Commutativity and collinearity: a historical case study of the interconnection of mathematical ideas. Part I

This two-part paper investigates the discovery of an intriguing and fundamental connection between the famous but apparently unrelated mathematical work of two late third-century mathematicians, a link that went unnoticed for well over 1500 years. In this, the first installment of the paper, we examine the initial chain of mathematical events that would ultimately lead to the discovery of this remarkable link between two seemingly distinct areas of mathematics, encompassing contributions by a variety of mathematicians, from the most distinguished to the relatively unknown.     

La preuve en mathématique

Proof and deductive method in mathematics have their origin in the classic model of exposition developed by Euclid in his famous book on Elements. The attitude of mathematicians towards this method has certainly evolved in the past centuries, but the relationship between understanding and acceptability of mathematical statements has not dramatically changed and still constitutes a characterising element of this discipline. This paper is aimed at explaining and discussing some aspects which may be considered at the origin of difficulties related to proof, in particular, it focusses on the tension between two poles, that of production and that of systematisation of mathematical knowledge.     

A partial introduction to financial asset pricing theory

We present an introduction to mathematical Finance Theory for mathematicians. The approach is to start with an abstract setting and then introduce hypotheses as needed to develop the theory. We present the basics of European call and put options, and we show the connection between American put options and backwards stochastic differential equations.     

Probabilistic expectation and rationality in classical probability theory

Throughout the 18th century, the notion of probabilistic expectation was a matter of controversy among mathematicians. Despite its seminal role in the earliest formulations of mathematical probability, such as that of Huygens, expectation did not remain a fixed concept but underwent several striking shifts in definition. This paper argues that the conception of expectation was altered by mathematicians in a deliberate effort to capture the salient aspects of rational decision making. As the notion of rationality successively took on legal, economic, and then psychological overtones, the definition of probabilistic expectation followed suit.     

Confluences of agendas: Emigrant mathematicians in transit in Denmark, 1933–1945

The present paper analyses the confluence of agendas held by Danish mathematicians and German refugees from Nazi oppression as they unfolded and shaped the mathematical milieu in Copenhagen during the 1930s. It does so by outlining the initiatives to aid emigrant intellectuals in Denmark and contextualises the few mathematicians who would be aided. For most of those, Denmark would be only a transit on the route to more permanent immigration, mainly in the US. Thus, their time in Copenhagen would exert only temporary influence over Danish mathematics; but as it will be argued, the impacts of their transit would be more durable both for the emigrants and for the Danish mathematical milieu. It is thus argued that the influx of emigrant mathematicians helped develop the institutional conditions of mathematics in Copenhagen in important ways that simultaneously bolstered the international outlook of Danish mathematicians. These confluences of agendas became particularly important for Danish mathematics after the war, when the networks developed during the 1930s could be drawn upon.     

John Wallis and the French: his quarrels with Fermat,Pascal, Dulaurens,and Descartes

John Wallis, Savilian professor of geometry at Oxford from 1649 to 1703, engaged in a number of disputes with French mathematicians: with Fermat (in 1657–1658), with Pascal (in 1658–1659), with Dulaurens (in 1667–1668), and against Descartes (in the early 1670s). This paper examines not only the mathematical content of the arguments but also Wallis’s various strategies of response. Wallis’s opinion of French mathematicians became increasingly bitter, but at the same time he was able to use the confrontations to promote his own reputation.     

Geometric aspects of the Kapustin–Witten equations

This expository paper introduces the Kapustin?CWitten equations to mathematicians. We discuss the connections between the complex Yang?CMills equations and the Kapustin?CWitten equations. In addition, we show the relation between the Kapustin?CWitten equations, the moment map condition and the gradient Chern?CSimons flow. The new results in the paper correspond to estimates on the solutions to the Kapustin?CWitten equations given an estimate on the complex part of the connection. This leaves open the problem of obtaining global estimates on the complex part of the connection.     

A neural approach to concept representation suggests explanations for certain aspects of aphasia,alexia, and agraphia

Slightly more than 10,000 constraints on permissible connection values within a relatively simple nerve network are discussed and shown to enable modeling of certain aspects of concept representation and production of semantic errors associated with injury-related aphasia, agraphia, and alexia. The constraints considered include those that must be satisfied for novel and familiar examples of familiar categories to be appropriately named, for groups containing members of several categories to be named, for members of each category to be listed, for information to be correctly transferred from one concept to another, and for the severity of errors to be minimized when inappropriate names are selected. The model is startling because it involves only six learning rules that guide adjustment of connection strengths to values which generally satisfy the 10,000 constraints, and because network connections are so strictly regulated—with several hundred constraints simultaneously limiting the range of values permitted for each connection—that it seems reasonable to ask whether any qualitatively different explanations for these aspects of concept representation are possible. This study describes a simulation rather than a mathematical model. Its inclusion in a journal of mathematical modeling is intended to spur the interest of mathematicians. so that an appropriate mathematical analysis can be conducted.     

