庞加莱 关于数学创造(翻译)

庞加莱关于数学创造（翻译）

庞加莱关于数学创造（翻译）

2010-11-15 15:37:21昨天刘未鹏在日志中提到庞加莱关于

数学创造的文章，顺手译成了中文，如下：

Mathematical Creation

数学创造

How is mathematics made? What sort of brain is it that can compose the propositions and systems of mathematics? How do the mental processes of the geometer or algebraist compare with those of the musician, the poet,

the painter, the chess player? In mathematical creation which are the key elements? Intuition? An exquisite sense of space and time? The precision of a calculating machine?

A powerful memory? Formidable skill in following complex logical sequences? A supreme capacity for concentration? 数学是由什么构成的？哪种类型的大脑能够创造数学的定

理和系统？几何学家或是代数学家的思维活动和音乐家、是人画家和象棋选手有什么不同？数学创造中，哪些是关键元素？直觉？空间和时间的精确感觉？机器一般的计算准确性？超强的记忆力？复杂逻辑推导的超强技能？极好的注

意力？

The essay below, delivered in the first years of this century as a lecture before the Psychological Society in Paris, is the most celebrated of the attempts to describe what goes on in the mathematician's brain. Its author, Henri Poincaré, cousin of Raymond, the politician, was peculiarly fitted to undertake the task. One of the foremost mathematicians of all time, unrivaled as an analyst and mathematical physicist, Poincaré was known also as a brilliantly lucid expositor of the philosophy of science. These writings are of the first importance as professional treatises for scientists and are at the same time accessible, in large part, to the understanding of the thoughtful layman.

下面的文章，是本世纪（20世纪）初期在巴黎心理学会上做的一次报告，是描述关于数学家大脑如何运转的最著名的尝试。该报告的作者，亨利·庞加莱（是政治家雷蒙德的表亲），是承担该任务的极其适当的人选。作为历史上最重要的数学家之一，一位无可匹敌的分析和数学物理学家，庞加莱同样擅长对科学哲学作出清晰准确的阐释。这份报告对于科学家们而言是一极其重要的专业论述，同时在很大程度上，也可以为非专业人士所理解。

Poincaré on Mathematical Creation

庞加莱关于数学创造

The genesis of mathematical creation is a problem which should intensely interest the psychologist. It is the activity

in which the human mind seems to take least from the outside world, in which it acts or seems to act only of itself and on itself, so that in studying the procedure of geometric thought we may hope to reach what is most essential in man's mind...

数学创造的起源是一个引起心理学家强烈兴趣的问题。这种创造活动似乎是涉及外部世界最少的、大脑内部发生的活动，仅仅、或者看起来仅仅依赖于和作用于它自身，因此，研究几何思考的过程，我们也许希望能够深入到人类思维的最本质。

A first fact should surprise us, or rather would surprise us if we were not so used to it. How does it happen there are people who do not understand mathematics? If mathematics invokes only the rules of logic, such as are accepted by all normal minds; if its evidence is based on principles common to all men, and that none could deny without being mad, how does it come about that so many persons are here refractory?

一个简单事实可能会使我们惊讶，因为我们对此并不熟悉。为什么有人不能理解数学？如果数学仅仅唤起的是逻辑法

则，那么正常的思维应该都能接受它；如果它基于所有人都了解的通常法则，除非神经错乱否则不能否认，为什么这么多人不能掌握数学呢？

That not every one can invent is nowise mysterious. That not every one can retain a demonstration once learned may also pass. But that not every one can understand mathematical reasoning when explained appears very surprising when we think of it. And yet those who can follow this reasoning only with difficulty are in the majority; that is undeniable, and will surely not be gainsaid by the experience of secondary-school teachers.

不是每个人都能创造数学，这一点毫不神奇。不是每个人学习了某个定理证明以后都能灵活运用它，这也毫不为怪。然而不是每个人都能够领会数学分析思维，即使反复解释也不能——仔细思考这一点就让人感到惊异了。而且能跟上分析思维的人，大多数也感到困难，这是不可否认的；相信每个当过中学教师的人都对此深有体验。

And further: how is error possible in mathematics? A sane mind should not be guilty of a logical fallacy, and yet there are very fine minds who do not trip in brief reasoning such as occurs in the ordinary doings of life, and who are incapable of following or repeating without error the

mathematical demonstrations which are longer, but which after all are only an accumulation of brief reasonings

wholly analogous to those they make so easily. Need we add that mathematicians themselves are not infallible?...

此外：数学怎么会产生错误？理智的思维不会犯下逻辑错误，然而在日常生活中不会被简单分析所难倒的人，在重复较长的数学证明时往往难免出错；然而数学证明终归不过是一系列简短分析的集合而已，分开来看的话它们看似如此简易。我们是不是应该加一句，数学家们并不是不会犯错的？

As for myself, I must confess, I am absolutely incapable even of adding without mistakes... My memory is not bad, but it would be insufficient to make me a good

chess-player. Why then does it not fail me in a difficult piece of mathematical reasoning where most

chess-players would lose themselves? Evidently because

it is guided by the general march of the reasoning. A mathematical demonstration is not a simple juxtaposition

of syllogisms, it is syllogisms placed in a certain order, and the order in which these elements are placed is much more important than the elements themselves. If I have the feeling, the intuition, so to speak, of this order, so as to perceive at a glance the reasoning as a whole, I need no

longer fear lest I forget one of the elements, for each of them will take its allotted place in the array, and that without any effort of memory on my part.

