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The Culture of Research Mathematics in 1860s Prussia: Adolph Mayer and the Theory of the Second Variation in the Calculus of Variations

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Research in History and Philosophy of Mathematics

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Abstract

The paper examines the intellectual culture of higher mathematics in Prussia and more broadly in Germany in the middle of the nineteenth century. There was at this time a strong ethos of pure mathematics in which the subject was pursued more for its intrinsic interest than for its utility or practical applications. This outlook was reflective of the prominence of neohumanism in German culture of the period. In the case of mathematics, it led to a higher degree of logical sophistication in the elaboration of theories. The work of Adolph Mayer in the calculus of variations at Prussia’s University of Königsberg is presented as a case study that illustrates the outlook and underlying values of German higher mathematics in the second half of the century. The self-consciously theoretical character of this mathematics distinguished it in a qualitative way from the style and mentality of the Enlightenment masters of analysis a century earlier. Our study provides evidence for the basic historicity of the development of analysis from 1750 to 1870.

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Notes

  1. 1.

    This essay originated in response to Josipa Petrunic’s call to explore how epistemic cultures in mathematical practice and theory are identified, and how they interact. A workshop on this subject was held at the University of Toronto in the summer of 2012. I am grateful to Hardy Grant for comments on an earlier draft of the paper, and to the referees for their comments.

  2. 2.

    In the present essay, the word “culture” is used not as a formal theoretical concept of sociology, but rather in its everyday sense as the common system of beliefs and values that members of society bring to projects of literary, artistic, and intellectual achievement. Within history of science, our understanding of this word is consistent with the way it is used in an article such as Paul Forman’s (1971) “Weimar culture, causality and quantum theory, 1918–1927: adaptation by German physicists and mathematicians to a hostile intellectual environment.” For a study that articulates the notion of a culture in science on a more explicit theoretical level, see Karin Knorr Cetina’s Epistemic Cultures: How Sciences Make Knowledge (1999).

  3. 3.

    Gerstell (1975) provides an account of mathematics education in Prussia in the first part of the nineteenth century.

  4. 4.

    See Turner (1971) and Schubring (1981, 2005) and the references contained therein.

  5. 5.

    For a standard account of this development, see Joseph Ben-David’s “German scientific hegemony and the emergence of organized science,” (Ben-David 1971, Chapter 7).

  6. 6.

    For a wide-ranging discussion of historical and philosophical notions of pure and applied mathematics, see Ian Hacking’s “Applications,” Chapter 5 of Why Is There a Philosophy of Mathematics at All (2014). Hacking (p. 153) distinguishes the philosophical meaning of pure as that which does not involve any experiential input from the more common meaning of pure as that which is independent of practical purposes. Authors such as du Bois-Reymond and Crelle seem to have understood pure mathematics in the second sense. What Jakob Fries called applied mathematics would certainly count as pure in the second sense. The theory of the second variation in the calculus of variations belongs to pure mathematics in either sense, while a subject such as Hamilton–Jacobi theory in dynamics—cultivated as an abstract theory—would evidently belong to pure mathematics in the second sense.

  7. 7.

    Pyenson (1983) in his book Neohumanism and the Persistence of Pure Mathematics in Wilhelmian Germany maintains that this outlook hindered developments in physics education at the end of the century, although his thesis has been challenged by Rowe (1985).

  8. 8.

    Schubring cites Leonard Nelson (ed.), “Vier Briefe von Gauss und Weber an Fries,” Abhandlungen der Fries’schen Schule V. 1 (1906), 431–440, p. 437. Concerning Fries 1822 book, Gauss provided the following advice to a skeptical student: “Young man, if after three years of intense study you have progressed to where you understand and appreciate this book, you can leave the university with the conviction that you have made use of your time better by far than the majority of your fellow students.” Quoted in Gregory (1983, p. 186).

  9. 9.

    For more information on the Jacobi school (see Klein 1926, pp. 112–115).

  10. 10.

