说起我与美食的渊源，那得追溯到相当远的从前。4岁吃螺蛳，5岁吃鱼，9岁包饺子，12岁上灶台，至今父亲仍然会不时得提起当年照片里我扒着螺蛳碗吃得满嘴油的样子。幼时喜文，高中也是学的文科，在这么多年诗词歌赋文言骚句的浸淫下，我至今记忆犹新最复杂的字居然也是与吃有关——爨，意为灶台，引申为做饭。字虽是记得，然而这字出自哪一篇文却是半分印象也没有了。

家里的口味是江浙惯有的精细清淡，薄盐少油。因是傍水长大，自小喜爱鱼虾螃蟹，莲藕菱角，茼蒿水芹。妈妈手巧，北方的各式菜品也是样样拿手值得称道，我在一旁耳濡目染耳闻目睹故而也渐渐耳熟能详。随着我渐渐长大，走过很多地方，各地美食也吃了不少，喜欢泰国的冬阴功汤，墨西哥的 burrito，荷兰的 Gouda 奶酪，意大利的 gelato，新疆的烤肉串，山东的馒头，苏州的酒酿，扬州的煮干丝，上海的小笼包⋯⋯

食而优则烹。

>>>

2009年冬天，Ridley 开着他的小 Dodge 带领我和 Ruben 一起去 H-Mart 扫货，两个男生在琢磨泰国菜，我则是准备做饺子。没有饺子的冬天是不完整的，在飘雪的夜晚几个朋友围炉吃饺子才是冬天里最温馨的画面。馅料是最传统的猪肉白菜。因为我不喜欢机器切割的饺子皮，所以只能自己和面擀皮。没有合适的擀面杖，只好用擀 pizza 面皮的 rolling stick 擀皮，自然不指望成品出来会是标准的边缘薄中间厚，却没想到擀出来是更令人无奈的边缘厚中间薄。

因为我动作太慢，两个男生游戏打腻了以后自顾在沙发上睡去。当第一锅饺子下锅后，很快香味飘散开来，贪睡的 Ridley 和 Ruben 竟然循着香味醒了。饺子出锅以后两个人迫不及待地开始狼吞虎咽，不顾刚出锅的灼人温度就往嘴里塞，烫了舌头在一旁哇哇乱叫，颇具喜感。Kathryn 喜欢锅贴蘸酱油，而且是指明要龟甲万的酱油。

[只有擀 pizza 用的擀面杖，勉强也擀一下饺子皮吧]

[饺子刚煮出来就被哄抢一空，好不容易把这么一点最后的锅贴保住拍了张大头照，拍完立马就被搜刮走]

>>>

2010年春天，我忽然想做馒头。在 Safeway 翻找了半天遍寻不着酵母，只好自己发面头。用白酒和了一点面，置于碗中覆上湿布发了整整三天面头才最终发好。因为面和得不多，就没有加苏打粉，蒸出来的馒头居然也没有酸味，松松软软，白白嫩嫩。八个小小的馒头被 Zack 抢去了一半。

[想做馒头，却没有酵母，只好遵从古法用白酒发面头]

[白白胖胖的馒头]

>>>

Laurel 生日的时候我做了一份双皮奶。虽然用的是全脂牛奶，但是毕竟是加工过的，脂质已经被过滤掉了许多，蒸出来的牛奶已经形成不了奶皮。为了去腥，在蒸制前泡了一点柠檬皮在牛奶里面。柠檬汁是不能加的，里面的酸性成分会与牛奶中的钙质结合形成不能被消化的沉淀物。双皮奶很像西点里的牛奶布丁，只不过布丁中添加的是全蛋，双皮奶中的是蛋白。剩下的蛋黄还可以与牛奶一起做凤凰奶糊。

[蓝莓双皮奶]

>>>

2010年夏天，Ridley 和 Ruben 两个小孩想吃日式 mochi，无奈，只得从命。没有豆沙卖，只好买了红豆自己做豆沙。红豆相比绿豆黄豆非常难泡开也非常难熟。红豆先泡三天再煮三小时才算是初步能看，之后要加油加糖熬半个小时才能把红豆泥熬干成为红豆沙。之后是用糯米粉做 mochi 皮，蒸熟的的糯米很黏手，Ruben 的 mochi 包得很不像样子，豆沙总是漏出来，糯米皮总也包不住。我和 Ridley 笑到直不起腰。

Ice cream mochi 更是难得的美味，不过我还不具有把冰淇淋包入糯米皮的能力，只有以后寻得时间再试试了。

[Ruben 和 Ridley 要求吃红豆 mochi，没有红豆沙卖便自己动手丰衣足食]

>>>

2010年秋天，我想做蛋糕。既然是做蛋糕，那么就要打蛋白霜了。三个蛋清与蛋黄分别分离开，加糖粉打发，经过另胳膊无比酸疼的半个小时慢慢变成充满了空气的霜状物。这时与添加了蛋黄和黄油的面粉糊搅拌均匀就可以入模进烤箱了。

其实，普通中筋面粉与植物油都是可以做蛋糕的，如果口味不挑剔的话跟低筋面粉黄油做出的蛋糕口味并没有太大区别。若不是对食材火候之类无比追求的老饕，就不要费心专门去寻找贵得要死的低筋面粉与黄油了。只要不是用氢化植物油，花生油玉米油甚至是橄榄油都是可以的。至于氢化植物油，还是少吃为妙，里面含有不能为人体分解也不能被排出的反式脂肪酸，只会附着在血管壁上加重心脏的负担。

[做蛋糕一定要打蛋白，只有打出绵密的蛋白才能做出松软可口的蛋糕]

[黄油用量减了三分之二，所以烤出来有裂痕]

>>>

2011年春天，偶然在 H-Mart 找到了鸡爪和猪蹄，果断买回家来卤。这些东西 Ruben 是不会碰的，即使我百般诱惑他也是坚定意志岿然不动。他都不知道自己错过了怎样的美味呢。不过想想也对，推己及人，如果一个韩国姑娘劝我吃蚕蛹，我也是不敢的，更不肖说是柬埔寨的烤狼蛛之类让我毛骨悚然的另类“美食”了。

鸡爪和猪蹄都很好卤，酱油辣椒随便洒一点就会满屋飘香。用高压锅煮的话会很快很省电，但是文火慢炖出来的猪蹄才有值得不顾形象囫囵饕餮的可口味道。富含胶质的食物对女生皮肤也是有好处的。各种动物皮肤都是美颜圣品，驴皮制的阿胶，鱼皮，猪皮等等都很美味，但是，不知为何我不喜欢鸡皮。

我喜欢烧肉，但是自己不喜吃肉。每每都是一帮吃货来我家蹭饭点名要红烧肉。南方的做法是生抽调味老抽上色，北方会用砂糖炒出那种油亮赤红的颜色。相比较而言，北方的做法更得我心。许是因为现在的肉都是大规模养殖场里的速成品，少了肉味寡淡平凡，一点点的糖可以把少少的肉味放大成绝顶的美味。猪小排红烧的最好不过的了，不同于五花肉的肥腻，小排瘦肉居多却也不乏肥肉，烧出来肥瘦相间可口美味。

[小火卤制的鸡爪]

[红烧排骨，装在 rubermaid 里面带给 Marina ]

>>>

一个人的时候往往偷懒，饭菜置于一个碗碟里可以少洗一些碗。想必很多会做饭的孩子都是不喜欢洗碗的，我也是如此，宁愿做饭也不愿意洗碗。几位吃货来我家总是自觉自愿地承担起洗碗的任务。

我烧菜不喜放油，油盐相比，油腻的食物是第一顶顶不喜欢，其次才是高盐的食物。油腻的食物会加重胆的负担，有胆结石的人最不可吃油腻食物，否则只能在痛昏死后被送上手术台切除胆囊。更重要的是，荤腥多了会胖。女孩子最要不得的就是胖。胖胖的小男生会被形容为可爱，胖胖的小女生却会被嘲笑。我们的社会对女性外表过于苛责，要瓜子脸、大眼睛、高鼻梁、樱桃口、苹果颊、白皮肤、长头发、瘦高个、小蛮腰、竹竿腿、青葱指、贝壳甲，还要聪明贤惠会家事，相夫教子样样行。很遗憾，以上几条我无一达标，唯一拿得出手的也只有这还过得去的厨艺了。

高盐的食物会使血液粘稠血压增大，对心血管是负担，对肾脏更是折磨。除非的从事体力劳动，否则还是少一些盐比较好。同样需要注意的还有不可食烫。曾经看过一个报道，太行山区的人民喜欢蹲在家门口吃刚出锅滚烫的面条，每家吃面的时候都可以与邻家遥遥相望远远地打个招呼，结果却是太行山区的高食管癌发病率。食管的神经分布稀疏，即使我们的食物很烫或很凉，迟钝的食管也不会察觉，各种不良刺激积少成多就会导致癌变。

[芹菜＋胡萝卜＋火鸡肉糜，土豆＋鸡丁，这样的组合光是颜色就够让人食指大动了。本来想做咖喱土豆鸡，临时起意放下锅的是却是酱油，截然不同的风味却一样好吃。]

>>>

酒酿可谓是我毕生的爱好了，喜欢那种甜腻的带着一丝酒味的奇特味道，也许这是江南人特有的喜好。父亲是做酒酿的高手，小的时候就喜欢一边看父亲做酒酿一边追问什么时候可以吃。因为太喜欢，父亲许诺，将来我怀孕和坐月子的时候他会天天为我做酒酿，这该是多么幸福的时光啊。

酒酿做起来并不复杂，只要保持手和容器的干净无油就好，混入了杂菌糯米会霉变，粘了油花的糯米也成不了酒酿。做酒酿最好是用玻璃或是搪瓷的容器，但是究竟这是为什么父亲也说不出个所以然。没有像家里那样的搪瓷缸，也没有玻璃碗，我只能拿最不合适的塑料制品当作容器。好在糯米不负众望地发酵完全，甜味酒味一丝不落。

下次有时间还可以做老北京的奶酪，酒酿加牛奶蒸制再晾凉，味道丝毫不逊色于乳酸菌发酵的酸奶。

Ruben 专门到我家来品尝酒酿。最为奇特的是，在满满一碗酒酿之后，Ruben 竟然点名要吃臭豆腐。这甜与臭的组合也只有他才想得出来了。

[刚做好的酒酿，跟超市里卖的兑了很多水的酒酿口味很不同]

[本来糯米的中间是有帮助发酵的洞的，但是这个洞自己填上了，并且还因为充满气体鼓了起来]

