本原方程研究的早期历史：从伽罗瓦到若尔当

伽罗瓦开启了本原方程可解性问题的研究，但并未完成，且其中包含一个错误定理。若尔当将可解本原方程的研究转化为可解本原置换群的研究，为伽罗瓦的结果补充了证明，揭示了伽罗瓦定理的错误。伽罗瓦的错误定理引致了可解方程的错误分类，若尔当对此提出了解决方案。通过讨论伽罗瓦研究本原方程的背景和动机，若尔当对伽罗瓦错误的发现与修正，以及伽罗瓦工作的发展，梳理1860年代以前从伽罗瓦到若尔当本原方程研究的早期历史，探析其影响和意义。该研究对19世纪代数学发展的主题转向影响深远，以此可以剖见数学被创造出来的一种方式。     

对埃米尔·阿廷《伽罗瓦理论》的历史分析

用抽象代数来处理伽罗瓦理论由阿廷完成,他重视线性代数方法,成功绕过本原元素定理而证明了伽罗瓦理论的基本定理。文章在已有研究的基础上,分析了阿廷成功解读伽罗瓦理论的多种因素及其著作《伽罗瓦理论》的背景、结构及定位。在伽罗瓦理论的抽象化历程中,阿廷受到戴德金、希尔伯特、施泰尼茨等人的影响,这说明代数学在19世纪和20世纪发展环境的变化是阿廷《伽罗瓦理论》成功的前提。最后,文章调查了《伽罗瓦理论》对现代伽罗瓦理论专著及代数学教材的影响。     

解读神秘的蛊术

湘西的“蛊术”和泰国的“降头术”被称为东南亚两大邪术。在湘西凤凰苗区为女人的专利，称为蛊婆或“草鬼婆”。虽然谁也拿不出具体的证据，但从外人甚至本地人都对蛊婆的存在深信不疑，甚至有“无益不成寨”的说法。     

历史与哲学视野中的"实验"——伽里森的《实验如何终结》与哈金的《表象与介入》之比较

近年来,以哈金为代表的新实验主义开始关注科学实验中的实践活动本身,开辟了科学哲学界新的研究进路.同时,以伽里森为首的一些科学史家也将目光投向了实验过程和科学实践.本文从伽里森和哈金的两部代表作:<实验如何终结>(HOW Experiments End)与<表象与介入>(Representing and Intervening)出发,着重分析两者时科学实验实践中实验的独立性、稳定性、情境性与实在性及实验与理论之间的相互作用的研究成果,并探讨了两者科学实验思想的异同之处.     

《决疑数学》--一部拉普拉斯概率论风格的著作

通过详细的考证，发现《决疑数学》的唯一原本是《大英百科全书》第八版（1859年）中伽罗威所著的《概率论》一文.借助《决疑数学》的原本，以拉普拉斯概率理论发展历史的大背景作为基本参照系，考察《决疑数学》各章的内容安排、风格体系以及观点等方面，说明《决疑数学》是一部具有拉普拉斯概率论风格的著作.最后，通过对《决疑数学》译文的考察，发现其用汉字代替西方数字和符号的翻译风格增加了概率论内容和方法表述的难度，也限制了《决疑数学》在中国概率论传播和普及中的作用.     

纳粹间谍用假钞买情报

1943年的一天夜里,在德国驻土耳其大使馆宿舍里响起了一阵电话铃声,莫伟·齐什被他的领导——大使冯·巴本打来的电话叫醒,冯·巴本告诉他,有急事,让他去一趟. 莫伟·齐什原是柏林市的记者,很早就加入了德国纳粹党.为了报偿他的忠诚,他被派往土耳其首都安卡拉担任德国大使馆的商务参赞,并兼任德国使馆情报处的负责人.由于他的工作出色受到希特勒的赞赏.……莫伟·齐什来到大使家,出门迎接他的大使说:“客厅里有一个人带来了重要情报,因为你负责这方面的工作,所以把你找来.”走进客厅的莫伟·齐什看到一位个子很矮、精神严肃的人.此人自我介绍说:“我叫迪伯罗,阿尔巴尼亚人.     