对于我自己，我承认，我甚至连做加法都不会不犯错。我的记忆力不坏，但也不足以让我成为一个好的象棋选手。为什么令大多数象棋选手感到吃力的数学题，却不会难倒我呢？显然是因为它是由普通的分析步骤构成的。数学证明并不是简单的演绎法的排列，它是由演绎法按特定顺序排列而成，而且排列顺序比元素本身更为重要。如果我对这个顺序产生某种感觉，或称直觉，只需要一眼就能感知到推理的整体，那么我不会担心我忘记其中的一个元素，因为每个元素都是特定方式放置在这个阵列中的，而不需要我用记忆去牢记。We know that this feeling, this intuition of mathematical order, that makes us divine hidden harmonies and relations, cannot be possessed by every one. Some will not have either this delicate feeling so difficult to define, or a strength of memory and attention beyond the ordinary, and then they will be absolutely incapable of understanding higher mathematics. Such are the majority. Others will have this feeling only in a slight degree, but they will be gifted with an uncommon memory and a great power of attention. They will learn by heart the details one

after another; they can understand mathematics and sometimes make applications, but they cannot create. Others, finally, will possess in a less or greater degree the special intuition referred to, and then not only can they understand mathematics even if their memory is nothing extraordinary, but they may become creators and try to invent with more or less success according as this intuition is more or less developed in them.

我们知道，这种感觉，这种数学顺序的直觉，带给我们隐含的和谐和关系的神圣感，并不能被每个人所掌握。有些人既缺乏这种难以被定义的精妙感觉，又缺乏超乎常人的记忆和专注的程度，以至完全不能把握高等数学。大多数人都是如此。有些人会产生少许这类感觉，但他们天生拥有与众不同的记忆力和注意力。他们会记住一个接一个的细节，他们能够理解数学，并且作出应用，但他们不能创造。其他人，最终或多或少地能够掌握一些这种直觉，这样他们不知能够理解数学，并且即使他们的记忆力并不超凡，也能变成创造者，并伴随着这种直觉的多少，获得相应程度的成功。

In fact, what is mathematical creation? It does not consist in making new combinations with mathematical entities already known. Anyone could do that, but the combinations so made would be infinite in number and

most of them absolutely without interest. To create consists precisely in not making useless combinations and in making those which are useful and which are only a small minority. Invention is discernment, choice.

事实上，什么是数学创造？它并不意味着对已知的数学事实重新组合。任何人都可以做到重新组合，但这种组合的数量是无限的，并且大多数毫无价值。创造，意味着不制造无用的组合，而仅制造那些少量且有用的。创造即鉴别，即选择。It is time to penetrate deeper and to see what goes on in the very soul of the mathematician. For this, I believe, I can do best by recalling memories of my own. But I shall limit myself to telling how I wrote my first memoir on Fuchsian functions. I beg the reader's pardon; I am about to use some technical expressions, but they need not frighten him, for he is not obliged to understand them. I shall say, for example, that I have found the demonstration of such a theorem under such circumstances. This theorem will have a barbarous name, unfamiliar to many, but that is unimportant; what is of interest for the psychologist is not the theorem but the circumstances.

现在可以挖掘得更深一些，来看看数学家的灵魂深处是什么了。我相信，我可以通过回忆自己的经历来进行充分说明。

不过我将仅限于讲述我写第一本关于Fuchsian函数论文集的情况。我请求听众原谅：我将会使用某些技术术语，但你们无需感到害怕，因为你们并不需要理解。比如，我会这么说，基于某种情况，我得到了这个定理的证明。这个定理可能会有一个奇怪的名字，大部分人会感到不习惯，但并不重要；你们心理学家感兴趣的并不是定理，而是情况本身。For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. By the next morning I had established the existence of a class of Fuchsian functions, those which come from the hypergeometric series; I had only to write out the results, which took but a few hours.

我挣扎了15天来证明不可能存在像后来我称之为Fuchsian 函数那样的函数。当时我一无所知；每天我坐在书案前，工作一到两个小时，尝试大量的组合却一无所获。一个晚上，和我平日的习惯不同，我喝了黑咖啡，没有睡觉。大量的思

绪汹涌，我感到它们相互碰撞直到契合，也就是说，慢慢地稳定下来。第二天早上之前，我已经建立好一类Fuchsian 函数的存在性证明，这些函数来自于超几何序列；我只需要把结果写下来即可，前后花了不过几个小时。

Then I wanted to represent these functions by the quotient of two series; this idea was perfectly conscious and deliberate, the analogy with elliptic functions guided me. I asked myself what properties these series must have if they existed, and I succeeded without difficulty in forming the series I have called theta-Fuchsian.

后来我想用两个序列的商来表达这些函数；想法很清晰而且经过深思熟虑，因为椭圆方程的相似性指引着我。我问自己，如果这些序列存在，它们会有怎样的属性，并且我毫无困难地成功构建了这些序列，后来我称之为theta-Fuchsian。Just at this time I left Caen, where I was then living, to go on a geologic excursion under the auspices of the school of mines. The changes of travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had

used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience's sake I verified the result at my leisure.

就是这时期，我离开了卡昂，这个我生活的地方，受我所在学院的资助进行一次旅行。旅途上的经历让我暂且忘记了数学工作。到达古特昂司后，我们上了一辆公共马车以去到另外的什么地方。突然，当我刚登上马车阶梯的刹那，一个想法来临，之前没有任何想法为之做准备，即我用来定义Fuchsian函数的变换实际上与非欧几何中的是等价的。我没有去证明这个想法；我几乎没有时间，因为一上马车我就参与了另一场已经开始的谈话，但我对自己的想法确信无疑。回到卡昂以后，我在良心催促下利用空余时间完成了证明。Then I turned my attention to the study of some arithmetical questions apparently without much success and without a suspicion of any connection with my preceding researches. Disgusted with my failure, I went to spend a few days at the seaside, and thought of something else. One morning, walking on the bluff, the idea came to

me, with just the same characteristics of brevity, suddenness and immediate certainty that the arithmetic transformations of indeterminate ternary quadratic forms were identical with those of non-Euclidean geometry.