>>>

最后放一张发芽红薯的全裸照。到处都是新冒出的芽，我想了半天这红薯究竟是该吃了该扔了还是该下楼找个地方埋起来种了。

作为一只吃货，我毕生的梦想大概就是食遍天下美食。可惜 Katie 现在去了秘鲁，否则我们可以一起烹饪。美女擅烘培，我擅烹煮；美女做甜点，我做主菜；最重要的是，美女会酿 ginger beer。

每年冬天放假的时候，很多吃惯了食堂自己从不开伙的孩子便到我家来蹭饭。几个人围着桌子一边看柯南一边大快朵颐，蓦然觉得其实自己离家也没有多遥远，即使远在万里之外仍然有很多温馨可爱的朋友关心着自己，虽然大多更惦记的是我的厨艺哈。

今年的冬天，我们再聚首吧。

。

在大学中，无论什么专业，微积分都是必修课程。文科的孩子为此叫苦不迭，理科的孩子在一旁捂嘴偷笑，工科的孩子一边撇嘴一边想：什么嘛，我们要学的比这个难得多了。

我一直认为，数学教育演变成为大多数人负担的现状，就算不是完全也有一大部分应该归咎于我们教授数学的方法以及教学载体。找好老师？老师大多不会教。找好课本？课本多数不靠谱。

当我还是大一新生的时候，也学过微积分，也为此困惑过：为什么这个定理要是这样的，为什么这种解法要那样思考。我带着满腹的疑问去问老师，老师却是满脸的不耐，翻来覆去只有一句话：你把这个方法背下来就可以了。我问的不是该如何解题，而是为什么要这样思考啊，亲爱的老师。很多时候，老师只是告诉我们算式的结果，却不指点学习数学真正的意义。数学变成了纯粹的题海，只有会做题会解题才是有价值的，而数学本身的美好却被再三忽略。课本上有的只是定理推论证明和练习，闭口不谈这些定理方法诞生的意义，也绝不涉及数学在实际应用中的延伸。

这样被狭隘化的数学，究竟还能走多远？

许多人觉得数学不好，其实是因为数学教得不好。很多简单的东西一被收录进课本里就变得繁复到面目可憎，那些本来可以更简洁更直观的解释被排斥在外，成为小聪明，成为旁门左道。的确，不可否认，在最速降线的问题上，Johann Bernoulli 给出的最冗长艰涩的证明最终推动了变分的发展，但是这不妨碍 Jacob Bernoulli 运用光学原理写就的证明成为最简明扼要的答案。Diophantus 成为数学史上的丰碑，很大程度上并不在于他的数学理论多么超越多么完备，而是在于他将 Pythagoras 等前人的数学表达得更通俗更具体。受到他的启发，门外汉 Fermat 才能在近两千年后以律师的身份荣登业余数学之王。

为什么会有负数，虚数？这些数看起来远离生活，自然数可以用来数苹果，那么负数是数什么用的呢？虚数又如何数呢？在初中的时候，老师简单明了地引入了负数这个概念，紧接着介绍了各种运算法则，留下一教室的孩子们面面相觑，负数究竟是什么数？其实，负数的诞生只是为了简化加法和减法的运算。在负数之前，加法和减法的独立的运算，各自有运算法则，这为大部分还是文盲的中世纪欧洲人在记账方面带来很多不便，因此，负数应运而生，加法与减法由此统一成为广义的加法，减去一个数等于加上此数的负数。紧接着，乘法与除法也由分数统一成为广义的乘法。数学之美中很重要的一条就是广义原理推及个例。

至于虚数，更为抽象。讲负数的时候，初中的孩子们尚且可以通过数轴具象化负数这种数：负数与正数相对于原点对称。到了高中接触复数的时候：为什么 i 也能是一个数？为什么有虚数这个概念？为什么虚数能表示旋转？要回答这些问题还要回到群论和 Galois 。17 岁的 Galois 证明了一元四次方程有通解，18 岁的时候又证明了一元五次或更高指数多项式的对应方程无实数通解。一个 n 次多项式却不能有 n 个根，这实为遗憾。虚数的诞生使这个遗憾不再，一个 n 次多项式的对应方程总是有 n 个复数根，整个数系一下子就圆满了。然后，复数却不能简单地反应到数轴上，一是因为实数轴已经完善，另一个原因是：究竟怎样的操作能够实现重复两步得到相反数？从 1 到 －1，在负数领域，我们可以通过相对于原点对称来完成取反，怎样才能把这简单的一步化成两步呢？数学家们给出的答案是把 1 绕原点逆时针旋转 90 度，接着再旋转 90 度，就自然与 －1 重叠了。因此，复数也有了 z = r(cosθ + i sinθ) 的表达方式。至此，复数也可以用来解释几何问题了。

有一个经典的用复数求解的几何题：在任一凸四边形的四条边上各做一个正方形，证明连接相对正方形中心的线段相互垂直且等长。

若是用平面几何证，怕是要动用相切圆之类的知识，辅助线也是必不可少的。而一旦用复数证，这个看似繁琐的几何题立刻就简单了许多。

>>>

证明：

设 O 为原点，凸四边形的四条边分别为 2a, 2b, 2c, 2d 四个复数所表示。引入2这个因子是为了以后计算的方便。

既然四边形是闭合的，故 2a + 2b + 2c + 2d = 0，所以 a + b + c + d = 0。

因为 p 是正方形的中点，所以 p 点处在 2a 所代表的边长的垂直平分线上，那么，从原点出发，要先走一个 a 的距离，逆时针旋转 90 度后再走一个 a 距离才能到 p，所以 p = a(1 + i) 。

同理，q = 2a + b(1 + i)；

。。。r = 2a + ab + c(1 + i)；

。。。s = 2a + 2b + 2c + d(1 + i)。

所以，连接 q, s 的线段 A 的复数表达为 A ＝ s - q = (b + 2c + d) + i(-b + d); 连接 p, r 的线段 B 的复数表达为 B ＝ r - p = (a + 2b +c) + i(-a + c)。

我们需要证明的是 A 与 B 垂直且等长，两者可以通过一个命题来证明： B ＝ iA，即 A + iB = 0。这个命题与将 A 旋转 90 度后得到 B 等价。

随即，证明变成了更为简单的基本复数运算：A + iB = (a + b + c + d) + i(a + b + c + d) = 0。问题得证。

Q.E.D

<<<

曾经，我无比期待过线性代数，能让 Euler 费解的 Cramer 悖论应该是有着特殊魅力的。可惜，在 方程组 -> 行列式 -> 矩阵运算 -> ... -> 线性空间 -> isomorphism 如此顺序的学习下，线性代数在我脑海中完全演变成为解线性方程组的工具，我百思不得其解 Cramer 悖论与我所学的线性代数的关系。直到学至 isomorphism ，我才恍然大悟，其实线性空间和映射才是真正的研究对象，线性方程组仅仅是线性代数广大范围的一隅，顶多算一个例子出现在关于线性空间某个章节的角落里。

不只学习顺序，更让人哭笑不得的是关于矩阵的定义。很长时间内，在我眼中矩阵就是一堆排列整齐的数，我既不知道这些数为什么要排列成这个阵势，也不直到为什么矩阵的乘法可以如此定义。若是数字方阵可以相乘，那么国庆军演的方阵是不是也能乘一乘？当然，这只是玩笑。后来才明白，原来矩阵相乘其实是做线性映射。正是因为如此定义乘法，乘得的新矩阵才能够表示所做的线性映射。还有就是特征值和特征向量，英文里的名字更加蛋疼：eigenvalue 和 eigenvector。虽然已经很久不回顾线性代数，我仍然可以在经济数学的教材里看到二位，却自始至终不知道定义这两个蛋疼值的意义何在。它们的数学意义我无从知晓，它们的经济学意义更是摸不着头脑。我所知道的，仅仅是 eigenvalue 与 eigenvector 两个名字来自德语，仅此而已。

大二的时候，我写了一篇关于 Mersenne 素数的论文。很多同学非常不解，认为数论是数学那么多分支里最没有意义的学科，而素数又是数系里最无趣的数。其实不然。Mersenne 素数是一种由 2 的素数次幂减 1 得到的素数。

第一个 Mersenne 素数是  M2 = 3。前四个看起来都很合理，分别为 M2 = 3, M3 = 7, M5 = 31, M7 = 127，但是自第五个 M13 = 8191 起数字一下子变得异常巨大。在计算机诞生以前，仅仅有 12 个素数被证明符合定义。虽然在信息飞速发展大型计算机层出不穷的今天，也仅仅一共找出了 47 个 Mersenne 素数。M47 有 12,978,189 位数，是货真价实的天文数字。许多人不禁要问：耗费那么多人力物力财力去找寻这么些大到无法想象的数究竟意义何在？我们知道这种数存在就够了，为什么要把它们找出来？这样的偏见来源于自数论创立以来两千多年未有实际用途的无奈事实。两千多年来，支撑着 Euclid 寻找完美数的，支撑着 Euler 证明 M31 是被 Mersenne 遗忘的素数的，恐怕绝不是研究数能带来多少名望多少利欲，而是数以及数学本身的魅力。数论这种最纯粹数学的实用价值直到现代计算机和密码学出现才开始渐渐崭露头角。寻找 Mersenne 素数这种蛋疼数的价值也是为了提高密码的保密性。

回到最初的那句话，“The first thing you should do is to always let ε > 0.” 出自我的分析教授。教授说，在证明前，第一个要提醒自己的就是 let ε > 0，无论要证的与 ε 是否相关，只要铭记这个概念，实数范围内的分析就不会有障碍。这句话成为那时支撑分析这门课程的框架。数列，无穷技术，函数等等各种数学表达仿佛都在这句短小精悍的话里变成 Mandelbrost 无穷推进却永无尽头的图像。现在，let ε > 0 已经成为我心中最重要的数学记忆。是分析让我对数学有了焕然一新的理解。随着对 Cantor 集 Weierstrass 函数的了解进一步加深，那些在题海中建立起的简陋直觉无不被一一摒弃，随之丢弃的还有那些陈旧的，诸如盲目相信课本，盲目跟随老师的观念。

现在，我将重拾abbott的书，没有了小P与胖J的支持，没有了教授的帮助，一个人单枪匹马单挑分析。夜深时分，看到旧日笔记中那道关于微积分里比值审敛法与根值审敛法的对比，看到自己辛苦忙了几个晚上证明根值审敛比比值审敛更具说服力的成果，忍不住在手背上写下这句话，忽然觉得，自此我才真正懂得什么是数学。

。

>>>

My mom took me for a trip in Xinjiang Province, the very northwest section of China, bordering Russia, Mongolia and some other countries. As soon as the plane landed in Ürümqi, we went straightforward to the Turpan Basin where there is lots of food. You all know me; there is no way I could resist the lure of food.