水果也能做电池?中学科技馆"手蓄电池"新体验

1780年的一天,意大利解剖学家伽伐尼在做青蛙解剖时,两手分别拿着不同的金属器械,无意中同时碰在青蛙的大腿上,青蛙腿部的肌肉立刻抽搐了一下,仿佛受到电流的刺激,他称之为"生物电".而意大利物理学家伏特在多次实验后则认为:伽伐尼的"生物电"之说并不正确.     

比萨斜塔经历"时光倒流"

相传1590年,伟大的意大利科学家伽俐略曾在比萨斜塔的顶层做过自由落体运动的实验,从此,人们同时记住了伽俐略和壮观且“斜而不倒”的比萨斜塔。然而,时间却使这座斜塔越来越斜了。20世纪80年代初,曾有人预言,此萨斜塔将在     

青年数学家、战士和人:E.伽罗瓦

伽罗瓦带着致命的枪伤,静静地躺在病床上。弟弟坐在一旁恸哭着。突然,伽罗瓦吃力地睁开眼睛,这双昔日充满自负和愤怒的激情的眼睛,闪着柔和的濒死的光。他低沉地说:“不要哭,我在二十岁的年纪死去,需要我的全部勇气。”在拒绝神父为他祈祷之后,伽罗瓦死去了。这一天,是公元1832年5月31日。在数学发展史上,这是一个令人诅咒的日     

伽莫夫的“隧道效应”理论和加速器人工核反应的实现

加速器的研制及其对重元素人工嬗变的作用时常有文章予以提及和介绍，但是对伽莫夫的”隧道效应“理论在其中所起的作用只是一提而过，没有做出详细介绍。该文经过多方面集史料和认真研究，对此做了比较详细的说明，并且对加速器的发明过程及其对重元素人工嬗变和质能等当定律的验证做了进一步的阐述。     

The Surveyability of Long Proofs

The specific characteristics of mathematical argumentation all depend on the centrality that writing has in the practice of mathematics, but blindness to this fact is near universal. What follows concerns just one of those characteristics, justification by proof. There is a prevalent view that long proofs pose a problem for the thesis that mathematical knowledge is justified by proof. I argue that there is no such problem: in fact, virtually all the justifications of mathematical knowledge are ‘long proofs’, but because these real justifications are distributed in the written archive of mathematics, proofs remain surveyable, hence good.     

The Role of Diagrams in Mathematical Arguments

Recent accounts of the role of diagrams in mathematical reasoning take a Platonic line, according to which the proof depends on the similarity between the perceived shape of the diagram and the shape of the abstract object. This approach is unable to explain proofs which share the same diagram in spite of drawing conclusions about different figures. Saccheri’s use of the bi-rectangular isosceles quadrilateral in Euclides Vindicatus provides three such proofs. By forsaking abstract objects it is possible to give a natural explanation of Saccheri’s proofs as well as standard geometric proofs and even number-theoretic proofs.     

Non-Monotonic Set Theory as a Pragmatic Foundation of Mathematics

In this paper I propose a new approach to the foundation of mathematics: non-monotonic set theory. I present two completely different methods to develop set theories based on adaptive logics. For both theories there is a finitistic non-triviality proof and both theories contain (a subtle version of) the comprehension axiom schema. The first theory contains only a maximal selection of instances of the comprehension schema that do not lead to inconsistencies. The second allows for all the instances, also the inconsistent ones, but restricts the conclusions one can draw from them in order to avoid triviality. The theories have enough expressive power to form a justification/explication for most of the established results of classical mathematics. They are therefore not limited by Gödel’s incompleteness theorems. This remarkable result is possible because of the non-recursive character of the final proofs of theorems of non-monotonic theories. I shall argue that, precisely because of the computational complexity of these final proofs, we cannot claim that non-monotonic theories are ideal foundations for mathematics. Nevertheless, thanks to their strength, first order language and the recursive dynamic (defeasible) proofs of theorems of the theory, the non-monotonic theories form (what I call) interesting pragmatic foundations.     

