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Egg-Forms and Measure-Bodies: Different Mathematical Practices in the Early History of the Modern Theory of Convexity

Published online by Cambridge University Press:  01 March 2009

Tinne Hoff Kjeldsen*
Affiliation:
Roskilde University, Denmark

Argument

Two simultaneous episodes in late nineteenth-century mathematical research, one by Karl Hermann Brunn (1862–1939) and another by Hermann Minkowski (1864–1909), have been described as the origin of the theory of convex bodies. This article aims to understand and explain (1) how and why the concept of such bodies emerged in these two trajectories of mathematical research; and (2) why Minkowski's – and not Brunn's – strand of thought led to the development of a theory of convexity. Concrete pieces of Brunn's and Minkowski's mathematical work in the two episodes will, from the perspective of the above questions, be presented and analyzed with the use of the methodological framework of epistemic objects, techniques, and configurations as adapted from Hans-Jörg Rheinberger's work on empirical sciences to the historiography of mathematics by Moritz Epple. Based on detailed descriptions and a comparison of the objects and techniques that Brunn and Minkowski studied and used in these pieces it will be concluded that Brunn and Minkowski worked in different epistemic configurations, and it will be argued that this had a significant influence on the mathematics they developed for those bodies, which can provide answers to the two research questions listed above.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2009