Xinjiang traditionally is dominated by the nomads. Their customs and preferences have lived till now, like grilling food, eating lamb, and drinking milk. Here, by "drinking milk", I mean unlike many other Asians who suffer from lactose intolerance as soon as they are off infancy, the Uyghur people live on diary products. 

[above: I can only tell you the ingredients, because I do not know how to translate their names in English. From left to right, chicken+potatoes, onions+carrots+tomatoes+green chillies, lamb+onions+carrots, the same as the very left dish.]

[above: This is a close up shot of the chicken+potatoes dish. When it is almost finished, the waiter will bring some noodles and you may mix them with the remanent in the plate; guaranteed delicious! It is quite spicy. Ruben, I am sure you will like it. Alert: it is not boneless chicken.]
 

To Katie: Have the most gargantuan amount of fun if you can in Lima! Ruben and I will try the brew first. We will let you know how it goes. We will send you pictures, write you instructions, and eat all the amazaki up. :P

To Ruben: We need to get together, make amazaki, cook food, drink beer, watch Conan, teach Donovan ridiculous words, and create verbose text messages. Ah, and at last, go to the library and renew my books for the I-cannot-remember-how-many times.

To Chris: Nothing, but safely bring Katie back. 

♥ you all :)

>>>

有一盒52张扑克牌，在经过彻底洗牌后反压置于桌面上。如果某人一次抽取17张牌，那么：

。。。1. 在第17张牌第一次抽到A是概率是多少？

。。。2. 在第17张牌第二次抽到A的概率是多少？

<<<

我随口报了几个算式，教授听了连连摇头，“No, no, no.” 我顿觉气苦，这看起来很简单的排列组合都不能轻易拿下，谈何攻克高等级的概率堡垒。其实这题看似容易，其实机关重重，回去我想了一个晚上，经过无数次推翻自己结论后，终于列出了准确的表达式。

明显可以看出，第一题是第二题的铺垫。我们可以这样思考：既然算的是前17张牌如何如何的概率，那么后面无论怎么抽牌无论抽到怎样的牌都是可以忽略不计的，所以，要关注的是前面的17张。且两问的样本空间是一样的，都是52张牌中选17张，且这17张牌有17!种排列方法。

那么，|S| = 

下面我们分开讨论两个问的事件。

第一问：

• 第一步：从4张A中为第17号牌位选1张
• 第二步：从48张非A牌中为前16个牌位选16张
• 第三步：排列16张牌

所以，|E1| = 

那么，P1 =  = 

...

第二问：

那么，P2 =  ＝ 

...

如果追加一问：

。。。3. 在这17张牌中有且仅有两个A的概率是多少？

在经过前两个问思考的基础上，这个问题就变得非常简单了：

...

其实，在看了我的解答后，教授曾高深莫测地跟我说，有更简单精妙的方法，之后就拒绝透露他所说的方法是如何地简单精妙了。回去之后，我百思不得其解，究竟是怎样的方法能够摒弃以上繁复的思维过程而一击即中此题要害。在跟Patrick的一次闲谈中，我们聊到所谓数学归纳法本质上究竟是一种严密的逻辑推理还是投机取巧的概率，忽然我就开窍了，正如很多幼儿漫画中那样，脑海上方一个大大的亮亮的小灯泡。至少，关于第一问“在第17张牌第一次抽到A是概率是多少”，我们的确是可以投机取巧的。

捷径就是：仅仅考虑4张A的位置，忽略其他所有的非A牌。

• 若52张牌都被抽取，那么其中会有4个牌位是A
• 若第一次抽到A是在第17张，那么剩下的3张A必然是在其后的35个牌位中

那么，P1 =  = 

如此一来，省去了许多思考那17个牌位中非A牌该如何如何的过程。虽然前一种方法是正确的，但是究竟颇费功夫。至于后面两问该如何投机取巧，就作为开放问题留在这里吧。

关于概率，有很多有意思的问题。排列组合扑克牌是概率中最简单最直接的。但是小小的52张牌却蕴含了无数精妙的理念。说起来，如果手里有一盒扑克牌，究竟有多少种排列这52张牌的方法呢？答案是52!，一个表达起来很简单却实际上庞大的数字，至于这个数有多少位数，我也不知道。

所以，一个成功的赌徒必然是概率大师，否则只能是他的运气好连上帝都嫉妒。下次去拉斯维加斯的时候，带上一本概率书吧，没准就能有意外惊喜。

关于扑克牌，就是这样，而概率的故事还长着呢。

那么，无限究竟是什么？

[part 1:      ]

早在人类开始计数的时候，“无限”的概念就自然地进入数学领域了。随着科学的进步和发展，数学沿着逻辑链条迅速增长，大量与无限密切相关的数学概念纷纷涌现。诸如：无穷级数、无穷序列、无限维数、无穷大量、无穷小量、可数集的势、连续统的势，等等，它们均以无限作为基本特征，在数学中构成一个庞大的体系。所以一般地说，搞数学的人对于无限的概念和无限的领域都是比较熟悉的。

但是，对于这样一个熟知的概念，它在数学家的心目中处于什么样的地位呢？这是一个耐人寻味的问题。纵览数学思想的形成和发展，我们发现了一个奇异的现象：历代数学家都非常不愿接触“无限”这个概念，几乎所有的数学无限都是在不得已的情况下产生的。最早的例子是欧几里得，他在《几何原本》中，为了回避“无限”一词费尽了心机。首先，他在叙述平行公理时，并不谈伸向无穷远的两根直线，也不直接给出两平行直线存在的条件，而是用“二直线能交于有限远处的条件”巧妙地给出了平行公理，避免了直接使用“无限”一词。接着，他又在定理的证明中，迟迟不肯使用平行公理，直到命题二十九，才不得不使用它。事实上，他非常不愿论及“无限远空间的事”，但是为了把他的体系推演下去，他又不得不论及它们。

另一个著名的例子是康托，他建立了一门崭新的学说——集合论，他用全新的、但是使人不安的方法处理无限。在那里，无限不再是某种增长的东西，而是通过一些数，以“完成了的无限”这种形式，把无限在数学中固定下来了。康托对于自己建立无穷集合论的过程，有这样一段表白：“这种无限的概念是和我所珍视的传统相违背的，和我自己的愿望更相违背，我是被迫接受这种观点的……”

进一步分析，我们还发现：历代数学家们都对无限怀有极大的敬畏心理。他们在整个数学研究中表现出的移情冲动，在无限的面前上升到最高峰。这种敬畏与宗教活动中人们对于上帝的崇拜或对于魔鬼的畏惧，如出一辙。

>>>

毕达哥拉斯说：无限标志着“恶”，它令人望而生畏。

帕斯卡说：“这些无限空间的永恒沉默使我恐惧。” 

康德说：无限“有着令人恐怖的崇高。” 

康托说：“所有这些特殊类型的无限，都是永恒的，它们都具有神性。” 

希尔伯特说：“无限这个概念既是我们最伟大的朋友也是我们心灵宁静的最大敌人……”

<<<

 

为什么会发生对于无限的回避和敬畏呢？这是一个异常神秘的问题。人们可以在哲学、人类学、心理学、社会学等各个领域中做出种种解释；但是，当我们剥去貌似神秘的表面现象，深究问题的实质时，一个令人难堪的结论显露了出来，那就是数学家在如此广泛地使用无限的时候，尚未弄清关于无限的一些最基本的问题。他们既不知道无限的定义，又不知道无限的属性；既无法步入无限，又无法回避无限；既不愿讨论无限，又忍受不了无限的诱惑；既怀疑无限的真实性，又在无限之中建立了庞大的数学体系……换言之，他们还不知道“无限是什么”，就冒然接触、或者说不得不接触无限了。这是一个事实。请听著名的数学哲学家T·丹齐克的一段叹词：

>>>

无限究竟是什么呢？……它是一种超乎自然的真理吗？是人类被造物主投到宇宙中来，赤裸和无知，唯有自由谋生的时候，造物主赐给他的少数几种天赋之一呢？还是这无限概念是试图达到最后的数的努力失败后在人的头脑中产生的呢？还是它只不过是人类无力做到用数来穷尽宇宙的一种表白呢？

<<<

 

这一串问句暴露了历代数学家对待无限的矛盾心境。

事实上，无限是人类从事“绝对性追求”的产物。应当指出，人具有追求绝对化的天性。诸如绝对的简单性、均匀性、齐次性、规则性、因果性，等等，都对人类有着极大的诱惑力。而无限性正是这种追求的最高表现。那么，这种追求是靠什么来完成的呢？它就是人类所特有的想象能力。古希腊哲学家芝诺有一句名言：“说过一遍的话，可以永远地重复。”这是人类对自身想象力的一种肯定，它成为我们认识无限的原动力和逻辑基础。但是，这个理论具有幻想的成份，或者说它纯粹是一种幻想。你怎么知道人的头脑可以对一种可能的动作做无限次重复呢？这种“幻想”在数学中的最基本表现就是计数过程。每一位数学家都确信：任何自然数都有一个后继数，因此我们可以从“一”一直点数下去，直至无限远处，永无穷尽。这是一种真实的活动吗？显然不是，谁也无法把一切数字穷举完毕。可以肯定地说，这仅仅是一种“幻想”！或者用数学家的话说：这是算术的基本假设！令人不安的是，素以严谨、完整和坚实而著称的数学体系，恰恰建筑在这个“假设”之上。对此，每一位数学家的心里都是十分清楚的，所以他们才如此回避和敬畏无限，陷入欲作不能、欲罢不忍的境地。请听希尔伯特的表白：

无限啊！从来没有别的问题这么深刻地打动人类的精神；也没有别的观念这么有效地刺激起人类的智慧；然而也没有别的概念能象无限概念这么需要澄清……

事实上，“无限的假设性”不但引起了数学家的不安，也引起整个人类的不安。可以肯定地说，任何科学和艺术活动都与无限的概念密不可分。而无限又是假设存在的，这无疑会导致人类内心的缺憾。为了弥补这种缺憾，人类在心灵深处建立起一座精神的高塔，以求与现实中的高塔达到平衡。这座“精神高塔”不是别的，就是宗教信仰！费尔巴哈曾明确指出：“宗教就是对无限的东西的意识。”徐复观也曾谈到：“人类对宗教的要求的主要内容之一，是要弥补现实中许多无可弥补的缺憾”。那么，这座“高塔”对于数学家产生了哪些影响呢？作为两个思想体系，数学无限与神学无限之间具有哪些神秘联系呢？无限的领域中，还有哪些待揭的谜团呢？这正是本文将要论述的内容。

