作者：William Lawvere
简介
美国数学家 William Lawvere，主要研究范畴论、拓扑学，他同时是一位马列主义者。他把黑格尔辩证法和毛泽东的矛盾论运用于范畴论、拓扑学的研究当中。在这篇文章中他探讨了资产阶级如何通过限制数学教育为神秘主义和迷信权威铺路
– Dr. William Lawvere, Inter-Disciplinary Conference, Windsor, February 10, 1996 –
博士在 1996 年温莎大学跨学科会议上的报告
For a long time calculus teachers have been demanding better textbooks. In recent months, after much fanfare, a few books claiming to respond to this demand have been launched. But many teachers are saying already that this "new" is very similar to the old that worked so poorly, except now with much more and elaborate and subtle attempts to persuade the student (and the teacher) that some kind of "understanding" is going on, even while all the more resolutely concealing the essential principles.
长久以来，微积分教师都想要更好的教材。近几个月又出现一批新书，大张旗鼓的宣传自己回应了这个需求。但许多教师表示这本“新书”依旧糟糕，无非用更加精心设计的话术说服师生，使其自以为产生了某种“理解”，结果却更顽固地掩盖基本原理。
Also in recent months, a journal for the guidance of college math teachers carried a lead editorial aiming to refute the "myth of scientific literacy". Under the banner of this "scientific literacy" a major curriculum-restructuring over the past decades had been justified, which many scientists judged to be a forced watering-down of content, aiming at producing students who had no understanding of science but who would be able to speak and write about it all the same. However now we are told that most students should not achieve even this "literacy", but only a much-reduced goal of "awareness". A current example (from the journal of a teachers union) of what such awareness means is the ability to just recognize terms such as "chaos theory" and "paradigm shift" and on that basis submit to whatever post-modernist reorganization is being proposed.
同样在最近几个月，一份指导大学数学教师的杂志发表了一篇社论，旨在驳斥“科学素养的迷思”。过去几十年来，各种课程在“科学素养”的旗帜下进行了大刀阔斧的改革。许多科学家认为这种改革刻意淡化教学内容，旨在培养对科学一无所知却能说会写的学生。结果我们现在得知，这些学生甚至连这种级别的“科学素养”都没必要达到，而只需要达到更水的目标“科学意识”就行了。关于这种“意识”的含义，目前的一个例子（来自一个教师联盟的期刊）是，学生只需有能力认出“混沌理论”和“范式转换”等术语，并基于此屈服于任何后现代主义重构就行了。
快乐教育
The claim that we are now in a "postmodern" and "playful" age is being put forward, by university centers for pedagogy, as a basis for replacing the learning of mathematics with "an inquiry into our habits". In an attempt to enlist some professors for this transformation, the old ecoterrorist claims, that mathematical thinking supports military technology and destruction of the environment, and that we "are" our mathematics, have been resurrected.
大学教育学中心声称我们正处于一个“后现代”和“玩乐”的时代，借此用“探究我们的习惯”来取代数学学习。为了争取一些教授参与这一转变，那种生态恐怖主义的陈词滥调，什么数学思维支持军事科技、环境破坏、什么我们“就是”我们的数学，又死灰复燃了。
田园派
Philosophical basis for these counter-reform "reforms" is also being offered in recent months. In U.S. and Canadian journals for college math teachers, it is revealed that mathematical ideas are social in character and that such social rituals constitute a third category of being, and we are assured that curriculum comes between the individual and the collective. At least two recent mathematics publications have praised the philosophers Kuhn, Lakatos, Feyerabend, and Popper, one saying that the philosophy of mathematics needs some great innovators in their mould, the other calling them exemplary "philosophers of Irrationalism ".
近几个月来，这些反改革的“改革”也获得了哲学背书。美、加面向大学数学教师的期刊“揭示”说，数学思想具有社会性，这种社会仪式构成了介于个人和集体之间的第三类存在，我们确信大学课程介于二者之间。最近，至少有两本数学刊物对库恩、拉卡托斯、费耶阿本德和波普尔等哲学家大加赞赏，其中一本说数学哲学需要他们这样的伟大创新者，另一本则称他们为 "非理性主义哲学家 "的典范。
The relation in 1996 between teaching of mathematics and philosophy of mathematics is only the latest stage in a long process. Three hundred years ago the Leibniz rule for the rate of change of a product, and Newton's theorem that the rate of change of the area under a curve is the height of the curve, were made explicit and publicized. This calculus was developed by Bernoulli and Euler, Cauchy and Maxwell into the universal instrument for designing engines, ships, electric power and communication systems, etc. Yet after three hundred years, most people, including most people who actually build those wonders, are still in no position to challenge these designs on their own grounds because knowledge of the instrument has been denied. Even more bizarre: the lack of comprehension is shared even by most people who have been through courses which the State has gone to considerable effort to provide.
