V.I. Arnold 论数学教育（转）
发信人: starrynight (T&J), 信区: AUTO
标 题: V.I. Arnold 论数学教育
发信站: 瀚海星云 (2003年11月25日19:31:24 星期二), 站内信件
谨以此文献给那些教给过我操蛋数学的老师们。
发信人: Atiyah (铁拳无敌孙中山), 信区: Mathematics
标 题: V.I. Arnold 论数学教育（试译）(转载)
发信站: 瀚海星云 (2003年07月04日08:24:08 星期五), 站内信件
【 以下文字转载自 Science 讨论区 】
【 原文由 Atiyah 所发表 】
地点： Palais de Découverte in Paris
时间 1997年3月7日.
数学是物理的一部分。物理学是一门实验科学，它是自然科学的一部分。而数
学是物理学中只需要花费较少的代价进行实验的那一部分。例如 Jacobi 恒等式（
保证三角形三条高交于一点）就是一个实验事实，正如同地球是圆的（即同胚于球
体）这样的事实一样。但是发现前者却要比发现后者需要较少的代价。
在20世纪中叶，人们试图严格地区分物理与数学。其造成地后果是灾难性的。
整整一代的数学家在对他们所从事的科学的另一半及其无知的情况下成长，当然，
对其他的科学就更无知了。这些人又开始把他们的丑陋的学院式的伪数学教给他们
的学生，接着这些丑陋的伪数学又被交给中小学校里的孩子们（他们完全忘记了
Hardy的警告：丑陋的数学在阳光下不可能总有藏身之处）。
既然那些从物理学中人为挖出来的学院式的数学既无益于教学，又对其他的科
学毫无用处，结果可以想见，全世界的人都讨厌数学家（甚至包括那些被他们教出
来的可怜的学校里的孩子们以及那些运用这些丑陋数学的人）。这些先天不足的数
学家被他们所患的低能症候群折腾的筋疲力尽，他们无能对物理学有个起码的了解。
令人们记忆犹新的由他们建造的一个丑陋建筑物就是“奇数的严格公理化理论”。
很显然，完全可能创造这样一种理论，使得幼稚的小学生们敬畏它的完美及其
内部构造的和谐（例如，这种理论定义了奇数个项的和以及任意个因子的乘积）。
从这种偏执狭隘的观点来看，偶数或者被认为是一类“异端”，或者随着时间流逝，
被用来作为该理论中几个“理想”对象的补充（为了遵从物理与真实世界的需要）。
很不幸的是，这种理论只是数学中一个丑陋而变态的构造，但却统治了我们的数学
教育数十年。它首先源自于法国，这股歪风很快传播到对数学基础的教学里，先是
毒害大学生，接着中小学生也难免此灾（而灾区最先是法国，接着是其他国家，包
括俄罗斯）。
如果你问一个法国的小学生：“2＋3等于几？”，他（她）会这样回答：“等
于3＋2，因为加法运算是可交换的”。他（她）根本不知道这个和等于几，甚至根
本不能理解你在问他（她）什么！
还有的法国小学生会这样定义数学（至少我认为很有可能）：“存在一个正方
形，但还没有被证明”。
据我在法国教学的经验，大学里的学生对数学的认识与这些小学生也差不多（
甚至包括那些在'高等师范学校'（ENS）里学习数学的学生－－我为这些显然很聪
明但却被毒害颇深的孩子们感到极度的惋惜）。
例如，这些学生从未见过一个抛物面，而且一个这样的问题：描述由方程xy=z^2
所给出的曲面的形状，就能使那些在ENS中研究的数学家们发呆半天；而如下问题：
画出平面上由参数方程（例如x = t^3 - 3t, y = t^4 - 2t^2）给出的曲线，对学
生来说是不可能完成的（甚至对大多数法国的数学教授也一样）。从微积分的入门
教科书直到Goursat写的课本，解这些问题的能力都被认为是每个数学家应具备的基
本技能。
那些喜欢挑战大脑的所谓“抽象数学”的狂热者们，把所有在数学中能与物理
和现实经常发生联系的几何统统排除在教学之外。由Goursat, Hermite, Picard等
人写的微积分教程被认为是有害的，最近差点被巴黎第6和第7大学的图书馆当垃圾
丢掉，只是在我的干预下才得以保存。
ENS的听完所有微分几何与代数几何课程的学生（分别被不同的数学家教的），
却既不熟悉由椭圆曲线 y^2 = x^3 + ax + b 决定的黎曼曲面，也不知道曲面的拓
扑分类（更别提第一类椭圆积分和椭圆曲线的群性质了，即 Euler-Abel 加法定理
）。他们仅仅学到了Hodge 构造以及 Jacobi 簇！
这样的现象竟然会在法国出现！这个国家可是为整个世界贡献了诸如 Lagrange ，
Laplace, Cauchy 以及 Poincaré, Leray 还有 Thom 这些顶级的伟大人物啊！对
我而言，一个合理的解释来自 I.G. Petrovskii, 他在1966年曾教导过我: 真正的
数学家决不会拉帮结派，只有弱者为了生存才会加入帮派。他们可以联结很多的方
面（可能会是超级的抽象，反犹太主义或者“应用的和工业上的”问题），但其本
质总是为了解决社会生存问题。
我在此向大家顺便提一下 L. Pasteur 的忠告：从来没有也决不会有任何所谓
的“应用科学”，而仅仅有的是科学的应用（十分有用的东东啊！）
长久以来我一直对 Petrovskii 的话心存疑虑，但是现今我越来越肯定他说的
一点没错。那些超级抽象活动的相当大的部分正在堕落到以工业化的模式无耻的掠
夺那些发现者的成果，然后再加以系统地组织设计使自己成为万能的推广者。就彷
佛美丽坚所在的新大陆不以哥伦布命名一样，数学结果也几乎从未以它们真正的发
现者来命名。
为避免被认为我在胡说八道，我不得不在此声明我自己的一些成果由于莫名其
妙的原因就被以上述方式无偿征用，其实这样的事情经常在我的老师（Kolmogorov,
Petrovskii, Pontryagin, Rokhlin）和学生身上发生。
M. Berry 教授曾经提出过如下两个原理：
Arnold 原理：如果某个理念中出现了某个人名，则这个人名必非发现此理念者
的名字。
Berry 原理：Arnold 原理适用于自身。