[part 2:      ]

数学与神学，本来处于人类文化的两极。表面上看它们不会有什么联系。但是，人类文化的形成具有整体性的特征，任何一个学科都不是孤立存在的。在它们发生和发展的过程中，总会产生一些影响和互渗，目前，此类问题已经引起人们的关注。

从第一节的论述中，我们已经看到：数学家在“无限”的面前，遇到了许多无法解释的问题。奇怪的是，这些问题经常与神秘主义的东西搅在一起。它们不但受到数学家的关注，而且也是神学家的热门话题。对于那些使数学家百思而不得其解的“无限”现象，我们从神学的典籍中，却可以轻易地找到微妙的解释。当然，这种解释对于科学而言，常常是荒谬的、无意义的、牵强的和零乱的，受到数学界的轻视和唾弃。但是，在文化人类学的宏观背景下，它们却是生动的、丰富的、具有启发性的和真实存在的文化现象，其中蕴含着许多人类智慧的精华。下面，我们仅就数学无限的三个最基本问题，探讨一下它们的神学解释。

[part 2.1: 无限的不可超越性]

我们知道，数学家有一个基本观念，即无限具有不可超越性。古希腊时期，这种观念占有统治地位。亚里士多德认为，无限是不完美的、未完成的、因而是不可思议的；它是不成形的、混乱的，只有那些限定而分明的东西才有其本性可言。欧几里得认为，关于在无限远空间所必然成立的事项的任何说法，它的具体意义总是含糊不清的，因为人的经验是有限的。这种思想极大地影响了古希腊数学的发展，使之在无穷量、无穷步骤、极限步骤等问题中裹足不前。例如他们用一正多边形来接近圆时，满足于使其差小于任一给定的量，但总留下一些量，以求这一步骤在直观上仍很清楚，避免涉及无穷小量。显然，无限的不可超越性是古典数学的信条。

有趣的是，在宗教神学中，无限的不可超越性也是神学家的信条。一段《圣经》故事中写道：从前，天下人的语言都是一样的，他们万众一心，没有征服不了的事。一次，他们在示拿试图通力建造一座城和一座塔，塔顶直抵天际（就是“无限”）。耶稣在天上看到了此事，非常不安，他不希望人类征服一切。于是下界变乱了人们的语言和口音，使众人分散在各地，塔也停工不造了。

这个故事告诉人们两个问题，一是说上帝不允许人类征服无限，任何超越无限的企图都是危险的。对于这一点，帕斯卡曾叹道：

>>>

我们燃烧着想要寻求一块坚固的基础地与一个持久的最后据点的愿望，以期在这上面建立起一座能上升到无穷的高塔；但是我们的整个的基础破裂了，大地裂为深渊。

<<<

显然，帕斯卡为数学界的同行们描绘了一幅比“变乱语言”更为可怖的图景，这无疑会在整个数学思想上留下深深的烙印。事实上，从芝诺悖论到康德的二律背反，无限已经使数学家产生极大的恐惧感。而最令人震惊的是十九世纪康托的工作，他竟使帕斯卡的预言成为现实。我们知道，康托在无穷集合论的研究中，提出“完成了的无穷”的概念，并且乘着思维的翅膀飞上“天际”（无限的境地），进行了许多定性、定量的研究。结果如何呢？一九零二年，罗素悖论的出现使数学大厦的整个基础断裂了，一系列怪现象、怪命题、怪结论纷纷出现，引起了数学界的极大恐慌。一时间众说纷纭，庞加莱说：“后一代将把集合论当作一种疾病，而人们已经从中恢复过来了。”外尔说：“康托关于超限数的等级是雾上之雾。”希尔伯特却反击道：“没有人能把我们从康托为我们创造的乐园中开除出去”。罗素也赞誉说：“康托的工作可能是这个时代所能夸耀的最伟大的工作。”最后，康托因受多重的打击精神崩溃，郁郁而死；数学家们至今仍在为修补“高塔”的裂缝而苦苦地工作。那么，在这里神学究竟处于什么样的地位？是一种启示？一种寓言？还是一种神秘的人生规则？这是耐人寻味的问题。

通天塔的故事还告诉人们：无限是存在的，但它仅属于上帝。这一观念反映在古代数学家的思想中，就变为：最末数是存在的，但是它不在人类力所能及的领域，因为它属于神。属于神的东西是不容亵渎的，所以在很长一段时间里，人们都以此作为回避无限的理由。每当数学因涉及无限而出现矛盾的时候，数学家就会把它交给上帝，以求安宁。既使在现代，当科学逐步摆脱了神学的束缚之后，“无限属于上帝”的观念仍在数学的思想深处占有一席之地。例如，康托就曾宣称他所发现的各类无穷具有神性；康托的好友科瓦列夫斯基甚至说：

>>>

对于康托……这些（连续统）势，是一种神圣的东西，从某种意义说，它们是通向无限的皇冠，通向上帝的皇冠的阶梯。

<<<

我们承认，这种表述大概主要产生于移情冲动，他们用宗教语言来倾诉科学发现的欢欣，这在当代科学家的思想中也是比较普遍的。但是，我们还应当看到，在这些表述的字里行间还浸润着深厚的宗教信仰，他们的心目中确实存在着一个统治着无限领域的上帝。不过，用爱因斯坦（Albert Einstein）的话说：“我信仰斯宾诺莎的那个存在于事物的有秩序的和谐中显示出来的上帝，而不信仰那个同人类的命运和行为有牵累的上帝。”17就是说，随着科学的发展，数学家心目中的上帝，并没有消失，只是发生了质的变化，已成为大自然规律的象征。

[part 2.2: 无限的不可确定性]

人们对于无限有这样一个定义，它表示一个没有穷尽、没有限度、超越时空和无始无终的过程。然而，这个过程是实在的，还是虚构的呢？如果说是实在的，那么谁曾经实践过任何一次无限的过程？如果说是虚构的，那么谁又能否定现实世界中诸如天体、数字以及万物的无限性呢？事实上，无限是一个不可确定的过程，或者说，它是不定型的。古希腊学者阿那克西曼德在探讨“世界本原”的问题时曾指出：无限是一种不定型的东西，它自身可以分化出许多对立的性质；冷热、干湿等等。这些性质的不同搭配构成了不同的物质，所以无限是万物的始基。

无限的不可确定性对于数学家的心理影响是巨大的，因为它直接地威胁着数学的严谨性和自由性。回顾早年的数学史，每当“不定型”的东西出现时，数学家都在极力地回避。实在躲不掉的时候，常常到神学中去寻找解释和安慰，因为“神语最出色地体现了能够用语言表达的绝对真理。”例如，毕达哥拉斯学派发现：正方形的对角线是不可用边来度量的。他们称这类数为阿洛贡即不可说。造物主的天工中出现了无法解释的破绽，应绝对保密，以免败露而将愤怒泄到人类身上。普罗克拉斯说：

>>>

听说，首先泄漏无理数的秘密者们终于悉数覆舟丧命。因为对不可说的和无定型的必须保密。凡揭露了或过问了这种生命的象征的人必定立遭毁灭，并万世受那永恒的波涛的摆布。

<<<

再如，印度人发明了零的概念。但是，当他们将零引入运算时，发现一个问题：某数除以零等于什么？显然这又是一个不定型的东西（无穷量）！对此，印度数学家婆塞羯罗解释说：“1／0象无限、不变的造物者一样，在旧世界毁灭或新世界创造的时候，在无数种类的生物生生死死的时候，凝然不变。”

可以说，在数学中将无限的不可确定性充分神秘化的最高表现者，就是帕斯卡。他有这样一段不同凡响的论述：

一加无限，并没有给无限增加任何东西，它不过是无限的尺度上再加一尽。有限消失在无限的面前，变成了纯粹的虚无。我们的精神在上帝之前便是如此；……我们虽然认识无限存在，但不知造它的性质。既然我们知道数目有限这种说法是谬误的，因而数目无限就是真确的了。但我们却不知道它是什么：说它是偶数既是错误的，说它是奇数也是错误的，因为加一之后它的性质并不改变；然而这是一个数，而一切数不是偶数便是奇数（这一点对一切有限数来说，都是真确的。）这样，我们就很可以认识到有一个上帝存在，而不必知道他是什么。

从上述例子中，我们可以深刻地体会到数学与神学两大思想体系的相互影响和渗透，它反映了人类文化的社会性和复杂性。从功过论，无限的不可确定性的神秘化，阻碍了数学思想的自由发展，也销蚀了数学家的穷研精神，使数学迟迟未能由静态进入动态。直至牛顿和莱布尼兹创立微积分学说，无穷量才堂而皇之地步入数学的殿堂。同时，他们也受到了来自神学的最猛烈的攻击。而攻击的焦点正是无限的不可确定性。请听当年大主教贝克莱的一段著名的檄文：

>>>

这些流率又是什么呢？是瞬息即逝的增量的速度。然而这些瞬息增量又是什么？既不是有限的量，又不是无限的量，也不是无。我们能不能把它们叫作死去数量的鬼魂呢？

<<<

 

回顾漫长的数学史，可以发现：无理数的不可确定性导致了第一次数学危机；流率或流差的不可确定性导致了第二次数学危机。而最大的数学危机发生于本世纪初，它是由集合论中的悖论的不可确定性而引发的。这时，神学已经失去了往日的光彩，只有在暗地里庆幸。但是，大数学家外尔在描述数学家们心理上的困惑时，仍然使用了神学的语言：“上帝是存在的，因为数学没有予盾；魔鬼也是存在的，因为我们不能证明这无予盾性。”

[part 2.3: 无限的敛散性]

以上我们介绍了无限的不可超越性和不可确定性。那么，在无限的面前，难道人类没有得到过任何可以把握的东西吗？不是的。事实上，数学家和哲学家都在有限的范围内发现了无限的一些基本特征。其中最重要的工作，就是对无限的分类。数学家把无限划分为发散的和收敛的两类，哲学家把无限划分为“恶的”和“真的”两类。它们之间存在着密切的相关性，并且深刻地反映了人类认识的宏观整体性。下面，我们采取对比的方法简要地介绍一下数学和哲学对于无限的分类过程，从中可以得到这样的结论：无限的领域并不是毫无路径可循的，随着人类认识的深化，无限的一些特征总会被发掘出来。但是，在这个过程中，神秘主义的东西始终是存在的，并且深深地影响着数学家和哲学家的思维。

