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Interactions Between Mathematics and Physics: The History of the Concept of Function—Teaching with and About Nature of Mathematics


In this paper, we discuss the history of the concept of function and emphasize in particular how problems in physics have led to essential changes in its definition and application in mathematical practices. Euler defined a function as an analytic expression, whereas Dirichlet defined it as a variable that depends in an arbitrary manner on another variable. The change was required when mathematicians discovered that analytic expressions were not sufficient to represent physical phenomena such as the vibration of a string (Euler) and heat conduction (Fourier and Dirichlet). The introduction of generalized functions or distributions is shown to stem partly from the development of new theories of physics such as electrical engineering and quantum mechanics that led to the use of improper functions such as the delta function that demanded a proper foundation. We argue that the development of student understanding of mathematics and its nature is enhanced by embedding mathematical concepts and theories, within an explicit–reflective framework, into a rich historical context emphasizing its interaction with other disciplines such as physics. Students recognize and become engaged with meta-discursive rules governing mathematics. Mathematics teachers can thereby teach inquiry in mathematics as it occurs in the sciences, as mathematical practice aimed at obtaining new mathematical knowledge. We illustrate such a historical teaching and learning of mathematics within an explicit and reflective framework by two examples of student-directed, problem-oriented project work following the Roskilde Model, in which the connection to physics is explicit and provides a learning space where the nature of mathematics and mathematical practices are linked to natural science.

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  1. 1.

    We use the term “explicit-reflective framework” in the sense of Abd-El-Khalick (2013).

  2. 2.

    For a more thorough history of the concept of function and generalized function, see Yuschkevich (1976) and Lützen (1982). A translation into English of a number of definitions of the concept of function from Bernoulli to Bourbaki can be found in Rüthing (1984).

  3. 3.

    A fine pedagogic introduction to Babylonian and ancient Greek astronomy can be found in Aaboe (2001).

  4. 4.

    Baron (1969) gives a fine introduction to the history of the calculus.

  5. 5.

    Here, we have set the wave velocity equal to 1.

  6. 6.

    In other cases, D’Alembert insisted that” the problem will be impossible” (D’Alembert 1747 §8).

  7. 7.

    In most other instances, D’Alembert was highly influenced by physics.

  8. 8.

    See Leçon 1.

  9. 9.

    For an account of the prehistory of the theory of distributions, see Lützen (1982).

  10. 10.

    For a description, analyses and discussion of the study program and the significance of the student-directed, problem-oriented project work for the learning and teaching of mathematics and science, see, e.g. Blomhøj and Kjeldsen (2009), Kjeldsen and Blomhøj (2009), and Andersen and Heilesen (2015).

  11. 11.

    For further details, see Kjeldsen and Blomhøj (2012).

  12. 12.

    For references, see Abd-El-Khalick (2013, p. 2088).


  1. Aaboe, A. (2001). Episodes from the early history of astronomy. New York: Springer.

  2. Abd-El-Khalick, F. (2013). Teaching with and about nature of science, and science teacher knowledge domains. Science & Education, 22(9), 2087–2107.

  3. Abd-El-Khalick, F., & Lederman, N. G. (2000). Improving science teachers’ conceptions of nature of science: A critical review of the literature. International Journal of Science Education, 22(7), 665–701.

  4. Albrechtsen, A., Christensen, K. G., & Hecksher, T. (2004). The dissemination of the theory of distributions after 1950. (In Danish). IMFUFA, text 428, Roskilde University.

  5. Andersen, A. S., & Heilesen, S. (2015). The Roskilde model: Problem-oriented learning and project work. Heidelberg, New York: Springer.

  6. Baron, M. E. (1969). The origins of the infinitesimal calculus. Oxford: Pergamon Press.

  7. Bernoulli, J. (1718). Remarques sur ce qu’on a donné jusqu’ici de solutions des problèmes sur les isopérimèters. Mémoires de l’Académie Royale des Sciences, 1718, 100. Page reference to Johan Bernoulli Opera Omnia Vol. 2, 235–69.

  8. Blomhøj, M., & Kjeldsen, T. H. (2009). Project organised science studies at university level: Exemplarity and interdisciplinarity. ZDM Mathematics Education, Zentralblatt für Didaktik der Mathematik, 41, 183–198.

  9. Bos, H. J. M. (1974). Differentials, higher-order differentials and the derivative in the Leibnizian calculus. Archive for History of Exact Sciences, 14, 1–90.

  10. Bourbaki, N. (1939). Théorie des ensembles (Facicule des résultats). Paris: Hermann.

  11. D’Alembert, J. L. R. (1747). Recherches sur la courbe que forme une corde tenduë mise en vibration. Histoire et Memoires de l’Académie Royale des Sciences de Berlin, 3, 214–219.

  12. D’Alembert, J. L. R. (1761). Recherches sur les vibrations des cordes sonores. Opuscules Mathématiques, 1, 1–64 and supplement 65–73.

  13. Dirichlet, J. P. G. L. (1829). Sur la convergence des series trigonometriques qui servent à représenter une function arbitraire entre des limites données. Journal für die reine und angewandte Mathematik, 4, 157–169. Dirichlet’s Werke 1, 117–132.

  14. Dirichlet, J. P. G. L. (1837). Über die Darstellung ganz willkürlicher Funktionen durch sinus- und cosinus-Reihen. Repertorium der Physik, 1, 152–174. Dirichlet’s Werke 1, 133–160.