不过，我们还是说回法国的数学教育上来。当我还是莫斯科大学数力系的一年
级新生时，集合论的拓扑学家 L.A. Tumarkin 教我们微积分，他在课堂上很谨慎地
一遍又一遍地讲述古老而经典的Goursat 版的法语微积分教程。他告诉我们有理函
数沿着一条代数曲线的积分可以求出来如果该代数曲线对应的黎曼面时一个球面。
而一般来说，如果该曲面的亏格更高这样的积分将不可求，不过对球面而言，只要
在一个给定度数的曲线上有充分多的double points 就足够了（即要求该曲线是
unicursal ：即可以将其实点在射影平面上一笔画出来）。
这些事实给我们造成多么深刻的印象啊（即使没有给出证明），它们给了我们
非常优美而正确的现代数学的思想，比那些长篇累牍的Bourbaki学派的论著不知道
好到哪里去了。说真的，我们在这里看到了那些表面上完全不同的事物之间存在着
令人惊奇的联系：一方面，对于相应的黎曼面上的积分与拓扑存在着显式的表达式，
而另一方面，在 double points 的个数与相应的黎曼面的亏格之间也有重要的联系。
这样的例子并不鲜见，作为数学中最迷人的性质之一，Jacobi曾指出：用同一
个函数就既可以理解能表示为4个数平方和的整数的性质，又可以描述一个单摆的运
动。
这些不同种类的数学对象之间联系的发现，就好比在物理学中电与磁之间联系
的发现，也类同于地质学上对美洲大陆的东海岸与非洲大陆的西海岸之间相似性的
发现。
这些发现对于教学所具有的令人激动的非凡意义是无法估量的。正是它们指引
着我们去研究和发现宇宙中和谐而精彩的现象。
然而，数学教育的非几何化以及与物理学的分离却割断了这种联系。例如，不
仅仅学习数学的学生而且绝大部分的代数几何学家都对以下提及的Jacobi事实一无
所知：一个第一类型的椭圆积分表示了相应的哈密顿系统中沿某个椭圆相曲线的运
动所走的时间。
我们知道一个 hypocycloid 就如同多项式环中的理想一样是无穷无尽的。但
是如果要把理想这个概念教给一个从未见过任何 hypocycloid 的学生，就好比把
分数的加法教给一个从来没有把蛋糕或苹果等分切割过（至少在脑子里切过）的学
生。毫无疑问孩子们将会倾向于同时分子加分子分母加分母。
从我的法国朋友那里我听说这种超级抽象的一般化正是他们国家的传统特色。
如果说这可能是一个世袭的缺陷，我倒不会不赞成，不过我还是愿意强调那个从
Poincaré那儿借来的“蛋糕与苹果”的事实。
构造数学理论的方式与其它的自然科学并没有什么不同。首先，我们要考虑一
些对象并对一些特殊的事例进行观察。接着我们试图要找到一些我们所观察到的结
果在应用上的限制，即寻找那些防止我们不正确地把我们所观察的结果扩展到更广
泛领域的反例。作为一个结果我们尽可能地明确提出那由经验得来的发现（如费马
猜想和庞加莱猜想）。这之后将是检验我们的结论到底有多可靠的困难的阶段。
就这一点来说，数学界已经发展出了一套特别的技术。这种技术，当被运用于
现实世界时，有时候很有用，但有时候也会导致自欺欺人。这样的技术被称为“建
模”。当构造一个模型时，要进行如下的理想化：某些只能以一定概率或一定的精
确性了解的事实，往往被认为是“绝对”正确的并被当作“公理”来接受。这种“
绝对性”的意义恰恰是，在把所有我们可以借助这些事实得出的结论称为定理的过
程中，我们允许自己依据形式逻辑的规则来运用这些“事实”。
显然在任何现实的日常生活中，我们的活动要完全依赖于这样的化减是不可能
的。原因至少在于所研究的现象的参数决不可能被绝对准确的知晓，并且参数的微
小变化（例如一个过程初始条件的微小改变）就会完全地改变结果。由于这个原因
我们可以说任何长期的天气预报都是不可能的，无论我们把计算机造的有多高级或
是记录初始条件地仪器有多灵敏，这永远也办不到。
与此完全一样的是，公理（那些我们不能完全确定的）的一个小小的改变虽是
容许的，一般来说，由那些被接受的公理推出的定理却将导出完全不同的结论。推
导的链（即所谓的“证明”）越长越复杂，最后得到的结论可靠性越低。复杂的模
型几乎毫无用处（除了对那些无聊的专写论文的人）。
数学建模的技术对这种麻烦一无所知，并且还不断地吹嘘他们得到的模型，似
乎它们真的就与现实世界吻合。事实上，从自然科学的观点看, 这种途径是显然不
正确的，但却经常导致很多物理上有用的被称为“有不可思议的有效性的数学”结
果（或叫做“Wigner原理”）。
我在此再提一下盖尔方德先生的一句话：还有另一类现象与以上Wigner所指的
物理中的数学具有相仿的不可思议的有效性，即生物学中用到的数学也是同样令人
难以置信的有效。
对一个物理学家而言，“数学教育所致的不易察觉的毒害作用”（F.Klein 原
话）恰恰体现在由现实世界抽离出的被绝对化了的模型，并且它与现实已不再相符。
这儿是一个简单的例子：数学知识告诉我们 Malthus 方程 dx/dt = x 的解是由初
始条件唯一决定的（也即相应的位于（t-x）－平面上积分曲线彼此不交）。这个数
学模型的结论显得与现实世界毫不相关。而计算机模拟却显示所有这些积分曲线在
t的负半轴上有公共点。事实上，具有初始条件 x(0) = 0 和 x(0) = 1的曲线在t=
-100 相交，其实在t=-100 时，你压根就不可能在两条曲线之间再插入一个原子。
欧式几何对这种空间在微小距离下的性质没有任何的描述。在这种情况下来应用唯
一性定理显然已经超出了模型所能容许的精确程度。在对模型的实际运用中，这种
情形必须要加以注意，否则可能会导致严重的麻烦。
我还想说的是，相同的唯一性定理也可解释为何在船只停泊码头前的靠岸阶段
必须得依靠人工操作：否则的话，如果行进的速度是距离的光滑（线性）函数，则
整个靠岸的过程将会耗费无穷长的时间。而另外可行的方法则是与码头相撞（当然
船与码头之间要有非理想弹性物体以造成缓冲）。顺便说一下，我们必须非常重视
这类问题，例如，登陆月球和火星以及空间站的对接――此时唯一性问题都会让我