数学家对于无限的分类产生于十七和十八世纪之间。当时，由于微积分的研究，数学中出现了大量的无穷级数。最初人们还没有认识到无穷级数的特殊性质，只把它们看作无限多个项的和。于是就轻率地把有理运算的性质应用于无穷级数，结果产生出许多奇异的答案。例如，无穷级数1－1＋1－1＋……，它既等于1，又等于1/2，又等于0，等等。这实际上是说1＝1/2＝0！何等荒谬的答案，它必然要引起数学家的关注和恐慌。但是，对此神学家却非常高兴，他们终于找到“上帝存在”的证明。意大利的僧侣数学家格兰迪认为，1/2＝0的结果正是上帝从虚无创造万有的象征，它还表明了上帝的公正和不偏不倚。23这种神秘主义的观点，对数学产生了很大的影响。直到十九世纪，人们还未能彻底地从无穷级数的困惑中挣脱出来。请听阿贝尔的一段表白：

>>>

发散级数真是魔鬼的发明，如果根据它们来作任何证明，简直是一种憾事。通过使用它们，人们就可以得出任何他所喜欢的结果，这就是些级数所以产生出这么多的谬误和这么多的悖论的原故……

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为了消除这些荒谬的结果，数学家们最终发现了无穷级数特殊性质：收敛性和发散性。只有那些收敛的级数才是可以确定的，我们可以运用有限的数学方法求出它的有限解；而发散的级数是不可确定的，它使一切数学方法失效。这是一个数学家不愿意接受的结论，正如M·克莱因所说：“总的说来，他们憎恨任何限制，甚至憎恨有必要去考虑一下收敛性的问题。”

当数学家受到发散级数困惑的时候，哲学界也开展了所谓“真无限”与“恶无限”的分类。哲学家黑格尔认为，那些发散级数是有限量的不断延伸或积累，它是不可把握的，是恶的无限。他举例说：“我们发现哈勒尔在一首著名的描写上帝的无限性的诗里，说道：

>>>

我们积累起庞大的数字，

一山又一山，一万又一万，

世界之上我堆起世界，

时间之上，我加上时间，

当我从可怕的高峰，

仰望着你，—以眩晕的眼：

所有数的乘方，

再乘以万千遍，

距你的一部分还是很远。”26

<<<

 

这种无限的增长，的确是可怖、可恶的。黑格尔还认为，只有那些收敛的级数才是真的无限。它们不是在有限之外，而是在有限之中；它们不是无限多的意思，也不是永远达不到的“彼岸”，而是现实中存在的东西。

黑格尔的论点具有重要的哲学意义，他对于数学界的影响是巨大的。人们可以以此为依据，剥掉无限的一些神秘主义的外衣，比较理智和清醒地面对无限了。但是，事情并没有就此结束，理智泯灭不了人类穷研的狂热。在人们还没有弄清为什么无限会有收敛与发散之分的时候，大量所谓真的无限和恶的无限却在现实世界中神秘地涌现出来。诸如，默比乌斯带和克莱因瓶，它们分别在二维和三维空间中构成了用有限表现无限的模型；巴赫的乐曲《音乐的奉献》（其中的逆行卡农）和埃舍尔的版画《瀑布》、《上升与下降》，它们分别在音乐和绘画艺术中使人的听觉和视觉产生了有限中的无限遐想。这些现象是否属于真无限的范畴呢？

而最大的恶无限大概要属元数学理论了，它是专为证明数学本身的严谨性而建立的。但是，它的严谨性又要由谁来证明呢？一定是“元元数学”了。霍夫斯塔特用“神灯”的故事形象地描述了这种困境：“阿基里斯清神灯满足他的上百个愿望，灯神只好请出元灯神，元灯神又从自己的长袍中取出金制的元元灯神，它可以满足元元希望……这个无限的过程通向上帝。上帝可以许诺无形的希望，包括任何希望的希望，但是这种希望是没有保障的。”

从这些例子中我们是否可以得到结论：当新的、更难以解释的“真无限”和“恶无限”大批出现的时候，神秘主义的东西又在人们的心灵中复活了。至于它们充任了什么角色，是值得深究的。本文仅指出了它们的存在性。

[part 3:      ]

本文简要地介绍了“数学无限”中的一些神秘主义问题。可以肯定地说，这种介绍是不够全面的，还有许多与无限相关的神秘主义现象和思想表现尚未论及。

值得强调的是，由于人类认识具有无限的特征，所以无限本身就象一个影子一样，时时地尾随着我们。虽然大哲学家康德等人一再劝说人们，不要去追求那些虚无飘渺的东西，否则会陷入思想的迷径永难出头。但是，别人可以在三维空间和永恒的时间中忍耐，数学家却不能，因为数学恰恰是建基在无限的假设之上。况且数学中还有许多问题尚无着落，数学家的思想经常处于迷惘之中。正如外尔所说：“关于数学最终基础和最终意义的问题还没有解决，或者根本就不能期望会有一个最后的客观回答。数学化很可能是人的一种创造性活动，像语言或音乐一样，具有原始的独创性，他的历史性决定不容许完全的客观的有理化。”外尔的话是深刻的，它本来可以把我们从无限的神秘性中彻底解脱出来。遗憾的是，它仅是一个幻想。数学家就是因为太爱幻想了，才引出了这么多的烦恼。请听J·布罗诺夫斯基的一段诗句：

>>>

在叙述数学的进展时，我既觉得兴致勃勃，又有局促不安的感觉，因为数学中幻想的成份太多，在人类的智力攀登中，数学不但是理性的阶梯，也是神秘思想的阶梯。

<<<

在果壳网看到一篇有爱的科普，自娱自乐，也期望可以娱乐到他人。

调戏果壳原文请戳>>>这里<<<

膜拜英语原文请戳>>>这里<<<

>>>

接下来大家将看到一篇著名而另类的论文，作者是 H. Petard（它是数学家 E. S. Pondiczery 的笔名，而 E. S. Pondiczery 本身其实也是一个笔名，它是数学家 Ralph P. Boas, Jr. 和同事们长期合作发表论文时虚构的一个数学家）。这篇文章于 1938 年发表在《美国数学月刊》上，详尽地向学术界介绍了与狩猎大型猎物相关的数学理论。从其他的同类文章中可以看出，Petard 的研究引来很多人为这篇文章添砖加瓦。尽管这一类文章并没有组成一个理论框架，但是它们确实能向有兴趣的读者在数学的发展分支方面进行了全面的介绍。

关于狩猎大型猎物的数学定理的研究

关于狩猎大型猎物的数学定理的研究

近些年来，这个鲜为人知的数学领域尚未得到我们认为应得的关注。在这篇论文里，我们将讨论一些相关的算法，希望这样能够吸引一些这个领域的其他工作者。我们所关注的内容应该限定在那些对理论有重要应用的方法上，而这些理论应该为数学家和物理学家所熟知，所以我们会忽略那些较为平凡的方法。

现在这个时期尤其适合于准备一下这个课题的清单，因为目前纯数学和理论物理的发展已经能够提供一些早期研究者难以想象的工具了。同时，一些优美而经典的技术也需要以现代的观念来重新理解它们的重要性。如同近些年来很多其他运用到数学的科学分支一样，对狩猎大型猎物的数学定理的研究将数学各个分支的定理奇妙地统一了起来。

为了叙述简便，我们不妨将提到的“大型猎物”限定为居住在撒哈拉沙漠上的狮子。显然，只需要在形式加以修改，我们在下文中列举的方法便可以自然地扩展到其他的野兽和其他的地域上。这篇论文分为三部分，分别讨论与数学、理论物理学和实验物理学相关的理论。

一、数学方法

1. 希尔伯特方法（或者称之为公理化方法）：我们将一个锁住的笼子放在沙漠的一个已知位置上，然后我们引入以下的逻辑系统：

3. 射影几何学方法：我们可以不失一般性地将整个沙漠看成是一个平面。我们将这个平面投影到一条线上，接着将这条线投影到笼子的一个内点。因此目标狮子便也被投影到这个内点上——也就是笼子里。

4. 波尔察诺-魏尔斯特拉斯方法：用一条南北走向的线将这个沙漠分成两部分。那么狮子不是在东边就是在西边，不妨设它在西边；再用一个东西方向的线分割狮子所在的部分，于是狮子不是在这部分的南边就是在北边……无限次地进行这个过程，每一步我们都会布下一个足够结实的围栏，而且所围区域的直径趋向于 0。于是这头狮子最终被包围在一个周长任意小的围栏里面了。

5. 集合论方法：沙漠是一个可分空间，所以它包含一个可数的稠密点集，可以以此构造一个以狮子为极限的子序列。接着我们沿着这个子序列悄悄地接近它，然后用合适的东西海扁它！

6. 皮亚诺方法：通过标准方法构造一条经过沙漠中每一点的连续曲线。我们已经知道，可以在任意短的时间内遍历这样的曲线。所以我们应该带上长矛，然后赶在狮子移动一个身长的距离之前飞速遍历整条曲线。

7. 拓扑学方法：我们发现一头狮子至少有着环的连通性。我们将沙漠变换到四维空间中，便可将其以扭结状态变换回三维空间中，这样它便无计可施啦。

8. 柯西积分法：让我们来考察一个“狮值”函数，设 ζ 为笼子，考虑下述积分：

其中 C 表示沙漠的边界。这个积分的值是 f(ζ)，也就是说有一个狮子在笼子里面。

9. 维纳-陶伯方法：我们从 L(-∞, ∞) 中选一个已被驯服的狮子 Lo ，它的傅里叶变换处处不为零，然后将它放到沙漠中。于是 Lo 便收敛于我们的笼子。根据维纳-陶伯定理（Wiener's General Tauberian Theorem），任意一个其他的狮子 L 也会收敛于同一个笼子。

二、理论物理学方法

10. 狄拉克方法：我们发现事实上野生狮子在撒哈拉沙漠中是观察不到的，因此如果沙漠中有狮子，那么他们一定是已经被驯服了的。在此我们将“抓住一个被驯服的狮子”作为一个练习留给读者。

11. 薛定谔方法：任意时刻一定有一个微小的正概率使得狮子在笼子中，守株待兔吧。

12. 核物理方法：将一头驯服了的狮子放进笼子里，对它和一头野狮子应用马约拉纳交换算符（Majorana exchange operator）。作为一个变型，假如你非要一头公狮子，我们可以在笼子里放入一头驯服了的母狮子，然后应用海森堡交换算符（Heisenberg exchange operator），它将连同自旋一并交换。

13. 相对论方法：我们在狮子周围撒下大量天狼星伴星作为诱饵。当狮子吃了足够多的时候，我们用一束光照射穿过沙漠——这束光在狮子周围会发生弯曲，于是它就会头昏眼花的，我们便能够悄无声息地接近它了。

三、实验物理学方法

14. 热力学方法：我们做一张半透膜——一张除了狮子别的东西都能透过去的半透膜，然后用它横扫整个撒哈拉大沙漠。

15. 原子裂变法：我们用慢中子辐射沙漠，于是狮子就带上了放射性，同时狮子会开始衰变。当衰变得差不多的时候，它便无力抗争了。

16. 磁光法：我们种下大量猫薄荷，并排列成透镜形状，这个透镜的轴向与地球磁场的水平切向平行。接着再将笼子放在透镜的一个焦点处。我们将已经磁化了的菠菜种满整个沙漠——我们都知道菠菜含有大量的铁。菠菜会被沙漠的食草动物吃掉，然后这些食草动物会被狮子吃掉。于是狮子们都被磁场转到和地球的磁场线平行的方向，然后他们便能被猫薄荷透镜聚焦到笼子里面。

<<<

这篇可爱的恶搞论文到此结束

偶然发现这么一个有意思的东西，一些math nerds用数学语言来诠释电影名称，趣味十足，同时对数学功底又有很高的要求。我那些不学数学的朋友们不停地要我解释为什么这部电影和这幅图配对那部电影和那幅图配对。在此我统一做出解释。

呵呵，什么时候我自己也能这么有创造力呢？

上图：

1). Matrix,。。。。。2). Signs,

3). Fireball,。。。。4). Mighty Ducks,

5). Cross of Iron, 。6). Social Network,

7). Sin City,。。。。8). Heat,

9). Inception,。。。10). Beauty and the Beast.

