Abstract
Boltzmann’s lectures on natural philosophy point out how the principles of mathematics are both an improvement on traditional philosophy and also serve as a necessary foundation of physics or what the English call “Natura Philosophy”, a title which he will retain for his own lectures. We start with lecture #3 and the mathematical contents of his lectures plus a few philosophical comments. Because of the length of the lectures as a whole we can only give the main points of each but organized into a coherent study. Behind his mathematics stands his support of Darwinian evolution interpreted in a partly Lamarckian way. He also supported non-Euclidean geometry. Much of Boltzmann’s analysis of mathematics is an attempt to refute Kant’s static a priori categories and his identification of space with “non-sensuous intuition”. Boltzmann’s strong attention toward discreteness in mathematics can be seen throughout the lectures. Part II of this paper will touch on the historical background of atomism and focus on the discrete way of thinking with which Boltzmann approaches problems in mathematics and beyond. Part III briefly points out how Boltzmann related mathematics and discreteness to music.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Ball, W. W. Rouse: 1960, A Short Account of the History of Mathematics, Dover, Publications Inc., New York.
Bell, E. T.: 1972, The Development of Mathematics, Dover Publications Inc., New York.
Bell, E. T.: 1965, Men of Mathematics, The Lives and Achievement of the Great Mathematicians from Zeno to Poincaré, Simon & Schuster, New York.
Blackmore, J. T. (ed.): 1995a, Ludwig Boltzmann: His Later Life and Philosophy, 1900-1906, Book One, A Documentary History, Kluwer Academic Publishers, Dordrecht/Boston/London.
Blackmore, J. T.: 1995b, Ludwig Boltzmann: His Later Life and Philosophy, 1900-1906, Book Two, the Philosopher, Kluwer Academic Publishers, Dordrecht/Boston/London.
Boltzmann, L.: 1871, '—Weitere Studien úber das Wärmegleichgewicht unter Gas-molektülen', Wiener Bericht 66, S.275-370, also in F. Hasenöhrl (ed.), 1909, Boltzmann, L.: Wissenschaftliche Abhandlungen, I. Band, reprinted in 1968, Chelsea Publishing Company, New York.
Boltzmann, L.: 1877, '—Ñber die Beziehung zwischen dem zweiten Hauptsatze der mechan-ischen Wärmegleichgewicht', Wiener Bericht 76, S. 373-435, also in F. Hasenöhrl (ed.), Ludwig Boltzmann: Wissenschaftliche Abhlandlungen, BandII, 164-223.
Boltzmann, L.: 1886, '—Der zweite Hauptsatz der mechanischen Wärmtheorie', in E. Broda (ed.), Populäre Schriften, Friedr. Vieweg & Sohn, Braunschweig/Wiesbaden (1979), p. 29.
Boltzmann, L.: 1897, '—Ñber die Unentbehrlichkeit of der Atomistik in der Naturwissen-shaft, in E. Broda (ed.): Populäre Schriften, Friedr. Vieweg & Sohn, Braunsch-weig/ Wiesbaden (1979), pp. 87-93 (shortened version of Boltzmann, Ludwig: 1925, Populäre Schriften, Auflage, Leipzig, pp. 141-61).
Boltzmann, L.: 1909, Wissenschaftliche Abhandlungen, I, II, III, F. Hassenöhrl (hrsg.), reprinted in 1968, Chelsea Publishing company, New York.
Boltzmann, L.: 1925 [1905], Populäre Schriften, Autgabe, Leipzig.
Boltzmann, L.: 1981, Vorlesngen über Gastheorie,BandI, II, in Roman Sexl (ed.), Ludwig Boltzmann Gesamtausgabe,BandI, Vorlesungen über Gastheorie, Akademische Drück und Verlagsanstalt, Graz.
Broda, E. (ed.): 1979, Ludwig Boltzmann, Populäre Schriften, Eingeleitet und ausgewählt von Engelbert Broda, Friedr. Vieweg & Sohn, Braunchweig/Wiesbaden.
Broda, E.: 1983, '—Ludwig Boltzmann als evolutionistischer Philosoph', Berichte zur Wissenschaftsgeschichte 6, 103-14.
Broda, E.: 1983, Ludwig Boltzmann, Man-Physicist-Philosopher, trans. Larry Gay and the author, Ox Bow Press, Woodgridge, Connecticut.
Broda, E.: 1986 [1955], Ludwig Boltzmann, Mensch-Physiker-Philosoph, Deuticke, Wien.
Cahan, D. (ed.): 1993, Hermann von Helmholtz and the Foundations of Nineteenth-Century Science, University of California Press, Berkeley.
Cantor, G.: 1955, Contributions to the Founding of the Theory of Transfinite Numbers, Translated, and provided with an introduction and notes by Philip E. B. Jordan, Dover Publications, Inc., New York.
