Mathematics as a Tool pp 69-90

Part of the Boston Studies in the Philosophy and History of Science book series (BSPS, volume 327)

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

Chapter

Abstract

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.

Keywords

Quasi-crystals Mathematization Interdisciplinarity “Mathematics as a tool” Modeling 

References

  1. Abraham, T. H. (2004). Nicolas Rashevsky’s mathematical biophysics. Journal of the History of Biology, 37, 333–385.CrossRefGoogle Scholar
  2. Andersen, H., & Wagenknecht, S. (2013). Epistemic dependence in interdisciplinary groups. Synthese, 190, 1881–1898.CrossRefGoogle Scholar
  3. Authier, A. (2013). Early days of X-ray crystallography. Oxford: Oxford University Press.CrossRefGoogle Scholar
  4. Axel, F., & Gratias, D. (Eds.). (1995). Beyond quasicrystals. Les Houches, March 7–18, 1994. Berlin/Heidelberg: Springer.Google Scholar
  5. Axel, F., Denoyer, F., & Gazeau, J. P. (Eds.). (2000). From quasicrystals to more complex systems. Les Houches, February 23–March 6, 1998. Springer.Google Scholar
  6. Bernal, J. D. (1964, July 28). The Bakerian lecture, 1962: The structure of liquids. Proceedings of the Royal Society of London. A: Mathematical and Physical Sciences, 280(1382), 299–322.CrossRefGoogle Scholar
  7. Blech, I. A., Cahn, J. W., & Gratias D. (2012, October). Reminiscences about a chemistry Nobel Prize won with metallurgy: Comments on D. Shechtman and I. A. Blech; Metall. Trans. A, 1985, vol. 16A, pp. 1005–1012. Metallurgical And Materials Transactions A, 43A, 3411–3414.Google Scholar
  8. Brecque, M. L. (1987/1988). Quasicrystals: Opening the door to Forbidden symmetries. MOSAIC, 18(4), 2–23.Google Scholar
  9. Burchfield, J. D. (1975). Lord Kelvin and the age of the Earth. London/Basingstoke: The Macmillan Press.CrossRefGoogle Scholar
  10. Burke, J. G. (1966). Origins of the science of crystals. Berkeley/Los Angeles: University of California Press.Google Scholar
  11. Bursill, L. A., & Lin, P. J. (1985, July 4). Penrose tiling observed in a quasi-crystal. Nature, 316, 50–51.CrossRefGoogle Scholar
  12. Coddens, G. (1988). A new approach to quasicrystals. Solid State Communications, 65(7), 637–641.CrossRefGoogle Scholar
  13. England, P., Molnar, P., & Richter, F. (2007a, January). John Perry’s neglected critique of Kelvin’s age for the Earth: A missed opportunity in geodynamics. GSA Today, 17, 4–9.CrossRefGoogle Scholar
  14. England, P., Molnar, P., & Richter, F. (2007b). Kelvin, Perry and the age of the Earth. American Scientist, 95(4), 342–349.CrossRefGoogle Scholar
  15. Epple, M., Kjeldsen, T. H., & Siegmund-Schultze, R. (Eds.). (2013). From “Mixed” to “Applied” mathematics: Tracing an important dimension of mathematics and its history (Oberwolfach reports 12). Oberwolfach: Mathematisches Forschungsinstitut.Google Scholar
  16. Galison, P. (1997). Image and logic. A material culture of microphysics. Chicago: The University of Chicago Press.Google Scholar
  17. Gelfert, A. (2014, May). Applicability, indispensability, and underdetermination: Puzzling over Wigner’s ‘Unreasonable effectiveness of mathematics’. Science & Education, 23(5), 997–1009.CrossRefGoogle Scholar
  18. Grattan-Guinness, I. (2008). Solving Wigner’s mystery: The reasonable (though perhaps limited) effectiveness of mathematics in the natural sciences. The Mathematical Intelligencer, 30(3), 7–17.CrossRefGoogle Scholar
  19. Hargittai, B., & Hargittai, I. (2012). Quasicrystal discovery: From NBS/NIST to Stockholm. Structural Chemistry, 23(2), 301–306.CrossRefGoogle Scholar
  20. International Union of Crystallography. (1992). Report of the Executive Committee for 1991. Acta Crystallographica Section A, A48, 922–946.Google Scholar
  21. Keller, E. F. (2002). Making sense of life. Explaining biological development with models, metaphors, and machines. Cambridge, MA/London: Harvard University Press.Google Scholar
