Number rules the universe.




    Reason is immortal, all else mortal.





    The highest form of pure thought is in mathematics”.






    Nature’s great book is written in mathematical symbol.




    Mathematics is the Queen of the Sciences,and Arithmetic the Queen of Mathematics.


                              ——高斯 Gauss




    Wir müssen wissen, wir werden wissen.

    We must know, we will know.






    Pure mathematics is,in its way,the poetry of logical ideas.





    The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience.





    Histories make men wise; poets, witty; the mathematics, subtle; natural philosophy, deep; moral, grave; logic and rhetoric, able to contend.


              ——培根(Francis Bacon)




     The only way to learn mathematics is to do mathematics.
           —— 哈尔莫斯(Paul Halmos)

   I believe in intuition and inspiration. Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution.



     Logic will get you from A to B. Imagination will take you everywhere.


    A true mathematician who is not also something of a poet will never be a perfect mathematician.

    Mathematics, rightly viewed, possesses not only truth, but supreme beauty, a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.
   < 拙译> 数学,当你正确地看待它时,不仅拥有真,而且拥有非凡的美 —— 一种犹如雕塑般冷峻而素朴的美,一种不引诱任何我们的较软弱天性的美,一种没有绘画和音乐那样富丽花俏的装饰的纯净之至的美,同时又能达到一种唯有最伟大的艺术才能表达的严格的完美。
               ——罗素(Bertrand Russell

   The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.
   < 拙译> 正如在诗歌中一样,在数学中同样能找到真正的欢乐的精神,升华的快感,那种超越凡俗接近神祗的美妙感觉——它是最高级的卓越成就的试金石。
               ——罗素(Bertrand Russell

    Besides language and music, it [mathematics] is one of the primary manifestations of the free creative power of the human mind, and it is the universal organ for world-understanding through theoretical construction. Mathematics must therefore remain an essential element of the knowledge and abilities which we have to teach, of the culture we have to transmit, to the next generation.
    < 拙译> 数学是除了语言与音乐之外,人类心灵自由创造力的主要表达方式之一,而且它是通过理论的构建理解宇宙万物的普适工具。因此,数学必须始终是我们得传授给下一代的知识和技能的要素和我们要传承给下一代的文化的要素。
                   ——外尔(Hermann Weyl)



    What we do may be small , but it has a certain character of permanence ; and to have produced anything of the slightest permanent interest , whether it be a copy of verses or a geometrical theorem , is to have done something utterly beyond the powers of the vast majority of men .
    < 拙译>我们所做的或许微不足道,但它具有某种永恒的特征;而创造了具有一丝一毫永久性兴趣的任何东西,不管是几行诗句也好,或者是一个几何定理也罢,已经是做了某件完全超出芸芸众生绝大多数人能力之上的事情。



            (A Mathematician's Apology)

    Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. "Immortality" may be a silly word, but probably a mathematician has the best chance of whatever it may mean.

    < 拙译> 当埃斯库罗斯被人遗忘之际阿基米德仍会被人铭记,因为语言会死亡而数学思想不会。“长生不死”是一个愚蠢的词,但不管它实际上何所指,数学家当属首选。



            (A Mathematician's Apology)



    If intellectual curiosity , professional pride , and ambition are the dominant incentives to research , then assuredly no one has a fairer chance of gratifying them than a mathematician. His subject is the most curious of all --- there is none in which truth plays such odd pranks. It has the most elaborate and the most fascinating technique, and gives unrivalled openings for the display of sheer professional skill. Finally , as history proves abundantly , mathematical achievement, whatever its intrinsic worth , is the most enduring of all.

     <拙译> 如果理智的好奇,职业的自豪和抱负心是研究工作的主要激励,那么可以肯定没有谁比数学家享有更好地满足这些欲望的机会了。他的研究对象是世上最激发好奇心的——没有哪一个领域里的真理爱跟人耍如此奇诡的把戏。它具有最完善精致和最有趣迷人的工具,它给人表现纯粹的职业技巧无可匹敌的良机。最后,正如历史充分证明的那样,数学上的成就,无论它的内在价值如何,是世间最永久恒常的。



            (A Mathematician's Apology)

    But on the whole the history of science is fair, and this is particularly true in mathematics. No other subject has such clear-cut or unanimously accepted standards, and the men who are remembered are almost always the men who merit it. Mathematical fame, if you have the cash to pay it, is one of the soundest and steadiest of investments.
<拙译> 但在总体上讲,科学的历史是公平的,对数学尤然。没有一门其他的学科象数学一样有着如此清晰明确和被一致公认的标准,而那些被铭记的人几乎都是当之无愧的。数学上的荣誉,如果你有支付得起的现金的话,是最合理最稳固的投资之一。




            (A Mathematician's Apology)



——谢国芳(Roy Xie)    



1. 导论






 这一美妙的定理最早由公元四世纪左右的古希腊数学家帕普斯(Pappus)发现,所以被称为帕普斯定理。表面看上去它平凡无奇,所涉及的是几何中最简单不过的元素——点和直线,但真正要动手证明它却感到出奇地困难,简直是“无从下手”,“无计可施”(倘若你不知道梅涅劳斯定理的话 [2]),沮丧之余有一种十分“诡异”和“神秘”的感觉,觉得此定理简直有点“莫测高深”,“莫名其妙”。














我们将用数学中一个普遍有效的思想方法证明帕普斯定理(菲尔兹奖得主Timothy Gowers在论如何自主地发现三次方程的解法(How to Discover for Yourself the Solution of the Cubic)中讲到了同样的思想方法)


二、 第二步是“化一般为特殊”,设法把一般情形归化为上面这个已经解决的特殊情形,具体的办法是考虑某种变换,只要该变换是保线性的(即把直线变为直线),我们就“大功告成”了。









【注5】 这表明不计点的顺序的改变(即变换下标1,2,3),总共就只有上述这两种特殊情形。特殊情形Ⅰ是只有一对平行线的情形(或者说一个交点是无穷远点),特殊情形Ⅱ是有三对平行线(或者说三个交点都是无穷远点)的情形,恰有二对平行线(或者说两个交点是无穷远点,一个交点是有穷点))的情况是不可能出现的。用射影几何的语言说,通过两个无穷远点的直线就是无穷远直线 ,帕普斯定理保证了第三个交点也在这条无穷远直线 上,即也是一个无穷远点。由此我们看到,只有把无穷远点包括在内,即把欧氏平面拓展为射影平面,帕普斯定理才有一个统一完美的表述。但局限在欧氏平面内,我们只能把这两种特殊情形单独列出来,分别加以讨论。






  3.      特殊情形的证明


从射影几何的观点看,这一最特殊的情形就蕴含了一般:我们只要证明对这一最特殊的情形帕普斯定理成立(这时候三个交点都为无穷远点,它们所共的线就是无穷远直线 ),就可以推知对一般情形帕普斯定理也成立,因为只要作一个透视投影,把三个交点中的两个变为无穷远点,就把一般情形转化为了这一最特殊的情形。




【注7】 但另一方面,在一个更大的变换群下保持不变的东西会更少,这意味着一个越大的变换群所决定的几何学的内容会越贫乏(因此射影几何的内容比欧几里得几何贫乏)。







 未完待续(to be continued)



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