Klein vs Mehrtens:
restoring the reputation of a great modern

J. Bair; P. Blaszczyk; P. Heinig; M. Katz; J. P. Schafermeyer; D. Sherry

doi:10.15330/ms.48.1.189-219

Historian Herbert Mehrtens sought to portray the history of turn-of-the-century mathematics as a struggle of modern vs countermodern, led respectively by David Hilbert and Felix Klein. Some of Mehrtens' conclusions have been picked up by both historians (Jeremy Gray) and mathematicians (Frank Quinn). We argue that Klein and Hilbert, both at Gottingen, were not adversaries but rather modernist allies in a bid to broaden the scope of mathematics beyond a narrow focus on arithmetized analysis as practiced by the Berlin school. Klein's Gottingen lecture and other texts shed light on Klein's modernism. Hilbert's views on intuition are closer to Klein's views than Mehrtens is willing to allow. Klein and Hilbert were equally interested in the axiomatisation of physics. Among Klein's credits is helping launch the career of Abraham Fraenkel, and advancing the careers of Sophus Lie, Emmy Noether, and Ernst Zermelo, all four surely of impeccable modernist credentials. Mehrtens' unsourced claim that Hilbert was interested in production rather than meaning appears to stem from Mehrtens' marxist leanings. Mehrtens' claim that [the future Brigadefuhrer] Theodor Vahlen... cited Klein's racist distinctions within mathematics, and sharpened them into open antisemitism" fabricates a spurious continuity between the two figures mentioned and is thus an odious misrepresentation of Klein's position.

1. J. Bair, P. B laszczyk, R. Ely, V. Henry, V. Kanovei, K. Katz, M. Katz, T. Kudryk, S. Kutateladze, T. McGaffey, T. Mormann, D. Schaps, D. Sherry, Cauchy, infinitesimals and ghosts of departed quantifiers, Mat. Stud., 47 (2017), №2, 115-144.

2. J. Bair, P. B laszczyk, P. Heinig, M. Katz, J. Schafermeyer, D. Sherry, Felix Klein vs Rowe, Gray, and Quinn: Restoring the reputation of a great modern, 2018. (in preparation)

3. T. Bascelli, E. Bottazzi, F. Herzberg, V. Kanovei, K. Katz, M. Katz, T. Nowik, D. Sherry, S. Shnider, Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow, Notices of the American Mathematical Society, 61 (2014), №8, 848-864.

4. T. Bascelli, P. B laszczyk, A. Borovik, V. Kanovei, K. Katz, M. Katz, S. Kutateladze, T. McGaffey, D. Schaps, D. Sherry, Cauchys infinitesimals, his sum theorem, and foundational paradigms, Foundations of Science, (2018).

5. P. Benisch, To Vanquish the Dragon, Feldheim, Jerusalem, New York, 1991.

6. L. Bieberbach, PersNonlichkeitsstruktur und mathematisches Schaffen, Unterrichtsblatter fur Mathematik und Naturwissenschaften, 40 (1934), 236-243.

7. L. Bieberbach, Zweihundertfunfzig Jahre Differentialrechnung, Zeitschrift fur die gesamte Naturwissenschaft, I (1935), 171-177.

8. P. Blaszczyk, V. Kanovei, K. Katz, M. Katz, S. Kutateladze, D. Sherry, Toward a history of mathematics focused on procedures, Foundations of Science, 22 (2017), №4, 763-783. See http:// arxiv.org/abs/1609.04531

9. G. Carrier, R. Courant, P. Rosenbloom, C. Yang, H. Greenberg, Applied mathematics: what is needed in research and education, SIAM Rev., 4 (1962), №4, 297-320.

10. E. Confalonieri, Beitrage zur Geschichte der mathematischen Werke von Karl Weierstrass, Teil III. 2013, self ed. online at https:// books.google.co.il/books?id=rQh1AgAAQBAJ&source=gbs_navlinks_s.

11. L. Corry, David Hilbert and the axiomatization of physics (1898.1918). From Grundlagen der Geometrie to Grundlagen der Physik. Archimedes: New Studies in the History and Philosophy of Science and Technology, 10. Kluwer Academic Publishers, Dordrecht, 2004.

