应用物理微积分中英文对照外文翻译文献

中英文对照外文翻译

中英文对照外文翻译

牛顿与莱布尼兹创立微积分之解析

摘要:文章主要探讨了牛顿和莱布尼兹所处的时代背景以及他们的哲学思想对其创立广泛地应用于自然科学的各个领域的基本数学工具———微积分的影响。

关键词:牛顿;莱布尼兹;微积分;哲学思想

Abstract:This paper mainly discusses the background of the times of Newton and Leibniz, and their philosophy of its founder is widely used in various fields of natural science basic mathematical tools --- calculus.

Key words: Newton; Leibniz; calculus; philosophical thought

今天,微积分已成为基本的数学工具而被广泛地应用于自然科学的各个领域。恩格斯说过:“在一切理论成就中,未有象十七世纪下半叶微积分的发明那样被看作人类精神的最高胜利了,如果在某个地方我们看到人类精神的纯粹的和唯一的功绩,那就正是在这里。”[1] (p. 244) 本文试从牛顿、莱布尼兹创立“被看作人类精神的最高胜利”的微积分的时代背景及哲学思想对其展开剖析。

一、牛顿所处的时代背景及其哲学思想

“牛顿( Isaac Newton ,1642 - 1727) 1642 年生于英格兰。??,1661 年,入英国剑桥大学,1665 年,伦敦流行鼠疫,牛顿回到乡间,终日思考各种问题,运用他的智慧和数年来获得的知识,发明了流数术(微积分) 、万有引力和光的分析。”[2] (p. 155)

1665 年5 月20 日,牛顿的手稿中开始有“流数术”的记载。《流数的介绍》和《用运动解决问题》等论文中介绍了流数(微分) 和积分,以及解流数方程的方法与积分表。1669 年,牛顿在他的朋友中散发了题为《运用无穷多项方程的分析学》

的小册子,在这里,牛顿不仅给出了求一个变量对于另一个变量的瞬时变化率的普遍方法,而且证明了面积可以由求变化率的逆过程得到。因为面积也是用无穷小面积的和来表示从而获得的。所以牛顿证明了这样的和能由求变化率的逆过程得到(更精确地说,和的极限能够由反微分得到) ,这个事实就是我们现在所讲的微积分基本定理。这里“,牛顿使用的是无穷小方法,把变量的无限小增量叫做“瞬”,瞬是无穷小量,是不可分量, 或是微元, 牛顿通过舍弃“瞬”求得变化率。”[3] (p. 199) 1671 年牛顿将他关于微积分研究的成果整理成《流数法和无穷级数》(1736) ,在这里,他认为变量是连续运动产生的,他把变量叫做流,变量的变化率叫做流数。牛顿更清楚地陈述了微积分的基本问题:已知两个流之间的关系,求它们流数之间的关系,以及它的逆问题。《流数法和无穷级数》是一部较完整的微积分著作。书的后半部分通过20 个问题广泛地介绍了流数法各无穷级数的应用。1676 年,牛顿写出了《求曲边形的面积》(1704) ,在这里,牛顿的微积分思想发生了重大变化,他放弃了微元或无穷小量,而采用了最初比和最后比的方法。

1687 年牛顿发表了它的划时代的科学名著《自然哲学的数学原理》,流数术(即微积分) 是其三大发现之一。正如爱因斯坦所说的:“牛顿啊??你所发现的道路在你的那个时代是一位具有最高思维能力和创造能力的人所发现的唯一道路,你所创造的概念即使在今天仍然指导着我们的物理学思想”。[4 ] (p. 192)

牛顿生活的时代正是英国发生变化的时代,当时英国发生了国内战争,资产阶级和贵族的阶级妥协,使英国资产阶级革命明显的带上了不彻底性。当时的英国资产阶级正在为现存的剥削阶级的一切上层建筑做永恒存在的论证,因此绝对化的思想成为占统治地位的主导思想,它也影响到当时的自然科学家们把形而上学的思想方法绝对化。牛顿的思想也受到了英国资产阶级革命不彻底性的影响,因而牛顿也往往不能从自然界本身或事物的本身来寻找最初的原因,而借助于外来的推动力。

牛顿在30 岁以前发现了微积分,并建立了经典力学体系,而他的后半生在自然科学的研究上几乎一事无成。这是由于在资本主义产生和形成的时期,资产阶级曾经向宗教神学发起冲击,帮助科学从神学中解放出来。但是当资产阶级的地位巩固以后,阶级斗争逐渐激化之时,资产阶级就逐渐衰退,他们就抓住各种各样的宗教信念作为奴役人民的思想武器。牛顿受其影响很大,其前半生由于自发的唯物主义的思想倾向,使他获得了巨大成就,而后半生则完全沉迷于神学的研究。