  15. Euler, L. (1748a). Intorductio in Analysin Infinitorum (2 volumes). Lausanne: Bousquet. Euler’s Opera Omnia (1) 8–9.

  16. Euler, L. (1748b). Sur la vibrations des cordes. Mémoirse de l’Académie des Sciences de Berlin, 4, 1748 (publ. 1750), 69–85. Euler’s Opera Omnia (2) 10, 63–77.

  17. Euler, L. (1755). Institutiones Calculi Differentialis. St. Petersburg: Academiae Imperialis Scientiarium. Euler’s Opera Omnia (1) 10.

  18. Euler, L. (1763). De usu functionum discontinuarum in analysi. Novi Commentarii Academiae Scientiarum Petropolitanae, 11, (1763, publ. 1768), 67–102. Euler’s Opera Omnia (1) 23, 74–91.

  19. Euler, L. (1765). Eclaircissements sur le movement des cordes vibrantes. Micellanea Taurenencia, 3 (1762–1765, publ 1766), 1-26. Euler’s Opera Omnia (2) 10, 377–396.

  20. Fourier, J. (1822). Théorie analytique de la chaleur. Paris: Firmin Didot.

  21. Godiksen, R. B., Jørgensen, C., Hanberg, T. M., & Toldbod, B. (2003). Fourier and the Concept of a Functionthe transition from Euler’s to Dirichlet’s concept of a function. (In Danish). IMFUFA, text 416, Roskilde University.

  22. Hankel, H. (1870). Untersuchungen über die unendlich oft oscillierenden und unstetigen Funktionen. Mathematische Annalen, 20, 63–112.

  23. Kjeldsen, T. H., & Blomhøj, M. (2009). Integrating history and philosophy in mathematics education at university level through problem-oriented project work. ZDM Mathematics Education, Zentralblatt für Didaktik der Mathematik, 41, 87–104.

  24. Kjeldsen, T. H., & Blomhøj, M. (2012). Beyond motivation—History as a method for the learning of meta-discursive rules in mathematics. Educational Studies in Mathematics, 80, 327–349.

  25. Lagrange, J. L. (1806). Leçons sur le calcul des fonctions (2nd ed.). Paris: Courcier.

  26. Laugwitz, D. (1992). Das letzte Ziel ist immer die Darstellung einer Funktion: Grundlagen der analysis bei Weierstraß 1886, historische Wurzeln und Parallelen. Historia Mathematica, 19, 341–355.

  27. Leibniz, G. W. (1673). De linea ex lineis numero infinitis. Acta Eruditorum 1692. Leibniz’s Mathematische Schriften, 5, 266–269.

  28. Lützen, J. (1982). The prehistory of the theory of distributions. New York: Springer.

  29. Lützen, J. (2011a). The physical origin of physically useful mathematics. Interdisciplinary Science Reviews, 36(3), 229–243.

  30. Lützen, J. (2011b). Examples and reflections on the interplay between mathematics and physics in the 19th and 20th century. In K-H. Schlote & Schneider (Eds.), Mathematics meets physics: A contribution to their interaction in the 19th and the first half of the 20th century (pp. 17–41). Frankfurt am Main: M. Deutsch, pp. 17–41.

  31. Lützen, J. (2013). The interaction of physics, mechanics and mathematics in Joseph Liouville’s research. In E. Barbin, & R. Pisano, (Eds.), The dialectic relation between physics and mathematics in the XIXth Century. (pp.79-96). Dordrecht: Springer (History of Mechanism and Machine Science, Vol. 16).

  32. Monna, A. F. (1972). The concept of function in the 19th and 20th centuries, in particular with regard to the discussions between Baire, Borel and Lebesgue. Archive for History of Exact Sciences, 9, 57–84.

  33. Rutherford, F. J. (1964). The role of inquiry in science teaching. Journal of Research in Science Teaching, 2, 80–84.

  34. Rüthing, D. (1984). Some definitions of the concept of function from Joh. Bernoulli to N. Bourbaki. The Mathematical Intelligencer, 6, 72–77.

  35. Schwartz, L. (1950/51). Théorie des distributions. Vol. 1 1950, Vol. 2 1951. Paris: Hermann.

  36. Sfard, A. (1991). On the dual nature of mathematical conception: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1–36.

  37. Sfard, A. (2008). Thinking as communicating. Cambridge: Cambridge University Press.

  38. Von Neumann, J. (1927). Mathematische Begründung der Quantenmechanik. Göttinger Nachrichten, 1927, 1–57.

  39. Weierstrass, K. F. W. (1872). Über continuierliche Funktionen eines reellen Argumente, die für keinen Werth des letzteren einen bestimmten Differentialquotienten besitzen. Weierstrass’ Mathematische Werke II (pp. 71–74). Berlin: Mayer und Müller.

  40. Wigner, E. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics, 13, 1–14. Page numbers refer to the reproduction in Mathematics: People, Problems, Results I-III, ed. Campbell, D.M. and Higgens.

  41. Yuschkevich, A. P. (1976). The concept of function up to the middle of the 19th century. Archive for History of Exact Sciences, 16, 37–85.

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Kjeldsen, T.H., Lützen, J. Interactions Between Mathematics and Physics: The History of the Concept of Function—Teaching with and About Nature of Mathematics. Sci & Educ 24, 543–559 (2015).

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  • Function Concept
  • Mathematical Knowledge
  • Mathematical Concept
  • Mathematical Practice
  • Epistemological Understanding