12. L. Corry, Modern Algebra and the Rise of Mathematical Structures. Second edition. Birkhauser Verlag, Basel, 2004.

13. L. Corry, Axiomatics, empiricism, and Anschauung in Hilberts conception of geometry: between arithmetic and general relativity. The architecture of modern mathematics, 133-156, Oxford Univ. Press, Oxford, 2006.

14. L. Corry, Axiomatics Between Hilbert and the New Math: Diverging Views on Mathematical Research and Their Consequences on Education, The International Journal for the History of Mathematics Education, 2 (2007), 21-37.

15. R. Courant, Felix Klein, Jahresbericht der Deutschen Mathematiker-Vereinigung, 34 (1926), 197-213.

16. S. Eilenberg, S. MacLane, General theory of natural equivalences, Trans. Amer. Math. Soc., 58 (1945), 231-294.

17. A. Fraenkel, Einleitung in die Mengenlehre, Dover Publications, New York NY, 1946. Originally published by Springer, Berlin, 1928.

18. J. Gray, Platos ghost. The modernist transformation of mathematics, Princeton University Press, Princeton, NJ, 2008.

19. I. Hacking, Why is there philosophy of mathematics at all?, Cambridge University Press, Cambridge, 2014.

20. M. Heidegger, Der Satz vom Grund. Pfulligen: Neske, 1957.

21. M. Heidegger, The Principle of Reason. Translated by Reginald Lilly. Indiana University Press, Bloomington and Indianapolis, 1991 (translation of [20]).

22. A. Hermann, Weltreich der Physik Von Galilei bis Heisenberg, Bechtle, 1980.

23. A. Hermann, Schonbeck, C. Technik und Wissenschaft, VDI-Verlag, Dusseldorf, 1991.

24. D. Hilbert, Grundlagen der Geometrie, Festschrift zur Feier der Enthullung des Gauss. Weber Denkmals in Gottingen, Teubner, Leipzig, 1899.

25. D. Hilbert, S. Cohn-Vossen, Anschauliche Geometrie. Die Grundlehren d. math. Wiss. in Einzeldarstell. mit bes. Berucksichtigung d. Anwendungsgebiete. 37, Berlin, Julius Springer, 1932.

26. E. Jaensch, Uber die Grundlagen der Menschlichen Erkenntnis. J.A. Barth, Leipzig, 1931. 27. V. Kanovei, K. Katz, M. Katz, T. Mormann, What makes a theory of infinitesimals useful? A view by Klein and Fraenkel, Journal of Humanistic Mathematics, 8 (2018), №1, 108.119. See http:// scholarship.claremont.edu/jhm/vol8/iss1/7/

28. M. Katz, L. Polev, From Pythagoreans and Weierstrassians to true infinitesimal calculus, Journal of Humanistic Mathematics, 7 (2017), №1, 87.104. See https:// arxiv.org/abs/1701.05187

29. F. Klein, Lectures on Mathematics delivered from 28 august to 9 september 1893 before members of the congress of mathematics held in connection with the world fair in Chicago at Northwestern University, reported by Alexander Ziwet, New York Macmillan, London 1894.

30. F. Klein, The arithmetizing of mathematics, Translated by Isabel Maddison, Bull. Amer. Math. Soc., 2 (1896), №8, 241-249.

31. F. Klein, Elementary mathematics from an advanced standpoint. V.I. Arithmetic, Algebra, Analysis, translation by E.R. Hedrick and C.A. Noble [Macmillan, New York, 1932] from the third German edition [Springer, Berlin, 1924]. Originally published as Elementarmathematik vom hoheren Standpunkte aus, Leipzig, 1908.

32. F. Klein, Vorlesungen uber die Entwicklung der Mathematik im 19. Jahrhundert, Fur den Druck bearbeitet von R. Courant und O. Neugebauer. Grundlehren Band 24, Springer, 1926.

33. M. Kline, Why Johnny CanЃft Add: The Failure of the New Math., New York: Vintage Books, 1973.

34. H. Kramer, Modernism and its enemies.Ѓh In The New Kriterion Reader: The first five years. Events and controversies of the 1980s in art, architecture, music, literature, theater, poetry, dance, intellectual life, and the academy, 8.14. Edited, with an introduction, by Hilton Kramer. New York: The Free Press, Macmillan, 1988.

