（俄语）原作者: V. I. Arnold
俄译英：A. V. Goryunov
英译中：Azure OpenAI GPT-3.5-turbo-16k (2023-08-01-preview)
On teaching mathematics
by V. I. Arnold
This is an extended text of the address at the discussion on teaching of mathematics in Palais de Découverte in Paris on 7 March 1997.
这是1997年3月7日在巴黎探索宫举行的有关数学教学讨论会上的演讲的扩展文本。
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.
数学是物理学的一部分。物理学是一门实验科学，是自然科学的一部分。数学是物理学中实验成本较低的部分。
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.
雅可比恒等式（它强制一个三角形的高度在一个点相交）与地球是圆的（即，同胚于一个球）一样，都是实验事实。但是它可以用更少的费用来发现。
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).
在二十世纪中叶，有人试图将物理学和数学分开。结果证明是灾难性的。整整一代数学家在不了解自己学科的一半的情况下成长起来，当然也对其他科学一无所知。他们首先开始向学生教授他们丑陋的学院派伪数学，然后传给了学童们（忘记了哈代的警告，即丑陋的数学在太阳下没有永久的位置）。
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.
由于与物理学脱节的学术数学既不适合教学，也无法应用于其他科学领域，结果导致了对数学家的普遍厌恶 - 无论是来自贫困学生（其中一些后来成为部长）还是用户。
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).
这座丑陋的建筑是由受过教育不足的数学家们建造的，他们因自卑感而筋疲力尽，无法使自己熟悉物理学。它让人想起了严谨的奇数公理理论。显然，可以创造这样的理论，并使学生们钦佩所得到结构的完美和内在一致性（例如，定义了奇数个项的和以及任意数量因子的乘积）。从这种教派观点来看，偶数可以被宣布为异端邪说，或者随着时间的推移，可以引入一些“理想”对象来补充理论（以满足物理学和现实世界的需求）。
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).
不幸的是，像上面那样扭曲的数学构建在数十年的数学教学中占主导地位。这种变态的现象起源于法国，迅速传播到数学基础教学中，首先是大学生，然后是各个学科的学生（首先是法国，然后是其他国家，包括俄罗斯）。
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!
对于问题“2 + 3等于多少”，一个法国小学生回答：“3 + 2，因为加法是可交换的。”他不知道这个和是多少，甚至无法理解被问到的是什么！
Another French pupil (quite rational, in my opinion) defined mathematics as follows: "there is a square, but that still has to be proved".
另一个法国学生（在我看来相当理性）将数学定义如下：“有一个正方形，但还需要证明”。
Judging by my teaching experience in France, the university students' idea of mathematics (even of those taught mathematics at the École Normale Supérieure - I feel sorry most of all for these obviously intelligent but deformed kids) is as poor as that of this pupil.
根据我在法国的教学经验来看，大学生对数学的理解（即使是在高等师范学校学习数学的学生——我最为遗憾的是这些明显聪明但受到扭曲的孩子们）与这个学生一样糟糕。
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).
例如，这些学生从未见过一个抛物面，而一个由方程xy = z^2给出的表面形式的问题让ENS的数学专业学生们陷入了困惑。对于学生们来说，在平面上绘制由参数方程给出的曲线（如x = t^3 - 3t，y = t^4 - 2t^2）是一个完全不可能的问题（甚至对于大多数法国数学教授来说也是如此）。
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.
从l'Hospital的第一本微积分教材（“理解曲线的微积分”）开始，大致上直到Goursat的教材，解决这类问题的能力被认为是每个数学家工艺技能的必要组成部分（连同乘法口诀的知识）。
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).
"抽象数学"的智力挑战者们将所有与物理和现实最常联系的几何学从教学中排除了。高尔萨、埃尔米特和皮卡尔的微积分教材最近被巴黎6和7大学（朱西厄）的学生图书馆丢弃，因为它们被认为过时且有害（只有在我的干预下才得以保留）。
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!
曾经上过微分几何和代数几何课程（由受人尊敬的数学家讲授）的ENS学生，事实上对于椭圆曲线y^2 = x^3 + ax + b的黎曼曲面以及曲面的拓扑分类一无所知（更不用提一阶椭圆积分和椭圆曲线的群性质，即欧拉-阿贝尔加法定理）。他们只学到了霍奇结构和雅可比变量！
How could this happen in France, which gave the world Lagrange and Laplace, Cauchy and Poincaré, 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.
