[笔记] 重读《古今数学思想》序言有感
莫里斯·克莱因 Morris Kline 教授,1908-05-01 ~ 1992-06-10
https://mathshistory.st-andrews.ac.uk/Biographies/Kline/Kline.jpeg
https://mathshistory.st-andrews.ac.uk/Biographies/Kline/
Morris Kline. Mathematical Thought from Ancient to Modern Times [M]. New York, Oxford, Oxford University Press, 1972.
(美)克莱因(M.Kline)著, 张理京,张锦炎译, 北京大学数学系数学史翻译组译. 古今数学思想[M]. 上海: 上海科学技术出版社, 1979-1981. (汉译本共计 4 册)
《古今数学思想》系统、全面、深入地讲解了核心数学的古代史、近代史和 1930 年代之前的现代部分。
不可替代的常销著作。
非常适合广大理工科师生、数学爱好者、科学史研究工作者。
一、 Preface 序言摘录
If we wish to foresee the future of mathematics our proper course is to study the history and present condition of the science. HENRI POINCARé
如果我们想预见数学的未来,正确的路线是研究(学习)这门科学的历史和现状。——亨利·庞加莱
另一种说法( MacTutor History of Mathematics ):
The true method of foreseeing the future of mathematics is to study its history and its actual state. 预测数学未来的真正方法是研究数学的历史和实际状态。
https://mathshistory.st-andrews.ac.uk/Biographies/Poincare/
Beyond these considerations, the importance of many of the more recent developments cannot be evaluated objectively at this time. The history of mathematics teaches us that many subjects which aroused tremendous enthusiasm and engaged the attention of the best mathematicians ultimately faded into oblivion. One has but to recall Cayley's dictum that projective geometry is all geometry, and Sylvester's assertion that the theory of algebraic invariants summed up all that is valuable in mathematics. Indeed one of the interesting questions that the history answers is what survives in mathematics. History makes its own and sounder evaluations.
【机器翻译】除了这些考虑之外,目前还不能客观地评价最近许多事态发展的重要性。数学史告诉我们,许多激发了极大热情并吸引了最优秀数学家注意的学科最终都被遗忘了。我们只能回忆一下Cayley的格言,即投影几何就是所有的几何,以及Sylvester的断言,即代数不变量理论总结了数学中所有有价值的东西。事实上,历史回答的一个有趣的问题是什么在数学中幸存下来。历史做出了自己的、更加合理的评价。
对照:华罗庚的“早发表,晚评价。”——1978年中国数学会成都会议。这是科学的规律,即科学工作要经过历史检验才能逐步确定其真实价值,也是他人生的体验,具有现实意义。
https://www.cas.cn/spzb1/JNHLGSS25ZN/HLG25YL/201006/t20100611_2879861.html
I hope that this work will be helpful to professional and prospective mathematicians. The professional man is obliged today to devote so much of his time and energy to his specialty that he has little opportunity to familiarize himself with the history of his subject. Yet this background is important. The roots of the present lie deep in the past and almost nothing in that past is irrelevant to the man who seeks to understand how the present came to be what it is. Moreover, mathematics, despite the proliferation into hundreds of branches, is a unity and has its major problems and goals. Unless the various specialties contribute to the heart of mathematics they are likely to be sterile. Perhaps the surest way to combat the dangers which beset our fragmented subject is to acquire some knowledge of the past achievements, traditions, and objectives of mathematics so that one can direct his research into fruitful channels. As Hilbert put it, "Mathematics is an organism for whose vital strength the indissoluble union of the parts is a necessary condition."
【机器翻译】我希望这项工作将有助于专业和未来的数学家。如今,专业人士不得不将大量的时间和精力投入到自己的专业上,以至于他几乎没有机会熟悉自己学科的历史。然而,这一背景很重要。现在的根源深深植根于过去,过去几乎没有什么与试图理解现在是如何形成的人无关。此外,尽管数学已经发展成数百个分支,但它是一个统一体,有它的主要问题和目标。除非各种专业对数学的核心有贡献,否则它们很可能是贫瘠的。也许对抗困扰我们这个支离破碎的学科的危险的最可靠的方法是了解数学过去的成就、传统和目标,这样人们就可以将自己的研究引导到富有成效的渠道中。正如希尔伯特所说:“数学是一个有机体,其生命力的必要条件是各部分不可分割的结合。”
The usual courses in mathematics are also deceptive in a basic respect. They give an organized logical presentation which leaves the impression that mathematicians go from theorem to theorem almost naturally, that mathematicians can master any difficulty, and that the subjects are completely thrashed out and settled. The succession of theorems overwhelms the student, especially if he is just learning the subject.