1996 年的数学教学与数学哲学之间的关系只是一个漫长过程中的最新阶段。三百年前，关于乘积的变化率的莱布尼兹法则和关于曲线下面积的变化率等于曲线的高度的牛顿定理被明确提出并公布于众。这种计算方法【即微积分，英文直译就叫计算】被伯努利和欧拉、柯西和麦克斯韦发展为设计发动机、船舶、电力和通信系统等的通用工具。然而在三百年后的今天，大部分人——包括大部分实际建造这些奇迹的人，仍无法凭自己的理解来质疑这些设计，因为他们被剥夺了掌握这种工具的知识。更离谱的是：这种欠缺同样存在于大部分国家花心思教育的人身上。
There do exist professors who believe that most students are incapable and unwilling to learn any serious subject. But already around 1750 in Milan, Italy, Maria Agnesi wrote and printed a textbook, based on the premise that all Italian youth could and should learn calculus. Her enlightened vision is yet to be realized (in Italy or elsewhere). What happened?
确实有一些教授认为，大多数学生无力也不愿学习任何严肃学科。但早在 1750 年左右，意大利米兰的玛丽亚-阿格尼西（Maria Agnesi）就编写印刷了一本教科书，其初衷是所有意大利青年都能够也应该学习微积分。然而她的这一开明设想至今仍未实现（无论在意大利还是别的地方）。这是怎么回事？
After the French revolution had introduced the decimalized system, it was necessary to recalculate the trigonometric and logarithmic tables used in construction, navigation, etc. A few years ago the following true story about this was widely publicized by a giant multinational computer corporation: An accomplished engineer called Prony was appointed to organize the large task of calculation. Borrowing from Adam Smith, he divided the personnel into three levels: Level A consisted of a few mathematicians who were able to invent appropriate formulas, Level B of a somewhat larger group of people who were able to convert the formulas into algorithms, but a much larger group C of men actually carried out the algorithms by adding and multiplying. The point which the computer corporation found worthy of resurrecting two hundred years later was this: "It was found that the work went more smoothly if those in group C knew no mathematics."
法国大革命引入十进制后，建筑、航海等领域使用的三角表和对数表需要重新计算。几年前，一家跨国计算机巨头公司广泛宣传了下面这个真实的故事：一位名叫 Prony 的杰出工程师被任命组织这项庞大的计算任务。他借用亚当·斯密的说法，将人员分为三个级别：A 级由若干数学家组成，他们能够发明适当的公式；B 级由稍多的人组成，他们能将公式转化为算法；但实际由人数更多的 C 组用加法和乘法执行算法。这家公司认为值得在两百年后复活的观点是这样的：“我们发现，如果 C 组的人不懂数学，工作会更顺利。”
科学共同体底层民工从事机械的、无意识的符号生产
In Britain in the 1830's millions of people were in motion, demanding democracy. Among the measures for quelling the demands of this Chartist movement, the Privy Council created for the first time a system of state-supported schools and teacher-training and inspection, to "introduce order and discipline into the working-class population when older methods of wielding authority had broken down." (Encyclopedia Brittanica) The education provided by these new schools is consciously two- tiered: for example one of the defining documents requires that "Arithmetic is the Logic of the poor". In the school systems modeled on that, it is often apparent that while teachers are struggling to teach enough, administrative expenditures and regulations have the larger aim of insuring that we don't teach too much.
