Bernhard Riemann（1826-1866）:Turning Points in the Conception of Mathematics

读后感（伪）

刚刚浏览过一遍Detlef Laugwitz的这本书。能感觉到作者作为Gottingen后辈对前辈高人的景仰之情。但是这本书并不是一本充满陈词滥调的流水簿子，而是脚踏实地地整理了Riemann一生的贡献，确实值得一读。

书中有很多有趣的事件以及相关的评论:

1.p9.传说Riemann早慧，一个例子就是Riemann的（中学）老师借给Riemann一本Legendre的《数论》，Riemann在几天之内就读完了这本859页的大书。Laugwitz的书中有这样一段叙述：

Two years later, during the oral part of the Abitur, he (Riemann) demonstrated his familiarity with the book in spite of the fact that he had not consulted it in the interim. Number theory held a special attraction for him. Schmalfuss does not mention Gauss' Disquisitiones. Incidentally, in 1859, when he was writing his paper on prime numbers, Riemann seems to have forgotten Legendre, because he does not mention Legendre's prime number formula but does mention the almost equivalent formula of Gauss.

也就是说，Riemann应当是知道Legendre是素数分布公式的提出者之一，但他在1859年的著名论文里只字未提Legendre，这让人感到很奇怪。

2.Dedekind轶闻：p31
...His(Dirichlet) wife Rebecka, sister of Felix Mendelssohn-Bartholdy, with whom, incidentally, Dirichlet was on very good terms, tried to bring to Gottingen some of the atmosphere of the Berlin salons.
...The musical Dedekind seems to have been a very popular figure in the social life of Gottingen and sometimes complained of an excessive number of invitations. On 14 February 1856 he mentions a "gigantic party of 60-70 persons" ("Riesengesellschaft von 60-70 Personen") at the Dirichlets',
at which he tirelessly provided music for the dancers.（可以作为数学家乐感不弱的一个证据？）

Laugwitz接着评论道：
Of course, as the son of a respected and well known Braunschweig family Dedekind could easily establish connections in Gottingen.Not so Riemann. But after Dirichlet's move to Gottingen, Riemann,whom he regarded highly, would have likewise found all doors open had his makeup been different.

Riemann这种性格，或许也是他短寿的原因之一？（Dedekind在1916年去世，比Riemann多活了50年，而两人仅差5岁）

3.Felix Klein

Klein的名字在本书中频频出现。Klein本人对Riemann极为推崇，对Riemann有过很多赞许之词。不过这些溢美之词也是Laugwitz的重点批评对象。

p150.

It is with a measure of surprise that one reads his frequent allusions to intuition and perception, which give the impression that these were admissible means of proof for Riemann.
And remarks like the following are not likely to show Riemann to have been
a rigorous mathematician:

"As for logical rigor, it is out of the question that we should set the same requirements
for Riemann's works as for the productions of Weierstrass. Rather,
Riemann is effective by virtue of his wealth of ideas and the fullness of his
viewpoints, which always hit the essential."

Laugwitz的批评很中肯：

There is a misunderstanding here that is reflected in the imprecise use of the word "rigor." A proof can and should be rigorous, but what is meant by rigorous assumptions or definitions?

Klein对Riemann数学直觉有很高的评价，但Klein同时也忽视了Riemann是一位计算高手，Riemann的成果不都是靠直觉打下基础的（p178-179）：

Klein had an odd view of Riemann. Siegel's cautious criticism of Klein applies as well to other mathematicians who were alive in 1932. Apropos of Riemann's prime number paper Klein said in 1894: "Riemann must have very often relied on his intuition". This is certainly not meant in a negative sense, for Klein prized intuition. But in the case of Riemann's paper, what was obviously required in addition to mathematical imagination was particularly stubborn effort.

Siegel就此对Klein作出这样的评论（p177）：

Summing up, Siegel objects strongly to the impression that one could share
with Felix Klein and Edmund Landau as long as one was unaware of the
notes on the zeta function: "The fairytale that Riemann found the results of
his mathematical paper by means of 'great general' ideas, without using the
formal tools of analysis, is probably not as widespread now as it was in Klein's
lifetime."

