什么是数?

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Henry今年暑假做了实习,8月中才从MIT回到多伦多。可他人还没有到多伦多,他母亲就给我布置了一项任务:帮Henry学习抽象代数。这是我的拿手菜,想当年在湖南师大学习这门课时,全年级只有本人考试得了满分;可惜现在愿意学习这门课的人不多了,难得Henry还想在任课教授退休前把该门课程Take了。

抽象代数研究数的结构,也就是数的运算法则;内容包括群论,环论,域的扩张,伽罗华理论。各种群的构造是无数数学家们长期研究的结果,绝不是某个人一时半会的即兴之作。能够领略到运算的本质,才不枉为人一世;相较那些处处问“这对我的现世的吃喝拉撒有什么用”的人,根本就是活在不同的层次。

我给Henry只上了5次课,也就是介绍了一点概要而已。结束时,他问到:究竟什么是数?这个问题早就困扰我多年了:在学习实变函数时就意识到了其不完善,但一直无法解决。在我报考硕士研究生的面试中,邵教授曾经问过我一个问题:什么是积分?我当时脱口而出:就是连续求和。教授说:说下去。可我仔细一想,才发现没有了下文。

几年前,当我悟出了数的本质时,这一切才变得明朗起来。我在博文《数是上天给人的启示》中,写了初步想法;之后一直在为细节奋斗。除了正整数是上帝创造的,其它都是人的工作:数学就是人造的世界,是对客观世界的反映。数字是静止的,但是我们取到任何一个单个数字的概率都是0,只有作为集合的数才能被人类所感知。明白了这点,Gauss引进的模运算就很自然了,理想数作为一个集合也就顺理成章了。数字只有在运算中才能有存在的意义,正如世界是不停地在改变,而所谓的改变,体现在数字上就是运算。

为了把我的领悟告知世人,我花钱买了一给Web Hosting,逐步把我的想法写出来。你可以到网站mattermatics.com去看看,不过需要我给你一个密码才行。网站远非完善,等我有空余时间了,才能去写一点。如果你想学会点什么本领,还得跟我联系,让你做点练习才行。

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