牛顿继承了培根的经验主义传统,特别重视实验和归纳推理的作用,他曾断言,自然科学只能从经验事实出发解释世界。这在当时对打击经院哲学的崇尚空谈、妄称神意来歪曲自然界是起过积极作用的。但是“,牛顿却拘泥于经验事实,片面强调归纳的重要性。只有大量的感性材料,一切停留在事物的现象上,单独依靠归纳的方法是得不出系统的普遍性的理性认识来的。在分析和综合、演绎和归纳的问题上,形而上学使牛顿陷入了矛盾。”[5] (p. 123)

二、莱布尼兹所处的时代背景及其哲学思想

“莱布尼兹(Gottfried Wilhelm Leibniz ,1646 - 1716)生于德国。??,1672 年赴巴黎,在那里接触到惠更斯等一些数学名流,引其进入了数学领域,开始微积分的创造性工作。”[2] (p. 165)

1684 年莱布尼茨发表了数学史上第一篇正式的微积分文献《一种求极限值和切线的新方法》。这篇文献是他自1673 年以来的微积分研究的概括与成果,其中定义了微分,广泛地采用了微分符号dx、dy ,还给出了和、差、积、商及乘幂的微分法则。同时包括了微分法在求切线、极大、极小值及拐点方面的应用。两年后,又发表了一篇积分学论文《深奥的几何与不

变量及其无限的分析》,其中首次使用积分符号“∫”,初步论述了积分(或求积) 问题与微分求切线问题的互逆问题。即今天大家熟知的牛顿- 莱布尼茨公式∫ba f ( x) dx = f ( b) - f ( a) ,为我们勾画了微积分学的基本雏形和发展蓝图。

“牛顿建立微积分是从运动学的观点出发,而莱布尼兹则从几何学的角度去考虑,所创设的微积分符号远远优于牛顿符号,并有效地促进了微积分学的发展。”[6] (p. 120) 牛顿发现微积分(1665 - 1666 年) 比莱布尼茨至少早了9 年,然而莱布尼茨公开发表它的微积分文章比牛顿早3 年。据莱布尼茨本人提供的证据说明他是在1674 年形成了微分的思想与方法。如果说,牛顿建立微积分主要是从运动学的观点出发,而莱布尼兹则是从哲学的和几何学的角度去考虑,特别是和巴罗的“微分三角形”有密切关系,莱布尼兹称它为“特征三角形”。巴罗的微分三角形对莱布尼兹有着重要启发,对微分三角形的研究,使他意识到求切线和求积问题是一对互逆的问题。莱布尼兹第一个表达出微分和积分之间的互逆关系。

莱布尼兹的许多研究成果和思想的发展,都包含在从1673 年起写的但从未发表过

的成百页的笔记中。1673 年左右,他看到求曲线的切线的正问题和反问题的重要性,他完全相信反方法等价于通过求和来求面积和体积。1684 年,莱布尼兹发表第一篇微分学论文《一种求极大、极小和切线的新方法,它也适用于分式或无理量,以及这种新方法的奇妙类型的计算》,对他以往的研究作了初步整理,叙述了微分学的基本原理,认为函数的无限小增量是自变量无限小变

化的结果,且把这个函数的增量叫做微分,用字母d表示。1675 - 1676 年间,他从求曲边形面积出发得到积分的概念, 给出微积分基本定理∫baf( x) dx = f ( b)- f ( a) 。1686 年莱布尼兹发表积分学论文《潜在的几何与分析不可分和无限》。1693 年,他给出了上述定理的一个证明。以上这些都发表在《教师学报》上。将微分和积分统一起来,是微积分理论得以建立的一个重要标志。莱布尼兹出生在德国路德派诸侯与天主教诸侯之间的对立而引起的“三十年战争”结束前。为了改变宗教纷争的局面,莱布尼兹立志要发现一种新的天主教和路德教都能适合的关于实体的学说,以成为两派教会得以联合的哲学基础。虽然莱布尼兹的意图是不可能实现的,但他后来却因此提出了一种与笛卡尔不同的实体学说———单子论。