Why Do Informal Proofs Conform to Formal Norms?

Kant discovered a philosophical problem with mathematical proof. Despite being a priori, its methodology involves more than analytic truth. But what else is involved? This problem is widely taken to have been solved by Frege’s extension of logic beyond its restricted (and largely Aristotelian) form. Nevertheless, a successor problem remains: both traditional and contemporary (classical) mathematical proofs, although conforming to the norms of contemporary (classical) logic, never were, and still aren’t, executed by mathematicians in a way that transparently reveals why these proofs—written in the vernacular to this very day—succeed in conforming to those norms.     

Towards a Theory of Mathematical Argument

In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent with argument assessment in non-mathematical contexts. I demonstrate this claim by considering the assessment of axiomatic or deductive proofs, probabilistic evidence, computer-aided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator’ view of proofs because it places derivations—which may be thought to invoke formal logic—at the center of mathematical justificatory practice. However, when the notion of ‘derivation’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument.     

矩阵力学与波动力学等价性证明的历史分析

矩阵力学和波动力学是量子力学的两种早期形态.前者由海森堡、约当、玻恩提出,后者由薛定谔提出.1926年,薛定谔给出二者之间等价性的证明.随后,泡利、狄拉克、约当、冯·诺伊曼等人或是为等价性证明给出新的版本,或是给出了间接的论证.这两种力学现在已被表述为海森堡绘景和薛定谔绘景,而数学家近来又从量子化问题出发对其等价性做出...     

Indexed Dendrograms on random Dissimilarities

This paper studies the random indexed dendograms produced by agglomerative hierarchical algorithms under the non-classifiability hypothesis of independent identically distributed (i.i.d.) dissimilarities. New tests for classifiability are deduced. The corresponding test statistics are random variables attached to the indexed dendrograms, such as the indices, the survival time of singletons, the value of the ultrametric between two given points, or the size of classes in the different levels of the dendogram. For an indexed dendogram produced by the Single Link method on i.i.d. dissimilarities, the distribution of these random variables is computed, thus leading to explicit tests. For the case of the Average and Complete Link methods, some asymptotic results are presented. The proofs rely essentially on the theory of random graphs.     

《四元玉鉴》的清代版本及《假令四草》的校勘研究

该文通过对《四元玉鉴》不同版本的对校及对清代有关该书的多种史料的研究，较为全面地阐述了《四元玉鉴》在清朝的流传及版本情况，指出罗士琳《四元玉鉴细草》本中《假令四草》的术，草均已经校改，并对其校改者进行初步推断。     

Objects and Processes in Mathematical Practice

In this paper it is argued that the fundamental difference of the formal and the informal position in the philosophy of mathematics results from the collision of an object and a process centric perspective towards mathematics. This collision can be overcome by means of dialectical analysis, which shows that both perspectives essentially depend on each other. This is illustrated by the example of mathematical proof and its formal and informal nature. A short overview of the employed materialist dialectical approach is given that rationalises mathematical development as a process of model production. It aims at placing more emphasis on the application aspects of mathematical results. Moreover, it is shown how such production realises subjective capacities as well as objective conditions, where the latter are mediated by mathematical formalism. The approach is further sustained by Polanyi’s theory of problem solving and Stegmaier’s philosophy of orientation. In particular, the tool and application perspective illuminates which role computer-based proofs can play in mathematics.     

The generation of random,binary unordered trees

Several techniques are given for the uniform generation of trees for use in Monte Carlo studies of clustering and tree representations. First, general strategies are reviewed for random selection from a set of combinatorial objects with special emphasis on two that use random mapping operations. Theorems are given on how the number of such objects in the set (e.g., whether the number is prime) affects which strategies can be used. Based on these results, methods are presented for the random generation of six types of binary unordered trees. Three types of labeling and both rooted and unrooted forms are considered. Presentation of each method includes the theory of the method, the generation algorithm, an analysis of its computational complexity and comments on the distribution of trees over which it samples. Formal proofs and detailed algorithms are in appendices.