随后，我把注意力转回到一些算术问题的研究上，这些问题一直没有进展，而且显然和我之前的研究没有丝毫联系。吸取了之前失败的教训，我到海边去了几天，并且考虑了一些其他事情。一天早晨，正在悬崖边散步，我忽然有了主意，想法和上次一样具有同样简短、突然的性质，而且几乎立即就能肯定，算术问题中的三元不定二次型变换等价于非欧几何中的变换。

Returned to Caen, I meditated on this result and deduced the consequences. The example of quadratic forms showed me that there were Fuchsian groups other than those corresponding to the hypergeometric series; I saw that I could apply to them the theory of theta-Fuchsian series and that consequently there existed Fuchsian functions other than those from the hypergeometric series, the ones I then knew. Naturally I set myself to form all these functions. I made a systematic attack upon them and carried all the outworks, one after another. There was one, however, that still held out, whose fall would involve that of

the whole place. But all my efforts only served at first the better to show me the difficulty, which indeed was something. All this work was perfectly conscious.

回到卡昂，我对此结果进行思考，并得到了一些结论。二次型的例子表明存在Fuchsian群而无须使用超几何序列的方法；我意识到可以对此应用theta-Fuchsian序列的理论，并得出Fuchsian方程而无须引入超几何序列。很自然地，我着手构建这些方程。我系统性地研究他们并且得到一个接一个的结论。然而，有一个问题迟迟不能解决，并且结果很可能影响到全局。因此我的努力一开始似乎效果很好，却只是引入了更加困难的问题。这些工作都是在我有意识地状态下完成的。

Thereupon I left for Mont-Valérien, where I was to go through my military service; so I was very differently occupied. One day, going along the street, the solution of the difficulty which had stopped me suddenly appeared to me. I did not try to go deep into it immediately, and only after my service did I again take up the question. I had all the elements and had only to arrange them and put them together. So I wrote out my final memoir at a single stroke and without difficulty.

I shall limit myself to this single example; it is useless to

multiply them...

后来我去了瓦勒里昂山服兵役；因此我忙着完全不相干的事情。有一天，正沿街走着，难题的解突然出现在我面前。我当时并没有立即深入，直到服完兵役，我才重新拾起该问题。我有了所有的元素，只需要重新安置它们即可、因此我一次性就写完了最后的论文集，没有遇到任何困难。

我只用这一个例子，因为多说无益。

Most striking at first is this appearance of sudden illumination, a manifest sign of long, unconscious prior work. The role of this unconscious work in mathematical invention appears to me incontestable, and traces of it would be found in other cases where it is less evident. Often when one works at a hard question, nothing good is accomplished at the first attack. Then one takes a rest, longer or shorter, and sits down anew to the work. During the first half-hour, as before, nothing is found, and then all of a sudden the decisive idea presents itself to the mind... 一开始，最令人震惊的是福至心灵的瞬间，反映了长时间、无意识的预备工作。数学创造中，无意识工作的重要性对我而言是无可争议的，并且可以在其他例子中，无意识工作不那么明显的情况下找到它的痕迹。通常，一个人思考困难问题时，最初的工作不会带来太好的结果。休息一会儿，或者

更长时间，然后重新坐下来工作。在开始半小时，像以前一样，没什么发现，然而突然决定性的想法呈现在思想面前。There is another remark to be made about the conditions

of this unconscious work; it is possible, and of a certainty it is only fruitful, if it is on the one hand preceded and on the other hand followed by a period of conscious work. These sudden inspirations (and the examples already cited prove this) never happen except after some days of voluntary effort which has appeared absolutely fruitless and whence nothing good seems to have come, where the way taken seems totally astray. These efforts then have not been as sterile as one thinks; they have set agoing the unconscious machine and without them it would not have moved and would have produced nothing...

关于无意识工作的条件还有另一个说法：它是可能的，并且某种程度上富有成效，仅当之前和之后都进行了长期有意识的工作。这些突然的灵感（之前的例子已经表明了）发生，仅当之前大量的努力失败，并且没有好的结果发生，而采取的方法又是不确定的。这些努力并不像我们认为的毫无作用；它们启动了无意识的机器，否则这架机器永远不会运转起来，也不会有所成就。

Such are the realities; now for the thoughts they force upon

us. The unconscious, or, as we say, the subliminal self plays an important role in mathematical creation; this follows from what we have said. But usually the subliminal self is considered as purely automatic. Now we have seen that mathematical work is not simply mechanical, that it could not be done by a machine, however perfect. It is not merely a question of applying rules, of making the most combinations possible according to certain fixed laws. The combinations so obtained would be exceedingly numerous, useless and cumbersome. The true work of the inventor consists in choosing among these combinations so as to eliminate the useless ones or rather to avoid the trouble of making them, and the rules which must guide this choice are extremely fine and delicate. It is almost impossible to state them precisely; they are felt rather than formulated. Under these conditions, how imagine a sieve capable of applying them mechanically?

这就是现实；现在我们来分析一下数学家们的想法。无意识，或称之为，潜意识的自我在数学创造中占有重要地位；这和我们之前所说是一致的。然而通常潜意识的自我被认为是完全自发的。现在我们看到数学工作并不是简单的机械运算，一架无论如何精密的仪器是不能胜任数学工作的。它也不仅

仅是应用规则的问题，不是根据某些固定法则做出最可能的组合。这样得到的组合可能数量中毒，但毫无用处、繁冗不堪。创造者的真实工作包括在这些组合中进行挑选以减少无用的，避免制造无用结论的麻烦，并且借以挑选的法则必须极其精致准确。几乎不可能准确陈述这些法则；我们更多地是感觉而不是规定。在这些条件下，怎能想象机械地来应用这些法则呢？

A first hypothesis now presents itself; the subliminal self is in no way inferior to the conscious self; it is not purely automatic; it is capable of discernment; it has tact, delicacy; it knows how to choose, to divine. What do I say? It knows better how to divine than the conscious self, since it succeeds where that has failed. In a word, is not the subliminal self superior to the conscious self? You recognize the full importance of this question...

