles by Red Bee Media Ltd

2019-04-06

［独立博客文摘 - 卢昌海］欧几里得与《几何原本》

用 “知之甚少” 已不足以形容我们对这位留下《几何原本》 (The Elements) 及其他数种著作，被尊为 “几何之父” (Father of Geometry) 的伟大先贤的生平了解之贫乏。

而阿基米德之所以出现在对欧几里得生活年代的界定中，乃是因为他在《论球和圆柱》 (On the Sphere and the Cylinder) 一书中提到过《几何原本》

普罗克洛斯提到过托勒密一世 (Ptolemy I) 跟欧几里得的一段广为流传的对话，前者问学习几何有无捷径，欧几里得答曰 “在几何中没有 ‘御道’ (royal road)”。由于托勒密一世的都城是亚历山大港，对话被认为发生在亚历山大港——但这虽能说明欧几里得是当时亚历山大港的知名几何学家，却也并不等同于在亚历山大港教过书。

对欧几里得的生平了解为何会如此贫乏？在两千多年后的今天恐已很难得到确凿回答了，有一种猜测认为欧几里得是历史上最早的科学专才之一，将精力完全投入了数学之中，从不参与任何政治性或事务性的活动，而后两者是那个时代的人物青史留名的重要渠道，因而欧几里得几乎是 “自绝” 于历史。著名美籍比利时裔现代科学史学家、科学史作为一门现代学科的创始人乔治·萨顿 (George Sarton) 曾经感慨道，对欧几里得以及其他某些先贤生平的这种无知 “是寻常而非例外的，人们记住了暴君和独裁者，成功的政治人物，富豪 (起码一部分富豪)，却忘记了自己最大的恩人”。

《几何原本》是经受时光洗礼流传至今的最早的数学专著之一，不过也被一些人视为是若干更早的数学专著失传的 “罪魁祸首”，因为在题材上被《几何原本》涵盖到的数学专著在跟这部伟大著作竞争时，大都落败陨灭了——而且更 “糟糕” 的是，如我们在后文将会看到的，《几何原本》在题材上的涵盖面偏偏是相当广的，并不限于几何。

对尽可能接近原始的版本的追索也从一个侧面显示了欧几里得的厉害：因为追索所得的 “欧几里得版” 与包括 “赛翁版” 在内的若干其他版本的相互比对，显示出了后者的诸多缺陷，比如引进了不必要的公设 (postulate)，忽视了必要的公理 (common notion)，等等。赛翁等人就像如今那些篡改金庸武侠的导演和编剧一样，虽有传播之功，却并没有与原作者同等的造诣，从而产生出的是更差而不是更好的版本。

漫长的时光抹去了大量线索，使我们很难用足够确凿的方式判定他们在原创与继承之间的比例分配。比如上文提到过的美籍比利时裔科学史学家萨顿就对很多先贤的 “……之父” 头衔存有疑虑——其中也包括了对欧几里得 “几何之父” 头衔的疑虑。瑞典哲学家安德斯·韦德伯格 (Anders Wedberg) 在其三卷本的《哲学史》 (A History of Philosophy) 的开篇也曾表示，那些被我们视为伟大原创者的哲学家有可能只是记叙先辈成果的有天赋的表述者。

普罗克洛斯曾评价道，《几何原本》的内容虽部分来自前人，但欧几里得将很多不严格的证明纳入了严整有序、无可怀疑的框架。从数学史的角度讲，这一评价是中肯的，《几何原本》的重要性与其说是罗列了大量旧定理或证明了若干新定理，不如说是示范了公理化体系的巨大威力，将数学证明的严密性推上了前所未有的高度。从这一角度讲，欧几里得与《几何原本》所享的盛誉是实至名归的。很多距离欧几里得时代不太遥远的古代学者也对《几何原本》做出了很高评价，而欧几里得除《几何原本》之外流传于世的其他著作也显示出了跟那样的评价相称的水准。

作为一部示范了公理化体系巨大威力的著作，《几何原本》一开篇——即第 1 卷——就展开公理体系，不带一个字的多余铺垫，直接就列出了 23 个定义， 5 条公设和 5 条公理。这是迥异于柏拉图和亚里士多德，乃至迥异于一切哲学著作的风格。