们头痛。
不幸的是，在现代数学的教科书里，即使是较好的一类课本里，对这种令人崇
拜的定理所隐藏的危险的事例或探讨都只字没有。我甚至已经形成了这样的印象，
那些学院派的数学家（对物理知识都一窍不通）都对公理化形式的数学与建模的主
要差异习以为常，而且他们觉得在自然科学中这是很普遍的，只是需要用后期的实
验来控制理论推演。
我想用不着去提什么初始公理的相对特征，人们也都不会忘记在冗长的论述里
犯逻辑错误是在所难免的（彷佛宇宙射线或量子振动所引发的计算崩溃）。每一个
还在工作的数学家都知道，如果不对自己有所控制（最好是用事例），那么在10页
论述之后所有公式中的记号有半数都会出问题。
与这样的谬误相抗的技术也同样存在于任何实验科学里，而且应该教给每一个
大学低年级的学生。
试图创造所谓的纯粹推导式的公理化数学的做法，使得我们不再运用物理学中
的研究方法（观察-建模－模型的研究－得出结论－用更多的观察检验模型）取而代
之的是这样的方法：定义－定理－证明。人们根本不可能理解一个毫无动机的定义，
但我们却无法阻止这些有罪的“代数－公理学家”。例如，他们总是想用长乘规则
的手段来定义自然数的乘积。但用这种方法乘法的交换性却难以证明，不过从一堆
的公理中仍有可能推导出这样的定理。而且完全可能逼着那些可怜的学生们来学习
这个定理以及它的证明（其目的不外乎是提升这门学科以及教授它的人的社会地位
）。显然，这种定义和这样的证明对教学和实际工作有百害而无一益。
理解乘法交换性的唯一可能的方式，打个比方就是分别按行序和列序来数一个
方阵里士兵的人数，或者说用两种方式来计算长方形的面积。任何试图只做不与物
理和现实世界打交道的数学都属于宗派主义和孤立主义，这必将损毁在所有敏感的
人们眼中把数学创造视为一项有用的人类活动的美好印象。
我将再揭示几个这样的秘密（可怜的学生们对此很有兴趣）。
一个矩阵的行列式就是一个平行多面体的（定向的）体积，这个多面体的每条
边就对应矩阵的列。如果学生们得知了这个秘密（在纯粹的代数式的教育中，这个
秘密被仔细地隐藏了起来），那么行列式的整个理论都将成为多线性形式理论的一
部分。如果用别的方式来定义行列式，则任何敏感的人都将会永远恨死了诸如行列
式，Jacobi式，以及隐函数定理这些鬼东西。
一个群又是什么东东呢？代数学家们会这样来教学：这是一个假设的集合，具
有两种运算，它们满足一组容易让人忘记的公理。这个定义很容易激起一个自然的
抗议：任何一个敏感的人为何会需要这一对运算？“哦，这种数学去死吧”――这
就是学生的反应（他很可能将来就成为了科学强人）。
如果我们的出发点不是群而是变换的概念（一个集合到自身的1－1映射），则
我们绝对将得到不同的局面，这也才更像历史的发展。所有变换的集合被称为一个
群，其中任何两个变换的复合仍在此集合内并且每个变换的逆变换也如此。
这就是定义的关键所在。那所谓的“公理”事实上不过是变换群所具有的显然
的性质。公理化的倡导者所称的“抽象群”不过是在允许相差同构（保持运算的一
一映射）意义下的不同集合的变换群。正如 Cayley证明的，在这个世界上根本就没
有“更抽象的”群。那么为什么那些代数学家仍要用抽象的定义来折磨这些饱受痛
苦的学生们呢？
顺便提一句，在上世纪60年代我曾给莫斯科的中小学生们讲授群论。我回避了
任何的公理，尽可能的让内容贴近物理，在半年内我就教给了他们关于一般的五次
方程不可解性的Abel 定理（以同样的方式，我还教给了小学生们复数，黎曼曲面，
基本群以及代数函数的monodromy 群）。这门课程的内容后来由我的一个听众
V. Alekseev 组织出版了，名为The Abel theorem in problems.
一个光滑流形又是什么东东呢？最近我从一本美国人的书中得知庞加莱对此概
念并不精通（尽管是由他引入的），而所谓“现代的”定义直到上世纪20年代才由
Veblen给出：一个流形是一个拓扑空间满足一长串的公理。
学生们到底犯了什么罪过必须经受这些扭曲和变形的公理的折磨来理解这个概
念？事实上，在庞加莱的原著《位置分析》（Analysis Situs）中，有一个光滑流
形的绝对清晰的定义，它要比这种抽象的玩意儿有用的多。
一个欧式空间R^N 中的k-维光滑子流形是一个这样的子集，其每一点的一个邻
域是一个从R^k到R^(N-k)的光滑映射的图象（其中R^k 和 R^(N - k) 是坐标子空间
）。这样的定义是对平面上大多数通常的光滑曲线（如 圆环 x^2 + y^2 = 1）或
三维空间中曲线和曲面的直接的推广。
光滑流形之间的光滑映射则是自然定义的。所谓微分同胚则是光滑的映射且其
逆也光滑。
而所谓“抽象的”光滑流形就是欧式空间的允许相差一个微分同胚意义下的光
滑子流形。世界上根本不存在所谓“更抽象的”有限维的光滑流形（Whitney 定理
）。为什么我们总是要用抽象的定义来折磨学生们呢？把闭二维流形（曲面）的分
类定理证给学生们看不是更好吗？恰恰是这样的精彩定理（即任何紧的连通的可定
向的曲面都是一个球面外加若干个环柄似的把手）使我们对现代数学是什么有了一
个正确的印象，相反的是，那些对欧式空间的简单的子流形所做的超级抽象的推广，
事实上压根没有给出任何新的东东，不过是用来展示一下那些公理化学者们成就的
蹩脚货。
对曲面的分类定理是顶级的数学成就，堪与美洲大陆或X 射线的发现媲美。这
是数学科学里一个真正的发现，我们甚至难以说清到底所发现的这个事实本身对物
理学和数学哪一个的贡献更大。它对应用以及对发展正确的世界观的非凡意义目前
已超越了数学中的其他的“成就”，诸如对费马大定理的证明，以及对任何充分大
的整数都能表示成三个素数和这类事实的证明。为了出风头，当代的数学家有时候
总要展示一些“运动会式的”成就，并声称那就是他们的学科里最后的难题。可想
而知，这样的做法不仅无助于社会对数学的欣赏，而且恰恰相反，会使人们产生怀
疑：对于这样的毫无用处的跳脱衣舞般的问题，有必要耗费能量来做这些（彷佛攀