“单子论是莱布尼兹哲学的核心内容。莱布尼兹认为一切事物都由单子这种精神的实体构成的,这种‘单子’既非物质的而又具有一定的质,它是精神性的,莱布尼兹就把它比之于灵魂。只有精神的单子才是真实的存在的实体,从单子是不可分的,即没有部分的“单纯”实体这一点出发,莱布尼兹就推论出它的一系列特征:单子没有部分,它就不能以自然的方式通过各部分的组合而产生,或通过各部分的分解而消灭,因此它的生灭只能出于上帝的突然创造或毁灭;单子没有部分,就不能设想有什么东西可以进入其内部来造成变化,这样,单子就成了各自独立或彻底孤立的东西,各单子之间不能有任何真正的相互作用或影响。单子之间没有量的差异, 而只有质的不同。”[7] (p. 85)

总之,莱布尼兹的基本观点是唯心主义的,也是形而上学的。他把宇宙的秩序都归因于上帝的预先决定。他肯定许多必然真理并非来自经验,他认为不但认识的对象都是由精神性的“单子”所构成。而且认识的主体也只能作为精神实体的心灵这种“单子”。他把一切发展变化都归因于上帝的“前定”,实际也就否定了真正的发展,这是他的观点的消极的一面。但另一方面,莱布尼兹的哲学也有积极方面,它的哲学中含有丰富的辩证法思想,他肯定实体本身就具有力,因而是能动的,实质上肯

定了物质与运动不可分的思想,他试图解决“不可分的点”和“连续性”的矛盾问题,接触到了个别与全体、间断性与连续性的对立统一问题,对促进理性和经验的辩证结合做出了一定的贡献。

三、牛顿、莱布尼兹创立微积分之比较

牛顿和莱布尼兹用各自不同的方法,创立了微积分学。如果说牛顿接近最后的结论要比莱布尼兹早一些,那么莱布尼兹发表自己的结论要早于牛顿。虽然牛顿的微积分应用远远超过莱布尼兹的工作,刺激并决定了几乎整个十八世纪分析的方向,但是莱布尼兹成功地建立起更加方便的符号体系和计算方法。两位微积分的奠基人,一位具有英国式的处事谨慎,治学严谨的风度,一位具有德国人的哲理思辨心态,热情大胆。由于阴阳差错的时代背景, 过分追求严谨的牛顿迟迟未将自己的发现发表,让莱布尼茨抢了一个发表的头筹。

牛顿和莱布尼兹的哲学观点的不同导致了他们创立微积分的方法不同。牛顿坚持唯物论的经验论,特别重视实验和归纳推理。他在研究经典力学规律和万有引力定律时,遇到了一些无法解决的数学问题,而这些数学问题用欧几里德几何学和16 世纪的代数学是无法解决的,因此牛顿着手研究新的以求曲率、面积、曲线的长度、重心、最大最小值等问题的方法———流数法。“牛顿的研究采用了最初比和最后比的方法。他认为流数是初生量的最初比或消失量的最后比。初生量的最初比就是在初生的瞬间的比值,消失量的最后比就是量在消失的瞬间的比值。”[4 ] (p. 180) 这个解释太模糊了,算不上精确的数学概念,只不过是一种直观的描述。最初比和最后比的物理原型是初速度与末速度的数学抽象,在物体作位置移动的过程中的每一瞬间具有的速度是自明的,牛顿就是从这个客观事实出发提出了最初比和最后比的直观概念。这样他就给出了极限的观点。

莱布尼兹的微积分创造始于研究“切线问题”和“求积问题”,他从微分三角形认识到:求曲线的切线依赖于纵坐标之差与横坐标之差的比值;求曲边图形的面积则依赖于在横坐标的无限小区间上的纵坐标之和或无限薄的矩形之和。莱布尼兹认识到求和与求差运算是可逆的。莱布尼兹用无穷小的思想给出了微积分的基本定理,并发展成为高阶微分。莱布尼兹的无穷小是分阶的,这源于他哲学中的单子论思想。“莱布尼兹在单子论中指出:不同的单子其知觉

的清晰程度是不一样的,并从一种知觉向另一种知觉过渡和变化,发展就是由单子构成的事物,由低级向高级的不同等级的序列。”[6] (p. 91) 可以说,莱布

尼兹的无穷小的分阶正是和它的客观唯心论的哲学体系中那个不同层次的单子系统是相对应的。莱布尼兹在微积分的研究过程中,连续性原则成为其工作的基石,而连续性原则是扎根于他哲学中无限的本质的思想。

牛顿和莱布尼兹创立微积分的相同点有:从不同的角度创立了一门新的数学学科,使微积分具有广泛的用途并能应用于一般函数;用代数的方法从过去的几何形式中解脱出来;都研究了微分与反微分之间的互逆关系。