现在我们可以得到第一个假设；潜意识的自我并不比意识层面上的自我表现要差；潜意识并不是自动的；它能够鉴别；它手法老练，技巧精湛；它知道如何挑选，它发掘。“发掘”的意思是，它知道如何超越意识层面的自我，因为它能完成有意识不能完成的事情。一句话而言，是否潜意识的自我比有意识的自我更好？至少，你现在应该了解这个问题的重要性。

Is this affirmative answer forced upon us by the facts I have just given? I confess that, for my part, I should hate to accept it. Re-examine the facts then and see if they are not compatible with another explanation.

但，是否我刚才举的实例就能确定无误地说明问题呢？我承认，在我的方面，我不喜欢这个结论。让我们重新检查这个例子，看看能不能得到其他解释。

It is certain that the combinations which present themselves to the mind in a sort of sudden illumination, after an unconscious working somewhat prolonged, are generally useful and fertile combinations, which seem the result of a first impression. Does it follow that the subliminal self, having divined by a delicate intuition that these combinations would be useful, has formed only these, or has it rather formed many others which were lacking in interest and have remained unconscious?

可以确定的是，在潜意识进行一段时间的工作后，就会产生灵光一现的想法，并且通常是相当有用的结论组合，就像是第一印象得出的结论。这是由于潜意识的自我产生的微妙直觉，判断出哪些组合是有用的，因此只采用了这些组合吗？还是同样产生了很多其他组合，而由于其无用性而停留在潜意识中？

In this second way of looking at it, all the combinations would be formed in consequence of the automatism of the subliminal self, but only the interesting ones would break into the domain of consciousness. And this is still very mysterious. What is the cause that, among the thousand products of our unconscious activity, some are called to pass the threshold, while others remain below? Is it a simple chance which confers this privilege? Evidently not; among all the stimuli of our senses, for example, only the most intense fix our attention, unless it has been drawn to them by other causes. More generally the privileged unconscious phenomena, those susceptible of becoming conscious, are those which, directly or indirectly, affect most profoundly our emotional sensibility.

用第二种角度来看，潜意识自动将所有组合产生，但只有最令人感兴趣的能突破意识的界限。这仍是非常神秘的。是什么愿意，使得我们的潜意识活动产生的千百种产品，只有少数能突破界限，而大多数停留在意识层面之下？仅仅是因为机会的原因，让它们得到了这些特权？显然不是；例如就我们的感觉而言，受到的所有的刺激，除非是特殊原因，只有那些最强烈的能够吸引我们的注意力。一般而言，那些比较特殊的潜意识活动，那些可能跃而成为意识层面的，是那些

直接或间接能深刻影响我们情感认知的活动。

It may be surprising to see emotional sensibility invoked apropos of mathematical demonstrations which, it would seem, can interest only the intellect. This would be to forget the feeling of mathematical beauty, of the harmony of numbers and forms, of geometric elegance. This is a true esthetic feeling that all real mathematicians know, and surely it belongs to emotional sensibility.

看到情感认知能唤起适当的数学证明，也许令人惊异；因为我们本认为这仅仅关系到智力。那是因为我们忘了这其中存在对数学之美的感知，对数和形式的协调感，以及几何优雅性的体验。这是所有真正数学家应当了解的美感，并且显然属于情感认知的范畴。

Now, what are the mathematic entities to which we attribute this character of beauty and elegance, and which are capable of developing in us a sort of esthetic emotion? They are those whose elements are harmoniously disposed so that the mind without effort can embrace their totality while realizing the details. This harmony is at once a satisfaction of our esthetic needs and an aid to the mind, sustaining and guiding. And at the same time, in putting under our eyes a well-ordered whole, it makes us foresee

数学专业英语

数学专业英语课后答案

2.1数学、方程与比例 词组翻译 1.数学分支branches of mathematics，算数arithmetics，几何学geometry，代数学algebra，三角学trigonometry，高等数学higher mathematics，初等数学elementary mathematics，高等代数higher algebra，数学分析mathematical analysis，函数论function theory，微分方程differential equation 2.命题proposition，公理axiom，公设postulate，定义definition，定理theorem，引理lemma，推论deduction 3.形form，数number，数字numeral，数值numerical value，图形figure，公式formula，符号notation(symbol)，记法/记号sign，图表chart 4.概念conception，相等equality，成立/真true，不成立/不真untrue，等式equation，恒等式identity，条件等式equation of condition，项/术语term，集set，函数function，常数constant，方程equation，线性方程linear equation，二次方程quadratic equation 5.运算operation，加法addition，减法subtraction，乘法multiplication，除法division，证明proof，推理deduction，逻辑推理logical deduction 6.测量土地to measure land，推导定理to deduce theorems，指定的运算indicated operation，获得结论to obtain the conclusions，占据中心地位to occupy the centric place 汉译英 （1）数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。 Mathematics comes from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches. （2）如果没有运用数学，任何一个科学技术分支都不可能正常地发展。 No modern scientific and technological branches could be regularly developed without the application of mathematics. （3）符号在数学中起着非常重要的作用，它常用于表示概念和命题。 Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often. （4）17 世纪之前，人们局限于初等数学，即几何、三角和代数，那时只考虑常数。Before 17th century, man confined himself to the elementary mathematics, i. e. , geometry, trigonometry and algebra, in which only the constants were considered. （5）方程与算数的等式不同在于它含有可以参加运算的未知量。 Equation is different from arithmetic identity in that it contains unknown quantity which can join operations. （6）方程又称为条件等式，因为其中的未知量通常只允许取某些特定的值。Equipment is called an equation of condition in that it is true only for certain values of unknown quantities in it. （7）方程很有用，可以用它来解决许多实际应用问题。

数学专业英语课文翻译(吴炯圻)第二章 2.