岩似的）练习吗？
曲面的分类定理应该被包含在高中数学的课程里（可以不用证明），但不知为
什么就连大学数学的课程里也找不到（顺便一下，在法国近几十年来说有的几何课
程都被禁止）。
在各个层次上，数学教育由学院的特征转回到表述自然科学的重要性的特征，对
法国而言是一个及其热点的问题。使我感到很震惊的是那些最好的也是最重要的条
理清晰的数学书，在这儿几乎都不为学生们所知（而依我看它们还没有被译成法语
）。这些书中有Rademacher和Touml写的《Numbers and figures》；Hilbert和
Cohn-Vossen写的《plitz, Geometry and the imagination》；Courant和Robbins
写的《What ismathematics?》；Polya写的《How to solve it》和《Mathematics
and plausiblereasoning 》； F. Klein写的《Development of mathematics in
the19th century 》。
我清晰地记得在学校时，Hermite 写的微积分教程（有俄语译本）给我留下了
多么强烈的印象。我记得在其最开始的一篇讲义中就出现了黎曼曲面（当然所有分
析的内容都是针对复变量的，也本该如此）。而积分渐进的内容是通过黎曼曲面上
道路形变的方法来研究（如今，我们称此方法为Picard-Lefschetz 理论;顺便提一
下，Picard是Hermite的女婿――数学能力往往是由女婿来传承：Hadamard-P. Levy
-L. Schwarz-U. Frisch这个王朝就是巴黎科学院中另一个这样的范例）。
由Hermite 一百多年前所写的所谓的“过时的”教程（也许早就被法国大学的
学生图书馆当垃圾扔掉了）实际上要比那些如今折磨学生们的最令人厌烦的微积分
课本现代化的多。
如果数学家们再不睡醒，那么那些对现代的（最正面意义上的）数学理论仍有
需要，同时又对那些毫无用处的公理化特征具有免疫力（这是任何敏锐的人所具有
的特征）的消费者们会毫不犹豫的将这些学校里的受教育不足的学究们扫地出门。
一个数学教师，如果至今还没有掌握至少几卷Landau 和 Lifshitz 著的物理学
教程，他（她）必将成为一个数学界的希罕的残存者，就好似如今一个仍不知道开
集与闭集差别的人。
--------------------
On teaching mathematics
by V.I. Arnold
Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.
The Jacobi identity (which forces the heights of a triangle to cross at one point) is an experimental fact in the same way as that the Earth is round (that is, homeomorphic to a ball). But it can be discovered with less expense.
In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy's warning that ugly mathematics has no permanent place under the Sun).
Since scholastic mathematics that is cut off from physics is fit neither for teaching nor for application in any other science, the result was the universal hate towards mathematicians - both on the part of the poor schoolchildren (some of whom in the meantime became ministers) and of the users.
The ugly building, built by undereducated mathematicians who were exhausted by their inferiority complex and who were unable to make themselves familiar with physics, reminds one of the rigorous axiomatic theory of odd numbers. Obviously, it is possible to create such a theory and make pupils admire the perfection and internal consistency of the resulting structure (in which, for example, the sum of an odd number of terms and the product of any number of factors are defined). From this sectarian point of view, even numbers could either be declared a heresy or, with passage of time, be introduced into the theory supplemented with a few "ideal" objects (in order to comply with the needs of physics and the real world).