这怎么可能发生在法国呢？法国给世界带来了拉格朗日和拉普拉斯、柯西和庞加莱、勒雷和汤姆。在我看来，I.G.彼得罗夫斯基在1966年给出了一个合理的解释：真正的数学家不会结成团伙，但是弱者需要团伙才能生存。他们可以因为各种原因联合起来（可能是超抽象性、反犹太主义或者“应用和工业”问题），但本质上都是为了解决社会问题——在更有文化素养的环境中生存。
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!).
顺便提醒一下，我要引用路易·巴斯德的警告：从来没有过，也永远不会有所谓的“应用科学”，只有科学的应用（非常有用的应用！）。
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.
在那个时候，我对彼得罗夫斯基的话有些怀疑，但现在我越来越确信他是多么正确。超抽象活动的相当一部分实际上只是将发现者的发现无耻地工业化，然后系统地归属给后来者和泛化者。就像美国不以哥伦布的名字命名一样，数学结果几乎从不以发现者的名字命名。
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:
为了避免被错误引用，我必须说明一下，我的个人成就出于某种未知原因从未被以这种方式侵占，尽管我的老师（科尔莫戈洛夫、彼得罗夫斯基、庞特亚金、罗赫林）和我的学生总是遭遇这种情况。M. Berry教授曾经提出了以下两个原则：
The Arnold Principle. If a notion bears a personal name, then this name is not the name of the discoverer.
阿诺德原则。如果一个概念带有一个人的名字，那么这个名字不是发现者的名字。
The Berry Principle. The Arnold Principle is applicable to itself.
贝瑞原则。阿诺德原则适用于自身。
Let's return, however, to teaching of mathematics in France.
让我们回到法国的数学教学。
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).
当我是莫斯科大学力学与数学系的一年级学生时，微积分课程由集合论拓扑学家L.A.图马金教授讲授，他认真地讲述了法国式的古典微积分课程，采用了古尔萨特版本。他告诉我们，有理函数沿着代数曲线的积分可以进行，如果相应的黎曼曲面是一个球面，而且一般情况下，如果曲面的层次更高，则无法进行积分。而要使曲线具有球面性质，只需要在给定次数的曲线上有足够多的双重点（这将使曲线成为单连通曲线：可以用一笔画在投影平面上的实点来表示）。
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.
这些事实如此引人入胜，以至于（即使没有任何证据）它们比整套布尔巴基著作更能给出关于现代数学的更好、更正确的理解。事实上，在这里我们发现了一种奇妙的联系存在于看似完全不同的事物之间：一方面，存在着对积分和相应黎曼曲面的拓扑的明确表达式，另一方面，存在着相应黎曼曲面的双重点数和层次的联系，这在实域中也表现为唯一性。
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.
雅可比指出，作为数学最迷人的特性，它能够用同一个函数控制整数作为四个平方数之和的表示以及钟摆的真实运动。
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.
这些对不同数学对象之间的联系的发现可以与物理学中电与磁的联系的发现或地质学中美洲东海岸与非洲西海岸的相似性的发现相媲美。
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.
这些发现对于教学的情感意义难以估量。正是它们教会我们去寻找和发现宇宙和谐的奇妙现象。
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.
数学教育的去几何化和与物理学的脱离断开了这些联系。例如，不仅学生，而且现代代数几何学家大体上都不知道这里提到的雅可比事实：第一类椭圆积分表示相应哈密顿系统中沿椭圆相轨道的运动时间。
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.
重新表述关于电子和原子的著名言辞，可以说一个内旋曲线和多项式环中的理想一样无穷无尽。但是教给从未见过内旋曲线的学生理想是多么荒谬，就像教给从未将蛋糕或苹果切成相等部分（至少在心里）的孩子分数相加一样。难怪孩子们更喜欢将分子加到分子上，将分母加到分母上。
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 Poincaré.
