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数学家Erdos(厄多斯)的故事(共78则)

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一个数学家就是一台把咖啡转化为数学定理的机器。--P. Erdos(厄多斯)

Erdos(1913-1996)是当代最伟大的数学家之一,他一生中同485位合作者发表过1475篇数学论文,涉及数学的许多领域。下面的这些故事主要来自他的传记《数字情种》,当然也有一些从别处搜集来的故事。

前面所引的那段话暗示了一天工作19个小时以上(在古稀之年依然如此)的Erdos对兴奋剂的依赖。1979年,他的朋友 Graham与Erdos打赌,只要他一个月之内不服用安非他明,Graham就输给他500美元。Erdos成功地赢得了这次打赌,但他对Graham 说:"你帮我证明了我不是一个瘾君子,但在这段时间内我一无所成。我早晨起来就盯着一张白纸发呆,毫无主意,跟普通人没有什么区别。你因此使数学的发展滞后了一个月。"所以他随后便重新开始服用兴奋剂。 

--2--

SF创造我们就是为了拿我们的痛苦取乐,我们死得愈早,他的计划就愈早落空。--Erdos

Erdos有一套他自己的语言。比如说,他称上帝为"SF"(Supreme Fascist,最大的法西斯份子),因为他总是折磨Erdos,藏起他的眼镜,偷走他的匈牙利护照,甚至把持着各种古怪的数学问题的解答不让他发现。 Erdos并不敬畏上帝,死亡对于Erdos的唯一意义就是使他无法再进行数学研究,所以他经常说:"在坟墓里有的是时间休息。"

Erdos从事数学研究的方式很独特。他总是游历于世界各地的大学和研究所,登门拜访那里的数学家,向对方宣布:"我的头脑敞开着。"然后他们便开始讨论数学问题,一连持续几天,直到双方都厌倦了为止。他从不在一个城市里连续呆上一个月,他的座右铭是:"另一个屋顶,另一个证明。(Another roof, another proof.)"

--3--

出生的不幸是什么时候降临在你头上的?--Erdos 常用的问候语

除了SF之外,Erdos还有很多独特的术语。他称小孩为ε,称孙辈为ε^2,称桥牌打得不好的人为o(x),称女人为"主人",男人为"奴隶",结婚为 "被俘虏了",离婚为"自由了",音乐为"噪声",酒精为"毒药",进行一次数学讲座为"布道",非数学家为"不值一提的人",美国"Sam",苏联为 "Joe"。

Martin Gardner第一次见到他的时候,他问Gardner:"你是什么时候到的?"Gardner连忙看表,这时Graham小声告诉他,对于Erdos来说,这句话的意思是:"你是什么时候出生的?"

--4--

一个数学家必须是在每个星期都有一些新的研究工作才成为数学家。--Erdos

Erdos见到他的同行时,总喜欢问:"你昨天有什么新的发现?"可事实上并不是每个数学家都能像他那样每年发表50多篇论文!

一次,他在巴黎演讲后,一位法国数学家问他关于某位有爵士头衔的英国数学家的近况。他回答:"这个可怜的家伙两年前就已死去了。"另外一位法国数学家却说:"不可能,上个月我还在罗马见过他。"Erdos答:"你应该明白我的意思,我是指他这两年没有搞出什么新东西来。"

在Erdos的术语里,"死了"是指这个人不再进行数学研究,"离开了"才是真正的生理学意义上的死亡。

--5--

我只要拿张纸,坐下,就能思考。--Erdos

在很多张Erdos的照片上,他都是低着头的。这种姿势很容易让人以为是在打盹,但Erdos声称他是在思考问题。

1939年,Erdos曾听过M.Kac在Princeton研究所做的一个报告。事后Kac这样回忆道:"在我报告的大部分时间内,他都快要睡着了。报告的内容跟他的兴趣毫不沾边。在将近结尾时我大致说了一下我在素因子方面遇到的困难。一提起数论,Erdos马上就精神起来,让我再解释一下到底困难在哪。不到几分钟,报告还没有结束,Erdos就打断我的讲话并宣布问题解决了!"