数学专业英语课文翻译(吴炯圻)第二 章 2. 数学专业英语3—A 符号指示集一组的概念如此广泛利用整个现代数学的认识是所需的所有大学生。集是通过集合中一种抽象方式的东西的数学家谈的一种手段。集，通常用大写字母：A、B、C、进程运行·、X、Y、Z ；小写字母指定元素：a、 b 的c、进程运行·，若x、y z.我们用特殊符号x∈S 意味着x 是S 的一个元素或属于美国的x如果x 不属于S，我们写xS.≠当方便时，我们应指定集的元素显示在括号内；例如，符号表示的积极甚至整数小于10 集{2,468} {2，，进程运行·} 作为显示的所有积极甚至整数集，而三个点等的发生。点的和等等的意思是清楚时，才使用。上市的大括号内的一组成员方法有时称为名册符号。涉及

到另一组的第一次基本概念是平等的集。DEFINITIONOFSETEQUALITY。两组A 和B，据说是平等的如果它们包含完全相同的元素，在这种情况下，我们写A = B。如果其中一套包含在另一个元素，我们说这些集是不平等，我们写 A = B。EXAMPLE1。根据对这一定义，于他们都是构成的这四个整数2，和8 两套{2,468} 和{2,864} 一律平等。因此，当我们用来描述一组的名册符号，元素的显示的顺序无关。动作。集{2,468} 和{2,2,4,4,6,8} 是平等的即使在第二组，每个元素 2 和 4 两次列出。这两组包含的四个要素2,468 和无他人；因此，定义要求我们称之为这些集平等。此示例显示了我们也不坚持名册符号中列出的对象是不同。类似的例子是一组在密西西比州，其值等于{M、我、s、p} 一组单词中的字母，组成四个不同字母M、我、s 和体育3 —B 子集S.从给定的集S，我们

数学专业英语2-10翻译

Although dependence and independence are properties of sets of elements, we also apply these terms to the elements themselves. For example, the elements in an independent set are called independent elements. 虽然相关和无关是元素集的属性，我们也适用于这些元素本身。 例如，在一个独立设定的元素被称为独立元素。 If s is finite set, the foregoing definition agrees with that given in Chapter 8 for the space n V . However, the present definition is not restricted to finite sets. 如果S 是有限集，同意上述定义与第8章中给出的空间n V ，然而，目前的定义不局限于有限集。 If a subset T of a set S is dependent, then S itself is dependent. This is logically equivalent to the statement that every subset of an independent set is independent. 如果集合S 的子集T 是相关的，然后S 本身是相关的，这在逻辑上相当于每一个独立设置的子集是独立的语句。 If one element in S is a scalar multiple of another, then S is dependent. 如果S 中的一个元素是另一个集中的多个标量的，则S 是相关的。 If S ∈0,then S is dependent. 若S ∈0，则 S 是相关的。 The empty set is independent. 空集是无关的。 Many examples of dependent and independent sets of vectors in V were discussed in Chapter 8. The following examples illustrate these concepts in function spaces. In each case the underlying linear space V is the set of all real-valued function defined on the real line. V 中的向量的相关和无关设置的许多例子是在第8章讨论。下面的例子说明这些概念在函数空间。在每个 基本情况下，线性空间V 是实线定义的所有实值函数集。 Let 1)(),(sin )(,cos )(32221===t u t t u t t u for all real t. The Pythagorean identity show that 0321=-+u u u , so the three functions 321,,u u u are dependent. 321,,u u u 是相关的。 Let k k t t u =)(for k=0,1,2,…, and t real. The set ,...},,{210u u u S = is independent. To prove this, it suffices to show that for each n the n+1 polynomials n u u u ,...,,10 are independent. A relation of the form ∑=0k k u c means that

数学专业英语第二版-课文翻译-converted

2.4 整数、有理数与实数 4－A Integers and rational numbers There exist certain subsets of R which are distinguished because they have special properties not shared by all real numbers. In this section we shall discuss such subsets, the integers and the rational numbers. 有一些R 的子集很著名，因为他们具有实数所不具备的特殊性质。在本节我们将讨论这样的子集，整数集和有理数集。 To introduce the positive integers we begin with the number 1, whose existence is guaranteed by Axiom 4. The number 1+1 is denoted by 2, the number 2+1 by 3, and so on. The numbers 1,2,3,…, obtained in this way by repeated addition of 1 are all positive, and they are called the positive integers. 我们从数字 1 开始介绍正整数，公理 4 保证了 1 的存在性。1+1 用2 表示，2+1 用3 表示，以此类推，由 1 重复累加的方式得到的数字 1,2,3，…都是正的，它们被叫做正整数。 Strictly speaking, this description of the positive integers is not entirely complete because we have not explained in detail what we mean by the expressions “and so on”, or “repeated addition of 1”. 严格地说，这种关于正整数的描述是不完整的，因为我们没有详细解释“等等”或者“1的重复累加”的含义。 Although the intuitive meaning of expressions may seem clear, in careful treatment of the real-number system it is necessary to give a more precise definition of the positive integers. There are many ways to do this. One convenient method is to introduce first the notion of an inductive set. 虽然这些说法的直观意思似乎是清楚的，但是在认真处理实数系统时必须给出一个更准确的关于正整数的定义。有很多种方式来给出这个定义，一个简便的方法是先引进归纳集的概念。 DEFINITION OF AN INDUCTIVE SET. A set of real number s is cal led an i n ductiv e set if it has the following two properties: (a) The number 1 is in the set. (b) For every x in the set, the number x+1 is also in the set. For example, R is an inductive set. So is the set . Now we shall define the positive integers to be those real numbers which belong to every inductive set. 现在我们来定义正整数，就是属于每一个归纳集的实数。 Let P d enote t he s et o f a ll p ositive i ntegers. T hen P i s i tself a n i nductive set b ecause (a) i t contains 1, a nd (b) i t c ontains x+1 w henever i t c ontains x. Since the m embers o f P b elong t o e very inductive s et, w e r efer t o P a s t he s mallest i nductive set. 用 P 表示所有正整数的集合。那么 P 本身是一个归纳集，因为其中含 1，满足(a)；只要包含x 就包含x+1, 满足(b)。由于 P 中的元素属于每一个归纳集，因此 P 是最小的归纳集。 This property of P forms the logical basis for a type of reasoning that mathematicians call proof by induction, a detailed discussion of which is given in Part 4 of this introduction.