从我的法国朋友那里听说，对超抽象概括的倾向是他们传统的民族特质。我并不完全不同意这可能是一种遗传病的问题，但我想强调一下，我是从庞加莱那里借用了蛋糕和苹果的例子。
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).
建立数学理论的方案与任何其他自然科学的方案完全相同。首先，我们考虑一些对象并在特殊情况下进行观察。然后，我们尝试找到我们观察的适用范围的限制，寻找可能阻止我们将观察无理扩展到过于广泛事件范围的反例（例如：连续奇数1、3、5、7、9的分割数为奇数个自然数和的数量依次为1、2、4、8、16，但接下来是29）。
As a result we formulate the empirical discovery that we made (for example, the Fermat conjecture or Poincaré conjecture) as clearly as possible. After this there comes the difficult period of checking as to how reliable are the conclusions .
因此，我们尽可能清晰地阐述了我们所做的经验性发现（例如，费马猜想或庞加莱猜想）。之后，就进入了一个困难的阶段，需要检查结论的可靠性。
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.
在数学领域，已经发展出一种特殊的技术。这种技术在应用于现实世界时，有时很有用，但有时也可能导致自欺欺人。这种技术被称为建模。在构建模型时，我们进行了以下理想化：某些只有一定概率或一定准确度的事实被认为是“绝对”正确的，并被接受为“公理”。这种“绝对性”的意义在于我们允许自己根据形式逻辑的规则使用这些“事实”，并将我们从中推导出的一切都宣布为“定理”。
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.
很明显，在任何现实生活活动中，完全依赖这样的推论是不可能的。原因至少是，所研究现象的参数从来都不是绝对准确的，而且参数的微小变化（例如，过程的初始条件）可以完全改变结果。比如说，因为这个原因，可靠的长期天气预报是不可能的，无论我们发展多少计算机和记录初始条件的设备。
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.
就像我们无法完全确定的公理的微小变化一样，通常情况下，它能够导致完全不同于从已接受的公理推导出的定理所得出的结论。推导链（"证明"）越长越复杂，最终结果的可靠性就越低。
Complex models are rarely useful (unless for those writing their dissertations).
复杂的模型很少有用（除非是为了那些写论文的人）。
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").
数学建模技术的本质是忽略这个问题，并以一种方式来描述你的演绎模型，仿佛它与现实完全一致。尽管从自然科学的角度来看，这种方法显然是错误的，但事实上，它在物理学中经常导致有用的结果，这被称为“数学在自然科学中的难以置信的有效性”（或“维格纳原则”）。
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.
在这里，我们可以加上I.M. Gel'fand的一句话：存在着另一种现象，它与Wigner所指出的数学在物理学中的难以置信的有效性相媲美，这就是数学在生物学中同样难以置信的无效性。
"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.
"数学教育的微妙毒害"（用F. Klein的话来说）对于物理学家来说，就在于绝对化的模型与现实分离，不再与之进行比较。这里有一个简单的例子：数学告诉我们，马尔萨斯方程dx/dt = x的解是由初始条件唯一确定的（即在（t，x）平面上对应的积分曲线不相交）。这个数学模型的结论与现实关系不大。计算机实验表明，所有这些积分曲线在负t半轴上有共同点。确实，例如，具有初始条件x（0）= 0和x（0）= 1的曲线在t = -10和t = -100几乎相交，你无法在它们之间放入一个原子。这样小的距离下的空间特性根本无法用欧几里得几何来描述。在这种情况下应用唯一性定理显然超出了模型的准确性。在实际应用模型时必须尊重这一点，否则可能会面临严重的麻烦。
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.
然而，我想指出的是，同样的唯一性定理解释了为什么船只靠泊码头的最后阶段需要手动操作：如果在驾驶过程中，接近速度被定义为距离的平滑（线性）函数，那么靠泊的过程将需要无限长的时间。另一种选择是与码头发生碰撞（通过适当的非理想弹性物体进行减震）。顺便说一下，在将第一批降落器降落在月球和火星上以及与空间站对接时，这个问题必须得到认真解决 - 在这里，唯一性定理对我们起了反作用。
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.
不幸的是，即使在较好的现代数学教科书中，也很难找到这样的例子，也很少讨论过过度崇拜定理的危险。我甚至有一种印象，学院派数学家（对物理学知之甚少）相信公理化数学与自然科学中常见的建模之间存在根本性差异，后者总是需要通过实验来对推论进行后续验证。
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.