【机器翻译】通常的数学课程在基本方面也是骗人的。它们给出了一个有组织的逻辑演示,给人的印象是数学家几乎很自然地从一个定理到另一个定理,数学家可以掌握任何困难,并且主题已经完全解决了。一连串的定理让学生不知所措,尤其是在他刚刚学习这门学科的时候。
The history, by contrast, teaches us that the development of a subject is made bit by bit with results coming from various directions. We learn, too, that often decades and even hundreds of years of effort were required before significant steps could be made. In place of the impression that the subjects are completely thrashed out one finds that what is attained is often but a start, that many gaps have to be filled, or that the really important extensions remain to be created.
【机器翻译】相比之下,历史告诉我们,一门学科的发展是由各个方向的结果一点一点形成的。我们也了解到,往往需要几十年甚至数百年的努力才能采取重大步骤。人们发现,所取得的成就往往只是一个开始,许多空白必须填补,或者真正重要的延伸仍有待创造,而不是完全解决这些问题的印象。
The polished presentations in the courses fail to show the struggles of the creative process, the frustrations, and the long arduous road mathematicians must travel to attain a sizable structure. Once aware of this, the student will not only gain insight but derive courage to pursue tenaciously his own problems and not be dismayed by the incompleteness or deficiencies in his own work. Indeed the account of how mathematicians stumbled, groped their way through obscurities, and arrived piecemeal at their results should give heart to any tyro in research.
【机器翻译】课程中精心设计的演示并没有显示出创造性过程中的挣扎、挫折感,以及数学家为了获得一个相当大的结构所必须走的漫长而艰巨的道路。一旦意识到这一点,学生不仅会获得洞察力,还会获得勇气,顽强地追求自己的问题,而不会因为自己工作的不完整或不足而沮丧。事实上,数学家们是如何跌跌撞撞,在模糊中摸索,并零碎地得出结果的,这本书应该会让任何研究新手感到振奋。
二、 感慨
2.1 短期时间评价阻碍科学创新
基于短期时间窗口的科学评价指标不但没有鼓励创新、促进科学原创的发展,反而阻碍了科学创新。追求短期效益的评价方式有可能蒙蔽科学政策的双眼,最终削弱科技发展的潜力。
那些基于短期引文窗口的文献计量指标事实上将科技政策导向了阻碍科技创新的歧途。这些指标的广泛使用有可能使新颖的科学研究及其从事者埋没于各类排名中,阻碍科学发展的进程。基金资助机构、人员聘用机构、科学评估机构都应采取更丰富、多元的评价方式保证科学创新的长远发展。
https://news.sciencenet.cn/htmlnews/2018/8/416516.shtm
2.2 短期雇佣合同、有偏见的选拔程序、不一致的激励:阻碍了高质量研究
顶刊 Nature Human Behaviour 在 2023-02-08 发文《Quality research needs good working conditions 高质量研究需要良好的工作条件》,重申:
High-quality research requires appropriate employment and working conditions for researchers. However, many academic systems rely on short-term employment contracts, biased selection procedures and misaligned incentives, which hinder research quality and progress.
高质量的研究需要研究人员的适当就业和工作条件。然而,许多学术体系依赖于短期雇佣合同、有偏见的选拔程序和不一致(背离科技初衷)的激励,这阻碍了研究质量和进展。
Current selection procedures fall into the trap of Goodhart’s law 5: “When a measure becomes a target, it ceases to be a good measure”. When researchers are required to pursue third-party funding or to publish in highly desirable journals, genuine scientific progress and high research quality are not necessarily incentivized. Instead, the incentive system can promote attention to ‘hot topics’ in mainstream research areas, and disincentivize the thorough, painstaking process of corroborating and documenting research to establish robust knowledge. This structure may reward researcher decisions that game the system: for instance, by focusing on ‘low-hanging fruit’, avoiding high-risk projects or even by (intentionally or unintentionally) using questionable research practices to succeed 6 — all of which are ultimately detrimental for genuine scientific progress.