牛顿和莱布尼兹创立微积分的不同点主要有:牛顿继承了培根的经验论,对归纳特别青睐。牛顿的微积分明显带着从力学脱胎而来的物理模型的痕迹,以机械运动的数学模型出现,其中的基本概念,如初生量、消失量、瞬、最初比和最后比等概念都来自机械运动,是机械运动瞬间状态的数学抽象。他建立微积分的目的是为了解决特殊问题,强调的是能推广的具体结果。而莱布尼兹强调能够应用于特殊问题的一般方法和算法,以便统一处理各种问题。莱布尼兹在符号的选择上花费了大量的时间,发明了一套富有提示性的符号系统。他把sum(和) 的第一个字母S 拉长表示积分,用dx 表示x 的微分,这套简明易懂又便于使用的符号一直沿用至今。

牛顿认为微积分是纯几何的自然延伸,关心的是微积分在物理学中的应用。经验、具体和谨慎是他的工作特点,这种拘束的做法,使他没有能尽情发挥。而莱布尼兹关心的是广泛意义下的微积分,力求创造建立微积分的完善体系。他富于想象,喜欢推广,大胆而且有思辩性,所以毫不犹豫地宣布了新学科的诞生。

牛顿和莱布尼兹都是他们时代的科学巨人。微积分之所以能成为独立的学科并给整个自然科学带来革命性的影响,主要是靠了牛顿与莱布尼兹的工作。从牛顿和莱布尼兹创立微积分的过程中可以看出:当巨人的哲学的沉思变成科学的结论时,对科学发展的影响是深远的。

参考文献:

[1 ]恩格斯. 自然辩证法[M] . 北京:人民出版社,1971.

[2 ]李文林. 数学史概论[M] . 北京:高等教育出版社,2003.

[3 ]杜瑞芝. 数学史辞典[M] . 山东:山东教育出版社,2000.

[4 ]吴文俊. 世界著名科学家传记[M] . 北京:科学出版社,1994.

[5 ]C H 爱德华. 微积分发展史[M] . 北京:北京出版社,1989.

[6 ]M 克莱因. 古今数学思想[M] . 上海:上海科学技术出版社,1979.

[7 ]费尔巴哈. 对莱布尼兹哲学的叙述、分析和批判[M] . 北京:商务印书

馆,1985.

Newton and Leibniz created calculus resolution Abstract:This paper mainly discusses the background of the times of Newton and Leibniz, and their philosophy of its founder is widely used in various fields of natural science basic mathematical tools --- calculus.

Key words: Newton; Leibniz; calculus; philosophical thought

T oday, calculus has become the basic mathematical tools are widely used in various fields of natural sciences. Engels said: "in all theoretical achievements, not like the invention of the first half of the 17th century calculus as be seen as the highest victory for the human spirit, somewhere, we see the human spirit pure and only merit, it is in here. "[1] (p. 244) This paper tries from Newton, Leibniz founded the historical background and philosophy" is seen as the highest victory of the human spirit "calculus its Expand profiling

Firstly,Newton’s background of the times and philosophy

"Newton (Isaac Newton, born in 1642 - 1727) was born in 1642 in England. ??, in 1661, he got into the University of Cambridge, in1665, London was popular in plague, Newton returned to the countryside, all day he was thinking about a variety of issues, used his wisdom and several years the knowledge, invented the number of flow cytometry (calculus), the analysis of the gravity and light. "[2] (p. 155)

On may 20, 1665, Newton's manuscript had the record of “the number of streams surgery”. , , other papers in the number of streams (differential) and integration, and solution flow equation with integral table. In 1669, Newton circulated entitled "the use of an infinite number of

equations analytics booklet in his friends, here, Newton was not only seeking a variable universal method for the instantaneous rate of change in another variable, but also proved that the area can be obtained by seeking the inverse process of the rate of change.Because the area is also used to represent the thus obtained. Newton proved this can be obtained by the inverse process of seeking the rate of change (more precisely, and the limit can be obtained by the anti-differential), the fact is that we are now talking about fundamental theorem of calculus. where, "Newton used the infinitesimal method, called" instantaneous "infinitely small increment of the variable, instantaneous infinitesimal, is not component or infinitesimal, Newton obtained abandon the" instantaneous "rate of change." [3 (p. 199)