35. T. Lessing, Zeitschrift fur experimentelle Padagogik psychologische und pathologische Kinderforschung, Band 9 (1909), 225-237.

36. A. Lewis, Review of Mehrtens [41] for MathSciNet, 2002. See http:// www.ams.org/mathscinetgetitem? mr=1770117

37. J.-P. Marquis, From a geometrical point of view. A study of the history and philosophy of category theory, Logic, Epistemology, and the Unity of Science, 14, Springer, Dordrecht, 2009.

38. H. Mehrtens, Die Gleichschaltung der mathematischen Gesellschaften im nationalsozialistische Deutschland, Jahrbuch Uberblicke Mathematik, 18 (1985), 83-103.

39. H. Mehrtens, Moderne-Sprache-Mathematik. Eine Geschichte des Streits um die Grundlagen der Disziplin und des Subjekts formaler Systeme, Suhrkamp Verlag, Frankfurt am Main, 1990.

40. H. Mehrtens, The social system of mathematics and National Socialism: a survey. In Math worlds, 219-246, SUNY Ser. Sci. Tech. Soc., SUNY Press, Albany, NY, 1993.

41. H. Mehrtens, Modernism vs. counter-modernism, nationalism vs. internationalism: style and politics in mathematics 1900-1950, Chapter 23 in LEurope mathЃLematique/Mathematical Europe. Histoires, mythes, identites/History, myth, identity. Edited by Catherine Goldstein, Jeremy Gray and Jim Ritter. Editions de la Maison des Sciences de lHomme, Paris, 1996, 517-529.

42. H. Mehrtens, H. Bos, I. Schneider, eds. Social History of Nineteenth Century Mathematics. Springer Science+Business Media, New York, 1981.

43. G. Mittag-Leffler, Une Page de la Vie de Weierstrass. Extrait dЃfune communication plus etendue, Compte Rendu du Deuxi`eme Congr`es International des Mathematiciens, Proc`es-Verbaux et Communications, Gauthier-Villars, Paris, 1902, 131-153.

44. M. Ohse, R. von Thadden, G. Trittel, Gottingen: Geschichte einer Universitatsstadt: Von den Anfangen bis zum Ende des Dreissigjahrigen Krieges. Band 3. Von der preussischen Mittelstadt zur sudniedersachsischen Grostadt 1866-1989. Vandenhoeck & Ruprecht, 1987.

45. C. Phillips, In accordance with a emore majestic order: the new math and the nature of mathematics at midcentury, Isis, 105 (2014), №3, 540-563.

46. F. Quinn, A revolution in mathematics? What really happened a century ago and why it matters today, Notices Amer. Math. Soc., 59 (2012), №1, 31-37.

47. B. Riemann, Fragmente philosophischen Inhalts. In Bernhard Riemanns gesammelte Mathematische Werke und Wissenschaftlicher Nachlass, Herausgegeben unter Mitwirkung von R. Dedekind von H. Weber, Leipzig, Druck und Verlag von B. G. Teubner, 1876.

48. A. Robinson, J. Laurmann, Wing theory. Cambridge, at the University Press, 1956.

49. A. Robinson, Non-standard analysis, Nederl. Akad. Wetensch. Proc. Ser. A, 64 = Indag. Math., 23 (1961), 432-440 [reprinted in Selected Papers see Robinson [51], pages 3.11].

50. A. Robinson, Formalism 64, In 1965 Logic, Methodology and Philos. Sci. (Proc. 1964 Internat. Congr.) pp. 228.246, North-Holland, Amsterdam, 1965.

51. A. Robinson, Selected papers of Abraham Robinson. V.II. Nonstandard analysis and philosophy. Edited and with introductions by W.A.J. Luxemburg and S. Korner, Yale University Press, New Haven, 1979.

52. D. Rowe, Jewish mathematics at Gottingen in the era of Felix Klein, Isis, 77 (1986), №3, 422-449.

53. D. Rowe, The philosophical views of Klein and Hilbert, The intersection of history and mathematics, 187.202, Sci. Networks Hist. Stud., 15, Birkhauser, Basel, 1994.