当前的选择程序落入古德哈特定律的陷阱5:“当一项措施成为目标时,它就不再是一项好的措施”。当研究人员被要求寻求第三方资助或在非常理想的期刊上发表时,真正的科学进步和高研究质量不一定会受到激励。相反,激励系统可以促进对主流研究领域中“热门话题”的关注,并抑制彻底、艰苦的验证和记录研究以建立可靠知识的过程。这种结构可能会奖励那些与系统博弈的研究人员的决定:例如,关注“低垂的果实”,避免高风险项目,甚至(有意或无意地)使用有问题的研究实践来取得成功6——所有这些最终都不利于真正的科学进步。
2.3 数学是个不可分割的整体
Hilbert: "Mathematics is an organism for whose vital strength the indissoluble union of the parts is a necessary condition."
希尔伯特:“数学是一个有机体,各部分不可分割的结合是其生命力的必要条件。”
何止数学啊!
1933年2月17日,普朗克在柏林为德国工程师协会所做演讲中说:
“科学是内在的整体。它被分解为单独的部门不是取决于事物的本质,而是取决于人类认识能力的局限性。实际上存在着由物理学到化学,通过生物学和人类学到社会科学的连续的链条。”
Denn die Wissenschaft bildet nun einmal sachlish genormmen eine innerkich geschlossene Einheit. Ihre Trennung nach verschiedenen Fächern ist ja nicht der Natur der Sache Begründet, sondern entspingt nur der Begrenztheit des menschlishen Fassungs, vermögens, welche zwangsläufig zu einer Arbeitsheilung führt. In der Tat zieht sich ein kontinuierliches Band von der Physik und Chemie über die Biologie und Anthropologie bis zu den sozialen und Geisteswissenschaften, ein Band, das sich an keiner Stelle ohne Willkür durchschneiden läßt.
2.4 阿诺德 1995:俄罗斯的有机体数学观
One other characteristic of the Russian mathematical tradition is the tendency to regard all of mathematics as one living organism. In the West it is quite possible to be an expert in mathematics modulo 5, knowing nothing about mathematics modulo 7. One's breadth is regarded as negative in the West to the same extent as one's narrowness is regarded as unacceptable in Russia.
俄罗斯数学传统的另一特点是倾向于全面地把数学看成一个充满活力的有机体。西方学界有可能一个人只是数学上某一方面的专家,而对相邻分支一无所知。一个学者涉猎较广在西方学界被看成一大缺点,而恰恰在俄罗斯一个学者研究领域太窄被看成同样程度的不足。
https://blog.sciencenet.cn/blog-107667-1224424.html
https://www.ams.org/journals/notices/199704/index.html
2.5 学史明理、学史增信
现实事物的发展,呈现出螺旋式上升或波浪式前进的特点或特征。
数学的发展也不例外:
“课程中精心设计的演示并没有显示出创造性过程中的挣扎、挫折感,以及数学家为了获得一个相当大的结构所必须走的漫长而艰巨的道路。一旦意识到这一点,学生不仅会获得洞察力,还会获得勇气,顽强地追求自己的问题,而不会因为自己工作的不完整或不足而沮丧。事实上,数学家们是如何跌跌撞撞,在模糊中摸索,并零碎地得出结果的,这本书应该会让任何研究新手感到振奋。”
三、 对“P对NP”的直接影响
庞加莱说:“如果我们想预见数学的未来,正确的路线是研究(学习)这门科学的历史和现状。”
1987年前后拜读《古今数学思想》。
“不仅会获得洞察力,还会获得勇气,顽强地追求自己的问题,而不会因为自己工作的不完整或不足而沮丧”,“让任何研究新手感到振奋”。
此外,还核对了下述具体内容:
(1)ZF, Zermelo–Fraenkel set theory, 策梅洛-弗兰克尔集合论
(2)康托定理(幂集公理),康托无穷基数序列
(3)连续统假设 CH
(4)无穷的麻烦
(5)哥德尔不完全性定理
……
参考资料:
[1] 闵超. 短效评价阻碍科学创新[N]. 中国科学报, 2018-08-14 第7版 视角
https://news.sciencenet.cn/dz/dzzz_1.aspx?dzsbqkid=30743
https://news.sciencenet.cn/dz/upload/2018813234921836.pdf
https://news.sciencenet.cn/htmlnews/2018/8/416516.shtm
[2] 李江,2018-05-28,短效评价阻碍科学创新 精选
https://blog.sciencenet.cn/blog-1792012-1116106.html
[3] Rima-Maria Rahal, Susann Fiedler, Adeyemi Adetula, Ronnie P.-A. Berntsson, Ulrich Dirnagl, Gordon B. Feld, Christian J. Fiebach, Samsad Afrin Himi, Aidan J. Horner, Tina B. Lonsdorf, Felix Schönbrodt, Miguel Alejandro A. Silan, Michael Wenzler & Flávio Azevedo. Quality research needs good working conditions [J]. Nature Human Behaviour, 2023-02-08
doi: 10.1038/s41562-022-01508-2
https://www.nature.com/articles/s41562-022-01508-2
[4] 来昕,2023-02-12,当今科研界的短期雇佣方式已经成了高质量和创新性科研的最大阻碍! 精选
https://blog.sciencenet.cn/blog-3518723-1375836.html
[5] 王媛. 为一段流浪的名人名言找到故乡[J]. 图书馆建设, 2019, (2): 158-162. 发布日期: 2019-03-26.