19 世纪 30 年代的英国，数百万人在运动中要求民主。为平息宪章派运动的要求，枢密院首次建立了一个由国家支持的学校体系，并对教师进行培训和检查，以“在旧的行使权力的方法已经失效的情况下，将秩序和纪律引入工人阶级。”（《不列颠百科全书》）这些新学校提供的教育有意识地分为两级：例如，其中一份界定文件要求“算术是穷人的逻辑学【Logic，大写指这门数学学科而非一般意义的逻辑】”。在以此为蓝本的学校制度中，显然一面是教师尽力教足知识，一面是财政和规章旨在确保我们别教太多。
In the period before 1848 there was optimism for the possibility of general enlightenment. For example, the Danish physicist Oersted, who discovered an important principle relating electricity and magnetism, set up an institution to make it known to all. The German mathematician Grassmann, who in 1844 published a new theory and method in geometry which now is becoming widely used by physicists, was actually a high-school teacher who insisted that his new dialectical philosophy was at least as important, since it was directed explicitly at assisting students to learn and understand. However, in the 1870's, when one of his followers published a book showing in detail how Grassmann's methods could be used to teach not only geometry, but also to introduce calculus in high-school, he received a very scathing review and condemnation for suggesting such an upset in the Prussian order of things; the author of the review, Felix Klein, was later the official representative of that Empire at the World's Fair held in conjunction with the opening of Rockefeller's University of Chicago.
1848 年之前，人们对普遍启蒙的可能性持乐观态度。例如，丹麦物理学家奥斯特，电与磁关联的发现者，成立了一个机构向所有人宣传这一发现。德国数学家格拉斯曼在 1844 年发表了几何学的新理论和新方法，现在已被物理学家广泛使用，但他实际上是一名中学教师，他坚持认为他的新辩证哲学至少同样重要，因为它明确旨在帮助学生学习和理解。可到了 1870 年代，当他的一名后继者出版了一本书，其中详细介绍了如何使用格拉斯曼的方法不仅可以教授几何，还可以把微积分引入高中时，他收到了一篇非常尖刻的评论和谴责，因为他提出了这样一个颠覆普鲁士秩序的建议；这篇评论的作者费利克斯·克莱因（Felix Klein）后来成为普鲁士帝国的官方代表，参加了与洛克菲勒的芝加哥大学开学典礼同时举行的世界博览会。
这个几何学的新理论和新方法其实就是向量空间、线性代数、矢量微积分。1861 年格拉斯曼还首次给出了算术公理，并熟练使用数学归纳法。但这些对当时的数学界而言都太过超前而不被接受，致使格拉斯曼转向研究语言学。
Following the Privy Council's lead there has been developed a body of technique, a sort of prize-fighter's technique for occasionally appearing to give in to the demands for reform while actually thereby directing our energies to serve an opposite aim. For example, forty years after Grassmann's death the Prussian establishment decided to make him "a great German soul" but he was portrayed in pragmatist journals as a philosophical idealist. In 1908 Lenin defended Grassmannn's materialist philosophy from this unwarranted distortion, and also remarked, in connection with some proposals to introduce higher mathematics into the schools, that it was surely not being done in order to deepen and broaden the knowledge of science, but rather to provide a basis for the promotion of idealist philosophy.
效仿英国枢密院的带头示范，一套阳奉阴违的技巧被发展起来了。例如，格拉斯曼去世四十年后，普鲁士当局决定把他塑造成“伟大的德国灵魂”，但他在实用主义刊物上却被描绘成一个哲学理想主义者。1908 年，列宁为格拉斯曼的唯物主义哲学辩护，使其免受这种无端歪曲。列宁还在谈到将高等数学引入学校的一些建议时指出，这种歪曲肯定不是为了深化和拓宽科学知识，而是为推广唯心主义哲学铺路。
熟悉吗？“当伟大的革命家在世时，压迫阶级总是不断迫害他们，以最恶毒的敌意、最疯狂的仇恨、最放肆的造谣和诽谤对待他们的学说。在他们逝世以后，便试图把他们变为无害的神像，可以说是把他们偶像化，赋予他们的名字某种荣誉，以便“安慰”和愚弄被压迫阶级，同时却阉割革命学说的内容，磨去它的革命锋芒，把它庸俗化。”——列宁《国家与革命》
Indeed, the popularizers of pragmatism and the organizers of collegiate math teaching were closely associated for many years, and the leading circles of philosophy, such as the Gifford Lectures in Scotland and the Silliman Lectures in Yale, began to systematically misuse mathematics and especially their audience's ignorance of mathematics. The Prime Minister of the British empire (who was nicknamed "bloody Balfour" for his suppression of the Irish and later would be famous for his declaration in support of Zionism) who wrote several books on philosophy, was also known for his Education Act which reorganized the high schools. Balfour stated in one of his Gifford lectures "I wish I were a mathematician".