Unfortunately, it was an ugly twisted construction of mathematics like the one above which predominated in the teaching of mathematics for decades. Having originated in France, this pervertedness quickly spread to teaching of foundations of mathematics, first to university students, then to school pupils of all lines (first in France, then in other countries, including Russia).
To the question "what is 2 + 3" a French primary school pupil replied: "3 + 2, since addition is commutative". He did not know what the sum was equal to and could not even understand what he was asked about!
Another French pupil (quite rational, in my opinion) defined mathematics as follows: "there is a square, but that still has to be proved".
Judging by my teaching experience in France, the university students' idea of mathematics (even of those taught mathematics at the 蒫ole Normale Sup閞ieure - I feel sorry most of all for these obviously intelligent but deformed kids) is as poor as that of this pupil.
For example, these students have never seen a paraboloid and a question on the form of the surface given by the equation xy = z^2 puts the mathematicians studying at ENS into a stupor. Drawing a curve given by parametric equations (like x = t^3 - 3t, y = t^4 - 2t^2) on a plane is a totally impossible problem for students (and, probably, even for most French professors of mathematics).
Beginning with l'Hospital's first textbook on calculus ("calculus for understanding of curved lines") and roughly until Goursat's textbook, the ability to solve such problems was considered to be (along with the knowledge of the times table) a necessary part of the craft of every mathematician.
Mentally challenged zealots of "abstract mathematics" threw all the geometry (through which connection with physics and reality most often takes place in mathematics) out of teaching. Calculus textbooks by Goursat, Hermite, Picard were recently dumped by the student library of the Universities Paris 6 and 7 (Jussieu) as obsolete and, therefore, harmful (they were only rescued by my intervention).
ENS students who have sat through courses on differential and algebraic geometry (read by respected mathematicians) turned out be acquainted neither with the Riemann surface of an elliptic curve y^2 = x^3 + ax + b nor, in fact, with the topological classification of surfaces (not even mentioning elliptic integrals of first kind and the group property of an elliptic curve, that is, the Euler-Abel addition theorem). They were only taught Hodge structures and Jacobi varieties!
