A Rational Belief: The Method of Discovery in the Complex Variable

The importance of mathematics in the context of the scientific and technological development of humanity is determined by the possibility of creating mathematical models of the objects studied under the different branches of Science and Technology. The arithmetisation process that took place during the nineteenth century consisted of the quest to discover a new mathematical reality in which the validity of logic would stand as something essential and central. Nevertheless, in contrast to this process, the development of mathematical analysis within a framework that largely involves intuition and geometry is a fact that cannot go unnoticed amongst the mathematics community, as we shall show in this paper through the research made by Bernhard Riemann on complex variables.     

拉普拉斯概率论的衰落

作为概率论历史中一个重要发展阶段的拉普拉斯概率论在20世纪初期走到了它的尽头。从19世纪初开始的关于概率论在法律判决中应用的争论、柯西的“新数学”的兴起及其对最小二乘法的责难、对拉普拉斯概率论基础的批评以及公理化概率论的产生等四个方面的讨论,展示了拉普拉斯概率论衰落的历史轮廓。     

Ambiguities of Fundamental Concepts in Mathematical Analysis During the Mid-nineteenth Century

In this paper we consider the major development of mathematical analysis during the mid-nineteenth century. On the basis of Jahnke’s (Hist Math 20(3):265–284, 1993) distinction between considering mathematics as an empirical science based on time and space and considering mathematics as a purely conceptual science we discuss the Swedish nineteenth century mathematician E.G. Bj?rling’s general view of real- and complexvalued functions. We argue that Bj?rling had a tendency to sometimes consider mathematical objects in a naturalistic way. One example is how Bj?rling interprets Cauchy’s definition of the logarithm function with respect to complex variables, which is investigated in the paper. Furthermore, in view of an article written by Bj?rling (Kongl Vetens Akad F?rh Stockholm 166–228, 1852) we consider Cauchy’s theorem on power series expansions of complex valued functions. We investigate Bj?rling’s, Cauchy’s and the Belgian mathematician Lamarle’s different conditions for expanding a complex function of a complex variable in a power series. We argue that one reason why Cauchy’s theorem was controversial could be the ambiguities of fundamental concepts in analysis that existed during the mid-nineteenth century. This problem is demonstrated with examples from Bj?rling, Cauchy and Lamarle.     

一部汇精集萃、推陈出新的扛鼎之作——读《中国科学技术史·数学卷》有感

《中国科学技术史·数学卷》在中国数学史研究的广度和深度上都有新的拓展,是对截止到21世纪初中国数学史研究成果的最新全面总结.同时,该书又从总体上对中国数学史研究提出了一些新观点和新方法,其中特别是尊重原始文献一项,应该引起数学史研究者的足够重视.     

学科交叉与分子生物学的革命和发展 --纪念DNA双螺旋模型建立50周年

20世纪生物学最激动人心的大事就是50年前DNA双螺旋结构的发现。作为纪念，本文从历史的角度阐明这场分子生物学革命的方法论动因：物理化学技术在分子生物学中的运用。化学在生物学中的运用随着有机化学的发展自然而然地进行，然而，物理学方法在生物学中的运用却需要理论的勇气和胆量。玻尔、薛定谔等对物理化学方法的倡导，洛克菲勒基金会对应用物理学方法研究分子生物学问题的资助等促使大批物理学家向生物学转移。同位素示踪方法、X射线衍射分析、超速离心技术等的应用推动着分子生物学的革命性发展。20世纪末和新的世纪，计算技术、应用数学和信息科学方法代替物理学方法成为推动分子生物学发展的新方法。     

古为今用:美国学者眼中数学史的教育价值

数学史与数学教育关系(HPM)是兴起于20世纪70年代的一个数学教育研究领域.从19世纪末开始,美国数学史家和数学教育家们对数学历史的教育功能多有阐述,成为促进HPM创立并深入发展的重要推动力量.这些功能包括:激发学生的学习兴趣、改变学生的数学观、使数学人性化、让学生从原始文献汲取数学家的原始思想和社会文化信息、帮助学生更好地理解和欣赏数学、增强学生的自信心、通过历史可以了解学生学习数学的困难和认知过程、为教材编写提供借鉴等等.这些功能对于我们今天运用科学史于科学教育的实践具有一定的借鉴和指导意义,也让我们树立这样的信念:数学史并不是无用的学问.     