注:Mark Kac,波兰裔美籍数学家,主要研究概率论与数学物理,有以他名字命名的Feynman-Kac公式。

--6--

匈牙利人的问题在于,每次战争我们都站错了队。--Erdos

Erdos于1913年3月26日出生于布达佩斯的一个匈牙利化了的犹太人家庭。匈牙利和犹太,这两个来自东方的古老的民族,曾经孕育了无数的伟人。八十三年后,当Erdos与世长辞时,Gordon Raisbeck在给Erdos的表妹的吊唁信中写道:

"你听说过两个哲学家讨论存在地外生命之可能性的典故吗?其中一人说,如果外星人存在,那么他们的智力水平就有可能超过我们人类,因此我们可以预期他们已经访问地球了。但他又说:'但他们有没有留下什么痕迹呢?'另外一位哲学家俯身过来时对他耳语道:'嘘!这里我们自称匈牙利人。' 正是有了像Erdos这样杰出的人,才成就了这样的典故。"

--7--

我告诉我母亲,如果你用100减去250,你便得到-150.--Erdos回忆他4岁时的经历

如果说这个世界上真的有天才,那Erdos无疑是其中之一。在蹒跚学步时,他就研究日历,计算母亲还有多久才能放假回家。(那时他父亲在战争中被俄国人俘虏,母亲在学校教书,他是由一名家庭女教师抚养。)3岁时,他便能心算3位数的乘法,4岁时便"发现"了负数。同样在4岁时,他就算出一些诸如乘火车去太阳需多长时间之类荒唐可笑的问题来做游戏。他问她母亲的朋友们多大,然后立即心算出她们已经活了多少秒。

在上高中之前,他母亲一直不让他上学,因为害怕他会得传染病。

他的英语是跟他父亲学的,--他父亲在集中营里为了打发时间,便用一本书自学了英语。

所以Erdos的英语发音一直很糟糕,因为他的父亲从来没有听过地道的英语。

--8--

这世上有这么多的丑恶现象,我都不敢说,假如上帝存在的话,是不是也是圣洁的。--Erdos

1919年,Miklós Horthy在匈牙利建立了欧洲第一个法西斯政权,并发起了一场血腥的排犹运动。几万犹太人被迫离开匈牙利,其中包括E.Teller,J.von Neumann, L.Szilard, E.Wigner. 这四个人都去了当时的科学圣地--德国,若干年后,他们又都逃到了美国,并参与了曼哈顿工程。

Erdos一家并没有离开。那时犹太人经常在光天化日之下遭到殴打甚至屠杀,(Wigner就曾遭暴徒袭击,)Erdos的母亲曾对Erdos说:"你知道现在犹太人实在是太难了,我们是不是要去洗礼?"六岁的Erdos回答道:"那好,你可以做你想做的,可我还会和原来一样。"

这就是Erdos,从不会放弃自己原则的Erdos.

综观Erdos的一生,他藐视任何法西斯权威,无论是武装暴徒,还是不学无术的大学官僚、美国移民局官员、匈牙利秘密警察、FBI、洛杉矶交警,甚至上帝。

注:Edward Teller,著名物理学家,被称为"氢弹之父"。

John von Neumann,20世纪最伟大的数学家之一,在纯粹数学和应用数学的许多领域中都有着无与伦比的贡献。仅"电子计算机之父"这一称谓便足以使他流芳百世。

Leo Szilard,著名物理学家、生物学家,核链式反应的提出者及专利拥有者。1939年8月2日,在他和Wigner、Teller的建议下, Einstein给罗斯福总统写信,呼吁美国抢在纳粹德国之前研制成原子弹,--事实上,那封著名信件就是Szilard起草的。

Eugene Wigner,20世纪最重要的理论物理学家之一,获1963年Nobel物理奖。 他还是Dirac的大舅子。

--9--

我不明白为什么我们初次见面时他会那样--说的全是数的平方和各种各样的证明。在我渐渐了解他之后,我发现他并不是那种喜欢过分张扬的人。……所以我真的不能解释为什么他总是在说他所知道的所有证法。--A.Vázsonyi对Erdos的回忆

1930年,17岁的Erdos第一次见到14岁的Vázsonyi时,前者对后者说的第一句话是:

"给我举出一个4位数。"

Vázsonyi答道:"2,532."

"它的平方是6,411,024. 对不起,我老了,否则我会立即告诉你它的立方。"

Erdos又问道:"你知道毕达哥拉斯定理的多少种证法?"