事实上，实用主义的普及者和大学数学教学的组织者多年来一直保持着密切的联系，哲学界的领军圈子，如苏格兰的 Gifford 讲座和耶鲁大学的 Silliman 讲座，早就开始系统地滥用数学，尤其是滥用听众对数学的无知。大英帝国首相贝尔福（因镇压爱尔兰人而被冠以 "血腥贝尔福 "的绰号，后来又因支持犹太复国主义的宣言而闻名于世）写过多部哲学著作，也因其重组中学的《教育法案》而闻名于世。贝尔福在他的一次吉福德演讲中说：“我希望我是一名数学家。”
大言不惭
Indeed being known as a mathematician became a road to historical recognition as a philosopher. For example Bertrand Russell's opinions on everything became sought-for and he even eventually received the Nobel Prize, partly because of his notoriety as a mathematician; through clever wordplay he devised a new branch of philosophy known as "foundations of mathematics" whose only role is to give mathematics permission to exist, and which must be written in symbols different from the usual mathematical ones. The latter ruse he had learned from Peano, whose followers had proudly produced a high-school text written entirely in symbols, in order to dispel any false idea that with "numbers", "lines", or "space" we are really referring to anything; amazingly, this work was advertised as a clarification of Grassmann.
事实上，以数学家的身份闻名已成为历史上承认哲学家身份的一条道路。例如，伯特兰-罗素（Bertrand Russell）对任何事物的观点都受到追捧，他甚至最终获得了诺贝尔奖，部分原因就在于他作为数学家的恶名；通过巧妙的文字游戏，他设计了一个被称为 "数学基础 "的新哲学分支，其唯一作用就是为数学的存在颁布许可证，还非得用不同于通常数学的符号来书写——这是他从皮亚诺那里学来的，皮亚诺的追随者曾自豪地编写了一本完全用符号书写的中学课本，以消除任何错误的想法，即我们真的是在用 "数"、"线 "或 "空间 "指代任何东西；令人惊讶的是，这部著作被宣传为对格拉斯曼的澄清。
Perhaps the best-known 20th century figure who consciously guided his actions by the pragmatic philosophy was Mussolini; but probably as important was John Dewey, whose teachings and organizations had a tremendous influence on education throughout the world. He was occasionally quite clear about the direction of his reform; for example, in China in 1919 he gave a course at a college for teachers, in which he enunciated his infamous principle:
20 世纪自觉以实用主义哲学指导行动的最著名人物或许是墨索里尼，但同样重要的可能是约翰-杜威，他的学说和组织对全世界的教育产生了巨大影响。他有时对自己的改革方向十分明确；例如，1919 年他在中国的一所师范学院开设了一门课程，阐述了他臭名昭著的原则：
"Teach the child, not the science"!
"教孩子，而不是教科学"！
多研究些问题，少谈些“主义”！不愧是有其师必有其弟。
Of course conscientious teachers through the centuries have done both: the acquisition of some portion of the accumulated knowledge of humanity (science) is the purpose of the child's presence in school, but the teacher endeavors to guide this acquisition with due regard for each child's particular situation. Why then Dewey's prohibition of the teaching of the knowledge? In China he compared the alleged "authoritarianism" of science with the recently-overthrown imperial regime, and since then the broad-brush charge of "authoritarianism" has been used thousands of times as a pretext to eliminate from school systems the teaching of the deductive aspect in geometry, of the grammatical parts of speech, of the diagramming of sentences, etc. Indeed, now many college teachers of mathematics analyze that a large part of the difficulties which are had by students fresh from high-school are not due to mathematics itself, but due to their first real encounter with the requirement that ordinary language be used in a precise way. Dewey's powerful principle of "teaching the child, not the science" has many corollaries, such as the anti-child theory that "learning is fun", and ultimately the logic whereby jokes take the place of reasoning; certainly the principle includes the injunction often addressed to pupils: "Say it in your own words". This injunction is very attractive to teachers, who know that understanding requires a conscious act by the individual. However, the whole atmosphere of the school often mandates that "in your own words" should mean "as imprecisely as possible" thus destroying the acquisition of concepts in any usable form.