甚至不提初步公理的相对性质，也不能忘记在长篇论证中逻辑错误的必然性（比如由宇宙射线或量子振荡引起的计算机故障）。每个从事数学工作的人都知道，如果不加以控制（最好通过示例），那么在大约十页之后，公式中一半的符号都会出错，分母中的二会找到自己的位置进入分子中。
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.
对抗这类错误的技术与任何实验科学中的实验或观察一样，都是通过外部控制来实现的，应该从学校的初级阶段就开始教授给所有学生。
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.
试图创造“纯粹”的演绎公理数学导致了物理学中所使用的方案（观察-模型-对模型的研究-结论-通过观察进行测试）的被拒绝，并被定义-定理-证明的方案所取代。理解一个没有动机的定义是不可能的，但这并不能阻止那些犯罪的代数学家-公理化者。例如，他们会毫不犹豫地通过长乘法规则来定义自然数的乘积。这样一来，乘法的交换性变得难以证明，但仍然可以从公理中推导出它作为一个定理。然后可以迫使学生学习这个定理及其证明（目的是提高科学和教学人员的地位）。显然，这样的定义和证明只会对教学和实际工作造成伤害。
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.
只有通过按照军队的行列重新计算士兵的数量，或者通过计算矩形的面积两种方式，才能理解乘法的交换律。任何试图在数学中摒弃物理和现实干扰的做法都是教条主义和孤立主义，这会在明智人士眼中破坏数学作为一种有用的人类活动的形象。
I shall open a few more such secrets (in the interest of poor students).
我将公开一些类似的秘密（为了贫困学生的利益）。
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.
矩阵的行列式是由其列构成的平行六面体的（有向）体积。如果学生们被告知这个秘密（这个秘密在纯粹的代数教育中被小心隐藏），那么行列式的整个理论就变成了多线性形式理论中的一个清晰的章节。如果行列式被以其他方式定义，那么任何明智的人都会永远厌恶所有的行列式、雅可比行列式和隐函数定理。
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).
什么是群？代数学家教导我们，群是一个满足一系列容易被遗忘的公理的集合，其中包含两种运算。这个定义引发了自然的质疑：为什么任何明智的人都需要这样的运算对呢？“哦，该死的数学”——学生得出结论（也许将来成为科学部长）。
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.
如果我们从一个转换的概念开始（一个将集合映射到自身的一对一映射），而不是从群体开始，我们会得到完全不同的情况，这是历史上的情况。如果一个集合的一组变换包含它们连续应用的结果以及每个变换的逆变换，那么这个集合就被称为一个群体。
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?
这就是全部的定义。所谓的“公理”实际上只是变换群的（显而易见的）性质。公理化者所称的“抽象群”只是考虑到同构的各种集合上的变换群（同构是保持运算的一对一映射）。正如Cayley证明的那样，世界上没有“更抽象”的群。那么为什么代数学家还要用抽象的定义来折磨学生呢？
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.
顺便说一下，在1960年代，我教莫斯科的学生群论。避免所有的公理化，尽可能地与物理学保持接近，在半年的时间里，我讲解了阿贝尔定理，即关于用根式解一般五次方程的不可解性（在此过程中，我还教授了复数、黎曼曲面、代数函数的基本群和单值函数的单值群）。这门课程后来由其中一位听众V. Alekseev出版成书，名为《问题中的阿贝尔定理》。
What is a smooth manifold? In a recent American book I read that Poincaré 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.
什么是光滑流形？最近我读了一本美国的书，书中提到庞加莱并不熟悉这个（由他自己引入的）概念，而“现代”定义只是在20世纪20年代末由维布伦提出的：流形是满足一系列公理的拓扑空间。
For what sins must students try and find their way through all these twists and turns? Actually, in Poincaré's Analysis Situs there is an absolutely clear definition of a smooth manifold which is much more useful than the "abstract" one.
学生们为了什么罪过必须在这些曲折中找到自己的道路呢？实际上，在庞加莱的《分析拓扑学》中，有一个绝对清晰的光滑流形定义，比那个“抽象”的定义更加有用。
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.