"1种。"Vázsonyi说。

"我知道37种。你知道位于一条直线上的点不能构成可数集合吗?"继而Erdos给出了一种证明方法,然后他说他必须跑了。

六十七年后,Vázsonyi仍然清晰地记得当时的情形:

"当Erdos说他必须'跑'时,他像只大猴子一样驼着背,侧着身子,摇摆着双臂,沿着街一溜儿小跑,引得行人时时回头观望。……当他年纪稍大之后,他的步态不太像猴了,但仍有些奇怪。他老是走得很快,甚至发展到会向一面墙直冲过去,然后突然止步,猛然掉头,再往回跑。有一次他没能止住脚步,一下子撞到墙上,弄伤了自己。"

--10--

如果你是一只耗子,你就知道了。--Erdos这样解释他把一只小猫称为法西斯的原因

尽管受到各种反犹法律的限制,Erdos还是得以在17岁那年进入布达佩斯的Pázmány Péter大学学习。他经常与他的朋友们在公园或广场里讨论数学问题和时事。

当时的法律禁止群众聚会,经常会有警察过来盘问他们。他们感觉密探无所不在,不能够畅所欲言。所以从那时起,Erdos开始使用一些自己独有的隐语。他用 "长波人"来代表共产主义者,因为红色光的波长较长;相应地,用"短波人"来代表法西斯主义者。

政治并不能干扰Erdos和他的朋友们对数学的兴趣。Vázsonyi回忆说:"他(Erdos)沉浸素数的世界里,素数是他的一切,他与素数之间似乎有着某种奇妙的密切关系。"大学一年级时,Erdos便证明了Bertrand猜想:"在n和2n之间总存在一个素数,其中n是大于1的整数。"这个猜想最初是在1848年由Chebyshev证明的,但Erdos的证明比Chebyshev的要简单得多。

20岁时,Erdos便宣称要用古希腊时期流传下来的Eratosthenes筛法来给出素数定理的初等证明,若干年后他果然做到了。这是后话。

注:Joseph Louis Francois Bertrand,19世纪法国数学家,在数论、概率论、几何、微分方程等领域均有贡献。

Pafnuty Lvovitch Chebyshev,19世纪俄国数学家,在数学许多领域有开创性工作,是俄国数学的奠基人。

Eratosthenes,与Archimedes同时代的学者,曾任亚历山大图书馆馆长。

--11--

我是犹太人,而匈牙利当时是个半法西斯国家。--Erdos谈及他离开匈牙利的原因

早在12岁的时候,Erdos就清楚地知道,由于政治上的原因,他迟早得离开匈牙利。1934年,他获得了博士学位,随后便到英国的Manchester 大学进行他的博士后研究。

在Erdos抵达英国的第二天,发生了两件对于他来说颇不寻常的事情,其中一件是这样的:

Erdos以前从来没有给面包片抹过黄油,在家里都是他母亲或佣人给他抹的。他回忆道:"我清楚地记得,那一次我刚去英国学习,茶点的时候上了面包。要是承认自己从来没有抹过黄油,那太让我难堪了,于是我就试了试。还不是太难。"

就在10年前,当他11岁的时候,他第一次给自己系了鞋带。

--12--

最好的数学既是美的,同时又是严肃的。--G.H.Hardy

在Erdos抵达英国的第二天,还发生了另外一件事,可能对他的影响更大,那就是他遇见了那个时代最为纯粹的数学家--Godfrey Harold Hardy.

数学家有很多种类型,像Erdos这样的可谓凤毛麟角。(不过好象公众心目中的数学家都是Erdos这样子的。)Hardy可谓是跟Erdos截然相反的一类数学家。

Erdos每天工作19个小时,Hardy的生活则有规律得多:他每天只花4个小时(上午9点到下午1点)用于数学研究,下午打板球和网球,晚上与 B.Russell,C.P.Snow, G.E.Moore, A.N.Whitehead, J.M.Keynes, G.M.Trevelyan, E.M.Forster, L.Strachey, L.Woolf人热烈交谈。

Erdos只对数学感兴趣,Hardy的兴趣则广泛得多。有一年,他在给朋友的明信片里谈了他新年的6项打算:"1.证明Riemann假设;2.不能让 211队在奥威尔举办的板球决赛阶段第四局比赛中出局;3.找到足以让公众信服的证明上帝不存在的论据;4.成为登上珠穆朗玛峰的第一人;5.成为苏联、大不列颠及德意志联合王国的首任总统;6.谋杀Mussolini."