54. D. Rowe, Review: Bernhard Riemann, Uber die Hypothesen, welche der Geometrie zu Grunde liegen, Hrsg. von Nurgen Jost, Klassische Texte der Wissenschaft. Springer 2013. David Hilbert, Grundlagen der Geometrie (Festschrift 1899), Hrsg. von Klaus Volkert, Klassische Texte der Wissenschaft. Springer 2015. Jahresber. Dtsch. Math.-Ver., 119 (2017), 169-186.

55. D. Schlimm, The correspondence between Moritz Pasch and Felix Klein, Historia Math, 40 (2013), №2, 183-202.

56. R. Schwarzenberger, The Importance of Mistakes: The 1984 Presidential Address, The Mathematical Gazette, 68, №445 (Oct., 1984), 159-172.

57. S. Segal, Mathematicians under the Nazis. Princeton University Press, Princeton, NJ, 2003.

58. M. Stone, The revolution in mathematics, Amer. Math. Monthly 68 (1961), 715-734.

59. R. Stower, Erich Rothacker: sein Leben und seine Wissenschaft vom Menschen, Band 2 von Bonner Schriften zur Universitats- und Wissenschaftsgeschichte, V&R unipress, 2012.

60. A. Stubhaug, The mathematician Sophus Lie. It was the audacity of my thinking, Translated from the 2000 Norwegian original by Richard H. Daly. Springer-Verlag, Berlin, 2002.

61. R. Vinsonhaler, Teaching Calculus with Infinitesimals, Journal of Humanistic Mathematics, 6 (2016), №1, 249-276.

62. H. Weyl, Die Idee der Riemannschen Flache, B.G.Teubner, 1913.

63. H. Weyl, Axiomatic versus constructive procedures in mathematics, With commentary by Tito Tonietti, Math. Intelligencer, 7 (1985), №4, 10-17, 38.