http://lib.cqvip.com/Qikan/Article/Detail?id=7001668228&from=Qikan_Search_Index
http://tsgjs.org.cn/CN/abstract/abstract486.shtml
[7] S. H. Lui, An Interview with Vladimir Arnol'd [J], NOTICES OF THE AMS, 1997, 44(4): 432-438
https://www.ams.org/journals/notices/199704/index.html
[8] 张涵信,周恒,童秉纲. 从流体力学发展看科研成果评价[N]. 科学时报, 2011-08-03 第A1版(要闻)
https://news.sciencenet.cn/sbhtmlnews/2011/8/247224.html
对湍流的研究一开始就有两种不同的着重点,一种是从理论推理出发,以简单的湍流为研究对象,再试图推广至复杂的湍流问题。另一种则是根据实际的需求,从观察物理现象出发,找出现象的物理本质和关键问题,提出简化模型,进行分析计算。
80多年的实践证明,第一种方法虽然发表的论文数以万计,但仍不能解决实际应用的需求;第二种方法虽然开始时也要在一定程度上依靠经验,但其将研究对象选定为管流、边界层流等实际存在的湍流,在实验和计算上做了许多工作,既解决了实际湍流计算中的众多需求,还同时使对湍流本质的研究达到了新的高度。
相关链接:
[1] 2023-05-28,[好书推荐] 克莱因的《古今数学思想》 Mathematical Thought from Ancient to Modern Times
https://blog.sciencenet.cn/blog-107667-1389718.html
[2] 2023-07-31,[小汇报] 对我有“冲击力”的几本书(科普、专著)(1):“P对NP”问题、“反思经典电磁理论”
https://blog.sciencenet.cn/blog-107667-1397378.html
[3] 2018-08-27,科普经典书目推荐:乔治·伽莫夫的《从一到无穷大》
https://blog.sciencenet.cn/blog-107667-1131373.html
[4] 2023-07-02,[好书推荐] 王宪钧教授的《数理逻辑引论》
https://blog.sciencenet.cn/blog-107667-1393769.html
[5] 2022-10-25,[讨论] 丘成桐老师:“你就发现书里面很多是错的”
https://blog.sciencenet.cn/blog-107667-1360906.html
[6] 2020-06-05,[求证] 爱因斯坦12岁时看的几何学书,是什么语言的?
https://blog.sciencenet.cn/blog-107667-1236596.html
[7] 2017-03-01,[资源] 数学百科全书和手册
https://blog.sciencenet.cn/blog-107667-1036897.html
[8] 2020-3-20,破除论文“SCI至上”:弗拉基米尔·阿诺德1995年的几句话
https://blog.sciencenet.cn/blog-107667-1224424.html
[9] 2023-02-14,Zenas 公理:2023年再遇 Nature 知音“Stop the peer-review treadmill. I want to get off”
https://blog.sciencenet.cn/blog-107667-1376250.html
[10] 2023-02-12,[新闻] “原创三大杀手:同行评议、短期考核、没有时间”被顶刊论文证实
https://blog.sciencenet.cn/blog-107667-1375905.html
[11] 2023-01-06,Zenas 公理:2023年《Papers and patents are becoming less disruptive over time》里的图示
https://blog.sciencenet.cn/blog-107667-1370782.html
[12] 2022-09-14,[惊呆了] 比 Zenas 公理还狠:应废除出版前的同行评议。Prepublication peer review should be abolished.
https://blog.sciencenet.cn/blog-107667-1355299.html
[13] 2021-07-24,Zenas 公理:2021年《Dynamics of Disruption in Science and Technology》里的图示
https://blog.sciencenet.cn/blog-107667-1296808.html
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