虽然我远没有看完Paul Richard Halmos(1916~2006)的所有数学教材,但我确实看了几本话题比较大众化的——因而同类教材也比较多的——从而在比较的基础上可以支撑我对Halmos的偏爱。所以我仍然能够直接说,我特别喜欢Halmos的数学书。
我看过他写的集合论、测度论、有限维向量空间和希尔伯特空间。这些书除了测度论之外都是很薄的。
至少我还不完整地看过他的自传——I want to be a mathematician。这个自传的开头就先明确地反醒了作者的人格特质,读来发现与我本人十分贴近,这足以支持我偏爱他的著作。这段话对其他读者来说也可以说明Halmos很可能是总能写出好教材的数学家。
I like words more than numbers, and I always did.
Then why, you might well ask, am I a mathematician? I don’t know. I can see some of the reasons in the story of my life, and I can see that chance played a role at least as big as choice; I’ll try to tell about all that as we go along. I do know that I wasn’t always sure what I wanted to be.
The sentence I began with explains the way I feel about a lot of things, and how I got that way. It implies, for instance, or in any event I men for it to imply that in mathematics I like the conceptual more than the computational. To me the definition of a group is far clearer and more important and more beautiful than the Cauchy integral formula. Is it unfair to compare a concept with a fact? Very well, to me the infinite differentiability of a once differentiable complex function is far superior in beauty and depth to the celebrated Campbell–Baker–Hausdorff formula about non-commutative exponentiation.
The beginning sentence includes also the statement that I like to understand mathematics, and to clarify it for myself and for the world, more even than to discover it. The joy of suddenly learning a former secret and the joy of suddenly discovering a hitherto unknown truth are the same to me—both have the flash of enlightenment, the almost incredibly enhanced vision, and the ecstacy and euphoria of released tension. At the same time, discovering a new truth, similar in subjective pleasure to understanding an old one, is in one way quite different. The difference is the pride, the feeling of victory, the almost malicious satisfaction that comes from being first. “First” implies that someone is second; to want to be first is asking to be “graded on the curve”. I seem to be saying, almost, that clarifying old mathematics is more moral than finding new, and that’s obviously silly—but let me say instead that insight is better without an accompanying gloat than with. Or am I just saying that I am better at polishing than at hunting and I like more what I can do better?
P. Halmos (1985), I want to be a mathematician, Springer
de Gennes (1976)的Journal de Physique Lettre,就是这一例子。他认为弹性的逾渗跟电导率的逾渗在模型上是等价的,因此弹性逾渗的临界指数就是电导率逾渗的临界指数。冯奚乔的论文是最早打破这一认识的论文之一。这也是他的学位论文工作。今天我们知道弹性逾渗与其他输运性质的逾渗是不同的,但仍然有很多open questions。特别是从弹性(线性弹性体)或粘度(Navier–Stokes流体)推广至流变学上的一般认识:粘弹性后,能否仍然去讨论“粘弹性逾渗”与纯几何的连接性逾渗的差别?从研究论文来看,我们不幸地看到,非但在冯奚乔以及大量其他人的工作之后,许多不假思索的实验家仍然按de Gennes (1976)的认识去粗略解读自己的实验数据;就算粘弹性临界现象也已经被大量报道之后,大量不称职的实验家也仍满足于弹性逾渗与几何逾渗临界点重合的假定、de Gennes (1976)的假定、超标度假定……试图反过来去下关于凝胶结构的结论。
这种完全无视理论认识的最新进展,对“什么未知、什么已知”的无知,对“实验家的当前任务是什么”问题的轻视,不幸地占据了大部分论文版面,影响了一代又一代的年轻人,从而淹没了正确的研究主线。90年代之后的科学界,进入了庸俗的publish or perish狂躁派对。冯奚乔的工作,包括逾渗的工作,或者在理论物理上稍微严肃些、形式些的兴趣,都显得太枯燥、太难引起大量波澜和impact,从而使实验的投入和“产出”不成正比,被迫变成了“隐学”。现在你再提起,就会被人评价一句:你这个问题太老了,现在没人感兴趣。
英语的typography,释义是the visual components of the written words。前面提到的“语言风格”风格问题,都是独立于“视觉呈现”的。如果说“语言风格”是论文润色机构的任务,那么“视觉呈现”——排版——就是期刊出版社的任务。在当前的学术圈现状中,比起语言风格,作者在排版上需要承担的部分相对要少一些。但是,重视排版至少有以下必要性:
磅(point)是出版用的单位体系,在这一单位体系12进制的,可能较常碰到的还有pica,12 picas = 1 points。在今天出版单位体系表示的各类尺寸,已经跟真实世界的物理尺寸建立了联系:1 inch = 72 points。
磅和英寸的换算关系
在屏幕显示方面,还会涉及到物理尺寸与像素之间的对应关系问题。例如,在一些专业作图软件中,字号既可表示为磅,又可表示为像素。但前者是使打印出来的东西的一致性所应采用的单位(因为它是物理长度单位,跟厘米英寸一样),但像素则是为了使屏幕显示出来的东西有一致性(看你怎么定义一致性)所应采用的单位。因为屏幕的最小显示元素是像素,无法在小于1像素之内做什么文章。所以在低分辨屏时代大家很精确地计算像素,因而习惯了像素单位。磅和像素之间的换算是一笔乱帐。大部分软件可能是按照1 inch = 96 px的惯例,来决定你设定了多少磅或多少英寸的东西,在显示屏上以多少像素显示给你看。你心里应该清楚,你不是在做网页设计或者软件界面设计,你的目标是打印出来之后的结果,所以软件的屏幕显示给你看到的只是“预览”。你应该抓住不放的是物理长度尺寸。