当然，几个世纪以来，有良知的教师在这两方面都做得很好：获取人类积累的部分知识（科学）是儿童在校学习的目的，但教师要努力根据每个儿童的具体情况引导他们获取知识。那么，杜威为什么禁止传授知识呢？在中国，他将所谓的科学 "专制主义 "与刚刚被推翻的帝国政权相提并论，自那时起，"专制主义 "这一笼统的指控被成千上万次地用作借口，从学校系统中取消几何中的演绎法、语法部分、句子分析图等教学。事实上，现在许多大学数学教师都分析说，刚从高中毕业的学生遇到的困难，很大一部分并不是数学本身造成的，而是因为他们第一次真正接触到必须准确使用普通语言的要求。 杜威的“教孩子，而不是教科学”这一强有力的原则有许多推论，例如“学习是有趣的”这一反儿童理论，以及最终以笑话代替推理的逻辑；当然，这一原则还包括经常对学生发出的指令：“用你自己的话说出来”。这条命令对教师很有吸引力，因为他们知道理解需要个人的有意识行为。然而，学校的整体氛围往往要求“用自己的话”意味着“尽可能不精确”，从而破坏了以任何可用形式获得概念。
Already in the early 1900's the tradition, that school should be devoted to learning the accumulated human knowledge, was being eroded in another way: The steel city of Gary, Indiana was built whole at Rockefeller's demand, the factories, workers' homes, sidewalks, and school system. To minimize the free time of the sons and daughters of the workers, extra-curricular activities at the schools were declared essential to the development of "the child". This Dewey-endorsed school system was studied by administrators from all over the world who journeyed to observe it. The whole Dewey program was styled "progressive" education, illustrating through the use of this term the fundamental tenet of pragmatist epistemology: truth is what you can get away with.
早在 20 世纪初，学校应该致力于学习人类积累的知识这一传统就已经以另一种方式受到侵蚀：在洛克菲勒的要求下，印地安那州的 Gary 钢铁城建立了完整的工厂、工人住宅、人行道和学校系统。为了尽量减少工人子弟的空闲时间，学校的课外活动被宣布为 "儿童 "发展的必要条件。世界各地的管理者纷纷前往观摩，对杜威认可的这一学校制度进行研究。杜威的整个计划被称为 "进步 "教育，通过使用这一术语，说明了实用主义认识论的基本信条：真理就是你能得到的东西。
In 1915 the U.S. mathematical organization split into two, one devoted primarily to the promotion of research, and the other supposedly devoted to the promotion of college teaching. The latter maintained and deepened its ties with pragmatic philosophy in 1921 when, at a meeting at Wellesly College, the widow of the leading publicizer of pragmatism, Paul Carus (whose stated aim was to promote religion on the basis of recent science), gave several thousand dollars to finance a series of monographs. At the same meeting the president of the organization gave his address entitled: "The religion of a mathematician", consisting of principles such as "since we know infinitesimals, we must also know our own insignificance; since we believe in infinity, we must also believe in an almighty; since we can imagine the fourth dimension, we can also imagine heaven, etc." This policy of the organization has never been repudiated, and its publications, which aim to give guidance to college teachers, have over the years refined to a precise art a writing style similar to that of the Scientific American, i.e. it is presumed that the readers will not advance from a lower to a higher level, and hence under the guise of "popularization", all concepts are made sufficiently imprecise so as to be unusable by anybody. The deliverer of that presidential address was the author of one of the few texts on the history of mathematics then available in English.
1915 年，美国数学组织一分为二，一个主要致力于促进研究，另一个据称致力于促进大学教学。1921 年，后者与实用主义哲学保持并加深了联系，在 Wellesly 学院的一次会议上，实用主义的主要宣传者 Paul Carus（其公开目的是在最新科学的基础上宣扬宗教）的遗孀捐赠了几千美元，用于资助一系列专著的出版。在同一次会议上，该组织的主席发表了题为 "数学家的宗教 "的演讲，其原则包括 "既然我们知道无穷小，我们也必须知道自己的渺小；既然我们相信无穷大，我们也必须相信全能者；既然我们可以想象四维空间，我们也可以想象天堂等等"。该组织的这一政策从未被摒弃，其出版物旨在为大学教师提供指导，多年来已将一种类似于《科学美国人》的写作风格提炼为一种精确的艺术，即假定读者不会从较低层次上升到较高层次，因此在 "大众化 "的幌子下，所有概念都变得足够不精确，以至于任何人都无法使用。该主席还是当时屈指可数的几本英文版数学史之一的作者。
作者曾经在大学中未经许可教授数学史