Cercignani, C.: 1998, Ludwig Boltzmann, The Man Who Trusted Atoms, Oxford University Press, Oxford.
Cohen, R. S. and Y. Elkana (ed.): 1921, Hermann von Helmholtz, Epistemological writings, The Paul Hertz/Moritz Schlick Centenary Edition of 1921, with Notes and Commentary by the editors, Newly Translated by Malcolm F. Lowe, 1977, Reidel Publishing Company, Dordrecht, Holland.
Dauben, J. W.: 1990, Georg Cantor, His Mathematics and Philosophy of the Infinite, Princeton University Press, Princeton, New Jersey.
Edwards, H. M.: 1995, 'Kronecker on the Foundations of Mathematics', in Jaakko Hintikka (ed.), From Dedekind to Gödel, Essays on the Development of the Foundations of Mathematics, Kluwer Academic Publication, Dordrecht, pp. 45-52.
Fasol-Boltzmann, I. M. (ed.): 1990, Ludwig Boltzmann Principien der Naturfilosofi, Lectures on Natural Philosophy 1903-1906, Springer-Verlag, Berlin.
Hasenöhrl, F. (ed.): 1909, Ludwig Boltzmann Wissenschaftliche Abhandlungen, Bände, I, II, III, reprinted in 1968, Chelsea Publishing Company, New York.
Heinzmann, G. (Réuni par): 1986, Poincaré, Russel, Zermelo et Peano,Textesdela discussion (1906-1912) sur les Fondaments des Mathématiques: des antinomies à la prédicativité, Librairie scientifique et Technique, Albert Blanchard, Paris.
Hintikka, J. (ed.): 1995, From Dedekind to Gödel, Essays on the Development of the Foundations of Mathematics, Kluwer Academic Publishers, Dordrecht.
Höflechner, W. (ed.): 1994, Ludwig Boltzmann, Leben und Brief, Akademische Druck und Verlaganstalt, Graz.
Höfler, A.: 1906, 'Ludwig Boltzmann als Mensch und als Philosoph', Süddeutsche Monatshefte, Vol. 3, Jahrgang, Heft 10, Oktober 1906, Stuttgart, pp. 1-4.
[Höfler, Alois]: 1913, Rückblick auf die ersten fünfundzwanzig Vereinsjahre der Philosoph-ischen Gesellschaft an der Universität zu Wien, im Selbstverlag der philosophyschen Gesellschaft, Wien, 1-47.
Kastler, A.: 1973, 'Le Concept d'atome depuis cent ans', Journal de Physique, Colloque C10, Supplément au Nos. 11-12, Tome 34, Novembre-Decembre, C10-43.
Kline, M.: 1972, Mathematical Thought from Ancient to Modern Times, Vols. 2 and 3, Oxford University Press.
Koenigsberger, L.: 1965, Hermann von Helmnoltz, trans. Frances A. Welby, Dover Publications Inc., New York.
Kremer, R. L.: 1993, 'Innovation through Synthesis: Helmholtz and Color Research', in D. Cahan (ed.), Hermann von Helmholtz and the Foundations of Nineteenth-Century Science, University of California Press, Berkeley, pp. 205-58.
Kuhn, T. S. (conducted by): 1962, Interview with: Phillip Frank, Frank's home, Cambridge, Massachusetts, July 16, 1962, 25-1-000 to 824.
Kuhn, T. S.: 1978, Black-body Theory and the Quantum Discontinuity 1894-1912, Clarendon Press, New York.
Mach, E.: 1976, in B. McGuinness (ed.), Knowledge and Error, Reidel, Dordrecht.
Radnitzky, G. and W. W. Bartley, III (eds.): 1993, Evolutionary Epistemology, Rationality, and the Sociology of Knowledge, Open Court, La Salle.
Sexl, R. (ed.): 1981, Ludwig Boltzmann Gesamtausgabe, Band 1, Vorlesungen über Gastheorie, Akademische Druck und Verlagsanstalt, Graz.
Sexl, R. (ed.): 1982, Ludwig Boltzmann Gesamtausgabe, Band 2, Vorlesungen über Max-wells Theorie der Elektricität und des Lichtes, Akademische Druck und Verlagsanstalt, Graz.
Sexl, Roman and J. Blackmore (ed.): 1982, Ludwig Boltzmann Gesamtausgabe, Band 8, Ausgewählte Abhandlungen, International Tagung Anlässlich des 75 Jahrestages Seines Todes 5-8 September 1981, Akademische Druck und Verlagsanstalt, Graz.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Tanaka, S. Boltzmann on Mathematics. Synthese 119, 203–232 (1999). https://doi.org/10.1023/A:1005203606773
Issue Date:
DOI: https://doi.org/10.1023/A:1005203606773