《几何原本》对公理和公设的区分跟亚里士多德的著作是明显相似的，即公设是指单一学科——对《几何原本》而言是几何——独有的 “真理”，公理则是适用于所有科学的 “真理”。

亚里士多德并且明确指出，并非所有真命题皆可被证明，必须将某些明显为真却无法证明的命题作为推理的起点，这是公理和公设的起源，也是其之所以必要的根本原因。一般认为，亚里士多德的这些观点对欧几里得是有一定影响的。不过，亚里士多德虽对公理和公设作出过区分，却不曾对具体的——即几何领域的——公设做过论述，《几何原本》所列的公设也因此被某些研究者，比如前文提到过的希腊数学史专家希斯，视为是欧几里得的原创。

《几何原本》所列的定义用现代公理体系的要求来衡量，只是一种形象化的努力，提供的是直观理解，作为教学说明不无价值，细究起来却往往会陷入逻辑困境——之所以如此，其实跟并非所有真命题皆可被证明相类似，因为对一个概念的定义势必会用到其它概念，就像对一个命题的证明势必会用到其它命题一样。原因既然类似，解决方法其实也就呼之欲出了，那就是必须引进一些不加定义的概念，就像必须引进不加证明的公理和公设一样，这也正是现代公理体系所走的路子。在现代公理体系中，基本概念是不加定义的，对其的全部限定来自公理体系本身 (当然，现代公理体系也并不排斥定义，但那通常是针对次级概念，所起的作用则是简化叙述)。《几何原本》没有走这样的路子，有可能是欧几里得没有意识到形象化定义的缺陷，但也不排除是出于教学考虑。事实上，关于《几何原本》的一个有趣但没有答案的问题乃是：它究竟是欧几里得写给同行的学术专著，还是写给学生的授课讲义？倘是后者，则对概念作一些逻辑上虽非无懈可击，但有助于直观理解的形象化描述不失为有益的选择。

在《几何原本》所构建的公理体系中，另一个可圈可点之处是对定义与存在性做出了一定程度的区分，从而避免了视所定义的概念为自动存在这一并非显而易见的错误。对定义与存在性的区分虽然连现代人也时常会稀里糊涂，历史却相当悠久，可回溯到欧几里得之前，从而并非欧几里得的独创。事实上，芝诺的悖论给人的一个重大启示便是：哪怕最直观的概念，其存在性也并非不言而喻。自那以后，对定义与存在性的区分就引起了像柏拉图和亚里士多德那样的先贤的注意，比如亚里士多德在《后分析篇》 (Posterior Analytics) 中就明确表示，定义一个客体不等于宣告它的存在，后者必须予以证明或作为假设。欧几里得的命题 1 和命题 46 属于对存在性予以证明，公设 1 和公设 3 则系将存在性作为假设，都可纳入亚里士多德的阐述。

说到对定义与存在性的区分，还有一点值得补充，那就是欧几里得对存在性的很多证明是所谓的 “构造性证明” (constructive proof)，也就是通过直接给出构造方法来证明存在性。在数学中，这是最强有力，从而也最没有争议的存在性证明。

事实上，《几何原本》中的 “几何” 一词有可能是后人添加的——比如 1570 年出版的第一个英文版名为《The Elements of Geometry》 (可译为《几何原理》或《几何基础》)， 1607 年出版的前 6 卷的中文版名为《几何原本》。但该书的希腊文书名 “Στοιχεῖα” 其实只对应于 “Elements”，其含义据普罗克洛斯所言，乃是证明之起点，其他定理赖以成立之基础，类似于字母在语言中的作用 (这个出自普罗克洛斯本人的比喻颇有双关之意，因为在希腊文里，字母恰好也是 “Στοιχεῖα”)。从这一含义来讲，《几何原本》的希腊文书名只对应于 “原理” 或 “基础”，起码在字面上不带 “几何” 一词。

亚里士多德在《形而上学》一书中就界定 “Elements” 为几何中其他命题所共同依赖的命题。考虑到亚里士多德是一位很可能对欧几里得有过重大影响的先贤，他对 “Elements” 一词的界定很可能意味着 “Elements” 这一书名从一开始就隐含了 “几何” 之意，而后人将 “几何” 一词显明化，则或可视为在 “Elements” 一词的本身含义扩张之后对原始含义的回溯。