Mathematics itself is often said to have made more advances in the 20th century than in all previous centuries. The advances include not only the solution and formulation of difficult problems with geometrical and other content, but also (indispensably to that) the development of unifying concepts which are of great simplifying and clarifying value. An opportunity seemed to present itself around 1960 to disseminate that simplifying and clarifying value to a vastly larger number of students. The occasion, as I understand it, was the following: The U.S. ruling circles, fresh from rejoicing that their friend Khrushchev had succeeded in overthrowing the socialist system and was transforming it into a pseudo-socialist system, suddenly realized that they were thereby also faced with a rival superpower. This implied a certain shift of the boundary between the B and C Levels on the Prony scale, a readjustment of the line between "arithmetic" and "logic" on the Privy Council's anti-Chartist plan: more students would have to learn more math and science in order to counter the Sputnik threat. Whatever the precise details of the background, the opportunity was presented around 1960 to have university researchers directing summer schools for eager high-school teachers, to have writing teams producing new text books for pupils and teachers, etc. The challenge was taken up enthusiastically by many professionals in the spirit of letting those concepts, which had proved to be so enlightening for them, also serve to enlighten everyone. Of course such an undertaking requires several years of pupil feedback and text revision (and new mathematical research!) in order to become successful. But that stage was never reached because the movement was discredited; the enthusiastic professionals had underestimated the preparation of the opposition. By an artful confusion of the meanings of words like "foundation", the foundationalist trend (which had become entrenched since the days when Bertrand Russell started at the London School of Economics - LSE) insisted that the texts must be written in their idiosyncratic notation. And the professional schools of pedagogy (housed since Dewey in ivory towers remote from the actual scientific departments) took leadership of the movement from the bewildered scientists, insuring its destruction.
人们常说，数学本身在 20 世纪取得的进步超过了以往所有世纪。这些进步不仅包括解决和阐述关于几何与其他内容的难题，而且还包括（不可或缺的）统一概念的发展，这些概念具有极大的简化和澄清价值。1960 年前后似乎出现了一个机会，可以向更多的学生传播这种简化和澄清的价值。据我所知，当时的情况如下：美国统治集团刚刚为他们的朋友赫鲁晓夫成功推翻社会主义制度并将其转变为伪社会主义制度而欢欣鼓舞，突然意识到他们也因此面临着一个超级宿敌。这意味着 Prony 所衡量的 B、C 间界限的一定偏移，枢密院反宪章派计划中“算术”与“逻辑”间界限的校正：为了应对人造卫星【代指苏联的军备与科技竞赛】的威胁，更多的学生必须学习更多的数学和科学知识。无论背景的具体细节如何，1960 年前后，大学研究人员为热心的中学教师指导暑期班、写作团队为学生和教师编写新课本等机会出现了。许多专家热情地接受了这一挑战，他们的精神是让那些对他们有启迪作用的概念也能启迪每一个人。当然，这项工作需要经过几年的学生反馈和文本修订（以及新的数学研究！）才能取得成功。但这一阶段从未达到，因为这场运动已经失去了信誉；热情的专业人员低估了反对派的准备。由于巧妙地混淆了 "基础 "等词的含义，基础主义思潮（自伯特兰·罗素在伦敦经济学院（LSE）就读时就已根深蒂固）坚持认为，课文必须用他们特立独行的符号来书写。而教育学专业学校（自杜威以来就设在远离实际科学部门的象牙塔中）则从困惑的科学家手中夺取了运动的领导权，确保其毁灭。
It would seem that an obvious way to improve math teaching would be to give more examples and more applications. This is correct, of course, but to give that as a demand and stop there was again to underestimate what we are up against. The 1970's and 80's saw the publication of many "applied" calculus texts in which explicit principles were subordinated to problems watered-down and distorted beyond usability from various fields. But as many professors in those various fields understand, math is theory. What a student needs of math in a field of application such as chemistry, business management, etc. is to know math as well as possible in order that the applied concepts be approachable with little math-related mystery and in order that mastery of appropriate new methods can be partly self-guided. Isolated, particular methods learned mechanically and then forgotten, and especially, half-baked attempts to teach an alleged application instead of the explicit principles of calculus, can only negatively affect the students' ability to apply math.