数学专业英语课后答案

2.1数学、方程与比例 词组翻译 1.数学分支branches of mathematics，算数arithmetics，几何学geometry，代数学algebra，三角学trigonometry，高等数学higher mathematics，初等数学elementary mathematics，高等代数higher algebra，数学分析mathematical analysis，函数论function theory，微分方程differential equation 2.命题proposition，公理axiom，公设postulate，定义definition，定理theorem，引理lemma，推论deduction 3.形form，数number，数字numeral，数值numerical value，图形figure，公式formula，符号notation(symbol)，记法/记号sign，图表chart 4.概念conception，相等equality，成立/真true，不成立/不真untrue，等式equation，恒等式identity，条件等式equation of condition，项/术语term，集set，函数function，常数constant，方程equation，线性方程linear equation，二次方程quadratic equation 5.运算operation，加法addition，减法subtraction，乘法multiplication，除法division，证明proof，推理deduction，逻辑推理logical deduction 6.测量土地to measure land，推导定理to deduce theorems，指定的运算indicated operation，获得结论to obtain the conclusions，占据中心地位to occupy the centric place 汉译英 （1）数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。 Mathematics comes from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches. （2）如果没有运用数学，任何一个科学技术分支都不可能正常地发展。 No modern scientific and technological branches could be regularly developed without the application of mathematics. （3）符号在数学中起着非常重要的作用，它常用于表示概念和命题。 Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often. （4）17 世纪之前，人们局限于初等数学，即几何、三角和代数，那时只考虑常数。 Before 17th century, man confined himself to the elementary mathematics, i. e. , geometry, trigonometry and algebra, in which only the constants were considered. （5）方程与算数的等式不同在于它含有可以参加运算的未知量。 Equation is different from arithmetic identity in that it contains unknown quantity which can join operations. （6）方程又称为条件等式，因为其中的未知量通常只允许取某些特定的值。Equipment is called an equation of condition in that it is true only for certain values of unknown quantities in it. （7）方程很有用，可以用它来解决许多实际应用问题。

数学专业英语(Doc版).10

学专业英语－How to Write Mathematics? How to Write Mathematics? ------ Honesty is the Best Policy The purpose of using good mathematical language is, of course, to make the u nderstanding of the subject easy for the reader, and perhaps even pleasant. The style should be good not in the sense of flashy brilliance, but good in the se nse of perfect unobtrusiveness. The purpose is to smooth the reader’s wanted, not pedantry; understanding, not fuss. The emphasis in the preceding paragraph, while perhaps necessary, might see m to point in an undesirable direction, and I hasten to correct a possible misin terpretation. While avoiding pedantry and fuss, I do not want to avoid rigor an d precision; I believe that these aims are reconcilable. I do not mean to advise a young author to be very so slightly but very very cleverly dishonest and to gloss over difficulties. Sometimes, for instance, there may be no better way t o get a result than a cumbersome computation. In that case it is the author’s duty to carry it out, in public; the he can do to alleviate it is to extend his s ympathy to the reader by some phrase such as “unfortunately the only known proof is the following cumbersome computation.” Here is the sort of the thing I mean by less than complete honesty. At a certa in point, having proudly proved a proposition P, you feel moved to say: “Not e, however, that p does not imply q”, and then, thinking that you’ve done a good expository job, go happily on to other things. Your motives may be per fectly pure, but the reader may feel cheated just the same. If he knew all abo ut the subject, he wouldn’t be reading you; for him the nonimplication is, qui te likely, unsupported. Is it obvious? (Say so.) Will a counterexample be suppl ied later? (Promise it now.) Is it a standard present purposes irrelevant part of the literature? (Give a reference.) Or, horrible dictum, do you merely mean th at you have tried to derive q from p, you failed, and you don’t in fact know whether p implies q? (Confess immediately.) any event: take the reader into y our confidence. There is nothing wrong with often derided “obvious”and “easy to see”, b ut there are certain minimal rules to their use. Surely when you wrote that so mething was obvious, you thought it was. When, a month, or two months, or six months later, you picked up the manuscript and re-read it, did you still thi nk that something was obvious? (A few months’ripening always improves ma nuscripts.) When you explained it to a friend, or to a seminar, was the someth ing at issue accepted as obvious? (Or did someone question it and subside, mu ttering, when you reassured him? Did your assurance demonstration or intimida