1). 22矩阵的通式；

2). Sines，读音与signs相同；

3). Ball的定义；

4). sup{A}的定义是取A集的最小上界，能取到的是A集中的最大元素或者是A的补集中比A所有元素都大的最小元素；

5). 表示叉乘，所以是cross，Fe是铁元素；

6). 不解释；

7). 正弦曲线，缩写为sin，正弦下面是一个城市的缩影，所以是city；

8). 微分方程中的热传导方程，学工程或者物理的应该很熟悉；

9). 猜的，答案不确定；

10). 上面的等式的欧拉恒等式，被誉为“the gold standard of mathematical beauty”，下面的666来自《圣经 启示录》中“number of the man”，后引申为“number of the beast”.

上图：

1). Snakes on a Plane,。2). Independence Day, 

3). Mean Girls,。。。。。4). Paths of Glory,

5). 8 Mile,。。。。。。。6). Matrix: Revolutions,

7). Field of Dreams,。。8). Alpha and Omega,

9). Thirteenth Floor,。。10). Sum of All Fears.

1). x,y是Cartesian plane，上面4条蛇；

2). 概率论中，若A,B两个事件是independent，那么P(A∩B)=P(A)P(B)，否则P(A∩B)=P(A∪B)－P(A)－P(B)；

3). 统计中sample mean的公式；

4). 三条路径都将glory串了起来；

5). 换算一下，8英里；

6). 22矩阵中rotation矩阵；

7). Field定义；

8). ∧是Boolean algebra中的and；

9). Floor函数；

10). 求和公式。

上图：

1). Manhattan,。。2). Joan of Arc,

3). Detour,。。。。4). Inception,

5). Odd Couple,。。6). In the Loop,

7). Star War,。。。8). Exorcist,

9). Transformers,。10). Absolute Power.

1). Man-hat-tan；

2). Arc length公式；

3). D to R，读一遍就好；

4). 一个方程套三层，和Inception异曲同工；

5). 两个都是奇数，所以是一对奇数；

6). Java语言中的循环；

7). 红光vs蓝光；

8). ∨是Boolean algebra中的or，ex-or-cist；

9). 三个变量，其中最后的一个是对工程学举足轻重的Laplace transformation；

10). P(A)是A集的power set，power外面加绝对值。

猜电影到此为止。

对我来说，分析却是我上过最有意义的课了。我不会说这是我最喜欢的课，事实上，多变量微积分要比分析有趣得多，任何和工程挂钩的数学都是富有趣味的。分析比不上多变量微积分的乐趣，甚至连数论都不如，但是分析的确是目前我感触最深的一门课。可以这么说，分析开拓了我对数学的理解，我能想到的所有关于实数集的颠覆性理念都来自分析。

比如说：

有这样一个集合，它有上界和下界，它的长度是0，但是它包涵了无数个元素，并且这个无限是不可数的，每个包涵其中的元素都不是孤点，因为在每个元素的左右都可以找到无数多个属于这个集合其他元素，因此，这个集合是稠密的。存在有多少个实数，这个集合里就有和实数一样多的元素。当然，这个神奇的集合有一个名字，叫做Cantor集。这个集合的构成说复杂不复杂，说简单也不简单。

Cantor集的定义如下：给定闭区间[0,1]，把这个区间分成相等的三段，去掉中间段即(1/3,2/3)，把剩下的两段中每一段都按刚才的方法再进行操作，重复以上动作，一直挖洞挖下去。在第二次操作后，剩下的区间是[0,1/9]∪[2/9,1/3]∪[2/3,7/9]∪[8/9,1]，第三次操作后区间将由8段构成。

集合的通式是这样的：

虽然说Cantor集长度是0，但是它包涵了无限多个元素，举例来说，1/4和1/13都在这个集合里。

其实Cantor集尚不是我最喜欢的例子。十八十九世纪的时候，西欧的数学家们研究函数，他们想定义函数，并且想知道连续和可微分之间的关系。现在，学微积分的孩子应该都是这样做题的：连续函数是可微分的，所以连续函数是可求导的。事实上，Weierestrass在1872年首次给出了一个在定义域上连续却完全不可微分的函数：

既然是连续的，那么这个函数是没有断点的，而它又完全不可微分，所以函数图像应该完全没有平滑的曲线。下图是这个函数在[0, 2π]上的图像。

Weierestrass函数开启了分析领域的大门，自此，各种各样反常理的实数函数被逐一发现。

至此，我都还没有给出明确的关于连续函数的定义。那么，究竟什么是连续函数？

那么，存在一个在有理数上连续在无理数上不连续的函数吗？存在一个在有理数上不连续在无理数上连续的函数吗？

答案是：第一个，不存在；第二个，存在。

有这么一个函数，它在任何一个有理数上都是不连续的，而在任意一个无理数上都是连续的，它有一个名字：Thomae's Function。

方程的图像很有趣，和它的定义一样有趣。

而证明这个函数在有理数上不连续在无理数上连续才是最有趣的部分。上个月，我和Patrick想了整整两个晚上最终完成了全部的证明。

证明构建于一个重要定理的基础上。

至于这个定理该如何证明，我在这里就不多做解释了，完整的证明比较长，因为4个部分都需要证明。要引用连续函数极限邻域等等定义。

在这里我仅仅展示关于Thomae's Function连续性的证明。自己的证明，手写再扫描，语言已经尽可能地详尽。

关于Thomae's Function，关于实分析，就是这样。

。

凯恩斯是一个睿智而博学的人，从二十世纪三十年代美英法三国起草布雷顿森林体系开始，他的观点初露头角，至布雷顿森林会议，思想上已完全成熟自成一家的凯恩斯与美方代表怀特二分天下。他提出的超主权货币提案虽然没有被通过，却为1967年尼克松在IMF提议建立世界上第一个超主权货币SDR打下基础。

凯恩斯的理论和他的《通论》影响了整整一个二十世纪。《通论》中第十二章对于国际汇率波动和长期预期的阐述尤其精彩。此后，几代的经济学家将其完善，引申出Behavioralist理论以及克鲁格曼的J Curve，为更好地预测货币价值以及稳定货币市场做出不可磨灭的贡献。在此我节选了整个第十二章，勉励自己，也算是为明年的毕业论文找灵感。

>>>

The General Theory by John Maynard Keynes (1936)
_____________________________________________________________

[p.147]

Chapter 12

THE STATE OF LONG-TERM EXPECTATION

 。

I

We have seen in the previous chapter that the scale of investment depends on the relation between the rate of interest and the schedule of the marginal efficiency of capital corresponding to different scales of current investment, whilst the marginal efficiency of capital depends on the relation between the supply price of a capital-asset and its prospective yield. In this chapter we shall consider in more detail some of the factors which determine the prospective yield of an asset.

The considerations upon which expectations of prospective yields are based are partly existing facts which we can assume to be known more or less for certain, and partly future events which can only be forecasted with more or less confidence. Amongst the first may be mentioned the existing stock of various types of capital-assets and of capital-assets in general and the strength of the existing consumers' demand for goods which require for their efficient production a relatively larger assistance from capital. Amongst the latter are future changes in the type and quantity of the stock of capital-assets and in the tastes of the consumer, the strength of effective demand from time to time during the life of the investment under consideration, and the changes in the wage-unit in terms of money which may occur during its life. We may sum up the state of psychological expectation which covers the [p.148] latter as being the state of long-term expectation; as distinguished from the short-term expectation upon the basis of which a producer estimates what he will get for a product when it is finished if he decides to begin producing it to-day with the existing plant, which we examined in Chapter 5.


II

It would be foolish, in forming our expectations, to attach great weight to matters which are very uncertain.[1] It is reasonable, therefore, to be guided to a considerable degree by the facts about which we feel somewhat confident, even though they may be less decisively relevant to the issue than other facts about which our knowledge is vague and scanty. For this reason the facts of the existing situation enter, in a sense disproportionately, into the formation of our long-term expectations; our usual practice being to take the existing situation and to project it into the future, modified only to the extent that we have more or less definite reasons for expecting a change.

The state of long-term expectation, upon which our decisions are based, does not solely depend, therefore, on the most probable forecast we can make. It also depends on the confidence with which we make this forecast on how highly we rate the likelihood of our best forecast turning out quite wrong. If we expect large changes but are very uncertain as to what precise form these changes will take, then our confidence will be weak.

The state of confidence, as they term it, is a matter to which practical men always pay the closest and most anxious attention. But economists have not analysed it carefully and have been content, as a rule, to discuss [p.149] it in general terms. In particular it has not been made clear that its relevance to economic problems comes in through its important influence on the schedule of the marginal efficiency of capital. There are not two separate factors affecting the rate of investment, namely, the schedule of the marginal efficiency of capital and the state of confidence. The state of confidence is relevant because it is one of the major factors determining the former, which is the same thing as the investment demand-schedule.

There is, however, not much to be said about the state of confidence a priori. Our conclusions must mainly depend upon the actual observation of markets and business psychology. This is the reason why the ensuing digression is on a different level of abstraction from most of this book.

For convenience of exposition we shall assume in the following discussion of the state of confidence that there are no changes in the rate of interest; and we shall write, throughout the following sections, as if changes in the values of investments were solely due to changes in the expectation of their prospective yields and not at all to changes in the rate of interest at which these prospective yields are capitalised. The effect of changes in the rate of interest is, however, easily superimposed on the effect of changes in the state of confidence.