看起来改进数学教学的一个显然方法就是举出更多的实例与应用。这当然没错，但若仅限于此，那又低估了我们面临的挑战。20 世纪七八十年代出版了许多“应用”微积分教材，其中明确的原理从属于被淡化和扭曲的问题，超出了各个领域的可用性。但正如这些不同领域的许多教授所理解的那样，数学就是理论。在化学、商业管理等应用领域，学生对数学的需求是尽可能地了解数学，以便在应用概念时不至于感到莫名其妙，并大致能通过自学掌握恰当的新方法。填鸭式教学教的那些孤立的、特殊的、学了就忘的方法，尤其是教授所谓的应用而非微积分明确原理的草率尝试，只会损害学生应用数学的能力。
The demand for better calculus text books in the English-speaking world thus began, for the above and many other reasons, to become more threatening. The initial response of the publishers (that they would never change their policy of offering the next year an exact copy of the competing text which had made most profit the previous year, only with more colors) was met by deserved contempt. Some of the more recent offerings are the result of multimillion dollar government intervention.
因此，由于上述及其他种种原因，英语世界对更好的微积分教科书的需求开始变得更具威胁性。出版商最初的反应（他们永远不会改变他们的政策，在次年提供于上一年市场竞争中获利最多的课本的相同副本，只是带上更多色彩）受到了应有的蔑视。最近推出的一些则是政府数百万美元干预的结果。
Contrary to the portrait of the professor who views teaching and research as inimical to each other, many see them as mutually supportive.
一反认为教学、科研互相对立的形象，许多教授认为二者是相辅相成的。
Many of the ideas, which have led to long and fruitful development by researchers, actually arose from attempts to explain matters more clearly to students. For example, attempts in the 1960's to provide a clearer, simpler, yet rigorous base for understanding calculus led to a new trend of research in the foundations of topology, logic, and analysis within which many innovative papers and over a dozen books have now been produced. On the other hand, research leads from time to time to new synthesizing concepts, which clarify matters enormously for the researchers, who then struggle to find ways to spread this clarification to students. For example research into the mathematical foundations of continuum thermomechanics and the constitutive relations of materials led to new, more direct, ways of dealing with infinitesimals, function spaces, and extensive quantities, which are now being taught to undergraduates in some places.
研究人员长期以来富有成果的许多想法，实际上都是为了更清楚地向学生解释问题而产生的。例如，20 世纪 60 年代，人们试图为理解微积分提供一个更清晰、更简单但又严谨的基础，这导致了拓扑学、逻辑学和分析学基础研究的新趋势，在这一领域中，现已发表了许多创新论文，出版了十几本著作。另一方面，研究工作也不时带来新的综合概念，这些概念极大地澄清了研究人员的问题，而研究人员则努力寻找向学生传播这些澄清的方法。例如，对连续热力学和材料构成关系的数学基础的研究导致了处理无穷小量、函数空间和广义量的新的、更直接的方法，有些地方正在向本科生教授它们。
Research and teaching are of course different aspects of endeavor, but as long as they are still alive, they have an orientation in common, a commitment to tirelessly combating the absence of knowledge.
研究和教学当然是不同方面的工作，但只要它们还活着，它们就有一个共同的方向，那就是致力于同知识欠缺作不懈斗争。
History shows that the teachers, yearning for a greater opportunity to participate in the creation and dissemination of enlightenment, will not be satisfied by waiting for this or that establishment entity to provide it. Not only would the fulfillment of these needs remain forever a mere policy objective; our enthusiasm would continue to be used as the engine for the spread of still more pseudo-knowledge and pessimism. The problem can be taken up for solution, without million-dollar grants, both by devising teaching materials which reflect the actual historical development of a given field (neither repeating some entrenched hundred-year old false summation, nor succumbing to the ultrarevolutionary post-modernist degeneration), as well as by making explicit the philosophy which emerges from actual research developments of the recent decades. Collective effort is necessary, however, to concentrate such materials and to disseminate them so as to serve the needs of society as a whole.
历史表明，教师们渴望有更多的机会参与启蒙教育的创造和传播，但他们不会满足于等待这个或那个机构来提供这种机会。【因为等待】不仅会使这种渴望的满足永远停留于单纯的政策目标【即永远无法满足】，我们的热情还将继续被用作传播更多伪知识与悲观主义的动力。要解决这个问题，不需要数百万美元的拨款，既可以通过编写反映特定领域实际历史发展的教材（既不重复某些根深蒂固的百年错误总结，也不屈服于极端革命的后现代主义堕落），也可以通过阐明从最近几十年的实际研究发展中产生的哲学。然而，有必要通过集体努力来集中这些材料并加以传播，以满足整个社会的需要。
(Hardial Bains Resource Centre Archives)
本文基于 DeepL 翻译，有润色。