	Klein vs Mehrtens: restoring the reputation of a great modern
Author	J. Bair, P. Blaszczyk, P. Heinig, M. Katz, J. P. Schafermeyer, D. Sherry j.bair@ulg.ac.be; pb@up.krakow.pl; heinig@ma.tum.de; katzmik@macs.biu.ac.il; jpschaefermeyer@gmail.com; David.Sherry@nau.edu 1) J. Bair, HEC-ULG, University of Liege, Belgium; 2) Institute of Mathematics, Pedagogical University of Cracow, Poland; 4) Department of Mathematics, Bar Ilan University, Ramat Gan, Israel; 5) Berlin, Germany; 6) Department of Philosophy, Northern Arizona University, Flagstaff, AZ 86011, USA
Abstract	Historian Herbert Mehrtens sought to portray the history of turn-of-the-century mathematics as a struggle of modern vs countermodern, led respectively by David Hilbert and Felix Klein. Some of Mehrtens' conclusions have been picked up by both historians (Jeremy Gray) and mathematicians (Frank Quinn). We argue that Klein and Hilbert, both at Gottingen, were not adversaries but rather modernist allies in a bid to broaden the scope of mathematics beyond a narrow focus on arithmetized analysis as practiced by the Berlin school. Klein's Gottingen lecture and other texts shed light on Klein's modernism. Hilbert's views on intuition are closer to Klein's views than Mehrtens is willing to allow. Klein and Hilbert were equally interested in the axiomatisation of physics. Among Klein's credits is helping launch the career of Abraham Fraenkel, and advancing the careers of Sophus Lie, Emmy Noether, and Ernst Zermelo, all four surely of impeccable modernist credentials. Mehrtens' unsourced claim that Hilbert was interested in production rather than meaning appears to stem from Mehrtens' marxist leanings. Mehrtens' claim that [the future Brigadefuhrer] Theodor Vahlen... cited Klein's racist distinctions within mathematics, and sharpened them into open antisemitism" fabricates a spurious continuity between the two figures mentioned and is thus an odious misrepresentation of Klein's position.
Keywords	arithmetized analysis; axiomatisation of geometry; axiomatisation of physics; formalism; intuition; mathematical realism; modernism; Felix Klein; David Hilbert; Karl Weierstrass
DOI	doi:10.15330/ms.48.2.189-219
Reference	1. J. Bair, P. B laszczyk, R. Ely, V. Henry, V. Kanovei, K. Katz, M. Katz, T. Kudryk, S. Kutateladze, T. McGaffey, T. Mormann, D. Schaps, D. Sherry, Cauchy, infinitesimals and ghosts of departed quantifiers, Mat. Stud., 47 (2017), №2, 115-144. 2. J. Bair, P. B laszczyk, P. Heinig, M. Katz, J. Schafermeyer, D. Sherry, Felix Klein vs Rowe, Gray, and Quinn: Restoring the reputation of a great modern, 2018. (in preparation) 3. T. Bascelli, E. Bottazzi, F. Herzberg, V. Kanovei, K. Katz, M. Katz, T. Nowik, D. Sherry, S. Shnider, Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow, Notices of the American Mathematical Society, 61 (2014), №8, 848-864. 4. T. Bascelli, P. B laszczyk, A. Borovik, V. Kanovei, K. Katz, M. Katz, S. Kutateladze, T. McGaffey, D. Schaps, D. Sherry, Cauchys infinitesimals, his sum theorem, and foundational paradigms, Foundations of Science, (2018). 5. P. Benisch, To Vanquish the Dragon, Feldheim, Jerusalem, New York, 1991. 6. L. Bieberbach, PersNonlichkeitsstruktur und mathematisches Schaffen, Unterrichtsblatter fur Mathematik und Naturwissenschaften, 40 (1934), 236-243. 7. L. Bieberbach, Zweihundertfunfzig Jahre Differentialrechnung, Zeitschrift fur die gesamte Naturwissenschaft, I (1935), 171-177. 8. P. Blaszczyk, V. Kanovei, K. Katz, M. Katz, S. Kutateladze, D. Sherry, Toward a history of mathematics focused on procedures, Foundations of Science, 22 (2017), №4, 763-783. See http:// arxiv.org/abs/1609.04531 9. G. Carrier, R. Courant, P. Rosenbloom, C. Yang, H. Greenberg, Applied mathematics: what is needed in research and education, SIAM Rev., 4 (1962), №4, 297-320. 10. E. Confalonieri, Beitrage zur Geschichte der mathematischen Werke von Karl Weierstrass, Teil III. 2013, self ed. online at https:// books.google.co.il/books?id=rQh1AgAAQBAJ&source=gbs_navlinks_s. 11. L. Corry, David Hilbert and the axiomatization of physics (1898.1918). From Grundlagen der Geometrie to Grundlagen der Physik. Archimedes: New Studies in the History and Philosophy of Science and Technology, 10. Kluwer Academic Publishers, Dordrecht, 2004. 