How could this happen in France, which gave the world Lagrange and Laplace, Cauchy and Poincar? Leray and Thom? It seems to me that a reasonable explanation was given by I.G. Petrovskii, who taught me in 1966: genuine mathematicians do not gang up, but the weak need gangs in order to survive. They can unite on various grounds (it could be super-abstractness, anti-Semitism or "applied and industrial" problems), but the essence is always a solution of the social problem - survival in conditions of more literate surroundings.
By the way, I shall remind you of a warning of L. Pasteur: there never have been and never will be any "applied sciences", there are only applications of sciences (quite useful ones!).
In those times I was treating Petrovskii's words with some doubt, but now I am being more and more convinced of how right he was. A considerable part of the super-abstract activity comes down simply to industrialising shameless grabbing of discoveries from discoverers and then systematically assigning them to epigons-generalizers. Similarly to the fact that America does not carry Columbus's name, mathematical results are almost never called by the names of their discoverers.
In order to avoid being misquoted, I have to note that my own achievements were for some unknown reason never expropriated in this way, although it always happened to both my teachers (Kolmogorov, Petrovskii, Pontryagin, Rokhlin) and my pupils. Prof. M. Berry once formulated the following two principles:
The Arnold Principle. If a notion bears a personal name, then this name is not the name of the discoverer.
The Berry Principle. The Arnold Principle is applicable to itself.
Let's return, however, to teaching of mathematics in France.
When I was a first-year student at the Faculty of Mechanics and Mathematics of the Moscow State University, the lectures on calculus were read by the set-theoretic topologist L.A. Tumarkin, who conscientiously retold the old classical calculus course of French type in the Goursat version. He told us that integrals of rational functions along an algebraic curve can be taken if the corresponding Riemann surface is a sphere and, generally speaking, cannot be taken if its genus is higher, and that for the sphericity it is enough to have a sufficiently large number of double points on the curve of a given degree (which forces the curve to be unicursal: it is possible to draw its real points on the projective plane with one stroke of a pen).
These facts capture the imagination so much that (even given without any proofs) they give a better and more correct idea of modern mathematics than whole volumes of the Bourbaki treatise. Indeed, here we find out about the existence of a wonderful connection between things which seem to be completely different: on the one hand, the existence of an explicit expression for the integrals and the topology of the corresponding Riemann surface and, on the other hand, between the number of double points and genus of the corresponding Riemann surface, which also exhibits itself in the real domain as the unicursality.
Jacobi noted, as mathematics' most fascinating property, that in it one and the same function controls both the presentations of a whole number as a sum of four squares and the real movement of a pendulum.
These discoveries of connections between heterogeneous mathematical objects can be compared with the discovery of the connection between electricity and magnetism in physics or with the discovery of the similarity between the east coast of America and the west coast of Africa in geology.
The emotional significance of such discoveries for teaching is difficult to overestimate. It is they who teach us to search and find such wonderful phenomena of harmony of the Universe.
The de-geometrisation of mathematical education and the divorce from physics sever these ties. For example, not only students but also modern algebro-geometers on the whole do not know about the Jacobi fact mentioned here: an elliptic integral of first kind expresses the time of motion along an elliptic phase curve in the corresponding Hamiltonian system.
Rephrasing the famous words on the electron and atom, it can be said that a hypocycloid is as inexhaustible as an ideal in a polynomial ring. But teaching ideals to students who have never seen a hypocycloid is as ridiculous as teaching addition of fractions to children who have never cut (at least mentally) a cake or an apple into equal parts. No wonder that the children will prefer to add a numerator to a numerator and a denominator to a denominator.
From my French friends I heard that the tendency towards super-abstract generalizations is their traditional national trait. I do not entirely disagree that this might be a question of a hereditary disease, but I would like to underline the fact that I borrowed the cake-and-apple example from Poincar?
The scheme of construction of a mathematical theory is exactly the same as that in any other natural science. First we consider some objects and make some observations in special cases. Then we try and find the limits of application of our observations, look for counter-examples which would prevent unjustified extension of our observations onto a too wide range of events (example: the number of partitions of consecutive odd numbers 1, 3, 5, 7, 9 into an odd number of natural summands gives the sequence 1, 2, 4, 8, 16, but then comes 29).
As a result we formulate the empirical discovery that we made (for example, the Fermat conjecture or Poincar?conjecture) as clearly as possible. After this there comes the difficult period of checking as to how reliable are the conclusions .
At this point a special technique has been developed in mathematics. This technique, when applied to the real world, is sometimes useful, but can sometimes also lead to self-deception. This technique is called modelling. When constructing a model, the following idealisation is made: certain facts which are only known with a certain degree of probability or with a certain degree of accuracy, are considered to be "absolutely" correct and are accepted as "axioms". The sense of this "absoluteness" lies precisely in the fact that we allow ourselves to use these "facts" according to the rules of formal logic, in the process declaring as "theorems" all that we can derive from them.
It is obvious that in any real-life activity it is impossible to wholly rely on such deductions. The reason is at least that the parameters of the studied phenomena are never known absolutely exactly and a small change in parameters (for example, the initial conditions of a process) can totally change the result. Say, for this reason a reliable long-term weather forecast is impossible and will remain impossible, no matter how much we develop computers and devices which record initial conditions.