III

The outstanding fact is the extreme precariousness of the basis of knowledge on which our estimates of prospective yield have to be made. Our knowledge of the factors which will govern the yield of an investment some years hence is usually very slight and often negligible. If we speak frankly, we have to admit that our basis of knowledge for estimating the yield ten years hence of a railway, a copper mine, a textile factory, the goodwill of a patent medicine, an Atlantic [p.150] liner, a building in the City of London amounts to little and sometimes to nothing; or even five years hence. In fact, those who seriously attempt to make any such estimate are often so much in the minority that their behaviour does not govern the market.

In former times, when enterprises were mainly owned by those who undertook them or by their friends and associates, investment depended on a sufficient supply of individuals of sanguine temperament and constructive impulses who embarked on business as a way of life, not really relying on a precise calculation of prospective profit. The affair was partly a lottery, though with the ultimate result largely governed by whether the abilities and character of the managers were above or below the average. Some would fail and some would succeed. But even after the event no one would know whether the average results in terms of the sums invested had exceeded, equalled or fallen short of the prevailing rate of interest; though, if we exclude the exploitation of natural resources and monopolies, it is probable that the actual average results of investments, even during periods of progress and prosperity, have disappointed the hopes which prompted them. Business men play a mixed game of skill and chance, the average results of which to the players are not known by those who take a hand. If human nature felt no temptation to take a chance, no satisfaction (profit apart) in constructing a factory, a railway, a mine or a farm, there might not be much investment merely as a result of cold calculation.

Decisions to invest in private business of the old-fashioned type were, however, decisions largely irrevocable, not only for the community as a whole, but also for the individual. With the separation between ownership and management which prevails to-day and with the development of organised investment markets, a new factor of great importance has entered in, which sometimes facilitates investment but sometimes adds [p.151] greatly to the instability of the system. In the absence of security markets, there is no object in frequently attempting to revalue an investment to which we are committed. But the Stock Exchange revalues many investments every day and the revaluations give a frequent opportunity to the individual (though not to the community as a whole) to revise his commitments. It is as though a farmer, having tapped his barometer after breakfast, could decide to remove his capital from the farming business between 10 and 11 in the morning and reconsider whether he should return to it later in the week. But the daily revaluations of the Stock Exchange, though they are primarily made to facilitate transfers of old investments between one individual and another, inevitably exert a decisive influence on the rate of current investment. For there is no sense in building up a new enterprise at a cost greater than that at which a similar existing enterprise can be purchased; whilst there is an inducement to spend on a new project what may seem an extravagant sum, if it can be floated off on the Stock Exchange at an immediate profit.[1] Thus certain classes of investment are governed by the average expectation of those who deal on the Stock Exchange as revealed in the price of shares, rather than by the genuine expectations of the professional entrepreneur.[2] How then are these highly significant daily, even hourly, revaluations of existing investments carried out in practice? [p.152]


IV

In practice we have tacitly agreed, as a rule, to fall back on what is, in truth, a convention. The essence of this convention though it does not, of course, work out quite so simply lies in assuming that the existing state of affairs will continue indefinitely, except in so far as we have specific reasons to expect a change. This does not mean that we really believe that the existing state of affairs will continue indefinitely. We know from extensive experience that this is most unlikely. The actual results of an investment over a long term of years very seldom agree with the initial expectation. Nor can we rationalise our behaviour by arguing that to a man in a state of ignorance errors in either direction are equally probable, so that there remains a mean actuarial expectation based on equi-probabilities. For it can easily be shown that the assumption of arithmetically equal probabilities based on a state of ignorance leads to absurdities. We are assuming, in effect, that the existing market valuation, however arrived at, is uniquely correct in relation to our existing knowledge of the facts which will influence the yield of the investment, and that it will only change in proportion to changes in this knowledge; though, philosophically speaking, it cannot be uniquely correct, since our existing knowledge does not provide a sufficient basis for a calculated mathematical expectation. In point of fact, all sorts of considerations enter into the market valuation which are in no way relevant to the prospective yield.

Nevertheless the above conventional method of calculation will be compatible with a considerable measure of continuity and stability in our affairs, so long as we can rely on the maintenance of the convention.

For if there exist organised investment markets and if we can rely on the maintenance of the convention, an investor can legitimately encourage himself with the [p.153] idea that the only risk he runs is that of a genuine change in the news over the near future, as to the likelihood of which he can attempt to form his own judgment, and which is unlikely to be very large. For, assuming that the convention holds good, it is only these changes which can affect the value of his investment, and he need not lose his sleep merely because he has not any notion what his investment will be worth ten years hence. Thus investment becomes reasonably 'safe' for the individual investor over short periods, and hence over a succession of short periods however many, if he can fairly rely on there being no breakdown in the convention and on his therefore having an opportunity to revise his judgment and change his investment, before there has been time for much to happen. Investments which are "fixed" for the community are thus made 'liquid' for the individual.

It has been, I am sure, on the basis of some such procedure as this that our leading investment markets have been developed. But it is not surprising that a convention, in an absolute view of things so arbitrary, should have its weak points. It is its precariousness which creates no small part of our contemporary problem of securing sufficient investment.

V

Some of the factors which accentuate this precariousness may be briefly mentioned.

(1)  As a result of the gradual increase in the proportion of the equity in the community's aggregate capital investment which is owned by persons who do not manage and have no special knowledge of the circumstances, either actual or prospective, of the business in question, the element of real knowledge in the valuation of investments by whose who own them or contemplate purchasing them has seriously declined.

(2)  Day-to-day fluctuations in the profits of existing [p.154] investments, which are obviously of an ephemeral and non-significant character, tend to have an altogether excessive, and even an absurd, influence on the market. It is said, for example, that the shares of American companies which manufacture ice tend to sell at a higher price in summer when their profits are seasonally high than in winter when no one wants ice. The recurrence of a bank-holiday may raise the market valuation of the British railway system by several million pounds.

(3)  A conventional valuation which is established as the outcome of the mass psychology of a large number of ignorant individuals is liable to change violently as the result of a sudden fluctuation of opinion due to factors which do not really make much difference to the prospective yield; since there will be no strong roots of conviction to hold it steady. In abnormal times in particular, when the hypothesis of an indefinite continuance of the existing state of affairs is less plausible than usual even though there are no express grounds to anticipate a definite change, the market will be subject to waves of optimistic and pessimistic sentiment, which are unreasoning and yet in a sense legitimate where no solid basis exists for a reasonable calculation.

(4)  But there is one feature in particular which deserves our attention. It might have been supposed that competition between expert professionals, possessing judgment and knowledge beyond that of the average private investor, would correct the vagaries of the ignorant individual left to himself. It happens, however, that the energies and skill of the professional investor and speculator are mainly occupied otherwise. For most of these persons are, in fact, largely concerned, not with making superior long-term forecasts of the probable yield of an investment over its whole life, but with foreseeing changes in the conventional basis of valuation a short time ahead of the general public. They are concerned, not with what an investment is [p.155] really worth to a man who buys it 'for keeps', but with what the market will value it at, under the influence of mass psychology, three months or a year hence. Moreover, this behaviour is not the outcome of a wrong-headed propensity. It is an inevitable result of an investment market organised along the lines described. For it is not sensible to pay 25 for an investment of which you believe the prospective yield to justify a value of 30, if you also believe that the market will value it at 20 three months hence.

Thus the professional investor is forced to concern himself with the anticipation of impending changes, in the news or in the atmosphere, of the kind by which experience shows that the mass psychology of the market is most influenced. This is the inevitable result of investment markets organised with a view to so-called 'liquidity'. Of the maxims of orthodox finance none, surely, is more anti-social than the fetish of liquidity, the doctrine that it is a positive virtue on the part of investment institutions to concentrate their resources upon the holding of 'liquid' securities. It forgets that there is no such thing as liquidity of investment for the community as a whole. The social object of skilled investment should be to defeat the dark forces of time and ignorance which envelop our future. The actual, private object of the most skilled investment to-day is 'to beat the gun', as the Americans so well express it, to outwit the crowd, and to pass the bad, or depreciating, half-crown to the other fellow.

This battle of wits to anticipate the basis of conventional valuation a few months hence, rather than the prospective yield of an investment over a long term of years, does not even require gulls amongst the public to feed the maws of the professional; it can be played by professionals amongst themselves. Nor is it necessary that anyone should keep his simple faith in the conventional basis of valuation having any genuine long-term validity. For it is, so to speak, a game of Snap,[p.156] of Old Maid, of Musical Chairs a pastime in which he is victor who says Snap neither too soon nor too late, who passed the Old Maid to his neighbour before the game is over, who secures a chair for himself when the music stops. These games can be played with zest and enjoyment, though all the players know that it is the Old Maid which is circulating, or that when the music stops some of the players will find themselves unseated.

Or, to change the metaphor slightly, professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view. It is not a case of choosing those which, to the best of one's judgment, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fifth and higher degrees.

If the reader interjects that there must surely be large profits to be gained from the other players in the long run by a skilled individual who, unperturbed by the prevailing pastime, continues to purchase investments on the best genuine long-term expectations he can frame, he must be answered, first of all, that there are, indeed, such serious-minded individuals and that it makes a vast difference to an investment market whether or not they predominate in their influence over the game-players. But we must also add that there are [p.157] several factors which jeopardise the predominance of such individuals in modern investment markets. Investment based on genuine long-term expectation is so difficult to-day as to be scarcely practicable. He who attempts it must surely lead much more laborious days and run greater risks than he who tries to guess better than the crowd how the crowd will behave; and, given equal intelligence, he may make more disastrous mistakes. There is no clear evidence from experience that the investment policy which is socially advantageous coincides with that which is most profitable. It needs more intelligence to defeat the forces of time and our ignorance of the future than to beat the gun. Moreover, life is not long enough; human nature desires quick results, there is a peculiar zest in making money quickly, and remoter gains are discounted by the average man at a very high rate. The game of professional investment is intolerably boring and over-exacting to anyone who is entirely exempt from the gambling instinct; whilst he who has it must pay to this propensity the appropriate toll. Furthermore, an investor who proposes to ignore near-term market fluctuations needs greater resources for safety and must not operate on so large a scale, if at all, with borrowed money a further reason for the higher return from the pastime to a given stock of intelligence and resources. Finally it is the long-term investor, he who most promotes the public interest, who will in practice come in for most criticism, wherever investment funds are managed by committees or boards or banks.[1] For it is in the essence of his behaviour that he should be eccentric, unconventional and rash in the eyes of average opinion. If he is successful, that will only confirm the general belief in his rashness; and if [p.158] in the short run he is unsuccessful, which is very likely, he will not receive much mercy. Worldly wisdom teaches that it is better for reputation to fail conventionally than to succeed unconventionally.