12. L. Corry, Modern Algebra and the Rise of Mathematical Structures. Second edition. Birkhauser Verlag, Basel, 2004. 13. L. Corry, Axiomatics, empiricism, and Anschauung in Hilberts conception of geometry: between arithmetic and general relativity. The architecture of modern mathematics, 133-156, Oxford Univ. Press, Oxford, 2006. 14. L. Corry, Axiomatics Between Hilbert and the New Math: Diverging Views on Mathematical Research and Their Consequences on Education, The International Journal for the History of Mathematics Education, 2 (2007), 21-37. 15. R. Courant, Felix Klein, Jahresbericht der Deutschen Mathematiker-Vereinigung, 34 (1926), 197-213. 16. S. Eilenberg, S. MacLane, General theory of natural equivalences, Trans. Amer. Math. Soc., 58 (1945), 231-294. 17. A. Fraenkel, Einleitung in die Mengenlehre, Dover Publications, New York NY, 1946. Originally published by Springer, Berlin, 1928. 18. J. Gray, Platos ghost. The modernist transformation of mathematics, Princeton University Press, Princeton, NJ, 2008. 19. I. Hacking, Why is there philosophy of mathematics at all?, Cambridge University Press, Cambridge, 2014. 20. M. Heidegger, Der Satz vom Grund. Pfulligen: Neske, 1957. 21. M. Heidegger, The Principle of Reason. Translated by Reginald Lilly. Indiana University Press, Bloomington and Indianapolis, 1991 (translation of [20]). 22. A. Hermann, Weltreich der Physik Von Galilei bis Heisenberg, Bechtle, 1980. 23. A. Hermann, Schonbeck, C. Technik und Wissenschaft, VDI-Verlag, Dusseldorf, 1991. 24. D. Hilbert, Grundlagen der Geometrie, Festschrift zur Feier der Enthullung des Gauss. Weber Denkmals in Gottingen, Teubner, Leipzig, 1899. 25. D. Hilbert, S. Cohn-Vossen, Anschauliche Geometrie. Die Grundlehren d. math. Wiss. in Einzeldarstell. mit bes. Berucksichtigung d. Anwendungsgebiete. 37, Berlin, Julius Springer, 1932. 26. E. Jaensch, Uber die Grundlagen der Menschlichen Erkenntnis. J.A. Barth, Leipzig, 1931. 27. V. Kanovei, K. Katz, M. Katz, T. Mormann, What makes a theory of infinitesimals useful? A view by Klein and Fraenkel, Journal of Humanistic Mathematics, 8 (2018), №1, 108.119. See http:// scholarship.claremont.edu/jhm/vol8/iss1/7/ 28. M. Katz, L. Polev, From Pythagoreans and Weierstrassians to true infinitesimal calculus, Journal of Humanistic Mathematics, 7 (2017), №1, 87.104. See https:// arxiv.org/abs/1701.05187 29. F. Klein, Lectures on Mathematics delivered from 28 august to 9 september 1893 before members of the congress of mathematics held in connection with the world fair in Chicago at Northwestern University, reported by Alexander Ziwet, New York Macmillan, London 1894. 30. F. Klein, The arithmetizing of mathematics, Translated by Isabel Maddison, Bull. Amer. Math. Soc., 2 (1896), №8, 241-249. 31. F. Klein, Elementary mathematics from an advanced standpoint. V.I. Arithmetic, Algebra, Analysis, translation by E.R. Hedrick and C.A. Noble [Macmillan, New York, 1932] from the third German edition [Springer, Berlin, 1924]. Originally published as Elementarmathematik vom hoheren Standpunkte aus, Leipzig, 1908. 32. F. Klein, Vorlesungen uber die Entwicklung der Mathematik im 19. Jahrhundert, Fur den Druck bearbeitet von R. Courant und O. Neugebauer. Grundlehren Band 24, Springer, 1926. 33. M. Kline, Why Johnny CanЃft Add: The Failure of the New Math., New York: Vintage Books, 1973. 34. H. Kramer, Modernism and its enemies.Ѓh In The New Kriterion Reader: The first five years. Events and controversies of the 1980s in art, architecture, music, literature, theater, poetry, dance, intellectual life, and the academy, 8.14. Edited, with an introduction, by Hilton Kramer. New York: The Free Press, Macmillan, 1988. 35. T. Lessing, Zeitschrift fur experimentelle Padagogik psychologische und pathologische Kinderforschung, Band 9 (1909), 225-237. 36. A. Lewis, Review of Mehrtens [41] for MathSciNet, 2002. See http:// www.ams.org/mathscinetgetitem? mr=1770117 37. J.-P. Marquis, From a geometrical point of view. A study of the history and philosophy of category theory, Logic, Epistemology, and the Unity of Science, 14, Springer, Dordrecht, 2009. 38. H. Mehrtens, Die Gleichschaltung der mathematischen Gesellschaften im nationalsozialistische Deutschland, Jahrbuch Uberblicke Mathematik, 18 (1985), 83-103. 39. H. Mehrtens, Moderne-Sprache-Mathematik. Eine Geschichte des Streits um die Grundlagen der Disziplin und des Subjekts formaler Systeme, Suhrkamp Verlag, Frankfurt am Main, 1990. 