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References

Blaschke, Wilhelm. 1940. “Hermann Brunn.” Obituary. Jahresbericht der Deutschen Mathematiker Vereinigung 50:163166.Google Scholar
Bonnesen, Tom and Fenchel, Werner. 1934. Theorie der konvexen Körper. Berlin: Julius Springer Verlag.Google Scholar
Brunn, Karl Hermann. 1887. Ueber Ovale und Eiflächen. Inaugural-Dissertation. Munich: Akademische Buchdruckerei von F. Straub.Google Scholar
Brunn, Karl Hermann. 1889. Ueber Curven ohne Wendepunkte. Habilitationsschrift, Munich: Theodor Ackermann.Google Scholar
Brunn, Karl Hermann. 1894. “Referat über eine Arbeit: Exacte Grundlagen für eine Theorie der Ovale.” Sitzungsberichte der Mathematik und Physik 24:93111.Google Scholar
Brunn, Karl Hermann. 1913. “Autobiography.” In Geistiges und Künstlerisches München in Selbstbiographien, 3943. Munich: Max Kellers Verlag.Google Scholar
Daston, Lorraine J. 1986. “The Physicalist Tradition in Early Nineteenth Century French Geometry.” Studies in History and Philosophy of Science 17 (3):269295.CrossRefGoogle Scholar
Dirichlet, Gustav Lejeune. 1850. “Über die Reduction der positiven quadratischen Formen mit drei unbestimten ganzen Zahlen.” Journal für die reine und angewandte Mathematik 40:209227.Google Scholar
Epple, Moritz. 1997. “Styles of Argumentation in Late 19th Century Geometry and the Structure of Mathematical Modernity.” In Analysis and Synthesis in Mathematics, edited by Otte, Michael and Panza, Marco, 177198. Dordrecht, Boston, London: Kluwer.CrossRefGoogle Scholar
Epple, Moritz. 1999. Die Entstehung der Knotentheorie. Kontexte und Konstruktionen einer modernen mathematischen Theorie. Braunsweig/Wiesbaden: Friedr. Vieweg & Sohn.CrossRefGoogle Scholar
Epple, Moritz. 2004. “Knot Invariants in Vienna and Princeton during the 1920s: Epistemic Configurations of Mathematical Research.” Science in Context 17:131164.CrossRefGoogle Scholar
Fenchel, Werner. 1983. “Convexity through the Ages.” In Convexity and Its Applications, edited by Gruber, Peter M. and Wills, Jörg M., 120130. Basel, Boston, Stuttgart: Birkhäuser.CrossRefGoogle Scholar
Gardner, Richard J. 2002. “The Brunn-Minkowski Inequality.” Bulletin (New Series) of the American Mathematical Society 39 (3):355405.CrossRefGoogle Scholar
Gauss, Carl Friedrich. 1863. Collected Works. Göttingen: Königliche Gesellschaft der Wissenschaften zu Göttingen.Google Scholar
Goldman, Jay R. 1998. The Queen of Mathematics. Wellesley, MA: A. K. Peters.Google Scholar
Goldstein, Catherine, Schappacher, Norbert, and Schwermer, Joachim, eds. 2007. The Shaping of Arithmetic after C. F. Gauss's Disquisitiones Arithmeticae. Berlin, Heidelberg, New York: Springer.CrossRefGoogle Scholar
Gruber, Peter. 1990. “Zur Geschicthe der Konvexgeometrie und der Geometrie der Zahlen.” In Ein Jahrhundert Mathematik 1890–1990, edited by Fischer, Gerd. Friedrich Hirzebruch, Winfried Scharlau and Willi Törnig, 421–455. Festschrift zum Jubiläum der DMV. Barunschweig-Wiesbaden: Deutsche Mathematiker-Vereinigung, Friedr. Vieweg & Sohn.Google Scholar
Gruber, Peter. 1993. “History of Convexity.” In Handbook of Convex Geometry, edited by Gruber, P. M. and Wills, Jörg M., 315. Elsevier Science Publishers.Google Scholar
Hashagen, Ulf. 2003. Walther von Dyck (1856–1934): Mathematik, Technik und Wissenschaftsorganisation and der TH München. Stuttgart: Franz Steiner Verlag Stuttgart.Google Scholar
Hermite, Charles. 1850. “Extraits de letters d. M. Ch. Hermite a M. Jacobi sur différents objects de la théorie des nombres.” Journal für die reine und angewandte Mathematik 40:261315.Google Scholar
Kjeldsen, Tinne Hoff. 2000. “A Contextualized Historical Analysis of the Kuhn-Tucker Theorem in Nonlinear Programming: The Impact of World War II.” Historia Mathematica 27:331361.CrossRefGoogle Scholar
Kjeldsen, Tinne Hoff. 2001. “John von Neumann's Conception of the Minimax Theorem: A Journey through Different Mathematical Contexts.” Archive for History of Exact Sciences 56:3968.CrossRefGoogle Scholar
Kjeldsen, Tinne Hoff. 2003. “New Mathematical Disciplines and Research in the Wake of World War II.” In Mathematics and War, edited by Booss-Bavnbek, Bernhelm and Høyrup, Jens, 126152. Basel, Boston, Berlin: Birkhäuser Verlag.CrossRefGoogle Scholar
Kjeldsen, Tinne Hoff. 2006. “The Development of Nonlinear Programming in Post War USA: Origin, Motivation, and Expansion.” In The Way Through Science and Philosophy: Essays in Honour of Stig Andur Pedersen, edited by Andersen, H. B., Christiansen, F. V., Jørgensen, K. F., and Hendricks, V., 3150. London: College Publications.Google Scholar