    For biographical information on Mayer (see Thiele 1999).

  11. 11.

    For biographical information on Richelot (see Cantor 1889).

  12. 12.

    See Goldstine (1980) and Fraser (1994).

  13. 13.

    For accounts of these developments. See Goldstine (1980) and Fraser (1996, 2003).

  14. 14.

    Noteworthy is the way in which the operational character of the variational process δ enters into the reasoning. See Fraser (1994).

  15. 15.

    For a history of Jacobi’s work in mathematical dynamics (see Nakane and Fraser 2002).

  16. 16.

    Spitzer came from a Jewish family in Mikulov in Moravia. The second variation in the calculus of variations was his first area of research in mathematics. He is perhaps best known for his writings on the Laplace transform, where he championed the priority of Laplace and became involved in a dispute with his Vienna contemporary Joseph Petzval (see Deakin 1981). Spitzer’s career was at the Vienna Handelsschule and he also taught at the Polytechnische Institut in Vienna.

  17. 17.

    One gets a sense for the theoretical character of mathematics in Mayer’s time in the theses that he included as part of the public defense of his Habilitation. Such theses were often only distantly connected to the subject of the Habilitation. Including among the eight theses presented by Mayer were the following: in purely analytic problems, geometric considerations are insufficient; infinitesimals may lead to new mathematical theorems but cannot be part of their proof; Lagrange’s multiplier method in the calculus of variations is in need of a rigorous and scientific foundation; in optics the ether must be taken to be incompressible; and the law of inertia is a pure truth of experience.

  18. 18.

    Mayer’s (1866) Beiträge zur Theorie der Maxima und Minima der einfachen Integrale seems to have had rather limited distribution, and is not widely cited in the literature on the calculus of variations. There are copies at the major German university libraries. There is a copy in the British Library, but no copy in the Bibliothèque nationale de France. In the United States, no copies exist in the libraries of Harvard and the Library of Congress. While the University of Chicago was only founded in 1890, it acquired many older books in the calculus of variations (an important field of mathematical research at Chicago into the middle of the twentieth century), although Mayer’s book was not among them. There are copies at Yale and Princeton, although it is not indicated in their catalogues when the book was acquired. Of course, every major university possesses Crelle’s journal, in which Mayer’s (1868) article appeared. Today Mayer’s Beiträge is available online at Google Books.

  19. 19.

    For a critical account of the outlook and mentality that d’Alembert, Euler, Lagrange, and other eighteenth-century figures brought to their work in analysis (see Ferraro 2008).

  20. 20.

    The biographies of Mayer in the Dictionary of Scientific Biography, the St. Andrews MacTutor website, and Wikipedia are based on the obituaries by VonderMühll (1908) and Liebmann (1908). Both of these authors mention Mayer’s Habilitation without providing any details concerning its contents or significance.

  21. 21.

    For a detailed history of this development (see Thiele 2007); Fraser (2009) provides an English-language essay review of Thiele’s book.

References

  • Ben-David, Joseph. 1971. The Scientist’s Role in Society A Comparative Study. Prentice-Hall. Englewood Cliffs, New Jersey.

    Google Scholar 

  • Bolza, Oscar. 1909. Vorlesungen über Variationsrechnung. Teubner. Leipzig and Berlin.

    Google Scholar 

  • Cantor, Moritz. 1889.“Richelot, Friedrich Julius,” in Allgemeine Deutsche Biographie, V. 28, 432. Duncker & Humblot, Leipzig.

    Google Scholar 

  • Clebsch, Rudolf Alfred. 1858a. “Ueber die Reduktion der zweiten Variation auf ihre einfachste Form.” Journal für die reine und angewandte Mathematik 55, 254-273.

    Article  MathSciNet  Google Scholar 

  • Clebsch, Rudolf Alfred. 1858b. “Ueber diejenigen Probleme der Variationsrechnung, welche nur eine unäbhangige Variable enthalten.” Journal für die reine und angewandte Mathematik 55, 335-355.

