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风雨飘摇中的数学

2010年11月18日 发表评论 阅读评论

注:这是自然辩证法老师要求写的一篇读后感。老师给了比较多的书目,叫我们选一本读,我选了莫利斯·克莱因《数学:确定性的丧失》一书,读完后,有点感触,所以把读后感也发到这里。顺便感谢下老师,虽然可我只上周上过一节,但上后才发现,上的和我想象的政治课完全不一样。老师讲的很生动,后面放的纪录片也很好,哎,真不应该翘那么多课的。

以下是读后感正文(有部分内容直接引自《数学:确定性丧失一书》):

风雨飘摇中的数学

《数学:确定性的丧失》读后感

每天,太阳总是准时的升起,准时的降落;每年,春夏秋冬,四季轮回。对于这一切自然现象,我们毫无条件的相信,对其确定性也毫不怀疑:我们不会担心明天太阳不再升起,明年春天不会回来。

同样的,就像对于自然,我们对于数学,以及数学模型化的物理规律,甚至远至建立在物理规律的发明创造,都毫不犹豫的相信。我们相信,1+1必然等于2,我们相信,路程等于速度乘以时间,我们相信,当我们按下开关时,灯就会如我们所愿的打开或关闭。仿佛一切的一切,都是那么确定,都是毋庸置疑的。

然而,但我看过莫利斯·克莱因(1908.5.1—1992.5.10)所著的《数学:确定性的丧失》一书,我才发现,我们所相信的一切的一切,其根基是如此的脆弱:我们现在对世界,对自然的认识,都是建立在数学的基础上;而数学本身的基础,却是如此的脆弱,整个数学,包括我们建立在数学之上对世界的认识,都处于风雨飘摇之中。

数学发展史以及三次危机

数学的发展史,就是一部追求真理的探索史,也是一部不断解决各种危机的历史。

真理的起源-古希腊的数学以及第一次危机

最早创建数学是古埃及和古巴比伦的一些先驱者们,但数学作为一个独立知识体系却是起源于古希腊。数学最早来源于经验,是经验的抽象,升华,和体系化。柏拉图,阿基里德这些伟大的思想者,通过对大量经验总结,对前人的数学知识的整理,以及独立深邃的思考,建立了一套较完备的整数理论以及几何理论。和整数理论相伴的,人们毫不怀疑的以为,自然界的任何数都可以通过自然数和自然数的比(即可约)来表示。直到出现第一次数学危机-无理数的出现。

大约公元前5世纪,不可通约量的发现导致了毕达哥拉斯悖论并最终导致了无理数的提出。当时的主流数学学派-毕达哥拉斯学派认为:宇宙间一切事物都可归结为整数或整数之比,毕达哥拉斯学派的一项重大贡献是证明了勾股定理,但由此也发现了一些直角三角形的斜边不能表示成整数或整数之比(不可通约)的情形,如直角边长均为1的直角三角形就是如此。这一悖论直接触犯了毕氏学派的根本信条,导致了当时认识上的"危机",从而产生了第一次数学危机。到了公元前370年,这个矛盾被毕氏学派的欧多克斯通过给比例下新定义的方法解决了,并最终导致了无理数的出现并且承认。

第一次数学危机对古希腊的数学观点有极大冲击。这表明,几何学的某些真理与算术无关,几何量不能完全由整数及其比来表示,反之却可以由几何量来表示出来,整数的权威地位开始动摇,而几何学的身份升高了。危机也表明,直觉和经验不一定靠得住,推理证明才是可靠的,从此希腊人开始重视演译推理,并由此建立了几何公理体系,这不能不说是数学思想上的一次巨大革命!

真理的繁荣-欧洲的数学复兴以及第二次危机

和文艺复兴相伴的是,整个自然科学的复兴,尤其是数学的复兴。在17世纪到19世纪这段时间里,欧洲出了许多数学巨星,包括牛顿,莱布列次,高斯等等。这些巨星的出现,让人们感觉这个世界是可以通过数学模型完美的描述的。有了这些数学知识,人们可以预测哈雷彗星下一次到来的时间,人们预测出了天王星的存在。这一切的一切,展示了数学的强大,我们仿佛已经可以通过数学掌控一切自然现象。然而,强大的数学在解决复杂的天文,物理问题一展神威之时,它本身却陷入了困境。负数,复数的出现,尤其是无穷小的出现,无不让数学陷入困境,出现了第二次数学危机。

1734年,英国哲学家、大主教贝克莱发表《分析学家或者向一个不信正教数学家的进言》,矛头指向微积分的基础–无穷小的问题,提出了所谓贝克莱悖论。他指出:牛顿在一些典型的推导过程中,第一步用了无穷小量作分母进行除法,当然无穷小量不能为零;第二步牛顿又把无穷小量看作零,去掉那些包含它的项,从而得到所要的公式,在力学和几何学的应用证明了这些公式是正确的,但它的数学推导过程却在逻辑上自相矛盾.