  22. Keys, A. S., & Glotzer, S. C. (2007). How do quasicrystals grow? Physical Review Letters, 99(235503), 1–4.Google Scholar
  23. Knorr Cetina, K. (1999). Epistemic cultures. How the sciences make knowledge. Cambridge, MA/London: Harvard University Press.Google Scholar
  24. Kragh, H. (2015). Mathematics and physics: The idea of a pre-established harmony. Science & Education, 24(5), 515–527.CrossRefGoogle Scholar
  25. Kuhn, T. S. (1962/1996). The structure of scientific revolutions (3rd ed.). Chicago/London: The University of Chicago Press.Google Scholar
  26. Levine, D., & Steinhardt, P. J. (1984, December 24). Quasicrystals: A new class of ordered structures. Physical Review Letters, 53(26), 2477–2480.CrossRefGoogle Scholar
  27. Lidin, S. (1991). Quasicrystals: Local structure versus global structure. Materials Science and Engineering, A134, 893–895.CrossRefGoogle Scholar
  28. Mackay, A. L. (1982). Crystallography and the Penrose pattern. Physica, 114A, 609–613.CrossRefGoogle Scholar
  29. Mackay, A. L. (1987). Quasi-crystals and amorphous materials. Journal of Non-Crystalline Solids, 97–98, 55–62.CrossRefGoogle Scholar
  30. Morgan, M. S., & Morrison, M. (Eds.). (1999). Models as mediators. Perspectives on natural and social science. Cambridge: Cambridge University Press.Google Scholar
  31. Patera, J. (Ed.). (1998). Quasicrystals and Discrete Geometry (Fields Institute monographs). Providence: American Mathematical Society.Google Scholar
  32. Penrose, R. (1974). The rôle of aesthetics in pure and applied mathematical research. Bulletin of the Institute of Mathematics and Its Applications, 10, 266–271.Google Scholar
  33. Penrose, R. (1979, March). Pentaplexity: A class of non-periodic tilings of the plane. The Mathematical Intelligencer, 2(1), 32–37.CrossRefGoogle Scholar
  34. Perry, J. (1895). On the age of the Earth. Nature, 51, 224–227, 341–342, 582–585.Google Scholar
  35. Reade, T. M. (1878, April). The age of the world as viewed by the geologist and the mathematician. The Geological Magazine. New Series, 5(4), 145–154.Google Scholar
  36. Schlote, K.-H., & Schneider, M. (Eds.). (2011). Mathematics meets physics. A contribution to their interaction in the 19th and the first half of the 20th century. Studien zur Entwicklung von Mathematik und Physik in ihren Wechselwirkungen. Frankfurt am Main: Verlag Harri Deutsch.Google Scholar
  37. Schwarzschild, B. (1985, February). “Forbidden fivefold symmetry may indicate quasicrystal phase”. Physics Today. News: Search & discovery, pp. 17–19.Google Scholar
  38. Senechal, M. (1995). Quasicrystals and geometry. Cambridge: Cambridge University Press.Google Scholar
  39. Senechal, M. (2006, September). What is a quasicrystal? Notices of the AMS, 53(8), 886–887.Google Scholar
  40. Senechal, M., & Taylor, J. (1990). Quasicrystals: The view from Les Houches. The Mathematical Intelligencer, 12(2), 54–64.CrossRefGoogle Scholar
  41. Senechal, M., & Taylor, J. (2013). Quasicrystals: The view from Stockholm. The Mathematical Intelligencer, 35(2), 1–9.CrossRefGoogle Scholar
  42. Shechtman, D., & Blech, I. A. (1985, June). The microstructure of rapidly solidified Al6Mn. Metallurgical Transactions, 16A, 1005–1012.CrossRefGoogle Scholar
  43. Shechtman, D., et al. (1984, November 12). Metallic phase with long-range orientational order and no translational symmetry. Physical Review Letters, 53(20), 1951–1953.CrossRefGoogle Scholar
  44. Steinhardt, P. J. (2013). Quasicrystals: A brief history of the impossible. Rend. Fis. Acc Lincei, 24, S85–S91.CrossRefGoogle Scholar
  45. Takakura, H., et al. (2007, January). Atomic structure of the binary icosahedral Yb-Cd quasicrystal. Nature Materials, 6, 58–63.CrossRefGoogle Scholar
  46. Wigner, E. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics, 13(1), 1–14.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Section for History and Philosophy of Science, Department of Science EducationUniversity of CopenhagenCopenhagenDenmark

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