A Cauchy-Dirac Delta Function

The Dirac δ function has solid roots in nineteenth century work in Fourier analysis and singular integrals by Cauchy and others, anticipating Dirac’s discovery by over a century, and illuminating the nature of Cauchy’s infinitesimals and his infinitesimal definition of δ.     

数学真理的发展及其对自然观演变的启示

19世纪下半叶以来，数学与自然科学各自的发展及其相互关系呈现出许多新的特点。特别是20世纪以来诞生的各种数学新理论，正在逐步地改变着数学真理的传统观念。数学真理与自然法则的关系变得日益复杂和深化了。数学新的真理性质对自然观的变革产生了深远的影响。     

从“古为今用”到“丝路精神”:吴文俊数学史观的形成与演变

吴文俊的数学史观,来自他对中国数学史的独创性研究。在"古为今用"思想的引领下,开辟了数学机械化的新领域,让中国古代数学为世界数学作出新贡献;"古证复原"原则的确立,开启了中国数学史研究的新时代;"两种数学主流"思想的提出,确立了中国传统数学在世界数学发展史上的地位。更为重要的是,2002年,吴文俊指出"丝路精神"的核心价值是"知识交流与文化融合"。因此,"古为今用""古证复原""两种主流"和"丝路精神"构成了吴文俊数学史观的核心要素,是指引新时代中国数学史研究的伟大旗帜。     

从部分尖端技术的发展看影响科技发展的因素

从我国局部尖端技术与西方发达国家相比较的差距来看,近五十年可分为三个阶段,并且差距呈现“减小—增大—再减小”的趋势。分析差距增减的原因,总结经验教训应处理好五个方面的关系,即：主观与客观、理论与实际、基础与尖端、自主与外援、个体与群体。影响科技发展的四个要素为：时代、基础、合力与潜力。只要我们处理好五个关系,充分发挥四个要素的作用,在新世纪将能够屹立干世界科技先进之林。     

鲁津:苏联早期杰出的数学家

鲁津是苏联20世纪二、三十年代最有影响的数学家之一。他是现代实变函数论的主要奠基人。在他的努力下形成的鲁津学派使苏联跻身于世界数学强国之林。鲁津学派涌现出很多世界级别的数学家。但是,鲁津案使他处于命运的低谷。     

"后基础主义"时代数学哲学研究的若干趋势——从波普尔、拉卡托斯到P.欧内斯特

数学基础主义之后,批判理性主义和经验主义(包括拟经验主义)成为某些数学哲学家热衷的选择。虽然对柏拉图主义和实在论有某种解构价值,但两者却混淆了数学与其他科学的本质区别。相比较而言,关于数学的社会建构主义观点虽然有不少真知灼见,但却陷入了某种形式的单一决定论的泥潭。     

心理逻辑经典实验的认知思考——认知科学背景下逻辑学与心理学的融合发展

19世纪末弗雷格创立的现代逻辑主张排除逻辑学中的一切心理因素,这一观点使得逻辑学与心理学渐行渐远。直到1975年认知科学产生之后,逻辑学和心理学才实现了真正意义上的融合发展,产生了心理逻辑(学)这一新的学科。众多逻辑学家和心理学家进行了大量心理逻辑实验来证明这一观点:人们在进行逻辑推理时要受到心理因素的影响。本文介绍了马库斯和里普斯在1979年所作的心理逻辑的经典实验,并对实验结果进行了认知层面的思考,在此基础上得出结论:认知(思维)的过程是逻辑和心理共同作用的过程,在认知科学蓬勃发展的背景下,逻辑学和心理学必将融合发展。     

An eighteenth-century medical-meteorological society in the Netherlands: an investigation of early organization, instrumentation and quantification. Part 1

