Abstract
During the mathematics lesson dealing with imaginary numbers, Törless, the young hero of Robert Musil’s novel, The Confusion of Young Törless, is truly amazed. Imaginary numbers are impossible: numbers which, put to the square, give a negative number cannot exist. Still, these imaginary numbers seem to be used to reach quite definite and concrete results. This looks to Törless as if mathematics could make you walk steadily on a bridge which had only a beginning and an end and nothing in between, as if the bridge were complete.
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Notes
Robert Musil, Die Verwirrungen des Zöglings Törless. Hambourg: Rowohlt Verlag, 1957. Quotation translated from the French edition: Robert Musil, les Désarrois de l’élève Törless, translation by Philippe Jaccottet. Paris: Editions du Seuil, 1984, p. 119.
Ilse M. Fasol-Boltzmann (Ed.), Ludwig Boltzmann Principien der Naturfilosofr. Lectures on Natural Philosophy 1903–1906 with Two Essays by Stephen G. Brush and Gerhard Fasol. Berlin, Heidelberg: Springer-Verlag, 1990. Boltzmann was then replacing Mach, a victim of a stroke, at one of the two chairs of philosophy.
Ludwig Boltzmann, “[3. Vorlesung]”, in: Ilse M. Fasol-Boltzmann, loc. cit.,pp. 83–85. When bracketed the name of the lecture refers to the preparatory notes for the corresponding lecture.
Ludwig Boltzmann, Populäre Schriften. Leipzig: Johann Ambrosius Barth, 1905. Partly translated in: Ludwig Boltzmann, Theoretical Physics and Philosophical Problems, edited by Brian McGuiness, translated by Paul Foulkes, Vienna Circle Collection, vol. 5. Dordrecht: D. Reidel Publishing Compagny, 1974. Whenever possible, citations and references will refer to the English edition.
Ludwig Boltzmann, “On the indispensability of atomism in Natural Science (Über die Unentbehrlichkeit der Atomistik in der Naturwissenschaften)” and “More on atomism (Nochmal über Atomistik)”,in: Ludwig Boltzmann, loc. cit.,resp. pp. 41–53 and pp. 54–56.
Ludwig Boltzmann, “On the indispensability of atomism in Natural Science”, ibid.,p. 43 and p. 50; “More on atomism”, ibid.,p. 55. See also, Ludwig Boltzmann, “On the Development of the Methods of Theoretical Physics in Recent Times (Über die Entwicklung der Methoden der theoretischen Physik in neuerer Zeit)”, ibid.,pp. 77–100, especially p. 97.
See Boltzmann’s account of the definition of velocity in his Lectures on the Principles of Mechanics. Ludwig Boltzmann, Vorlesungen ueber die Principe der Mechanik,2 vols. Leipzig: Johann Ambrosius Barth, 1897 and 1904. Vol. 1, § 3, pp. 10–13. Ludwig Boltzmann, Lectures on the Principles of Mechanics (excerpts), in: Ludwig Boltzmann, Theoretical Physics and Philosophical Problems, loc. cit.,pp. 223–254, p. 231–233.
And this, also in physics, as it appears in one of Boltzmann’s first scientific articles. Ludwig Boltzmann, “Über die Integrale lineare Differentialgleichungen mit periodischen Koeffizienten”, in: Ludwig Boltzmann, Wissenschaftliche Abhandlungen, Fritz Hasenöhrl (Ed.), 3 vols. Leipzig: Johann Ambrosius Barth, 1909. Vol. 1, pp. 43–48.
Ludwig Boltzmann, “On the indispensability of atomism in Natural Science”, in:Ludwig Boltzmann, Theoretical Physics and Philosophical Problems., loc. cit.,p. 43.
Ludwig Boltzmann, “More on atomism”, ibid. The actual finite difference equation doesn’t appear explicitly in the text, but has the advantage of condensing in one line Boltzmann’s explanations. The same line of thought is already at the basis of Boltzmann’s discretization of energy in the famous article where he demonstrates the so called “H Theorem”. See Ludwig Boltzmann, “Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen”, in:Ludwig Boltzmann, Wissenschaftliche Abhandlungen, loc. cit.,vol. 2, 1872, pp. 316–402. It is also clearly displayed in the way Boltzmann accounts for his results in: Ludwig Boltzmann, Lectures on Gas Theory,translated by Stephen G. Brush. Berkeley, Los Angeles: University of Califomia Press, 1964, vol. 2, § 81. For more detailed comments on these matters, see Nadine de Courtenay, Science et épistémologie chez Ludwig Boltzmann - La liberté des images par les signes,Thése. Paris, 1999. Chap. 5 and épilogue.
Ludwig Boltzmann, “On the Indispensability of Atomism in Natural Science”, in: Ludwig Boltzmann, Theoretical Physics and Philosophical Problems, loc. cit.,p. 43.