在《几何原本》的煌煌 13 卷中，内容分布大体是这样的：第 1~~4 卷主要为平面几何，但间杂了数的理论——比如第 2 卷给出了乘法对加法的分配律等，并求解了若干代数方程；第 5~~6 卷为比例理论及相似理论，但同样间杂了数的理论，且关于数有很深刻的洞见；第 79 卷以对数学分支的现代分类观之，是对几何与数的相对比例的的逆转——转入了以数为主的数论范畴，其中包括了对素数有无穷多个等重要命题的证明 (第 9 卷命题 20)；第 10 卷延续了以数为主的局部 “主旋律”，对 “不可公度量” (incommensurable)——也就是无理数——做了详细讨论；第 1113 卷重返几何，但由平面走向立体，以对包括 “柏拉图正多面体” (Plato solid) 在内的诸多立体几何话题的探讨结束了全书。

第 5 卷所间杂的关于数的 “很深刻的洞见”。这一卷关于数的介绍，可以说是继毕达哥拉斯学派发现无理数之后，希腊数学在数的理论上的再次推进。这次推进虽未像发现无理数那样发现新类型的数，却具有很高的系统性，加深了关于数的理解，也因此赢得了后世数学家的敬意。比如科学巨匠艾萨克·牛顿 (Isaac Newton) 的老师艾萨克·巴罗 (Isaac Barrow) 曾将这一卷所构筑的比例理论称为整部《几何原本》中最精妙的发明，认为 “没什么东西比这一比例学说确立得更牢固，处理得更精密”。 19 世纪的英国数学家阿瑟·凯莱 (Arthur Cayley) 也表示 “数学中几乎没什么东西比这本奇妙的第 5 卷更美丽”。

用希腊数学史专家希斯的话说， “在欧几里得对相同比值的定义与戴德金的现代无理数理论之间存在着几乎巧合般的严格对应”。这个跨越两千多年时光的 “严格对应” 正是《几何原本》第 5 卷所间杂的关于数的 “很深刻的洞见”，那样的洞见当然是美丽的——智慧上的美丽。

《几何原本》中的数的理论在第 7~9 卷得到了进一步发展。这几卷被科学史学家萨顿称为 “第一部数论专著” (first treatise on the theory of number)。从这个意义上讲，欧几里得不仅是最著名的几何学家，也是第一部数论专著的作者，堪称 “通吃” 了当时的数学领域。不过关于《几何原本》中的数，有一个微妙之处值得一提，那就是《几何原本》对 “数” (number) 和 “量” (magnitude) 作了一个如今看来并无必要的区分，其中 “量” 本质上是线段长度，可以表示无理数[注四]， “数” 则由单位长度积聚而成，本质上是整数，相互间的比值则是有理数。这种区分造成了一定的繁琐性，比如 “数” 的比值与 “量” 的比值本该是统一的，《几何原本》中的定义——前者为第 7 卷定义 20，后者为第 5 卷定义 5——却很不相同，给后世的诠释者带来过不小的困扰，可以说是《几何原本》的一个缺陷。

与其他各卷相较，《几何原本》的第 10 卷是命题最多的，共有 115 个命题，约占全书命题总数的 1/4。在这些命题中，很值得一提的是命题 1，即 “给定两个不相等的量，若从较大的量中减去一个大于其一半的量，再从余量中减去大于其一半的量，如此连续进行，则必能得到一个比较小的量更小的量。” 由于 “较小的量” 是任意的，因此由这一命题所得到的是任意小的量，这是所谓 “穷竭法” (method of exhaustion) 的基础，在一定程度上也是微积分思想的萌芽。 “穷竭法” 后来被阿基米德 (Archimedes) 用于计算很多形体的面积和体积，欧几里得本人也在《几何原本》的第 12 卷中用它证明了一系列重要命题，比如圆的面积正比于直径的平方，球的体积正比于直径的立方，圆锥的体积是与它同底等高的圆柱体积的 1/3，等等，是《几何原本》的重要亮点之一。