Kjeldsen, Tinne Hoff. 2008. “From Measuring Tool to Geometrical Object: Minkowski's Development of the Concept of Convex Bodies.” Archive for History of Exact Science 62 (1):5989.CrossRefGoogle Scholar
Klee, Victor, ed. 1963. “Convexity.” Proceedings of Symposia in Pure Mathematics, volume VII, Providence, Rhode Island: American Mathematical Society.Google Scholar
Klein, Felix. 1908. Elementarmathematik vom höheren Standpunkte. Leipzig: B. G. Teubner.Google Scholar
Klein, Felix. [1908] 1939. Elementary Mathematics from an Advanced Standpoint. English translation of the third German edition of Klein 1908. New York: Maximillan.Google Scholar
Kline, Morris. 1972. Mathematical Thought from Ancient to Modern Times. Oxford, New York: Oxford University Press.Google Scholar
Minkowski, Hermann. 1887. “Über einige Anwendungen der Arithmetik in der Analysis.” Probationary lecture, 85–88 in Schwermer 1991.Google Scholar
Minkowski, Hermann. [1891a] 1911. “Über die positiven quadratischen Formen und über kettenbruchähnliche Algoritmen.” In Gesammelte Abhandlungen, vol. I, 243260. Leipzig, Berlin: B. G. Teubner.Google Scholar
Minkowski, Hermann. [1891b] 1911. “Über Geometrie der Zahlen.” In Gesammelte Abhandlungen, vol. I, 264265. Leipzig, Berlin: B. G. Teubner.Google Scholar
Minkowski, Hermann. [1893] 1911. “Über Eigenschaften von ganzen Zahlen, die durch räumliche Anschauung erschlossen sind.” In Gesammelte Abhandlungen, vol. I, 271277. Leipzig, Berlin: B. G. Teubner.Google Scholar
Minkowski, Hermann. 1896. Geometrie der Zahlen. B. G. Teubner.Google Scholar
Minkowski, Hermann. [1897] 1911. “Allgemeine Lehrsätze über die konvexen Polyeder.” In Gesammelte Abhandlungen, vol. II, 103121. Leipzig, Berlin: B. G. Teubner.Google Scholar
Minkowski, Hermann. [1901a] 1911. “Über die Begriffe Länge, Oberfläche und Volumen.” In Gesammelte Abhandlungen, vol. II, 122127. Leipzig, Berlin: B. G. Teubner.Google Scholar
Minkowski, Hermann. [1901b] 1911. “Über die geschlossenen konvexen Flächen.” In Gesammelte Abhandlungen, vol, II, 128130. Leipzig, Berlin: B. G. Teubner.Google Scholar
Minkowski, Hermann. 1903. “Volumen und Oberfläche.” In Gesammelte Abhandlungen, vol, II, 230276. Leipzig, Berlin: B. G. Teubner.Google Scholar
Minkowski, Hermann. 1910. Geometrie der Zahlen. Leipzig: B. G. Teubner.Google Scholar
Minkowski, Hermann. 1911. “Theorie der konvexen Körper, insbesondere Begründung ihres Oberflächenbegriffs.” In Gesammelte Abhandlungen, vol, II, 131229. Leipzig, Berlin: B. G. Teubner.Google Scholar
Minkowski, Hermann. 1911. Gesammelte Abhandlungen. Leipzig, Berlin: B. G. Teubner.Google Scholar
Minkowski, Hermann. [1889] 1973. Briefe an David Hilbert. Berlin, Heidelberg, New York: L. Rüdenberg, H. Zassenhaus.Google Scholar
Reid, Constance. 1970. Hilbert. Berlin, Heidelberg, New York: Springer Verlag.CrossRefGoogle Scholar
Rheinberger, Hans-Jorg. 1997. Towards a History of Epistemic Things: Synthesizing Proteins in the Test Tube. Stanford: Stanford University Press.Google Scholar
Rowe, David. 2004. “Making Mathematics in an Oral Culture: Göttingen in the Era of Klein and Hilbert.” Science in Context 17:85129.CrossRefGoogle Scholar
Schwermer, Joachim. 1991. “Räumliche Anschauung und Minima positive definiter quadratischer Formen.” Jahresbericht der Deutschen Mathematiker-Vereinigung 93:49105.Google Scholar
Schwermer, Joachim. 2007. “Reduction Theory of Quadratic Forms: Towards Räumliche Anschauung in Minkowski's Early Work.” In The Shaping of Arithmetic after C. F. Gauss's Disquisitiones Arithmeticae, edited by Goldstein, Catherine, Schappacher, Norbert, and Schwermer, Joachim, 483504. Berlin, Heidelberg, New York: Springer.CrossRefGoogle Scholar
Scharlau, Winfried. 1977. “A Historical Introduction to the Theory of Integral Quadratic Forms.” Conference on Quadratic Forms. Edited by Crzech, G.. Queen's University, Kingston, Ontario, Canada.Google Scholar
Scharlau, Winfried and Opolka, H, Hans. 1985. From Fermat to Minkowski. New York: Springer Verlag.CrossRefGoogle Scholar
Steiner, Jakob. 1882. Gesammelte Werke. Vol. II. Berlin: G. Reimer.Google Scholar
Strobl, Walter. 1985. “Aus den wissenschaftlichen Anfängen Hermann Minkowskis.” Historia Mathematica 12:142156.CrossRefGoogle Scholar
Thompson, A. C. 1996. Minkowski Geometry. Cambridge, MA: Cambridge University Press.CrossRefGoogle Scholar
Toepell, M. 1996. Mathematiker und Mathematik an der Universität München 500 Jahre Lehre und Forschung. Munich: Institut für Geschichte der Naturwissenschaften.Google Scholar