    Article  Google Scholar 

  • Deakin, Michael A. B. 1981. “The development of the Laplace transform, 1737-1937: I. Euler to Spitzer, 1737-1880.” Archive for History of Exact Sciences 25 (1981), pp. 343-390.

    Article  MathSciNet  Google Scholar 

  • Delaunay, Charles. 1841. “Thèse sur la distinction des maxima et des minima dans les questions qui dépendent de la méthode des variations.” Journal de Mathématiques pures et appliqées 6, 209-237.

    Google Scholar 

  • Ferraro, Giovanni. 2008. The Rise and Development of the Theory of Series up to the Early 1820s. In the series Sources and Studies in the History of Mathematics and Physical Sciences. Springer Science and Business Media, New York.

    Google Scholar 

  • Forman, Paul. 1971. “Weimar Culture, Causality and Quantum Theory, 1918-1927: Adaptation by German physicists and Mathematicians to a Hostile Intellectual Environment,” Historical Studies in the Physical Sciences 3 (1971), 1-115.

    Article  Google Scholar 

  • Fraser, Craig. 1994. “Calculus of variations.” In Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Ed. I. Grattan-Guinness (Routledge, 1994), V.1, pp. 342-350.

    Google Scholar 

  • Fraser, Craig. 1996. “Jacobi’s Result (1837) in the Calculus of Variations and its Reformulation by Otto Hesse (1857). A study in the changing interpretation of mathematical theorems,” in Jahnke et Knoche (1996), pp. 149-172.

    Google Scholar 

  • Fraser, Craig. 2003. “The calculus of Variations: A Historical Survey,” in Jahnke (2003), 355-384.

    Google Scholar 

  • Fraser, Craig. 2009. “Sufficient Conditions, Fields and the Calculus of Variations,” Historia Mathematica 36 (2009), 420-427.

    Article  Google Scholar 

  • Friedman, Michael and Nordmann, Alfred (Eds.). 2006. The Kantian Legacy in Nineteenth-century Science. MIT Press. Cambridge, MA.

    Google Scholar 

  • Fries, Jakob Friedrich. 1822. Die mathematische Naturphilosophie. C. F. Winter. Heidelberg.

    Google Scholar 

  • Gerstell, Marguerite. 1975. “Prussian Education and Mathematics,” American Mathematical Monthly 82, pp. 240-245.

    Article  MathSciNet  Google Scholar 

  • Goldstine, Herman H. 1980. A History of the Calculus of Variations from the 17th through the 19th Century. Springer-Verlag. New York and Berlin.

    Google Scholar 

  • Gregory, Frederick. 1983. “Neo-Kantian Foundations of Geometry in the German Romantic Period.” Historia Mathematica 10, 184-201.

    Article  MathSciNet  Google Scholar 

  • Hacking, Ian. 2014. Why Is There Philosophy of Mathematics At All? Cambridge University Press. Cambridge, UK.

    Google Scholar 

  • Hesse, Ludwig Otto 1857. “Über die Criterien des Maximums und Minimums der einfachen Integrale.” Journal für die reine und angewandte Mathematik 54, 227-273.

    Article  Google Scholar 

  • Husserl, Edmund 1882. Beiträge zur Theorie von Variationsrechnung. Unpublished dissertation University of Vienna. In 1983 a French translation of Husserl’s dissertation was published under the title Contributions à la théorie du calcul des variations. Trans. Mlle Devouard. Edited by J. Vauthier. No. 65 of Queen’s Papers in Pure and Applied Mathematics (Eds. A. J. Coleman et al). Kingston, Ontario, Canada.

    Google Scholar 

  • Jacobi, Carl Gustav 1837. “Zur Theorie der Variations-Rechnung und der Differential-Gleichungen.” Journal für die reine und angewandte Mathematik 17, 68-82.

    Article  MathSciNet  Google Scholar 

  • Jahnke, H. Niels. 2003. A History of Analysis. American Mathematical Society.