40. H. Mehrtens, The social system of mathematics and National Socialism: a survey. In Math worlds, 219-246, SUNY Ser. Sci. Tech. Soc., SUNY Press, Albany, NY, 1993. 41. H. Mehrtens, Modernism vs. counter-modernism, nationalism vs. internationalism: style and politics in mathematics 1900-1950, Chapter 23 in LEurope mathЃLematique/Mathematical Europe. Histoires, mythes, identites/History, myth, identity. Edited by Catherine Goldstein, Jeremy Gray and Jim Ritter. Editions de la Maison des Sciences de lHomme, Paris, 1996, 517-529. 42. H. Mehrtens, H. Bos, I. Schneider, eds. Social History of Nineteenth Century Mathematics. Springer Science+Business Media, New York, 1981. 43. G. Mittag-Leffler, Une Page de la Vie de Weierstrass. Extrait dЃfune communication plus etendue, Compte Rendu du Deuxi`eme Congr`es International des Mathematiciens, Proc`es-Verbaux et Communications, Gauthier-Villars, Paris, 1902, 131-153. 44. M. Ohse, R. von Thadden, G. Trittel, Gottingen: Geschichte einer Universitatsstadt: Von den Anfangen bis zum Ende des Dreissigjahrigen Krieges. Band 3. Von der preussischen Mittelstadt zur sudniedersachsischen Grostadt 1866-1989. Vandenhoeck & Ruprecht, 1987. 45. C. Phillips, In accordance with a emore majestic order: the new math and the nature of mathematics at midcentury, Isis, 105 (2014), №3, 540-563. 46. F. Quinn, A revolution in mathematics? What really happened a century ago and why it matters today, Notices Amer. Math. Soc., 59 (2012), №1, 31-37. 47. B. Riemann, Fragmente philosophischen Inhalts. In Bernhard Riemanns gesammelte Mathematische Werke und Wissenschaftlicher Nachlass, Herausgegeben unter Mitwirkung von R. Dedekind von H. Weber, Leipzig, Druck und Verlag von B. G. Teubner, 1876. 48. A. Robinson, J. Laurmann, Wing theory. Cambridge, at the University Press, 1956. 49. A. Robinson, Non-standard analysis, Nederl. Akad. Wetensch. Proc. Ser. A, 64 = Indag. Math., 23 (1961), 432-440 [reprinted in Selected Papers see Robinson [51], pages 3.11]. 50. A. Robinson, Formalism 64, In 1965 Logic, Methodology and Philos. Sci. (Proc. 1964 Internat. Congr.) pp. 228.246, North-Holland, Amsterdam, 1965. 51. A. Robinson, Selected papers of Abraham Robinson. V.II. Nonstandard analysis and philosophy. Edited and with introductions by W.A.J. Luxemburg and S. Korner, Yale University Press, New Haven, 1979. 52. D. Rowe, Jewish mathematics at Gottingen in the era of Felix Klein, Isis, 77 (1986), №3, 422-449. 53. D. Rowe, The philosophical views of Klein and Hilbert, The intersection of history and mathematics, 187.202, Sci. Networks Hist. Stud., 15, Birkhauser, Basel, 1994. 54. D. Rowe, Review: Bernhard Riemann, Uber die Hypothesen, welche der Geometrie zu Grunde liegen, Hrsg. von Nurgen Jost, Klassische Texte der Wissenschaft. Springer 2013. David Hilbert, Grundlagen der Geometrie (Festschrift 1899), Hrsg. von Klaus Volkert, Klassische Texte der Wissenschaft. Springer 2015. Jahresber. Dtsch. Math.-Ver., 119 (2017), 169-186. 55. D. Schlimm, The correspondence between Moritz Pasch and Felix Klein, Historia Math, 40 (2013), №2, 183-202. 56. R. Schwarzenberger, The Importance of Mistakes: The 1984 Presidential Address, The Mathematical Gazette, 68, №445 (Oct., 1984), 159-172. 57. S. Segal, Mathematicians under the Nazis. Princeton University Press, Princeton, NJ, 2003. 58. M. Stone, The revolution in mathematics, Amer. Math. Monthly 68 (1961), 715-734. 59. R. Stower, Erich Rothacker: sein Leben und seine Wissenschaft vom Menschen, Band 2 von Bonner Schriften zur Universitats- und Wissenschaftsgeschichte, V&R unipress, 2012. 60. A. Stubhaug, The mathematician Sophus Lie. It was the audacity of my thinking, Translated from the 2000 Norwegian original by Richard H. Daly. Springer-Verlag, Berlin, 2002. 61. R. Vinsonhaler, Teaching Calculus with Infinitesimals, Journal of Humanistic Mathematics, 6 (2016), №1, 249-276. 62. H. Weyl, Die Idee der Riemannschen Flache, B.G.Teubner, 1913. 63. H. Weyl, Axiomatic versus constructive procedures in mathematics, With commentary by Tito Tonietti, Math. Intelligencer, 7 (1985), №4, 10-17, 38.
Pages	189-219
Volume	48
Issue	2
Year	2017
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Klein vs Mehrtens: restoring the reputation of a great modern