欧几里德空间R^N中的一个光滑的k维子流形是其子集，在每个点的邻域内都是R^k到R^(N - k)的光滑映射的图像（其中R^k和R^(N - k)是坐标子空间）。这是对平面上最常见的光滑曲线（比如圆x^2 + y^2 = 1）或三维空间中的曲线和曲面的直接推广。
Between smooth manifolds smooth mappings are naturally defined. Diffeomorphisms are mappings which are smooth, together with their inverses.
在光滑流形之间，自然定义了光滑映射。微分同胚是光滑映射及其逆映射的组合。
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)?
一个“抽象”的光滑流形是指在一个欧几里得空间中考虑的光滑子流形，其等价于一个微分同胚。世界上没有“更抽象”的有限维光滑流形（惠特尼定理）。为什么我们要继续用抽象的定义折磨学生呢？证明一下关于闭合二维流形（曲面）的显式分类定理不是更好吗？
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.
正是这个奇妙的定理（例如，它表明任何紧致连通的定向曲面都是一个带有若干手柄的球体），给出了对现代数学的正确印象，而不是那些超抽象的对欧几里得空间中天真子流形的一般化描述，实际上并没有提供任何新的东西，而被公理化者们作为成就来呈现。
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.
分类表面定理是一项顶级的数学成就，可与发现美洲或X射线相媲美。这是数学自然科学的真正发现，甚至很难说这个事实本身更多归功于物理学还是数学学科。就其对应用和正确世界观发展的意义而言，它远远超过了数学中的“成就”，如费马最后定理的证明或任何足够大的整数可以表示为三个质数之和的事实的证明。
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.
为了宣传，现代数学家有时将这些体育成就呈现为他们学科的最高境界。可以理解的是，这不仅不有助于社会对数学的赞赏，反而引起了对于在这些无人需要和期望的奇特问题上浪费精力（类似攀岩的）锻炼的必要性的健康怀疑。
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).
分类表面定理本应该包含在高中数学课程中（可能不包括证明），但由于某种原因，甚至在大学数学课程中也没有包含（顺便说一下，在法国，几十年来几乎所有的几何学都被排除在外）。
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 Töplitz, 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.
数学教学在法国的各个层次上的回归，从学术闲谈到呈现自然科学的重要领域，是一个特别热门的问题。我很惊讶的是，这里的学生几乎不了解最好、最重要的方法论数学书籍（而且在我看来，这些书籍似乎没有被翻译成法语）。其中包括Rademacher和Töplitz的《数与图形》，Hilbert和Cohn-Vossen的《几何与想象》，Courant和Robbins的《什么是数学？》，Polya的《如何解决问题》和《数学与合理推理》，以及F. Klein的《19世纪数学的发展》。
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.
我还记得在我上学的那些年里，埃尔米特的微积分课程给我留下了深刻的印象（这门课程在俄语中有翻译！）。
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).
黎曼曲面在其中出现，我想，在最早的几堂课中（当然，所有的分析都是复数的，正如它应该是的）。通过在黎曼曲面上的路径变形来研究积分的渐近行为，这些路径变形是在分支点的运动下进行的（现在，我们会称之为皮卡-勒夫谢茨理论；顺便说一下，皮卡是埃尔米特的女婿 - 数学能力经常通过女婿传递：哈达玛德王朝 - P. 列维 - L. 施瓦茨 - U. 弗里施是巴黎科学院的另一个著名例子）。
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.
一百年前赫尔米特的“过时”课程（可能现在已经从法国大学的学生图书馆中被抛弃）比现在那些最无聊的微积分教科书要现代得多，而现在的学生们却被这些教科书所折磨。
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.
如果数学家们不醒悟过来，那么保持对现代数学理论的需求以及对无用公理闲谈的免疫力（任何明智之人都具备的特质）的消费者最终将拒绝在学校和大学使用未受良好教育的学究式服务。
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.
一个数学老师，如果连兰道和利夫希茨的课程中的一些内容都没有掌握，那么他将成为一个过时的人，就像现在有人不知道开集和闭集的区别一样。
V.I. Arnold
Translated by A.V. GORYUNOV