数学专业英语课文翻译(吴炯圻)第二章2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12

数学专业英语

3—A 符号指示集 一组的概念如此广泛利用整个现代数学的认识是所需的所有大学生。集是通过集合中一种抽象方式的东西的数学家谈的一种手段。 集，通常用大写字母：A、B、C、进程运行·、X、Y、Z ；由小写字母指定元素：a、 b 的c、进程运行·，若x、y z.我们用特殊符号x∈S 意味着x 是S 的一个元素或属于美国的x如果x 不属于S，我们写xS.≠当方便时，我们应指定集的元素显示在括号内；例如，由符号表示的积极甚至整数小于10 集{2,468} {2，4.6，进程运行·} 作为显示的所有积极甚至整数集，而三个点等的发生。点的和等等的意思是清楚时，才使用。上市的大括号内的一组成员方法有时称为名册符号。 涉及到另一组的第一次基本概念是平等的集。 DEFINITIONOFSETEQUALITY。两组A 和B，据说是平等的（或相同的）如果它们包含完全相同的元素，在这种情况下，我们写A = B。如果其中一套包含在另一个元素，我们说这些集是不平等，我们写A = B。 EXAMPLE1。根据对这一定义，由于他们都是由构成的这四个整数2，4.6 和8 两套{2,468} 和{2,864} 一律平等。因此，当我们用来描述一组的名册符号，元素的显示的顺序无关。 动作。集{2,468} 和{2,2,4,4,6,8} 是平等的即使在第二组，每个元素2 和4 两次列出。这两组包含的四个要素2,468 和无他人；因此，定义要求我们称之为这些集平等。此示例显示了我们也不坚持名册符号中列出的对象是不同。类似的例子是一组在密西西比州，其值等于{M、我、s、p} 一组单词中的字母，组成四个不同字母M、我、s 和体育 3 —B子集 S.从给定的集 S，我们可能会形成新集，称为.的子集例如，组成的那些正整数小于 10 整除 4 （集合{8 毫米}）的一组一般是的所有甚至小于 10.整数集的一个子集，我们有以下的定义。子集的定义。A 一组据说是B，集的一个子集，我们写A B每当A的每个元素也属于B.我们还说包含B A或B包含。关系称为集。A和B的声明并不排除可能性，B。事实上，我们可能B A和B A，但只有当A和B都具有相同的元素发生这种情况。换句话说，A = B当且仅当B和B A。这一命题是上述定义的平等和包容的直接后果。如果A和B，但A≠B，然后我们说的就是你的真子集我们表明这通过编写B.在所有的应用程序集理论，我们有一套固定事先，S，我们只关心这给定组的子集。底层的设置的不同而有所不同从一个应用程序，到另一台；它将转交作为每个特定的话语的通用组。符号{X∣X∈S和X满足P}，将指定的所有元素X在S中满足该属性集体育当通用设置为我们所指的id的理解，我们省略参照以S，我们只需写{X∣X满足P}。这读取 '"集的所有这种x满足p。' "在此方法中指定的设置说笔下定义的属性，例如，所有正实数的一组可以被指定为{X∣X大于 0} ；通用集S，在这种情况下理解为所有实数集。当然，这封信x是个笨蛋，并可由任何其他方便的符号替换。因此，我们可以写{x∣x大于0} = {y∣y大于0} = {t∣t大于 0} 等等。它有可能设置为不包含任何元素。这套被称为空集或无效设置，并将由symbolφ表示。我们会考虑φto是每一集的一个子集。有些人觉得很有用的一套类似于一个容器（例如，一个袋子或框）包含某些对象，其元素。空集则类似于一个空的容器。为了避免逻辑的困难，我们必须区分元素x和集{x}的唯一元素是x,(A box with a hat in it is conceptually distinct from the hat itself.)尤其是，空的setφis集合{φ}不相同。事实上，空设置φcontains没有元素而集{φ}有一个元素φ（一个框，其中包含一个空框不是空的）。组成一个元素的集合，有时也称为一个元素集。 2.4 整数、有理数与实数 4－A Integers and rational numbers There exist certain subsets of R which are distinguished because they have special properties not shared by all real numbers. In this section we shall discuss such subsets, the integers and the rational numbers. 有一些 R 的子集很著名，因为他们具有实数所 不具备的特殊性质。在本节我们将讨论这样的子集，整数集和有理数集。 To introduce the positive integers we begin with the number 1, whose existence is guaranteed by Axiom 4. The number 1+1 is denoted by 2, the number 2+1 by 3, and so on. The

数学专业英语第二版的课文翻译

1－A What is mathematics Mathematics comes from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches. And in turn, mathematics serves the practice and plays a great role in all fields. No modern scientific and technological branches could be regularly developed without the application of mathematics. 数学来源于人类的社会实践，比如工农业生产，商业活动，军事行动和科学技术研究。反过来，数学服务于实践，并在各个领域中起着非常重要的作用。没有应用数学，任何一个现在的科技的分支都不能正常发展。From the early need of man came the concepts of numbers and forms. Then, geometry developed out of problems of measuring land , and trigonometry came from problems of surveying . To deal with some more complex practical problems, man established and then solved equation with unknown numbers ,thus algebra occurred. Before 17th century, man confined himself to the elementary mathematics, . , geometry, trigonometry and algebra, in which only the constants are considered. 很早的时候，人类的需要产生了数和形式的概念，接着，测量土地的需要形成了几何，出于测量的需要产生了三角几何，为了处理更复杂的实际问题，人类建立和解决了带未知参数的方程，从而产生了代数学，17世纪前，人类局限于只考虑常数的初等数学，即几何，三角几何和代数。The rapid development of industry in 17th century promoted the progress of economics and technology and required dealing with variable quantities. The leap from constants to variable quantities brought about two new branches of mathematics----analytic geometry and calculus, which belong to the higher mathematics. Now there are many branches in higher mathematics, among which are mathematical analysis, higher algebra, differential equations, function theory and so on. 17世纪工业的快速发展推动了经济技术的进步，从而遇到需要处理变量的问题，从常数带变量的跳跃产生了两个新的数学分支-----解析几何和微积分，他们都属于高等数学，现在高等数学里面有很多分支，其中有数学分析，高等代数，微分方程，函数论等。Mathematicians study conceptions and propositions, Axioms, postulates, definitions and theorems are all propositions. Notations are a special and powerful