Erdos是天生的流浪者,Hardy则憎恶旅行,尤其是害怕坐船。如果Hardy不得不乘船出海,他就会给朋友寄一张明信片,宣称自己已经证明了 Riemann假设。他认为这样一来上帝就不会让他在旅途中死去,因为上帝--他"个人的敌人"--是不会让他享有证明Riemann假设的荣誉的。

Erdos其貌不扬,看上去还有些神经质,Hardy则是出了名的美男子,十足的绅士。

当然,他们也有一些共同点,比如说他们两人都不太喜欢上帝。Hardy拒绝涉足任何带有崇拜色彩的地方,为了迁就他,剑桥大学特意在校规里加了一条,使他 "可以豁免某些职责,不参加礼拜。"他们两人也都终身未婚。他们都喜欢与人合作:Hardy与Little-wood及Ramanujan的合作至今仍被人津津乐道,Erdos则有多达485名合作者。

他们两人更重要的共同点是:他们都是纯而又纯的数学家,而且也都是第一流的数学家。

注:Godfrey Harold Hardy,剑桥分析学派后期的领袖人物,主要研究调和分析与解析数论。

Bertrand Russell,20世纪最著名、最重要、最卓越的思想家之一,曾获1950年Nobel文学奖。

Charles Percy Snow,英国小说家、物理学家和政府官员,曾任剑桥大学行政领导人。

George Edward Moore,英国实在论哲学家,著有《伦理学原理》、《哲学研究》等。

Alfred North Whitehead,英国数学家、哲学家,著有《数学原理》(与Russell合著)、《过程与实在》、《观念的历险》等。

John Meynard Keynes,有史以来最伟大的经济学家之一。

George Macaulay Trevelyan,英国自由主义历史学家,曾任剑桥大学三一学院院长。

Edward Morgan Forster,英国小说家、散文家、社会和文学评论家,著有小说《霍华兹恩

德》、《印度之行》等。

Lytton Strachey,英国传记作家、批评家,著有《维多利亚女王时代名人传》等。

Leonard Woolf,英国文学家、出版家、政界人士、记者和国际主义者,他还是Virginia Woolf的丈夫。

Georg Friedrich Bernhard Riemann,19世纪德国数学家,在数学的许多领域内均有天才的开拓性工作,对20世纪数学的发展有着难以估量的深远影响。

John Edensor Littlewood,英国数学家,与Hardy密切合作达35年,共同撰写了100篇论文。

Srinivasa Aaiyangar Ramanujan,天才的印度数学家,对无穷级数和数论有着惊人的直觉。

--13--

明年哥廷根见!--Erdos理想中的犹太人祝酒辞

到英国后,Erdos开始了他的流浪生涯。他频繁来往于英国的各个大学之间,从没有连续七天呆在同一个城市。那时候世界数学的中心是哥廷根大学,尽管希特勒的兴起已使哥廷根开始衰落。Erdos一直想去哥廷根,但作为犹太人的他始终没有达成这个心愿。

他每年要回布达佩斯三次,看望双亲和老朋友。有一次,在他回家期间,Vázsuoni正研究一个图论问题,并找到了这个问题的必要条件。他回忆道:"我几乎天天与Erdos见面。但我犯了一个致命的错误--我在电话里把自己的发现告诉了他。我称这个错误是致命的,是因为他在20分钟后就回电告诉了我证明充分性的方法。'该死的,'我想,'现在我只好和他合作写这篇论文了。'这个著名的Erdos数'1'究竟给我带来了什么,我几乎一无所知。"

"Erdos数"是数学界流传的一个典故。即给每一个数学家赋予一个Erdos数:Erdos本人的Erdos数是0;曾与Erdos合作发表过文章的人的Erdos数是1;没有与Erdos合作发表过文章,但与Erdos数为1的人合作过的是2;……自然,不属于以上任何一类的就是∞.