    Google Scholar 

  • Jahnke, H. Niels and Michael Otte (Eds.). 1981. Epistemological and Social Problems of the Sciences in the Early Nineteenth Century. D. Reidel Publishing Company. Dordrecht, Holland; Hingham, MA.

    Google Scholar 

  • Jahnke, H. Niels and Norbert Knoche (Eds.) 1996. History of Mathematics and Education, Volume 11 of the series “Studien zur Wissenschafts-, Sozial und Bildungsgeschichte der Mathematik”. Vandenhoeck. Göttingen.

    Google Scholar 

  • Jordan, Camille. 1896. Cours d’analyse de l’École polytechnique V. 3: “Calcul intégral. Équations différentielles.” Paris.

    Google Scholar 

  • Klein, Felix. 1926. Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert Teil 1. Berlin.

    Google Scholar 

  • Knorr Cetina, Karin. 1999. Epistemic Cultures: How Sciences Make Knowledge. Harvard University Press. Cambridge, Massachusetts.

    Google Scholar 

  • Lagrange, Joseph L. 1797. Théorie des fonctions analytiques. Paris. The second edition appeared in 1813 and is reprinted as Oeuvres 9.

    Google Scholar 

  • Lagrange, Joseph L. 1801. Leçons sur le calcul des fonctions. Paris. Reissued in 1804 in Journal de l’École Polytechnique, 12 cahier, tome 5.

    Google Scholar 

  • Lagrange, Joseph L. 1806. Leçons sur le calcul des fonctions. Nouvelle edition. Paris. This edition includes additions and “un traité complèt du calcul des variations.” Reprinted as Oeuvres 10.

    Google Scholar 

  • Lagrange, Joseph L. 1867-1892. Oeuvres de Lagrange. 14 volumes. Paris: Gauthier-Villars.

    Google Scholar 

  • Liebmann, Heinrich. 1908. “Adolf Mayer †,” in Jahresbericht der Deutschen Mathematiker-Vereinigung. V. 17, 355-362.

    Google Scholar 

  • Lindelöf, Lorenz and François Moigno. 1861. Leçons de calcul différentiel de calcul intégral. Tome quatrième. - Calcul des variations. Authored by Lindelöf and revised in collaboration with Moigno. Paris.

    Google Scholar 

  • Mainardi, Gaspare. 1852. “Sul Calculo dell variazioni.” Annali di scienze mathematiche e fisiche 3 (1852), 149-192.

    Google Scholar 

  • Mayer, Adolph. 1866. Beiträge zur Theorie der Maxima und Minima der einfachen Integrale. Habilitationsschrift. B. G. Teubner. Leipzig.

    Google Scholar 

  • Mayer, Adolph. 1868. “Ueber die Kriterien des Maximums und Minimums der einfachen Integrale.” Journal für die reine und angewandte Mathematik 69, 238-263.

    Article  MathSciNet  Google Scholar 

  • Mayer, Adolph. 1886. “Begründung der Lagrange’sche Multiplicatorenmethode in der Variationsrechnung.” Mathematische Annalen V. 26, 74-82.

    Google Scholar 

  • Mayer, Adolph. 1904 and 1906. “Über den Hilbertschen Unäbhangigkeitsatz in der Theorie des Maximums und Minimums der einfachen Integralen.” Mathematische Annalen, V. 58, 235-248, V.62, 335-350.

    Google Scholar 

  • Mehrtens, Herbert, Henk Bos and Ivo Schneider (Eds.). 1981. Social History of Nineteenth Century Mathematics. Birkhäuser. Boston, Basel and Stuttgart.

    MATH  Google Scholar 

  • Nakane, Michiyo and Fraser, Craig G. 2002. “The Early History of Hamilton-Jacobi Dynamical Theory, 1834-1837,” Centaurus 44, 1-67.

    Article  MathSciNet  Google Scholar 

  • Pulte, Helmut. 2006. “Kant, Fries, and the Expanding Universe of Science” in Friedman and Nordmann (Eds.), 101-122.