直到19世纪,柯西系统地发展了极限理论。柯西认为无穷小量应该是要怎样小就怎样小的量,因此本质上它是变量,而且是以零为极限的量,至此柯西澄清了前人的无穷小的概念,另外实数理论,集合论的建立,也把无穷小量从束缚中解放出来,第二次数学危机得以基本解决。

真理的完备之路-第三次数学危机

进入19世纪,极限理论,集合论的提出和完善,使得人们以为数学理论已经走向完美,但是,就在人们为数学庆功之时,几个悖论的提出,让人们对数学的信心跌入低谷,数学的确定性受到了前所未有的挑战,数学真正走到了风雨飘摇的边缘。数学理论遇到了第三次的危机。

数学史上的第三次危机,是由1897年的突然冲击而出现的,到现在,从整体来看,还没有解决到令人满意的程度。这次危机是由于在康托的一般集合理论的边缘发现悖论造成的。由于集合概念已经渗透到众多的数学分支,并且实际上集合论成了数学的基础,因此集合论中悖论的发现自然地引起了对数学的整个基本结构的有效性的怀疑。

真理确定性的丧失

1897年,福尔蒂揭示了集合论中的第一个悖论。两年后,康托发现了很相似的悖论。1902年,罗素又发现了一个悖论,它除了涉及集合概念本身外不涉及别的概念。罗素悖论曾被以多种形式通俗化。其中最著名的是罗素于1919年给出的,叫做“理发师悖论”。理发师宣布了这样一条原则:他给所有不给自己刮脸的人刮脸,并且,只给村里这样的人刮脸。当人们试图回答下列疑问时,就认识到了这种情况的悖论性质:"理发师是否自己给自己刮脸?"如果他不给自己刮脸,那么他按原则就该为自己刮脸;如果他给自己刮脸,那么他就不符合他的原则。

罗素悖论使整个数学大厦动摇了。无怪乎弗雷格在收到罗素的信之后,在他刚要出版的《算术的基本法则》第2卷末尾写道:"一位科学家不会碰到比这更难堪的事情了,即在工作完成之时,它的基础垮掉了,当本书等待印出的时候,罗素先生的一封信把我置于这种境地"。于是终结了近12年的刻苦钻研。

承认无穷集合,承认无
穷基数,就好像一切灾难都出来了,这就是第三次数学危机的实质。尽管悖论可以消除,矛盾可以解决,然而数学的确定性却在一步一步地丧失。现代公理集合论的大堆公理,简直难说孰真孰假,可是又不能把它们都消除掉,它们跟整个数学是血肉相连的。所以,第三次危机表面上解决了,实质上更深刻地以其它形式延续着。

数学家们认识到他们的工作面临一个更严峻的危机。当然,他们不打算袖手旁观,坐视几个世纪的努力毁于一旦。因为坚固性依赖于为推理所选定的基础,很明显,只有重建全部基础才能挽大厦于将倾。在重建后的数学基础上,逻辑和数学公理都必须加强,因而建造者们决定将基础打得更深些。不幸的是,在以什么方式,在什么地方加强基础这一点上,他们没能达成一致。每个人都认为自己能确保坚实性,每个人都想按自己的方式重建。结果是产生了一片既谈不上巍然,又无坚实基础,无规无矩,四处展翼的危房,每一翼都自称数学的唯一殿堂,每间房屋都藏有数学思想的奇珍异宝。这一切,让数学变得愈加不确定,前途愈加不明。

数学将往何处去

第三次数学危机导致了数学发展到这样一个阶段:逻辑主义、直觉主义、形式主义和集合论公理化主义都各执一见,都觉得自己的理论可以更好的解决危机,更接近真理。哪一种可以被合适地称之为数学,人们各执己见,而且,每一种数学结构都有某种程度上截然不同的上层建筑。因此,直觉主义者在他们应该接受什么作为基本的、合理的直觉问题上意见不一致:仅仅只有整数还是也包括一些无理数?排中律只适用于有限集合还是也可用于可数集?还有构造性方法的概念问题。逻辑主义者则单一地依赖逻辑,而且对于可约性公理、选择公理及无穷公理还怀有疑虑。集合论公理化主义者则可以沿几个不同方向中的任一个前进,这取决于他们对选择公理和连续统假设的取舍。甚至形式主义者也能遵循不同的路径,其中一些有别于将用于建立相容性的元数学原理。

逻辑主义者、形式主义者和集合论公理化主义者都依赖于公理化的基础。在本世纪头几十年中,这种基础被拥为建立数学的可以选用的基础。但是哥德尔的理论表明,没有一个公理体系可以包含属于任何一种结构的所有真理, 勒尔海姆-斯科伦定理则表明每一个体系包含的真理比预计的要多。只有直觉主义者才能不在乎公理化的方法所提出的问题。所有这些关于哪个基础最好这一问题的不一致和不确定性以及缺乏相容性的证明,就像达摩克里斯之剑一样悬在数学家头上。无论一个人接受哪种数学哲学,他都冒着自相矛盾的危险。

对数学大厦这几种相互抵触的方法揭示了这样一个主要事实:不是只有一种而是有很多种数学,数学这个词应从多种意义上进行理解,也许可用于任何一种方法。哲学家桑塔亚那曾经说过, “不存在什么上帝,上帝也是凡夫俗子。”今天,我们可以很肯定的说,不存在这样一种被普遍接受的数学理论,数学发展的方向愈加不确定了,数学,正处于风雨飘摇状态之中。

不过无论如何,人类对数学的探索,对真理的追求之路会一直走下去:也许我们无法真正掌握真理,或者真正的真理并不存在,但是我们可以让我们的认识接近真理,让我们的认识更接近自然的规律。

路漫漫其修远兮,吾将上下而求索!



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  2. 2010年11月19日03:28 | #2

    @grapeot

    也许因为看了你那篇“Inspiration Fragments”才选的“不确定性的”

    btw:我是直觉党

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