--14--

Wir Mathematiker sind alle ein biβchen meschugge. (我们数学家都有些疯狂。)--E.Landau曾对Erdos说的一句话

在常人眼里,数学家往往是些怪人。其实数学家多半还是很正常的,只是当他们思考起数学问题时,脑海里就容不得别的东西了。

Harold Davenport的妻子Anne Davenport回忆说:"有一次在三一学院,我想大概是在30年代,Erdos和我丈夫在一个公共场所思考了一个多小时而彼此一言不发。最后, Harold说:'这不是0,而是1',这才打破了长时间的沉寂。然后就是如释重负、一片欢欣了。他们周围的人都以为他们疯了。的确,他们是疯了。"

注:Edmund Landau,德国数学家,主要研究数论与复变函数,曾在柏林大学和哥廷根大学任教授。Erdos没去过哥廷根,Landau却访问过剑桥,他们可能就是在剑桥见面的。据说Landau怀疑Littlewood这个人的存在性,所以专程去英国看了一下。这个故事还有另外一种版本,说Wiener在访问剑桥时遇见了Littlewood,便说:"哈!原来还真有你这么一个人!

我还以为Littlewood是Hardy发表那些写得不太好的文章时所用的化名呢!"Wiener对这个笑话感到非常苦恼,专门在自传里辟了谣。

Harold Davenport, 剑桥学派的数论专家,曾任伦敦数学会主席,有各种论著196种。

--15--

人们研究你是因为你是一个数学家,而不是因为你研究Leibniz.--Erdos对Godel如是说

1938年9月3日发生了捷克事件。那天Erdos正在布达佩斯,他当晚便匆匆赶回英国,几周后又去了美国,到位于Princeton的高等研究所(Institute for Advanced Study)工作。

高等研究所是30年代初由Bamberger兄妹捐资兴建的,首任所长为教育家Abraham Flexner. 这个研究所聘请的第一批研究员是:Alexander, Einstein, Morse, Veblen, von Neumann, Weyl.

Oppenheimer曾称高等研究所是一座疯人院,--许多年以后他自己成为这家疯人院的院长。在这里,Erdos肯定算不上是天才,但却仍然显得很怪异,因为一年半以后他就因为过于"uncouth and unconventional"而被解聘了。不过天真的Erdos却把在研究所的这段日子称为他在数学上最有成果的时期。

当时的高等研究所拥有世界上最出色的一批数学家,比如说Godel. 那时候Godel的兴趣已经转向了形而上学,所以Erdos常与他吵得不可开交。

注:Gottfried Wilhelm Leibniz,17世纪德国百科全书式的学者,微积分的创始人之一。

Kurt Godel,奥地利裔美籍数学家、逻辑学家。

James Waddell Alexander,美国拓扑学家。

Harold Marston Morse,美国数学家,Morse理论的创立者。

Oswald Veblen,美国几何学家和拓扑学家。

Claude Hugo Hermann Weyl,Hilbert的得意门生,20世纪最伟大的数学家之一。

Robert Oppenheimer,美国物理学家,曼哈顿计划的主持人,被称为"原子弹之父"。

--16--

我知道Einstein绝不信奉人格化的上帝,因为我问过他。--Erdos

如同那个时代曾待在Princeton的其他人一样,Erdos曾有幸与Einstein交往。有一次在Einstein家吃午饭时,Erdos把自己最钟爱的素数理论讲给Einstein听。Einstein对这一理论的评价很高,但他对细节并不感兴趣。

他们两人主要讨论的话题还是政治。Erdos一向很关心政治,经常就世界政治和人类的普遍问题发表一些悲观言论。

原子弹的问世使Einstein再次成为世界关注的焦点,--尽管没有Einstein也能造出原子弹。Erdos曾问Einstein:"40年前你想到过你的质能方程会在你的有生之年得到应用吗?"Einstein说:"我没料到。我曾想也许最终会得到应用,但没料到会那么快。"

其实像Einstein或者Erdos这样的科学家并不很关心自己的理论能否得到应用。一个极端的例子是Erdos曾经拜访过的Hardy,他就宣称最好的理论是毫无用处的,比如数论和相对论,--不幸的是他举的这两个例子都错了。