Klein认为Riemann的成就是因Gottingen“几何学氛围”的影响而造就的(p213-214.)：

In his lectures on the history of mathematics (Klein 1926, 249) Felix Klein
says that Riemann "could not have listened to many lectures of the by then
70-year-old Gauss, who lectured little anyway. The shy young student was certainly unable to establish human relations with Gauss; after all, Gauss taught
very reluctantly, took little interest in most of his students, and was quite
unapproachable. We must nevertheless call Riemann a student of Gauss, in
fact, the only true student of Gauss who grasped his inner (!) ideas, as we
are now gradually getting to know them in outline from the Nachlass".

Klein, who was one of the most enthusiastic propagators of Riemann's complex
analysis, goes on to say: "Riemann's close relation to Gauss in his scientific
ideas is quite wonderful and almost puzzling for us." (p. 249). "The only explanation for this is that the Gottingen atmosphere was then saturated with these geometric ideas, and exerted
its uncontrollable but powerful pressure on the very gifted and receptive
Riemann. It is of utmost importance what spiritual environment a person enters;
it influences him far more strongly than the facts and concrete knowledge
that are offered to him!"

Laugwitz的评论相当不客气：

The present conception of the history of science makes it impossible for us
to be satisfied with Klein's necromancy. Klein had access in Gottingen to all
the sources we have relied on; he made incomplete use of them.

Riemann除了受到Gauss的影响，Jacobi，Dirichlet，W.Weber等人对Riemann也有巨大影响，而并非Klein的一句“Gottingen的气氛”所能概括。

Riemann's other teachers, Jacobi, Dirichlet, and W. Weber, who shared with
Gauss some of the main contemporary research interests and maintained contact
with him, exerted palpable rather than "atmospheric" influence on him.
The claim of Gottingen mathematicians, in Klein's time as well as later, that
"Extra Gottingam non est vita (mathematica)" was a self-generated myth. It
seems that Berlin and Italy agreed better with the "sensitive" Riemann than
did Gottingen m and not just meteorologically.

作者还有一个有力的论据来反驳Klein的观点(p218.)：

There is one more thing we must confront Klein with. Dedekind was exposed
to the same Gottingen atmosphere as Riemann and yet his orientation turned
out to be very different from Riemann's.（一个主攻代数，另一个是分析+几何）

作者对Klein还有一段有趣的评论:(p249.)

At first Klein's program(Erlangen Program) was intended as a classification principle for existing geometries, and since the young man did not as yet know everything
in existence, this classification helped him. But later the principle became a
dogma, and a geometry became the invariant theory of a group. We know of
similar developments. In the middle of the 20th century Bourbaki formulated a
classification principle intended for the whole of mathematics. Klein brought
together groups and geometry, and Bourbaki algebraic and topological structures.
We realize the fruitfulness of bringing conceptually distinct things under
one roof but also the dangers of dogmatism.（作者的观点是：数学大一统也是要冒风险的？）

4.Riemann的博士论文

Gauss对Riemann博士的评语(p118.)：

"The paper submitted by Herr Riemann is a concise testimony to its author's thorough and
penetrating studies of the area to which the subject treated therein belongs;
of a diligent and ambitious, truly mathematical spirit of investigation, and
of praiseworthy and fertile independence. The report is prudent and concise,
and in places even elegant; nevertheless, most readers might well wish for
even greater transparency of arrangement in some of the parts. Taken in its
entirety, it is a solid and valuable work which not only meets the requirements
usually set for test papers for the attainment of the doctorate but exceeds
them by far."

Laugwitz的评论：

If one has a certain amount of experience with evaluations and forgets for a
moment that here the princeps mathematicorum is writing about a person destined to become probably the most distinguished of his students, then one gets the following impression. The referee recognizes that the author has penetrated deep into a highly specialized field and has done this with great diligence, independently,and without the referee having to suggest the topic to him. There is no mention of the author's new ideas, of the solution of problems, or of new
methods, but it is recognized that he may well be showing signs of independent research activity. The presentation is terse, elegant only in spots, and on the whole not clear enough.