In exactly the same way a small change in axioms (of which we cannot be completely sure) is capable, generally speaking, of leading to completely different conclusions than those that are obtained from theorems which have been deduced from the accepted axioms. The longer and fancier is the chain of deductions ("proofs"), the less reliable is the final result.
Complex models are rarely useful (unless for those writing their dissertations).
The mathematical technique of modelling consists of ignoring this trouble and speaking about your deductive model in such a way as if it coincided with reality. The fact that this path, which is obviously incorrect from the point of view of natural science, often leads to useful results in physics is called "the inconceivable effectiveness of mathematics in natural sciences" (or "the Wigner principle").
Here we can add a remark by I.M. Gel'fand: there exists yet another phenomenon which is comparable in its inconceivability with the inconceivable effectiveness of mathematics in physics noted by Wigner - this is the equally inconceivable ineffectiveness of mathematics in biology.
"The subtle poison of mathematical education" (in F. Klein's words) for a physicist consists precisely in that the absolutised model separates from the reality and is no longer compared with it. Here is a simple example: mathematics teaches us that the solution of the Malthus equation dx/dt = x is uniquely defined by the initial conditions (that is that the corresponding integral curves in the (t,x)-plane do not intersect each other). This conclusion of the mathematical model bears little relevance to the reality. A computer experiment shows that all these integral curves have common points on the negative t-semi-axis. Indeed, say, curves with the initial conditions x(0) = 0 and x(0) = 1 practically intersect at t = -10 and at t = -100 you cannot fit in an atom between them. Properties of the space at such small distances are not described at all by Euclidean geometry. Application of the uniqueness theorem in this situation obviously exceeds the accuracy of the model. This has to be respected in practical application of the model, otherwise one might find oneself faced with serious troubles.
I would like to note, however, that the same uniqueness theorem explains why the closing stage of mooring of a ship to the quay is carried out manually: on steering, if the velocity of approach would have been defined as a smooth (linear) function of the distance, the process of mooring would have required an infinitely long period of time. An alternative is an impact with the quay (which is damped by suitable non-ideally elastic bodies). By the way, this problem had to be seriously confronted on landing the first descending apparata on the Moon and Mars and also on docking with space stations - here the uniqueness theorem is working against us.
Unfortunately, neither such examples, nor discussing the danger of fetishising theorems are to be met in modern mathematical textbooks, even in the better ones. I even got the impression that scholastic mathematicians (who have little knowledge of physics) believe in the principal difference of the axiomatic mathematics from modelling which is common in natural science and which always requires the subsequent control of deductions by an experiment.
Not even mentioning the relative character of initial axioms, one cannot forget about the inevitability of logical mistakes in long arguments (say, in the form of a computer breakdown caused by cosmic rays or quantum oscillations). Every working mathematician knows that if one does not control oneself (best of all by examples), then after some ten pages half of all the signs in formulae will be wrong and twos will find their way from denominators into numerators.
The technology of combatting such errors is the same external control by experiments or observations as in any experimental science and it should be taught from the very beginning to all juniors in schools.
Attempts to create "pure" deductive-axiomatic mathematics have led to the rejection of the scheme used in physics (observation - model - investigation of the model - conclusions - testing by observations) and its substitution by the scheme: definition - theorem - proof. It is impossible to understand an unmotivated definition but this does not stop the criminal algebraists-axiomatisators. For example, they would readily define the product of natural numbers by means of the long multiplication rule. With this the commutativity of multiplication becomes difficult to prove but it is still possible to deduce it as a theorem from the axioms. It is then possible to force poor students to learn this theorem and its proof (with the aim of raising the standing of both the science and the persons teaching it). It is obvious that such definitions and such proofs can only harm the teaching and practical work.
It is only possible to understand the commutativity of multiplication by counting and re-counting soldiers by ranks and files or by calculating the area of a rectangle in the two ways. Any attempt to do without this interference by physics and reality into mathematics is sectarianism and isolationism which destroy the image of mathematics as a useful human activity in the eyes of all sensible people.
I shall open a few more such secrets (in the interest of poor students).
The determinant of a matrix is an (oriented) volume of the parallelepiped whose edges are its columns. If the students are told this secret (which is carefully hidden in the purified algebraic education), then the whole theory of determinants becomes a clear chapter of the theory of poly-linear forms. If determinants are defined otherwise, then any sensible person will forever hate all the determinants, Jacobians and the implicit function theorem.
What is a group? Algebraists teach that this is supposedly a set with two operations that satisfy a load of easily-forgettable axioms. This definition provokes a natural protest: why would any sensible person need such pairs of operations? "Oh, curse this maths" - concludes the student (who, possibly, becomes the Minister for Science in the future).
We get a totally different situation if we start off not with the group but with the concept of a transformation (a one-to-one mapping of a set onto itself) as it was historically. A collection of transformations of a set is called a group if along with any two transformations it contains the result of their consecutive application and an inverse transformation along with every transformation.