--17--

他在Princeton的大街上走来走去,挥舞着双手,旁若无人地比划着。--Louise Straus对Erdos的回忆

同Einstein相比,他的数学助手Ernst Straus与Erdos之间有着更多的共同语言。每当他在自己的宿舍里住腻的时候,他就跑到Straus的家里呆上若干天。

那时Straus夫妇经常在半夜里听到一声巨响,那是Erdos在开窗户,--他从没想到应该轻轻地把窗户放下。他不会使用淋浴器,不会把水龙头关上。他到公用电话亭里彻夜不停地往里面塞硬币,给世界各地的数学家打电话,宣称自己在Straus家里,然后和他们讨论问题。他还邀请附近的一些朋友到这里来,不管主人是否同意。

许多年以后,Straus说:"Einstein曾对我说过……对于一个科学家而言,首要任务是解决核心问题,而不为其他问题所动--无论那些问题多难,多么具有诱惑力。Erdos完全违背了Einstein的这一番话,但他却取得了成功。他几乎痴迷于他所遇到的每一个难题,并成功解决了其中的大部分。"

--18--

你应该找一份实际的正经工作。--Halmos对Erdos的劝告

P.R.Halmos曾记载过Erdos在Princeton时的另一个故事。Hurewicz曾经提出过这样一个问题:Hilbert空间中有理点集合的维数是多少?Erdos听到这个问题后,便问"Hilbert空间"和"维数"各是什么意思。有人把定义告诉他,很快Erdos便得出了答案。这是 Erdos对一个他几乎一无所知的领域作出的贡献!

注:Paul Richard Halmos,匈牙利裔美籍数学家,Springer-Verlag出版社主编。著有《测度论》等。

Witold Hurewicz,波兰裔美籍数学家,维数理论和同伦论的开创者。

David Hilbert,德国人,历史上最有影响的数学家之一。他使哥廷根成为当时世界上数学

家朝拜的圣地。

--19--

Erdos在任何时刻都集中精神思考。--S.M.Ulam
那时候Erdos手头很拮据,只能靠大学发给他的访问津贴度日。被高等研究所解聘后,他一度失去了生活来源,幸好Ulam向他伸出了援助之手。Ulam是 Erdos在剑桥结识的朋友,当他得知Erdos的窘境后,便邀请Erdos到他工作的Wisconsin-Madison大学来访问。
许多年以后,Ulam在他的自传中回忆道:"Erdos身材中等偏下,极度的神经质,他几乎总是'上窜下跳',或者挥舞着他的双臂。他的眼神表明他总是在思考着数学问题,只有在他就一些还模糊不清的世界时事政治和人类的普遍问题发表相当悲观的言论时才会被打断。如果他突然产生了一个有意思的想法,就会一下子跳起来,挥舞双臂,然后再坐下来。"
注:Stanislaw Marcin Ulam,波兰裔美籍数学家。早年研究拓扑,后因参与曼哈顿工程,兴趣遂转向应用数学。他是Monte Carlo方法的创始人之一。

--20--

他(Erdos)具有神秘的技能,可以运用最深奥、抽象的数学工具来预测原子弹的反应。--Ulam
1943年,Ulam到Los Alamos参与原子弹的研制。他极力劝说Erdos也加入他们的行列,Erdos本人也很愿意为消灭法西斯而出力。于是Erdos给他的同胞 E.Teller写信,申请加入曼哈顿工程。但Erdos在信中特别强调他战后要回匈牙利,所以理所当然地被取消了资格。
Erdos就是总喜欢和权威对着干,而且根本不遵守各种保密规则。他还给在Los Alamos的P.Lax寄了一张明信片:"亲爱的Peter,我的间谍告诉我Sam正在造原子弹,告诉我,这是真的吗?"
还有一次,Erdos和包括Lax在内的几个匈牙利人一起吃晚饭。席间他们一直用匈牙利语交谈,Erdos却突然用英语大声问:"原子弹的研制进展如何? "
注:Peter Lax,匈牙利裔美籍犹太数学家,主要研究泛函分析、偏微分方程和应用数学。曾任美国数学会主席、美国原子能委员会计算和应用数学中心主任、纽约大学 Courant数学研究所所长、纽约大学Courant数学和计算实验室主任。1987年获Wolf奖。他还是北京大学荣誉研究博士。

 

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