Riemann就博士论文与Gauss谈过话，但没有得到有价值的指导，Laugwitz做出了这样的评论：

So far, no one has been able to find any indication that Gauss had discussed with Riemann the contents of his paper or had given him any hints or suggestions. Riemann would have reported such things. After all, he mentioned the rather disappointing conversation with Gauss which comes
down, more or less, to this: right now I happen to be writing on a related topic, but your paper has not interested me enough that I should immediately and eagerly plunge into it.

Some (e.g., Remmert (1991, Band 2, 158)) think that the old Gauss was "chary of praise" ("lobkarg"). But against this is the fact that a few years earlier he had praised young Eisenstein to the skies. We will make no guesses about the great Gauss' admittedly baffling behavior toward Riemann.

感慨一下，老高斯的学生真难当！

p.s.作者对Gauss的动机做了一点猜测(p249.)

In conclusion let us recall once more Klein's raptures over the mystical influence
of the Gottingen atmosphere. We see things very differently now. It
is certainly true that in complex analysis Riemann went on from the work of
Gauss. But while Gauss openly expressed the highest praise for Eisenstein,
there is hardly any proof that he ever commended Riemann for carrying forward
his, that is Gauss', work. The reason for this is clear. Eisenstein stayed
completely within the frame of Gauss' image of mathematics, which was dominated
above all by the algorithmically graspable, whereas Riemann's transition
to the conceptual, to manifolds, to properties of functions rather than to functional
expressions, made Gauss uncomfortable.（不知这种推测有几分真确？）

:: 他在1859年的著名论文里只字未提Legendre，这让人感到很奇怪

我猜测可能是这样几个原因：

1. Riemann的文章不是历史综述，因此只是有选择地提及历史上的工作；
2. Gauss与Riemann的关系很近，因而被提及（Goldschmidt也被提及，或许因为是师兄？）；
3. Legendre的公式比较难看。

Laugwitz对Felix Klein评语的评价作为指正无疑是很有价值的，作为批评(尤其是 "Klein had access in Gottingen to all the sources we have relied on; he made incomplete use of them"那样的批评)则恐有些苛求。Klein并非史学家，数学家谈历史所依据的通常只是自己的经历、经验以及对史料的有限且往往是随意而非特意的涉猎。后人读这种数学家谈历史的资料时，应该将之视为是该数学家对历史的一种看法，而非历史研究。评价的标准也相应地应该有所不同（当然，如前所述，这并不妨碍指出该数学家看法的不足之处）。

我个人觉得，Laugwitz只是在陈述事实。曾经在网上看到Yau的一篇演讲稿，对Riemann也有高度评价。但是他说Riemann曾计算过zeta函数的前100万个(!)零点，这就是罔顾事实了。恐怕这说明人为了支撑自己的观点，可能会对历史真实进行或多或少的修改，这也和Klein《数学在19世纪的发展》英译本序的一句话相合：Klein对部分数学家的评论有时显得“小气”“尖酸刻薄”，而失去了一定的公正性。

英文不好，不知道有没有中文译本？

好像还没有中文译本。

:: 但是他说Riemann曾计算过zeta函数的前100万个(!)零点，这就是罔顾事实了

Yau会这么说？这太奇怪了，简直有违常识，恐怕是口误或文稿记录/翻译的错误。

http://www.tsinghua.edu.cn/publish/news/4205/2012/20120529125628568247264/20120529125628568247264_.html

黎曼几天时间看完那本800多页的书，可信度到底有多高？总感觉不太可能，这实在太惊人了…………

不知老Yau咋搞的，即便几百万个是口误，那“精确到20位”也不对，唉……

楼上的问题Laugwitz也想到了，Laugwitz指出，Riemann的中学老师对Riemann的中学时代的记录，是为纪念这位已经去世的优秀学生而写的：

Two letters, by Schmalfuss and Seffer respectively, shed a great deal of light
on Riemann as a gymnasium student in Lfineburg (1842-1846). But they must
be interpreted with caution, for they were meant to be used in an obituary for
a former student who later acquired great fame (N. 849-853).

我记得Michael Spivak 的 A comprehensive introduction to differential geometry 第二卷有黎曼的演讲的英文翻译，可以参考。