数学专业英语课后答案.doc 2

2.1 数学、方程与比例 （1）数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。 Mathematics comes from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches. （2）如果没有运用数学，任何一个科学技术分支都不可能正常地发展。 No modern scientific and technological branches could be regularly developed without the application of mathematics. （3）符号在数学中起着非常重要的作用，它常用于表示概念和命题。Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often. （4）17 世纪之前，人们局限于初等数学，即几何、三角和代数，那时只考虑常数。 Before 17th century, man confined himself to the elementary mathematics, i. e. , geometry, trigonometry and algebra, in which only the constants were considered. （5）方程与算数的等式不同在于它含有可以参加运算的未知量。 Equation is different from arithmetic identity in that it contains unknown quantity which can join operations. （6）方程又称为条件等式，因为其中的未知量通常只允许取某些特定的值。Equipment is called an equation of condition in that it is true only for certain values of unknown quantities in it. （7）方程很有用，可以用它来解决许多实际应用问题。 Equations are of very great use. We can use equations in many mathematical problems. （8）解方程时要进行一系列移项和同解变形，最后求出它的根，即未知量的值。To solve the equation means to move and change the terms about without making the equation untrue, until the root of the equation is obtained, which is the value of unknown term. 2.3 集合论的基本概念 （1）由小于10 且能被 3 整除的正整数组成的集是整数集的子集。 The set consisting of those positive integers less than 10 which are divisible by 3 is a subset of the set of all integers. （2）如果方便，我们通过在括号中列举元素的办法来表示集。 When convenient, we shall designate sets by displaying the elements in braces. （3）用符号?表示集的包含关系，也就是说，式子 A ? B 表示 A 包含于B。 The relation ?is referred to as set inclusion; A?B means that A is contained in B. （4）命题 A ? B 并不排除 B ? A 的可能性。 The statement A?B does not rule out the possibility that B?A.

数学专业英语第二版2.2课文翻译

2.2 A 为什么要研究几何？ 我们为什么研究几何？开始此文本研究的学生也许会问，"什么是几何。什么可以预料从这项研究获得？" 许多居于领导地位的学术机构承认，所有学习这个数学分支的人都将得到确实的收益，许多学校把几何的学习作为入学考试的先决条件，从这一点上可以证明。 几何学起源于很久以前巴比伦人和埃及人测量他们被尼罗河洪水淹没的土地，希腊语几何来源于geo ，意思是“土地”，和metron, 意思是"度量值"。早在公元前2000 年，我们发现这些民族的土地测量者利用几何知识重新确定消失了的土地标志和边界。 几何是研究由线所组成的图形的科学。几何的学习是成功工程师、科学家、建筑师和制图员培训的重要部分。木匠、机械师、采石者、艺术家和设计师在他们的职业中都应用几何的知识。在这门课程中，学生将学到大量几何图形，例如线条、角、三角形、圆和许多种设计和模式。 所得的几何研究的最重要的目标之一是使学生在他的听力、阅读和思维更审慎。学习几何他远离盲目接受语句和思想的实践领导和教想清楚与批判前形成的结论。学习几何使他被领导远离语句和思想的盲目接受的做法，教导形成结论之前，考虑清楚和审慎。（学生通过几何的学习而达到的最重要目标是：在听，读，和思考时变得更加审慎。在学习几何的过程中，他们不再盲目地接受一些陈述和思想，而是在得出结论之前学会了清楚和审慎的思考。） 学习几何的学生可以获得许多其他不太直接的利益。这些人当中必须包括训练英语的精确使用和分析新情况与新问题时直达基本要素，以及利用毅力、创意和逻辑推理来解决问题的能力。（这些人当中必须包括训练英语的精确使用和分析新情况与新问题时直达基本要素的能力，以及利用毅力、创意和逻辑推理来解决问题。）大自然的创作将作为一种几何研究的副产品。一种鉴赏能力属于几何形成的规律和审美在人们的作品中。学生也应该发展数学与我们的文化和文明数学家贡献的意识。 2.2B 一些几何术语 1.立体和平面。立体是一个三维图形。立体的常见示例是立方体、球体、柱体、圆锥和棱锥。立方体有光滑、平整的六个面。这些面被称为平面曲面，或简称平面。平面曲面是二维的，有长度和宽度。黑板或桌面的表面都是平面曲面的一个例子。 2、线条和线段。我们都很熟悉线，但很难定义这术语。一条线可由在一张纸上移动的钢笔或铅笔标记表示。一条线，可以被看做是一维的，即只有长度。尽管我们绘制一条直线时会赋予它宽度和厚度，但是当考虑线时，只考虑痕迹的长度。点没有长度、宽度和厚度，但标记了一个位置。我们熟悉铅笔尖，针尖这样的表达。我们可以用一个小圆点来表示一个点，在它旁边用打印体大写字母来命名，如图2-2-1中的点A。 直线用大写字母标志它上面的两个点或者旁边的一个小写字母来命名。图2-2-2 的直线是读"直线AB "或者“直线l"。直线向两个方向无限延伸，没有终点。线上两点间的部分被称为一条直线段。直线段用两个端点命名。因此，图2-2-2，我们称为AB 是线l 的一条直线段。当不引起混淆时，表达"直线段AB 通常被线段AB代替，或者简称直线AB。 有三种线：直线、折线和曲线。弯曲的线条或，简单地说，曲线是指其中没有任何部分是直的折线是由连起来的直线段构成，如图2-2-3的ABCDE。 3.圆的部分。平面上的闭曲线当其中每点到一个固定点的距离均相当时叫做圆。固定点称为圆心。图2-2-4，O 是ABC 中心，或简单的O.A连接圆心到圆周上点的直线段称为圆的半径。OA，OB，以及OC都是圆O的半径。经过圆心并且两个端点在圆周上的直线段被称为圆的直径。直径等于两个半径。连接圆周上两点的任意直线段被称为弦。图2-2-4 圆的弦是ED。 从这个定义很明显直径是弦。圆周的任何部分都是一条弧线，如弧AE，其中由AE 表示。