In 1671 Newton made his results on the study of calculus organized into flow method and the infinite series (1736), where, he believed that the variable was continuous motion generated variable called flow the rate of change of a variable was called the number of streams. Newton stated more clearly the basic problems of the calculus: the relationship between the two streams were known, found the relationship between the number of their flow, as well as its inverse problem. "The number of streams law and infinited series" was a more complete calculus book. The second half of the book passed by 20 problems introduced a number of streams infinite series application of the law.In 1676, Newton wrote "seeking curved edge-shaped area" (1704), where the thinking of Newton's calculus significant changed, he gave up a micro or infinitesimal, while used the initial and The last ratio method. In 1687 Newton published its landmark scientific books "mathematical principles of natural philosophy, the number of flow cytometry (Calculus) was one of the three major findings. As Einstein said: "Newton ?? road you have found in your era is the one with the highest of thinking and creative skills of people who found the only way you created the concept even today still guiding our thinking in physics ". [4] (p. 192)

Newton lived in an age Britain was changing times, when the British brake the civil war, the bourgeoisie and the aristocracy class compromised, made the bourgeois revolution in Britain not have thoroughness. The British bourgeois superstructure to do

eternal argument for all the existence of the exploiting classes , therefore the absolute ideological become the dominant ideology of the dominant, it also affected the natural scientists absolute metaphysical way of thinking . Newton's ideas have also been a bourgeois revolution in Britain did not thoroughly Thus Newton often can not find the initial the cause from the nature or the things of itself ,and by means of the external driving force.

Before the age of 30, Newton discovered calculus, and established Classical Mechanics. However, he spended almost nothing on the study of the natural sciences. This was in capitalism and the formation period, the bourgeoisie had to theology attack to help liberate science from theology. But, When the bourgeois entrenched, the gradual intensification of the class struggle, the bourgeoisie gradually declined, they seized a wide variety of religious beliefs as slavery ideological weapon of the people. Newton influenced by its former life due to the ideological tendency of spontaneous materialism, he was a great achievement, the latter half was completely addicted to the study of theology.

Newton inherited the tradition of Bacon's empiricism, special attention to the role of experimentation and inductive reasoning, he asserted the natural science can only explain the world starting from the empirical facts. This was for advocating talked against scholasticism, misuse the divine to distort the nature played a positive role. However, Newton was rigidly adhere to the empirical facts, one-sided emphasis on the importance of induction. Emotional material, all stuck in the phenomenon of things, relying solely on the inductive method was to have no system of universal rational knowledge to. on the issue of analysis and synthesis, deductive and inductive metaphysics Newton caught in a contradiction. "[5] (p. 123)

Secondly,Leibniz’sbackground of the times and philosophy

Leibniz (Gottfried Wilhelm Leibniz, 1646 - 1716) was born in Germany. ??, 1672 in Paris, where exposured to the Huygens some mathematical celebrities, to lead into the field of mathematics, the beginning of the calculus creativework [2] (p. 165)

In 1684, Leibniz published the first official history of mathematics calculus literature, "a

demand limit and tangent method. This literature of summarized results of the study of calculus since 1673, which defined the differential, extensively used the differential symbol dx, dy, and also gived the sum, commossion,accumulated, and quotient the power of the rules of differentiation. Including the application of the differential method in the tangent line, great minima and inflection. Two years later, he published a a calculus thesis "esoteric geometry and not Variables and their infinite analysis ", which for the first time using the integral symbol ∫" preliminary discusses the points (or quadrature) problem with differential quadrature tangent reciprocal problem. Well-known Newton - Leibniz formula ∫ BA f (x) dx = f (b) - f (a), gave us an outline of the basic shape of the calculus and the blueprint for the development.

Newton's calculus was the departure from the point of view of kinematics, Leibniz from geometry viewed to consider the creation of calculus symbol was far better than Newton's symbol, and effectively promoted the development of the calculus of[6] (p. 120) Newton discovered calculus (1665 - 1666) at least nine years earlier than Leibniz, Leibniz, however, published its calculus article three years earlier than Newton. According to the evidence be provided by the Leibniz himself that he formed in 1674, the ideas and methods of the differential. If the Newton calculus was the departure from the point of view of kinematics, Leibniz was from the point of view of philosophy and geometry to consider, especially closely related to the differential triangle and Barrow, Leibniz Martinez called it a "characteristic triangle". Barrow's differential triangle Leibniz has inspired the differential triangle, made him aware of the tangent line and quadrature problem is a pair of mutually inverse problem. Leibniz first express the inverse relationship between the differential and integral.

Leibniz’s many research and development of ideas ware included in hundreds of pages of notes written since 1673, but unpublished. In 1673, he saw the importance of the tangent to the curve of the positive and inverse problems he does believe that the anti-method is equivalent to the area and volume by summing the requirements. In 1684, Leibniz published his first differential calculus papers "a seeking of maximum, minimum, and a new method of tangent, it also applied to fractional or unreasonable