Henrik Kragh Sørensen

Abel and his mathematics in contexts

Abstract200 years ago, on August 5, 1802, Niels Henrik Abel was born on Finnøy near Stavanger on the Norwegian west coast. During a short life span, Abel contributed to a deep transition in mathematics in which concepts replaced formulae as the basic objects of mathematics. The transformation of mathematics in the 1820s and its manifestation in Abel’s works are the themes of the author’s PhD thesis. After sketching the formative instances in Abel’s well-known biography, this article illustrates two aspects of the transformation which concern the introduction of concept based mathematics and the related shift in standards of mathematical rigor. Furthermore, the article outlines some of the many bicentennial celebrations in Norway and gives a short, thematic introduction to the literature on Abel and his work.

“The End of Proof”? The Integration of Different Mathematical Cultures as Experimental Mathematics Comes of Age

In this paper, the recent emergence of a professed “experimental” culture in mathematics during the past three decades is analysed based on an adaptation of Hans-Jorg Rheinberger’s notion of “experimental systems” that mesh into experimental cultures. In so doing, I approach the question of how distinct mathematical cultures can coexist and blend into a common understanding that allows for cultural convergence while preserving heterogeneity.

Appropriating Role Models for the Mathematical Profession: Biographies in the American Mathematical Monthly Around 1900

During the first decade of its existence, the American Mathematical Monthly regularly published short biographies of mathematicians. When read as appropriations of past lives, these biographies can be analysed to provide new insights into the images of mathematics, and of American mathematics in particular, held by groups of authors and welcomed by the readership of the Monthly. Thus, the approach in this paper is a “meta-biographical” one in which biographies are analysed not for their content about past mathematicians but for their appropriation and framing in the context where they were produced. This approach brings forward new insights into the professionalization of mathematics in the United States, the promotion of different disciplines, and the efforts of individuals to cast themselves as following in the footsteps of some of the greatest heroes of mathematics.

Cultivating the Herb Garden of Scandinavian Mathematics: The Congresses of Scandinavian Mathematicians, 1909–1925

As a reaction to the changed political landscape in Scandinavia following the dissolution of the union between Norway and Sweden in 1905, the prominent Swedish mathematician Gosta Mittag-Leffler extended ‘a brotherly hand,’ calling for Scandinavian colleagues to meet for a congress of mathematicians in Stockholm in 1909. This event became the first in a series of biannual meetings which proved to be an important institution for Scandinavian mathematics. During the first decades after 1909, the congresses would form and consolidate themselves through the construction of a new Scandinavian identity for mathematicians which developed alongside and in relation to both international and national contexts and developments. In this paper, we shall demonstrate that these meetings served a complex set of agendas at the individual, national, and international level. In particular, they reflect a changing conception of cooperation in science for mutual cultural gain combined with a flexible institutionalisation that allowed the Scandinavian mathematicians to use the congresses for various diplomatic ambitions. We base our analyses of the Scandinavian Congresses of Mathematics on the notion of a shared ‘conational’ identity developed adjacent to national identities. We then analyse the formation, consolidation, delineation, and reflections of this institution in order to demonstrate how the efforts to unite Scandinavian mathematicians were contingent on and influenced by simultaneous currents of internationalisation and shared history, culture, and language in the Scandinavian region.

Facilitating Source-Centered History of Mathematics

We present and discuss initiatives to develop source-centered teaching materials in history of mathematics for upper secondary education, aiming at meeting the objective of the Danish curriculum to make history of mathematics relevant. To this end we present the design template for such multi-purpose materials we developed, which allows devising materials neither too superficial nor too specialized, and we address the constraints on and affordances of historical sources in adapting to teaching objectives. It includes differentiation and scalability for using historical sources, and provides opportunity for interdisciplinary teaching, another requirement for Danish upper secondary education. We also report on (i) the recent application of our design approach to develop such source-centered materials in collaboration with small groups of dedicated teachers, and (ii) students’ positive response to the inquiry-driven teaching based on this material.

Shaping Mathematics as a Tool: The Search for a Mathematical Model for Quasi-crystals

Although the use of mathematical models is ubiquitous in modern science, the involvement of mathematical modeling in the sciences is rarely seen as cases of interdisciplinary research. Often, mathematics is “applied” in the sciences, but mathematics also features in open-ended, truly interdisciplinary collaborations. The present paper addresses the role of mathematical models in the open-ended process of conceptualizing new phenomena. It does so by suggesting a notion of cultures of mathematization, stressing the potential role of the mathematical model as a boundary object around which negotiations of different desiderata can take place. This framework is then illustrated by a case study of the early efforts to produce a mathematical model for quasi-crystals in the first two decades after Dan Shechtman’s discovery of this new phenomenon in 1984.

Studying appropriations of past lives: using metabiographical approaches in the history of mathematics

B iographies of mathematicians may be written for any number of more or less explicitly stated reasons: The biographer might have the ambition to present ‘accurate’, ‘factual’ or ‘objective’ accounts of past lives; but in reality, his biography is only an ‘imperfect sketch’ essentially framed by the choices he has made, as one mathematical biographer has observed (see Halsted 1895, 106; see also Sørensen 2016, 88, 93). These choices are, in turn, subject to the availability of sources, to the expertise and interest of the biographer, and to the context in which the biography is intended to be read. Thus, it is possible for different authors in different contexts to write multiple biographies of the same protagonist; and collections of biographies about different people can serve both as a dictionary with dates of birth and death, and as corpuses that present past lives to modern readers for them to learn something. The biographical genre has developed substantially over the centuries with inputs from various practices. From obituaries and depictions of the Saints, hagiographic characteristica such as the framing in turns of positive traits worthy of emulation have entered and endured in biographies (see also France 2002). And many other intellectual contexts have added analytical perspectives such as philosophical, psychological, social, economical, and biological frames for understanding the lives of those worthy of interest. Additional dimensions are added when the biographee is a scientist or a mathematician whose work and professional context and values may be directly accessible only to small readerships (for just some discussions on scientific biography, see, for example, Nye 2006; Porter 2006; S€ oderqvist 2007). Yet, when the identities (professional or otherwise) of the biographee, the biographer, and the intended reader align, biographies can become valuable entry points into studying these identities and the contexts in which they are formed. Although both deal with the past, history and biography are distinct endeavours in that they seek to attain different objectives, as Francis Bacon (1561–1626) argued (see also Caine 2010, 9ff). Biographies are essentially focused on people of perceived importance, and their purpose is often to make important individuals understandable and familiar to present readers. Other historical accounts, by comparison, either chronologies or narrations, seek to represent and understand complex events and dynamics by unravelling causes and contexts.

What's Abelian about abelian groups?

The association of names to mathematical concepts and results (the creation of eponyms) is often a curious process. For the case of abelian groups, we will be taken on a quick, guided tour of the life of Niels Henrik Abel, elliptic functions, a curve called the lemniscate, the construction of the regular 17-gon, and a particular class of solvable equations before we can begin to appreciate how Abels name was attributed to a concept (groups) not yet invented in his lifetime. Therefore, I will have to address how ‘group theory’ was done before it was even invented. As the story unfolds, indications of a broader development in mathematics in the early nineteenth century will emerge. In that century, large parts of analysis underwent transformations from a predominantly formula-centred approach to a more conceptual one, and our story features important examples of how the processes of generalization functioned.