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A Problem-Oriented Multiple Perspective Way into History of Mathematics – What, Why and How Illustrated by Practice

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The Richness of the History of Mathematics

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Abstract

This chapter is written with students in mind. It introduces and describes a problem-oriented multiple perspective approach to history of mathematics, which is a methodology to history of mathematics that is based on an action-oriented conception of history. It is explained how this approach is an open approach to history of mathematics in the sense that the research is driven by a question-answer strategy where the decisive factors for the development have not been decided beforehand, and it is clarified in what sense this approach moves beyond the internal/external division in the historiography of mathematics. The approach is illustrated by three examples from the history of twentieth century mathematics. The first is focused on the invention of the concept of a general convex body, and is a case that can be seen as an exemplar of the move of mathematics into an autonomous enterprise, which is an aspect of the twentieth century mathematics. The second case is concerned with the influence of WWII in the development of mathematical programming. It is an example of how conditions, or urgencies, in society might influence the development of mathematics together with more internal motivated driving forces. The third example deals with Nicolas Rashevsky’s early development of mathematical biology. This case demonstrates how conditions within the sciences, and in society, have a significant influence on what kind of research is being developed, and how mathematical modelling can function as a research tool at the frontier of science. As such, the chapter is an attempt to lay out, present and explain the theoretical perspective and methodology for a problem-oriented multiple perspective approach to history of mathematics and illustrate its strengths and versatility through the three examples.

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Notes

  1. 1.

    I have borrowed the term ‘multiple perspective’ from Bernard Eric Jensen, who is a general historian, i.e. not a historian of mathematics or science (Jensen 2003). The problem-orientation is inspired from the Roskilde Model of problem-oriented learning and project work (Andersen and Heilesen 2015), see Andersen and Kjeldsen (2015a, b) for the theoretical foundation and a review of the key concepts.

  2. 2.

    Basically, internalism accounts represent the point of view that mathematics develops without any influences from outside of mathematics whereas externalism considers factors from outside of mathematics in historical accounts.

  3. 3.

    See e.g. Bonnesen and Fenchel (1934), Klee (1963).

  4. 4.

    See Hashagen (2003, p. 245).

  5. 5.

    See Kjeldsen (2008, p. 60) and Kjeldsen (2009, pp. 85–86).

  6. 6.

    See Toepell (1996) and Hashagen (2003).

  7. 7.

    For details, see Kjeldsen (2008, 2009).

  8. 8.

    See the webpage of the society http://www.mathopt.org/

  9. 9.

    Similar ideas were published by the Russian economist and mathematician Leonid V. Kantorovich in 1939, see Kantorovich (1939/1960) and Leifman (1990).

  10. 10.

    See for example Owens (1989), Schweber (1988), Zachary (1997).

  11. 11.

    For an elaboration of that, see Kjeldsen (2000, 2001, 2003, 2009).

  12. 12.

    See e.g. Leonard (1992) and Mirowski (1991).

  13. 13.

    The first duality result for nonlinear programming is due to Werner Fenchel (1953), see Kjeldsen (forthcoming-a).

  14. 14.

    See Kjeldsen (2000).

  15. 15.

    Rashevsky was neither the first one to suggest this analogy nor the first person to attempt to introduce mathematical and physical methods into biology, see e.g. (Israel 1993; Keller 2002).

  16. 16.

    For Rashevsky’s relationship with the Rockefeller Foundation, see Abraham (2004). For a scientific biography of Rashevsky, see Shmailov (2016).

  17. 17.

    See Kjeldsen (2019). For a discussion of the significance of the case of Rashevsky for teaching in mathematics education, see Kjeldsen (2017), Kjeldsen (forthcoming-b), Jessen and Kjeldsen (2021).

  18. 18.

    The case was used in connection with philosophical ideas regarding the growth of mathematics of how new objects are introduced into mathematics and how we are able to reason with new objects, see Kjeldsen and Carter (2012).

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark

    Tinne Hoff Kjeldsen

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  1. Tinne Hoff Kjeldsen
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Correspondence to Tinne Hoff Kjeldsen .

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Editors and Affiliations

  1. SPHERE UMR 7219, Université Paris 7 Paris Cité, Paris cedex 13, France

    Karine Chemla

  2. Faculty of Philosophy, Universidad de Sevilla, Sevilla, Sevilla, Spain

    José Ferreirós

  3. Department of Mathematics, University of Michigan, Ann Arbor, MI, USA

    Lizhen Ji

  4. School of Mathematics, University of Wuppertal, Wuppertal, Germany

    Erhard Scholz

  5. Institute for Advanced Studies in History of Science, Northwest University, Xi’an, Shaanxi, China

    Chang Wang

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Kjeldsen, T.H. (2023). A Problem-Oriented Multiple Perspective Way into History of Mathematics – What, Why and How Illustrated by Practice. In: Chemla, K., Ferreirós, J., Ji, L., Scholz, E., Wang, C. (eds) The Richness of the History of Mathematics. Archimedes, vol 66. Springer, Cham. https://doi.org/10.1007/978-3-031-40855-7_1

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