Ludwig Boltzmann, “Models”, in: Ludwig Boltzmann, Theoretical Physics and Philosophical Problems, loc. cit.,pp. 213–220, p. 215.
See also Nadine de Courtenay, oc. cit.,chap. 5.
On this subject, see for instance, Morris Kline, Mathematical Thought from Ancient to Modern Times, 3 vols. New York, Oxford: Oxford University Press, 1990. Especially vol. 2, chap. 26 and vol. 3, chaps. 40, 43.
Ilse M. Fasol-Boltzmann (Ed.), /oc. cit.,“[12. Vorlesung]”, pp. 96–98, p. 97. The phrase appears in the preparatory notes for the lesson n12, but recurs throughout the Lectures in different forms. See in particular lectures n° 11 and 12.
Ludwig Boltzmann, Lectures on the Principles of Mechanics (excerpts), in: Ludwig Boltzmann, Theoretical Physics and Philosophical Problems, loc. cit., p. 233. Felix Klein, Lectures on Mathematics. New York, London: Macmillan and Co., 1894, p. 48.
Ludwig Boltzmann, Lectures on the Principles of Mechanics (excerpts), in: Ludwig Boltzmann, Theoretical Physics and Philosophical Problems, loc. cit.,p. 233. In fact, the continuous differentiable functions constitue the exception!
Ilse M. Fasol-Boltzmann (Ed.), /oc. cit.,p. 77. Note taken on the back of Boltzmann’s notebook for the Lectures.
Ilse M. Fasol-Boltzmann (Ed.), /oc. cit.,“6. Vorlesung”, pp. 174–180, and “7. Vorlesung”, pp. 180–186.
Ilse M. Fasol-Boltzmann (Ed.), /oc. cit.,“8. Vorlesung”, pp. 187–193; “9. Vorlesung”, pp. 194200; “10. Vorlesung”, pp. 201–205.
The lecture was delivered in 1903, but there is no evidence that Boltzmann knew about Russell’s disclosure of the paradoxes the very same year. Although his diagnosis seems right, his argumentation doesn’t fall in with the now standard account.
David Hilbert, “On the infinite”, English translation by S. Bauer-Mengelber, in: Jean van Heijenoort (Ed.), From Frege to Gödel. A Source Book in Mathematical Logic, 1879–1931. Cambridge, London: Cambridge University Press, 1967, pp. 369–392, p. 371.
A fuller argumentation of this essential feature of Boltzmann’s thought would require, in addition to what follows, a detailed analysis of the article he used to consider as his single genuine “philosophical article”: “On the question of the objective existence of processes in inanimate nature (Über die Frage nach der objectiven Existenz der Vorgänge in der unbelebten Natur)”,in:Ludwig Boltzmann, Theoretical Physics and Philosophical Problems, loc. cit.,pp. 57–76. For such an analysis, see Nadine de Courtenay, /oc. cit,chap. 3 and chap. 4.
Ilse M. Fasol-Boltzmann (Ed.), loc. cit., “5. Vorlesung”, pp. 168–174, p. 173.
Ibid., “[5. Vorlesung]”, pp. 87–89, p. 87.
Letters from Boltzmann to Franz Brentano, in: Walter Höflechner, Ludwig Boltzmann Leben und Briefe, Publikationen aus dem Archiv der Universität Graz, vol. 30. Graz: Akademische Druck-u. Verlagsanstalt Graz, 1994. Letters dated from Vienna, 26 December 1904, pp. 380382 and Vienna, 4 January 1905, pp. 383–384.
Ilse M. Fasol-Boltzmann (Ed.), loc. cil.,notes taken on the 31 October 1904, p. 111.
Ibid., “11. Vorlesung”, pp. 205–212, p. 209.
Ibid., “12. Vorlesung”, pp. 212–221, p. 216.
lbid., “14. Vorlesung”, pp. 230–240, p. 236.
Ibid., “4. Vorlesung”, pp. 162–168. Boltzmann’s criticism is not exactly directed against the axiomatizations of Peano and Hilbert, but against the ‘demonstrations’ of the properties of the elementary operations Helmholtz develops, in a kindred spirit, out of ‘definitions’ in his famous article entitled “Numbering and Measuring”.
Ibid., “6. Vorlesung”, p. 175. In this lecture, Boltzmann argues, more precisely, against the view that integer numbers are a priori. But the former lectures allow to direct the argument against any inclination to account for integer numbers by “sharp definitions”.
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De Courtenay, N. (2002). The Role of Models in Boltzmann’s. In: Heidelberger, M., Stadler, F. (eds) History of Philosophy of Science. Vienna Circle Institute Yearbook [2001], vol 9. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1785-4_9
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