This is all the definition there is. The so-called "axioms" are in fact just (obvious) properties of groups of transformations. What axiomatisators call "abstract groups" are just groups of transformations of various sets considered up to isomorphisms (which are one-to-one mappings preserving the operations). As Cayley proved, there are no "more abstract" groups in the world. So why do the algebraists keep on tormenting students with the abstract definition?
By the way, in the 1960s I taught group theory to Moscow schoolchildren. Avoiding all the axiomatics and staying as close as possible to physics, in half a year I got to the Abel theorem on the unsolvability of a general equation of degree five in radicals (having on the way taught the pupils complex numbers, Riemann surfaces, fundamental groups and monodromy groups of algebraic functions). This course was later published by one of the audience, V. Alekseev, as the book The Abel theorem in problems.
What is a smooth manifold? In a recent American book I read that Poincar?was not acquainted with this (introduced by himself) notion and that the "modern" definition was only given by Veblen in the late 1920s: a manifold is a topological space which satisfies a long series of axioms.
For what sins must students try and find their way through all these twists and turns? Actually, in Poincar?s Analysis Situs there is an absolutely clear definition of a smooth manifold which is much more useful than the "abstract" one.
A smooth k-dimensional submanifold of the Euclidean space R^N is its subset which in a neighbourhood of its every point is a graph of a smooth mapping of R^k into R^(N - k) (where R^k and R^(N - k) are coordinate subspaces). This is a straightforward generalization of most common smooth curves on the plane (say, of the circle x^2 + y^2 = 1) or curves and surfaces in the three-dimensional space.
Between smooth manifolds smooth mappings are naturally defined. Diffeomorphisms are mappings which are smooth, together with their inverses.
An "abstract" smooth manifold is a smooth submanifold of a Euclidean space considered up to a diffeomorphism. There are no "more abstract" finite-dimensional smooth manifolds in the world (Whitney's theorem). Why do we keep on tormenting students with the abstract definition? Would it not be better to prove them the theorem about the explicit classification of closed two-dimensional manifolds (surfaces)?
It is this wonderful theorem (which states, for example, that any compact connected oriented surface is a sphere with a number of handles) that gives a correct impression of what modern mathematics is and not the super-abstract generalizations of naive submanifolds of a Euclidean space which in fact do not give anything new and are presented as achievements by the axiomatisators.
The theorem of classification of surfaces is a top-class mathematical achievement, comparable with the discovery of America or X-rays. This is a genuine discovery of mathematical natural science and it is even difficult to say whether the fact itself is more attributable to physics or to mathematics. In its significance for both the applications and the development of correct Weltanschauung it by far surpasses such "achievements" of mathematics as the proof of Fermat's last theorem or the proof of the fact that any sufficiently large whole number can be represented as a sum of three prime numbers.
For the sake of publicity modern mathematicians sometimes present such sporting achievements as the last word in their science. Understandably this not only does not contribute to the society's appreciation of mathematics but, on the contrary, causes a healthy distrust of the necessity of wasting energy on (rock-climbing-type) exercises with these exotic questions needed and wanted by no one.
The theorem of classification of surfaces should have been included in high school mathematics courses (probably, without the proof) but for some reason is not included even in university mathematics courses (from which in France, by the way, all the geometry has been banished over the last few decades).
The return of mathematical teaching at all levels from the scholastic chatter to presenting the important domain of natural science is an espessially hot problem for France. I was astonished that all the best and most important in methodical approach mathematical books are almost unknown to students here (and, seems to me, have not been translated into French). Among these are Numbers and figures by Rademacher and T鰌litz, Geometry and the imagination by Hilbert and Cohn-Vossen, What is mathematics? by Courant and Robbins, How to solve it and Mathematics and plausible reasoning by Polya, Development of mathematics in the 19th century by F. Klein.
I remember well what a strong impression the calculus course by Hermite (which does exist in a Russian translation!) made on me in my school years.
Riemann surfaces appeared in it, I think, in one of the first lectures (all the analysis was, of course, complex, as it should be). Asymptotics of integrals were investigated by means of path deformations on Riemann surfaces under the motion of branching points (nowadays, we would have called this the Picard-Lefschetz theory; Picard, by the way, was Hermite's son-in-law - mathematical abilities are often transferred by sons-in-law: the dynasty Hadamard - P. Levy - L. Schwarz - U. Frisch is yet another famous example in the Paris Academy of Sciences).
The "obsolete" course by Hermite of one hundred years ago (probably, now thrown away from student libraries of French universities) was much more modern than those most boring calculus textbooks with which students are nowadays tormented.
If mathematicians do not come to their senses, then the consumers who preserved a need in a modern, in the best meaning of the word, mathematical theory as well as the immunity (characteristic of any sensible person) to the useless axiomatic chatter will in the end turn down the services of the undereducated scholastics in both the schools and the universities.
A teacher of mathematics, who has not got to grips with at least some of the volumes of the course by Landau and Lifshitz, will then become a relict like the one nowadays who does not know the difference between an open and a closed set.
V.I. Arnold
Translated by A.V. GORYUNOV
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