研究四色问题的意义及理论构想

四色问题又称四色猜想,是世界近代三大数学难题之一．1976年两位美国数学家Appel与Haken借助计算机给出了一个证明．时至今日,四色问题的正确性早已得到数学界所承认．但是围绕它的非计算机证明,在近几十年来涌现出了各种不同的研究成果．一方面丰富了图论的内容,另一方面又促进了图的染色理论的发展．本文从研究四色问题的意义出发;揭示了四色问题所隐藏的深刻规律,在此基础上提出了一个比四色问题更具有广泛意义的理论构想．主要目地为四色问题的非计算机证明提供一个研究方向．     

Karamata functions and differential equations: Achievements from the 20th century

This paper presents the major achievements of the 20th century regarding Karamata functions and the theory of differential equations, made mostly by V. Mari?, M. Tomi?, E. Omey, J.L. Geluk. The connection between these notions was first noticed by V.G. Avakumovi? (1910–1990). Slowly and regularly varying functions were introduced by J. Karamata (1902–1967). A group of mathematicians from the Karamata School of classical mathematical analysis were pioneers in research on these functions and their role in the theory of differential equations. Special attentions is given to the study of the Thomas–Fermi, Emden–Fowler and Friedmann equations, as well as the classical second order linear differential equations.     

What do mathematicians mean by proof? A comparative-judgement study of students’ and mathematicians’ views

We present a study in which mathematicians and undergraduate students were asked to explain in writing what mathematicians mean by proof. The 175 responses were evaluated using comparative judgement: mathematicians compared pairs of responses and their judgements were used to construct a scaled rank order. We provide evidence establishing the reliability, divergent validity and content validity of this approach to investigating individuals’ written conceptions of mathematical proof. In doing so, we compare the quality of student and mathematician responses and identify which features the judges collectively valued. Substantively, our findings reveal that despite the variety of views in the literature, mathematicians broadly agree on what people should say when asked what mathematicians mean by proof. Methodologically, we provide evidence that comparative judgement could have an important role to play in investigating conceptions of mathematical ideas, and conjecture that similar methods could be productive in evaluating individuals’ more general (mathematical) beliefs.     

Qualities of examples in learning and teaching

In this paper, we theorise about the different kinds of relationship between examples and the classes of mathematical objects that they exemplify as they arise in mathematical activity and teaching. We ground our theorising in direct experience of creating a polynomial that fits certain constraints to develop our understanding of engagement with examples. We then relate insights about exemplification arising from this experience to a sequence of lessons. Through these cases, we indicate the variety of fluent uses of examples made by mathematicians and experienced teachers. Following Thompson’s concept of “didactic object” (Symbolizing, modeling, and tool use in mathematics education. Kluwer, Dordrecht, The Netherlands, pp 191–212, 2002), we talk about “didacticising” an example and observe that the nature of students’ engagement is important, as well as the teacher’s intentions and actions (Thompson avoids using a verb with the root “didact”. We use the verb “didacticise” but without implying any connection to particular theoretical approaches which use the same verb.). The qualities of examples depend as much on human agency, such as pedagogical intent or mathematical curiosity or what is noticed, as on their mathematical relation to generalities.     

The algebra of marriage: an episode in applied group theory

In 1949, André Weil contributed a mathematical appendix to Claude Lévi-Strauss's landmark book, The elementary structures of kinship. In this appendix, Weil (one of the Bourbaki mathematicians) used group-theoretic techniques to model Australian marriage systems. Weil's paper marked the beginning of mathematical anthropology. This essay describes Weil's analysis of marriage systems and traces the uneasy history of the application of group theory to kinship studies.     

MATHEMATICS LEARNING TRAJECTORIES: CLASS,CAPITAL AND CONFLICT

This is the story of Marie and Edward as they approach the time of transfer from the primary to the secondary school. They both consider themselves to be successful mathematicians and have shared common classroom experiences throughout their time in the primary school. However, as they approach this critical relocation point it becomes clear that their future mathematical careers are set on two distinct trajectories. This paper explores the impact of the class-formed family habitus on them as learners of mathematics, and describes how family transferred cultural capital propels one of them whilst notions of conflict threaten to restrict the progress of the other.     

Metaphoric and metonymic signification in mathematics

This paper explores the roles of metaphor and metonymy in making sense of the ambiguities inherent in representation of mathematical constructs. Using the metaphors of “chains of signification” and “descent into meaning” for metonymies and metaphors respectively, these literary figures are discussed with regard to their use in mathematics. Synonymy, homonymy and polysemy are viewed as explanatory constructs in an analysis of ways in which metaphor and metonymy aid learners and mathematicians alike in making sense of mathematical ideas and resolving ambiguities.     

返璞归真 大道至简

介绍几位著名数学家有关数学教育的论述,并在深入反思中国传统数学分析教材的的基础上,就数学分析课程改革提出三点建议：数学分析课要回归微积分平实自然的本来面貌;教材内容要精简,贯彻少而精的基本原则;适当讲一点人文,讲一点历史.     

Viribus unitis! shall be our watchword: the first International Congress of Mathematicians,held 9–11 August 1897 in Zurich

Georg Cantor voiced the need for opportunities facilitating international mathematical cooperation as early as in 1888. A decade and efforts by a number of mathematicians later, the first International Congress of Mathematicians marked the beginning of an era where personal relations between mathematicians were considered to be of great importance. Furthermore, it set the standards for future congresses. As well as giving an overview of the pre-history and the organization of the congress, I look at a wider historic context, conjecture on the reasons why it was held in Zurich and why such a great emphasis was placed on the social aspect. This paper is a slightly modified version of the talk given at the BSHM Research in Progress Meeting held 3 March 2012 in Oxford.