以时间的延绵而论，欧几里得的《几何原本》可以跟此前的古希腊原子论及亚里士多德的逻辑鼎足而三，以体系的恢宏而论，则远远超过了亚里士多德的逻辑，更绝非在很长时间里只具抽象意义的古希腊原子论可比。

《几何原本》成了一个巨大典范，小到以诸如 “证毕” (öπερ ’έδει δεîξαι，其拉丁文缩写是如今几乎每个中学生都熟悉的 Q.E.D.) 表示证明结束的习惯，大到以公理化体系作为理论构筑和表述的基本手段，都被广泛模仿。在《几何原本》的模仿者中，包括了科学家——比如牛顿、哲学家——比如伊曼努尔·康德 (Immanuel Kant)、神学家——比如托马斯·阿奎纳 (St. Thomas Aquinas)，等等。至于数学家，则不仅仅是模仿者，而且早已程度不等地习惯了以公理化手段为数学理论的 “标配”。

《几何原本》及其后继作品还对许多著名学人的个人成长起到了近乎 “第一推动力” 的作用。比如爱因斯坦在晚年自述中回忆道： “12 岁时，我经历了另一种性质完全不同的惊奇：是在一个学年开始时，我得到一本关于欧几里得平面几何的小书时经历的。那本书里有许多断言，比如三角形的三条高交于一点，虽然一点也不显而易见，却可以如此确定地加以证明，以至于任何怀疑都似乎是不可能的。这种明晰性和确定性给我留下了难以形容的印象。”

英国哲学家伯特兰·罗素 (Bertrand Russell) 也在自传中回忆道： “11 岁时，我开始在哥哥的指导下学习欧几里得，这是我一生最重大的事件之一。我从未想到过世上竟有如此有滋味的东西。当我学了第五个命题之后，哥哥告诉我那被普遍认为是困难的，但我却一点也没觉得困难。这是我第一次意识到我也许有一些智慧。” 😄

可以毫不夸张地说，哪怕《几何原本》的所有内容都出自前人，将之整理成如此严整有序、恢宏深邃的逻辑体系——这被史学界公认为是欧几里得的贡献——也足以使欧几里得成为数学史乃至科学史上最伟大的教师，使《几何原本》成为数学史乃至科学史上最伟大的教科书。

2019-03-26

LeetCode - Algorithms - 44. Wildcard Matching

Only mark. It’s an example of dynamic programming.
I don’t know how to solve it, what’s more, I can’t understand the answer of others, so I run code of others on leetcode.

Java

by XN W

class Solution {
    public boolean isMatch(String s, String p) {
        int pl = 0;  
        for (int i = 0; i < p.length(); i++) {  
             if (p.charAt(i) != '*') {  
                  pl++;  
             }  
        }  
        if (pl > s.length()) {  
             return false;  
        }  
        if (s.length() == 0 && pl == 0) {  
             return true;  
        }
        boolean[][] store = new boolean[2][s.length() + 1];  
        store[0][0] = true;  
        store[1][0] = false;   
        for (int ps = 1; ps <= p.length(); ps++) {
             if (p.charAt(ps - 1) == '*') {  
                  store[1][0] = store[0][0];  
             } else {  
                  store[1][0] = false;  
             }  
             for (int ss = 1; ss <= s.length(); ss++) {  
                  if (p.charAt(ps - 1) == '?'  
                            || p.charAt(ps - 1) == s.charAt(ss - 1)) {  
                       store[1][ss] = store[0][ss - 1];  
                  } else if (p.charAt(ps - 1) != '*') {  
                       store[1][ss] = false;  
                  } else {  
                       store[1][ss] = store[0][ss - 1] || store[1][ss - 1]  
                                 || store[0][ss];  
                  }  
             }  
             for (int i = 0; i <= s.length(); i++) {  
                  store[0][i] = store[1][i];  
             }  
        }  
        return store[1][s.length()];      
    }
}

Submission Detail

by XN W

1808 / 1808 test cases passed.
Runtime: 7 ms, faster than 100.00% of Java online submissions for Wildcard Matching.
Memory Usage: 37.7 MB, less than 93.46% of Java online submissions for Wildcard Matching.

ref

Wildcard Matching

I hate this algorithms!! Seriously!!!

recursive way
DP way