In many areas of the eighteenth century was a starting point for the quantification of science. It was a period in which the mania for collecting led to the first attempts in systematization and classification. This penchant for collecting was not limited to natural history specimens or curiosities. Due in part to the development of mathematical and physical instruments, which became more widely available, scholars were confronted with the informative value of numbers. On the one hand, sequences of measurements appeared to be the key to the advancement of scientific knowledge, yet on the other hand the mathematical apparatus to deal with these data was still largely lacking. As a result of this the first meteorological networks organized in the eighteenth century all became bogged down in the large amount of information that was collected but could not be processed properly. This development is illustrated in a case study of an early Dutch meteorological society, the Natuur-en Geneeskundige Correspondentie Soci?teit (1779-1802). What were the factors that triggered this interest in the weather in the Netherlands? What were the goals and expectations of the contributors? What were their methodological strategies? Which instruments were used to measure which meteorological parameters? How was the stream of numbers generated by these measurements organized, collected and interpreted? An analysis of this process reveals that limits on the advancement of meteorology were not only imposed by eighteenth-century Dutch Republic and the lack of a proper theoretical insight were also crucial factors that eventually frustrated the breakthrough of meteorology as an academic science in the Netherlands. This breakthrough was only achieved in the second half of the nineteenth century.     

试论数学哲学的文化转向及其影响

数学文化研究是数学哲学研究与发展的一个新的方向,开辟了数学哲学研究的全新视角。本文从文化哲学、科学文化哲学以及数学哲学的后现代发展历程中梳理了数学哲学的文化转向,以及所产生的影响。     

整体主义社会理论的本体论承诺之变迁

自19世纪末至现在,整体主义社会理论在本体论承诺上经历了几次变迁。迪尔凯姆确立了最强的形式;50年代,布洛德贝克退缩到个体主义立场;90年代,金凯德和索耶退缩到依随性,但是,艾梅克等人在突现论基础上建立了非还原的层级本体论。     

Taming the Infinite

For over two thousand years thought about the infinite was dominated by Aristotelian hostility to the idea that the infinite could be a legitimate object of mathematical study. Then Cantor's work late in the nineteenth century seemed to overturn this orthodoxy. However, by highlighting ways in which infinitude still could not be brought under the control of mathematicians, Cantor's work may in fact have reinforced the orthodoxy.     

从在国际顶尖基础数学期刊发表论文情况看我国数学近几十年来的发展

为了通过中国大陆地区在基础数学顶尖科研期刊发表论文的情况,了解我国数学的发展和现状,本文选取了10个国际最权威的基础数学期刊,分为两类,对中国数学家在这些刊物发表论文的情况进行分析。统计结果表明我国在第一类期刊发表的论文数量很少,这表明我国数学的研究水平还有待提高,原创性研究方面还有待努力;但是近30年来,我国在第二类期刊发表的论文数量显著地逐年增加,这显示我国数学研究团队的数量和实力都有了很好的发展。     

论中国逻辑与中国传统数学

中国传统数学受到了中国逻辑尤其是墨家逻辑的影响,形成了自己的推理方法体系--以"以类合类"为方法论基础,以"类"和"分类"为推理的核心成分,以"推类"为主导推理范式.这一方法体系,在相当长的时期内使中国传统数学处于世界的领先地位,因此,可以下这样的结论:中国逻辑不是阻碍中国传统数学进一步发展的因素,中国传统数学自身逻辑思路也不是制约中国传统数学发展的原因.     

Mathematical Naturalism: Origins, Guises, and Prospects

During the first half of the twentieth century, mainstream answers to the foundational crisis, mainly triggered by Russell and Gödel, remained largely perfectibilist in nature. Along with a general naturalist wave in the philosophy of science, during the second half of that century, this idealist picture was finally challenged and traded in for more realist ones. Next to the necessary preliminaries, the present paper proposes a structured view of various philosophical accounts of mathematics indebted to this general idea, laying the ground for a desirable integration of their strenghts.