    Google Scholar 

  • Pyenson, Lewis. 1983. Neohumanism and the Persistence of Pure Mathematics in Wilhelmian Germany. Memoirs of the American Philosophical Society, Vol. 150. American Philosophical Society. Philadelphia.

    Google Scholar 

  • Rowe, David E. 1985. “Felix Klein’s “Erlanger Antrittsrede.” A Transcription with English Translation and Commentary.” Historia Mathematica 12, 123-141.

    Article  MathSciNet  Google Scholar 

  • Scharlau, Winfried. 1981.”The Origins of Pure Mathematics,” in Jahnke and Otte (1981), pp. 331-347.

    Google Scholar 

  • Schubring, Gert. 1981. “The Conception of Pure Mathematics as an Instrument in the Professionalization of Mathematics,” in Mehrtens et al. (1981), pp. 111-143.

    Google Scholar 

  • Schubring, Gert. 2005. Conflicts Between Generalization, Rigor, and Intuition: Number Concepts Underlying the Development of Analysis. Springer. New York.

    Google Scholar 

  • Scriba, Christoph J. 1973. “Jacobi, Carl Gustav Jacob”. Dictionary of Scientific Biography V. 7, Ed. Charles C. Gillispie, pp.50-55. Charles Scribner’s Sons. New York.

    Google Scholar 

  • Spitzer, Simon. 1854-1855. “Über die Kriterien des Grössten and Kleinsten bei den Problemen der Variationsrechnung”, Sitzungsberichte der Mathematisch-Naturwissenschaften Classe der Kaiserlichen Akademie der Wissenschaften. Part One appears in volume 13 (1854), pp. 1014-1071. Part Two appears in volume 14 (1855), pp. 41-120. Vienna.

    Google Scholar 

  • Thiele, Rüdiger. 1999. “Adolph Mayer 1839-1908,” in Reiner Groβ and Gerald Wieners (Eds.), Sächsische Lebensbilder Band 4, 211-227. Verlag der Sächsischen Akademie der Wissenschaften zu Leizig.

    Google Scholar 

  • Thiele, Rüdiger. 2007.Von der bernoullischen Brachistochrone zum Kalibrator-Konzept : ein historischer Abriss zur Entstehung der Feldtheorie in der Variationsrechnung (hinreichende Bedingungen in der Variationsrechnung). Brepols Publishers. Turnhout, Belgium.

    Google Scholar 

  • R. Stephen Turner. 1971. “The Growth of Professorial Research in Prussia, 1818 to 1848 — Causes and Context,” Historical Studies in the Physical Sciences V. 3, 137–182.

    Article  Google Scholar 

  • VonderMühll, Karl. 1908. “Zum Andenken an Adolph Mayer (1839-1908)”, in Mathematischen Annalen V. 65, 433-434

    Google Scholar 

  • von Escherich, Gustav Ritter. 1899. “Die zweite Variation der einfachen Integrale,” Sitzungsberichte der Österreichische Akademie der Wissenschaften, V. 108.

    Google Scholar 

  • Weierstrass, Karl. 1927. Vorlesungen über Variationsrechnung. Edited by Rudolf Rothe. Akademische Verlagsgesellschaft. Leipzig.

    Google Scholar 

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  1. Institute for the History and Philosophy of Science and Technology, University of Toronto, Toronto, ON, Canada

    Craig Fraser

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  1. Mathematical, Information and Computer Sciences, Point Loma Nazare University, San Diego, CA, USA

    Maria Zack

  2. Department of Philosophy, McGill University, Montreal, QC, Canada

    Dirk Schlimm

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Fraser, C. (2018). The Culture of Research Mathematics in 1860s Prussia: Adolph Mayer and the Theory of the Second Variation in the Calculus of Variations. In: Zack, M., Schlimm, D. (eds) Research in History and Philosophy of Mathematics. Proceedings of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-90983-7_8

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