(5) So far we have had chiefly in mind the state of confidence of the speculator or speculative investor himself and may have seemed to be tacitly assuming that, if he himself is satisfied with the prospects, he has unlimited command over money at the market rate of interest. This is, of course, not the case. Thus we must also take account of the other facet of the state of confidence, namely, the confidence of the lending institutions towards those who seek to borrow from them, sometimes described as the state of credit. A collapse in the price of equities, which has had disastrous reactions on the marginal efficiency of capital, may have been due to the weakening either of speculative confidence or of the state of credit. But whereas the weakening of either is enough to cause a collapse, recovery requires the revival of both. For whilst the weakening of credit is sufficient to bring about a collapse, its strengthening, though a necessary condition of recovery, is not a sufficient condition.

VI

These considerations should not lie beyond the purview of the economist. But they must be relegated to their right perspective. If I may be allowed to appropriate the term speculation for the activity of forecasting the psychology of the market, and the term enterprise for the activity of forecasting the prospective yield of assets over their whole life, it is by no means always the case that speculation predominates over enterprise. As the organisation of investment markets improves, the risk of the predominance of speculation does, however, increase. In one of the greatest investment markets in the world, namely, New York, the [p.159] influence of speculation (in the above sense) is enormous. Even outside the field of finance, Americans are apt to be unduly interested in discovering what average opinion believes average opinion to be; and this national weakness finds its nemesis in the stock market. It is rare, one is told, for an American to invest, as many Englishmen still do, 'for income'; and he will not readily purchase an investment except in the hope of capital appreciation. This is only another way of saying that, when he purchases an investment, the American is attaching his hopes, not so much to its prospective yield, as to a favourable change in the conventional basis of valuation, i.e. that he is, in the above sense, a speculator. Speculators may do no harm as bubbles on a steady stream of enterprise. But the position is serious when enterprise becomes the bubble on a whirlpool of speculation. When the capital development of a country becomes a by-product of the activities of a casino, the job is likely to be ill-done. The measure of success attained by Wall Street, regarded as an institution of which the proper social purpose is to direct new investment into the most profitable channels in terms of future yield, cannot be claimed as one of the outstanding triumphs of laissez-faire capitalism which is not surprising, if I am right in thinking that the best brains of Wall Street have been in fact directed towards a different object.

These tendencies are a scarcely avoidable outcome of our having successfully organised 'liquid' investment markets. It is usually agreed that casinos should, in the public interest, be inaccessible and expensive. And perhaps the same is true of stock exchanges. That the sins of the London Stock Exchange are less than those of Wall Street may be due, not so much to differences in national character, as to the fact that to the average Englishman Throgmorton Street is, compared with Wall Street to the average American, inaccessible and very expensive. The jobber's 'turn', the high [p.160] brokerage charges and the heavy transfer tax payable to the Exchequer, which attend dealings on the London Stock Exchange, sufficiently diminish the liquidity of the market (although the practice of fortnightly accounts operates the other way) to rule out a large proportion of the transactions characteristic of Wall Street.[1] The introduction of a substantial Government transfer tax on all transactions might prove the most serviceable reform available, with a view to mitigating the predominance of speculation over enterprise in the United States.

The spectacle of modern investment markets has sometimes moved me towards the conclusion that to make the purchase of an investment permanent and indissoluble, like marriage, except by reason of death or other grave cause, might be a useful remedy for our contemporary evils. For this would force the investor to direct his mind to the long-term prospects and to those only. But a little consideration of this expedient brings us up against a dilemma, and shows us how the liquidity of investment markets often facilitates, though it sometimes impedes, the course of new investment. For the fact that each individual investor flatters himself that his commitment is 'liquid' (though this cannot be true for all investors collectively) calms his nerves and makes him much more willing to run a risk. If individual purchases of investments were rendered illiquid, this might seriously impede new investment, so long as alternative ways in which to hold his savings are available to the individual. This is the dilemma. So long as it is open to the individual to employ his wealth in hoarding or lending money, the alternative of purchasing actual capital assets cannot be rendered sufficiently attractive (especially to the man who does [p.161] not manage the capital assets and knows very little about them), except by organising markets wherein these assets can be easily realised for money.

The only radical cure for the crises of confidence which afflict the economic life of the modern world would be to allow the individual no choice between consuming his income and ordering the production of the specific capital-asset which, even though it be on precarious evidence, impresses him as the most promising investment available to him. It might be that, at times when he was more than usually assailed by doubts concerning the future, he would turn in his perplexity towards more consumption and less new investment. But that would avoid the disastrous, cumulative and far-reaching repercussions of its being open to him, when thus assailed by doubts, to spend his income neither on the one nor on the other.

Those who have emphasised the social dangers of the hoarding of money have, of course, had something similar to the above in mind. But they have overlooked the possibility that the phenomenon can occur without any change, or at least any commensurate change, in the hoarding of money.

VII

Even apart from the instability due to speculation, there is the instability due to the characteristic of human nature that a large proportion of our positive activities depend on spontaneous optimism rather than on a mathematical expectation, whether moral or hedonistic or economic. Most, probably, of our decisions to do something positive, the full consequences of which will be drawn out over many days to come, can only be taken as a result of animal spirits of a spontaneous urge to action rather than inaction, and not as the outcome of a weighted average of quantitative benefits multiplied by quantitative probabilities. Enterprise [p.162] only pretends to itself to be mainly actuated by the statements in its own prospectus, however candid and sincere. Only a little more than an expedition to the South Pole, is it based on an exact calculation of benefits to come. Thus if the animal spirits are dimmed and the spontaneous optimism falters, leaving us to depend on nothing but a mathematical expectation, enterprise will fade and die; though fears of loss may have a basis no more reasonable than hopes of profit had before.

It is safe to say that enterprise which depends on hopes stretching into the future benefits the community as a whole. But individual initiative will only be adequate when reasonable calculation is supplemented and supported by animal spirits, so that the thought of ultimate loss which often overtakes pioneers, as experience undoubtedly tells us and them, is put aside as a healthy man puts aside the expectation of death.

This means, unfortunately, not only that slumps and depressions are exaggerated in degree, but that economic prosperity is excessively dependent on a political and social atmosphere which is congenial to the average business man. If the fear of a Labour Government or a New Deal depresses enterprise, this need not be the result either of a reasonable calculation or of a plot with political intent; it is the mere consequence of upsetting the delicate balance of spontaneous optimism. In estimating the prospects of investment, we must have regard, therefore, to the nerves and hysteria and even the digestions and reactions to the weather of those upon whose spontaneous activity it largely depends.

We should not conclude from this that everything depends on waves of irrational psychology. On the contrary, the state of long-term expectation is often steady, and, even when it is not, the other factors exert their compensating effects. We are merely reminding ourselves that human decisions affecting the future, whether personal or political or economic, cannot [p.163] depend on strict mathematical expectation, since the basis for making such calculations does not exist; and that it is our innate urge to activity which makes the wheels go round, our rational selves choosing between the alternatives as best we are able, calculating where we can, but often falling back for our motive on whim or sentiment or chance.


VIII

There are, moreover, certain important factors which somewhat mitigate in practice the effects of our ignorance of the future. Owing to the operation of compound interest combined with the likelihood of obsolescence with the passage of time, there are many individual investments of which the prospective yield is legitimately dominated by the returns of the comparatively near future. In the case of the most important class of very long-term investments, namely buildings, the risk can be frequently transferred from the investor to the occupier, or at least shared between them, by means of long-term contracts, the risk being outweighed in the mind of the occupier by the advantages of continuity and security of tenure. In the case of another important class of long-term investments, namely public utilities, a substantial proportion of the prospective yield is practically guaranteed by monopoly privileges coupled with the right to charge such rates as will provide a certain stipulated margin. Finally there is a growing class of investments entered upon by, or at the risk of; public authorities, which are frankly influenced in making the investment by a general presumption of there being prospective social advantages from the investment, whatever its commercial yield may prove to be within a wide range, and without seeking to be satisfied that the mathematical expectation of the yield is at least equal to the current rate of interest, though the rate which the public [p.164] authority has to pay may still play a decisive part in determining the scale of investment operations which it can afford.

Thus after giving full weight to the importance of the influence of short-period changes in the state of long-term expectation as distinct from changes in the rate of interest, we are still entitled to return to the latter as exercising, at any rate, in normal circumstances, a great, though not a decisive, influence on the rate of investment. Only experience, however, can show how far management of the rate of interest is capable of continuously stimulating the appropriate volume of investment.

For my own part I am now somewhat sceptical of the success of a merely monetary policy directed towards influencing the rate of interest. I expect to see the State, which is in a position to calculate the marginal efficiency of capital-goods on long views and on the basis of the general social advantage, taking an ever greater responsibility for directly organising investment; since it seems likely that the fluctuations in the market estimation of the marginal efficiency of different types of capital, calculated on the principles I have described above, will be too great to be offset by any practicable changes in the rate of interest. 

Footnotes:

[p.148] 1 -  By "very uncertain" I do not mean the same thing as "very improbable".  Cf. my Treatise on Probability, chap. 6, on "The Weight of Arguments". [back to text]

[p.151] 1 -  In my Treatise on Money (vol. ii, p.195) I pointed out that when a company's shares are quoted very high so that it can raise more capital by issuing more shares on favourable terms, this has the same effect as if it could borrow at a low rate of interest.  I should now describe this by saying that a high quotation for existing equities involves an increase in the marginal efficiency of the corresponding type of capital and therefore has the same effect (since investment depends on a comparison between the marginal efficiency of capital and the rate of interest) as a fall in the rate of interest.  [back to text]

[p.151] 2 -  This does not apply, of course, to classes of enterprise which are not readily marketable or to which no negotiable instrument closely corresponds.  The categories falling within this exception were formerly extensive.  But measured as a proportion of the total value of new investment they are rapidly declining in importance. [back to text]

[p.157] 1 -  The practice, usually considered prudent, by which an investment trust or an insurance office frequently calculates not only the income from its investment portfolio but also its capital valuation in the market, may also tend to direct too much attention to short-term fluctuations in the latter. [back to text]

[p.160] 1 -  It is said that, when Wall Street is active, at least a half of the purchases or sales of investments are entered upon with an intention on the part of the speculator to reverse them the same day.  This is often true of the commodity exchanges also. [back to text]

关于《通论》凯恩斯以及汇率，就是这样。